A 566742 ARTES LIBRARY 18371 SCIENTIA VERITAS OF THE UNIVERSITY OF MICHIGAN | E-PLURIBUS UNUM TUEBOR SI-QUAERIS PENINSULAM AMOENAM CIRCUMSPICE DEPARTMENT OF ENGINEERING RAILROAD ENGINEERS' FIELD-BOOK AND EXPLORERS' GUIDE. 2164 ESPECIALLY ADAPTED TO THE USE OF RAILROAD ENGINEERS ON LOCATION AND CONSTRUCTION, AND TO THE NEEDS OF THE EXPLORER IN MAKING EXPLORATORY SURVEYS. BY, C H. C. GODWIN. SECOND, REVISED EDITION. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1894. Copyright, 1890, BY JOHN WILEY & SONS. DRUMMOND & NEU, Electrotypers, 1 to 7 Hague Street, New York, FERRIS BROS., Printers, 326 Pearl Street, New York, 1 Rulars Jano 16-42 PREFACE. 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. Speaking 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 iv PREFACE. Logarithmic Tables. The method I employ myself on such work is to take with me into camp the logarithmic portion of Chambers' Mathematical Tables-which I have had bound in pocket-book form-giving the logarithms of numbers up to 108,000 and of trigonometrical functions to 7 places of deci- mals in this way, high accuracy, when it is wanted, can be obtained much more readily and efficiently than by any table which could reasonably be inserted in a book suitable for pocket use; and as the logarithmic tables are rarely wanted outside the tent, they form a sort of stay at-home counter- . part to the Field-book itself. Table IX is inserted solely for convenience in the reduction of indices, barometric formula, etc., and a few like operations, in which the use of logarithms is more or less essential. COLORADO, January, 1889. H. C. GODWIN. INTRODUCTION. THE Contents of this Field-book are divided mainly into four parts: Part I. Dealing with Railroad Location. Part II. Dealing with Railroad Construction. Part III. Dealing with Reconnoissance and Exploratory Surveys. Part IV. Giving various General Information. To these are added a Short Appendix and a Set of Tables, comprising those generally required for Field use. Although Part III should, from its nature, take precedence over Parts I and II, since Reconnoissance is usually the first step towards Location, yet the subject of Exploratory Survey- ing is here treated too fully-in comparison with Parts I and II-to warrant its being regarded merely as an Introduction to them. I have therefore considered it a special subject, and accordingly given to it a subsequent position. ▼ 3 اسم 3 CONTENTS. RAILROAD LOCATION. GENERAL CONSIDERATIONS. SEC. 1. Conditions of Economical Location. } 2. Train Resistances. 3. 4. 5. 6. 7. Rolling Resistance. Resistance due to Oscillation and Concussion. Atmospheric Resistance.. · Resistance due to Curvature. Resistance due to Gravity. 8. Diagram of Resistance. Limiting Velocity on any Grade.. 9. Propelling Force of Locomotive. Coefficient of Adhesion. Sliding Friction. Limiting I.H.P.. Weight on Driving-wheels. Grate-surface. 10. Diagram of Propelling Force.... 11. · Limiting Speed for any given I.H.P.. Internal Frictional Resistances. Back-pressure, Wire-drawing, etc.. Diameters of Driving-wheels. 12. I.H.P. required at any given speed. Most Economical Speed Limiting Grade.. 13. Weight of Locomotives and Rolling-stock. 14. Resistance due to Inertia.. Rotative Energy of the Wheels.. 15. Resistance caused by Application of Brakes.. Automatic Brakes Hand Brakes.. 16. Initial Velocity... 17. Height corresponding to Velocity. Table of Heights corresponding to Velocity. • PAGE 2 2 1 Q Q 0 0 ** 3 3 4 4 5 8 8 8 8 9 9 9 10 10 11 11 11 • D 12 12 13 13 13 14 14 15 15 15 16 17 17 • 17 18 V 18. Assumption of Mean Resistance and Mean Propelling Force.. 19. Graphic Method of solving Dynamical Problems... 20. Examples... . . vi CONTENTS. SEC. 21. Rise and Fall…….. Profile of Velocities.. PAGE 19 22. Effects of Rise and Fall. 23. 24. Maximum Grade.. 25. Economy of Locomotive.. 26. Compensation for Curvature. 27. Compensation for Brakes.. 28. Broken Grades.... Momentum Grades.. 29. Danger of breaking Train and Derailment. 30. Work done on Grades. 31. Pusher-grades Table of Pusher-grades 32. Maximum Curvature. Safe Speed on Curves. 33. Short Tangeuts... 1 Location of Curves.. • 34. Table of Work done against Resistances. 35. Cost of Work done against Resistance 36. Cost per Train-mile………. 37. Economy of Construction. 20 20 21 22 22 23 23 2 2 7 * 2 * * * 23 24 24 26 26 = 26 26 27 27 28 COST OF OPERATING. 8888888 28 28 29 30 30 30 31 38. Cost of Operating Pusher-grades 39. To test Relative Cost of Various Routes.. 40. Effect of Alterations in Alignment.. 41. To estimate Effect of Ditto... RECEIPTS. 42. Deviating to catch Way-business 31 COST OF CONSTRUCTION. 43. Average Cost of Track....... Average Cost of General Construction. Average Cost of One Mile of Track..... Cost of Trestle-work, Trusses, Tunnels, etc.. 8888 32 32 33 33 INSTRUMENTS. 44. Transit Adjustments... 45. Remarks.. 46. Stadia. Adjustments 47. Compass.. Remarks... 48. Magnetic Variation.. Chart of Magnetic Variation.. 49. Dumpy Level……. 50. Y Level... · • · 36 39 42 *****& 34 34 42 42 43 断断​代 ​44 45 45 CONTENTS. vii SEC. 51. Correction for Curvature and Refraction.. 52. Hand Level PAGE 46 47 THE SURVEY. 53. Reconnoissance and Preliminary Surveys.. Running the Line to Grade... Table of Grades and Grade Angles 54. Transit Work..... 48 49 50 51 55. Latitudes and Departures. 52 • 56. Azimuth Observations 57. ... A. Maximum Elongation of Polaris B. Observation of y Cassiopeia and Polaris.. C. Observation of Alioth and Polaris... 58. Convergence of Meridians 59. Simple Triangulations... Offsetting the Transit-line.. 60. Levelling. • Precision of a Line of Levels 61. Taking Topography 62. Contour Lines. Locating by means of Contour Lines.. 63. Levels and Curvature... 54 55 56 57 58 60 61 61 62 62 64 64 66 64. Equations.. 65. Value of Topography.. 66 67 68 68 ► • 66. Tangents and Curves 67. Selection of Curves by Eye. 68. Balance of Cuts and Fills.. 69. Establishing the Grades.. · Rough Estimation of Grading. 70. Estimating by Centre Heights.. CURVES. 71. Radius and Degree of Curves. 72. Corrections for 50-foot Chords 73. Length of Curves...... 74. Nomenclature and Symbols.. 75 Fundamental Formulæ.. • PROBLEMS IN SIMPLE CURVES. 76. To lay out a curve by deflection angles.. To find corrected length of any sub-chord. Example.... 77. To locate a curve when the apex is inaccessible.. 78. To locate a curve by offsets from a tangent... Ditto if the apex, P.C., etc., are inaccessible.... 79. To locate a curve by offsets from the chords produced. 贪 ​? 80. To locate a curve by ordinates from a long chord... Example.... Ditto by mid-ordinates. 69 69 69 71 71 72 74 74 75 * 2 2 2 2 78 78 78 81 82 83 85 87 87 88 viii CONTENTS. SEC. 81. To pass a curve through a fixed point, I being given.. 82. To run a tangent from a curve to any fixed point... 83. To connect two curves by a tangent.. PAGE 89 90 90 84. Given a curve joining two tangents, to change the P.C. so that the curve may end in a parallel tangent....... 85. To transfer a curve both at its P.C. and P.T. to parallel tangents. 86. Given a curve joining two tangents, to change R and the P.C. so that the new curve may end in a parallel tangent at a point opposite to the original P.T.. 87. Given a curve, to find R of another curve, which, from the same P.C., will end in a parallel tangent.. 88. Given a curve joining two tangents, to change R and the P.C. so that the curve may end in the same P.T., but with a change in direction.. 91 92 92 93 93 COMPOUND CURVES. 89. Locating compound curves.. 90. To locate a C.C. when the P.C.C. is inaccessible. 91. Given a simple curve ending in a tangent, to connect it with a parallel tangent by means of another curve.. • == 94 94 95 4 95 92. To connect a curve with a tangent by means of another curve of given radius.. 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 its radius.... 97 94. To connect two curves already located by means of another curve of given radius.... 88 98 95. To locate any portion of a C.C. from any station on the curve.. 99 TRANSITION CURVES. 96. Advantages of Transition Curves. 97. 98. 99. Method I. . . . . Method II……. Method III. 100. Vertical Curves. CONSTRUCTION. 100 100 104 105 10% 101. Division of the Subject...... 109 A. SETTING OUT WORK. 102. Clearing Right of Way, etc.. 109 103. Location of Culverts, etc.. 104. System of Drainage.-Ditches 105. Checking Benchmarks and Alignment. 106. Cross-sectioning..... • 109 110 111 111 Setting Slope-stakes 112 Points at which Cross-sections should be taken.. • ... 114 CONTENTS. ix SEC. PAGE 107. Reference Points.... 108. Staking out Borrow-pits. 109. Staking out Foundation-pits for Culverts 115 115 110. Setting out Bridge Foundations.. 111. Setting out Trestlework. 112. Setting out Tunnels. 113. Giving "Grade" and centres... Increase in Gauge on Curves. 115. Inspecting the Grading. Shrinkage and Increase. 114. Difference of Elevation on Curves. Effect on the Dump and on Trestles.. 116. Running Track-centres and setting Ballast-stakes. 117. Permanent Reference-points 118. Turnouts and Crossings. 119. Locating by Offsets. • • 120. Example... 121. Turnouts and Crossings on Curves. 122. Curving Rails. 123. Expansion of Rails. 115 116 117 118 120 121 121 123 124 124 125 125 125 127 129 129 132 132 B. THE ESTIMATING OF LABOR AND MATERIAL. 124. The Cost of Earthwork and Rockwork removed by Carts.. 125. Ditto, by other means.. 133 136 126. Overhaul... 136 127. The Calculation of Earthwork. 137 • Areas of Cross-sections... 139 128. 129. The Pyramid. Wedge, and Prismoid. The Prismoidal Formula.. 139 140 130. The Method of Average End-areas 143 • Prismoidal Corrections... 143 131. 132. The Method of Equivalent Level Sections. The Method of Centre-heights..... 146 147 133. 134. Earth-work Tables Correction for Curvature. 147 148 135. 136. Contents of the Toe of a Dump. General 149 · • 152 137. Timber-work 152 Table of Board Measure. 138 Iron-work……. 139. Weight of Bolts, Nuts, and Bars.. Railroad Spikes.. Angle-bars and Bolts per mile. Weight of Rails per mile.. Ballast and Ties per mile.. • · • • 151 Fractions of an Inch in Decimals of a Foot.. • · • 153 153 • + 153, 154 • • 154 155 155 156 X CONTENTS. SEC. EXPLORATORY SURVEYING. 140. Introduction…... INSTRUMENTS. 141. The Sextant.-Adjustments, etc.. 142. 143. 144. 145. Use of the Sextant.. Parallax Supplementary Arc. Observing Horizontal Angles. Eliminating Instrumental Errors. 146. The Artificial Horizon. 147. The Chronometer. 148. Barometers.. 149. 150. 151. 152. Barometric Formulæ. Reduction of Errors of Gradient. Taking Readings.. Diurnal and Annual Gradient.. 153. The Cistern Barometer. 154. 155. 156. • To fill a barometer.. Reading the barometer. Cleaning the barometer. 157. The Aneroid Barometer……….. 158. Elevation Scales.. EXPLORATORY SURVEYS. PAGE 157 157 159 159 161 161 162 162 164 165 166 168 169 • 169 170 170 171 • 171 • 172 173 159. Division of the Subject.... 160. To find the distance apart, etc., of two inaccessible points.. 161. The "Three-point Problem". 162. Positions fixed by bearings. 163. • • Positions fixed by intersection.. 164. Obtaining Heights of Mountains trigonometrically 165. 166. 167. Refraction of the Air.. Reciprocal Angles.. · • By Depression of the Sea Horizon.. 168. Observing Altitudes and Depressions…….. With a Sextant and Artificial Horizon. With a Transit... 169. Measurement of a Base……. Correction for Temperature. Reduction to Sea-level... · 170. Example of Triangulating on Exploratory Surveys.. 174 175 · 176 178 178 178 180 180 181 181 181 182 182 182 183 183 184 185 · 185 . 185 186 186 186 171. To measure a horizontal angle without an instrument……. 172. To measure a vertical angle without an instrument.. 173. Measurement of Distance by Sound. • • • 174. Measurement of Time by Vibrations... 175. Direct Measurement and Compass Courses. Odometers and Pedometers. Estimating the Rate of Progress. CONTENTS. X1 Ex • 197. 198. By Lunar Culminations.. 199. By Lunar Distances. 200. 177. 178. 179. 180. 181. SEC. 176. Astronomical Observations.. Solar Time………. Equation of Time. Sidereal Time….….. Right Ascension and Declination. Correcting for Longitude, etc.. 182. Hour-angle... 183. Examples 184. Refraction. 185. Parallax. ... 186. 87. 188. Correcting for Semi-diameter. Augmentation. Dip. Summary of Corrections • · • 189. Latitude.—By Meridian Altitudes. 190. 191. 192. 193. Remarks By Transits across the Prime Vertical By an Altitude out of the Meridian. By Double Altitudes... :94. By an Altitude of Polaris at any time • 195. Longitude.-Local Time, by an Altitude of a Star. Local Time, by Equal Altitudes of a Star 196. Local Time, with a Transit. PAGE 187 187 188 189 189 • • 190 191 192 194 195 197 197 . 198 198 199 201 202 · . 203 205 206 : 06 • 208 208 209 210 By Jupiter's Satellites 201. To test the chronometer rate.. • 202. To set the transit in the meridian. 203. Interpolation by Successive Differences. 214 214 214 215 204. "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 versâ………. 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 211. Development of a Spherical Surface. 212. Example.. 213. Star Map. Star Tables.. 221 221 • 2-22 225 226, 227 [ 228 228 229 229 • 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. xii CONTENTS. SEO. 218. Transverse Strength of Rectangular Beams... 219. Natural Slopes of Earth.. 220. Weight of Earths, Rocks, etc., per cubic yard. 221. Weight of Timber and Metals per cubic foot.. 222. Mortar, Cement, and Concrete... 223. Notes on Timber.—Selection of Trees.. • PAGE 229 230 230 291 231 231 224. Defects of Timber. 232 · 225. Felling Timber.. 233 226. Seasoning and preserving Timber. 233 227. Decay of Timber. 234 228. 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... 211 234. 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 64 " III. Tangential Offsets at 100 feet.. IV. Mid-ordinates to 100-foot Chords. 259 259 · V. Long Chords.... 260 VI. Mid-ordinates to Long Chords. 263 ་ เส 66 44 XI. "L Secants and Cosecants. 66 XII. 66 . XIII. σε VII. Minutes in Decimals of a Degree. VIII. Squares, Cubes, Square and Cube Roots. IX. Logarithms of Numbers.-1 to 1000. X. Natural Sines and Cosines.. Tangents and Cotangents. 264 265 282 285 294 309 Versines and Exsecants... 321 EL 66 XIV. Cubic Yards per 100 feet, in terms of Centre-height.... 345 XV. Cubic Yards per 100 feet, in terms of Sectional Area... 350 XVI. Mutual Conversion of Feet and Inches into Meters and Centimeters.. XVII. Mutual Conversion of Miles and Kilometers. XVIII. Length of 1' arcs of Latitude and Longitude.. XIX. Mutual Conversion of Mean and Sidereal Time. XX. Mutual Conversion of Time and Degrees.. 354 355 355 356 • 358 PART I. RAILROAD 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- cation 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 Operating Expenses Receipts - (Ⓡ ing) ( Interest on Capital spent on Construct. Equipment, etc. = Profits 20 RAILROAD 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. TRAIN RESISTANCES. 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 lbs. per ton (2000 lbs.) of weight of train in the former case, while with Passenger Coaches or Loaded Cars it only amounts to about 4 lbs. 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 lbs. 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 the RAILROAD LOCATION. 3 1 수 ​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 lbs. instead of 20 lbs. 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 lb. 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 lbs. 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 lb. per train running through still air at a velocity of 1 mile per hour, and increases as the square of the speed: thus, c.g., at 40 m. p. h. it amounts to about 480 lbs. 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 RAILROAD LOCATION. resistance, (b) it may be ascertained with tolerable accuracy by allowing 0.03 lb. 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 lbs. at 40 m. p. h. = 480 480 960 lbs. (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 3.7 lbs. 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 ordinary 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 lbs. 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 lb. per ton. Thus, e.g., on a 5° curve it amounts at a velocity of 35 m. p. h. to about 2 lbs. 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 for 3 (Fig. 1) to the force of gravity AC, which enters into the question; or, what is the same thing, the ratio of ab to ac. 6 B A FIG. 1. a But since, in dealing with ordinary inclines, we may con- sider ac = cb, we may say that AB ab AC cb' 3 I WAS ** so that the resistance caused by gravity per ton (2000 lbs.) equals in lbs. 20 × rate per cent of the grade. Thus on a 2.5 p. c. up- grade the gravity resistance equals 50 lbs. 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 ordinates to the resistance line for the 10° curve, and add 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 a0.25 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 +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. Resistance 2.00 1.50 Grade Grade Total Resistance 0.25 Grade on' Curve Total Resistance on 10. Curve -Level Evek Tangent, Total Resistance on 10.0 10 20 Oscillation Concussion Rolling Atmospheric Registance 40 50 CO Kiles per hour. 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 lbs. (6) 1 -M. IN PA Line D C Line G Resistance Levél Tange Propelling Force 10 20 30 40 50 60 Miles per hour. DIAGRAM II. PROPELLING FORCE OF LOCOMOTIVE IN LBS. PER TON. Locomotive 500 I. II. P. Engine and Tender = 60 tons. f = 0.2 10 Cars, 20 tons each. SCALE, 1 inch vert. = 10 lbs. (7) 8 RAILROAD LOCATION. In order to find the Limiting Velocity 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 m. p. h. 9. Next comes the consideration of the counteracting force, namely: THE PROPELLING FORCE OF THE LOCOMOTIVE. - The Coefficient of Adhesion, i.c., 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, 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 ད 1:|: 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 lbs.) on the drivers, coefficient of adhesion, f = 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 ƒ = 0.2, the above formula becomes I. H. P. = 7NV (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 any 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 RAILROAD 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 lbs. per ton (2000 lbs.), 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-f 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: 2000 Wf Make OA = 36.9 lbs. per ton. Tot. Weight of Train Then draw Aa = I. H. P. 5.9 Wf 17.6 miles per hour, 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 O equal to 40A, the offset equals 4Aa; at a distance equal to OA, the offset equals 2A, 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 cngine 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 lbs. per ton (2000 lbs.) 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 KAILROAD 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, For instance, it by reason of its higher adhesive qualities. Thus would probably tax the engine with large 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. 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 Diagramı 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 f, 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 lbs.) by the resist- ance in lbs. per ton (taken from Diagram I) and multiplying the product by .003 V (V being 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 V3, and the tractive force nearly as V2. The total amount of steam used theoretically, on a run, is nearly proportional to V. 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 RAILROAD LOCATION. 13 velocity. To ascertain the Limiting Grade which it is possible to work, we find that an engine and tender weigh- ing 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. Heavy Passenger Engine. Consolidation Engine. Decapod Engine... No. of Drivers. Weight in tons on each Driver. 4 512 8 10 6 7 2000 lbs.) (1 ton Weight in tons, engine and tender, with fuel and water. 55 75 95 Box car, empty, weight 10 tons. 66 loaded, Flat empty, 20 8 Passenger car, empty, weight 20 tons loaded, 25 Drawing room car, Sleeping-car, weight, 35 30 to 45 Length 34 feet. CC 34 50 50 to 60 feet. 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 lbs. 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 no frictional resistances. It is found by experiment that a force of 1 lb. acting on a weight of 32.2 lbs. (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 RAILROAD 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 lbs. per ton of train equals 1.518 V 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 1.6 V F= t to 1.7 V t where F = constant accelerating force in lbs. per ton (2000 lbs.) 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 S = 44 Vt, where S 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 1 formula given above for Fapplies 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 lbs. 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. I 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 lbs. per ton, and thus by our previous formula gives a value for t = 0.53 minutes, from which we can obtain S1400 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 which would be required to stop it if going at 30 m. p. h. in the same time.. 16. So far we have dealt only with a change of velocity from Rest to V, or from V to Rest. Suppose, however, in the former case that the train, instead of being at rest, before the accelerating force F is applied, has an Initial Velocity (v). The formulæ given in section 14 then become changed, F in 16 RAILROAD LOCATION. this case varying from about F = 1.6 (V v) t to 1.7 (V — v) t and S = 44 ( V + v)t. And just as the previous formulæ 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 × 20 = 32 lbs. per ton, which gives a resistance equivalent to a +1.6 p. c. grade. Problems such as the above, where the value of Fis 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 lbs. 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 for a 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 formulæ, we can, by finding the value of Swhen F = 2000, obtain for any given velocity the corresponding vertical height. RAILROAD LOCATION. 17 1 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. 0 1 C? 2 3 4 10 5 6 77 со 8 9 10 3.5 4 3 5.1 5.9 20 14.1 15.5 17.0 18.6 30 31.7 33.8 36.0 40 56 3 59.2 62.1 50 88 O 91.5 95.1 60 126.7 131 0135.3 70 172.5 177.4 182 5 6.9 79 20.2 22.0 23.8 38.3 40.7 43.1 45.6 65.1 68.2 71.3 74.5 77.8 98.9 102.7 106.5 110.4 114.4 118.4122.5 139 7 144.2 148.7 153.3 158.0 162.8167.6 187.6 192.8 198 0 203.3208.7 214.2219.7 90 10.2 11.4 12.7 25.7 27.6 29.6 48.2 50.8 53.5 81.2 84.6 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 m. p. h. to 25 m. p. h. in a distance of 1000 ft. we have Height corresponding to 40 m. p. h. ..25 56.3 feet = 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 formulæ similar in effect to those given above, obtaining approximate values of S. 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 RAILROAD 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. = 0Q X Take any ordinate NQ, and make PQ NM Similarly take other ordinates, and thus fix other positions of the point P. Draw the curve OPD through these points. Then, if (as in : 20 Diag. II) 1 inch vertical = 10 lbs., and 1 inch horizontal miles per hour, the area (shown shaded in Diag. II) enclosed by the curve OPD, the line OH, 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 ordinates of 20 and 34 m. 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 Opq, 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 in 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 lbs. 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 ordinate from OH to the curve EG, for the * All measured in inches on the diagram. RAILROAD LOCATION. 19 zero. line of propelling force then coincides with OH-i.e., equals 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. B -21.m.p.h. C ·27—” >> A 10.m.p.h. 40. Datum 80. 70. D 40. 50.th.p.h. E רד F 190 40. £ FIG. 2. In Fig. 2 let ABCDEF 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. rr B = 56 - 40 = 16; .. Vat B21 m. p. h. C=16+10=26; · (( C = 27 • D = 40 E = 50 • F= 86 - 30 = 56; F = 40 • D = 26 + 30 = 56 ; E=5630 86; 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 specd: 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- locity 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 1 1 - 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 Fa 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 certain 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 lbs. 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 lbs. 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 lbs. per ton,--i.e., 12 lbs., in accordance with Sec. 3, and a small margin of 3 lbs. 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. (See Note H, Appendix.) 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. 26. Compensation for Curvature.-From Diagram I we see that at 10 miles per hour the resistance for each degree of curvature is about 1 lb. per ton, i.e., equivalent to a +0.5 p. c. grade, and that at about 30 m. 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 LOCATION. 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 m. 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 RAILROAD LOCATION. train is small compared with the Total Energy to be expended on them. 4. The nearer the uniform grade approaches the "Maxi- inum 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 IIeights —e.g., an increase in elevation of 20 feet reduces the velocity from 30 to 18 miles 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 the 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 C 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 II in Fig. 4 has to be reached, the work done in hauling the train along the uniform grade EII 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 A 100. 80 480. 150. 300. B 250. FIG. 3. grades where a lighter engine could have hauled the train equally well. If each of the divisions EF, FG, and GH were 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 H E FIG. 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 II 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 POSSIBLE WITH— Grade worked by one Engine. Net Load of Train in tons. 1 Pusher. 2 Pushers. Level. 2675 0.38 0.74 0.2 1758 0.75 1.26 0.5 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 cou- 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 Carves, it is diffi- 66 RAILROAD LOCATION. 27 } ' 1 ! 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 D 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 RAILROAD LOCATION. LROAD TABLE SHOWING COMPARATIVE VALUES OF RESISTANCES AS REGARDS WORK DONE Items. Distance. Curvature. Rise and Fall. 1 mile 5280 feet. 600° 25.0 feet. 1° Curvature 8.8 10 0.041 1 foot Rise and Fall.. 211.2 เ 24° 1.0 66 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 OPERATING. 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. 5.86 1° 0.166 เ 1 foot Rise and Fall. 35.2 (jo 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: Motive Power.... Train Expenses....{ Road Repairs General.... • { Oil, Fuel, Waste, Water. Driver, Fireman. Repairs. Train Hands. Repairs and Renewals to Cars. Track, Road-bed, Structures. 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. 1 foot.. 1° Curvature.. 1 foot Rise and Fall.. $350 $5,833.33 0.066 0.39 2.33 1.10 6,50 38.88 J 2 } 1 This assumes that each " daily" train only runs 350 days in the year, which makes a sort of allowance for Sundays, "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 × $388 = $776 in lowering (only one foot) such a summit as C in Fig. 3; but if Chad 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) × 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 an 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 from to as much as if it had been run, 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 × 5 × $280 = $5,600 $280 Work not done, 30 × 2,100 4 $7,700 Total..... 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 × 1000 × $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: H C L=1+ + 150 900' where 7 = actual length in miles, H C total rise and fall in feet, 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 J * Y י 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. RECEIPTS. 42. The Receipts usually vary from about 1.5 to 2.0 the cost of operating; and it is not often that the locating-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 $2000 per annum. 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 locate 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 lbs )... Angle-bars, per lb. Bolts, Spikes. '' Ties (in place), each. - Ballast--Gravel, p. cu. yd.. Broken Stone, p. cu. yd..... Track-laying per mile.. $25 00 to $45 00 02 03 03" 05 02" 04 20" 50 25 75 75 1 50 250 00 500 00 Then we have the following, according to circumstances: Solid Rock, per cu. yd.... Loose Rock or Hard Pan, per cu. yd.. り ​J Earth, per cu, yd 1st Class Masonry, per cu. yd... 2d 16 66 3d 64 " Dry rubble Riprap, per cu. yd. .. Iron erected in bridge-work, per lb... Timber in Trestles, per M. เ "L ་་ Culverts, • Log Culverts, per M. Piling driven, per lin, ft $0 75 to $2 00 35 75 10" 50 10 00 “ 30 00 7.00" 10 00 5.00" 7 00 2.00" 5 00 1 00 เ 2.00 08 45.00 80 04 25.00" แ 15 00 25 00 10 00 4. 20 00 Grubbing, per Station... Clearing, per acre Overhaul, p. cu. yd. per Sta. Fencing. per mile of track. Telegraph line-Single wire. 300 00 175 00 .. 66 25 12.00 " 20.00" 01 " 15 20 00 30 00 02 800 00 250 00 RAILROAD 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 lbs. p. yd.) 710 Angle-bars, 20 lbs. each { 1420 Bolts, 7 kegs, 200 lbs. each 5670 lbs. Spikes, 38 kegs, 150 lbs. each 2640 Ties. Ballast, 3667 cu. yds. Gravel. Track-laying. • $3,862 00 355 CO 56 00 171 00 924 00 · -1,831 00 375 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 Trestle work 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 lin. 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 Secs. 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 RAILROAD LOCATION. 35 if on pointing to C and then reducing to the ground we again strike A, this adjustment is correct. verti- But suppose the first time the cal" plane had struck the ground at B, and then on reversing, instead of striking Bagain, it cuts through some point D. Mark a point E between D 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. adjustment should then be correct. This C DIE IA B FIG. 5. D. To make the line of collimation perpendicular to the horizontal axis.-Having levelled up the instrument at O, in Fig. 6, point the telescope to some object C. Turn the telescope over and mark the point A, at a distance 40 B A D E C FIG. 6. equal to about OC, where it strikes the ground in the opposite direction. By making A0 OC we then obtain a correct adjustment for the line of collimation, even though the object- slide is defective; that is the only reason for making A0 and OC about the same length. Reverse, and again point to C; if on turning the telescope over once more it again strikes A, this adjustment is correct. But if instead of intersecting A it cuts through some other point D, then mark a point E between D and B, distant from D by one quarter of DB, and by means of the capstan-headed screws move the diaphragm so as to bring the intersection of the cross-hairs to coincide with E. This adjustment should then be correct. This is liable to throw out adjustment B slightly, so watch that at the same time. E. To make the long bubble-tube parallel to the line of collimation.-Level up the instrument and clamp the vertical arc. By means of the tangent-screw of the vertical arc bring the bubble to the centre of the tube. Then if the : 36 RAILROAD LOCATION. small bubble-tubes were sufficiently sensitive to render the vertical axis, when the instrument is levelled up, truly verti- cal, all points cut by the line of collimation equally distant from the instrument would have the same elevation. But it is more satisfactory to obtain a truly vertical axis by means of the long bubble-tube itself, on account of its greater sensitive- uess; thus: Level up as accurately as possible by the small tubes, and then treat the long bubble-tube as if it were one of the smaller tubes, putting it into a temporary state of adjust- ment A, by means, not of the screws at the ends of the bubble- tube, but by aid of the tangent-screw of the vertical arc, and then by its means obtain a truly vertical axis. Then take the readings on two points A and B equally distant from the instrument and in opposite directions; next move the transit to a point about in the same straight line as A and B, but at as short a distance beyond either of them, say A, as the instrument can be focussed to read and level up by the small tubes. Take the reading at A, say 3.43; then if B were previously found to be 1.84 feet higher than A, the telescope should read 1.59 on B if this adjustment were correct. If we do not read this, the screws at the end of the long bubble-tube must be so altered as to bring the bubble to the centre when the instrument reads 1.59. On again pointing to A, the difference between A and B should then be almost 1.84. If it is not quite 1.84, proceed as before until the adjustment is correct. By moving the instrument into the same line as A and B, as above, we avoid the necessity of levelling up this vertical axis again by means of the long bubble-tube. Besides the above adjustments, some instruments have a means of Centring the Eye piece and also of Adjusting the Object-Slide. (See Note C, Appendix.) 45. Remarks.-Another way of performing adjustment C is by means of an object and its reflection in still water, or even in a plate of syrup. A star at night does well for this, but it is advisable to select one as nearly east or west as pos- sible, as its motion in azimuth is then a minimum. If at any time adjustment C is not correct, we can obtain true results by "reversing," as in Fig. 5, and remembering that half-way between the two points so found is the correct point. This latter remark applies also to adjustment D. It is a good plan to reverse on a back-sight every few sights, as it RAILROAD LOCATION. 377 takes practically no extra time and at once detects if anything is wrong. By taking a point half-way between two points, as D and B in Fig. 6, we can do good work with an instrument in which this adjustment is very far from correct. As regards adjustment E:-If we had a level handy, it is much more convenient to level two points with it; or if there is a sheet of still water at hand, two pegs driven down to its surface do equally well. To ascertain the Index-error of the vertical circle in instruments where it cannot be corrected for instrumentally, set the vertical axis truly vertical, as explained under adjustment E, then level up the telescope and observe the readings on the vertical arc. If they are at zero, there is no index-error; but if not, the difference between the readings and zero is the index-error. If the transit has a Striding-level attached, adjustment C may then be more accurately performed by means of it-whether the striding-level is in adjustment itself or not, for it is only the difference of the readings that is required. To make adjustment C then proceed thus: Level up by the small bubble- tubes and point the telescope towards the north; take the read- ings of the bubble on the glass, both at its east and west end; then reverse the striding-level, end for end, and take the read- ings a second time: one quarter of the difference between the sum of the two east readings and the sum of the two west readings equals the number of divisions on the tube that the bubble must be moved by means of the pivot-screws in order to make the "horizontal axis" level, that end being too high the sum of whose readings is the greater. If the striding-level is in adjustment, we have only to screw up the "horizontal axis" so as to agree with it. We can, of course, adjust the striding-level by placing it on the pivots already levelled, and bringing the bubble to the centre of the tube. Lighting the cross-hairs, when the instrument has no lantern attached, can be effected by fastening a piece of bright tin--or even white paper-over and partly in front of the object-glass, so as to cast the reflection of a light on the ground into the tube of the telescope; but the reflector must not obstruct more than half of the field of the object-glass. A piece of tin or paper with a 4-inch hole in the centre of it, fastened at a suitable angle over the object-glass, answers very well. In moving the diaphragm when the telescope bas an invert- 38 RAILROAD LOCATION. ing eye-piece, it has to go in the opposite direction to what ap- pears to be the right one. If working with an instrument the graduation of which is faulty, read each angle in different parts of the circle. The graduations can always be tested by reading with both verniers on various parts of the circle. In observing an angle, if we take the mean result obtained by both verniers, we eliminate errors due to eccentricity of the vertical axis and the graduated circle, as well as reduce the errors of graduation. When great accuracy is required in reading an angle the best method to use is BORDA'S REPETITION, which slightly reduces the errors of observation, while it diminishes those of gradua- tion in inverse order to the number of times the angle is re- peated. The process is thus: Clamp the vernier-plate to zero, and read the angle by both verniers according to the usual method. Then, keeping the vernier-plate clamped, point the telescope again to the first object, and proceed as before through any number of repetitions. At the end of the final angle read the verniers, adding 360° for each complete revolution which has been made, and divide the total angular measurement by the number of times the angle was repeated. The quotient is the required angle. In this way, provided there is no play about the tangent-screws, an angle can be read with confidence to a few seconds by a very inferior instrument. (See Note I.) In ordinary work, if sure of the correct centring of the verti cal axis and also of the graduation itself, there is no need to read by both verniers; but it is advisable to read always by the same vernier if only one is used. An instrument correct according to the adjustments given above gives correct results when dealing with objects distant from it by the amount OC in Fig. 6, but if there is defective cen- tring of the object-slide-not to be confounded with eccentricity of the optical axis of the object-glass-it will not give correct results in dealing with objects at distances from it greater or less than OC. This can always be tested by ranging points in a 'straight" line for a thousand feet or so, beginning as near to the instrument as the focus will permit. Then, if, on ranging the same points in again from the other end, they do not coin- cide, one balf the difference between the points is the error in alignment. In this way, even with a bad instrument, a straight line can be run. We can of course also run a straight RAILROAD LOCATION. 39 line with an instrument which has a defective object-slide, if in proper adjustment, by taking back-sights and fore-sights equal in length to OC, Fig. 6, so that then the object-slide will occupy the same position as it did when the line of collimation was adjusted. If the object-slide works correctly, then, although the object- glass may be badly centred,-i.e., its optical axis will not co- incide with the centre of the telescope tube,-if the line of col- limation is in correct adjustment for one distance it will give correct results at all distances. Parallax is caused by the focus of the object-glass and that of the eye-piece not coinciding at the cross-hairs. To correct for it, shift the eye-piece in and out until the cross-hairs are seen distinctly. Then point the telescope to some distant object and move the object-glass in and out until the image of the object is seen sharp and clear, coinciding apparently with the cross-hairs. STADIA. 46. Transits used on location should be fitted with adjustable stadia-hairs. These are usually adjusted to read 1 foot on a rod at a distance from the centre of the instrument equal to (100 feet + distance from object-glass in its mean position to the centre of the instrument + focal length of the object-glass), usually making a distance in all of about 101.25 feet. And since the stadia-hairs should be placed so as to be equidistant from the ordinary horizontal hair, at a distance of 101.25 feet the distance read between each pair of adjacent hairs should be 0.50 feet. If the hairs are not adjustable, but are fastened to the ordinary diaphragm, then the measurements on the rod must be regulated to suit the hairs, remembering that the apex of the angle sub- tended by the distance read on the rod is not at the centre of the instrument, but at a point in front of the object-glass by a dis- tance equal to the focal distance of the object glass, which is usually 1.25 feet in front of the centre of the instrument. If the hairs are unadjustable, and we wish to use an ordi- nary levelling-rod to read on, then suppose at 101.25 feet wc read 0.88 feet between the stadia-hairs, we must divide every reading in feet by 0.88 in order to obtain the distance in terms of 40 RAILROAD LOCATION. 100 feet. Thus if at a certain point we read 4.40 on the rod, the distance will be 500 feet, or 501.25 feet from the centre of the instrument. To find the focal-length of the object-glass, focus it for a distant object; the distance from the cross-hairs to the object-glass then equals the focal-length. On sloping ground, if the rodman is careful about holding the rod perpendicular to the line of sight, swaying it slowly to and fro so as to permit of the minimum reading being taken, then if the centre hair reads somewhere about 5 feet on the rod (i.e., the height of the instrument above the ground) we have only to multiply the distance as read on the incline by the cosine of the inclination, in order to obtain the true horizontal distance. But if really correct work is wanted, it is best to have a bub- ble-tube attached to the rod so that it can be held vertically, and then correct for the inclination as follows: / E E E E FIG. 7. 1 I I --G In Fig. 7 the distance EF = AB cos? FEC, EF being in terms of 100 feet, and FEC being the angle of inclination as measured to C, the ordinary horizontal hair of the instrument, assuming that the stadia-hairs are equidistant from C, and that 1 foot on the rod corresponds with 100 feet in distance. In order to reduce this to the centre of the instrument, we should of course add to EF the amount 1.25 X cos FEC, but for ordinary inclinations we may assume this correction to equal 1 foot. Thus, if FEC 30°, and AB 6.00, then = RAILROAD LOCATION. 41 E' F = 1 ft. + (6 × .75)= 451 feet. To obtain the height HG in Fig. 7, the best way is to make CII on the rod equal to the height of the point E above the ground, say 5 fect. Then HG = EF tan FEC. Thus in the above example HG 260 feet. The following table gives the VALUES OF COS FEC, where FEC is the inclination angle: INCLINA- TION. 0' 10' 20' 30' 40' 50' 0° 1.0000 1.0000 1.0000 .9999 .9999 9998 .9997 .9996 .9995 .9993 .9992 .9990 2 .9988 .9986 .9983 .9981 .9978 .9976 .9978 .9969 .9966 .9963 .9959 .9955 .9951 .9947 .9943 .9938 .9934 .9929 .9924 .9919 .9914 .9908 9902 .9897 6 .9891 .9885 .9878 9872 .9865 .9858 7 .9851 .9844 9837 .9830 .9822 .9814 8 9806 .9798 .9790 .9782 .9773 .9764 9 .9755 .9746 .9737 .9728 .9718 .9708 10 .9698 .9688 .9678 .9668 .9657 .9647 11° .9636 .9625 .9614 .9603 .9591 .9580 12 .9568 .9556 .9544 .9532 .9519 .9507 13 .9494 .9481 .9468 .9455 .9442 .9428 14 .9415 .9401 .9387 .9373 .9359 .9345 15 .9330 .9315 .9301 .9286 .9271 .9256 16 .9240 .9:225 .9209 .9193 .9177 .9161 17 .9145 .9129 .9112 .9096 .9079 .9062 18 .9045 .9028 .9011 .8993 .8976 .8958 19 8940 8922 .8904 .8886 .8867 .8849 20 .8830 .8811 .8793 .8774 .8754 .8735 21° .8716 .8696 .8677 .8657 8637 .8617 22 .8597 8576 .8556 .8536 .8515 .8494 23 .8473 .8452 .8431 .8410 .8389 .8367 24 .8346 .8324 .8302 .8280 .8258 .8236 25 .8214 .8192 .8169 .8147 .8124 .8101 26 .8078 .8055 .8032 .8009 .7986 .7962 27 .7939 .7915 .7892 .7868 .7844 .7820 28 .7796 7772 .7747 .7723 .7699 .7674 29 .7650 .7625 .7600 .7575 .7550 .7525 30 .7500 .7475 .7449 .7424 .7398 .7373 312 7347 .7322 .7296 7270 7244 .7218 32 .7192 .7166 7139 .7113 .7087 .7060 33 .7034 .7007 .6980 .6954 .6927 .6900 34 .6873 .6846 .6819 .6792 .6765 .6737 35 .6710 .6683 .6655 .6628 .6600 .6573 A COCO CE CO 36 .6545 .6517 .6490 .6462 .6434 .6406 37 .6378 .6350 .6322 6294 .6266 .6238 38 6210 .6181 .6153 .6125 .6096 .6068 39 .6039 .6011 .5982 .5954 .5925 .5897 40 .5868 .5839 .5811 .5782 .5753 .5725 42 RAILROAD LOCATION. THE COMPASS. 47. The adjustments of the Compass are as follows: A. To make the needle swing horizontally.-Level the compass, then by means of the slide-piece on the needle regulate its centre of gravity so that it will swing horizontally. B. To straighten the needle.-See if both ends of the needle point to exactly opposite graduations while the compass is being turned completely around. If so, the needle is straight and the pivot is properly centred. But if not, the error will arise from either one or both of these not being correct. Turn the compass until some graduation, say 90°, comes precisely to the northern end of the needle, and bend the pivot until they do. Then turn the compass until the opposite 90° is at the north end of the needle. Mark the place where the southern end of the needle then points. Take off the needle and bend it until its southern end points half-way between 90° and the point already marked, while its northern end is kept at the opposite 90° by slightly moving the compass around. The needle will then be straight, although it will not intersect opposite degrees on account of the eccentricity of the pivot. C. To centre the pivot.-Turn the compass around until a place is found where the opposite ends of the needle cut opposite degrees. Then turn the compass quarter-way around, or through 90°. If the needle then cuts opposite degrees, the pivot is in adjustment; but if not, bend the pivot until it does. The needle should then cut opposite degrees while being turned completely around. Remarks. If the magnetism of the needle gets weak, it may be renewed as follows: Cover the needle with a thin film of oil, and then with the north pole-the end marked with a line across it-of an ordinary magnet rub the south end of the needle, beginning at the centre and working outwards towards the end; similarly rub the north end of the needle with the south pole of the magnet. After doing this a few times the magnetism should be sufficiently restored. Reading both ends of the needle corrects for eccentricity of the pivot if the needle is straight; it also of course reduces the errors of graduation. Should the glass cover become electrified, as it will if but slightly rubbed, so that the needle sticks to the under side of it and will not " traverse" properly, touching the glass in RAILROAD LOCATION. 43 several places with the moistened finger, or breathing on it, will remove the electricity. A compass when left standing for any considerable time should always have its needle free, in order to prevent loss of magnetic power. Of course when carried it should always be clamped. Iu taking a compass reading, not only must all iron and steel substances be kept well away, but metal magnifiers of all sorts are liable to cause a slight deflection, owing to the possi- bility of impurity in the material of which they are composed. Magnifiers coated with nickel are especially bad, since nickel itself is a decidedly magnetic metal. Since the magnetic attraction varies in different places, adjustment A, if correct in one place, will probably want looking to if the instrument is taken anywhere else. MAGNETIC VARIATION. 48. By referring to the Chart of Magnetic Variation, we sec that in North America the variation is both towards the east and the west. The line of no variation" which separates these two divisions is found to be constantly shifting west- wards at an average rate of about 4' per annum. This causes a gradual increase in all variations to the west, and a corre- sponding decrease in all variations to the east; and changes similar to these are going on all over the globe. Besides this secular variation we have diurnal and annual variations, but for practical field purposes these latter may be ignored. The former of them is such that the needle attains its extreme westerly position at about 2 P.M. cach day, and its extreme easterly position at about 8 A.M.; while the latter shows itself generally by a slight increase in variations west, and decrease in variations east, during the summer. The chart here given is more as a matter of interest than for any real use in the field. If the variation at any place is wanted accurately, usually the only satisfactory way is to take it directly by observation, as shown in Sec. 57. For very rough work, however, an idea of the amount of variation can be obtained from the chart by interpolating by eye. The lines of no variation" are shown thicker than the others. 160 190 160. 140 120 100 80 60 40 20 20 40 35 70% (0 . 55 W. تھا 60 50- 25, E. 200 E. 30:E 40 15 E. 30 10 E. 20 6E. E. 10 0 5 E. 10 20 9°E. 30 4.5° W.. 10° /W 35°-W 25° W. 20 O 508 W. .... 03 100 120 140 160 2 W. ·W. -60 40 ❤ 50 15E. 20 E. 60 25 E. 5 E. 25 W. ·20° W. 10 W. 15 W 5 W. 30 160 180 30% E W. 160 140 120 100 80 30% W. 10 W. 15 W. 209 W. 25 ว 35 W. 00 W. O 40° W. W. 45° W. W. W. 69 50 40 CHART SHOWING LINES OF EQUAL MAGNETIC VARIATION. 20 55 W. 0 20 40 60 80 100 120 ¡75' AF 140 50 40 30 20 10 0 Lil -10 -20 30 40 50 20°- E. E. 60 160 Eng.by Am. Bk. Note Co. N.Y, RAILROAD LOCATION. 45 THE LEVEL. 49. We will first take the DUMPY LEVEL, which usually needs only two adjustments. A. To make the bubble-tube perpendicular to the vertical axis.-This is done in just the same way as with one of the small bubble-tubes in adjustment A of the transit. B. To make the line of collimation parallel to the bubble-tube.-This is done in a similar way to the adjust- ment of the long bubble-tube of a transit, already described, except that in this case it is the line of collimation that has to be made parallel to the bubble-tube, so that now it is the cross-hairs that have to be moved. In this case of course there is no necessity to set up the instrument in about the same line as A and B" as there was in the case of the transit. " Another way of performing this adjustment is by the method of “reciprocal observations," as given for the Hand- level in Sec. 52. The remarks which applied to the telescope of a transit ap- ply with equal force to the telescope of a level; more especially the remark on the running of a straight line if the object-slide is badly centred. If the level has a means of adjusting the eye-piece and object-slide, see Note C, Appendix. 50. The Y Level has three adjustments as follows: A. To make the line of collimation coincide with the axis of the telescope.-Open the clips of the Y's. To adjust the vertical hair, mark the intersection of the cross-hairs on some fixed object, and revolve the telescope in its Y's so that the level will be upside down; and then if the intersection falls to one side of the object, one half the error must be cor- rected for by the capstan-headed screws. To adjust the hori- zontal hair, turn the telescope over as before, and if the inter- section of the hairs strikes above or below the object, correct as before for one half the error. B. To make the bubble-tube parallel to the line of collimation.-This adjustment consists of two parts. First, bring the bubble to the centre and then revolve the telescope in its Y's through about 20°; if the bubble then runs to one end, half the error must be corrected for by the horizontal screws at the end of the tube, raising or lowering as may be required. 46 RAILROAD LOCATION. For the second part of this adjustment, place the telescope over a pair of opposite levelling-screws, open the clips, and bring the bubble to the centre of the tube. Reverse the telescope end for end in its Y's; if the bubble is not then in the centre, one half the error must be corrected for by the vertical screws at the end of the bubble-tube. On levelling-up and again re- versing, this adjustment should be found to be correct. Swing the telescope half- C. To make the axis of the telescope perpendicular to the vertical axis.-Level up. Place the telescope over a pair of opposite levelling-screws. way round on its vertical axis. If then the bubble has left the centre, bring it half-way back by means of the large cap- stan-headed nuts of the Y's. Then place the telescope over the other pair of levelling-screws, and if necessary proceed as before. This adjustment should then be correct. Remarks.--As with the transit, if the object-slide of a level is defective the line of collimation when adjusted is only cor- rect for back-sights and fore-sights of equal length with the distance of the object on which the line of collimation was adjusted. In levelling, whenever possible, keep the fore-sights and back-sights of equal length: if so, accurate work can be done with an instrument thoroughly out of adjustment, for then the actual height of the instrument itself is of no importance. If, as in levelling uphill, it is necessary to take extremely short forc-sights, they should be counteracted by short back sights --if not at the time, as soon afterwards as possible. 51. There is no need to allow for CURVATURE OF THE EARTH OR REFRACTION in sights under 700 feet, and then, if taking fore-sights and back-sights of about the same length, the corrections would counteract each other; so that it is only in taking an extremely long fore-sight or back-sight, which is not counteracted by a more or less equal sight in the opposite direction, that we need apply corrections for curvature or refraction. For CURVATURE the correction in feet amounts to 0.67L", where L= length of sight in miles, and is to be subtracted from the reading on the rod: this being simply the tangential RAILROAD LOCATION. 47 offset for a curve-see Sec. 78-with radius equal that of the earth. For REFRACTION, on an average, it amounts in feet to 0.1L2, which is an experimental quantity, and is to be added to the reading on the rod. So that, taking the two together, we may say that the correction in feet amounts to 0.57L2, and is to be subtracted from the reading on the rod, the eleva- tion as taken or given by the level being always too low. This is equivalent to about .0002 ft. at 100 feet; so that, since it increases as the square of the distance at say, 1200 feet, it will equal .0002 × 122.03 foot. The following table gives. the JOINT CORRECTIONS FOR CURVATURE AND REFRACTION, worked out by the above formula, and is useful in ascertain- ing the elevation of the surrounding country : stanc Distance in Miles. Correction in Ft. || Distance in Miles. Correction in Ft. 1 0.57 5 14.25 10 57.0 15 128 20 228 25 356 30 513 40 912 50 1,425 60 2,052 80 3,648 100 5,700 Thus from the table, if the level gives a point on a distant mountain, say 30 miles off, the elevation of that point will be equal to the elevation of the instrument + 513 feet. 52. The Hand-Level.-The only adjustment necessary as a rule with this instrument is to make the line of collimation parallel to the bubble-tube. To do this, sight from a point A to a point B, as in Fig. 8, and then back again from B to A. A D B C-- FIG. 8. If the level is in adjustment the two sights should coincide at A. But suppose C is the point struck instead of A; then D, 48 RAILROAD LOCATION. a point half-way between A and C, will be on a level with B: therefore the hair must be adjusted on the line BD. A handier way is of course to adjust it by means of another level, or a sheet of water. THE SURVEY. 53. The object of the following notes is not to show the mode of conducting location,-which of course can only be picked up by actual experience in the field,-but merely to give solutions of the various mathematical and instrumental problems which arise in the course of the work. In the case of Exploratory Surveys, the instruments used and the problems which arise being usually entirely different from those which come into question in ordinary location, they will be considered separately in Part III. A Reconnoissance survey, as generally understood, may also be classed with the above, or it may take the form of a rough preliminary survey, a compass perhaps being substituted for the ordinary transit. As regards Compass-surveys, there is among engineers a strong prejudice against them, but in a country tolerably free from local attraction a compass-line is surely correct enough for preliminary work; for though by it accuracy cannot be obtained at any one point, its errors are not accumulative, but in a great measure counteract each other, so that the line as a whole should give very fair results. Another method of performing rough work is the Stadia process, by means of which very good results have often been obtained, the engineering staff consisting merely of an engi- neer and a rodman, the only instruments used being a rod and a small transit with bubble-tube and stadia attachment. Com- paring compass and stadia work, the former is usually more suitable in timber and the latter in open country. The term Preliminary survey is variously used, sometimes indicating a mere reconnoissance, but more generally a survey the object of which is to obtain accurate topography, in order by its means to select the final location. regards the degree of accuracy to be employed in preliminary work, it of course depends in what way the results are to be As RAILROAD LOCATION. 49 used; but it is generally best to run the transit-line of the "preliminary" with as much accuracy as is attainable under the circumstances. If this is done we then have a line on which we can at all points depend, and which we can use as a base for other lines, knowing, if we branch off from it at one point, the exact course we must make to strike it again at any given station. We will therefore suppose in the follow- ing notes that the final location is to be selected by the aid of an accurately run preliminary line, topography having been taken on either side of the transit line to a distance of from, say, 100 to 600 feet, according to the nature of the ground. On Preliminary Surveys, by means of a hand-level and prismatic-compass, the engineer-in-charge, keeping ahead of the party, is generally able to ascertain approximately where the line will go, and then the transit-man has merely to follow more or less the route indicated, being guided by the con- sideration of running the line as much as possible to a con- stant rate of grade. If the line, however, is being run to the maximum grade-or any other rate of grade which it is the wish of the engineer to maintain-along a continuous trans- verse slope, such as a mountain-side, the transit-man can choose the line tolerably well for himself, since he only has to select his stations so as to maintain the required rate, which he can do by means of the vertical arc. But in selecting these points he has to bear in mind the probable amount of curva- ture which there will be between the station where the instru- ment is standing and the place at which the front picket is to be set, and allow for it in setting the picket. (See Sec. 26.) Thus, suppose he is running the line to a 1.5 p. c. grade, and that he estimates the distance to the picket to be about 500 feet, and the probable total curvature in that distance to be 15°, then the grade-angle, instead of being 51', as in the fol- lowing table, will be 48. If he has stadia-hairs in his instru- ment,—as he ought to have,—he can read off the distance with sufficient accuracy on the picket itself, and in this way form his estimates more closely. The difference in distance along the straight course and along the probable location must also be allowed for where the deviation is great. The following is a 50 RAILROAD LOCATION. Feet per Station. Feet per Inclina- Mile. tion. TABLE OF GRADES AND GRADE-ANGLES. Feet per Station, Feet per Inclina- Mile. tion. Feet per Station. Feet per Inclina- Mile. tion. O .01 .528 21 .51 26.928 17 32 1.01 53.328 31 43 .02 1.056 41 .52 27.456 17 53 1.02 53.856 35 04 .03 1.584 1 02 .53 27.984 18 13 1.03 54.384 35 24 .04 2.112 1 23 .54 28.512 18 31 1.04 54.912 35 45 .05 2.640 1 43 .55 29.040 18 54 1.05 55.410 36 05 .06 3.168 2 04 .56 29.563 19 15 1.06 55.968 36 26 .07 3.696 2 24 .57 30.096 19 36 1.07 56.496 36 47 08 4.224 2 45 58 30.624 19 56 1.08 57.024 37 08 09 4.752 3 06 591 31.152 20 17 1.09 57.552 37 28 .10 5.280 3 26 .00 31.680 20 38 1.10 58.080 37 49 .11 5.808 3 47 .61 32.208 20 58 1.11 58.608 38 09 .12 6.336 4 08 62 32.736 21 19 1.12 59.136 38 30 .13 6.864 4 28 .63 33 264 21 39 1.13 59.664 38 51 .14 7.392 4 49 .64 33.792 22 00 1.14 60.192 39 11 .15 7.920 5.09 .65 34.320 22 21 1.15 60.720 39 32 .16 8.448 5 30 .66 34.848 22 41 1.16 61.248 39 53 .17 8.976 5 51 .67 35.376 23 02 1.17 61.776 40 13 18 9.504 6 11 .68 35.904 23 23 1.18 62.304 40 34 .19 10.032 6 32 .69 36.432 23 43 1.19 62.832 40 54 .20 10.560 653 .ΤΟ 36.960 24 041.20 63.360 41 15 .21 11 088 7.13 .71 37.488 24 24 1.21 63.888 41 35 .22 11.616 7 34 72 38.016 24.45 1.22 64.416 41 56 23 12.144 7 51 .73 38.544 25 06 1.23 64.944 42 17 24 12.672 815 .74 39.072 25 26 1.24 65.472 42 38 25 13.200 8 36 .75 39.600 25 47 1.25 66.000 42 58 26 13.728 8.56 .76 40.128 26 08 1.26 66.528 43 19 .27 14.256 9 17 .77 40.656 26 28 1.27 67.056 43 39 .28 14.784 938 .78 41.184 26 49 1.28] 67.584 44.00 .29 15.312 9.58 .79 41.712 27 09 1.29 68.112 44 21 .30 15.840 10 19 .80 42.240 27 30 1.30 68.640 44 41 .31 16.368 10 39 .81 42.768 27 51 1.31 69.168 45 02 .32 16.896 11 00 82 43.296 28 11 1.32 69.696 45 23 .33 17.424 11 21 .83 43.824 28 32 1.33 70.224 45 43 .34 17.952 11 41 .841 44.352 28 53 1.34 70.752 46 04 .35 18.480 12 02 .85 44.880 29 13 1.35 71.280 46 24 .36 19.008 12 23 .86 45.408 29 34 1.36 71.808 46 45 .37 19.536 12 43 .87 45.936 29 54 1.37 72.336 47 06 38 20.064 13 04 .88 46 464 30 15 1 38 72.864 47 26 .39 20.592 13 24 .89 46.992 30 36 1.39 73.392 47 47 .10 21.120 13 45 .90 47.520 30 57 1.401 73.920 48 08 41 21.648 14 06 .91 48.048 31 17 1.41 74.448 48 28 42 22.176 14 26 92 48.576 31 38 1.42 74.976 48 49 43 22.704 14 47 .93 49.104 31 58 1.43 75.504 49 09 41 23.232 15 08 .94 49.632 32 19 1.44 76.032 49 30 .49 .50 .45 23.760 .46 24.288 24.816 .47 .48 25.344 25.872 26.400 15 28 .95 50.160 32 39 1.45 76 560 49 51 15 49 .96 50.688 33 00 1.46 77.088 50 11 16 09 .97 16 30 .98 16 51 .99 52.272 17 11 1.00 52.800 51.216 33 21 1.47 77.616 50 32 51.744 33 41 1.48 78.144 50 52 34 02 1.49 78.672 51 13 31 23 1.50 79.200 51 34 RAILROAD LOCATION. 51 Station. ad Jod TABLE OF GRADES AND GRADE-ANGLES.—Continued. Feet per Inclina- Mile. tion. Feet per Station. Feet per Inclina- Mile. tion. Feet per Station. Feet per Inclina- Mile. tion. O " 1.51 79.728 51 541.91 100.848 1 05 39 3.55 187.440 2.01 59 1.52 80.256 52 15 1.92 101.376 1 06 00 B.60 190.080 2 03 42 1.53 80 784 52 36 1.93 101.904 1 06 20 3.65 192.720 2 05 25 1.54 81.312 52 56 1.94 102.432 1 06 41 B.70 195.360 2 07 08 155 81.840 53 17 1.95 102.960 1 07 02 2.75 198.000 2.08 51 11.56 82.368 53 37 1.96) 103.488 1 07 22 3.80 200.640 2 10 34 1.57 82.896 53 58 1.97 104.016 1 07 43 3.85 3.85) 203.280 2 12 17 1.58 83.424 54 19 1.98 104.514 1 08 043.90 205.920 2 14 00 1.59 83.952 1.60 84.480 54 39 1.99] 55 00 2.00 105.072 1 08 243.95 208.560 2 15 43 105.600 1 08 454.00 211.200 2 17 26 1.61 $5.008 55 212.05 108.240 1 10 28 4.10 216.480 2 20 52 1.62) 85.536 55 41 2.10 110.880 1 12 11 4.20 221.760 2 24 18 1 63 86 064 56 02 2.15 113.520 1 13 54 4.30 227.040 2 27 44 1.64 86.592 56 22 2.20 2.20 116.160 1 15 37 4.40 232.320 2 31 10 1.65 87.120 56 43 2.25 118 800 1 17 204.50) 237.600 2 34 36 1 66 87.6-48 57 04 2.30 121.440 1 19 03 4.60 242.880 2 38 01 167 88.176 57 24 2.35 124.080 1 20 46 4.70 248.100 2 41 27 1.68 88.704 57 45 2.40 126.720 1 22 29 4.80 253.440 2 44 53 1.69 89.232 58 06 2.45 129.360 1 24 12 4.90 258.720 2 48 19 1 70 89 760 58 262.50 132.000. 1 25 56 5.00 264.000 2 51 45 1.71 90.288 58 47 2.55 134.640 1 27 39 5.10 269.280 2 55 10 1.72 90.816 59 07 2.60 137.280 1 29 22 5.20 274.560 2 58 36 1.73 91.344 59 28 2.65 139.920 1 31 05 5.30 279.840 3 02 09 1.74 91.872 59 49 2.70 142.560 1 32 48 5.40 285.120 3 05 27 1.75 92.400 1 00 09 2.75 145.200 1 34 31 5.50 290.400 3 08 53 1.76 92.928 1 00 30 2.801 147.840 1 36 14 5.60 295.680 3 12 19 1.77 93.456 1 00 51 2.851 150.480 1 37 57 5.70 300.960 3 15 44 1 78 93.984 1 01 11 12.90 153.120 1 39 40 15.80 306.240 3 19 10 1.79 94.512 1 01 32 1/2 95| 155.760 1 41 23 5.90 311.520 3 22 36 1 80 95.040 1 01 52 3.00 158.400 143 06 [6.00 316.800 3 26 01 1.81 95.568 1 02 13 3.05 161.040 1 44 49 6.10 322.080 3 29 27 1.82 96.096 1 02 34 3.10] 163.680 1 46 32 6.20 327.360 3 32 52 1.83 96.624 1 02 54 8.15 166.320 1 48 15 6.30 332.640 3 36 18 1.84 97.152 1 03 15 3 20 168.960 1 49 58 6.40 337.920 3 39 43 1.85 97.680 1 03 35 3.25 171.600 1 51 41 16.50 843.200 3 43 08 1.86 98.208 1 03 56 3.30 174 240 1 53 24 5.60 348.480 3 46 34 1 87 98.736 1 04 17 3.35 176.880 1 55 07 6.70 353.760 8 49 59 1.88 99.264 1 04 378.40] 179.520 1 56 50 6 80 359.040 3 53 24 1.89 99.792 1 04 58 3.45 182.160 1 58 33 6.90 364.320 3 56 50 1.90 100.320 1 05 19 3.50 184.800 2 00 16 7.00 369.600 4 00 15 When the running is tolerably easy, instead of taking a series of short courses, it is often better to insert a curve at once, selecting one which is likely—as near as can be guessed -to coincide with the probable final location; for in this way truer results can be arrived at than by a series of independent courses. 54. As regards the Instrument-work itself, the method of reading angles as so much "to the right or to the left" is " 52 RAILROAD LOCATION. decidedly feeble. The best way is to start with the verniers reading zero when the telescope is pointing towards the mag- netic, or still better, the true north; then the first augle read is the magnetic (or true) bearing of the first course. On mov- ing the instrument up to the front picket, the horizontal circle should be kept clamped, and the reading of the vernier again, when the instrument is next set up, constitutes a check on the former reading; for though there will probably have been some slipping of the plates, owing to the shaking while being carried from one station to the other, an error of a degree or so is easily detected. When the telescope is pointed to the back- sight the verniers should then read the same as they did at the other end of the line, and thus for the next course, on turning through the required angle, it will be its bearing-magnetic or true as the case may be-that is read. The compass-reading should also be taken for each course, at each end of the course, which thus forms an additional check on the work, and also detects local attraction. For if, when the instrument is set up, the needle does not on any course read the bearing correspond- ing with the vernier reading (if the zero corresponds with the magnetic north) or does not give the difference in the readings equal to the " variation," if the zero corresponds with the true north, if the work is correct, the cause is either the change in variation, or local attraction, or both these causes combined. If the instrument is a good one there is no need to read by more than one vernier. (See Sec. 45.) But it should usually be the same vernier that is read, and that vernier will then always be on the same side of the transit-line. If, however, the line of collimation, from some cause or other, such as a defective object-slide which cannot be remedied in the field, is unreliable, the error can be counteracted to a large extent by taking the bearings with the same vernier on op- posite sides of the line at alternate stations. 55. With the bearings taken as above, or in fact taken in any way, the most satisfactory method of plotting the work is by means of LATITUDES AND DEPARTURES. This method involves a little extra work, but its advantages over the ordinary protractor method-or even the method of "chords" or "natural tangents"-are so great as to make the few minutes extra time taken in preparing the notes time well spent. The main advantage of this method is that an error RAILROAD LOCATION. 53 made in plotting one station is not transmitted to the next, as in the ordinary methods, for each station is plotted entirely independent of the previous one; and thus of course we can plot any one part of the location on the plan in its right posi- tion, without having to work through from the beginning. Again, if we know the position of the point we are making for, we can, without keeping a continuous plot of the work, tell at any station how much we are off our direct route, and what course we ought to steer to strike the point we are mak- ing for. The method of keeping and plotting the notes is best shown as follows: 1600. 987.2 North & South Base. 518. 164.5 44.00 36.007 10.36 21.00 East & West Base. 897.2 FIG. 9. 1961.2 26.50 2352.5 2343.3 2857.5) Suppose Fig. 9 represents the first five courses of a prelimi- nary line, the notes for these courses will then be kept thus: Total Total Sta. Dist. Read. Lat. Bearing. Dep. Lat. Dep. 0 10.36 1036 1064 60° 90° N. 60º E. E. 518 897.2 0 1064. 518. 897.2 21 00 26.50 550 130° 950 S. 50° E. -353.5 421.3 518. 1961.2 30° N. 30° E. 822.7 475. 164.5 2382.5 36.00 800 -40° N. 40° W. 612.8 -514.2 987.2 2857.5 44.00 1600. 2343.3 Readings which give a westerly course should be considered negative; so also should latitudes south and departures west, as shown above. Then 54 RAILROAD LOCATION. and Latitude for any Sta. = Distance × Cosine of Bearing, Departure Total Latitude for any Sta. X Sine Total Latitude for preceding Sta. + Lat. for preceding Sta. Total Departure for any Sta. "" Total Departure for preceding Sta. + Dep. for preceding Sta. The term Latitude" is an abbreviation of "Difference of Latitude." The terms Cosines" and Sines" are more ap- propriate when the bearings are kept with no particular refer- ence to the true or magnetic meridian. By the aid of cross-section paper (if true to scale) we can plot the survey from the notes with only a straight-edge. Thus, e.g, to find the position of Sta. 26+ 50, we read off along the N. and S. base a distance to the north equivalent to 164.5 feet, and along the E. and W. base a distance to the east equivalent to 2382.5 feet; the intersection of the co- ordinates from these two points gives the position required. On a long plan, if we have the base-lines drawn straight, and points accurately scaled off along them at, say, every 1000 feet, there is very little chance of making an appreciable error in the plotting of the plan if the notes are correctly worked out. But although this method is undoubtedly the best, unless the notes are well checked, it is very liable to give rise to errors owing to arithmetical mistakes in the notes them- selves. But where good work is wanted, and in cases where probably the method of plotting by "chords" or "natural tangents" would otherwise have been used, the method of Latitudes and Departures, well checked, gives far better results, and probably takes no longer than the other ways. 56. The only way in which to feel sure that there are no appreciable mistakes in the transit-work is to check the bear- ing of the alignment every now and again by an observation for azimuth. This should be done, if possible, before starting the survey, or in any case as soon after as possible, and the notes then already taken reduced to their true bearings. By taking the magnetic pole as the standard of our bearings, we have no means of applying an accurate check to the work at a later period; but if we start with the vernier at zero, when the telescope is pointing to the true north, we can then check our course at any time on the survey. ! RAILROAD LOCATION. 55 Engineers generally fight rather shy of anything in connec tion with astronomical work; but considering that it is almost as easy to check the alignment by means of a star as by any known point on the Earth's surface,-and usually much more accurate,—it is a great pity that observations for azimuth are not used more frequently than they are. It is so much more satisfactory for the transit-man himself to know if he is doing good work; and considering that the transit-line is usually taken as the basis of all the plans to be afterwards constructed, every possible means of checking the work should be used. 57. The handiest methods of obtaining the true north are the following, one of which is applicable in most northern latitudes about every 6 hours, and can be applied without any knowledge at all of astronomical work: A. By a Maximum Elongation.-In Fig. 10 let Z represent the zenith, P S the pole, the Pole-star (Polaris). W Z S E P FIG. 10. Then the small circle round the pole shows the path and direction of the star's motion, the time taken in making the circuit being nearly 24 hours. Now the radius of this small circle in angular measure is only about equal to 14° (or 23 diameters of the sun), so that the apparent motion of the pole-star in azimuth (i.e., horizontally) will, when due east or west, be nothing at all, and for several minutes together when about east or west the motion will be inappreciable to ordinary railroad transits. Thus if we know about what time the star will be at its east or west elongation,-i.e., due east or due west,-and also the amount in azimuth by which when at those points it will be distant from the pole, we can, by setting the telescope on the star when at either of its elongations and applying the required correction in azimuth, obtain the direction of the true north. The following table shows approximately the times at which the elongations will occur. The amount of the correction in azimuth, which really equals the angle WZP (or EZP), may be found by solving the spherical right-angled triangle WPZ, the angle at W being 90°, the side WP being equal to 90°-the 56 RAILROAD LOCATION. declination" of the star. For Declinations of Stars see Table in Sec. 213. Thus we have Sin azimuth = cos (dec.) sec (lat.), PZ being the complement of the latitude of the place of ob- servation. Thus suppose in latitude 50° N., in January 1889, we have the telescope clamped on Polaris at its eastern clongation, the vernier reading 2°.05'; then the sine of the azimuth correc- tion = .0349, which gives a value for the correction of 2°.00, so that the telescope will be pointing due north when the vernier is set to read 0°.05'. (See note D, Appendix.) TIMES OF ELONGATIONS OF POLARIS. Month. 1st Day. 11th Day. 21st Day. Eastern. Western. Eastern. Western. | Eastern. Western. h m. Jan.. Feb... h. m. 0.39 P.M 0.31 A M. 10.36 A.M. 10.25 P.M. h. m. 11.59 A.M. h. m. 9.57 Mar. 8.46 6 8.34 66 8.07 9.45 7.55 9.18 h. m. h. m. 11.47 P.M. 11.20 A.M. 11.08 P.M. 66 9.06 "L 7.27 7.16 ( April. 6.44 6.32 66 6 05 เ 5.53 แ 5.26 5.14 66 May. 4.46 4.34 4.07 เ 3.55 เ 3.28 3.16 June 2.45 2.33 แ 2.05 แ 1.54 66 1.26 66 1.14 { July.. 0.47 เ 0.35 เ 0.04 Aug. 10.42 P.M 10.34 A.M. 10.03 P.M. 9.55 11.56 A.M.11.25 P.M.11.17 A M. 9.23" 9 15 Sep. 8.40 66 8.32 8 01 เ 7.53 Oct... 6.42 CC 6.34 เเ 6.03 5.55 Nov. Dec. 4.40 "C 4.33 * 4.01 3.53 06 (C 7.21 5.24 3 22 7.13 " C 5.16 เ 3.14 2.42 .. 2.34 2.03 1.55 1.24 1.16 Although the hour-angles from which the above times are calculated vary year by year and in different latitudes, they may be considered to be sufficiently correct between the years 1890 and 1900, and between latitudes 25° and 65° N. Where extreme accuracy is wanted, the time of observation may be calculated as in note D, Appendix. The above times increase by about 4 minutes every 10 years. But as these elongations occur only at intervals of 12 hours, more or less, it is well to have some other means of obtaining the true north, which can be used when the above method is inapplicable. The two following are similar to one another in principle, but occur about 12 hours apart, and from 5 to 7 hours from the time of the elongations given above. B. In Fig. 11 let P be the pole and S the Pole-star, and let A RAILROAD LOCATION. 57 " represent Alioth (e Ursa Majoris), and C represent the star Gamma" (y) Cassiopeia. The arrows and dotted lines show the paths and the directions of the The A motion of the three stars. positions of the stars in the figure are those which they would oc- cupy about the time of the western elongation of Polaris; but since the complete circuit occupies about 24 hours, we see that in about 6 hours C will be about vertically under S. When this occurs (i.e., when S and Care in the same vertical plane), clamp the tele- scope on Polaris, and wait through an interval of time which is to be found from the interval of 29 minutes 30 seconds for Jan. 1, 1889, by applying for any later date an annual correc- tion of After the lapse of this interval Polaris will be due north. 19 seconds. FIG. 11. (( C. The third method consists in making use of Alioth in a similar manner to that in which we have just made use of y Cassiopeia. But in this case, when Alioth is vertically below Polaris, Polaris will be nearly at its upper culmination" (or "transit," as its passage across the meridian is called), but his makes no difference in the mode of procedure. The inter- val to wait when using Alioth was, on Jan. 1, 1889, about 27 minutes, and increases annually by 17 seconds. To calculate the above intervals, see note E, Appendix; but for ordinary work the figures given above are sufficiently correct as far north as 70°, and as far south as A or Care visible at their lower culminations. The altitude at which C or A will be above the norizon when due north equals about Latitude of the place -30°; so that observations B and D cannot practically be used farther south than latitude 35° N. If, however, the instrument has a reflecting eye-piece, if either observation B or C is needed farther south than these limits, A and C can be used at their upper culminations, which will take place near the zenith, the intervals of time and modes of procedure will be the same as for the lower culminations. To obtain the azimuth of Polaris at any time see Sec. 202. 58 RAILROAD LOCATION. There can be no difficulty about finding these stars if it is remembered that the altitude of the pole-star is about equal to the latitude of the place; that the "pointers" pp, Fig. 11, point towards it; that A and C are each about 30° from the pole-star; C, A, and S being all three more or less in a straight line. The remarks made in Sec. 45 regarding the vertical axis, etc., should be carefully attended to. The times at which observa- tions B and C will occur can be found near enough by notic- ing the positions of the stars themselves. ،، In observation A the instrument should be " reversed" on the star at the elongation. In observations B and C, where the star's motion in azimuth is comparatively rapid, observe, say, 2 minutes before the star is due north, and then again 2 minutes after its transit: the mean result should then be taken. An error of about 2 minutes in time in observations B and C causes an error in azimuth of about 1'. The verticality of the two stars should be also tested by a reversal of the instrument. 58. In checking the line by an azimuth observation as already described, it must be borne in mind that the converg- ence of the meridians needs a very important correction in the bearings relatively to other points east or west of the place where the observation is taken. This may be best shown by means of Fig. 12. P. Let ONEF represent a sector of the northern hemisphere, and let A be the point on the earth's surface at which the survey was started, a continuous 'straight" line being run which had at A a bearing due west. After we have traversed a difference of longitude which is represented by the angle EOF (or the spherical angle N) and have arrived Nat C, we shall be considerably south of the point A, our line having taken the course AC in the figure: so that, if at C we take an observation for azimuth, we shall find our line to have a bearing considerably south of west; and similarly all straight lines run from A, either towards the east or west, have a tendency to run to the south; similarly in the southern hemisphere they would have a tendency to run to the north. Thus in order to run a line from A to a point B, keep- E B C F A FIG. 12. RAILROAD LOCATION. 59 ing in the same latitude the whole way, it becomes necessary to run it as a curve. (See Sec. 209.) Now the amount of this increase in bearing from the north is equal to the convergence of the meridians between the two places, so that in the case of A and B the difference in the bearings of the same straight line obtained by observation at each place will be represented by the angle BPA, which for ordinary work we may consider equal to the difference of longitude of the two places multiplied by the sine of their mean latitude. (See note F, Appendix.) Thus if in latitude 40° north we start a straight line from A due west and run it to C through 1° of longitude, the bearing obtained by observation at Cshould be S. 89°, 21′ W. But since it often needs some calculation to ascertain the difference of longitude, we can best proceed in ordinary work by finding from the following table the correction to be applied. Thus if in latitude 50° N. we have run a line which gives a total amount of easting or westing (i.e., Total Departure) equal to 60 miles, the amount of the correction to apply will be 60 × 1′ 02″ = 1° 02′. TABLE OF CORRECTION FOR CONVERGENCE FOR 1 MILE OF EASTING OR WESTING. Lat. Correction Lat. for 1 mile. Correction for 1 mile. Lat. Correction for 1 mile. 10° 9.18 270 26.52 44° 50".19 11° 10.13 28° 27.66 45° 52.00 120 11.07 29° 287.85 46° 53.83 13° 12.02 30° 30.03 470 55.67 14° 12.98 31° 31.26 48° 57.67 15° 13.96 3:20 32.49 49° 59.83 16° 14.93 33° 33.83 50° 1' 02.00 170 15/.92 34° 35.17 51° 1' 04".17 18° 16.91 35° 36" 50 52° 1'06.67 19° 17.93 36° 37.83 530 1' 09.17 20° 18.94 37° 39.17 542 1/ 11.67 21° 19'.98 38° 40".67 55° 1':14'.33 22° 21.02 39° 42.17 56° 1' 17.17 23° 27.10 40° 43.67 57° 1/ 20.00 24° 23/1 41° 45.17 58° 1/ 23.00 25° 24" 30 420 46.85 59° 1' 26" 25 26° 25.38 43° 48/.52 60° 1'30".00 This shows the necessity, when running a long continuous survey, of referring all bearings to an Initial Meridian, either 60 RAILROAD LOCATION. at the point from which the survey started, or at a point near its centre. The same remarks of course apply to magnetic courses to a certain extent, but in this latter case, on account of the constantly changing variation, such corrections are hardly practical. 59. When the transit-line crosses a river or ravine or some other obstruction over which it is difficult to obtain direct measurement, the best way to proceed is by Triangulating, using whichever of the methods shown in Fig. 13 is most ap- plicable to the case. B C D A H K L FIG. 13. The angles at A and Feach 90°, and at J, K, and L 60°; then AB AC tan C, DE DF sec D, GH = IG sin I cosec H, JK=JL = KL, where H 180° − (I + G). = If the ground on which we measure our base has a tolerably uniform slope in the direction of the base, it is better to take direct measurement along the surface of the ground and mul- tiply the distance so obtained by the cosine of the inclination to obtain the horizontal distance, than to "break-chain." Whatever difference in elevation there may be between two such points as A and B, if the base measurement is reduced to the horizontal, the distance as calculated for AB, from the angles observed with a transit, will also be the horizontal dis- RAILROAD LOCATION. 61 tauce. If the angles were observed with a sextant, of course this would not be the case. (See Sec. 144.) If, instead of encountering such obstructions as those given above, an obstacle which we are unable to see across presents itself, such as a huge detached rock on which we cannot set up the instrument, then perhaps as good a way as any to get round it is by offsetting the line so as to run past it on a paral- lel one, and then on the far side, by equal offsets, getting back on to the former line. If the obstacle, however, is too large to pass it well by this means, we can apply the equilateral tri- angle JKL (Fig. 13). This latter method is a good one to use whenever practicable: there is no calculation necessary in connection with it, the angles used are those most favorable to exact work, and where the obstacle can be seen over, a check can be applied by observing the angle at K. After having run the line a certain distance ahead, repre- sented by the amount L, it is often necessary to back-up" and start the line again from the instrument so as to strike a point a certain distance d on one side of the point where the first line struck; the correction C for this may be found thus: tan C d L For more on the subject of triangulation, etc., see Part III. 60. The LEVELLER'S WORK on preliminary location consists mainly in taking the elevation at every full station, and at any intermediate points where he may consider it ad- visable to do so. The best form of keeping notes on such work is the following: Sta. B.S. Int. F.S. H.I. Elevation. B.M. 4.25 195 4.8 106.60 102.35 101.8 +50 7.3 196 3.28 5.61 99.3 100.99 in which Elevation in any line H.I. — F.S. or and H.I. in any line = H.I. Int. } in same line, Elev. B.S. in preceding line. 62 RAILROAD LOCATION. C The Intermediate" column is sometimes omitted, but the insertion of it makes it easier to check each page by means of the difference of the sum of the Back-sights and Fore-sights. To apply this check between two stations, A and B for instance, which have been used as turning-points, add to- gether all the back sights between A and B (including the B.S. at A, but excluding it at B); then add together all the fore- sights (excluding the F.S. at A, but including it at B): the difference of these two sums should equal the difference in elevation of A and B. If the sum of the back-sights is greater than the sum of the fore-sights, B is higher than A; but if less, then lower. The levels should be worked out in the field whenever time permits, for reference on the work. The profile for each day's work should be made out when possible in the evening of the day on which the work was done. As regards the precision of a line of levels run as above, the probable error is usually assumed to vary as the square root of the distance. The limit on the British Ordnance Survey is 0.01 foot per mile; the U. S. Coast Survey requires a limit of 0.03 per mile. If we assume a limit of 0.05 per mile for rough work, the probable error for any distance equals 0.05 mile. Thus in 100 miles the probable error 0.50 ft. For more on the subject of levelling see Parts II and III. 61. The TOPOGRAPHER'S WORK consists principally in taking the ground slopes, with more or less accuracy, at every full station and at any intermediate points where he may con- sider it necessary, by means of which a contour plan may be constructed. To do this he obtains from the leveller the elevation of each station and plus station at which he has taken levels. There is a variety of methods in use of obtaining the slopes, and the advantage of each depends on the accuracy required, the nature of the country, and the vertical distance apart of the contour-lines. Where the slopes are steep and accurate work is wanted, a 10-foot slope-rod with clinometer gives very good results, but is a cumbersome sort of instrument to carry about. Where 5-foot contours are wanted, a hand-level is very con- RAILROAD LOCATION. 63 venient, since by considering the height of the eye above the ground to be 5 feet, the point corresponding to each contour- line is located at once by the level,-5 feet being an easy height to which to accommodate one's self,-and by pacing the distance between these points we have thus simply to enter the dis- tances in the notes through which each contour passes. By taking the alternate points selected in this way, this method is of course equally applicable to 10-foot contours. shows how this method is worked. Fig. 14 Elev. 1823.8 FIG. 14. Suppose, e.g., that for a certain station the topographer ob- tains from the leveller the elevation of 1823.8, and that he is taking 5-foot contours. Then, if the ground is as shown in Fig. 14, he proceeds as follows: The contour-line nearest to this elevation is that of 1825 feet, the plane of which passes about 1 ft. above the ground-level at the station, so that by standing at the point a he can estimate with his eye the amount of 1.2 feet, and thus find the point b which corresponds with the contour of 1825. Similarly, standing at b he finds c, and so on up the slope as far as he considers necessary. Then re- turning to a, he works in the same way on the lower side. If the distances are wanted accurately, he should have a man with a tape to assist; but as a rule, pacing, where it is practi- cable, gives good enough results. The only notes to be kept in this case are the distances out (right or left) to the respective contours. An Abney hand-level (with vertical arc) is also frequently used, and gives good results. All methods, however, which involve taking the angles of the slopes themselves necessitate extra work. One method of reducing this amount of labor is to have a set of scales for the various slopes, each made pro- portional to the cotangent of the inclination; but by the use 64 RAILROAD LOCATION. of cross-section paper and a small protractor we can probably do the work equally well and equally fast. The stadia method is often found very convenient for ob- taining topography where the above methods would fail to give good results. : But besides taking the contours, the topographer must also take note of the courses of streams, etc., on each side of the line within a distance (usually) of a few hundred feet. The bearings of these he can take with a small prismatic-compass. He should also be constantly on the lookout for anything which may be of service in making up the preliminary esti- mates, such as indications of the probable classification, the flood-marks of water-courses, etc. If the topographer does his work thoroughly, he usually has difficulty in keeping up with the transit and level; but this is rarely a disadvantage, as the chances are that there will be occasional "backing-up” to be done by the party ahead. 62. The GENERAL PLAN of the "preliminary" survey showing the alignment, topography, etc., is usually plotted to a scale of 400 feet to an inch, as in Figs. 15 and 16, thus agreeing with the horizontal scale of the profile. A a را FIG. 15. In Fig. 15 let abcdef represent a portion of the preliminary line as shown on the general plan, plotted to a scale of 400 feet to the inch; and let the line have been run to a + 1.25 p. c. grade, and the contours be given for every 5 feet vertical. Then if each station at which the instrument was set up was at grade," the grade-contour will pass through each of these points, but gradually rising from one contour to another, crossing them successively at distances of about 400 feet apart; so that if, as in Fig. 15, station a happens to fall on a contour-line, เ RAILROAD LOCATION. 65 the grade-contour will cut the next line above, 400 feet farther on, at c; and since the next station d is only 200 feet from c, it will be situated about half-way between two of the contours. Now this grade-contour is the line which, if adopted for the final location, would give no cuts or fills at all, so that it is the line which would render the cost of construction a minimum. The judgment of the engineer here comes in to decide how much it is advisable to deviate from this limit. So far the work has been more or less mechanical, for there are usually enough governing points along the route to decide within two or three hundred feet the course of the preliminary line; but fitting the final location on to the plan is quite another matter. Suppose that the engineer considers that the straight line AB (Fig. 15) is about where the final line should be located. Then the shaded portions in the figure show cuts and fills alternately— shaded vertically being "cut," and horizontally "fill" and the points where the line AB intersects the grade-contour will of course be the " grade-points." The amount of centre-cut and centre-fill can be read off at any point-not by scaling, but by counting the number of contour spaces there are be- tween the line AB and the grade-contour. Thus, e.g., at a point in AB opposite c, there are 23 contour spaces, equivalent to 12 feet vertical, so that at this point we should have a 123- ft. centre-cut. By taking in this way a few points here and there, the engineer can by means of Table XIV, form a fair idea of the number of cubic yards in each proposed cut or fill, making allowance of course where the surface-slope is steep, as shown in Sec 69. In this way, then, there is no great difficulty in obtaining a line which will make the cuts balance the fills, this being sim- ply a matter of a few trials Where curvature, however, is involved, it is not so much the question of balance as of the total amount of cut and fill, which needs consideration. By having the various curves drawn on a horn protractor, or on a piece of tracing cloth, the result of adopting any cer- tain curve can be seen at once by sliding it up and down over the plan. Then, again, a change of grade for a short distance may ap- pear advisable, which necessitates altering the grade-contour. The question of overhaul, too, has to be considered, and the avoidance as much as possible of long shallow cuts. The 66 RAILROAD LOCATION. probable classification, too, will of course affect the balance of cuts and fills. The advisability of raising the grade to avoid an expensive rock-cut also needs consideration. A little ex- perience, however, goes a long way, and the engineer usually finds that there is little doubt to a few feet as to where the line ought to go. 63. The main features of the final location having been de- termined as above, and drawn on the plan, the approximate position of the points of curvature, etc., can be taken off by scale, and the line thus located on the ground; any little alter- ations being made, the advantages of which have become apparent when the line is seen actually staked out. A fresh set of levels must of course be taken over the new alignment, and a profile constructed showing the rates of grade, etc., finally adopted. As regards compensating for curvature where transition curves are not used, the rate of grade should be changed at the P.C. and P.T. Many engineers, however, prefer making the change at the nearest full" station; it makes little difference, how- ever, which way is adopted. Bench-marks should be given at distances of a third of a mile apart or so, and guard-stakes set solidly beside the hubs. If the location is being "rushed," there is no need to fill in the transition curves, for that can be done equally well by the sec- tion-engineer when he takes over the work for construction. When these curves are omitted, however, it should be so shown ou the plan, as in Fig. 16. 64. It often happens that after the line is located a consider- able distance ahead an alteration in the alignment is deemed advisable, necessitating a shortening or lengthening of a cer- tain portion of the line. This causes a break in the “through- chainage." Such a break as this should, wherever possible, be referred to a point where there is a change of grade, or at least to a point on a tangent, so as to simplify the running of the grades and curves as much as possible. It should be in- dicated conspicuously in the notes and on the plans and pro- files in the form of an equation; the station on the line which comes first being read first. Thus if the left-hand side of the equation is the greater, it means that the line has been lengthened; but if the right-hand side be the greater, it has RAILROAD LOCATION. 67 been shortened by an amount equal to the difference of the two sides. 65. The method of locating described above is of course suitable only to rolling or mountainous country; but where there is any doubt as to whether or not it is better to take con- tours, the engineer may generally come to the conclusion that it is better to do so. There is among some engineers an idea that the time spent in taking the topography might have been ? Sta./1053 +30′ 44 6.45 10° Sta. 1048 +70 Sta. 1043+20 3.28 1820 FIG. 16. better used in running a series of trial lines. Of course in many cases this is true; but it must be remembered that a pre- liminary line with topography well taken to a distance on either side of, say, 500 feet (as is perfectly feasible in ordinary rolling country) covers a width of 1000 feet so completely, as to render the running of a trial-line within that area entirely needless; and that in order to settle the question absolutely as to the location through, say, a valley half a mile wide, two or at the most three lines run as above are all that can ever be required; while by the method of trial lines how many are needed before the engineer can feel satisfied that he has finally obtained as good a line as can be got? And then it is only the best of the trial-lines that is usually selected, which in all prob- 68 RAILROAD LOCATION. ability will be inferior to the line selected from the contour plan. Besides this, if topography is taken, the engineer can at any future time show evidence as to the advisability of having adopted the route which he finally selected. It is a duty he owes to himself as well as to the Railway Company to be able to prove that the location has been good, and how is he to do this if he has simply trusted to the correctness of his eye? 66. In country where the running is easy, one or two trial- lines usually show pretty closely where the final line ought to go, for the long courses may then be converted into tangents, and curves be substituted for the shorter ones as in Fig. 17. 1 C B A 1 FIG. 17. If the long courses predominate, it is usually better to get their location fixed first, and then join them by curves; but when the shorter ones are in excess, it is the curves that have to be first located, and the tangents made subservient to them. If the notes of the courses are kept by "Latitudes and De- partures," the exact curve necessary to replace such courses as ABC can be at once found according to Sec. 77. 67. An engineer with a good "eye" can often tell by mere- ly looking over the ground what degree of curve is wanted to fit the surface, i.e., where the difference between a 3° 30′ and a 3° 45′ makes very little difference. Table II, of Tangents and Externals, is a good guide to this in many cases. For instance, by getting into position near the apex of the re- quired curve, the engineer, with the aid of a hand-level and a prismatic compass, can often tell about how far from where he is standing the curve should pass. Thus, suppose he finds the angle of intersection to be about 40°, and that the curve should pass about 120 feet from the apex: he then finds from the Table that for an intersection-angle of 40° a 1° curve gives an external distance of 368 feet, therefore the RAILROAD LOCATION. 69 degree of curve which he wants will be found by dividing this by 120; thus a 3° 04' curve will probably suit the case. Where the APEX of a curve can be located without much trouble it is always better to do so; and of course this applies more especially to places where extreme accuracy in the centre- line is of importance; such as where bridge-work or trestling are required in the neighborhood of the curve. 68. The balancing of cuts and fills in comparatively level country is usually unadvisable, partly on account of the extra expense involved by the matter of over-haul, but mainly be- cause, though the dump should be kept as high as possible, cuts in such country, and especially long shallow ones, gener- ally add very considerably to the operating expenses. Thus the amount of borrow in such cases may often with economy be made very considerable. 69. On work of this sort the line is generally located first, and then the grades fixed by means of the profile. This is usually done by straining a piece of silk along the surface-line, by means of which the effect of adopting certain grades cor- responding with the various positions of the thread can at once be seen; and, judging by the depth of centre-cut or fill, a fair estimate can thus be made of the amount of excavation and embankment required. ---B---- A Grade Line FIG. 18. Where the work, however, is comparatively heavy, the fol- lowing method will be found to give considerably better results: Suppose the dotted surface-line in Fig. 18 to be part of a pro- 70 RAILROAD LOCATION. file on which we want to fix the grades so as to make the cuts and fills balance, and that in this case we wish to make a por- tion of cut A together with the whole of cut B sufficient to fill the hollow C. On a piece of tracing cloth, say 10 inches long, draw a straight heavy line which is to be the grade-line; then turn to Table XIV, and see what depth of cut is required to give 1000 cubic yards contents in a length of 100 feet. Thus if the cuts are to have a 20-foot base and slopes of 1 to 1, as in Fig. 18, the depth of cut required will be about 8.3 feet. Then draw the parallel line above the grade-line already drawn at this distance from it, according to the vertical scale of the pro- file (in Fig. 18 taken as 40 feet to an inch); and again above that line draw another, distant from the grade-line by an amount corresponding to the depth of cut required to give 2000 cu. yds. in a length of 100 feet; and then draw a third for 3000 cu. yds., and so on, as many as are required. Similar ly, on the lower side of the grade-line draw lines as above, suitable to the required base and slopes of the fill. Place the tracing-cloth over the profile, as in Fig. 18. If then the hori- zontal scale of the profile is 400 feet to an inch, take a "40" scale, and scale off along the horizontal dotted lines shown in the figure. One division of the scale then corresponds to 100 cu. yds. Thus, in order to make the cuts balance the fills (not allowing for shrinkage, etc.) the grade-line must be so placed that the sum of the horizontal dotted lines above it is equal to that of the lines below it. By sliding the tracing-cloth up and down, a balance can soon be obtained. By scaling off and adding the lengths of the lines together mentally, the con- tents of a cut or fill can be approximated to in a very few seconds; or the contents may be read off by means of the ver- tical divisions on the profile paper. Where there is a steep surface-slope, an allowance must of course be added to the results as obtained by the above method. The allowance which should be made for this depends, com- paratively speaking, very little on either the width of the road- bed or the depth of the cut or fill at the centre, but depends mainly on the slopes themselves; so that we may say roughly, that the following corrections are applicable to any ordinary depth of cut (or fill) or width of road-bed. Thus, if by the above method we make the contents of a certain cut to amount to 20,000 cu. yds., with side slopes of RAILROAD LOCATION. 71 1 to 1, if the average surface-slope is about 10°, a fair estimate of the contents will be given by 21,000 cu. yds. Surface-slope. Slope Ratio. 5° 10° 15° 20° 1 to 1 1 to 1 1 p. c. 5 p. c. 8 p. c. 17 p. c. 2 p. c. 8 p. c. 20 p. c. 45 p. c. As to the effect of shrinkage, it may generally be ignored in dealing with the balancing of cuts and fills. (See Sec. 113.) A simple rule in dealing with rock-work is to assume that 100 cu. yds. of rock in excavation make 150 cu. yds. in embank- ment. 70. It has been assumed so far that in estimating the amount of excavation and embankment the method of centre-heights is used. In the long run the results so obtained may generally be considered to give sufficiently close results for most pre- liminary estimates. But when the surface-slopes are such as to necessitate continued corrections being applied, the average slopes at the different stations may be jotted down by the leveller when taking the elevations and the quantities worked out according to Mr. Trautwine's method of equivalent level sections, or some similar process. CURVES. 71. Radius. Degree and Length of Curve.-Railroad curvature in Canada and the United States is expressed in F A グ ​E R B C FIG. 19. terms of the angle ACB, Fig. 19, which subtends a chord, AB, 72 RAILROAD LOCATION. 100 feet in length; and this angle is called the Degree of the curve, and equals D. In curves of small degree, i. e., of large radius, D varies very nearly inversely as the radius R. To convert D into R, we have in the right-angled triangle A EC D 50 sin 2 R and to convert Rinto D this becomes D R50 cosec 2 (1) (2) from which formula Table I has been calculated. From Equations 1 and 2 we see that R varies inversely as D sin and since it is only when is very small that its 2' D 2 sine may be considered to vary as the angle itself, it follows that although we may say that the radius of a 10' curve is one tenth that of a 1' curve, by considering the radius of a 10° curve to be one tenth that of a 1° curve, we should, ou accu- rate work, be led into an appreciable error. Thus by Equation 2, and instead of R of a 1° curve = 5729.65 feet, Rof a 10° curve = 573.69 feet, 572.96. 72. The general practice of setting out curves on railroad construction is by means of 50-foot Subchords, assuming that the angle subtended by any subchord at the centre Cis proportional to its length. Suppose, for instance, we wish to locate a 10° curve, we see from Fig. 19 that since AB = 100 feet, if we wish to substitute for it two separate equal chords AF and FB, they must each exceed 50 feet in length, and the length of each must equal AE cosec AFC. D Now AFC = 90°· and AE = 50; therefore 4 D Corrected 50-ft. chord 50 sec · (3) RAILROAD LOCATION. 73 Thus, instead of using 50-foot chords it is the lengths given in the following table which must be used in order that two of them may give the same curve for the same deflection-angle as would be given by a 100 foot chord: VALUES OF CORRECTED 50-FOOT CHORDS. Deg. Chord. Deg. Chord. Deg. Chord. Deg. Chord. 1° 50.000 6° 50.017 11° 50.057 16° 50.122 2° 50.001 No 50.024 120 50.068 17° 50.138 3° 50.004 8° 50.031 13° 50.080 18° 50.155 4° 50.007 9° 50.039 14° 50.093 19° 50.172 5° 50.012 10° 50.048 15° 50.107 20° 50.191 If the above corrections are not applied, the curve that is set out, instead of passing through the point A will pass through a at a distance from F 50 feet, and its radius r will equal CF instead of CF, and CF 25 sec CFA; therefore r = 25 cosec 4 (4) If we compare this equation with Equation 2, we see that the radius of a curve of any given value of D set out by 50-foot chords, according to the usual method, is exactly equal to half the radius of a curve whose degree D = set out by hundred- 2 foot chords. Thus the radius of a so called 10° curve, if set out by 50-foot chords, actually equals one half the radius of a 5° curve, i e., 573.14 feet, not 573.69 as intended. 76. To find the corrected length of any other subchord, see Sec. The corrections which we have just seen to be necessary to accurate work, practically in a distance of 100 fect amount to nothing at all, but often in the total length of a curve they mount up considerably. For instance, a 10° curve run in on location with a 100-foot chain, which should then of course be a true 10° curve, can- not be expected to "come out" well when tried on construc- tion with 50-foot chords; for if the curve is 900 feet long and 74 RAILROAD LOCATION. the instrument work and measurement absolutely correct, it will not close by 0.8 foot. 73. The length of a curve, in terms of 100-foot stations, as measured along 100-foot chords, may be at once found by dividing the total angle (C) at the centre, in degrees, by the degree of the curve. Thus if L = true length of curve, L = C I D D (nearly), (5) where I = angle of intersection. (See Eq. 7.) So that if the angle subtended at the centre of a 10° curve = 40°, the length of the curve along the chords = 400 feet; and this method, on account of its simplicity, is that usually adopted on railroad work for the measurement of curves. But the true length of the curve will of course be greater than this in the same ratio as the arc AFB in Fig. 19 exceeds the 100-foot chord AB. Now the angle at the centre of a circle which is subtended by an arc equal to the radius equals 180° π = 57°.29578, so that the true length of a curve is given by the equation L = CR 57.2958 IR 57.2958 (6) Thus if C = 40° and R = 573.686 feet (i.e., a 10° curve), L = 400.507 feet,-not 400 feet, as in the example above. Had this 10° curve been set out with corrected 50-foot chords, it would have measured (along the chords) 400.38 feet. Table IV gives the length of arcs of various curves sub- tended by 100-foot chords, from which the true length of a curve may be at once found. 74. Before proceeding to the more practical problems in connection with the setting out of curves in the field, it will be well to consider a few of the more important equations which form the groundwork on which these problems are built up. RAILROAD LOCATION. 75 First, as regards the nomenclature of the various parts, as shown in Fig. 20. P.C. T A 10 -D- C M E Y T P.T. R α P.C. Point of Curve. - = FIG. 20. Beginning of Curve. P.T. Point of Tangent. End of Curve. A = Apex. I = Intersection-angle. Central angle. D = Degree of Curve, if bd = 100 feet. T = Sub-tangent. E M External distance. Mid-ordinate to Long Chord. Y = Long Chord. C L = Length of Curve. Ꭱ Radius. These symbols will be maintained throughout this article on curves. 75. Now because Aa and Ab are tangents to the curve at a and b, therefore OaA and ObA must each equal 90°, and the angle aAb at the apex must equal 180° C; therefore I = C. . . Again, in the triangle b0d, since the angle at b 90° therefore the D Tangential Deflect.-Angle for a 100-foot chord 201 (7) 2010 } (8) 76 RAILROAD LOCATION. In the right-angled triangle AOɑ T - R tan therefore, by Equation 7, C 2 1 T = R tan (9) And if in this we substitute the value for R given in Equa- tion 2, this becomes I T = 50 tan Cosec D Rice (10) Again, E = R exsec AOɑ; therefore, by Equation 7, I ER exsec 2 And by combining Equations 9 and 11 we obtain E = T cot Lexsec /1/ I therefore I E = T tan 4 So also MR vers therefore, by Equation 7, I MR vers [2 And by combining Equations 11 and 13, we obtain E 1 M vers I 2011 exsec 2 therefore M = E cos حاجه I • . . (11) (12) (13) (14) , RAILROAD LOCATION. wry Again, by trigonometry, therefore Y = T cos Aab; 2 I Y = 27 cos And combining this with Equation 9, we obtain therefore I Y = 2R tan COS 14 1 Y = 2R sin Again, by combining Equations 13 and 16, we obtain therefore 2M 1 Y = sin I s 23 vers 102 I Y = 2M cot 4 (15) (16) (17) The above equations can readily be followed by referring to Secs. 231 and 232. The following table may be of assistance in selecting quick- ly the equations required. Thus, suppose we have 7 and Y given, and want R; we see at once that Equation 15 will give us I; and then, by Equation 9, we can obtain R. Use Given. Required. Given. Eq. Required. Use Eq. Ꭱ RA D 1 R, Y I. L D 5 M, Y Ι. Τ D 10 D. I D R 2 I, R I. T R 9 I, D I. E R 11 1, E I, M R 13 I. Y I. Y R 16 I, R D. L I 5 I, T R. T. 1 9 I, M D. T 1 10 I, R IZL7777EEE1 16 17 5 T 9 Τ 10 Τ 12 Τ 15 11 12 14 M B R. E I 11 I. E M 14 T. E I 12 I. Y M 17 R, M I B I. T I 15 E, M I 14 I, R Y 16 T, Y I 15 I, M Y 17 78 RAILROAD LOCATION. PROBLEMS IN SIMPLE CURVES. 76. To lay out a curve by deflection-angles.-In Fig. 20 we have already seen (Eq. 8) that the angle Abd D; but 2 suppose we measure off another 100-foot chord de: then dbe also D 2 (since boe 2D, which makes Obe = 90° D). Similarly, we might show that for any number of consecutive 100-foot chords the total deflection-angle would, for each one, D increase by the amount 2 But though the Total Deflection-angle from the tangent is proportional to the number of full stations when these are the only points given on the curve, as we have already seen in the case of 50-foot subchords, if we insert intermediate stations without correcting the lengths of the subchords, the degree of the curve increases at once. In order to find the corrected length of any subchord we may proceed thus: In Fig. 21 let ab represent a hundred- foot chord, then the angle acb = D; and let represent any subdivision of it corresponding with the length of any uncor- rected subchord: then the corrected length I will be given by Equation 16, when a Y C : D = 1 : 100. If we then insert this value of C in Equation 16, we obtain Y = 2R sin ᎠᏓ 200' . (18) FIG. 21. Y being the corrected length of the nominal subchord . In ordinary work, except where a sharp curve is run con- tinuously throughout with subchords, we may ignore this correction. Not taking the correction into account, the deflection for D any subchord is to 2 n4 as the length of the subchord is to 100 feet; so that for any subchord we have Deflect, in } = 0.3D × Length of Subchord in feet; (19) minutes RAILROAD LOCATION. 79 and this equation applies to a corrected subchord if we insert in it its uncorrected length. Thus for a 14-foot subchord on a 3° curve the deflection- angle is 0° 12'.6. Let us suppose that we are given a 3° Curve to the Right to locate from a P.C. at Sta. 421 +36, I being equal to 12" 30'. The length of the curve we find from Equation 5--since this is assumed as the standard method of measurement for rail- road curves--to be 416.7 feet, therefore the P.T. will be at Sta 425+ 52.7; then if we intend to use 50-foot subchords, our notes will be arranged as follows: 3' CURVE TO THE RIGHT. P.C. Sta. 421 + 36.0. P.T. Sta. 425 - 52.7. Length of curve = 416.7 feet. Intersection-angle = 12° 30'. Subtangent = 209.2 feet. Station. Distance. Deflection. Index. Remarks. 421 +36 0° 0' P.C. +50 14 0° 12'.6 0° 12′.6 422 50 0° 45' 0° 57'.6 +50 " 66 1° 42'.6 423 " li 2° 277.6 +50 (6 (C 3° 12.6 Hub. 424 66 3° 57'.6 +50 4° 42'.6 425 แ เ 5° 27'.6 + 50 + 52.7 << 6° 12.6 2.7 0° 02'.4 6° 15' P.T. The Index-reading at any station equals the sum of the de- flections up to that station; then since the Index-reading at the P.T. is represented by the angle Aba in Fig. 20, and Aba is I easily proved equal to therefore the Index-reading at the 2' P.T. must equal half the intersection-angle, thereby giving a check on the calculations. Having the notes worked out as above, set the transit up at the P.C. as in Fig. 22, and setting the index to zero, clamp the telescope on to a back-sight on the tangent (or on to the apex 80 RAILROAD LOCATION. if it has been put in); then for any station the vernier must read the angle given in the index-column for that station. But suppose that when we have reached Sta. 423 + 50 we are un- able to see any farther. Then set a hub (with a tack in it) at that station and a back-sight at the P.C. Set up over the hub, +50 422 P.C. + 50 424 423 +50 425 +50 +50 P.T. FIG. 22. and setting the vernier back to zero, clamp the telescope on to the back-sight and turn off the remaining deflections by mak- ing the readings for the respective stations the same as those given in the Index-column. Thus: (1) When pointing to any station, the vernier must always be set to read the Index-reading for that station. (2) When on the tangent at any station, the vernier must always be set to read the Index-reading for that station. By adhering to these two rules all possibility of error as re- gards the index-readings is avoided, and with the notes worked out as above we may locate the curve equally well from either end. In order to find the bearing of the tangent at any station with reference to the tangent at the P.C., we have simply to multiply the index-reading at that station by two. Thus, if in the above example the tangent at the P.C. lies north and south, the bearing of the curve at Sta. 423 + 50 will be N. 6° 25'.2 E. Usually in locating railroad curves there is no necessity to work out the deflections closer than to the nearer half-minute. In places where accurate measurement is difficult to obtain, and great exactness is wanted, as in giving centres for piers in the middle of a river, we can often do better work by using RAILROAD LOCATION. 81 Two Transits, one on either side of the stream, and fixing the points by intersection. (See Sec. 163.) 77. To locate a curve when the apex is inaccessible. P.C. a A b FIG. 23. -Suppose, as in Fig. 23, we have been unable to locate the apex of a proposed curve, but have connected the two tangents at a and b by the line ab. Then in the triangle Aub we know the distance ab and the angles at a and b; therefore we have ab sin b Aa= where A 180° (a+b). sin A We can then find the position of the P.C. For example, suppose Aa = 320 feet and I = 40°; then if we wish to connect the two tangents by a 5° curve, since the distance from A to the P.C. is given by Equation 9 (or Table II) = 417.2 feet, therefore the P.C. will be situated 97.2 feet back on the tangent from a. We can then locate the curve according to Sec. 76. N. and S. Base a A E. and W. Base FIG. 24. But suppose, instead of running a direct line ab, it is more 82 RAILROAD LOCATION. convenient to run a succession of courses as in Fig. 24. Then, if the position of the stations a and b has been worked out by " 'Lats and Deps." we can at once find the angles at a and b and the length ab. For instance, let Tot. Lat. of a = 1020 N. Tot. Lat. of b = 810 N. Tot. Dep. of a = 560 E. Tot. Dep. of b = 1430 E. Then the bearing of ab will be given by the angle at a in the triangle aeb; thus tan α = 1430 560 1020 810 4.143. S. 76° 26' E., and the length Therefore the bearing of ab ab = (1020 - 810) sec a = 895.2. Then if the bearing of the tangent at a = N. 80° E., and of the tangent at b = S. 60° E., we have in the triangle Aab, a = 23° 34′ and b = 16° 26′ from which we can find the position of the P.C. as above. If the notes have not been already worked out by Lats. and Deps. the position of b with reference to a can be most easily calculated by taking the tangent at a as the N. and S. base. 78. To locate a curve by offsets from a tangent.-Let C < 10+40 11 X B 12 13 14 FIG. 25. ab be a tangent to the curve at a. gential offset at any station is Now the value of the tan- t = R vers C. But C = ND where N = number of Stas. along the curve to t, therefore t = R vers ND. • (20) RAILROAD LOCATION. 83 Similarly, the distance along the tangent from a to the offset 7 equals = X-R sin ND. (21) Thus, for example, suppose a falls at Sta. 10+ 40, and we wish from this point to set out a 10° curve by offsets from the tangent at a; then at Sta. 11 t = R vers 6° = 3.14 feet, and the distance along the tangent at which this offset must be set off equals X= R sin 6° 59.95 feet. The values of t at distances along the curves from a, 100 feet apart, are given in Table III, calculated by Equation 20. A formula that often comes in handy in the field for com- puting tangential offsets, and which is usually true enough when X does not exceed 150 feet, is t = X2 (nearly). 2R Tangential offsets may often be made use of when, on ac- count of some obstacle or other, the method given in Sec. 76 cannot be used. By offsetting the tangent itself occasionally, as in Fig. 26, we can with ease run a curve past a succession of obstacles, and at the same time keep the offsets compara- tively short. FIG. 26. Another occasion on which this method can be used to ad- vantage is when the apex, P.C. and P.T. are inaccessible. Suppose, by way of example, that we have to locate a 10° curve in a position such as is represented in Fig. 27, the angle 84 RAILROAD LOCATION. of intersection having been found according to Sec. 77 to be, A FIG. 27. say, 90°, and the distance from A to some fixed accessible point e to be 723.7 feet: then ae will equal 150 feet. Suppose we are able to begin running in the curve at c, a point 200 feet along the curve from a: then the offset at c will equal 34.6 feet, at a dis- tance from a along the tangent of 196.2 feet or from e = 346.2 feet; and the offset at d, 300 feet along the curve from a, equals 76.9 feet at a tangential distance of 286.8 feet from a, or from e = 436.8 feet. 436.8 feet. Thus we have two points c and d fixed on the curve, by means of which we can locate any other part of the curve accessible to them, as shown in Sec. 76. Or, suppose we have such a case as that shown in Fig. 28, where we have run the curve ab round as far as d, but find that the P.T. is inaccessible, and yet wish to get on to the tan- gent without adopting the method given in Sec. 77. A con- venient method of doing this is to locate the apex A, if acces- sible, by setting off from e, the middle point of the curve, the external E, found by Equation 11; then we have one point on the tangent Ab. α E e f P α FIG. 28. Again, by running on the curve as far as is possible to d, we can there set the vernier to read the (Index-reading for b+ Diff. of Index between d and b) — 90° and set off df = t, RAILROAD LOCATION. 85 found by Equation 20 thus if the Index-reading for d 40° and for b = 60°, the vernier must read 60° 20° -- 90° -10°. The angle ND in Equation 20 equals of course the angle dob We thus have a second point f on the tangent Ab, and therefore we have its direction. Then by Equation 9, since Ab – T, we can, by triangulation, find the distance of A from some accessible point p on the tangent; then bp = Ap T. Or, since by Equation 21, fb R sin dob we can triangulate from p to f instead. = ― If A is inaccessible also, instead of proceeding as in Fig. 27, we might when at d set the vernier to the (Index-reading for b+ Diff. in Index between d and b), which will give a line dh parallel to the tangent at b. Thus the vernier must read 80°. We can then set off ph df, and thus obtain two points f and p in the direction of the tangent Ab; and since we know dh = fp by direct measurement, and fb by calculation, we thus have the distance bp. = Again, if we have an obstacle on the curve itself, we can run a tangent from some point on the curve which will clear it, and so connect the curve at the further side in a similar way to that shown in Fig. 27; or we might run a Long Chord past it and lay it off by ordinates as in Sec. 80. 79. To locate a curve by offsets from the chords pro- duced.-Let it be required to locate a 10° curve an by offsets સ 20 α e k J M FIG. 29. P from the chords produced, and let, for example, the length of the curve = 360 feet. In Fig. 29-exaggerated for the sake of 86 RAILROAD LOCATION. clearness-let ab, bg, and gi be 100-foot chords. then if eb is in the same straight line as ab and is equal to bg, the triangle beg is similar to the triangle obg; therefore bg: R = eg. bg. So that, calling the chord bg = c, and the chord deflection eg =d, we have C2 d = R' (22) but this value of d of course only holds good when the length of the preceding chord (as ab) is equal to c. Again, if fg therefore ab = leg. eg. eg, then the triangle bfg the triangle axb, Therefore, if t = the tangential offset, c² t 2R (23) a formula (already given in the last section in other terms) which holds good for any lengths of chord, provided the angle at x = 90°. When c = 100 feet, we also have the formula D t = 100 sin ཧྱ་ To find a tangent to the curve at any station, say i, we have only to set off the value of t = kg, obtained by Equation 23, at station g; ki will then be the tangent at i. In order to locate the curve we therefore proceed as follows: Measure ab = 100 feet; b will then be distant from aæ, the tangent produced, by an amount t = 8.72 feet. Set pickets at a and b, and range in the point e, 100 feet from b; then g will be distant from e by an amount d = 17.43 feet, and from b by 100 feet. Similarly we can locate i. But the 60-foot subchord in, since it is not equal to gi, can- not be located by a deflection from the chord gi produced, ac- cording to Equation 22. So we must find the tangent at i by setting off at g the amount kg 8.72 feet; then, having ob- tained the tangent at i, we can calculate the offset mn for the 60-foot chord in (by Equation 23), which equals 3.14 feet, and RAILROAD LOCATION. 87 In order to find the this brings us to the P.T. of the curve. direction of the tangent at n we may either set off at i the value of t for the chord in, or we may produce the chord in to q, making ng = in, and then from q set off an offset qp = mn; np will then be the direction of the tangent. Theoretically, we ought always to make the angle between a tangent and its offset = 90°, and between a chord produced and its offset = 90° 90° angle subtended by the chord at the centre; but in ordinary work there is no need to be particular about this. 80. To locate a curve by ordinates from a long chord. -Suppose, as in Fig. 30, we have two stations a and b given, we then have the length of the arc adb, and so we can find C by Equation 5. Now if d is the middle point on the curve the deflection-offset t from the tangent at d to a = M, the ordinate at d; therefore, by Equation 20, α W d e У Y IN FIG. 30. MR vers (24) M being the mid-ordinate to the long chord Y. The length of an ordinate from the chord to any other station e will be given by the equation m MR vers ND, (25) where N the number of stations measured along the curse from d to e; and the distance from the centre of the long chord at which m must be set off is given by y = R sin ND, (26) 10°, which is the same as Equation 21 for the value of X in Sec. 78. To take an example: Suppose a is at Station 2+20 and b at Station 640, then d will fall at Station 4+ 30. Let D then = 42°; and we can find Y either by direct measure- ment or by Equation 16 = 2R sin 21° 411.2 feet. Similarly by Equation 23 we find M = 38.1 feet. 88 RAILROAD LOCATION. If we then wish to set off an ordinate to Sta. 3.00, we have N=1.3; therefore y = R sin 13° 129.1 feet, and m, by Equation 24, 38.1 R vers 13° 23.4 feet. - - It is usually unnecessary to calculate the values of y, except perhaps when near the ends of the chord. Thus, in the above example, had we assumed y = 100N = 100N 130 feet, it would practically have made no difference in the position of Sta. 300. If we have the length of the chord Y given, we may ob- tain directly from it by means of Equation 16; or, con- C versely, when we know C we can obtain Y The lengths of Long Chords subtending arcs up to 6 stations are given in Table V; also the length of arcs subtended by 100-foot chords. Thus, if C = 20° and D = 10°; Y instead of being equal to 200 feet, really equals 200.254 feet, which is the result we should obtain if we used Equation 6 instead of Equa- tion 5 to obtain the value of L. The middle ordinate may also be correctly found thus : Y2 M=R - √ R² (27) 4' and any other ordinate m = M− R + √R² — y². (28) An approximate formula, which is really a corruption of Equa- tion 27, is M = Y2 8R (nearly). (29) It is sufficiently true, however, when Y is small, the error on a 20° curve, in the case of a 50-foot chord, only amounting to .002 foot. By comparing Equation 29 with Equation 23, we see that the mid-ordinate to a short chord may be con- sidered equal to one quarter the tangential offset at a distance along the tangent equal to the chord. A convenient method of locating small arcs is that shown RAILROAD LOCATION. 89 in Fig. 31, where, having found M by Equation 24 or 29, the mid-ordinate for the subchord ac may be considered equal to a d M FIG. 31. 4M, and the ordinate e of the sub-subchord de similarly equal to one quarter the ordinate at d. 81. To pass a curve through a fixed point, the angle of intersection being given.-Suppose we first find the A α P FIG. 32. position of the fixed point p (Fig. 32) with reference to Aa in terms of the distance Ap and the angle aAp: then and pAO = 90° - (aAp+1) I sin Apo = sin pAO sec therefore in the triangle ApO we have po equal R = Ap sin pAO cosec (pAO + ApO). Apo always exceeds 90°. 90 RAILROAD LOCATION. 82. To run a tangent from a curve to any fixed point. -Let p (in Fig. 33) be the fixed point, and a and b be any two h $ Ъ FIG. 33. points on the curve,-b, however, being on the side remote from p, yet as near to the probable situation of the tangent-point d as is possible. Then taking the chord ab as a base, the length of which is given by Equation 16, observe the angles at a and b in the triangle abp; then when apb 180° bpab sin a cosec apb, (a + b). Now if bh is the tangent at b, we know the angle hbp, and can thus find But by Euclid eb = 2R sin hbp. dp = Vop Xep. Thus by measuring off a distance bf - bp dp and offsetting to the curve, we find the required tangent-point d. 83. To connect two curves by a tangent.-First suppose, P α d a' C FIG. 34. On the as in Fig. 34, that both curves are of the same direction. curve of smaller radius R select a point p slightly more re- mote from the other curve than the tangent-point at a prob- ably is. On the curve of larger radius R' find a point p' which has its tangent parallel to the tangent at p. This may be done by running a trial-line to some station s; and then, by comparing the direction of the tangents at p and s, we find how far along the curve from s, p' will be situated. RAILROAD LOCATION. 91 Now if pd is the tangent at p, and cb is perpendicular to pp', we have pca = dpp' acb, and (R' R) vers dpp' sin acb = (nearly), pp' + (R' — R) sin dpp' pp' being obtained by direct measurement; and aa' = Pp' + (R' - 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. b a d P α FIG. 35. Then select p on the side of a towards the other curve. Then, as before, pca = dpp'-acb; but in this case. sin acb= and (RR) vers dpp' pp + (R' + R) sin dpp' (nearly), aa' = pp' + (R′ + R) sin dpp' (R' + R) sin ach. 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. Ъ 84. Given a curve joining 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 curve ab, instead of ending at b, will end in a parallel tangent, distant from the tangent at b by the amount e. 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 92 RAILROAD 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. 37, where ab is the given curve, and it is required to shift it to parallel tangents at each end, as at a' and b'. b ¿ a а 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 positious 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. V A α 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' (b' being directly opposite to b). Then if O is the centre of the curve ab and R its radius, and O' the cen- tre of the curve a'b' and R' its radius, by Equation 11, therefore and la' Ab Rexsec I, FIG. 38. and Ab' R' exsec I; bb' RR = exsec I' I aa' — bb' cot 2* 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 b'. Let O be the centre of the curve ab and R its radius, and O be the centre of the curve ab' and R' its radius. Then R R′ = 00'; therefore b e R- R' = vers I 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' I. Then if O is the centre of the curve ab and R its radius, and O' and R' are the centre and radius of the curve a'b, R vers I R' vers I'; therefore R vers I R' = vers I' and aa' = R sin I— R' sin I'. FIG. 40. a' 94 RAILROAD LOCATION. COMPOUND CURVES. 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, ench 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 each 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. q a FIG. 41. The Suppose, as in Fig. 41, p (the P. C. C.) is inaccessible. 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 f, and from it set off the offset fh= 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 find 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 a Ъ e cos C = 1 R — r' R 11 from which we can at once find the P.C.C. 2. Let be be the given curve, FIG. 42. 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, R グ ​a 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 measure e: then, since cos C = e 1 R – „' we can thus find the position of the P.C.C. P.C.C. 96 RAILROAD 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 d_at the R Ъ Ꭻ d FIG. 44. curvature aop aod. intersection of the tangent at b with the given curve ac, and ob- serve the angle of intersection at d = aod; then Ρ cos aop = R cos (aod) R γ 2' Thus p, the P.C.C., will be sit- uated at a distance along the curve from d represented by the 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 daod; then cos (aop) = R cos (aod) R + r P from which we can find p as above. a Do R P 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 R γ 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 taugent 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 the curve inwards to b', then to find p', the new position of the P.C.C., we have R b b P FIG. 46. ,2* α e cos o' = cos o + R — r but, B. If abp' were the given curve, and it were required to shift it outwards to b, then cos o cos o' - e 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 Ꮧ b е cos o' = cos o O- R-r ان R a But (B) if ap'b were the given FIG. 47. compound curve, and it were required to shift it outwards, then e cos o == cos o' + R 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. P q R C S FIG. 48. As in Fig. 48, let R 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 Then, since og tan (aqs) ab R— r' qs = ab cosec (aqs). 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 180° cqp = 180° (ogs+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 Rr, then qs + R + r 2 sin (aqp) = ab 2(op - R)* RAILROAD LOCATION. 99 95. To locate any portion of a compound curve from any station on the curve. 19 a 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, bc, 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. - - 3d. 3d; Now if d = the deflection from the taugent at a for Sta. b, then, if ab be produced to f, the angle fbc d + 2d Again, if the chord be be produced to g, the angle ecg = 2d + 3d 5. Then in the triangle abc, the angle at b = 180° and since the length of the chords can be found by Equation 16 (Sec. 74), we can find the side ac and the angles at a and c. Again, in the triangle ace, the angle at c = 180° (bea+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 io 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. or M 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.C. 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, parabole, 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 ab, 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 ɑ Po 2.00 2.50 1 5°30' 1.50 G-- a 1.00 0.50 0 1 FIG. 50. itself outwards by the amount ac, and also to throw the P.C. at a backwards by the amount oc, so that the point o beccines the new P.C. Now and ac = Y sin d R vers C, Med oc = Y cos 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., Y sin d and Y cos 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 RAILROAD LOCATION. left a distance of 3.06 feet, and the new P.C. 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 FROM THE TANGENT AT ANY STATION, AND THE VALUES OF C, Y sin d, AND Y cos d. 0 .50 1.00 1.50 2.00 2.50 3.00 Transit. 0° 15' 0° 15' Transit. 0° 37' 0° 30' 1° 10' 1° 52' 2° 45' 3° 47' 1° 071 1° 55' 2° 521 4° 00' 0° 521 0° 30' Transit. 0° 45' 1° 37 2° 40' 3° 521 1° 50' 1° 22' 0° 45' Transit. 1° 00' 2° 071 3° 25 3° 07' 2º 35' 1° 521 1° 00' Transit. 1° 15' 2° 37 4° 45' 4° 07' 3° 20' 2° 221/ 1° 15' Transit. 1° 30' 6° 421 6° 00' 5° 07' 4° 05' 2° 521/ 1° 30' Transit. C 0° 30' 1° 30' 3° 00' 5° 00' 7° 30' 10° 30' Y sin d 0.22 1.09 3.05 6.54 11.98 19.80 in feet. Y cos d 50.00 99.99 149.95 199.81 249.41 298.74 in feet. RAILROAD LOCATION. 103 TABLE B.-CHANGING 2° IN EVERY 50 FEET. 0 .50 1.00 1.50 2.00 2.50 3.00 Transit. 0° 30' 1° 15' 2° 20' 3° 45 5º 30' 7° 35' 0° 30' Transit. 1° 00' 2º 15' 3° 50 5° 45' 8° 00' 1° 45' 1° 00' Transit. 1° 30' 3° 15' 5° 20' 7° 45' 3° 40' 2° 45' 1° 30' Transit. 2° 00' 4° 15' 6° 50' 6° 15' 5° 10' 3° 45' 2° 00' Transit. 2° 30' 5° 15' 9° 30' 8° 15' 6° 40' 4° 45' 2° 30' Transit. 3° 00' 13° 25/ 12° 00' 10° 15' 8° 10' 5° 45' 3° 00' Transit C 1° 00' 3° 00' 6° 00' 10° 00' 15° 00' 21° 00' Y sin d in feet. 0.44 2.18 6.10 13.06 23.89 39.37 Y cos d 50.00 99.98 149.80 199.32 248.12 295.70 in feet. The stations located as above need only be considered as temporary 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 off the ordinate M (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. from Temp. Sta. M in feet. Dist. from Temp. Sta. M in feet. 2 ft. .011 10 ft. 4 .016 12 66 .035 .040 18 ft. .050 20 .052 6 .022 14 " .044 22 เ .054 8 เ .030 16 " .048 24" .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 RAILROAD 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 C. 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 a a FIG. 51. ს 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 original main curve ab. Offset ab inwards by an amount afe; then if R = radius of the terminal curve cd, we have e cos fod = 1 R — (r− e) from which we can find the position of d; and ca = R — (r — e) sin fod, RAILROAD LOCATION. 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 fd = 33.9 feet, d is the P.C.C., and ca = 2(r — e) sin fod, fod being found from the formula • cos fod = 1 − е 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 o, be the centre of the new main curve ced, whose radius radius R₁. And let 02, 02 be the centre of the terminal curves ac and db, whose radii R2. e ƒ d α 01 02 1. Given R, and R2. Then FIG. 52. and coid sin 2 ao₂c= 02 (R₂ — R) sin R₂-- R₁ R2 Ꭱ aob - cod 2 aob 2 106. RAILROAD LOCATION. Thus we obtain the position of the points c and d. 2. Given R₁ and ao₂c — bo₂d. Then aob co₁d R sin R₁ sin 2 2 R₂ aob co₁d sin sin 2 2 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 R₂ should equal about 3R. The distance aob ƒe = (R₂ — R₁) sin ao2c cosec (R − R₁). 2 Suppose, however, in substituting the 3-centre curve for the simple one, it is advisable for the points e and f to coincide as in Fig. 53. U 01 FIG. 53. 1. Given R, and R2, we then have 02 aob vers uo2C = (R― R₁) vers Ꭱ 2 R₂- R₁ Then a must be put back on the tangent to u, and aob - au = (R − R₁) vers 2 (cot UO2C 2 cot 4 aob). RAILROAD LOCATION. 107 2. Given R, and uo,c, we then have aob (R-R₁) vers 2 R2 = R₁ + vers uo₂C au being found as above. VERTICAL CURVES. 100. We have already considered the dangers which arise from sudden changes of grade (see Sec. 29). Where these changes are considerable, amounting to, say, 0.5 p. c. in the difference of grade, it is advisable to round off the angle at the junction of the two grades by means of vertical curves. On bridge-work this should be more especially attended to. Theoretically, the curve which should be applied is a parabola, and this happens also to be the simplest form of curve to insert in practice. ཉ་ e ак f d b FIG. 54. In Fig. 54 let ac and cb be two grades between which it is required to insert a vertical curve. Now cf2cd; therefore, if the letters a, b, and c stand respectively for the elevations at those points, cd 0102 с a + b 4 and the correction e at any other point is given by the equa- tion e cd. 12 (ac)2 ac and cb are usually made about 200 feet each. 108 RAILROAD LOCATION. Vertical curves are not usually inserted during location, or even shown on the location profile; but the corrections for them should be worked out before the cross-sectioning begins, and the grade as shown on the construction profile should be the corrected grade. * Note.-In dealing with deflection-angles and offsets of curves, the en- gineer-entirely ignorant of the Differential Calculus-may often save himself a considerable amount of labor by making use of the principle of Successive Differences, an application of which is given in Sec. 203, Part III. Thus, e.g., the deflection-angles given in Tables A and B, Sec. 97, may be calculated up to 300 feet merely by the application of the 2d differences, and may be extended considerably beyond that amount by using the 3d differences. More especially is this method applicable in calculating offsets to a curve which may be considered to vary as the Square of the tangential distance, for then their 2d differences will be constant. As an example of this, the values of (H given in Sec. 130,-varying as the square of (H- H'),-have for their 2d differ- ence 1.852, which does not change; therefore the differences of the differences of the values in the table increase regularly, the difference between any two values being greater than the preceding difference by this amount; thus the calculation of such a table as that is merely a matter of simple addition as soon as the 2d difference has been obtained. The engineer should be always on the lookout for this in the construc- tion of tables, etc. H')2 SL 27 X 6' ! PART II. CONSTRUCTION. 101. THE Field-work of engineering during Construction may be divided into two parts, the first (A) dealing with the setting out of the work, and the second (B) with the estimat- ing of the labor and material employed in its execution; and in this order it will be well to consider the subject. A. THE SETTING OUT OF WORK. 102. An engineer, when given a subdivision of a road to look after during its construction, often finds merely the centre- line staked out at every 100 feet,-with hubs indicated by Guard-stakes at the transit stations,-and bench-marks every half-mile or so apart. He is provided with a copy of the loca- tion profile and of the transit-notes and bench-marks, and with the notes and plans connected with any special features in the construction on his subdivision for which he will be held responsible—such as plans of bridge-sites, culverts, etc. If in a timber country, the first thing he has to do is to see to the Clearing of the Right of Way, which he does by marking out the limits-if the clearing is to be carried to the full width-by blazing the trees at distances of a hundred feet or so apart on either side of the centre-line, and inscribing the letter C. While the clearing is being done, he usually has time to examine the country along the line with an eye to the location of culverts and the size of openings necessary, and to make a closer examination of the probable classification of the cuts than the location party probably had the opportunity of doing. 103. In order to obtain a correct idea as to what size of openings may be necessary, he is guided by the flood-marks 109 110 RAILROAD CONSTRUCTION. along the water-courses; and if there is any doubt about these in the neighborhood of the line, he must follow them up until he finds some definite indication of the amount of flow, or else forms a more or less accurate estimate of it for himself, by an examination of its source. In selecting the points for culverts and the sizes required, the engineer must bear in mind the effect of drainage upon the natural well-defined water-courses: for instance, water that before the construction of ditches ran more or less broadcast over the country, -as is frequently the case in low marshy land, thereby perhaps in a dry season showing no indications of its existence at another time of the year, or which in a wet season may be simply indicated by a saturation of the soil, may, when conducted by ditches to the mouth of a culvert, present a very decided reality. Often too, by cutting a small ditch, two streams can be brought together at a less cost than would be involved by the construction of two separate culverts. For a masonry culvert is an expensive article in the first place, and the usual substitute -a timber one-a still more expensive article in the long run. When the dump is low, open wooden culverts are the best to use as temporary expedients, for any defects in them are readily visible, and masonry culverts can be built to replace them with very little trouble. For small openings piping does admirably, but should be well bedded; as a temporary sub- stitute for pipes, small plank culverts may be inserted, which may afterwards serve as a means of inserting the pipes them- selves. 104. A thorough system of drainage along each side of the road-bed should be one of the first points to which the attention of the engineer should be given, for it is often possible to greatly decrease the cost of construction by constructing ditches some little time before the commencement of the work. As regards the form and size of such ditches, it is usually sufficient to make them with slopes of 1 to 1, but with plenty of width in the base: as a rule, for each foot of water likely to be in the ditch there should not be less than three feet of base, and the rate of fall should be made as uniform as is compatible with the cost of construction. For small ditches, the rate of fall should not be less than 0.2 p. c. if possible; but a large ditch which is likely to have a depth of water of not less than RAILROAD CONSTRUCTION. 111 one foot wil draw tolerably well with a fall of only 0.1 p. c. Neither should the fall be so great as to permit scouring to any large extent. Small extra ditches are usually staked out with centre-stakes only, and the amount of excavation calculated from the centre- heights. But for larger ones slope-stakes should be set, and if the surface is irregular it must be properly cross-sectioned. 105. It is often the case that the cross-sectioning of the work has been done by a party detached from the main location party: if so, the engineer usually has time to check the bench- marks and insert new ones for himself at points which he may consider suitable. These B.M.'s should not be less than 10 stations apart; their positions should be such as to do away as much as possible with turning-points. They should be marked B.M., and the elevation of each inscribed on it. At each bridge-site there should be a bench-mark close at hand. It is a good plan also, if there is time, to check the alignment from the transit-notes. Any error discovered, either in the levels or the alignment, should be at once reported. For discrepan- cies arising in the checking of the alignment by using short chords, see Part I. 106. When, however, the subdivision engineer has the cross-sectioning to do himself, if the construction is being started at various points on his work almost simultaneously with his taking charge, he then has his time from the very first fully occupied in taking cross sections. The amount of work which this involves depends a good deal on the manner in which the grading is to be measured. If measured in excavation only, then it is merely the cuts that have usually to be cross-sectioned; but if measured in cut and fill, both must receive equal attention. In the former case, where borrowing has to be done, it is often necessary, however, to have the fills also cross-sectioned, for, owing to the impossi- bility of measuring the borrow-pits correctly, the work may have to be measured in the fills, and this must be borne in mind at the time of cross-sectioning. Also, to obtain a correct estimate of the over-haul it is necessary to have the fill connected with it cross-sectioned. At all points, too, where the question of the distribution of material is likely to arise, cross- sections of the fills are useful, but these need not be taken with 112 RAILROAD CONSTRUCTION. 君 ​the same accuracy as those required for the measurement of the work. To cross-section properly, five men are wanted besides the engineer, namely, a rodman, a man to carry stakes, another to drive them and another to mark them, and a tapeman,-for though the setting of slope-stakes is sometimes done separately from the cross-sectioning, it usually saves both time and expense to do both at once. Before starting to cross-section, the engineer will do well to construct a small table for each different width of road-bed and set of slopes which he is likely to use, giving the "distances out" to the slope-stakes for various amounts of side-heights. For though he rapidly acquire these after a little practice,- and should be checked in his calculations of them by the rodman,—still, by having a table before him, he saves con- siderable mental work and insures greater accuracy. He should also be provided with a small scratch-block. The best way to explain the method of cross-sectioning is by means of an example. B A C H Y FIG. 55. Let bBAC, in Fig. 55, represent a surface which we wish to cross-section. We first take the elevation at the centre A, which should correspond within a tenth or so with that given on the location profile. By subtracting the grade at the station from this elevation we thus have H, the centre cut at A. The rodman then goes to the left and holds the rod at some point b near where he judges the slope-stake will come. If on ob- taining the side-height for bit is found that the proper distance out from A for this height does not agree with the distance out as actually measured, other points must be tried until a point is obtained, such as B, where these two correspond. An error of only a few tenths in distance can be estimated for by eye without taking a separate reading to correct for it, so that two or three trials are usually all that are required to fix the RAILROAD CONSTRUCTION. 113 position for the slope-stake; and on comparatively level ground the point can be usually hit off by a good rodman at the first trial. Similarly on the right the point C must be fixed. If there are any decided irregularities in the surface, such as is represented at D, the elevations of such points must also be taken. The following rules give all that is required as regards the actual levelling : 1. When H.I. is above grade.—If the rod-reading exceed the difference in elevation of the H.I. and Grade, the excess = the fill; but if it is less, the deficiency the cut. Consequently, when the rod-reading = the difference of H.1. and Grade that point is a Grade-point. 2. When H.I. is below Grade, the rod-reading + the difference of H.I. and Grade the fill. Cut is always indicated by a positive, and Fill by a negative sign. The following is a good form for keeping the notes: Re- Sta. L. C. Ru. B.S. F.S. H.I. Elev. Grade. marks. 0.0 +3.0 1020 +1.0 1.3 102 30 101.0 100.00 7.0 - 1.0 14.5 +3.3 +1.0 1021 0.0 1.3 101.0 101.00 8.5 6.0 11.5 3.0 0.0 1022 2.0 2.3 11.5 7.0 100.0 102.00 +1. p. c. grade Roadbed 14' in fill, 20' in cut. Slopes 1½ to 1. - There is no need to work out the elevations in the field, but so doing in the office afterwards forms a useful check on the work, since H.I. F.S. (which of course is the elevation) should agree within a tenth or so with the sum of grade ± centre-height, F.S. representing the rod-reading at the centre. We see from the above that it is the Difference of H.I. and Grade which is the foundation of the calculation at each station, and this, when worked out for the next station after a turning- point, can be modified for the succeeding stations by merely adding or subtracting the difference in grade. Thus the cal- culation is simpler than it at first appears from the above rules. The slope-stakes should be marked S.S. on the outer sides 114 RAILROAD CONSTRUCTION. and the numbers of the stations on the inner. The centre- stakes should have the cut or fill marked on them. As to the points at which cross-sections should be taken, the rodman in selecting them should bear in mind that it is not necessarily the highest or lowest points that are re- quired, but those points which, when joined by straight lines, will give the contents as nearly as possible equal to the true volume. It is impossible as well as unnecessary to take account of many of the small irregularities which occur, but by a judicious selection of points these may to a considerable ex- tent be made to counteract each other. Where the contents are calculated by "average areas"-as is usually the case—we can easily find from Sec. 130 what limit should be adopted as regards the difference in centre-heights and widths between the slope-stakes of two cross-sections, in order that the error in the volume as calculated shall not exceed a certain amount. For exact work a difference of two feet between the centre- heights of two adjoining cross-sections is about the limit which should be allowed; but in ordinary practice we may say that a cross-section should be taken every 50 feet when the differ- ence in centre-height amounts to about 5 feet. This is, of course, mainly to reduce the errors which arise from using an approximate method of calculating the quantities, and not to take into consideration the irregularities of surface. To counteract as much as possible these latter, judgment in the selection of the cross-sections has a better effect than labor spent in obtaining a large number of cross-sections a few feet apart. They should also be taken whenever "grade" occurs on either the edge of the road-bed or in the centre; and when- ever a cross-section is taken where a grade-point falls in the road-bed its position must be obtained. For if a grade-point is the only point obtained at any station, it necessitates assum- ing centre- and side-heights afterwards in working out the con tents, in order to make use of that grade-point, so that it is much more satisfactory-and in the end involves no more work—to obtain these heights by direct measurement. There is of course no need to take cross-sections any closer together on a curve than on a tangent, as may be easily seen from Sec. 134. When in doubt as to the material in a certain cut, i.e., as to whether it is earth or rock, etc., it is best to cross-section it RAILROAD CONSTRUCTION. 115 for the usual earth-slopes and have it stripped to that width in one or two places; if then rock is encountered in a solid bed, the rest of the cut may be cross-sectioned for rock, and as soon as the rock is reached the earth trimmed off to its This of course proper slopes before the rock is worked. necessitates a cross-sectioning of the rock surface as well as of the original ground-surface, and these cross-sections should be taken at the same stations, so as to facilitate the calculation of the respective volumes of earth and rock. 107. The referencing of the P.C.'s and P.T.'s is a part of the engineer's work which must also be attended to before cou- struction begins. Reference-points should be placed, two on each side of the alignment, at angles of about 45° with it, and sufficiently distant to be free from all chance of disturb- ance during construction; the point referenced thus lies at the intersection of the two lines joining the opposite points. Sometimes, however, especially on side-hill work, it is neces- sary to place all the reference points on one side of the track, in which case the apex of the angle formed by the lines pass- ing through each pair of reference points is the point refer- enced. Each reference-point should be marked R.P. on a guard-stake set beside it, and the magnetic bearings and dis- tances of the points entered in the notes. 108. The Staking out of Borrow-pits consists in driving stakes at the corners of the proposed pits, and obtaining eleva- tions of the ground-surface so as to form the upper line of a set of parallel cross-sections of the pit, the lower line being ob- tained by taking levels immediately under those taken on the surface, when the excavation is completed. In order that the bottom levels may be properly connected with those taken The on the surface, reference-points must be established. simplest way of doing this is by driving hubs, say 10 feet back from the edge of the pit, in the line of each cross-section. By taking the cross-sections 27 feet apart, as is often done, there is some little labor saved in calculating the contents, since the mean of any two cross-sections in square feet equals the volume between them in cubic yards. A sketch plan of each pit should be made in the note-book, and properly lettered to accord with the notes. 109. Staking out Foundation-pits for Culverts, either masonry or timber, consists of setting stakes at the corners as 116 RAILROAD CONSTRUCTION. given by the foundation plan and marking on each stake the cut necessary. A sketch of each pit should be made in the note-book, and of course the amount of cut at each stake re- corded. When the foundation consists of timber, the pit should be low enough to insure the timber being at all times, if possible, kept under water, or at any rate moist; about 18 inches is the average depth for foundation-pits for wooden culverts on Railroad work. In staking out, it should also be remem- bered that the culverts should not have a fall of more than, say, 1 in 10, so that when the ground slopes transversely to a greater extent than this the culvert must be put on the skew so that its inclination will not exceed this amount. If the depth of the foundation-pit exceeds 4 or 5 feet, it should be staked out a foot wide all round to allow room for working. 110. Setting out Bridge-foundations.-When a bridge is on a tangent there is no difficulty about staking out the foundation-pits, that needs particular mention. The work is usually best done with a transit and tape from the centre-line, -an optical square comes in very handy for this, the offsets being obtained by scale or otherwise from the foundation plan. In this way there is less liability to make an error than in any other, since each point is set out independently of the previous oues. When the material is not likely to stand vertically, it should be given a slope sufficient to warrant its stability. If there is not room to admit of this, then of course the sides must be shored-up in some way, A When, however, the bridge is on a curve, if the span is short, it is from the tangent at the centre of the bridge that the offsets must be set off. In dealing, however, with bridges of comparatively long spans, the centre of the curve on the bridge will by no means coin- cide with the centre of the structure, as is shown by Fig. 56. R P FIG. 56. B M Now AB will be the centre- line of the bridge, where cb= ordinate at M to ab (see Equation 23, Sec. 80); so that the true centres of the piers RAILROAD CONSTRUCTION. 117 lie considerably outside the centre-line at those points. If any pier, as c, is inaccessible, c (its centre) may be located as follows: In the centre-line of the track take some accessible point P, and set off PB perpendicular to AB, making then will PBR (vers POM-vers bOM); Be = R(sin POM ab 2 C may then be located either by direct measurement from B, or by intersection. In setting out bridge-foundations great care should be given to a thorough system of referencing all important points, and the reference-points must be so selected as not to be ob- structed by staging or scaffolding during the progress of the work. 111. Setting out Trestlework.-In locating the position for the piles in low pile-bents, it is sufficient to locate the centre of each bent and then set off the positions for the piles by measuring out from the tangent at the centre, finding the angle by eye; if possible, the position of each pile should be marked with a stake. When piles are being driven on a curve by a floating pile- driver, in water too deep to drive stakes, the centre of each bent must be given by the intersection of the lines given by two transits, as in Sec. 76. If, however, the trestle is on a tangent, by placing pickets on either bank in line with each row of piles the centre for any pile can be given without the aid of an instrument; or pickets can be so set that the pile-driver can line itself in without the assistance of any one on the bank: the distances between the bents may be taken by measurement from one bent to the next. In the case of framed bents resting on sills, it is advisable to have the sills brought to a solid foundation at about an indicated elevation before the framing-bill is made out in this way a firmer foundation is often obtained at a cost of less labor than if the exact elevation for the sills was prescribed. The sills for each bent should then be accurately levelled and centred. : In dealing with high trestles, the transverse centre-line of 118 RAILROAD CONSTRUCTION. each bent should be referenced, the reference-points being 1: a considerable distance from the bent itself, so as the better to permit the line being carried to a high elevation in the structure if required. The length of the chords should be corrected according to Sec. 76. Where pony-bents are used they should be so skewed around as to conform with the contour of the ground; they must be accurately levelled before the sills are laid on. In giving points for "cut-offs" in piling out of reach, the pile should be blazed and a tack driven into it, the distance above the tack-which should be in full feet-being inscribed. The position of the tack is best found as follows: For ex- ample, let the difference of H.I. and grade 6.11 feet; then if the point of cut-off is 2 feet below grade, and it is wished to put in the tack so as to read "5 feet below cut-off," we must read on the rod 0.89 foot. The position of the tack is then at the foot of the rod. = 112. Setting out Tunnels.-This is work which often needs considerable time and care, in order that the results ob- tained may be satisfactory. Let Fig. 57 represent the section of a tunnel in course of construction. A C E B G FIG. 57. The first thing to do is to establish some point C in the alignment from which a good view-if possible-may be had of the mouths of any shafts which it may be required to sink, and also of two distant points A and B, also in the same straight line. If the instrument is then set up at C and the telescope clamped on to A, on reversing it the point B should be intersected. By repeated trials the three points A, B, and C are then established in the same straight line, and these points should be permanently marked. In order to obtain the centre-line of the tunnel, say at the left end, another point & in the same line as AB must be 4 RAILROAD CONSTRUCTION. 119 given, and the centre-line is then obtained by the production of AG. But suppose the work is to be carried on also from one or dropped" more shafts as EF, then the alignment has to be " from ED to the elevation of the tunnel at F, and in this There are three or operation the greatest care is necessary. four ways in which this can be done, but the following is that usually adopted for tunnel-work, as it admits of greater ac- curacy than the others, which are more suitable for simpler mining operations: Two instruments such as that shown in Fig. 58 should be firmly bolted on either side of the shaft as D and E, and near to its edge, both being lined in vertically over the centre-line of the tunnel. a S 20 Each instrument consists of a plate p-with a narrow verti- cal slit in it and scale s attached-which can be moved side- ways by means of the screws a and b, so that it can be set to any desired reading on the scale— the scale being read by a vernier v attached to the main body of the instrument. Having set these two instruments approximately in line, then, by a series of observations taken at different times,-so as to counteract as much as possible the varying conditions which affect each separate sight,-ascertain for each instrument the mean of the readings. Having then set the plates to give that reading, the centres of the vertical slits coincide with the mean alignment. FIG. 58. Two fine steel wires must then be carried from one slit to the other, each being placed against the vertical edge, so that they form two parallel lines, close together, across the shaft, one on cach side of the alignment. Midway between these two wires, and as near to the edge of the shaft as possible, but on opposite sides of it, two fine copper wires should be passed, long enough to reach down to the tunnel at F, and to the ends of these two heavy plumb-bobs should be attached. The wires should be enclosed in wooden tubes to protect them from cur- rents of air, falling water, etc. The plumb-bobs themselves should be immersed in buckets of water to lessen their oscilla- 1 RAILROAD CONSTRUCTION. 120 tions. Scales should then be placed so as to read these oscil- lations slightly above the plumb-bobs. The mean of these sets of readings then gives a point on the alignment, and from the two points so obtained the centre-line of the tunnel may be extended in either direction by first establishing a point in one direction, and then in the other; and these points can then be checked by observing whether all four are in the same straight line: if found to be correct, they should be perma- nently established. The levels may be dropped by means of a steel tape, with which the levelling-rod used has been pre- viously compared. The length of the tunnel may be found either by direct measurement (breaking-chain) or by triangulating. In locating a tunnel, it should be remembered that it is usually cheaper to open a cut at depths under 60 feet than to bore. In many clays, however, a cut of this depth would be barely practicable owing to the increase in the inclination of the slopes necessary on account of the depth itself, and in such cases the limit is considerably less than this. As regards the advisability of sinking shafts, it is mainly a question of the depth of shaft required, the need of ventilation, and the facilitating the transport of material. Where the depth is not excessive it is usually policy to sink several shafts in a long tunnel, and work from each independently, for the work is thereby considerably hastened, and after its completion the shafts themselves form admirable means of ventilation. Side-drifts, where they are possible, accomplish the same re- sults as shafts, and are usually to be preferred to them on ac- count of less risk to life and property during construction, and their convenience afterwards. Where the alignment has not to be carried to any great dis- tance from the points dropped to the bottom of a shaft as above described, it is better to sink the shaft a few feet on one side of the centre-line, and to reach the tunnel from it by means of a cross-heading. The centre line in the tunnel is best given by points on the roof from which plumb-lines can be hung when required. 113. Giving Grade and Centres forms a very large por- tion of the work to be done by the engineer during construc- tion. The giving of "grade" may be greatly facilitated by having stakes driven to grade, from which at any future time RAILROAD CONSTRUCTION. 121 the levels may be given with a hand-level-an instrument highly useful during railroad construction. To have to carry a heavy level for several miles just to give grade at two or three stations, as is frequently done, is absurd. By having a bubble-tube attached to the telescope of the transit a consider- able amount of trouble may also be saved, and with it the clevations can be given quite as correctly as are ever required on a railroad dump. In setting grade-stakes, allowance must be made in dealing with material which is likely to shrink in order to allow for it. The amount of the Shrinkage depends considerably on the pressure to which the material is subjected, consequently on the height of the fill: as an average, however, in earthy soils the linear contraction is about 19 p. c., so that a 10-foot fill should be "put up" 1 foot above grade. In dealing with wet or frozen soils greater allowance should be made, but with dry sandy material, less. The allowance also depends very largely on the manner in which the dump is constructed. A dump well trodden by horses usually shrinks very little, and in many such cases there is no need to allow for shrinkage at all; but where the work is put up by tipping or shovelling, double the allowance may in some cases be none too much. The increase in bulk in rock, as well as the shrinkage of earth, necessitates an allowance being made when arranging for the distribution of material. A good general rule for this is, that 10 yards of earth in excavation make 9 yards in em- bankment, and 10 yards of rock in excavation make 17 yards in embankment. As regards "giving centres" during construction, it should be seen that the slope-stakes are intact, and then by their means the centres for a cut or fill may be usually obtained from the cross-section notes, without the trouble of setting up the transit, with accuracy quite sufficient to enable the con- tractor to proceed with his work. 114. Difference of Elevation on Curves.-The centrif- ugal force brought into play by the inertia of the train when going round a curve must be counterbalanced by a more or less equal and opposite force in order to prevent the flanges of the outer wheels being pressed too severely against the rails. The simplest way of bringing a counteracting force into play 122 RAILROAD CONSTRUCTION. is to make use of a component of the weight itself, which H tional to H G' Z G F M FIG. 59. Centrifugal Force may be done by canting the track as in Fig. 59. Thus, if the force W, representing the weight of a car, be resolved into its rectangular compo- nents N (normal to the track) and F (parallel to the track), we see from Sec. 7 that F is propor- H being the difference in elevation of the rails, and G the gauge--or more strictly, the distance from centre to centre of rails. Now the value of the centrif- V2 ugal force in pounds equals where v = velocity in feet 32R' per second, and R the radius of the curve; so that when there is no tendency to tip over on either side--if we as- sume, as we may well do in practice, that F is the com- ponent parallel to the centrifugal force-we have H 22 G 32R therefore H Gv2 32R So that, substituting for R the value given in Sec. 71, and substituting F, velocity in miles per hour, for v, we have H = .00067 GV2 sin D; or, as an approximate formula, easy to remember, we have GV 2 H = 15R (nearly). If we take G 4' 8", we then have H= .0032 V2 sin D. 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′ 8″ gauge), gives the difference in elevation of the two rails in feet, at various speeds for different degrees of curvature. DEGREE OF CURVE. Vel. in m. p. h. 1° 2º 3° 4° 5° 6° 10 9° | 12° 16° 10 .006.011.017.023 .029 .034.040.051.069.091 20 .023.046.069.091 .114 .137.160.206.274.365 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 uncased by transition curves, the difference in elevation at the P.C. and P.T. must be at least equal to what it is at any other part 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.C. or 124 RAILROAD CONSTRUCTION. P.T., as the case may be. For a 3° curve it is usually sufficient to begin the difference in elevation about 100 feet back, and 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 " 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, and 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 en- 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. 116. 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, for 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 he 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.C.'s, and P.T.'s. They should be placed on the outer side of the curves, at right angies 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 formula 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. a F A B Then if G = gauge, N= number of the frog, F= "Frog angle," = Angle of Intersection at F, we have FIG. 60. R AF radius of turnout curve, frog distance, AB = length of switch-rail, D= degree of curve, F cot 2 F G N- tan = O 2 2 AF' AF=2GN, R=2GN², AFR+ (x + 1) = sin F, R AF cosec F 7) AB= √4GN² × Throw. The throw according to Sec. 78 AB2 2R 2012 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' 8" and a throw of 5". 1 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 9° 32' 56.50 338.98 10° 58' 16.8 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 3.000 values of AF and R by .637; and the above value of : 4.708 4.708 D must be multiplied by =1.57. AB is of course de- 3 : • 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- C.-- TF M -D B FIG. 63. Starting from the centre-line AD with a given frog number, we select a certain length n, expressing the length of the brauch 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 R vers nD, t and the distance along the track AD to this offset equals TR sin nD. 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 = 1 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 Ne t is set off from the point e, which point is found from the formula ce = (d We thus have the point N. 2t) cot nD. 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 d vers nD 2R' 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. S 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' 8". Then from the above formule we have nD = 1.3 × 9° 31′ = 12' 22', t 602.6 × vers 12' 22' 14 feet, T= 602.6 X sin 12 22 129 feet, ce = 12 × cot 12° 22′ = 54.7 feet, and y = 1.2 feet. The notes for the setting out of the crossing may then be arranged as follows: A 22.4 1.2 37.7 Headblock 2.4 75.3 129. 11. E M N 5-1.7 14. 129 129. 2.4 75.3 Headblock/22.1 37.7 1.2 B FIG. 61. 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, radius of the outer track A, R and r = radius of the turnout curve AC The length of AC may then be found in terms of nD, thus: d vers nD = R— r 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 B A --d- is 9°, therefore 7 637 feet. FIG. 65. 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 AC202.7 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 Then r 21 = radius of the inner track B, radius of branch CB, radius of branch AC. R a(i R− r₁+ vers BHC (R+ r) (r + r'₁) from which we can find the length of the branch BC; and afr J "1 - 99 vers BOA = and since the angle (R + r) (R + d − î'₁) ' AEC BOA+ BHC, we can thus find the length of the arc AC, and locate the crossing with the transit, starting from either end A or B. A E H B FIG. 66. In order to use frogs of the same number for tracks A and B, we must have the change of curvature at A equal to that at B. The positions of the frogs may be found according to Sec. 119. The positions of the frogs may be found according to Sec. 119. In the case of a Double Turnout the engineer can, by ap- plying the formulæ 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 mid-ordinate. (See Sec. 80.) LENGTH OF RAILS IN FEET. DEG. OF CURVE. 30 28 26 20 18 14 10 In. In. In. In. In. In. In. 10 .240 .192 .156 .096 072 .048 . 02 | 20 .456 .408 318 .204 .163 .096 .048 २० .696 .612 .528 312 .264 .144 .072 40 .948 .828 .720 .420 .348 .216 .108 50 1.19 1.03 .888 .528 420 .264 .132 Gº 1.40 1.22 1.06 .624 504 .312 .156 1.64 1.44 1.25 .782 .588 .360 .180 So 1.90 1.64 1.43 .$10 672 .408 .204 10° 2.35 2.05 1.73 1.04 852 .540 1264 1:ཀྭ¢ 2.83 2.47 2.15 1.26 1 02 .636 812 140 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 Rails.-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 at a temperature of 120° F. to about at a temperature of -40° F. This must be carefully attended to. 5 16 1 // 16 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 T- RAILROAD CONSTRUCTION. 133 1 中 ​ages $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 M N 4 + I' 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 RAILROAD CONSTRUCTION. 4. Spreading on the embankment.-The cost of this varies considerably, but may be said to average about 13 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, 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, 13 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 TOTAL COST, PLOUGHED AND SPREAD, IN CENTS. Length of Cu. yds. hauled Lead in per day per feet. cart. Light sandy soil. Common Strong Stiff clay or loam. heavy soil cementea gravel. 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 25.9 1800 9.1 23.1 24.8 26.3 27.3 2000 8.3 24 6 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 محمد I 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 forcu. 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 M N 6 + L' 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. Cost per cu. yd. for hauling and emptying. Total cost per per cart per cu. yd. day. 50 18.5 68 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 CONSTRUCTION. 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. A- 126. Estimating Overhaul.-It is common to allow an K -7 [M] C E D FIG. 67. represented by 7 in Fig. 67. B 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 the limit of free haul, 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 - l, and if S represents the contents of AD (or EB), then the amount of overhaul = S(L1). = Thus, for example, if L 1000 ft., l 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 7 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 EC. 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 2 137 RAILROAD CONSTRUCTION. Y す ​} # } 1 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 necessary, 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 "prismoid ;" 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 138 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 arca of course-taking for instance the triangle ABC— equals AB × 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, h, h', l, and l'are as shown in Fig. 55, Area H Н 2 B (l + l ' ) + = (k + h'), which is the formula most generally in use. 3. If the surface is horizontal, then this becomes B Area H H (3 + 1). 4. Or, if regularly inclined, B.h Area +th', 2 where h is the greater side-height, and 7 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 F D E H A C B L 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 ID + EJ (IJ)+ 2 EJ+FC 2 (JO) + FC+GK 2 (CK) + GK+HL 2 (KL) ID. IA 2 BL. HL 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 RAILROAD 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 arcas 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. C B D FIG. 69. The contents of such a pyramid as ABCD are found by the formula S- ABC X. AD 3 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 E A B G F FIG. 70. by using the formula for the rectangular wedge is immaterial; thus we may say that the contents Sarea ABCDE × AG 2 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 PRISMOIDAL FORMULA equal S (A+a+4M), 6 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 ɑ, 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 a M A C D FIG. 71. composed of two warped surfaces, BCcb and CDde, 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, B α M Ꮯ A C D FIG. 72. d 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. 3 143 RAILROAD CONSTRUCTION. 130. The method used nowadays almost entirely for the calculation of grading, is that of Average End-areas, which assumes that A+ = 4+ a L. 2 S 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 prismoidal 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 may be used, found by deducting the prismoidal formula from the formula for average end-arcas; and this correction, when the surface of each end-section is horizontal, equals in cubic yards C = (H ~ H')? · SL 27 × 6 where Hand H' are the end centre-heights in feet, s the slope- rates, and L the lengths of the solid in feet. Taking s = 13 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 = = 1. H H' 0 1 2 Co 3 4 5 6 -7 со 8 9 0 0 1 4 8 15 23 33 45 59 75 10 93 112 133 156 181 268 300 334 20 370 408 448 490 533 < 208 237 578 626 675 726 779 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 difference in centre-heights of 23 feet, if s 13 and there is no slope 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 1957 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, C = (H - H') (W - W') L 27 X 12 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 cach cross-section the value of (HH') and (WW'), 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-stakes, then C must be sub- tracted. RAILROAD CONSTRUCTION. 145 -100 FEET. TABLE OF THE VALUES OF C, WHEN L = H-H' in feet. W - W' in feet. 1 20 3 4 10 5 6 7 8 9 10 123 .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.1 .6 1.2 1.8 2.4 3.0 3.6 4.3 4.9 5.5 6.2 .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 1.8 3.6 5.5 7.4 9.2 11.1 12.9 14.8 16.6 18.5 2.1 4.3 6 5 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 13 4.0 80 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 31.5 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 19 5.8 11.7 17.6 23.4 29.3 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 0 27 8.3 16.6 24.9 33.2 28 8.6 29 56.1 64.1 72.1 41.5 49.9 58.3 66.6 17.2 25.8 34.5 43.1 51.8 60.5 8.9 17.8 26.8 35.7 44.7 53.7 62.7 80.5 89.5 30 9.3 18.5 27.7 37.0 46.3 55.6 64.9 74.1 83.3 92.6 48.1 80.2 74.9 83.3 69.1 77.7 86.4 71.6 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 rough 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 Without the on the principle of Equivalent Level Sectious. "" 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 (H-H') × (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, un- corrected; while on the next contract he takes very likely he bas 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-beights, 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-arcas, 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 any one 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 Secs. 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 RAILROAD 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 A CuB 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 Ry is to R, arcs A R Equivalent, Y α 20 Section B 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± y) R where x' represents the corrected area and ≈ 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 r', 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 Ra 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 ordinates 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. B FIG. 74. First, let us assume the surface of the ground to be level; then the simplest way to obtain correctly the contents of the toc is to consider each corner as a quarter of a cone; then if II 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 H³g²; so that the entire contents of the toe are given by the formula S = .523H³s² + .5BH2s; B being the width of the road-bed in feet. This formula is easily worked out by means of Table VIII. S must then be divided by 27 to reduce it to cubic yards. If s = 1, then the above equation becomes S = .75BII² + 1.17877³. 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 I is 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 H 8 = 8 (1 + 1). the value of S being obtained as above. The value of ʼn may be obtained quite well enough by plot- ting Hand 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 formulæ, 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 O Width in In. TABLE OF BOARD MEASURE. THICKNESS IN INCHES. 3750.4167.4583 5000 .5833 со 6667 9 10 11 12 14 16 7500 .8333 .9167 1.000 1.167 1.333 .7500 .8750 1.000 1 125 1.250 1.375 1.500 1.750 2.000 1.000 1.167 1.333 1.500 1.667 1.833 2.000 2 333 2.667 1 14 2 2 3 31 4 41 5 51 6 7 1 0833 1250.1667 .2083 .2500 .2917.3333| .1250 .1875 .2500 .3125 3750 .4063.5000 .1667 .2500 .3333 .4688 .4688 .5625 .5833.6667 2083 .3125 4167 .5208 .5208.6250 .7292 .8333 .8333 34 4 9 10 11 12 13 14 15 16 20 25 18 .5625 .6250 .6875 6250 .6875 .7500 .8333 .9167 .9375 1.042 1.146 .2500.3750 .5000 .6250 7500 .8750 1.000 1.125 1.250 1.375 2917.4375 .5833 .7292 8750 1.021 1.167 1.213 1.458 1.604 .3333 .5000] 6667.8333 1.000 1.167 1.333 1.500 1.667 1.833 .3750.5625 7500 .9375 1.125 1.313 1.500 1 688 1.875 2.063 .4167.6250 .8333 1.042 1.250 1.457 1.666 1.875 2.083 2.292 4583 .9167|1.146 .6875 .9167 1.146 1.375 1.603 1.833 2 063 2.292 12.521 .5000 7500 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 58 .8750 1.167 1.458 1.750 2.042 2.333 2.625 2.917 3.208 6667 1.000 1 333 1.667 2.000 2.333 2.667 3.000 3.333 3.667 7500 1.125 1.500 1.875 2.250 2 625 3.000 3.375 3.750 4.125 8333 1.250 1.667 2.083 2.500 2.917 B 333 8.750 4.167 4.583 .9167 1.875 1.833 2.292 2.750 3.208 3.666 4.125 4.583 15.042 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 1.083 1.625 2.167 2.708 3.250 13.792 4.333 4.875 5.417 5.958 1.167 1.750 2.333 2.917 3.500 4.083 4.667 5.250 5.833 6.417 1.250 1.875 2.500 3.125 3.750 4.375 5.000 5.625 6.250 6.875 1.333 2.000 2.667 3.333 4.000 4.667 5.333 6.000 6.667 7.333 1.500 2.250 3.000 3 750 4.500 5.250 6.000 6.750 7.500 8.250 1.667 2.500 3.333 4.167 5.000 5.833 6.667 7.500 8.333 9.167 } 1.250 1.458 1.667 1.875 2.083 2.292 2.500 2.917 3.333 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.500 4.000 1.750 2.042 2.333 2.625 2.917 3.208 3.500 4.083 4.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000 4.667 5.333 2.250 2.625 3.000 3.375 3.750 4 125 4.500 5.250 6.000 2.500 2.917 3.333 3.750 4.167 4.583 5.000 5 833 6.667 2.750 3.208 3.667 4.125 4.583 5.042 5.500 6.417 7.333 3.000 3.500 4.000 4.500 5.000 5.500 6.000 7.000 8.000 3.500 4.083 4 667 5.250 5.833 6.417, 7.000 8.167 9.333 4.000 4.667 5.833 6.000 6.667 7.333 8.000 9.333 10.67 4.500 5.249 6.000 6.750 7.500 8.250, 9.000 10.50 12.00 5.000 5.833 6 667 7 500 8.333 9.167 10 000 11.67 13 33 5.500 6.417 7.333 8.250 9.167 10.08 11.000 12.83 14.67 6.000 7.000 8.000 9.000 10.00 11.00 12.00 14.00 16.00 6.500 7.583 8.666 9.750 10.83 11.92 13.00 15.17 17.33 7.000 8.167 9.333 10.50 11.67 12.83 14.00 16.33 18.67 7.500 8.750 10.00 11.25 12.50 13.75 15.00 17.50 20.00 8.000 9.333 10.67 12.00 13.33 14 67 16.00 18.67 21.33 9.000 10.50 12.00 13.50 15.00 16.50 18.00 21.00 24.00 10.000 11.67 13.33 15.00 16.67 18.33 20.00 23.33 26.67 152 RAILROAD CONSTRUCTION. amounts to about 20 feet the difference in the results by the two methods is very considerable. 136. The original notes of the cross-sections should be copied on the left-hand pages of another note-book, and opposite them, on the right-hand pages, the sectional areas, contents, etc., should be entered as soon as worked out. A "Record” should also be kept, into which each separate item should be entered as soon as completed,-not in detail, but simply the total amounts; these notes then form the groundwork of the final estimate. The details are entered separately in note- books apportioned to each class of work. As regards taking notes for the monthly estimates, the simplest way is to walk over the work and sketch on the prog- ress profile the state of construction at the time. Another way, possibly more convenient in light work, is to note the percentage of the total amount which is done up to date. * The classification is often a matter of considerable difference of opinion, especially in the allowance for loose rock." All boulders, etc., exceeding the limit for loose rock must be carefully measured. When there is much of this to do, a good plan is to have a man especially to look after it on two or three subdivisions, who can also take the Force Account and give to the contractors any simple information they may require con- cerning the work. The subdivision engineers and their men are thus saved a very considerable amount of time and work. TIMBER-WORK. 137. Timber is usually measured in railroad structures in B. M. (Board Measure), the contract for culverts, etc., being let by the 1000 feet B. M. One foot B. M. 144 cubic inches, so that the B. M. of any given stick is found by multiplying together the width and thickness in inches and the length in feet, and dividing the result by 12. The first portion of this calculation and the division by 12 is accomplished by means of the table on page 151. In altering the length of trestle-posts, etc., to make allow- ance for the difference in elevation of the two rails, the follow- ing table will be found useful, as well as in many similar operations: RAILROAD CONSTRUCTION. 153 FRACTIONS OF AN INCHI IN DECIMALS OF A FOOT. In. 0 1 2 3 4 5 6 7 8 0 133 16 SONIA W WIN ON W -2-32 ان ایک 9 10 11 Foot.0833.1667.2500.3333.4167.5000.5833 6667.7500.8333.9167 0026.0859.1693.2526.3359.4193 .5026|.5859.6693.7526.8359.9193 .0052.0885.1719.2552.3385 219 .5052.5885.6719.7552.8385.9219 0078.0911.1745.2578.3411.4245 .5078.5911.6745 7578.8411.9245 .0104.0938.1771.2604.3438.4271.5104.5938.6771.7604.8438 9271 0130.0964.1797.2630.3464.4297 .5130.5964.6797.7630.8464.9297 0156.0990.1823.2656.3490 4323 .515.5990.68237656.8490.9323 .0182.1016.1849.2682.3516.4349 .5182.6016 6849.76828516.9349 0208.1042.1875.2708.3542.4375.5208.6042.6875,7708.8542 9375 0234.1068.1901.2734.3568.4401 5234 6068 6901 7734.8568 9401 0260.1094.1927.2760.3594.4427 260.6094 6927.7760.8594.9427 0286.1120.1953.2786 .3620.4453.5286|.6120|,6953|.7786|.8620|.9453 0313.1146.1979 2813.3646.4479 7313.6146.(979 7813.8646 9479 • 0339.1172.2005.2839.3672.4505 .5339.6172.7005.7839).8672.9505 0365.1198.2031.2865.3698 4531.5365.61987031.7865.8698.9531 0391.1224.2057.2891.3724.4557, 5391.6224 7057.7891.8724.9557 0417.1250.2083 2917.3750.4583.5417.6250.7083.7917.8750.9583 0443.1276.2109.2943.3776 4609 5443 6276 7169.7943.8776,9609 .0469.1302.2135.2969.3802 4635.5469 6302.7135.7969.802.9635 .0495.1328.2161.2995.38284661.5495.6328 7161.7995.8828.9661 .0521.1354.2188 3021.3854.4688 5521 6354 7188.8021.8854.9688 0547.1380.2214 3047.3880.4714.5547),(380| 7214 8047.8880.9714 .0573.1406.2240.3073 3906 4740.5573.6406], 7240.8073.8906 9740 .6599.1432 2266.3099.3932.4766.5599 6432.7266 8099.8932 9766 .0625.1458.2292.3125.3958.4792.5625 6458.7292.8125.8958.9792 .0651.1484.2318.3151.3984.4818 5651.6484.7318.8151.8984.9818 .0677.1510.2344.3177.4010 4844 5677 6510 7344 8177.9010 9844 .0703.1536.2370.3203.4036.4870′.5703 6536.7370.8203.9036.9870 .0729.1563.2396.3229.4063.4896 .5729 6563 7396.8229.9063.9896 .0755.1589.2422.3255.4089.4922.5755.6589.7422.8255.9089.9922 .0781.1615.2448.3281.4115.4948 .5781 6615.7448.8281.9115.9948 .0807.1641.2474.3307.4141 4974.5807.6641.7474.8307.9141.9974 31 32 0 1 2 3 4 5 6 7 8 9 10 11 For notes on the strength, etc., of timber, see Part IV. IRON-WORK, 138. In In estimating the weight of Bolts and Nuts the weight of the heads and nuts themselves may be taken from the following table, assuming them to be of ordinary propor- tion: Diameter of Bolt. Hex. Head and Nut Sq. Head and Nut.. 00,00 WEIGHT OF BOLT-HEAD AND NUT. سرات 1} 3 حراست ง C2 21/1 ༢ 1 13 13 13 lbs.lbs bs.bs lbs.lbs 'bs lbs. lbs. lbs. lbs lbs .017.057.128 .27 .43| .73|1.1| 2.9 3.8 5.6 8.8 17 5.6 .021.069.164.32 55 88 13 2.6 4.4 7.0 10.5 21 | 154 RAILROAD CONSTRUCTION. The weight of the shanks of the bolts may be found from the following table of the weight and strength of iron rods. If, however, the screw end is upset, with a consequent enlarge- ment of the nut and head, the usual allowance for the weight due to upsetting, and square head and nut, will be equal to about 13 diameters of additional length of the shank of the bolt. If the uut and head are hexagonal, 11 diameters are then sufficient. This allowance is suitable when the length of the upsetting equals about 6 diameters of the shank. Thus if we have a 1-inch bolt upset for 6', if 36" long and the head and nut square, its weight will be given by the weight of a 1-inch bar 49" long. WEIGHT AND STRENGTH OF ROUND WROUGHT- IRON BARS. Diam. Weight in Breaking Diam. in inches. lbs. per foot Strain in in Weight in lbs. per foot Breaking Strain in run. lbs. inches. run. lbs. عرين مرسم 1 .0414 550 13 TO 093 1240 12 .165 2:200 13 .258 3430 14 .372 4950 1 .506 6720 .661 8800 .837 11130 1.03 13750 1.25 16620 1.49 19780 1.75 23300 2.03 26880 2.33 30910 2.65 35170 -400 1+ 2300 −OOR CHAN -KILARINEPOE 3.35 42340 4.13 52200 5.00 63170 5.95 75260 6.99 88260 8.10 102370 9.30 117600 10.6 133700 12 0 142900 13.4 160400 14 9 178500 16.5 198000 18.2 218200 20.0 239400 23.8 285000 As a safe working strain one fifth of the above breaking strains may usually be taken. The two washers generally used to each bolt weigh together about the same as a length of shank 14 diameters; but if the bolt is upset, they then weigh about the same as a length = 22 diameters. Railroad Spikes.-The following table gives the weight, etc., of the spikes commonly used for fastening the rails to the ties: RAILROAD CONSTRUCTION. 155 י J Į Length Thick- in ness in inches. inches No. per keg of 150 lbs. No. per in Length Length Thick- ness in lb. keg of No. per No. per Ib. inches. inches. 150 lbs. } 4/2 400 2.66 705 4.70 488 3.25 390 2.60 295 1.97 1 6 5 257 1.71 6 aaaa & ST NYPG-N- 350 2.33 289 16 1.93 218 1.46 310 2.07 262 1.75 196 1.30 The following table gives the angle-bars and bolts neces- sary for 1 mile of track: Length of Rails in feet. No. of Angle- No. of Bolts. bars. Length of Rails in feet. No. of Angle- No. of Bolts. bars. 24 880 1760 27 782 1564 25 844 1688 26 812 1624 28888 751 1508 30 704 1408 The following table gives the weight of Rails required for 1 mile of track: Y Weight of Rail Weight Weight 7 per yard. per mile. of Rail per yard. Weight per mile. Weight of Rail per yard. Weight per mile. lbs. 40 45 48 50 52 COAST P2388 tons. lbs. lbs. tons. lbs. lbs. tons. lbs. 62 1920 56 88 0 65 102 320 0 1600 57 89 1280 68 106 1920 75 960 60 94 640 ΤΟ 110 0 78 1280 62 97 960 72 113 820 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 lbs. per yard by 1; the product is the weight required in tons of 2000 lbs. (the tons. in the table = 2240 lbs.) The weight of iron in lbs. per yard is given by multiplying its sectional area in inches by 10, assuming the iron to weigh 480 lbs. per cubic foot. Steel rails usually weigh about 490 lbs. per cubic foot. 156 RAILROAD CONSTRUCTION. 139. BALLAST AND TIES.-The following table gives the amount of ballast required per mile of road: Depth in inches. Top Width, Single Track. Top Width, Double Track. 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 4691 5085 5474 8996 9388 9780 30 6111 6600 7087 11490 11980 12470 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 No. of Ties. in inches. Centre to Centre in inches. No. of Ties. *22* 18 3520 27 2347 20 3168 30 2112 22 2899 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. INSTRUMENTS. 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 cne head. For astronomical work the 7 Į ¡ { 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 ARC OF 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 1. 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 may be 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 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 S 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 I, 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 SAR, where A is a point in the line of sight, found by producing 81 to its intersection. But suppose S' and R were the two objects between which the angle is to be observed, then a will be the 160 EXPLORATORY SURVEYING. Finally, apex of the angle measured. Finally, if S is situated at s, so that sI is parallel to RA, then the angle given by the sextant between 8 and R=0° (i.e., if there were no index-error the s reading should be zero), and if the reflection of R 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 RI and RA are sensibly parallel,- as was assumed in Adjustment C,-the question of instrumental S S H DIRET LITTUTE". a B 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 art ficial horizon there is another form of paral· lax which sometimes needs consideration due to the apex ▲ of the angle observed not coinciding with the artificial horizon. Let R be the image of a star S reflected in the artificial horizon Then if SA is parallel to SR, as is sensibly the case when deal ing with objects at a considerable distance from the instrument, the angle SAR may be considered equal to twice the angle SRB, i.c., 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 S is comparatively close at hand, then we cannot consider SAR = 2SRB, and consequently by dividing 66 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 so situated that the angle AS 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 b is parallel to RB (the surface of the artificial 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 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 SA, it becomes necessary either to apply a correction, or to arrange the positions of the horizon and the instrument so that the point 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 ARC, 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 cye 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 borizontal. 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 ab0, then Aa and Bb represent respectively the altitudes of A and B, and the complement of A B FIG. 76. the altitude of each equals its "zenith distance," AZ or BZ. Then in the spherical triangle ABZ, since we know all three sides, therefore (since ab = Z) COS ab 2 sin S sin (S-AB) sin AZ sin BZ where S AZ+BZ+AB 2 145. 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 EXPLORATORY SURVEYING. 163 • [ រ ፡ 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 eutering 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 land, 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- rate." The "shop- nometers they will probably gain on their 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 ou 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 five 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: D E Adm B 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 formule 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's law for the pressure of gases, to obtain the difference in the heights of A and C 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 EXPLORATORY 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, vy 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 formule 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 as the barometric coefficient. Various values are given for this coefficient, but probably that given by Regnault 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 H X=60384 log where II is the barometric reading at the lower station and h 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(T+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-64 900 1+ 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 1 1 — .0001(T”' — t')' 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 X=60384 log h (1 T+t64 1 + X (1+ 900 .0026 cos cos 2L 21) (T. 1 − 1 7). .0001 (T' t' 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- ter's is X=60384 log H 1 + T+t-64 900 If Hand h do not differ by more than about 3000 feet we may do away with the logarithms in the above equation, which thus becomes, approximately, — X = 52450 H h H+ h 1+ T+t-64 900 The error involved by this formula is inappreciable within the limits stated. By assuming (T+t) to equal 108° this formula becomes X = 55000 H-h H+h which is generally known as Belville's Formula and is con- venient for rough work. The table opposite gives the VALUES OF T t 900 (π +6 64). 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+t-64 T+t T+t64 T+t T+t-64 T+t T+t-64 900 900 900 900 20- -.0489° 66° +.00290 112° +.0533° 158° +.1044° 22 .0467 68 .0044 114 .0556 160 .1067 24 .0444 70 .0067 116 .0578 162 .1089 26 .0422 72 .0089 118 .0600 164 .1111 28 .0400 74 .0111 120 .0622 166 .1133 30 -.0378 76 +.0133 122 +.0644 168 +.1156 32 .0356 78 .0156 124 .0667 170 .1178 34 .0333 80 .0178 126 .0639 172 .1200 36 .0311 82 .0200 128 .0711 174 .1222 38 .0289 84 .0222 130 .0733 176 .1244 40 -.0267 +.0244 132 +.0756 178 +.1267 42 .0244 88 .0267 134 .0778 180 .1289 44 .0222 90 0289 136 .0800 182 .1311 46 .0200 92 .0311 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 .1400 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 6.1 .0000 110 .0489 154 .0511 156 .1000 200 .1511 .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 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 high; 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 in 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: Warın both the EXPLORATORY SURVEYING. 171 50 mercury and tube and filter in through a paper funnel-the hole of which does not exceed of an inch-to about 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 the 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- cering 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, the aneroid should always be held horizon- tally, for the weight of the parts themselves has a very considerable influence on the readings: a difference correspond- 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 ancroids 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 formulæ already given. BAROMETRIC AND ATMOSPHERIC HEIGHTS. in. Bar. Alt'de Bar. Alt'de. Bar. Alt'de. feet. in. feet. in. feet. Bar. Alt'de. Bar. Alt'de in. feet. in. feet. 21. 9900.1 .1 9768.3 23. .1 7375 1 7254.7 25. .1 5060.6 4949.8 27. 2924.4 29. 940.9 .1 2821.8 .1 845 4 .2 9637.1 .2. 7134.7 24839.5 22719.6 .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: .4 560 7 .5 9247.0 .6 9118.3 .78990.0 .8 8862.4 .9 8735.3 8608.9 22. .1 8483.0 .28357.7 .3 8233.0 24. .8 6426.0 .9 6309.6 6193.8 26. .5 6778.1 .5 4511.0 .5 2415.2 5 466.5 .6. 6660.2 .6 4402.3 .6 2314.4 .6 372.6 6542.8 .7 4294.0 .7 2214.0 279.0 .8 4186.3 .8 2114.0 .8 185.7 .9 4078.9 .9 2014.3 .9 92.7 3971.9 28. 1915.0 30. 0.0000 .1 6078.3 .1 3865.4 .1 1816.0 .1 92.5 .2 5963.4 .2 3759.3 .2 1717.4 184.7 .3 5848.9 .3 3653.6 31619.2 .3 276.6 8108.7 .5 7985.1 .6 7862.0 .4 5734.9 .5 5621.4 .43548.3 .1 1521.3 .4 368.2 .5 3443.4 .5 1423.7 .5 459.5 .6 5508.3 .6 3338.8 .6 1326.5 .6 - 550.6 .7 7739.4 .7 5395.7 .73234.6 .7 1229.6 .7 · 641.4 .8 7617.5 .8 5283.6 .8 3130.8 .8 1133.0 .8 781.9 .9 7495.9 .9 5171.9 .9 3027.4 .9 1036.8 .9 822.2 174 EXPLORATORY 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., iu 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 D FIG. 78. 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 tan K sin ADC sin CBD sin CAD sin BDC then CAB N tan ABC) = tan (45° 2 ACB 【) cot 2 then CAB = CAB+ ABC 2 CAB — ABC + 2 and sin BDC sin ACB ABCD sin CBD sin CAB If C can be ranged in line with A and B we can then find the position of A and B separately, as shown in Sec. 59; the difference of the distances so obtained gives the length of AB, and the bearing is obtained by direct observation. 176 EXPLORATORY SURVEYING. Suppose, however, that in Fig. 78 the length and direction of AB is known, and it is the distance CD which is required. Then observe the angles at C and D and obtain CAB as be- fore, but in this case the last formula becomes CD AB = siu CBD sin CAB sin BDC sin ACB This might be also solved by assuming a certain length for CD, and from it finding as above what the length of AB must be; then the true AB is to the value of AB so obtained as the true CD is to the assumed value of CD. C D B FIG. 79. If, as in Fig. 79, the lines AB and CD cross each other, the above formule apply equally well. 161. The problem known commonly as the "Three-point Problem" is probably the most useful method there is of establishing the position of any given point; it is as follows: D S E FIG. 80. Suppose, as in Fig. 80, we know the position of three points A, B, and C and wish to fix the position of the point S; we can do it by simply observing the angle ASB and BSC. EXPLORATORY SURVEYING. 177 Then, in order to obtain the position of S geometrically, pro- ceed as follows: Find D, the centre of the circle ABS (by setting off at A and Bangles equal to 90° ASB). Then draw the circle through the points A, B, and S. Similarly find the centre E and draw the circle BCS. Then S, the point of intersection opposite B, is the position required. When one of the angles is obtuse, set off its difference from 90° on the opposite side of the line joining the two objects to that on which the point of observation lies. When the angle ABC the supplement of the sum of the two angles, the position of S will be indeterminate by this method. S may often be obtained with sufficient accuracy instrument- ally by plotting the angles ASB and BSC on a piece of tracing- cloth, and sliding it over the plan until the required position is obtained. The "station-pointer" is an instrument much used for this purpose, especially in hydrographers' offices, where soundings are usually plotted in this way. If accuracy is required the position of S may be found ana- lytically thus, as given by Prof. Gillespie : Let AB = c; c; BC = a; ABC = B; ASB = S; and BSC S'. Also make T = 360° — S — S' — B, and let Then BAS U, and BCS V. = = c sin S' cot U cot T a sin S cos T +1); V=T- U; c sin U SB = or SB sin S a sin V sin S' c sin ABS a sin CBS SA = and SC SC = sin S sin S' Thus if ASB AB we find Then ABC 33° 45', BSC = 22° 30', 6000 ft. and BC 4000 ft., 104° 28′ 39″. U = 105° 08′ 10"; whence V 94° 08′ 11″. SA SB 10425.1 ft., 7101.9 ft., and SC: = - = 9342.9 ft. 178 EXPLORATORY SURVEYING. 162. The position of a point may also be fixed by observing the bearings from it of two known points, and may be found on the plan by drawing through those points the bear- ings so obtained; their intersection gives the point required. 163. Another common method of fixing the positions of B FIG. 81. outlying points is by intersection, as in Fig. 81, the position of the two points of observation A and B being known. 164. While on the subject of triangulation, it will be as well to consider the methods of obtaining the heights of mountains trigonometrically. In the first place, suppose we are able, as in Fig. 82, to ob- E A B C FIG. 82. tain two points A and B in the direction of C (a point the elevation of which we wish to obtain) both at the same eleva- tion, and to measure the distance between them; then CD= cot CAD AB cot CBD' If, however, the two points cannot be taken at the same level, but have to be taken such as E and A, observe the angle CEA, EXPLORATORY SURVEYING. 179 and at A the altitudes of C and E, either with an artificial horizon or with the vertical arc of a transit.. Then in the triangle EAC ACEA sin E cosec C, where the angle at C the sum of the altitudes of E and C (taken at A) the angle at E. Then CD = AC sin CAD. This would of course hold equally good if EA sloped the other way, but then Calt. of C from A alt. of A from E-angle at E. The correction for curvature and refraction given in Sec. 51 must be added to the height as obtained above. But suppose it is not convenient to obtain a base as above in the same direction as C. Then, as in Fig. 83, measure a C A B FIG. $3. basc AB (not necessarily level) and observe the angles CAB and CBA. Then in the triangle ABC AC AB sin B cosec C. Next observe the altitude of Cfrom A, i.c., the angle CAD: then CD = AC sin CAD. To the height so obtained, the correction for curvature and refraction given in Sec. 51 should be added. Suppose it is required to find the difference in elevation of two inaccessible points, the simplest way is to find the eleva- tion of each separately, as above, and subtract the one from the other. 180 EXPLORATORY SURVEYING. 165. In observing altitudes, the refraction of the air enters so largely into the question and varies so enormously according to the condition of the atmosphere, that every pre- caution must be taken to eliminate the errors due to it, where accurate work is wanted. A 不 ​B FIG. 84. Its nature is such that suppose A and B are two stations visible from each other, the line of sight between A and B, instead of being straight, follows a curved course as shown in Fig. 84, making the altitude as observed at A too great, by the amount F, which is termed the "angle of refraction." Similarly the depression of A as observed from B will be too small. Thus the tendency of refraction is to make objects appear at a higher elevation than they really are; so that in observing altitudes a correction for refraction should be always subtracted from the apparent altitude to obtain the true altitude. In ordinary work the corrections given in Sec. 51 for both curvature and refraction are sufficiently correct. But for highly accurate work-on which this article does not treat— various allowances and corrections must be made. Refraction diminishes with altitude and is slightly greater over water than land. It is generally at its maximum during the night, and at its minimum about noon; but it is steadier in the night than in the day time, and for this reason night work is usually as reliable as work done during the day. About sunrise and sunset are the worst times to observe alti- tudes, for not only is refraction then high in quantity, but also extremely variable. A day with the sky overcast is a good day on which to take an observation. Clear days are more subject to rapid changes than dull oues. (For Astronomical Refraction, sce Sec. 184.) 166. A method of eliminating to a great extent the effect of refraction in observing the difference of elevation of two EXPLORATORY SURVEYING. 181 stations A and B, is that of observing Reciprocal Angles. Thus in Fig. 84, at A, the altitude of B should be observed, and at B (when practicable) the depression of A. Half the difference of these angles will be the combined correction, and the tangent of half their sum, multiplied by the horizon- tal distance between them, will give the difference of level, after adding the correction for curvature of the earth given in Sec. 51 This method assumes that the coefficient of refrac- tion is the same at both A and B; therefore the angles should, if possible, be observed simultaneously, lest the refracting power of the air should change in the interval. (For the cor- rection for Refraction, see Sec. 51.) 167. To obtain the height of a mountain by the ob served depression of the sea horizon.-The depression of the horizon, or as it is commonly called at sea the “Dip,” taking R = the earth's mean radius of curvature in feet, equals in seconds therefore 2H D = 206265 R 63.8 WH; D VII 63.8 where H Height in feet. = A Thus, were it not for refraction, we could find the elevation of A (Fig. 85) by merely observing the dip D. But D' is the dip actu- ally observed; so that, taking re- fraction into account, the above formula becomes NII (nearly), D' 55 which can only be depended on to give approximate results. 168. In observing altitudes with a H R sextant and artificial horizon, as for instance in Fig. 84, the altitude of FIG. 85. B will be one half the altitude read on the arc. since it is the 182 EXPLORATORY SURVEYING. ‘double altitude” that is actually observed. To find a point Con the same level as the instrument the altitude can then be measured down from AB. To observe the depression of A from B with a sextant and artificial horizon, we must estab- lish some point-as far off as possible so as to reduce parallax --the altitude of which exceeds about 6°, and observe its alti- tude correctly, and then obtain the angle between it and the object whose depression we wish to find. At night a star may often be made use of for this purpose, allowance being made for its motion. This method may also be employed in reading altitudes which would otherwise need the use of a supplementary arc. (See Parallax, Sec. 142.) To read an altitude or depression with a transit, observe the altitude first in the usual way, then "reverse" and point the telescope to the object and read its supplement; the mean alti- tude so obtained is free from error due to the "horizontal axis" not being truly perpendicular to the "vertical axis" of the instrument. The errors of graduation and observation are also somewhat reduced.* 169. It is essential that a survey which consists of a series of triangulations should have an accurate base to start from. Sometimes in exploratory surveys the distance between two mountain peaks, or some prominent objects near the point at which the survey starts, is already known with sufficient ac- curacy to warrant the line joining them being accepted as a base, but more usually it is necessary to obtain the distance between such points from a base more or less accurately measured. For this purpose of course as level a piece of ground must be obtained as possible, and as there is often difficulty in find- ing such a site long enough for a base, it becomes necessary to start from a short base and then extend it by a series of tri- angulations, the angles of which fall, if possible, between the limits of 30° and 120°. As regards the MEASUREMENT OF A BASE for ordinary work we can consider a steel tape, properly tested at a given tem- perature, to be sufficiently accurate. The correction for tem- perature amounts to about .000007 of the length of the tape for every 1° Fah. Thus a 100-foot tape, tested at a temperature of 50° F., would give a result too long by about 3 feet in 2 miles at a temperature of 90° F. * Adjustment E, page 35, is also corrected for. EXPLORATORY SURVEYING. 183 Since all maps are made on the assumption that the linear measurements are reduced to the sea-level, in dealing with high altitudes the length of the base may be multiplied by h 1— (nearly), where h = elevation above sea-level, r = radius of the earth (see Sec. 206), in order to reduce it to sea-level. But this is a refinement which is usually only needed in work requiring great accuracy. (6 170. In making a regular triangulation survey, the angles of the main triangles are of course themselves observed; but in such work as exploratory surveys, where mountain peaks are selected as stations," such a method of procedure would, on account of the time and difficulty involved, be out of the question. A readier method of proceeding may be best shown by an example as in Fig. 86. It depends upon always having in view at any station at least two points whose positions are knoron. Suppose we have obtained, by triangulation or otherwise, the distance between and bearing of two conspicuous points A and B, and suppose our route lies along the dotted line abcd. α FIG. 86. At a, a point from which A and B are visible, we observe the bearings of A and B, and thus fix the position of a. Sup- pose that from a a distant mountain peak C is visible, we take the bearing of it also; then if we wish to fix the position of such apoint as b, from it we observe the bearings of Band C. When 184 EXPLORATORY SURVEYING. we get to c we locate its position by bearings from A and B; but suppose we can see A and B no farther, it then becomes necessary to establish two other points which we may use as we have already used A and B. A bearing to C will then lo- cate it. We also observe the bearing of D. When dis reached, we observe the bearings of C, c, and D, which fix its position and also the position of D. No simpler way of keeping a course can be had than this; and it has the enormous advantage over many of the methods in use, that it fixes the main topographical features bordering along the route at the same time as positions on the route itself. The explorer must be constantly on the lookout for points ahead on his probable route and in the neighborhood. The drawback to the method is its inaccuracy when worked by magnetic bearings alone. But if the points are well selected, an error of a degree or so in the bearings is really immaterial in work of this class, and the errors usually more or less counter- act each other. Besides, from time to time the courses and distances can be easily checked by the establishment of another base, and the work already done more or less corrected, and a fresh start made. If we keep three or more points in view we are able to apply the trigonometrical method given in Sec. 161, and thus do very accurate work so long as we are careful in establishing cor- rectly the positions of A, B, C, D, etc. In following along valleys, or in sight of a distant range of mountains, this method works admirably, and if a transit is at hand a check may be applied from time to time on the dis- tances and bearings with very little trouble. a A There is no need to apply any correction, however extensive the triangulations may be, for the curvature of the earth, since the spherical excess of a spherical triangle contain- ing 75.5 square miles is only 1"; so that in a triangle containing 4530 square miles the sum of the three B angles only exceeds 180° by 1'. FIG. 87. a 171. To measure horizontal angle without an instrument between two EXPLORATORY SURVEYING. 185 such points as A and B from 0, as in Fig. 87. Range in a and b with A and B, each distant from O by, say, 50 feet. Meas- ure ab, then sin AOB 2 ab 100 172. To measure a vertical angle without an instru- ment, probably the simplest way is to hold a pencil vertically out at arm's length and note the length subtended on it. Then if the distance from the eye to the pencil = 1 and p is the length subtended on the pencil, tan A P where A is the angle required. Similarly if I were the dis- tance of some object whose height II we wish to obtain, H= Lp 173. Distance across an open stretch of water can often be taken with sufficient accuracy by observing the time occupied by the passage of the report of a gun from one point to the other. This may be done in the day-time if there is a tele- scope handy to watch for the smoke, but otherwise the flash of course can be best seen at night. The velocity v, in feet per second, with which sound travels, depends greatly on the temperature; thus at 32° F., v = 1090; at 60° F., v = 1125; and at 100° F., v = 1175. By taking the mean of 3 or 4 shots, the distance may be obtained with confidence to a quarter of a mile. If the wind is blowing hard in the direction from which the sound comes, the velocity of the wind may be added to v. 174. We can observe an interval of time when a watch is not at hand by counting the vibrations of a stone tied to the end of a string. If from the centre of gravity of the stone (and the string) to the point of suspension is 39.1 inches, each vibration occupies one second. For any other length L, each vibration occupies N L seconds. 39.1 186 EXPLORATORY SURVEYING. The vibrations should be kept as small as possible so as to re- duce the resistance of the atmosphere. In this way a toler- ably long interval may be measured with a fair amount of confidence. The best way, however, is to compare the vibra- tions with a watch subsequently. B. BY DIRECT MEASUREMENT AND COMPASS COURSES. 175. By far the most convenient and accurate method of obtaining direct measurement on exploratory surveys is by means of an odometer, which answers the same purpose as the patent log at sea, only more efficiently; but unfortunately it necessitates the use of some wheeled vehicle, which is not always a convenient appendage to an exploring outfit. Pedometers answer well in country where the condition of the ground is comparatively regular and walking easy, but where the surface is much broken they are worse than useless, being misleading as well. The best means of then ascertaining the distance travelled is by estimating the rate of progress and keeping track of the time. The approximate rate may always be found by noting the time occupied in covering, say, 100 yards; then if t the time occupied in seconds, the velocity in miles per hour equals 200 V (nearly); t so that we have the following values of v for various values of t: t secs. V t บ t V t บ m. p. h. sees. m. p. h. secs. m. p. h. secs. m. p. h. 200 1 80 133 1.5 66 100 ૨ 2 50 0234 2.5 40 5 33 6 28 7 M2 25 8 22 9 20 10 As regards keeping the courses by compass, in open country, it is best to establish the bearing of some point ahead on the probable route and then to correct it by estimation, if, when abreast of that point, it should be found to be considerably to EXPLORATORY SURVEYING. 187. one side of the route taken. In timber country, the bearing of the sun being taken from time to time, it forms a highly useful guide when no distant landmarks are visible. At night the pole-star forms as good a guide as could be wished for. C. BY ASTRONOMICAL OBSERVATIONS. 176. Before attempting the solution of astronomical prob- lems in connection with the establishment of positions ou the earth's surface, it will be well to give a few explanations as briefly as possible regarding the fundamental principles in- volved, and definitions of the terms used. TIME. 177. Civil or Common Time is really what is termed in astronomical language Mean Solar Time, with this difference, that a civil day being reckoned from midnight to midnight, the corresponding astronomical day is reckoned from the noon of that day to the following noon, and is also counted con- tinuously up to 24 hours. Thus 4 A.M. on Jan. 10 would be stated in mean solar time as 16h 0 Jan. 9. Now the velocity with which the earth travels round the sun varies in different parts of its orbit. Owing to this cause and also to the obliquity of the ecliptic (sce Sec. 180) the sun's apparent motion is ir regular. Thus we find that the sun is apparently travelling faster in winter than its average rate, and in summer slower. It is simpler to consider the earth as stationary and the celestial bodies as revolving round it. In speaking of the velocity of the sun's motion, then, it is its motion among the stars-or on the star sphere-that is referred to, not its actual motion in the sky; the average rate of this motion is about 59′ per day and in a direction opposite to that in which the whole star sphere is apparently revolving, so that the motion of the sun in the sky is really slower than that of any given star, the result of which is that the star apparently revolves round the earth 366 times while the sun only makes 365 revolutions (nearly). Now, owing to the irregularity in the sun's motion, it is more convenient to substitute for the real sun a fetitions one, termed 188 EXPLORATORY SURVEYING. the "Mean Sun," which is imagined to make the same number of revolutions in the course of the year as the real sun, but always to maintain the same rate of motion. Thus it follows that the mean sun sometimes crosses the meridian-i.e., is due south-before, and sometimes after, the real or, as it is termed in the Nautical Almanac, the apparent sun. 178. The interval of time between the passage of these two suns across the meridian is called the Equation of Time, which when the mean sun is ahead of the apparent sun is con- sidered positive, and when the apparent sun is ahead, negative. Thus, since the mean sun is always south at mean noon, by adding or subtracting (as the case may be) the equation of time to or from 24 hours-subtracting 24 hours if necessary—we ob- tain the mean solar time at which the apparent sun is on the meridian, i.e., apparent noon. Thus, if for a certain day the equation of time is given as + 12m 04s, the apparent sun will be on the merilian 12m 04s after mean noon, or at 0h 12m 04s astronomical mean time. Had the equation been negative, ap- parent noon would have occurred at 23h 47m 56 mean astro- nomical time. Expressing the relative positions of the two suns in the form of an equation, we have Mean Time = Apparent Time ± Equation of Time. The mean time of that sun is the greater whose R. A. is the less. (See Sec. 180.) Day of Month. Jan. Feb. March. April. May. June. 1 11 + 4m 08 8 21 21 11 41 13m 54s 14 29 13 47 10 06 13m 283m 50 7 12 1 25 310 038 3 48 3 37 2m 249 0 36 +1 31 July. August. Sept. Oct. Nov. Dec. 1 3m 366m 048 11 5 15 21 6 05 +++ Om 138 10m 278 16m 198 4 56 +2 53 - 3 35 7 06 13 19 15 22 15 49 13 53 10m 39s - 6 23 - 1 31 The above values of the Equation of Time show approxi- mately the positions of the two suns relatively to each other throughout the year. These values change but little from year to year; and are sufficiently accurate to enable an engineer EXPLORATORY SURVEYING. 189 to find mean time to a few seconds whenever be may not have a Nautical Almanac at hand; or to correct the reading of a sun-dial, which of course gives apparent solar time, in order to reduce it to mean time. 179. Now the interval of time between the passage of a star across the meridian one day and its passage on the following day is equal to one Sidereal day; and since the sun makes only 365.242 revolutions to 366.242 of the stars, we have 23h 56 4.09 mean solar time, 24h 03m 56.55 sidereal time; A sidereal day or, A mean day or, in other words, To convert a sidereal interval of time into mean solar units, it has to be reduced at the rate of 9.830 seconds per hour;- while To convert a mean solar interval into sidereal units, it has to be increased at the rate of 9.856 seconds per hour. Sidereal time is reckoned from the "vernal equinox," or the moment at which the sun crosses from the southern to the northern hemisphere, and is thus, in a way, altogether inde- pendent of mean solar time; but if we know the moment at which the vernal equinox occurs in mean time, we thus have a means of connecting sidereal with mean time. But instead of having to start our calculations from the vernal equinox each time, the sidereal time of mean noon is given for every day in the year in the Nautical Almanac; so that To convert sidereal time into mean time, we have this rule: From the sidereal time given (increased if necessary by 24 hours) subtract the sidereal time at the preceding noon, and then reduce the result at the rate of 9.830 seconds per hour;—and, To convert mean time into sidereal time: Increase the mean time at the rate of 9.856 seconds per hour; the time thus ob- tained, added to the sidereal time at the preceding noon (subtracting 24 hours if necessary), gives the corresponding sidereal time. The Conversion of the Intervals may be greatly facilitated by means of Table XIX. DECLINATION AND RIGHT ASCENSION. 180. These are terms used to denote the positions of celestial bodies in the star sphere relatively to the equinoctial (which is really its “equator”) and a plane perpendicular to it passing 190 EXPLORATORY SURVEYING. through the vernal equinox; in the same way as terrestrial Latitudes and Longitudes give the positions of placcs on the earth's surface, relatively to the equator and the meridian of Greenwich. The plane of the earth's equator produced to the star sphere gives what is called the Equinoctial; and the Ecliptic, which is really the plane occupied by the earth's orbit, is inclined to the equinoctial at an angle of about 23° 27′ (slightly varying), which is termed the Obliquity of the Ecliptic. Instead, however, of expressing the Right Ascension of bodies as so many degrees E. or W. of the vernal equinox, it is more convenient to adopt the phraseology of sidereal time and denote the positions of bodies according to the interval of time at which they cross the meridian after the zero of sidereal time, i.c., the vernal equinox. Thus it follows that the sidereal time at which a body is on the meridian is given by its Right Ascension (R.A.), so that instead of speaking of the "sidereal time at preceding noon" as in the rules given in Sec. 179, wc might have said "the R. A. of the mean sun at preceding noon," for the sidereal time at noon is often so stated in almanacs. And if we know the sidereal time at mean noon, say at Greenwich, we can, by adding or subtracting the equa- tion of time (as the case may be) obtain the R.A. of the apparent sun at mean noon at Greenwich, and by correcting the sidereal time at mean noon at the hourly rate of +9.856 seconds, and also correcting the equation of time, we can find the sun's R. A. at any later hour. The Declination of a body, which is really its angular measure on the star sphere, north or south of the equinoctial, is considered positive when north, and negative when south. 181. But so far we have assumed, except in the case just mentioned above, that it has been unnecessary to correct either the equation of time, R.A. or Dec., as given in the almanac; but since these quantities are always varying, and they are only given for a certain hour at a certain place, when required for any other hour the values as given in the tables must be corrected-usually with sufficient accuracy by simple inter- polation-to reduce them to the time for which they may be required. And since every 15° of longitude rest is equivalent to 1 hour later and 15° east to 1 hour earlier, if in longitude 90' west of Greenwich we want the declination of the sun at EXPLORATORY SURVEYING. 191 4 P.M., and for noon on that day it was given in the almanac as +17° 40′, and at noon on the following day as + 18° 00', the declination at 4 P.M. in longitude 90° west (which is equivalent to 10 hours later) will be 17° 48'.3; and in the same way the R. A. and Equation of time must be corrected. In dealing with stars, daily and hourly corrections are un- necessary, since their Decs. and R.A.'s change but little in the course of the year (see Sec. 213); but in dealing with the moon, the change is so rapid as to necessitate a more accurate inter- polation than would be given by simple proportion as above. HOUR-ANGLE, ETC. 182. The "bour-angle" is a term which may best be ex- plained by means of Fig. 87. P co-lat. N co-alt. lat. co-dec. dec. S alt. Path of Star Equinoctial b a FIG. 87. Suppose a person stationed at A, on the earth's surface, ob- serves a star S at an altitude Sa above the horizon ab. Then if P is the celestial pole and Z the zenith, since he knows the declination of the star, if he also knows his latitude, he has the three sides of the spherical triangle PZS given by the comple- ments of these values; and this triangle, if PZb is the meridian of A, is generally known as the astronomical triangle, and the angle ZPS is the hour-angle, which, if expressed in time, is really the difference in R. A. of the star S and of a point on the meridian at the moment of the observation; or, in other words, it equals the difference between the R.A. of the star and the sidereal time at the moment. Thus if the hour-angle 192 EXPLORATORY SURVEYING. in sidereal time H and the local sidereal time T, we have, to convert the hour-angle into sidereal local time, T= H+R.A. (— 24 hours if necessary); and conversely, IIT (+ 24 hours if necessary) — R.A., which is the formula for obtaining the hour-angle when the body observed is either the moon, a planet or star; the R. A. being the R. A. of the body observed at the moment of obser- vation. In the case of the sun, in order to convert the hour- angle into mean local time, we have simply to reduce it to apparent time by dividing by 15 (as given below), and then apply the equation of time (corrected for the time of observa- tion) to reduce the apparent time to mean time; and the con- verse of this-to find the hour-angle when given the mean local time-is simply a reversal of the process, for the sun's apparent time is its hour-angle. The value of the hour-angle in angular measure, as ob- tained for instance by solving the astronomical triangle, must be subtracted from 360° when the star lies in east in order to give it its true value. Then in order to convert h into H, since 1 hour is equivalent to 15°, we have H (in hours) = 7 (in degrees), 15 and this equation of course holds good if for the words "hours" and "degrees" we substitute on both sides either the word "minutes" or "seconds." So that, for instance, if we obtain by an observation of a star in the cast a value for the hour-angle-as obtained from the astronomical triangle-of 40°, we have h = 320°; therefore H 21" 20m. - Table XX greatly facilitates the conversion of H into h, or vice versa. 183. The following examples serve to illustrate what has already been said. 1. At what hour will Arcturus culminate (i.e., be on the meridian) on Sept. 18, 1889, at Greenwich? From the Nautical Almanac we find that the sun's mean R.A. at mean noou at Greenwich on Sept. 18. 11b 50m 229.8, and also that the R. A. EXPLORATORY SURVEYING. 193 of Arcturus will then 14h 10m 37s.8; and since the R. A. of the star is really the sidereal time at which it culminates, we have merely to convert its R A. into mean time according to Sec. 182. Thus Arcturus will be on the meridian at 2h 20m 158 mean astronomical time, i.e., at 2h 20m 15º P.M. +10 2. What will be the R. A. of the apparent sun on Nov. 15, 1889, in longitude 90° W. at 4 P.M.? Since 4 P.M. in 90° W. occurs. 10 hours after mean noon at Greenwich, and from the Nautical Almanac we find the Sun's mean R. A. at mean noon on Nov. 1515h 39m 038.0. Since the correction for 10 hours X9s.8561m 38s.5, the Sun's mean R.A. corrected to date = 15h 40m 418.5. Similarly the equation of time corrected to date = 15m 089.3; and since the apparent sun is then ahead of the mean sun, the R.A. of the apparent sun for the date re- quired = 15h 39m 40$.5 – 0¹ 15m 08$.3 = 15h 24m 32s.2. 3. Find the Sun's declination at 8 A.M. July 22, 1889, in longitude 30° E. Now 8 A. M. at 30° E. occurs 6 hours before mean noon at Greenwich; and from the Nautical Almanac the declination at Greenwich at mean noon on July 22d = +20° 12′ 16″, which, corrected to 6 hours earlier, = + 20° 15′ 15″, which is the declination required. 4. Given 10 24 08s as the local astronomical mean time on Feb. 1, 1889, in longitude 60° W. to convert it into local sidereal time. According to Sec. 179, we must first convert this time into a sidereal interval by increasing it at the rate of 9.856 secs. per hour, which gives 10h 25m 50.5, and the sidereal time at mean noon 4 hours later than Greenwich mean noon 20h 48m 119.2, thus the local sidereal time (deducting 24 hours) 7h 14m 018.7. 5. Suppose on June 1, 1889, we observe Castor at 2h 30m 04 A.M. local time, in longitude 105° W. what is the hour-angle in angular measure? This in mean astron. time equals, May 31. Increase at rate of 9.856 per hour... Sidereal interval in sidereal time.... .14h 30m 048 2m 229.9 14h 32m 268.9 ་ Sidereal time at mean noon in 105° W. May 31.. 4h 37m 508.7 Sidereal local time of obs. T. R.A. of Castor... • Hour-angle H (subtracting 24 hours) Therefore Angular equivalent • • .19h 10m 178.6 7h 27m 328.7 2h 37m 500.3 ..39° 27' 35'" 194 EXPLORATORY SURVEYING. 6. Given the hour-angle of the apparent sun in the east, as obtained from the astronomical triangle, as 14° 29′ 10″ on June 14, 1889, in longitude 90° E., find the mean local time. Since the observation is in the east, h 345° 30′ 50″, which corre- sponds with 23h 02m 03s; therefore the observation occurred 23h 02m 03 apparent time after apparent noon on June 14; and at that moment the mean sun was ahead of the apparent sun by Om Om 10s, therefore the mean local time of observation = 23h 02m 13s June 14. REFRACTION, PARALLAX, SEMI-DIAMETER, AND DIP. 184. In Secs. 51 and 165 we have already considered the effect of Refraction when dealing with objects on the earth's surface. The same uncertainty exists in dealing with celestial objects as to the amount of the correction necessary to counter- Alt. Ref. Alt. Ref. Alt. Ref. Alt. Ref. || Alt. | Ref. / // Alt. Ref. Alt., Ref. O 0 00 33 00 2 30 16 23 6 30 752 12 20 4 16 30 1 38 0 05 32 11 235 16 04 6 407 41 12 40 4 09 31 1 35 0 10 31 22 2 40 15 45 6 507 31 13 004 03 32 1 31 0 15 30 36 2 45 15 27 7 007 21 13 20 3 57 33 1 28 0 20 29 50 2 50 15 09 7 10 7 12 13 40 351 34 124 0 25 29 06 2 55 14 52 7 207 08 14 00 3 46 35 1 21 0 30 28 23 3 00 14 35 7 30 654 14 20 3 40 36 1 18 66 0 35 27 41 3 05 14 19 7 40 6 46 1440 335 37 1 16 858 359 360 60 0 33 61 0 32 62 0 30 63 0.29 64 0 28 65 0 27 0 25 67 0 24 0 40 27 00 3 10 14 03 750 6 38 15 00 3 30 38 1 13 68 0 23 0 45 | 26 20 3 15 13 48 800 6 30 15 30 3 23 39 1 10 69 0 22 0 50 25 42 3 20 13 33 8 10 6 22 16 00 3 17 40 1 08 ΤΟ 021 0 55 25 053 25 13 19 8 20 6 15 16 30 3 11 41 1 05 71 0 20 1.00 24 29 3 30 13 05 8:30 6 08 17 00 3 05 42 1 03 72 | 0 19 1 05 23 51 3 40 12 39 8 40 6 01 17 30 2 59 431 01 73 0 17 1 10 23 20 3 50 12 14 8 50 5 55 18.00 2 54 44 0.59 74 0 16 1 15 22 47 4 00 11 50 9.00 | 5 49 18:30 2 49 45 0 57 75 0 15 1 2022 15 4 10 11 28 9 10 5 43 19 00❘ 2 44 46 0 55 76 0 14 1 25 21 44 4 20 11 07 1 30 21 15 4 30 10 47 9 30 9 20 5 37 5 31 19 30 2 40 47 0 53 77 0 13 20 00 2 36 48 0 51 78 0 12 1 35 20 46 4 40 10 28 9 40 5 26 20 30 2 32 490 50 79 0 11 1 40 20 18 450 10 10 9 505 20 21 00 2 28 50 0 48 80 0 10 1 45 19 51 5.00 953 10 00 5 15 21 30 2 24 51 046 81 0 09 1 50 19 25 5 10 9 37 10 15 5 08 22.00 2 20 52 0 45 82 0.08 1 55 18 59 5.20 9 21 10 30 5 00 23 00 2 14 53 0 43 83 0 07 2.00 18 35 5 30 9.07 10 45 4 54 24.00 | 2:07 54 0 41 84 0 06 205 18 11 5 40 8 53 11 00 4 47 25.00 2 02 55 0 40 85 0 05 2 10 17 48 5 50 8 39 11 15 4 41 26 00 1 56 56 0 38 86 0 04 2 15 17 26 6.00 8 27 11 30 4 35 27 00 1 51 57 0 37 87 0 03 2 2017 04 6 10 8 15 11 45 4 29 28 00 1 47 58 0.36 88 0 02 2 25 16 44 6 20 8 03 12 00 4 23 29 00 1 43 59 0 31 89 0 01 EXPLORATORY SURVEYING. 195 act the refractory power of the air, as we found to exist when the objects observed were near at hand; but in the case of Astronomical Refraction the altitude of the object is a much more important factor than in the previous case; for the lower the altitude not only the more obliquely do the rays pass through the successive layers of air, but the extent of atmos- phere which they have to traverse is greater than at a higher altitude. The preceding table of Mean Refractions, calculated for a barometer pressure of 29.6 inches and a temperature of 50° F., may be used at all times under ordinary circumstances, when dealing with celestial objects whose altitudes exceed 30°. At low altitudes the corrections given in the table should be corrected by multiplying them by the factors B and T, which make allowance respectively for the height of the Barometer and the Temperature of the air: thus True Refraction Mean Refraction X BX T. VALUES OF B. Bar. In. 28 28.5 29 29.5 30 30.5 31 B 0.946 0.963 0.980 0.997 1.014 1.031 1.017 VALUES OF T. 30° F. 10° F. 10° F. +30° F.+50° F. +70° F. +90° F. 0.960 0.925 Temp. Τ 1.180 1.130 1.082 1.038 1.000 The correction for refraction must of course be subtracted from the observed altitude. 185. The positions of all celestial bodies as given in the Nautical Almanac are calculated with reference to the Centre of the Earth; thus if, as in Fig. 88, an observer at A observes the altitude of the sun S to be the angle SAH, in order to re- duce this angle to the centre of the earth, i.e., to the angle SOh, he must add to it the angle ASO, which is termed the Purallactic angle. Now if S were just on the horizon, i.e, at H, then ΛΟ Radius of Earth sin AIO = HO Distance of Sun 196 EXPLORATORY SURVEYING. where AHO is termed the Horizontal Parallax, and is given in the Nautical Almanac. In the case of the sun it varies Z S E Equinoctial S Horizon HIS S P a SIM FIG. 88. from about 8".7 to 9".0. In order to reduce this to Parallax in Altitude, we have from the above figure therefore sin ASO = sin AHO sin SAZ; sin (Par. in alt.) sin (Hor. Par.) cos (alt.); = or, assuming the sines of small angles to be proportional to the angles themselves, Par. in alt. Hor. Par. × cos (alt.). Thus, at an altitude of 45°, Parallax in altitude = 6″, and at 60° = 4". In the case of the moon, since its distance from the earth compared with the radius of the latter makes it important what value of the radius is used, the Hor. Par. is given in the Nautical Almanac as Equatorial horizontal parallax, meaning that the value of the radius used is that at the Equator; thus for other latitudes the correction taken from the following table should be subtracted from it before applying the cor- rection for altitude, in order to obtain the value of the Hori- zontal parallax suitable for the latitude in question : LATITUDE. Eq. Hor. Par. 10° 53' 0.3 61' 0.4 20° 30° 40° 50° 60° 70° 80° 90° 17.2 27.7 4.4 6/.2 8".0 9.4 10.3 10.6 1.4 37.1 5/1 7.2 9.2 10.8 11.9 12 2 EXPLORATORY SURVEYING. 197 7 186. Correcting for Semi-diameter.—In taking an alti- tude of the sun, the upper or lower "limb" is generally ob- served, and the altitude so obtained corrected by the subtrac- tion or addition of the semi-diameter-obtained from the Nautical Almanac-to reduce it to the sun's centre. In observ. ing with an artificial horizon, the application of the correc- tion for semi-diameter can be avoided by bringing the reflec- tions to coincide. With either a transit or sextant a good way is to observe one limb and note the time, and immediately after observe the other limb and note the time; the mean alti- tude may then be considered to give the altitude of the sun's centre at the mean time. Similarly in observing the transit of the sun across any vertical plane we take the mean time of the passage of its east and west limbs. In observing the moon, we usually can only observe one limb; and in this case, on account of its proximity to the earth, it is necessary to apply a correction to the semi-diameter as given in the Nautical Almanac, which assumes the observer to be at the centre of the carth, in order to allow for the increase in its semi-diameter on account of his being nearer to it than the centre of the earth. This is termed correcting for the Augmentation of the Semi-diameter. The corrections are given in the following table · Semi-diam. APPARENT ALTITUDE. 10° 20° 30° 40° 50° 60° 70° 80° 90° 14' 30" 17' 0" 2/.4 4.7 6/.9 8.8 10.5 11.8-12.9 13.5 13.7 3.4 6.5 9.5 12.1 14.4 16.3 17.7 18.6 18.8 In finding the time occupied by the semi-diameter of the sun or moon in crossing the meridian, it must be remembered that it is only when the declination = 0° that (if the R. A. is not changing) the semi-diameter will travel across the plane at the rate of 15° to one sidereal hour (or 15° 2' 24" to one mean hour). At any other declination we have, as the rate of travel, 15° = 1 sid. hour X cos (dec.), on just the same principle as the length of a degree of longi- tude decreases as the cosine of the latitude. In the same way, 198 EXPLORATORY SURVEYING. it is only when the body is on the horizon that its semi-diam eter can be measured, without correction, by the horizontal circle of a transit, for as the altitude of the body increases, so also does the horizontal circle increase its reading in propor- tion to the secant of the altitude. The change in R.A. during the passage of the semi-diam- eter must of course be added to the time which it would have occupied had its R. A. been constant. 187. Dip. This is a correction only necessary when the sea- level is taken as the horizon, and is practically the same as that given in Sec. 167. It is to be subtracted from the observed altitude. The following are its approximate values, but re- fraction enters too largely into the question to enable accuracy to be obtained by the use of a sea-horizon : Height above Sea-level in feet, 5 10 20 30 40 50 0 60 75 2' 5" 3' 0" 4' 10" 5' 10" 6′ 0″ 6′ 40″ 7′ 20″ 8′ 10″ Dip, . Other values may be found from the values of Д, calculated according to Sec. 167. 188. We will now sum up the corrections (which we have already considered) necessary to apply in taking ordinary ob- servations. 1. Observation for Altitude. A. Using a sea horizon or level. If a Star. Observed Altitude (— Dip) ± Index-error Refraction = True Altitude. If the Sun, or a Planet. Observed Altitude (— Dip) ± Index-error- Refraction ± Semi-diameter + (IIor. Parallax X cos alt.) True Altitude. If the Moon. Observed Altitude ( — Dip) ± Index-error Refraction (Hor. Eq. Parallax corrected for latitude and converted into Par. in alt.) ± Semi- diameter, reduced for Augmentation - True Alti- tude. B. Using an artificial horizon. S In this case the double-altitude as read on the arc+or the Index-error must be divided by 2 in order to obtain the observed altitude, and then the other corrections—except of EXPLORATORY SURVEYING. 199 course for Dip, which only comes in when using a sea-hori- zon-applied as above. If the two reflections are brought to coincide, there will be no correction needed for semi-diameter; but a more perfect observation can usually be obtained by bringing the limb of one reflection in contact with the oppo- site limb of the other, in which case the semi-diameter must be corrected for as above. "Index-error" includes errors of any sort in connection with the instrument for which allowance must be made. 2. Observation for Azimuth. If a Star. Observed Azimuth = True Azimuth. If the Sun or a Planet. Observed Azimuth (Semi- diameter sec alt.) = True Azimuth. X If the Moon. Observed Azimuth ± Semi-diameter (rc- duced for Augmentation) × Sec. alt. True Azimuth. Having now considered all the corrections which need be applied in the case of ordinary field observations when using either a sextant or small portable transit, we will next consider the methods by which the latitude and longitude of a place may be established by astronomical observations. LATITUDE. 189. A. By a Meridian Altitude.-In Fig. 88, if for the moment we assume the observer to be at the centre of the earth, so as to do away with the idea of parallax, if PSH is the meridian and S the Sun, SE represents the Sun's Dec. N.: and if its declination did not change, since Sa indicates its path, we can easily see that its altitude would be greatest when on the meridian. But since its declination is always changing, the Sun attains its maximum altitude in the northern hemi- sphere when its declination is changing towards the north, after it has passed the meridian, and when changing towards the south, before it reaches the meridian. The difference between its meridian altitude and its maximum altitude does not ex- ceed at any season 1', so that in ordinary work the maximum altitude is assumed as being equal to the meridian altitude. In taking an observation of the moon with a sextant it is necessary to allow for this, especially about the time of the equinoxes, the difference between its meridian and maximum altitudes sometimes amounting to as much as 2′ 15″. 200 EXPLORATORY SURVEYING. When a transit is used to observe the meridian altitude, it is usually set in the meridian, so that no correction is then re- quired. For the amount of the correction, see Note G, Appendix. Now in Fig. 88, if Oh were the observer's horizon, the alti- tude of the Sun is represented by the angle SOh, Z is the Zenith, and the latitude of the place of observation is given by the angle ZE. Therefore the latitude of the place equals ES+SZ= Dec. N. + Zenith distance. And since the Zenith distance is the complement of the alti- tude, we are thus able, by means merely of the meridian alti- tude, to obtain the latitude; and this applies equally well to all celestial bodies, so that in the northern hemisphere, if, as Sin Fig. 88, the Dec. is N., then Lat. = Dec. N. + Zenith distance. (a) If declination is south, as S', Lat. Zenith distance - Dec. S. Dec. S.. (b) If the Star is above the Zenith, as S", C Lat. Dec. N. Zenith distance.. (c) If the Star is below the pole, as S'"', Lat. = Altitude + Co-declination. . (d) In the Southern Hemisphere the same formulæ apply, bear ing in mind that what is South in the southern hemisphere is equivalent to what is North in the northern. The altitude taken "below the pole" is of course the mini- mum altitude. The altitudes of S" and S" are observed in the north. Suppose, for instance, we observe the meridian altitude of Regulus on Mar. 17, 1889, to be 40° 16′ 40″. Now the declination of Regulus at that date = 12° 30′ 30″; so that we have EXPLORATORY SURVEYING. 201 Observed altitude of Regulus.. Correction for refraction... True altitude Therefore, zenith distance. · Declination of Regulus. 40° 16' 40" 1' 07' 40° 15′ 33″ 49° 44′ 27'' 12° 30' 30" Therefore, Latitude by Eq. (a).. 62° 14' 57'' N. Again, suppose on Feb. 8, 1889, in longitude 105° W., the meridian altitude of the sun's upper limb is observed to be Correction for refraction. . . . 48° 27' 20" 50" • • • + 5" 16' 15" parallax….. "semi-diameter. True altitude of sun's centre. Therefore, zenith distance... • Now the sun's declination S. at Greenwich at app. noon on Feb. 8. Correction for 7 hours later. • Sun's declination at date.. Therefore, Latitude by Eq. (b). 48° 10' 20' 41° 49′ 40″ • = 14° 49′ 30" 5' 36" 14° 43′ 54″ = 27° 05′ 46″ N. 190. It is always preferable to use a star instead of the sun or moon for a meridian altitude. The moon should only be used in thick weather, when the stars are invisible. In select- ing a star for the observation, the altitude should not be less than 30° if possible, on account of refraction. In order for a star to appear above the horizon on the meridian, the sum of the declination and co-latitude must exceed 0°, and the excess equals the true altitude, remembering that declination north is and south this gives a check before the observation is taken, preventing the wrong star being used. For stars be- low the pole as S" in Fig. 88, in order that the star may be visible above the horizon at its minimum altitude the latitude must exceed the co-declination, the excess being the true altitude. When using a transit, we may proceed in two ways: 1. By observing the maximum altitude and correcting ac cording to Sec. 189, and Note G, Appendix. 2. By setting the transit in the meridian, and then observ- ing the altitude of the passage. 202 EXPLORATORY SURVEYING. The meridian may best be obtained by an Elongation of Polaris as described in Sec. 57, or by the other methods de- scribed in Secs. 57 and 202. In taking meridian altitude it is well to observe a star in the north as well as a star in the south; the mean result is then tolerably free from instrumental errors. Polaris, either at its upper or lower transit, is a good star to use on account of its slow motion admitting of several alti- tudes being taken. B. By Transits across the Prime Vertical. 191. This is the most accurate method of obtaining the latitude, but necessitates the use of a transit. N co-lat. E co-alt. Path of Star Equinoctial Horizon co-dec. a FIG. 89. In Fig. 89 let PZE represent the meridian, Z the zenith, P the celestial pole, and S the body, the time of whose transit across the prime vertical-i.e., the vertical plane ZO, lying due east and west-we wish to observe, in order by it to obtain the latitude. Now in the spherical triangle ZPS the angle at P= the hour-angle 7 (see Sec. 182), and ZS = the co-alt. of the body when on the prime vertical, ZP the co-lat- itude, and PS the co-declination. Therefore, since Z = 90°, tan (lat.) = tan (dec.) × sec h. But in order to obtain 7, we must know the exact local time of the observation, which may be obtained according to Secs. EXPLORATORY SURVEYING. 203 195, etc. The longitude we need only know with sufficient accuracy to admit of correcting the sidereal time at mean noon, i.e., for ordinary work, to about 20 miles. This method of determining the latitude of a place admits of high precision, since an error of 1 second in the local time only causes an error of about 14 seconds in latitude, or about 170 feet. The passage of the star across the prime vertical should be observed both in the east and the west (or else another star used), and the mean result taken to eliminate errors. The altitude of a body when on the prime vertical is given by the equation sin (alt.) = sin (dec.) cosec (lat.); and the hour at which the observation occurs is given by the equation sec h = tan (lat.) cot (dec.). If the transit has three vertical hairs, which it should at least have for astronomical work, the star may be observed at, say, its eastern transit on the north side of the prime vertical upon the hair which is to the left of the collimation centre; then after reversing the instrument, the star may be observed again on the same hair. If the telescope is left in the last position until the star comes to its western transit, it is observed again on the same hair to the south of the prime vertical, and then reversing the telescope the star again crosses the same hair on the north side. Thus a latitude determination is arrived at free from instrumental errors and with the errors of observa- tion greatly reduced. It is best to select a star with as small a declination as possible, as its motion in azimuth will then be more rapid. C. By an Altitude out of the Meridian. 192. It often happens that just about the time when the sun or star is on the meridian suitable for obtaining the latitude according to method A, it becomes obscured by passing 204 EXPLORATORY SURVEYING. clouds. If, however, the local time is known approximately, P Z S E Equinoc/tial\ FIG. 90. the latitude can still be ob- tained in the following way: Suppose in Fig. 90 PZE is the meridian and S a star which has only a short time before crossed the meridian. Then in 'astronomical triangle Horizon the PZS, if we know ZS co-alt., "" PS= co-dec. and the hour-angle ZPS, we can at once, by solving the spherical triangle, find the side PZ = co-lat. But instead of using the common for- mulæ (as given in Sec. 233), the following will be found sim- pler : Make and tan A = cos ZPS × tan PS, cos B = cos A X cos ZS X sec PS. Then, if the six-o'clock circle and the prime vertical lie on the same side of S, as will always be the case when S is near the meridian, co-latitude = A − B ; but if Slies between them, we have co-latitude = A + B. But since this method is really only suitable for use within an hour or two of the meridian circle, it is the former of these two equations which is almost exclusively used. When the latitude and declination are of contrary signs, we then have simply Lat. = (A+B) — 90°. To use this method, it is necessary to know the value of the hour-angle with tolerable accuracy. This can be obtained by one of the methods given in Secs. 195, etc.; or in the case of a star it can easily be obtained by observing its altitude before reaching the meridian,―assuming that it is only cloudy about the time of the meridiau passage,-noting the time by an or- EXPLORATORY SURVEYING. 205 dinary watch; then on the other side of the meridian, if the moment is observed at which it again reaches the same alti- tude, half the interval (converted into a sidereal interval) hour-angle II (see Sec. 182). With the sun this is only appli- cable when its declination is changing but little, or when near the zenith. D. By double Altitudes. 193. The following are very convenient methods of ob- taining the latitude when the local time is not known. A. By too altitudes and the interval of time between them. In Fig. 91 let Z be the zenith, P the celestial pole, S and S' the two positions of the star at the moments at which the alti- tudes and times are observed. P Z S S time FIG. 91. Then the interval between the two observations in sidereal the hour-angle, which converted into angular meas- ure = SPS'. Then in the triangle PSS', SP=S'P = co- declination; thus we can find SS' and PS'S. Then in the triangle ZS'S, since we have the three sides we can find the angle ZS'S, which, subtracted from PS'S, gives the angle PS'Z. Then in the triangle PS'Z we have S'P, S'Z, and the angle PS'Z, from which we can find PZ = co-latitude. A good common watch is all that is required to observe the intervals. But instead of taking two altitudes of the same star, it is better to observe― B. By simultaneous altitudes of different stars.-The hour- angle is given by the difference in R. A. of the two stars, and the rest of the working is the same as above. When, how- ever, there is but one observer, so that the altitudes must be taken in succession, he must proceed thus: The altitude of one star must be taken, and the time noted by the watch; the 206 EXPLORATORY SURVEYING. altitude of the other star must then be taken, and the time again noted. After a short interval the altitude of the second star must again be taken, and the time noted. He thus finds the motion in altitude of the second star in a given time, from which, by proportion, he can find what its altitude was when the first star was observed. In both A and B the altitudes as observed must of course be reduced to the true altitudes in order to obtain SZ and S'Z. 194. On the last page of the Nautical Almanac for each year is given a Table for computing the latitude from an observed Altitude of Polaris at any time, the hour-angle being approx- imately known; and as full instructions accompany the table, these need not be repeated here. The local time being known, the hour-angle H is of course obtained as in Example 5, Sec. 183. LONGITUDE. 195. The simplest way of obtaining the longitude of a place is to find its correct local time, and compare it with a chronometer which gives Greenwich time; the difference be- tween the two times equals the difference of longitude: so that if we have a chronometer at hand keeping Greenwich time, obtaining the longitude is simply a matter of obtaining the local time. A. To obtain Local Time by an altitude of a star. If it were not for the slowness of the motion of a star when near the meridian, a convenient method of obtaining the local time would be to reduce its R. A. to mean time at the mo- ment of its maximum altitude, which would then be the mean local time of its transit. But in order to obtain a well-defined moment of observation, it is necessary for the motion in alti- tude to be as rapid as possible, and for this reason a star should be selected as near the prime vertical as possible. Sup- pose at a certain moment by the chronometer we observe the al- titude of a star S (see Fig. 90); then if the latitude is known, in the triangle PZS, since PZ = co-lat. co-lat. 1, PS = co-dec. d, and SZ = co-alt. = a, we have, by spherical trigonometry, = COS 201 sin s sin (s- α) a) sin d sin l EXPLORATORY SURVEYING. 207 where s = a + d + i 2 and h the hour-angle ZPS; if the dec- lination and the latitude are of opposite signs, d dec. +90°. Now the nearer S is to the prime vertical, the less is an ac- curate knowledge of the latitude essential, and the less does an error in altitude affect the result. Thus the body should be observed as nearly east or west as possible, and certainly not within an hour or two of its transit. The following table shows the errors in longitude in min- utes of arc involved by an error of 1 minute in latitude, when S is observed at different bearings in different latitudes. LATITUDE. Bearing. 10° 20° 30° 40° 50° 60° 70° 10° 5'.67 5'.76 6'.55 77.40 8/.82 11/.33 6'.03 20° 21.75 24.79 3'.17 3'.59 4/27 5'49 2/92 40° 17.19 1'.21 1'38 17.55 1'.85 27.38 14.27 60° 0'.58 0'.59 0'.67 0'.75 0'.90 1'.15 0'.62 80° 0'.18 0'.18 0'.20 0'.23 0/.27 0'.35 0'.19 Thus in latitude 30° if the bearing of a star when observed is 80' an error in latitude of 5 miles would only cause an error of about half a mile in longitude. An error in the altitude is of much more importance, as the following table, giving the errors in longitude in minutes caused by an error of one minute in altitude, shows: LATITUDE. Bearing. 10° 20° 30° 40° 50° 60° 70° 10° 5'.91 6'.25 6' 65 30° 27.03 24.17 2'30 50° 17.32 17.39 1'.51 90° 17.01 17.06 1'.15 7.50 8'.96 12'.17 2/64 3'14 3'.98 1.71 27.03 21.63 1/.31 1'.55 2'.00 2'.90 16′.87 5484 32.78 Since the accuracy of the altitude is of great importance, it is well to take several sights, say 3 or 5, within a minute or so of each other, and note the corresponding chronometer read- ings; the mean altitude may then be considered to correspond with the mean time. If the local time which was used in order to correct the sidereal time at noon for the assumed 208 EXPLORATORY SURVEYING. longitude is found to have been appreciably in error, allow ance must be made for this. In observing altitude for time, if great accuracy is desirable, it is well to observe both in the cast and the west; the mean result of the two sets is thus practically free from instrumen- tal errors. This method of course applies equally well to the sun as to a star; and since the co-declination is always a large arc, whatever error there may be in it, there will only be half that error in the half sum; and since the errors in these alti- tudes oppose one another, an error in the co-declination such as might arise from an error of two or three degrees in the longitude assumed to correct the sun's declination will not seriously affect the result. B. To obtain local time by equal altitudes of a star. 196. All that we have to do in this case is to observe the altitude of a star in the cast and note the time, then note the time when in the west it again descends to the same altitude. Half the interval between the two observations is the "mid- dle-time," which corresponds with the local sidereal time given by the star's R.A. Thus we have simply to convert the star's R. A. into mean local time and compare it with the middle-time by the watch to obtain the watch-error. By taking a set in the cast and a set in the west, since index or instrumental errors do not enter into the question at all, the mean altitude for the mean time should give a really good result. There is no necessity to apply a correction for refraction, unless the barometric pressure or temperature has changed considerably between the observations. C. To obtain local time with a transit. 197. The best way to proceed with a transit is to set it in the meridian and observe the time of transit of the sun or one or more stars; the correct local time is then found by merely converting the R. A. of the body at the time of its transit into mean time. 198. But so far in obtaining the longitude we have assumed that we have had at hand a chronometer rated to Greenwich time. But since little reliance can be placed on chronometers EXPLORATORY SURVEYING 209 when travelling across country, one of the following methods should be adopted as a check on the chronometer from time to time. TO OBTAIN THE LONGITUDE BY LUNAR CULMINATIONS. The principle on which this method of obtaining Green- wich time is based is as follows: In the Nautical Almanac the moon's R.A. is given for every hour during the year at Greenwich. If then in any other longitude we find the moon's R. A. at a certain moment, that moment will correspond with the time at Greenwich at which the moon would have the same R. A. as that which we ob- served. Thus, if the moon's R.A. in the Almanac at 6 P.M. were given as 8h, if in a certain longitude we find at exactly 10 P.M. local time the moon's R.A. to be 8h, we know we are in a longitude 4 hours ahead of Greenwich, i.e., 60° E. To obtain the R.A. of any body by observation, we have only to find the mean local time of its transit across the meridian and convert it into sidereal time, which is the R.A. required. Thus we proceed as follows: Find the correct local time by the watch. Set the transit in the meridiau. Observe the moment of transit of the moon's bright limb Again find the correct local time by the watch. The moon's semi-diameter, which is given for every 12 hours in the Almanac, must then be found and divided by 15 to re- duce it to equivalent time, which would then be the sidereal time occupied by its passage if its declination 0° and its R.A. were unchanging. But since its R. A. is always increas- ing, the passage of the semi-diameter will occupy a time longer than this by an amount which may be obtained from the Almanac by simple proportion, by seeing what the increase in R. A. is at the assumed Greenwich time of the observa- tion; the total time of the passage so obtained multiplied by the secant of the declination (see Sec. 186) then gives the time actually occupied in the passage; and this added to, or deducted from, the observed tin.e of transit of the limb, gives the time of transit of the moon's centre, which, converted into sidereal time, gives the moon's R. A. at the moment of obser- vation. 210 EXPLORATORY SURVEYING. It is well to take a set of observations for time before and after the moon's passage; and the instrument, if possible, should not have less than 3 vertical hairs, the passage across each of which may be observed and reduced to the centre hair.* Every possible precaution should be taken in this obser- vation, for the error of a second of time in observing the moon's limb, compared with the corrected watch time, —i.e., an error of 1 second in R. A.,-may easily cause an error in longitude of 5 miles. Thus by a single observation with a small transit we cannot depend on our longitude to within about 10 miles. But if the observer is stationed for 3 or 4 days at any one place, by taking the mean result of 3 or 4 observations he should be able to obtain the longitude with a probable error, say, not exceeding 4 or 5 miles. corre- sponding with an error in Greenwich time (in ordinary lati- tudes) of from 20 to 30 seconds. Having now obtained the moon's R. A., the next thing to do is to find the hour at Greenwich with which it corresponds. Since the moon's change in R. A. is usually rapid, and great accuracy is necessary, the ordinary method of simple interpo- lation will not apply here. The following formula may therefore be used instead: T-t= 60 (A a) d / T D+ 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 t; D d Increase in R. A. in 1 mean minute at time t; 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- * See Sec. 197. 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. 2 In Fig. 92 let M' and S' 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 S'Z the co-altitudes. The truc positions will differ from these by the differences in altitude MM' and SS': the moon, on account of the S correction for parallax exceeding St that for refraction, will be elevated above its apparent position; whilst the star, on account of re- fraction only, will be depressed below its obscrved position. s M M' FIG. 92. Now, if the apparent altitudes are observed at the time of observing the lunar distance S'M', we have the three sides of the triangle S'ZM', so that the angle at Z may be found trigo- nometrically. Then the two sides S'Z and M'Z, being cor- rected for refraction and parallax, give the sides of the cor- rected triangle SZM; 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: sin D COS H + H' cos C, 2 2 212 EXPLORATORY SURVEYING. where sin² C= cos 8 cos (8 ~ d) cos H cos I' H+H' cos h cos h' cos² where 8 = h + h' + d 2 and h Н h' app. alt. of star; true alt. of star; D true distance SM. app. alt. of moon's centre, H= true alt. of moon's centre, H d = app. distance S'M', An error of a minute or two in the altitude able difference in the distance. makes no appreci- 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 the 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 R.A. and the sidereal time at the moment + 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 (alt.) = sin L sin (E+ d) sin E where cot E = cot L cos h, and the hour-angle. If h exceeds 90° cos h is negative, h which will make cot E also negative; so that to avoid the use EXPLORATORY SURVEYING. 213 of supplements, it is simpler to say sin (alt.) = sin L sin (E — d) 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 bimself 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" 50m 04$.9. With a sextant this may best be done by observing the altitudes of 3 or 4 stars lying fairly cast 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 PZS 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 8 sin (s d) COS 2 sin a sin l a, d, and l being the complement of the altitude, declination and latitude respectively, and s the half sum of a, d, and 1. In the latter case we have cos a = cos d cos + sin d sin l cos h, from which we obtain sin Z = sin h sin d cosec a. hour-angle. (See Sec. 182.) When the latitude and declination are of opposite signs, d = dec. +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. interpolation formula is The ndı n(n 1) Fº = F+ + 1 1 X 2 n(n − 1) (n − 2) d₂+ 1 X 2 X 3 dз +, etc. For example, suppose we wish to find the moon's declina- tion at Greenwich at 2h 15m on Nov. 15, 1889. From the Nautical Almanac we find the 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- natious for, say, 3 or 4 following hours, as given in the Almanac. In the second column we put down the first differences of these (d₁) obtained by subtracting downwards and prefixing the proper algebraic sign. In the third column we place the second difference (da) (i.c., 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 = 1: so that now we have merely to insert the upper values in the columns for d₁, d₂, etc., and the above value of n, in order to find the declination at 2h 15m. 2 F di Dec. at 2h = 18° 17′ 4″ d2 7' 59" d3 3b 18° 09' 5" - 6" 8' 05' " +1″ 4h 18° 01' 0" - 5" 8' 10" 5h 17° 52′ 50″ Thus, therefore, Fa 18° 17′ 4" - 1' 59".8+.56" — .07"; Dec. at 2h 15m : = 18° 15' 04".75. In such a case as the above, as it happens, the simple method of interpolation would have given F" = 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 cau 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 scries 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 1 2056.20 feet 4 2056 16 feet 3 Then the most probable value of the result is given by 2056.32+(2056.20 × 4)+(2056.16 × 3) 1 +4+3 = 2056.20. 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 term 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 formulæ 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 Ꭱ 20890564-106960 cos 2L+ 228 cos 4L; EXPLORATORY SURVEYING. 219 and the radius of curvature in a direction perpendicular to the meridian equals in feet r = 20961932-35775 cos 2L+ 46 cos 4L. Thus at the Equator R = 20783832 feet, r = 20926203 feet; and at the poles R = 20890564 feet, 720961932 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 364609.11866.7 cos 2L + 4 cos 4L, and the length of a degree of longitude equals in feet d = 365542.5 cos L 311.8 cos 3L+0.4 cos 5L. 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 are given in Sec. 73, if we take the above value of the mean radius of curvature, we find the length of an arc on the earth's surface in feet equals = 6076n (nearly), where n the number of minutes in the arc; and the con- verse of this, n = (nearly), 6076 220 EXPLORATORY SURVEYING. enables us to convert any given distance into its equivalent in angular measure. If it is desirable to obtain the value of more accurately than by this means, we can do so by obtaining first the value of l 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 ", given in Sec. 206. Then if we call this latter value l', the length of an arc subtending 1' at the earth's centre, which makes an angle A with the meridian, equals I cos² A+ l' sin² 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 and 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+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 Acot 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 D, to find the latitude and longitude of D.-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 + 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 AND cos a cosec AD. The formulæ 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 pole, and L any given latitude; then, consider- ing the earth to be a sphere, the angle LPC = the latitude of L, so that PLLC 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 any point C to a parallel of latitude AC L from a straight line AB, tangent to AC at A.-We can do this by treating the parallel of latitude AC in Fig. 94 as a curve FIG. 93. P N 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 A N C right-angled spherical triangle ANB, and so find the latitude of B, if we know the differ- ence of longitude N, thus: tau (lat. B) = tan (lat. 4) 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, B 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. FIG. 94. 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.c., 59° 30' N.; thus AC about 10 × 60 × 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 = 10d. N. and S. Base. A Circle of Latitude of A. C E. and W. Base. S. 83.16 E. Direct Course, A. to B. FIG. 95. d 90 B Next we make AD = AC, and through D 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 CDA = 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 AC equally into 20 parts, and draw the meridians perpendicular to the curve at each of these points, i.c., 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, then 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 JAN. FEB. 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. DEC. 5.H + 4.H 6. H +10 +11 3.II ΛΟΝ 2. H 7.H 8.H #1 19* *12 14 13 16, 9.II MAR. 10. H 15 11. H 12. H 1. H 17 Co 100 25 125 23.H. 0. H +26 -10. P. M.— A 1 ► 18 13. H 20 Dec. +19 11. H 22. H 21. I AUG. * 24 23 H⭑03 H 61 MAY. 22 So N. Dec. H GI Equinoctial H⭑SI H LI 21+ H 91 JUNE. APR. 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 0 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.c., due soutb, at JULY. 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 23h 56m; 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. Mag. Dec. An. Var. R.A. An. Var. 183 2 y Cassiopeiæ • Alpherat, a Andromeda Polaris, a Ursa Minoris..| O "/ // h. m. s. 2.0+28 28 39 28 28 39+ 19.88 0 2 39 19.88 0 2 39 88 42 5918.90 1 18 08 2.0| 2.060 06 55 S. +3.09 23.15 19.56 0 50 01 +3.58 4 Algol, B Persei. 2.7 40 31 381 • 14.12 3 0 573.88 5 a Persei 2.0 49 27 55 13.10 3 16 24 4.26 6 Aldebaran, a Tauri.. 1.0 16 17 07 -7.52 4 29 33 3.44 7 8 a Arietis. 9 10 11 13 14 15 16 a Ürsæ Majoris. 17 y Ursa Majoris. 18 19 20 21 គន 24 a Cygni... 25 26 n Ursa Majoris. Arcturus, a Bootis. Spica, a Virginis Antares, a Scorpii. 22 Vega, a Lyræ. Altair, a Aquilæ. 23 Fomalhaut, a P. Aust... Markab, a Pegasi.. Capella. a Aurigæ.. Rigel, 8 Orionis Betelgeuze, a Orionis.. Sirius, a Canis Majoris.. 12 Castor, a Geminorum. Pollux, B Geminorum. Procyon, a Canis Minoris Regulus, a Leonis. - 1.0 16 33 52 1.7+ 32 07 53 ± 1.3 28 17 37 1.0+5 30 32 1.3 2.0 12 30 34 62 21 0 2.354 18 42 2.049 52 03 1.019 45 38 1.0 10 34 54 1.3 26 11 06 1.0! 45 53 03 +4.03 5 08 29 4.42 2.022 56 14 1.08 19 50 1.2723 8 17.7 2 0 55 3.37 4.40 5 09 12 2.88 0.95 5 49 10 3.25 4.71 6 40 15 2.64 7.55 7 27 31 3.84 - 8.41 7 38 31 3.68 8.99 7 33 29 3.14 17.47 10 02 28 3 201 - 19.36 10 56 52 3.75 20.03 11 47 59 3.18 18.08 13 43 10 2.37 ― 18.88 14 10 36 2.73 · 18.90 13 19 21 3.15 - 8.30 16 22 36 3.67 1.938 40 50 3.17 18 23 22 33 11 2.03 9.27 19 45 2.93 1.744 53 02 12.72 20 37 39 2.04 1.3 30 12 37 18.99 22 51 31 3.82 • 2.0 14 36 29 19.30 22 59 14 2.98 EXPLORATORY SURVEYING. 227 IN THE SOUTHERN HEMISPHERE WE ALSO HAVE- NAME. Mag. Dec. An. Var. R. A. An. Var. O h. m. s. S. ß Hydri 3.0 77 52 46 • Achernar, a Eridani.. 1.0 57 48 03 20.28 18.36 0 19 54 3.23 1 33 34 2.23 Canopus, a Argus 1.0 52 38 07 - 1.87 6 21 29 +1.33 B Argus 1.5 69 15 36 - 14.80 9 11 59 0.68 a Crucis.. 1.0 62 29 02 20.01 12 20 26 3.29 B Centauri.. 1.0 - 59 50 14 17.59 13 55 59 4.18 a Centauri 1.0 - — 60 22 47 15.38 14 32 05 +4.05 a Trianguli Aust. 2.0 68 49 21 7.16 16 36 55 + 6.30 a Ophiuchi. 2.0 a Gruis 2.0 +12 38 29 47 29 53 - 2.87 17 29 47 2.78 +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: 214. To find the Horse-power of Falling Water. H.P. = 0.00189 QII, where the number of cubic feet of water passing over the fall per minute, and I height of fall in feet. H 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 cross-section of the stream in square feet, and Q = cubic feet of water that pass per minute, the 5000 A Q = = T 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 lbs. equals where FW 8S F = fall of hammer in inches, W = weight of hammer in lbs., 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.. Sandy clay.. Sand. Gravel... Sandstone. Firm Rock • • 1.0 to 2.0 tons per sq. foot. 2.0 to 4.0 ،، 3.0 to 5.0 4.0 to 5.0 2.0 to 4.0 CC 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 lbs., f coefficient of modulus of rupture. ར Then W bd²ƒ 18L 18 WL 18 WL d = and b bf d²f For the values of ƒ sec following table. = For example, if b = 6″, d = 10", and L = 20 feet, if we take ƒ 10,000 lbs., by the above formula W = 16,666 lbs.; 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 lbs. 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 f= 10,000 lbs. and the 230 MISCELLANEOUS. load per foot run as equivalent to 4000 lbs., we have as the equivalent value of W, 30,000 lbs. 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" × 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 lbs. pcr foot run should usually be taken as the uniformly distributed load. In spans exceeding 15 feet 3500 lbs. is usually sufficient. These values take no account of the weight of the beams them- selves. VALUES OF ƒ. Material. Lbs. per sq. in. Ash. 12,000 to 14,000 Birch. Blue Gum · • Elm... 11,700 18,000 6000 to 9700 Material. Lbs. per sq. in. Red Pine. Spruce Brit. Oak Am. Red Oak…. 219. Natural Slopes of Earths. Material. Slope. Material. Slope. + 7100 to 9500 9900 to 12,300 12,000 10,600 Material. Slope. Gravel... 40° Dry Sand.. 38° Sand. Vegetable Earth.. 28° Compact Earth... 50° Ruble. 45° 220 Shingle Clay (drained) 45° 39° Clay (wet). 16° 220. Weight of Earths, Rocks, etc., per cubic yard. Weight in lbs. Weight in lbs. Weight in lbs. Material. Material. Material. per cu. yd. per cu. yd. per cu. yd. Sand.. 3360 Clay 3470 Quarts 4590 Gravel. 3360 Chalk 4030 Granite. 4700 Mud 2800 Sandstone.. 4370 Trap'. 4700 Marl... 2900 Shale.. 4480 Slate.. 4810 A cubic yard of water weighs about 1680 lbs. MISCELLANEOUS. 231 221. Weight of Timber and Metals per cubic foot. Material. Weight in lbs. per cu. ft. Weight in lbs. Weight in lbs. Material. Material. per cu. ft. per cu. ft. Elm, English. 35 Pine, red. 36 Iron, cast 450 Canadian Elm 45 white. 30 เ wrought 482 Maple 42 Teak... 50 Steel.... 490 English Oak. 48 Spruce.. 30 Copper. 550 American Oak 50 Larch.. 34 Lead. 710 222. Mortar, Cement, etc. (common mixtures). Mortar.-1 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 lins lime to 24 of burnt clay, ground together. Beton.-1 of hydraulic mortar to 13 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." (C 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 bickory. 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. (C '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 may be 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 lachrymans-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. Chemical.-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. When 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 bright-red 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. 229. Strength of Rope.-The table on following page gives some idea of the strength of ordinary Manilla Rope. 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. SIZE OF ROPE. Diam. in inches, Circum. Breaking- strength in lbs. Breaking- Diam. in in inches. inches. Circum. in inches. strength in lbs. مله را به 0.71 375 1.43 1,500 2.14 3,380 1 2.86 6,000 938 19 21 7.14 37,500 8.57 54,000 -'N 10.0 73,600 4 11.4 96,000 13 3.57 9,380 43 12.1 121,000 1/31/ 4.28 13,500 5 14.2 150,000 2 5.70 24,000 17.1 216,000 Wire Ropes.-The following table gives the strength of iron and cast-steel wire rope : TABLE OF IRON AND CAST-STEEL WIRE ROPE. BREAKING- BREAKING- SIZE OF Rope. SIZE OF RO PE. STRENGTH IN LBS. STRENGTH IN LBS. Diam. Circum. Diam. Circum. Iron. C. Steel. Iron. C. Steel. in In. in In. in In. in In. IRM T mg59 coke 1호 ​6,960 15,000 13 78,000 154.000 21 17,280 36,000 108,000 212,000 3! 32,000 66,000 2 130,000 250.000 4 54,000 101,000 21 6. F 148,000 310,000 1 11 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. X 3.14159 X .886226 Diameter Diameter Diameter Diameter2 X .7071 X 7851 Radius × 6.28318 = circumference. side of an equal square. = side of an in♬ ribed square. = area of circle. Circumference X .31831 = diameter. Circumference = 3.5449 V Diameter Length of arc = 1.1283 V circumference. 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. 1 Arc of 1' to rad. 1 = 0.000004848. Degrees in arc whose length = radius 57°.2957795. π = 3.1415926536; Log = 0.4971499. π MISCELLANEOUS. 237 H 231. PLANE TRIGONOMETRY.-In Fig. 96, if the angle GAE 90°; then in the right- angled triangle ABC, if AB = Radius www. G F D B unity, BC sin A; AF cosec A; AC= cos A; CE = versin A; DE = tan A; BII co-versin A; AD = C A; BD = exsec A; A E C GF cot A; BF co-exsec A. FIG. 96. Therefore BC AC BC sin A = cos A = tan ▲ = AB AB AC AB AB AC cosec A = sec A cot A = BC' AC BC' Thus, sin A 1 cosec A 1 1 cos A tan A = sec A 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 formule 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). AB² = AC² + BO²; Area of triangle B = 90° AC.BC = 2 Examples of Right-angled Triangles. 1. Given A = 30°, and AC = 100, find BC. A. BC We see above that tan A = therefore AC BC AC tan A = 57.73 238 MISCELLANEOUS. 2. Find the sine of 128°. = Since sin (180° — A) A) sin A, sin 128° = sin (180° 52°) = sin 52°, which from the tables we find = 0.788. Solution of Oblique-angled Triangles. a B с α A B A C FIG. 97. FIG. 98. sin A a sin B sin C. (1) с A B a b A+ B tan tan 2 a+b (2) ? 2 A + B, A − B A = + (3) પે 2 ર 2 A+ B A B B = 2 (4) 2 A+ B COS 2 c = (a + b) (5) A – B' COS 2 Let a+b+c = s; then 2 2(s—b) (s — c) vers A = · (6) bc A COS 201 s(s — a) bc (7) MISCELLANEOUS. 239 Area of triangle = √ s(8 − a) (s—b) (s — c). ab sin C. a² sin B sin C (8) (9) || . (10) 2 sin A = - A 180° — (B+C). . (11) The above formulæ are all that are required for the ordi- nary solution of plane triangles. Remarks.-Though such a formula as No. 2 simply men- tions A and B and their opposite sides, it holds equally well whether we substitute C for A, or C 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 a sin B 2. Given B, c, and b, find C. By Equation 1, c sin B sin C- b 3. Given A, B, and c, find a. By Equation 11, C = 180° (A+B); and by Eq. 1, c sin A a α= sin C 4. Given B, a, and c, find A and b. By Eq. 2, A C a с A+ C tan tan ; 2 a + c ૨ 240 MISCELLANEOUS. from which we obtain the value of A Ꮯ 2 and by Eq. 11, A+ C B 90° ૭ 2 2 therefore we can find A from Eq. 3. Then by Eq. 5, b = (a + c) 5. Given a, b, and c, find B. By Eq. 6, A+ C COS 2 A C COS 2 a) 2(s — α) (s — c) vers B = ac or, we might equally well have used Eq. 7. 232. The following general equations are worth noting: A A sin Atan A cos A = √ 1 — cos² ▲ = 2 sin A COS 9 cos A cot A sin A √ 1 — sin² A = 2 cos² 14--1 1; tan Asin A sec A = vers 2A sin 2A = exsec A cot 2' tan cot A = cos A cosec A sin 24 vers 24 2 exsec A А vers A 1 - cos A = 2 sin² = cos A exsec A; 2 A vers A exsec A = sec À – 1 = tan A tan 2 cos A MISCELLANEOUS. 241 233. Spherical Trigonometry. B C A b FIG. 99. a RIGHT ANGLED TRIANGLES.-In Fig. 99 let A= 90°; then sin b sin a sin B; tan ctan a cos B; cot = cos a tan B; tan csin b tan C; cosa cos b cos c; cos B = cos b sin C; tan a = COS tan b C sin c = tan b tan B sin b sin a = sin B cos B sin C = COS C = cos b cos a cos b sin b sin B = sin a' tan b tan c tan b cos C C: = tan C = tan B = tan a sin b sin c' COS C = cos C sin B cos B cos b = cot C cos a sin C 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 C 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. B a C A FIG. 100. Let ABC in Fig. 100 represent any oblique-angled spherical triangle; then sin A sin B sin C sin a sin b sin c (1) COS tan a + b 2 с ~ A B 2 tan COS A + B 2 (za) A ~ B sin a с 2 tan - tan (26) 2 A + B sin 2 α b COS tan A+ B 2 Ꮯ 2 = cot 2 a+b (300) COS a sin A B tan 2 ~ C 2 = cot (36) a + b sin 2 cos c = cos a cos b + sin a sin b cos C; (4) A sin (s— b) sin (s — c) sin (5) sin b sin c ᏅᏓ cos S cos (S A) siu 2 = V (6) sin B sin C a+b+c A + B + C where s = and S= p 2 2 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. A, B, and c, a, b, and C, or, given a, b, and C, A, B, and ɑ, A, B, and a, A, B, and a, སྐྱུ a and b; c; c; Borb; C ; こ ​"( 3 "" " 2a and 2b. 3a and 3b. 2a, 2b, and 3b. 4 1 1 and 2a. " c; 1 and 3a. a, b, and A, C ; 1 and 2a. (( a, b, and A, 1 and 3a. A, B, and c, C; 3a, 36, and 2b. 234. Measures of length and surface. MEASURE OF LENGTH. Miles. Furlongs. Chaius. 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 5 5 16.5 198 0.00056818 0.0045454 0.045454 10.181818 1 3 36 0.00018939 0.00151515 0.01515151 0.0606060 0.33333 0.000015783 0.000126262 0.001262626 0.00505050 0.0277777 0.083333 1 12 1 Sq. Miles. Acres. MEASURE OF SURFACE. 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 10.000009764 0.00625 0.0625 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. AVOIRDUPOIS. Ton. Cvrt. 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 0.083333 1 20 480 0.068571 0.004166 0.05000 1 24 0.0034285 0.0001736 0.002083333 0.0416666 1 1.215275 14.58333 291.6666 7000 0.00014285 1 MEASURE OF CAPACITY, Cub. Yard. Bushel. Pecks. Cub. Feet. Gallons. Cub. inch. 1 21.6962 0.03961 1 27 1.24445 100.987 4 201.974 46656 9.30918 2150.42 0.037037 0.009259 0.803564 0.25 0.107421 1 3.21425 7.4805 1728 0.31114 1 2.32729 537.605 0.133681 0.429684 1 231 0.000547 0.001860 0.004329 1 APPENDIX. NOTE A. (See Sec. 10.) Ir 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 lbs. per sq. in. The product divided by the diameter of a driver in inches gives the pro- pelling force in lbs., ignoring "internal frictional resistances. Theoretically, the mean effective cylinder-pressure in lbs. per sq. in. equals where P P+2.3P (Log S) S 15, absolute boiler-pressure in lbs. per sq. in. and S 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 lbs. per sq. in. and the stroke = 24 inches; if steam is cut off at 6 inches, the theoretical mean cylinder- pressure 59 lbs. per square inch, which at 50 m. p. h. will probably be reduced to about 36 lbs.: so that if the diameter of the piston = 16 inches, and of the driving-wheels 60 inches, the propelling force will equal 3680 lbs.; and if we deduct 10 p. c. from this for internal frictional resistances, the propelling force = 3200 lbs. 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 ordinates of the curves must be multiplied by (3600)2 2000 × 5280 X 32.240 (nearly) to reduce them to tons weight (2000 lbs.), in miles per hour units. Then, with the units selected, the equation of motion is d (OQ) = NQ – MQ. dt But if a is the space passed over, da OQ = dt d (02) = 09/21 (02), dt dx so that and therefore OQ. d (OQ) NQ - MQ dx, the graphic process giving the integral. But with the scales used in Diagram II, instead of multiplying the ordinates 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 : 'Hav- To Adjust the Object-slide of a Transit. 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. ι '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 collimation 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 h cos hcot (dec.) × tan (lat.), where the hour angle (sce 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 being at W or E. The angle h 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: A the difference in R. A. of the stars, d the declination of Polaris, Let and D the declination of the other star. APPENDIX. 249 Find p and m from the formulæ COS A sin D tan m = tan D' p = Cos m then find a from the formula cos a = p sin (d+ m). Then Z, the azimuth, is given by sin Z = sin A cos D cos d cos L sin a where 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 Then i cos D sin S sin A sin a L+ d COS h 2 Z~8 cot 2 tan L~ d ૭ 2 sin 2 where 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 sin convergence = sin diff. of long. 2 X sin (lat.). 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 G. (See Sec. 189.) The difference in altitude in seconds of arc, between the me- ridian altitude and the maximum altitude of a body, is equal to where a d2 4a' 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 precedes 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. NOTE H. (See Sec. 24.) Theoretically the train could just start and eventually attain the speed indicated, provided that the values of MN at all speeds lower than the given one are greater than the value of MN at the speed selected; but in order that the train may attain the speed required in a reasonable time, ample allowance must be made for inertia. NOTE I. (See Sec. 45.) Two very common causes of error in observing angles, which remain unaffected by the process of repetition, are (1) station-twist, due usually to some such cause as the action of the sun, which gives to the instrument a more or less steady motion in one direction, and (2) general instability of the support. The former may be eliminated by taking the mean of two sets of readings, one taken from left to right, and the other from right to left, and the latter by the application of a constant (c) obtained thus: Let A be the reading of some required angle; then if B is the reading of the residuary angle, C = 360° — (A+B) 2 for the residuary angle should of course equal 360° — A. TABLES. TABLE I.-RADII. Deg. Radius. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. || Deg. Radius. 0° 0' Infinite 1° 0' 5729.65 2° 0' 2864.93 3° 0' 1910.08 4° 0' 1432.69 1 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 1878.77 3 1+15.01 4 85943.7 4 5371.56 4 2772.53 4 1868.56 4 1409.21 5 68754.9 5 5288.92 5 2750.35 5 1858 47 5 1403.46 6 57295.8 6 5208.79 6 2728.52 6 1848.48 6 1397.76 7 19110.7 7 5131.05 72707.04 7 1838.59 77 1392.10 8 42971.8 3 5055.59 8 2685.89 8 1828.82 81386.49 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 13 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 20 17188.8 122 19 4351.67 19 2473.37 19 1727.75 19 1327.63 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 2:2 4192.47 22 2421.12 22 1702.10 22 1312.43 23 14946.7 23 4141.96 23 2404.19 23 1693.72 23 1307.45 24 14323.6 24 1092.66 24 2387.50 24 1685.42 24 1302.50 25 13751.0 25 4044 51 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 12732.4 27 3951.54 27 2338.78 27 1661.00 21 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 32 10743 01 32 3736.79 32 31 2276.84 2261.86 31 1629.52 31 1268.87 32 1621.84 32 1264.21 33 10417.5 33 3696 61 33 2247.08 33 1614.22 33 1259.58 34 10111.1 34 3657.29 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 37 9549.31 9291.29 36 3581.10 36 2203.87 36 1591.81 36 1245.89 38 9046.75 39 8814.78 40 8594.42 41 37 38 3508.02 39 3472.59 40 3437.87 8384.80 41 3403.83 3544.19 37 2189.84 37 1584.48 37 1241.40 38 2175.98 38 1577.21 38 1236.94 39 2162.30 40 39 1570.01 39 1232.51 2148.79 40 1562.88 40 1228.11 41 2135.44 41 1555.81 41 1223.74 42 8185.16 42 3370.46 42 2122.26 42 1548.80 42 1219.40 43 7994.81 43 3337.74 43 2109.24 43 1541.86 43 1215.30 44 7813.11 44 45 7639.49 45 3305.65 3274.17 44 2096.39 44 1534.98 44 1210.82 45 2083.68 45 1528.16 45 1206.57 46 7473.42 46 3243.29 46 2071.13 46 1521.40 46 1202.36 47 7314.41 48 7162.03 49 7015 87 50 6875.55 47 3212.98 48 3183.23 49 3154.03 50 3125.36 47 2058.73 48 2046.48 49 2034.37 50 2022.41 47 1514.70 47 1198.17 48 1508.06 48 1194.01 49 1501.48 49 1189.88 50 1494.95 50 1185.78 51 6740.74 52 6611.12 53 6486.38 51 3097.20 52 3069.55 53 3042.39 51 2010.59 52 1998.90 58 1987.35 51 1488.48 51 1181.71 52 1482.07 52 1177.66 53 1475.71 53 1173.65 51 6366.26 54 3015.71 54 1975.93 51 1469.41 54 1169.66 55 6250.51 55 2989.48 55 1964.64 55 1463.16 55 1165.70 56 6138.90 56 2963.71 56 1953.48 56 1456.96 56 1161.76 57 6031.20 57 2938.39 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 GO 5729.65 60 2864.9 59 1920.75 60 1910.08 59 1438.68 59 1150.11 60 1432.69 60 1146.28 252 TABLE I.-RADII. Deg. Radius. || Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 5° 0' 1146.28 6° 0' 955.36670819.020 8° 0' 716.779 9° 0' 637.275 1 636.099 2 | 634.928 1 1142.47 1 952.722 1 2 1138.69 2 950.093 817.077 2815.144 1 715.991 2 713.810 3 1134.94 3 947.478 3813.238 3 712.335 3 633.761 4 1131.21 4 944.877 4 811.303 4 710.865 4 632.599 5 1127.50 5 942.291 5 809.397 5 709.402 5 631.440 6 1123.82 6 939.719 6 807.499 6 707.945 6 630.286 7 1120.16| 7 937.161 7 805.611 7 706.493 î 629.136 8 1116.52 8 934.616 8 803.731 8 705.048 8 627.991 9 1112 91 9 932.086 9 801.860 10 1109 33 10 929.569 10 | 799.997 10 9 703.609 702.175 9 626.849 10 625.71 11 1105.76 11 927.066 11 798.144 12 1102.22 12 924 576 12 796.299 13 1098.70 13 922.100 13 794.462 14 1095.20 14 919.637 14 792.634 11 700.748 12 699.326 13 697.910 14 696.499 11 624.579 12 623.450 13 622.825 14 621.203 15 1091.73 15 917.187 15 790.814 15 695.095 15 620 087 16 1088.28 16 914.750 16 789.003 16 693.696 16 618.974 17 1084.85 18 1081.44 19 1078.05 20 1074.68 17 912.326 18 19 907.517 20 905.131 909.915 18 17 787.210 785.405 17 692.302 17 19 783.618 20 781.840 18 600.914 19 689.532 20 688.156 617.865 18 616.760 19615.660 20 614.563 21 1071.34 22 1008.01 23 1064.71 24 1061.43 25 1058.16 26 1054.92 27 1051.70 28 1048.48 21 902.758 22 900.397 21 780.069 22 778.307 23 898.048 24 895.712 25 893.388 23 776.552 24 774.806 25 773.067 26 891.076 26 771.336 27 888.776 27 769.613 78***** 21 686.785 22 685.419 21 613.470 22 612.380 23 684.059 23 611.295 24 682.704 24 610.214 25 681.354 26 | 680.010 25 609.136 ; 26 608 062 27 678.671 27 606.992 28 886.488 28 767.897 28 677.338 29 30 1045.31 1042.14 30 881.946 29 884.211 29 766.190 29 676.008 30 764.489 30 674.686 28 605.926 29 604.864 30 603.805 31 1039.00 31 879.693 32 1935.87 32 877.451 33 1032.76 33 875.221 34 1029.67 34 873.002 35 1026.60 35 870.795 36 1023.55 37 1020.51 38 1017.49 31 32 30 | 868.598 37 866.412 762.797 761.112 33 759.434 34 757.764 35 756.101 36 754.445 37 752.796 31 32 672.056 33 670.748 34 669.446 35 668.148 36 666.856 37 665.568 673.369 31 34 602.750 32 601.698 33 600.651 599.607 35 598.567 36 597.530 37 596.497 38 864.238 38 751.155 38 664.286 39 1014.50 39 862.075 39 749.521 40 1011.51 40 859.922 30 747.894 39 663.008 40 661.736 38 595.467 39 594.441 40 593.419 41 1008.55 41 857.780 41 746.274 42 1005.60 42 855.648 42 744.661 41 660.468 42 659.205 41 592.400 42 591.384 43 1002.67 43 853.527 43 743.055 43 657.947 43 590.372 44 999.762 44 851.417 44 741.456 44 656.694 44 589.364 45 996.867 45 849.317 45 739.864 45 655.446 45 588.359 46 993.9881 46 847.228 46 738.279 46 654.202 47 991.126 47 845.148 47 736.701 47 652.963 49 50 48 988.280 985.451 982.638 50 838.972 48 843.080 48 735.129 49 841.021 49 733.564 50 732.005 51 979.840 51 836.933 52 977.060 52 834.904 53 974.294 54 971.544 55 968.810 55 828.876 51 730.454 52 728.909 53 832.885 53 727.370 48 651.729 49 650.499 50 649.274 51 648.054 52 646.838 53 645.627 50 51 98 H83. 46 587.357 47 586.359 48 585.364 49 584 373 583.385 582.400 52 581.419 53 580 441 54 830.876 54 725.838 55 724.312 56 966.091 56 826.886 56 722.793 57 963.387 57 824.905 57 721.280 58 960.698 58 822.934 59 958.025 59 820.973 58 719.774 59 718.273 60 955.366 60 819.020| 60 716.779 8888 54 | 644.420 54 55 643.218 56 642.021 57 640.828 58 639.639 59 638.455 579.466 55 578.494 56 577.526 57 576 561 58 | 575.599 59 574.641 60 637.275 60 573.686 253 TABLE I.-RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 2 571.784 2 477.018 4 569.896 4 475.705 4 6 568.020 6 474.400! 8 566.156 8 473.102 10° 0' 573.686 12° 0' 478.339 14° 0' 410.275 16° 0' 359.265 18° 0' 319.623 2358.523 4 357.784 2 | 319.037 4 318.453 6 317.871 8 317.292 {{ 409.306 408.341 6 407.380 8406.424 6 357.048 8 356.315 10 564.305 10 471.810 10 405.473 10 355.585 10 316.715 12 562.466 12 470.526 12 14 560.638 14 469.249 14 404.526 403.583 12 354.859 12 316.139 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 | 400.782 553.447 22 464.209 22 399.857 24 551.678 24 462.966 24 398.937 26 549.920 26 461.729 26 398.020 28 548.174 28 460.500 28 397.108 20 351.981 22 351.269 24 350.560 26 349.854 28 | 349.150 30 | 516.438 30 459.276 30 396.200 30 348.450 32 | 544.714 34 543.001 36 541.298 36 455.646 38 539.606 38 454.449 40 537.924 40 453.259 42 536.253 42 452.073 32 458.060 34 456.850 32 395.296 32 347.752 20 313.860 22313.295 24 | 312.732 26 312.172 28 311 613 30 | 311.050 32 310.502 34 394.396 34 347.057 34 309.949 36393.501 38 392.609 36 346.365 36 309.399 38 345.676 38 308.850 40 391.722 40 344.990 40 308.303 42 | 390.838 42 344.306 42 307.759 41 46 534.593 532.943 48 531.303 50 529.673 44 450.894 46 449.722 44 389.959 46 389.084 44 343.625 44 307.216 46 48 448.556 50 447.395 48 388.212 342.947 48 342.271 46 306.675 48 306.136 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 54 340.260 54 304.531 56 524.843 56 58 523.252 58 443.951 442.814 56 384.765 56 339.595 56 304.000 58 383.913 58 338.933 58 303.470 11° 0' 521.671 13° 0' 441.684 15° 0' 2 520.100 2 440.559 2 382.220 2 337.616 4 518 539 4 439.440 4 381.380 4 336.962 383.065 17° 0' 338.273 19° 0' 302.943 2302.417 4 301.893 6 516.986 6 438.326 6 380.513 6 336.310 6 301.371 8 515.443 8 437.219 10 513.909 10 436.117 12 512.385 12 435.020 8 379.709 10 378.880 12 378.054 14 510.869 14 433.929 14 377.231 16 509.363 16 432.844 16 376.412 18 507.865 18 431.764 18 375.597 8 335.660 10 335.013 12 334.369 14 333.727 16 333.088 18 332.451 26 20 506.376 20 430.690 22 504.896 24 503.425 24 428.557 501.962 20 374.786 20 331.816 22 429.620 22 373.977 21 373.173 26 427.498 26 372.372 28 500.507 28 426.145 28 371.574 30 | 499.061 32 497.624j 30 425.396 32 424.354 31 496.195 34 423.316 30 370.780 32 369.989 34 369.202 22 331.184 24 330.555 26 329.928 28 329.303 30 328.689 8 300.851 10 300.333 12 299.816 14 299.302 16298.789 18 298.278 20 297.768 22 297.260 24296.755 26 296.250 28 295.748 30 295.247 32 328.061 32 294.748 34 327.443 34 294.251 36 494.774 36 422.283 36 368.418 36 326.828 36 293.756 38493.361 38 421.256 38 367.637 38326.215 38 293.262 40 491.956 40 420.233 40 366.859 40 325.604 40 292.770 42 490.559 42 419.215 42 366.085 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 46 364.547 46 323.786 46 291.303 48 486.417 48 416.192 48 363.783 48 323.184 48 290.818 50 485 051 50 415.194 50 363.022 50 322.585 50 290.331 52 483.694 52 414.201 52 362.264 52 321.989 52 289.851 54 482.344 54 413.212 54 361.510 54 321.394 54 289.371 56 481 001 56 412.229 56 360.758 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. Tan- Exter- gent. nal. Tan- Exter- Tan- Angle. gent. nal. Angle. Exter- gent. nal. I. T. E. I. T. E. I. T. E. 1° 50.00 .218 11 2 ૭ CECCE DECE 10′ 58.34 .297 66.67 .388 75.01 .491 30 83.34 .606 40 91.68 .733 100.01 .873 12 108.35 1.024 116.68 1.188 30 125.02 1.364 133.36 1.552 141.70 1.752 JAUNE CAUNE 10' 560.11 27.313 551.70 26.500 21° 1061.9 97.577 10 1070.6 99.155 20 568.53 28.137 20 1079.2 100.75 576.95 28.974 30 1087.8 102.35 585.36 29.824 593.79 30.686 602.21 31.561 22 610.64 32.447 619.07 33.347 627.50 34.259 635.93 35.183 644.37 30.120 CECON CHA 40 1096.4 103.97 50 1105.1 105.60 1113.7 107.24 10 1122.4 108.90 20 1131.0 110.57 30 1139.7 112.25 40 1148.4 1183.95 50 1157.0 115.66 3 150.04 1.964 13 652.81 37.070 23 1165.7 117.38 20 30 40 50 UNCEN 10 158.38 2.188 166.72 2.425 20 175.06 2.674 30 183.40 2.934 191.74 3.207 50 4 200.08 3.492 14 AR 10 661.25 38.031 669.70 39.006 678.15 39.993 40 686.6) 40.992 695.06 42.004 703.51 | 43.029 24 10 208.43 3.790 10 711.97 44.066 20 216.77 4.099 20 720.44 45.116 30 225.12 4.421 30 728.90 46.178 40 233.47 4.755 40 737.37 47.253 50 241.81 5.100 50 745.85 48.341 CAONE CECUE 10 1174.4 119.12 20 1183.1 120.87 30 1191.8 122.63 40 1200.5 | 124.41 50 1209.2 126.20 1217.9 128.00 10 1226.6 129.82 20 1235.3 131.65 30 1244.0 133.50 40 1252.8 135.35 50 1261.5 137.23 5 250.16 5.459 15 754.32 49.411 25 1270.2 139.11 6 GNCONA CLCEZ 10 258.51 5.829 20 266.86 6.211 30 275.21 6.606 40 283.57 7.013 50 291.92 7.432 300.28 7.863 16 10 308.64 8.307 20 316.99 8.762 30 325.35 9.230 40 333.71 9.710 50 342.08 10.202 CACNE CACUE 10 762.80 50.554 20 771.29 51.679 30 779.77 52.818 40 788.26 53.969 50 796.75 55.132 805.25 56.309 26 10 813.75 57.498 20 822.25 58.699 30 830.76 59.914 40 839.27 61.141 50 847.78 62.381 CACONE CACNE 10 1279.0 141.01 20 1287.7 142.93 30 1296.5 144.85 40 1305.3 146.79 50 1314.0 148.75 1322.8 150.71 10 1331.6 152.69 20 1340.4 154.69 30 1349.2 156.70 40 1358.0 158.72 50 1366.8 160.76 350.44 10.707 17 856.30 63.634 27 1375.6 162.81 8 A8898 ARAR 10 358.81 11.224 20 367.17 11.753 30 375.54 12.294 40 383.91 12.847 50 392.28 13.413 400.66 13.991 18 10 409.03 14.582 20 | 417.41 15.181 30 40 50 425.79 15.799 434.17 16.426 442.55 17.065 CACOE CAUNE 10 864.82 64.900 40 890.41 20 873.35 66.178 30 881.88 67.470 68.774 50 898.95 70.091 907.49 71.421 888 10 916.03 72.764 20 924.58 74.119 30 933.13 75.488 40 941.69 76.869 50 950.25 78.264 CAUNE CACON 10 1384.4 164.86 20 1393.2 166.95 30 1402.0 169.04 40 1410.9 171.15 50 1419.7 173.27 1428.6 175.41 10 1437.4 177.55 20 1446.3 179.72 30 1455.1 181.89 40 1464.0 184.08 50 1472.9 186.29 9 450.93 17.717 19 958.81 79.671 29 1481.8 188.51 10 CACLE SACKH 10 459.32 18.381 10 967.38 81.092 10 1490.7 190.71 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 85.1431 40 1517.4 197.53 50 492.88 21.161 50 1001.7 86.904 50 1526.3 199.82 501.28 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 40 30 526.48 534.89 24.138 30 1036.1 92.924 30 1562.1 209.12 24.913 40 1044.7 94.462 40 1571.0 211.48 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. Angle. Tan- Exter- gent. nal. Angle. Tan- Exter- gent. nal. Tan- Angle. Exter- gent. nal. I. T. E. I. T. E. I. T. E. 31° 1589.0 216.25 41° $2 CHACEN CACNE 10 1598.0 218.66 2142.2 10' 2151.7 387.38 51° 390.71 2732.9 618.39 20 1606.9 221.08 20 2161.2 394.06 30 1615.9 223.51 30 2170.8 397.43 40 1624.9 225.96 40 2180.3 400.82 50 1633.9 228.42 50 2189.9 404.22 1643.0 230.90 42 2199.4 407.64 52 10 1652.0 233.39 10. 2209.0 411.07 20 1661.0 235.90 20 2218.6 414.52 30 1670.0 238.43 30 2228.1 417.99 40 1679.1 240.96 40 2237.7 421.48 50 1688.1 243.52 50 2247.3 424.98 CACCH CACE 10' 2743.1 622.81 20 2753.4 627.24 30 2763.7 631.69 40 2773.9 636.17 2784.2 640.66 2794.5 645.17 2804.9 649.70 2815.2 654.25 2825.6 €58.83 2835.9 663.42 2846.3 668.03 33 1697.2 246.08 43 2257.0 428.50 53 2856.7 672.66 34 JAUNE JACEZ 10 1706.3 248.66 20 1715.3 251.26 30 1724.4 253.87 40 1733.5 256.50 50 1742.6 259.14 1751.7 261.80 44 1760.8 264.47 20 1770.0 267.16 30 1779.1 269.86 40 1788.2 272.58 278.05 A288 AR888 1843.3 289.20 50❘ 1852.5 292.02 1861.7 294.86 40 38 CACON 39 10 20 30 | 2057.2 358.11 50 40 40 50 2132 7 35 36 37 50 1797.4 275.31 1806.6 10 1815.7 280.82 20 1824.9 283.60 30 40 10 1834.1 286.39 1870.9 297.72 20 1880.1 300.59 30 40 1889.4 303.47 1898.6 306.37 50 1907.9 309.29 1917.1 312.22 47 10 1926.4 315.17 20 1935.7 318.13 30 1945.0 321.11 1954.3 324.11 50 1963.6 327.12 1972.9 10 1982.2 333.19 20 1991.5 336.25 30 2000.9 339.32 40 2010.2 342.41 50 2019.6 345.52 2029.0 348.64 49 2038.4 351.78 2047.8 354.94 40 | 2066.6 361.29 2076.0 364.50 2085.4 367.72 50 10 2094.9 870.95 20 2104.3 374.20 30 2113.8 377.47 2123.3 380.76 384.06 20 2451.8 502.54 30 2461.7 506.42 40 2471.5 510.33 50 2481.4 514.25 518.20 2491.3 10 2501.2 522.16 20 2511.2 526.13 30 2521.1 530.13 40 2531.1 534.15 50 2541.0 538.18 2551.0 542.23 2561.0 546.30 20 2571.0 550.39 30 2581.0 554.50 40 2591.1 558.63 50 2601.1 562.77 2011.2 10 2621.2 571.12 20 2631.3 575.32 30 2641.4 579.54 40 2651.5 583.78 50 2661.6 588.04 2671.8 592.32 10 2681.9 596.62 20 2692.1 600.93 30 2702.3 605.27 40 2712.5 609.62 50 2722.7 614 00 45 46 CACOON CAUNE JEUNE CEC08 10 2266.6 432.04 10 2867.1 677.32 20 2276.2 435.59 20 2877.5 681.99 30 2285.9 439.16 30 2888.0 686.68 40 2295.6 422.75 40 2898.4 691.40 50 2305.2 446.35 50 2908.9 698.13 2314.9 449.98 54 2919.4 700.89 10 2324.6 453.62 10 2929.9 705.66 20 2334.3 457.27 20 2940.4 710.46 30 2341.1 460.95 30 2951.0 715.28 40 2353.8 464.64 40 2961.5 720.11 50 2363.5 468.35 50 2972.1 724.97 2373.3 472.08 55 2982.7 729.85 10 2383.1 475.82 20 2392.8 479.59 30 2402.6 483.37 40 2412.4 487.17 50 2422.3 490.98 2432.1 494.82 56 10 2441.9 498.67 CACONE CACUZ 10 2993.3 734.76 20 3003.9 739.68 30 3014.5 744.62 3025.2 749.59 50 3035.8 754.57 3046.5 759.58 10 3057.2 764.61 20 3067.9 769.66 30 3078.7 774.73 40 3089.4 779.83 50 3100.2 784.91 57 3110.9 790.08 330.15 48 10 ARAGA ARAGO 58 CAUNE CACEE 10 3121.7 795.24 20 3132.6 800.42 30 | 3143.4 805.62 40 3154.2 810.85 50 3165.1 816.10 3176.0 821.37 10 3186.9 826.66 20 3197.8 831.98 30 3208.8 837.31 40 3219.7 842.67 50 3230.7 848.06 566.94 59 3241.7 853.46 9288 28-8 10 3252.7 858.89 20 3263.7 864.34 30 3274.8 869.82 40 3285.8 875.32 50 3296.9 880.81 60 3308.0 886.38 10 3319.1 891.95 20 3330.3 897.54 30 3341.4 903.15 40 3352.6 908.79 50 3363.8 914.45 256 TABLE II.—TANGENTS AND EXTERNALS TO A 1º CURVE. Angle. Tan- Exter- gent. nal. Angle. Tan- Exter- Tan- gent. nal. Angle. Exter- gent. nal. I. T. E. I. T. E. I. 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 1833.0 40 3420.1 943.12 40 4137.4 1337.7 50 3431.4 948.92 50 4150.1 1345.1 62 3442.7 954.75 72 4162.8 1352.6 82 CACONN 10 | 3154.1 960.60 20 3465.4 966.48 30 3476.8 972.38 40 3488.3 978.31 50 3499.7 984.27 CECOU 10 4175.6 1360.1 20 4188.5 1367.6 30 4201.2 1375.2 30 50 40 4214.0 1382.8 4226.8 1390.4 50 OTACONH CA 40 4951.5 1843.1 50 4966.1 1852.6 4980.7 1862.2 10 4995.4 1871.8 20 5010.0 1881.5 5024.8 1891.2 5039.5 1900.9 5054.3 1910.7 63 3511.1 990.24 73 4239:7 1398.0 83 5069.2 1920.5 64 CAUNE CAUNE 10 3522.6 996.24 10 4252.6 1405.7 20 3534.1 1002.3 20 4265.6 1413.5 30 3545.6 1008.3 30 4278.5 1421.2 40 3557.2 1014.4 10 4291.5 1429.0 50 3568.7 1020.5 50 1304.6 1436.8 3580.3 1026.6 74 4317.6 1444.6 10 3591.9 1032.8 10 4330.7 1452.5 20 3603.5 1039.0 20 4343.8 1460.4 30 3615.1 1045.2 30 4356.9 1468.4 40 3626.8 1051.4 40 4370.1 1476.4 50 3638.5 1057.7 50 4383.3 1481.4 65 3650.2 1063.9 75 4396.5 1492.4 85 66 ARS AG 10 3661.9 1070.2 20 3673.7 1076.6 30 3685.4 1082.9 40 3697.2 1089.3 50 3709.0 1095.7 3720.9 1102.2 76 10 3732.7 1108.6 20 3741.6 1115.1 30 3756.5 1121.7 40 3768.5 1128.2 50 3780.4 1134.8 CACON CACNU 10 4109.8 1500.5 20 4423.1 1508.6 30 4436.4 1516.7 40 4449.7 1524.9 40 50 4463.1 1533.1 50 4476.5 1541.4 86 10 4489.9 1549.7 20 4503.4 1558.0 20 30 | 4516.9 1566.3 30 40 4530.4 1574.7 40 50 4514.0 1583.1 50 ARGO ARO AR38 =28-8 10 5084.0 1930.4 20 5099.0 1940.3 30 5113.9 1950.3 40 5128.9 1960.2 50 5143.9 1970.3 84 | 5159.0 1980.4 10 5174.1 1990.5 20 5189.3 2000.6 30 5204.4 2010.8 40 5219.7 2021.1 50 5234.9 2031.4 5250.3 2041.7 10 5265.6 2052.1 20 5281.0 2062.5 30 5296.4 2073.0 5311.9 2083.5 5327.4 2094.1 5343.0 2104.7 5358.6 2115.3 5374.2 2126.0 5389.9 2136.7 5105.6 2147.5 5421.4 2158.4 67 3792.4 1141.4 777 68 CAUNE JAUNU 10 3804.4 1148.0 10 4557.6 4571.2 1600.1 1591.6 87 5437.2 2169.2 20 3816.4 1154.7 30 3828.4 1161.3 40 3840.5 1168.1 40 50 3852.6 1174.8 50 2898 20 4584.8 1608.6 30 4598.5 1617.1 4612.2 1625.7 4626.0 1634.4 10 3864.7 3876.8 1188.4 1181.6 78 4639.8 1643.0 88 10 4653.6 1651.7 20 3889.0 1195.2 20 4667.4 1660.5 20 30 3901.2 1202.0 30 4681.3 1669.2 A8898 ARA 10 5453.1 2180.2 20 5469.0 2191.1 30 5184.9 2202.2 40 5500.9 2213.2 50 5517.0 2224 3 5533.1 2235.5 10 5549.2 2246.7 5565.4 2258.0 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 2303 5 70 40 OTACON OTACON 10 3950.2 1229.7 20 3962.5 1236.7 30 3974.8 1243.7 40 3987.2 1250.8 50 3999.5 1257.9 4011.9 1265.0 80 10 4024.4 20 4036.8 1272.1 1279.3 20 30 4049.3 4061.8 50 4074.4 1286.5 1293.6 1300.9 HECNE CECUE 10 4737.2 1704.7 20 4751.2 1713.7 30 4765.3 1722.7 40 4779.4 1731.7 50 4793.6 1740.8 4807.7 1749.9 90 10 4822.0 1759.0 4836.2 1768.2 30 4850.5 1777.4 50 40 4864.8 4879.2 1786.7 1796.0 CACUE CECEE 10 5646.9 2315.0 20 5663.4 2326.6 30 5679.9 2338.2 40 5696 4 2349.8 50 5713.0 2361.5 5729 7 2373.3 10 5746.3 2385.1 20 5763.1 2397.0 30 5779.9 2408.9 40 5796.7 2420.9 50 5813.6 2432.9 257 TABLE II.-TANGENTS AND EXTERNALS TO A 1° CURVE. Tan- Ex- Angle. Tan- Ex- Tan- gent.ternal. Angle. gent.ternal, Angle. Ex- gent.ternal I. T. E. I. T. E. I. T. E. 91° 5830.5 2444.9 97 6476.2 2917.3 103 7203.2 3474.4 10 5847.5 2457.1 10 6495.2 2931.6 20 5864.6 2469.3 20 40 30 5881.7 2481.5 5898.8 2493.8 30 28 6514.3 2945.9 6533.4 2960.3 30 40 6552.6 2974.7 50 5916.0 2506.1 50 6571.9 2989.2 92 5933.2 2518.5 98 6591.2 3003 8 104 20 30 CEEES 10 5950.5 2531.0 10 6610.6 3018.4 5967.9 2543.5 20 6630.1 3033.1 5985.3 2556.0 30 6649.6 3047.9 40 6002.7 2568.6 10 6669.2 3062.8 50 6020.2 2581.3 50 6688.8 3077.7 CECUE CACNE 10 7224.7 3491.3 20 72-46.3 3508.2 7268.0 3525.2 40 7289.8 3542.4 50 7311.7 3559.6 7333.6 3576.8 10 7355.6 3594.2 20 7377.8 3611.7 30 7399.9 3629.2 40 7422.2 3646.8 50 7444.6 3661.5 93 6037.8 2594.0 99 6708.6 3092.7 105 7467.0 3682.3 94 30 CECEE CLEET 10 6055.4 2606.8 20 6073.1 2619.7 30 6090.8 2632.6 40 6108.6 2645.5 40 50 6126.4 2658.5 6144.3 2671.6 100 10 6162.2 2684.7 20 6180.2 2697.9 6198.3 2711.2 50 40 6216.4 2724.5 6234.6 2737.9 95 6252.8 2751.3 101° 10 6271.1 2764.8 20 6289.4 2778.3 30 6307.9 2792.0 40 6326.3 2805.6 50 6344.8 2819.4 96 6363.4 2833.2 102 10 6382.1 2847.0 20 6400.8 2861.0 20 30 6419.5 2875.0 28390 98898 8893 Fass 0728.4 3107.7 6748.2 3122.9 6768.1 3138.1 6788.1 3153.3 6950.6 50 6808.2 3168.7 6828.3 10 6848.5 3199.6 20 6868.8 3215.1 30 6889.2 3230.8 40 6909.6 3246.5 50 6930.1 10 6971.3 3294.1 20 6992.0 3310.1 30 7012.7 3326.1 40 7033.6 3342.3 3184.1 106 3262.3 CACUN CECUE 10 7489.6 3700.2 20 7512.2 3718.2 30 7534.9 3736.2 40 7557.7 3754.4 50 7580.5 3772.6 7603.5 3791.0 10 7626.6 3809.4 20 7649.7 3827.9 30 7672.9 3846.5 40 7696.3 3865.2 50 7719.7 3884.0 3278.1 107 7743.2 3902.9 50 7054.5 3358.5 7075.5 3374.9 108 7096.6 3391.2 7117.8 3407.7 30 7139.0 3424.3 40 6438.4 2889.0 40 7160.8 3440.9 50 6457.3 2903.1 50 7181.7 3157.6 CECCH CA&XZ 10 7766.8 3921.9 20 7790.5 3940.9 30 7814.3 3960.1 40 7838.1 3979.4 50 7862.1 3998.7 7886.2 4018.2 10 7910.4 4037.8 20 7934.6 4057.4 30 7959.0 4077.2 40 7983.5 4097.1 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. Cur. Cur. Cur. I. Cur. Cur. Cur. Cur. Cur. Cur. 001.003 .000 .011 30 .013 .025 .32 .39 20 98888 10° .03 .06 .09 .13 .16 .19 10° 20 .06 .13 .19 .26 30 .10 .19 .29 .39 .19 .59 40 .13 .26 .40 .53 .67 50 .17 .34 .004 .006 .007 .008 .017 .022 .028 .034 .038 .051 .065 .070.093 .117 .078 .141 .116.151 .151.189 .227 .340 .485 .168 .225 .283 .240 .321 .403 .332 .445 .558 .671 .450 .603.756 .910 .604 .809 1.015 1.221 .806 1.082 1.355 1.633 .721 1.086 1.456 1.825 2.197 80 40 .023.046 .51 .68 .85 1 02 50 .037.075 60 .21 .42 .63 .84 1.05 1.27 60 .056.112 ΤΟ .25 .51 .76 1.02 1.28 1.54 ΤΟ .080.159 80 .30 .61 .91 1.22 1.53 1.84 90 .36 100 43 .86 1.30 1 74 2 18. 2.62 80 .110 .220 .721.09 1.451.832.20 90 .149.299 100 .200 .401 110 .51 1.03 120 110 .62 1.25 1.93 2.52 3 16 3.81 120 1.56 2 08 | 2.61|3.14 .268.536 360 (See Note on page 259.) 258 TABLE III.-TANGENTIAL OFFSETS 100 FT. ALONG THE CURVE. Deg. of Curve. 0' 10' 20' 30' 40/ 50' 0° 0.000 0.145 0.291 0.436 0.582 0.727 10 0.873 1.018 1.164 1.309 1.454 1.600 20 1.745 1.891 2.036 2.181 2.327 2.472 ३० 2.618 2.763 2.908 3.054 3.199 3.345 40 3.490 3.635 3.781 3.926 4.071 4.217 5° 4.362 4.507 4.653 4.798 4.943 5.088 6° 5.234 5.379 5.521 5.669 5.814 5.960 70 6.105 6.250 6.395 6.540 6.685 6.831 8° 6.976 7.121 7.266 7.411 7.556 7.701 go 7.846 7.991 8.136 8.281 8.426 8.571 10° 8.716 8.860 9.005 9.150 9.295 9.440 110 9.585 9.729 9.874 10.019 10.164 10.308 120 10.453 10.597 10.742 10.887 11.031 11 176 13° 11.320 11.465 11.609 11.754 11.898 12.043 11° 12.187 12.331 12.476 12.620 12.764 12.908 15° 13.053 13.197 13.341 13.485 13.629 13.773 16° 13.917 11.061 14 205 14.349 14.493 14.637 17° 11.781 14.925 15.069 15.212 15.356 15.500 18° 15.643 15.787 15.931 16.074 16.218 16.361 19° 16.505 16.648 16.792 10.935 17.078 17.222 20° 17.365 17.508 17.651 17.794 17.937 18.081 21° 18.224 18.367 18.509 18.652 18.795 18.938 22° 19.081 19.224 19.366 19.509 19.652 19.794 23° 19.937 20.079 20.222 20.364 20.507 20.649 24° 20.791 20.933 21.076 21.218 21.360 21.502 TABLE IV.-MID-ORDINATES TO A 100-FT. CHORD. Deg. of 0 1 Curve. 0° 10° 20° ૨૭ 3 4 5 6 7 8 9 0.000 0.218 0.436 0.655 0.873 1 091 1.309 1.528 1.746 1.965 2.183 2.402 2.620 2.839 3.058 3.277 3.496 3.716 3.935 4.155 4.374 4 594 4.814 5.035 5.255 5.476 5.697 5.918 6.139 6.360 Note. As an example illustrating the use of Table II, suppose we require the value of T for a 5° curve, where I = 40° 20′. Then T= 2104.3 5 +.13420.99. : TABLE V.-LONG CHORDS. Actual LONG CHORDS. Degree Arc, of One Curve. Station. 2 Stations. 3 Stations. Stations. Stations. 4 5 6 Stations. 0° 10' 100.000 200.000 299.999 1 20 CECEE CEGE. 399.998 499.996 599.993 20 .000 199.999 299.997 399.992 499.983 599.970 30 .000 199.998 299.992 399.981 499.962 599.933 40 .001 199.997 299.986 399.966 499.932 599.882 50 .001 199.995 299.979 399.947 499.894 599.815 100.001 199.992 299.970 399.924 499.848 599.733 .002 199.990 299.959 399.896 499.793 599.637 .002 199.986 299.946 399.865 499.729 599.526 .003 199.983 299.932 399.829 499.657 599.401 10 .003 199.979 299.915 399.789 499.577 599.260 .004 199.974 299.898 399.744 499.488 599.105 2 100.005 199.970 299.878 399.695 499.391 3 10 20 30 40 50 23333 28838 598.934 10 .006 199.964 299.857 399.643 499.285 598.750 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 .010 199.939 299.756 399.389 498.778 597.862 100.011 199.931 299.726 399.315 498.630 597.604 .013 199.924 299.695 399.237 498.474 597.331 .014 199.915 299.662 399.154 498.309 597.043 .015 199.907 299.627 399.068 498.136 596.740 ·017 199.898 299.591 398.977 497.955 596.423 .019 199.888 299.553 398.882 497.765 596.091 4 100.020 199.878 299.513 398.782 497.566 595.744 Сл 5 40 CECES CECEE 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 .030 199.822 299.289 398.223 496.449 593.792 100.032 199.810 299.239 398.099 496.201 593.358 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 .041 199.756 299.023 397.559 495.123 591.476 50 .043 199.741 298.964 397.413 494.832 590.970 6 100.046 199.720 298.904 397.264 494.534 590.449 40 -I 7 20 30 40 50 CECUE CECUE 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 .056 199.662 298.648 396.623 493.257 588.221 50 .059 199.644 298.579 396.453 492.917 587.628 100.062 199.627 298.509 396.278 492.568 587.021 10 .065 199.609 298.438 396.099 492.212 586.400 .068 199.591 298.364 395.916 491.847 585.765 .071 199.572 298.289 395.729 491.474 585.115 .075 199.553 298.212 395.538 491.093 584.451 .078 199.533 298.134 395.342 490.704 583.773 8 100.081 199.513 298.054 395.142 490.306 583.081 40 9 30 40 50 ARAB AR838 10 .085 199.492 297.972 394.938 489.900 582.875 20 .088 199.471 297.888 394.731 489.486 581.651 30 .092 199.450 297.803 394.518 489.064 580.920 .095 199..428 50 099 199.406 100.103 199.383 297.716 394.302 297.628 394.082 297.538 393.857 488.634 580.172 488.196 579.409 487.749 578.633 10 .107 199,360 297.446 393.629 487.294 577.843 20 .111 199.337 297.352 393.396 486.832 577.039 .115 199.313 297.257 393.159 486.361 576.222 .119 199.289 297.160 392.918 485.882 575.390 .123 199.264 297.062 392.673 485.395 574.545 10 100.127 199.239 296.962 392.424 484.900 573.686 TABLE V.-LONG CHORDS. LONG CHORDS. Degree of Actual Arc, Curve. One Station. 2 Stations. 3 Stations. 4 Stations. 5 6 Stations. Stations. 10° 10/ 100.131 199.213 296.860 392.171 484.397 572.813 20 .136 199.187 296.756 391.914 483.886 571.926 11 CAUNE CE& 30 .140 199.161 296.651 391.652 483.367 571.027 40 .145 199.134 296.544 391.387 482.840 570.113 50 .149 199.107 296.436 391.117 482.305 569.186 100.154 199.079 296.325 390.843 481.762 568.245 10 .158 199.051 296.214 390.565 481.211 567.292 20 .163 199.023 296.100 390.284 480.653 566.324 30 .168 198.994 295.985 389.998 480.086 565.343 40 .173 198.964 295.868 389.708 479.511 564.349 50 .178 198.935 295.750 389.414 478.929 563.341 12 100.183 198.904 295.629 389.116 478.338 562.321 10 20 40 13 20 30 50 ERROR ERROR .188 198.874 295.508 388.814 477.740 561.287 .193 198.813 295.381 388.508 477.135 560.240 30 .199 198.811 295.259 388.197 476.521 559.180 .204 198.779 295.132 387.883 475.899 558.107 50 .209 198.747 295.004 387.565 475.270 557.020 100.215 198.714 294.874 387.243 474.633 555.921 10 .220 198.681 294.742 386.916 473.988 554.809 .226 198.648 294.609 386.586 473.336 553.684 232 198.614 294.474 386.252 472.675 552.546 40 237 198.579 294.337 385.914 472.007 · 551.395 .243 198.544 294.199 385.572 471.332 550.232 14 100.249 198.509 294.059 385.225 470.649 549.056 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 50 15 30 40 CACON OA 40 .274 198.364 293.483 383.801 467.840 544.226 .280 198.327 293.335 383.435 467.119 542.987 100.286 198.289 293.185 383.065 466.390 541.736 10 .292 198.251 293.034 382.691 465.654 540.472 20 .299 198.212 292.881 382.313 464.911 539.196 .306 198.173 292.726 381.931 464.160 537.908 .312 198.134 292.570 381.546 463.401 536.608 50 .319 198.094 292.412 381.156 462.635 535.296 16 100.326 198.054 292.252 380.763 461.862 533.972 17 C*CUE JEUNE 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 .361 197.846 291.430 378.737 457.886 527.171 100.368 197.803 291.261 378.320 457.069 525.776 10 .375 197.760 291.090 377.900 456.241 524.369 20 .382 197.716 290.918 377.475 455.413 522.950 30 .390 197.672 290.743 377.047 454.574 521.519 40 .397 197.628 290.568 376.615 453.728 520.078 50 .405 197.583 290.390 376.179 452.875 518.625 18 100.412 197.538 290.211 375.739 452.015 517.160 10 .420 197.492 290.031 375.295 451.147 515.685 20 .428 197.446 289.849 374.848 450.373 514.198 30 .436 197.399 289.665 374.397 449.392 512.699 40 .444 197.352 289.479 373.942 448.504 511.190 50 .452 197.305 289.292 373.483 447.608 509.670 19 100.460 197.256 289.104 373.021 446.706 508.139 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 .484 197.111 288.528 371.610 443.957 503.479 40 .493 197.062 288.333 371.133 443.028 501.905 50 .501 197.012 288.137 370.652 442.091 500.320 20 100.510 196.962 287.939 370.167 441.147 498.724 261 TABLE VI.-MID-ORDINATES TO LONG CHORDS. Degree 1 2 3 4 5 6 of Curve. Station. Stations. Stations. Stations. Stations. Stations. 0° 10' 1 2 3 CAUNE CAUNE CACNE CACUE .036 .145 .327 .582 .909 1.309 20 .073 .291 .654 1.164 1.818 2.618 30 .109 .436 .982 1.745 2.727 3.926 40 .145 .582 1.309 2.327 3.636 5.235 50 .182 .727 1.636 2.909 4.545 6.541 .218 .873 1.963 3.490 5.453 7.852 .255 1.018 2.291 4.072 6.362 9.160 20 .201 1.161 2.618 4.054 7.270 10.468 30 .327 1.309 2.945 5.235 8.179 11.775 .364 1.454 3.272 5.816 9.087 13.082 .400 1.600 3.599 6.398 9.994 14.389 .436 1.745 3.926 6.979 10.902 15.694 .473 1.891 4.253 7.560 11.809 17.000 20 .509 2.030 4.580 8.141 12.716 18.304 .515 2.181 4.907 8.722 13.623 19.608 .582 2.327 5.234 9.303 14.529 20.912 .618 2.472 5.561 9.883 15.435 22.214 .654 2.618 5.888 10.464 16.341 23.516 .691 2.763 6.215 11.044 17.246 24.817 .727 2.908 6.542 11.621 18.151 26.117 .763 3.054 6.808 12.204 19.055 27.416 .800 3.199 7.195 12.784 19.959 28.714 .836 3.345 7.522 13.363 20.863 30.012 4 .872 3.490 7.848 13.943 21.766 21.308 5 20 30 40 50 22800 288 10 .909 3.635 8.175 14.522 22.668 32.603 .945 3.781 8.501 15.101 23.570 33.896 30 .982 3.926 8.828 15.080 24.471 35.189 40 1.018 4.071 9.151 16.258 25.372 36.480 50 1.054 4.217 9.480 16.837 26.272 37.770 1.091 4.362 9.807 17.415 27.171 39.059 10 1.127 4.507 10.133 17.992 28.070 40.346 1.161 4.653 10.459 18.570 28.968 41.631 1.200 4.798 10.785 19.147 29.866 42.916 1.237 4.943 11.111 19.724 30.762 44.198 1.273 5.088 11.436 20.301 31.658 45.479 CO 6 1.309 5.234 11.762 20.877 32.553 46.759 7 20 30 40 50 JACUN CACUE 10 1.346 5.379 12.088 21.453 33.448 48.037 20 1.382 5.524 12.413 22.029 34.341 49.313 30 1.418 5.669 12.739 22.604 35.234 50.587 40 1.455 5.814 13.064 23.179 36.126 51.860 50 1.491 5.960 13.389 23.751 37.017 53.130 1.528 6.105 13.715 24.328 37.907 54.399 10 1.564 0.250 14.040 24.902 38.796 55.666 1.600 6.395 14.365 25.476 39.684 56.931 1.637 6.540 14.689 26.019 40.571 58.193 1.673 6.685 15.014 26.622 41.458 59.45£ 1.710 6.831 15.339 27.195 42.343 60.712 8 1.746 6.976 15.663 27.767 43.227 61.969 9 40 50 CAUNE CALLE 10 1.782 7.121 15.988 28.338 44.110 63.223 20 1.819 7.266 16.312 28.910 44.992 64.475 30 1.855 7.411 16.636 29.481 45.873 65.724 40 1.892 7.556 16.960 30.051 46.753 66.972 50 1.928 7.701 17.284 30.621 47.632 68.216 1.965 7.816 17.608 31.190 48.510 69.459 10 2.001 7.991 17.932 31.759 49.386 70.699 20 2.037 8.136 18.255 32.328 50.201 71.936 30 2.074 8.281 18.578 32.896 51.135 773.171 2.110 8.426 18 902 33.164 52.008 74403 2.147 8.571 19.225 34.031 52.880 75.632 10 2.183 8.716 19.548 34.597 53.750 76.859 262 TABLE VI.—MID-ORDINATES TO LONG CHORDS. Degree of Curve. 1 Station. 2 3 4 5 6 Stations. Stations. Stations. Stations. Stations. 10° 10' 11 CACCH SECUR 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 56.353 80.523 40 2.329 9.295 20.838 36.859 57.218 81.739 2.365 9.440 21.160 37.423 58.081 82.951 2.402 9.585 21.483 37.986 58.943 84.161 2.438 9.729 21.804 38.549 59.804 85.368 2.475 9.874 22.126 39.111 60.663 86.571 2.511 10.019 22.418 39.673 61.521 87.772 2.547 10.164 22.769 40.234 62.377 88.969 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 20 30 50 13 14 20 40 50 15 20 40 50 ARAGA ARAZA ARAGO ARADA 2.657 10.597 23.732 41.914 64.937 92.542 2.693 10.742 24.053 42.473 65.787 93.727 2.730 10.887 24.374 43.031 66.636 94.908 40 2.766 11.031 21.694 43.588 67.482 96.086 2.803 11.176 25.014 44.145 68.328 97.260 2.839 11.320 25.334 44.701 69.171 98.431 10 2.876 11.465 25.654 45.256 70.012 99.598 20 2.912 11.609 25.974 45.811 70.851 100.762 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 73.361 104.232 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 3.131 12.476 27.887 49.126 75.859 107.669 30 3.168 12.620 28.206 49.676 76.687 108.807 3.204 12.761 28.524 50.225 77.513 109.941 3.241 12.908 28.841 50.773 78.337 111.071 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 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 3.423 13.629 30.427 53.504 82.429 116.662 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 20 17 20 40 ARA8 A888 10 3.533 14.061 31.376 55.133 84.861 119.967 3.569 14.205 31.692 55.675 85.667 121.061 30 3.600 14.349 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 57.291 88.074 124.315 3.716 14.781 32.953 57.832 88.872 125.391 10 3.752 14.925 33.267 58.369 89.668 126.463 3.789 15.069 33.582 58.906 90.462 127.520 30 3.825 15.212 33.896 59.441 91.254 128.593 3.862 15.356 34.210 59.976 92.043 129.651 50 3.899 15.500 34.523 60.510 92.830 130.704 18 3.935 15.643 34.837 61.042 93.616 131.753 19 AR88 ARAGO 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 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 16 935 37.645 65.797 100.577 140.981 40 4.301 17.078 37.956 66.321 101.339 141.982 50 4.338 17.222 38.266 66.843 102.098 142.978 20 4.374 17.365 38.576 67.365 102.855 143.969 263 TABLE VII.-MINUTES IN DECIMALS OF A DEGREE. 0" 10" 15" 20" 30" 40" 45" 50" 0 .00000 00278 .00417 .00556 .00833 .01111 .01250 .01389 0 1 .01667 .01944 .02083 .02222 .02500 .02778 .02917 .03055 1 2 .03333.03611 .03750 .03889 .04167 .04444 .04583 .04722 2 3 .05000 .05278 .05417 .05556 .05833 .06111 .06250 .06389 4 .06667 .06944 .07083 .07222 .07500 .07778 .07917 .08056 .08333 .08611 .08750 .08889 .09167 .09-444 .09583 .09722 5 .10000 .10278 .10417 .10556 .10833 .11111 .11250 .11389 .11667 .11941 .12083 .12222 .12500 .12778 .12917 .13056 8 .13333 .13611 .13750 .13889 .14167 .14444 .14583 .14722 8 9 .15000 .15278 .15417 .15556 .15833 .16111 .16250 .16389 9 10 .16667 .16944 .17083 .17222 .17500 .17778 .17917 .18056 10 11 .18333 .18611 .18750 .18889 .19167 19411 .19583 .19722 11 12 .20000 .20278 .20417 20556 .20833 .21111 .21250 .21389 12 13 .21667 .21944 .22083 22222 22500 .22778 22917 .23056 13 14 .23333 23611 .23750 .23889 .24167 .21111 .24583 .24722 14 15 .25000 .25278 .25417 .25556 25833 .26111 .26250 .26389 15 16 .26667 .26944 .27083 .27222 27500 .27778 .27917 .28056 16 17 28333 28611 28750 28889 .29167 .29414 .29583 .29722 17 18 30000 .30278 .30417 .30556 30833 .31111 .31250 .31389 18 19 31667 .319-14 32083 32222 .32500 .32778 .32917 .33056 20 .33333 .33611 .33750 .33889 .31167 .34444 .34583 .34722 20 21 .35000 .35278 35417 .35556 35833 .36111 .36250 .36389 21 36667 .36941 .37083 .87222 .37500 .37778 .37917 .38056 22 .38333 .38611 .38750 .38889 .39167 .39444 .40000 .40278 .40417 .40556 .40833 D 41667 .41944 .42083 .42222 .42500 .43333 .43611 .43750 .43889 .44167 .39583 .39722 23 .41111 .41250 .41389 24 .42778 .42917 .43056 25 .41144 .44583 .44722 26 45000 .45278 .45417 .45556 .45833 .46111 46250 .46389 28 46667 46944 .47083 .47222 .47500 .47778 .47917 48056 28 29 48333 .48611 48750 .48889 .49167 .49444 .49583 .49722 29 30 .50000 .50278 .50417 .50556 .50833 .51111 .51250 .51389 30 31 51667 .51941 .52083 .52222 .52500 52778 .52917 .53056 31 32 .53333 53611 .53750 .53889 .54167 54444 .54583 .54722 32 222 72****AXRA X8 19 33 .55000 .55278 .55417 .55556 .55833 .56111 .56250 .56389 33 31 .56667 .56941 .57083 .57222 .57500 57778 .57917 .58056 34 36 35 .58333 .58611 .58750 .60000 .60278 .60417 .58889 .59167 .59444 .59583 .59722 35 .60556 .60833 .61111 .61250 .61389 36 41 42 43 41 5889 39¬ 37 .61667 .61944 .62083 .62222 .62500 .62778 .62917 .63056 38 .63333 .63611 .63750 .63889 .64167 64444 .64583,64722 39 .65000 .65278 .65417 .65556 .65833 .66111 66250 .66389 39 40 .66667 .66911 .67083 .67222 .67500 .67778 .67917 .68056 40 68333 .68611 .68750 .68889 .69167 .69444 69583 .69722 70000 .70278 .70117 .70556 70833 71111 .71250 .71389 42 .71667 .71944 72083 72222 72500 .72778 .72917 .73056 73333 .73611 .73750 .78889 45 75000 .75278 75417 75556 75833 • D • 46 .76667 .76944 77083 .74167 .74444 .76111 .77222 .77500 .77778 .74583 .74722 44 .76250 .76389 45 .77917.78056 46 47 .78333 .78611 .78750 .78889 .79167 79111 .79583.79722 47 48 .80000 .80278 .80417 .80556 .80833 .81111 .81250 .81389 48 49 .81667 .81944 .82083 .82222 .82500 .82778 .82917 .83056 49 50 .83333 .83611 .83750 .83889 .84167 .84444 .84583 .84722 50 51 .85000 85278 .85417 .85556 .85833 86111 .86250.86389 51 52 .86667 .86911 .87083 .87222 53 .88333 .88611 .88750 88889 51 .90000 .90278 .90417 .90556 • .87500 .87778 .89167 .89444 90833 .91111 .91250 .87917 .89588 .88056 .89722 52 *O TAARA*£*** *∞ ∞ ∞ ∞ 37 38 41 53 .91389 54 55 .91667 .91941 .92083 .92222 • 56 .93333 .93611 .93750 93889 • • 57 .95000 .95278 .95417 .95556 .92500 .92778 .92917 94167 94444 .94583 94722 56 .95833 .96111 .93056 55 .96250 .96389 57 58 .96667 .96944 .97083 .97222 59 .98333 .98611 .98750 .98889 .97500 .97778 .99167 .99414 .97917 .98056 58 .99583 .99722 59 னே 0" 10" 15" 20" 30" 40" 45" 50" 204 TABLE VIII.-SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1 1 1 1.0000000 1.0000000 1.000000000 4 8 1.4142136 1.2599210 .500000000 9 27 1.7320508 1.4422496 .333333333 16 64 2.0000000 1.5874011 250000000 25 125 2.2360680 1.7099759 .200000000 36 216 2 4494897 1.8171206 .166666667 49 343 2.6457513 1.9129312 .142857143 8 64 512 2.8284271 2.0000000 .125000000 9 81 729 3.0000000 2.0800837 .111111111 10 100 1000 3.1622777 2.1514347 .100000000 11 121 1331 3.3166248 2.2239801 .090909091 12 144 1728 3.4041016 2.2894286 .083333333 13 169 2197 3.6055513 2.3513347 .076923077 14 196 2744 3.7416574 2.4101422 .071428571 15 225 3375 3.8729833 2.4662121 .066666667 16 256 4096 4.0000000 2.5198421 .062500000 17 289 4913 4.1231056 2.5712816 .058823529 18 324 5832 4.2426407 2.6207414 055555556 19 361 6859 4.3588989 2.6684016 .052631579 20 400 8000 4.4721360 2.7144177 .050000000 21 441 9261 4.5825757 2.7589243 .047619048 484 10648 4.6904158 2.8020393 .045454545 529 12167 4.7958315 2.8438670 .043478261 576 13824 4.8989795 2.8844991 041666667 625 15625 5.0000000 2.9240177 .040000000 676 17576 5.0990195 2.9624960 .038461538 27 729 19683 5.1961524 3.0000000 .037037037 28 784 21952 5.2915026 3.0365889 .035714286 29 841 24389 5.3851648 3.0723168 .034482759 30 900 27000 5.4772256 3.1072325 .033333333 31 961 29791 5.5677644 3.1413806 .032258065 32 1024 32768 5.6568542 3.1748021 .031250000 33 1089 35937 5.7445626 3.2075343 030303030 34 1156 39304 5.8309519 3.2396118 .029411765 35 1225 42875 5.9160798 3.2710663 .028571429 36 1296 46656 6.0000000 3.3019272 .027777778 37 38 41 43 45 46 47 51 588 GER#99599 ANSRHHANKS 853 1369 50653 6.0827625 3.3322218 .027027027 1444 54872 6.1644140 3.3619754 .026315789 39 1521 59319 6.2449980 3.3912114 .025641026 40 1600 64000 6.3245553 3.4199519 .025000000 1681 68921 6.4031242 3.4482172 .024390244 42 1764 74088 6.4807407 3.4760266 .023809524 1849 79507 6.5574385 3.5033981 .023255814 44 1936 85184 6.6332496 3.5303483 .022727273 2025 91125 6.7082039 3.5568933 .022222222 2116 97336 6.7823300 3.5830479 .021739130 2209 103823 6.8556546 3.6088261 .021276600 48 2304 110592 6.9282032 3.6342411 .020833333 49 2401 117649 7.0000000 3.6593057 .020408163 50 2500 125000 7.0710678 3.6840314 .020000000 2601 132651 7.1414284 3.7084298 .019607843 52 2704 140608 7.2111026 3.7325111 .019230769 53 2809 148877 7.2801099 3.7562858 .018867925 51 2916 157464 7.3484692 3.7797631 .018518519 55 3025 166375 7.4161985 3.8029525 .018181818 56 3136 175616 7.4833148 3.8258624 .017857143 57 3249 185193 7.5498344 3.8485011 .0175-43860 58 3364 195112 7.6157731 3.8708766 .017241379 59 3481 205379 7.6811457 3.8929965 .016949153 60 3600 216000 7.7459667 3.9148676 .016666667 61 3721 226981 7.8102497 3.9364972 .016393443 62 3844 238328 7.8740079 3.9578915 .016129032 265 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 100 101 102 103 8188638 RECCFeetee 35ZBZ88688 REKAR 823 3969 250047 7.9372539 3.9790571 .015873016 4096 262144 8.0000000 4.0000000 .015625000 4225 274625 8.0622577 4.0207256 015384615 4356 287496 8.1240384 4.0412401 .015151515 4489 300763 8.1853528 4.0615480 014925373 • 4624 31-4432 8.2462113 4761 328509 8.3066239 4.0816551 4.1015661 .014705882 .014492754 4900 343000 8.3666003 4.1212853 014285714 • 5041 357911 8.4261498 4.1408178 .014084507 5184 373248 8.4852814 4.1601676 .013888889 5329 389017 8.5440037 4.1793390 .013698630 5476 405224 8.6023253 4.1983364 .013513514 5625 421875 8.6602540 4.2171633 .013333333 5776 438976 8.7177979 4.2358236 .013157895 5920 456533 8.7749641 4.2543210 .012987013 6084 474552 8.8317609 4.2726580 .012820513 6241 493039 8.8881944 4.2908404 .012658228 6400 512000 8.9442719 4.3088695 .012500000 6561 531441 9.0000000 4.3267487 .012345679 6724 551368 9.0553851 4.3114815 .012195122 6889 571787 9.1104336 4.3620707 .012048193 7056 592704 9.1651514 4.3795191 .011904762 225 614125 9.2195445 4.3968296 .01176-1706 7396 636056 9.2736185 4.4140049 .011627907 7569 658503 9.3273791 4.4310476 .011494253 7744 681472 9.3808315 4.4479602 .011363636 7921 704969 9.4339811 4.4647451 .011235955 90 8100 729000 9.4868330 4.4814047 .011111111 8281 753571 9.5393920 4.4979414 .010989011 8464 778688 9.5916630 4.5143574 .010809565 8649 804357 9.6436508 4.5306549 .010752688 8836 830584 9.6953597 4.5468359 .010638298 90:25 857375 9.7467943 4.5629026 .010526316 9216 884736 9.7979590 4.5788570 .010416667 9409 912673 9.8488578 4.5947009 .010309278 9604 941192 9.8994949 4.6104363 .010204082 9801 970299 9.9498714 4.6260650 .010101010 10000 1000000 10.0000000 4.6415888 .010000000 10:201 1030301 10.0498756 4.6570095 009900990 10404 1061208 10.0995049 4.6723287 009803922 10609 1092727 10.1188916 4.6875482 009708738 104 10816 1124864 10.1980390 4.7026694 .009615385 105 11025 1157625 10.2469508 4.7176940 .009523810 106 11236 1191016 10.2956301 4.7326235 .009433962 107 11449 12250-13 10.3440804 4.7474594 .009345794 108 11664 1259712 10.3923048 4.7622032 .003259259 109 11881 1295029 10.4103065 4.7768562 009174312 110 12100 1331000 10.4880885 4.7914199 .009090909 111 12321 1367631 10.5356538 4.8058955 .009009009 112 12544 1404928 10.5830052 4.8202845 .008928571 113 12769 1442897 10.0301458 4.8345881 0088-49558 114 12996 1481544 10.0770783 4.8188076 .008771930 115 13225 1520875 10.7238053 4.8629142 .008695652 116 13456 1560896 10.7703296 4.8769990 .008620690 117 13689 1601613 10.8166538 4.8909732 .008547009 118 13924 1643032 10.8627805 4.9048681 .008474576 119 14161 1685159 10.9087121 4.9186847 .008403361 120 14400 1728000 10.9544512 4.9324242 .008333333 121 14641 1771561 11.00 0000 4.9460874 .008264463 1.22 14884 1815848 11.0458610 4.9596757 .008196721 123 15129 1860867 11.0905365 4.9731898 .008130081 124 15976 1906624 11.1335287 4.9866310 .008064516 266 TABLE VIII.- Continued. ! No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 125 15625 1953125 11.1803399 5.0000000 .008000000 126 15876 2000376 11.2249722 5.0132979 .007936508 127 16129 2048383 11.269427 5.0265257 .007874016 128 16384 2097152 11.3137085 5.0396842 .007812500 129 16641 2146689 11.3578167 5.0527743 .007751938 130 16900 2197000 11.4017543 5.0657970 .007692308 131 17161 2248091 11.4455231 5.0787531 .007633588 132 17424 2299968 11.4891253 5.0916434 .007575758 133 17689 2352637 11.5325626 5.1044687 .007518797 134 17956 2406104 11.5758369 5.1172290 .007462687 135 18225 2460375 11.6189500 5.1299278 007407407 136 18496 2515456 11.6619038 5.1425632 .007352941 137 18769 2571353 11.7046999 5.1551367 .007299270 138 19041 2628072 11.7473401 5.1676493 .007246377 139 19321 2685619 11.7898261 5.1801015 .007194245 140 19600 2744000 11.8321596 5.1924941 .007142857 141 19881 2803221 11.8743421 5.2048279 .007092199 142 20104 2863288 11.9163753 5.2171034 .007042254 143 20449 2924207 11.9582607 5.2293215 .006993007 144 20736 2985984 12.0000000 5.2414828 .006944444 145 21025 3048625 12.0415946 5.2535879 .006896552 146 21316 3112136 12.0830460 5.2656374 006849315 147 21609 3176523 12.1243557 5.2776321 .006802721 148 21904 3241792 12.1655251 5.2895725 006756757 149 22201 3307949 12.2065556 5.3014592 .006711409 150 22500 3375000 12.2474487 5.3132928 .006666667 151 22801 3442951 12.2882057 5.3250740 .006622517 152 23104 3511808 12.3288280 5.3368033 .006578947 153 23409 3581577 12.3693169 5.3484812 .006535948 151 23716 3652264 12.4096736 5.3601084 .006493506 155 24025 3723875 12.4498996 5.3716854 .006451613 156 24330 3796416 12.4899960 5.3832126 .006410256 157 24649 3869893 12.5299641 5 3946907 .006369427 158 24964 3944312 12.5698051 5.4061202 006329114 · 159 25281 4019679 12.6095202 5.4175015 .006289308 160 25600 4096000 12.6491106 5.4288352 .006250000 161 25921 4173281 12.0885775 5.4401218 C06211180 162 26244 4251528 12.7279221 5.4513618 .006172840 163 26569 4330747 12.7671453 5.1625556 .006134969 164 26896 4410944 12.8062485 5.4737037 .006097561 165 27225 4492125 12.8452326 5.4848066 .006060606 166 27556 4571296 12.8840987 5.4958647 .006024096 167 27889 4057403 12.9228480 5.5068784 .005988024 168 28224 4741632 12.9614814 5.5178484 .005952281 169 28561 4826809 13.0000000 5.5287748 .005917169 170 28900 4913000 13.0394048 5.5396583 .005882353 171 29241 5000211 13.0766968 5.5504991 .005847953 172 29584 5088448 13.1148770 5.5612978 .005813953 173 29029 5177717 13.1529464 5.5720546 005780347 174 30276 5268024 13.1909060 5.5827702 .005747126 175 30625 5359375 13.2287566 5.5934447 .005714286 176 30976 5451776 13.2664992 5.6040787 .005681818 177 81329 5545233 13.3041347 5.6146724 .005649718 178 31684 5639752 13.3416641 5.6252263 .005617978 179 32041 5735339 13.3790882 5.6357408 .005586592 180 32400 5832000 13.4164079 5.6462162 .005555556 181 32761 5929741 13.4536240 5.6566528 .005524862 182 33124 6028568 13.4907376 5.6670511 .005494505 183 33489 6128487 13.5277493 5 6774114 .005464481 184 33856 6229504 13.5646600 5.6877340 .005434783 185 34225 6331625 13.6014705 5.6980192 .005405405 186 34596 6434856 13.6381817 5.7082675 .005376344 267 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 187 34969 6539203 13.6747943 5.7184791 .005347594 188 35344 6644672 13.7113092 5.7286543 .005319149 189 35721 6751269 13.7477271 5.7387936 .005291005 190 36100 6859000 13.7840488 5.7488971 .005263158 191 36481 6967871 13.8202750 5.7589652 .005235602 192 36864 7077888 13.8564065 5.7689982 .005208333 193 37249 7189057 13.8924440 5.7789966 .005181347 194 37636 7301384 13.9283883 5.7889604 .005154639 195 38025 7414875 13.9642400 5.7988900 .005128205 196 38416 7529536 14.0000000 5.8087857 .005102041 197 38809 7645373 14.0356688 5.8186479 .005076142 198 39204 7762392 14.0712473 5.8284767 .005050505 199 39601 7880599 14.1067360 5.8382725 .005025126 200 40000 8000000 14.1421356 5.8480355 .005000000 201 40401 8120601 14.1774469 5.8577660 .004975124 202 40804 8242408 14.2126704 5.8674643 .004950495 203 41209 8365427 14.2478068 5.8771307 .004926108 204 41616 8489664 14.2828569 5.8867053 .004901961 205 42025 8615125 14.3178211 5.8963685 .004878049 206 42436 8741816 14.3527001 5.9059406 .004854369 207 42849 8869743 14.3874946 5.9154817 .004830918 208 43264 8998912 14.4222051 5.9249921 .004807692 209 43681 9129329 14.4568323 5.9344721 .004784689 210 44100 9261000 14.4913767 5.9439220 .004761905 211 44521 9393931 14.5258390 5.9533418 .004739336 212 44944 9528128 14.5602198 5.9627320 .004716981 213 45369 9663597 14.5945195 5.9720926 .004694836 214 45796 9800344 14.6287388 5.9814240 .004672897 215 46225 9938875 14.6628783 5.9907264 .004651163 216 46656 10077696 14.6969385 6.0000000 004629630 217 47089 10218313 14.7309199 6.0092450 004608295 218 47524 10360232 14.7648231 6.0184617 .004587156 219 47961 10503459 14.7986486 6.0276502 .004566210 220 48400 10648000 14.8323970 6.0368107 004545455 221 48841 10793861 14.8660687 6.0459435 004524887 222 49284 10941048 14.8996644 6.0550489 .004504505 223 49729 11089567 14.9331845 6.0641270 .004484305 224 50176 11239424 14.9666295 6.0731779 .004464286 225 50625 11390625 15.0000000 6.0822020 .004444444 226 51076 11543176 15.0332964 6.0911994 .004424779 227 51529 11697083 15.0065192 6.1001702 .004405286 228 51984 11852352 15.0996689 6.1091147 .004385965 229 52441 12008989 15.1327460 6.1180332 .004366812 230 52900 12167000 15.1657509 6.1269257 .004347826 231 53361 12326391 15.1986842 6.1357924 .004329004 232 53824 12487168 15.2315462 6.1446337 .004310345 233 54289 12649337 15.2643375 6.1534495 .004291845 234 54756 12812904 15.2970585 6.1622401 .004273504 235 55225 12977875 15.3297097 6.1710058 .004255319 236 55696 13144256 15.3622915 6.1797466 .004237288 237 56169 13312053 15.3948043 6.1884628 .004219409 238 56644 13481272 15.4272486 6.1971544 .004201681 239 57121 13651919 15.4596248 6.2058218 .004184100 240 57600 13824000 15.4919334 6.2144650 *.004166667 241 58081 13997521 15.5241747 6.2230843 .004149378 242 58564 14172488 15.5563492 6.2316797 .004132231 243 59049 14348907 15.5884573 6.2402515 .004115226 244 59536 14526784 15.6204994 6.2487998 .004098361 2-15 60025 14706125 15.6524758 6.2573248 004081633 246 60516 14886936 15.6843871 6.2658266 .004065041 247 61009 15069223 15.7162336 6.2743054 .004048583 248 61504 15252992 15.7480157 6.2827613 .004032258 268 TABLE VIII.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 249 62001 15438249 15.7797338 6.2911946 .004016064 250 62500 15625000 15.8113883 6.2996053 .004000000 251 63001 15813251 15.8429795 6.3079935 .00398406+ 252 63504 16003008 15.8745079 6.3163596 003968251 253 64009 16194277 15.9059737 6.3247035 .003952569 251 64516 16387064 15.9373775 6.3330256 .003937008 650:25 16581375 15.9687194 6.3413257 003921569 65536 16777216 16.0000000 6.3496042 .003906250 257 660 19 16974593 16.0312195 6.3578611 .003891051 258 6656 17173512 16.0623784 6.3660968 .003875969 259 67081 17373979 16.0934769 6.3743111 003861004 260 67600 17576000 16.1245155 6.3825043 .003846151 261 68121 17779581 16.1554944 6.3906765 .003831418 262 68644 17984728 16.1864141 6.3988279 003816794 263 69169 18191447 16.2172747 6.4069585 .003802281 264 69696 18399744 16.2480768 6.4150687 .003787879 265 70225 18609625 16.2788206 6.4231583 .003773585 266 70756 18821096 16.3095061 6.4312276 .003759398 267 71289 19034163 16.3401346 6.4392767 .003745318 268 71824 19248832 16.3707055 6.4473057 .003731343 269 72361 19465109 16.4012195 6.4553148 .003717472 270 72900 19683000 16.4316767 6.4633041 .003703704 271 73441 19902511 16.4620776 6.4712736 .003090037 272 7398£ 20123648 16.4924225 6.4792236 .003676471 273 74529 20346417 16.5227116 6.4871511 .003663904 274 75076 20570824 16.5529454 6.4950653 .003649635 275 75625 20796875 16.5831240 6.5029572 003636364 276 76176 21024576 16.6132477 6.5108300 .003623188 277 76729 21253933 16.6433170 6.5186839 .003610108 278 77284 21484952 16.6733320 6.5265189 .003597122 279 77841 21717639 16.7032931 6.5343351 .003584229 280 78400 21952000 16.7332005 6.5421326 .003571429 281 78961 22188041 16.7630546 6.5499116 .003558719 282 79524 22425768 16.7928556 6.5576722 .003546099 283 80089 22665187 16.8226038 6.5654144 .003533569 281 80656 22906301 16.8522995 6.5731385 .003521127 285 81225 23149125 16.8819430 6.5808443 .003508772 286 81796 23393656 16.9115345 6.5885323 .003496503 287 82369 23639903 16.9410743 6.5962023 .003484321 288 82944 23887872 16.9705627 6.6038545 .003472222 289 83521 24137569 17.0000000 6.6114890 .003460208 290 81100 24389000 17.0293864 6.6191060 .003448276 291 84681 24642171 17.0587221 6.6267054 .003436426 292 85264 24897088 17.0880075 6.6342874 003424658 293 85849 25153757 17.1172428 6.6418522 .003412969 294 86-136 25412184 17.1464282 6.6493998 .003401361 295 87025 25672375 17.1755640 6.6569302 .003389831 296 87616 25934336 17.2046505 6.6644437 .003378378 297 88209 26198073 17.2336879 6.6719403 .003367003 298 88801 26463592 17.2626765 6.6794200 .003355705 299 89401 26730899 17.2916165 6.6868831 .003344482 300 90000 27000000 17.3205081 6.6943295 .003333333 301 90601 27270901 17.3493516 6.7017593 .003322259 302 91204 27543608 17.3781472 6 7091729 .003311258 303 91809 27818127 17.4068952 6.7165700 .003300330 304 92416 28094164 17.4355958 6.7239508 .003289474 305 930:25 28372625 17.4642492 6.7313155 .003278689 306 93636 28652616 17.4928557 6.7386641 .003267974 307 94219 28934443 17.5214155 6.7459967 .003257329 308 94864 29218112 17.5499288 6.7533134 .003246753 309 95481 29503629 17.5783958 6.7606143 .003236246 310 96100 29791000 17.6068169 6.7678995 .003225806 269 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 311 96721 30080231 17.6351921 6.7751690 .003215434 312 97344 30371328 17.6635217 6.7824229 .003205128 313 97969 30664297 17.6918060 6.7896613 .003194888 314 98396 30959144 17.7200451 6.7968844 .003184713 315 99225 31255875 17.7482393 6.8049921 .003174603 316 99856 31554496 17.7763888 6.8112847 .003164557 317 100189 31855013 17.8044938 6.8184620 .003154574 318 101124 32157432 17.8825545 6.8256242 .003144654 319 101761 32461759 17.8605711 6.8327714 .003134796 320 102400 32768000 17.8885438 6.8399037 .003125000 321 103041 33076161 17.9164729 6.8470213 003115265 3:22 103684 33386248 17.9413584 6 8541240 .003105590 323 104329 33698207 17.9722008 6 8612120 .003095975 324 104976 34012224 18.0000000 6.8682855 .003086420 325 105625 34328125 18.0277564 6.8753443 .003076923 326 106276 34645976 18.0554701 6.8823888 .003067485 327 106929 34965783 18.0831413 6.8894188 .003058104 328 107584 35287552 18.1107703 6.896-43-45 .003048780 329 108241 35611289 18.1383571 6.9034359 .003039514 330 108900 35937000 18.1659021 6.9104232 .003030303 331 109561 36264691 18.1934034 6.9173964 .003021148 332 110224 36594368 18.2208672 6.9243556 .003012048 333 110889 36926037 18.2482876 6.9313008 .003003003 334 111556 37259704 18.2756669 6.9382321 .002994012 335 112225 37595375 18.3030052 6.9451496 .002985075 336 112896 37933056 18.3303028 6.9520533 .002976190 337 113569 38272753 18.3575598 6.9589434 .002967359 338 114244 38614472 18.3847763 6.9658198 .002958580 339 111921 38958219 18.4119526 6.9726826 .002949853 310 115600 39304000 18.4390889 6.9795321 .002941176 341 116281 39651821 18.4661853 6.9863681 .002932551 342 116961 40001688 18.4932420 6.9931906 .002923977 343 117649 40353607 18.5202592 7.0000000 .002915152 344 118336 40707584 18.5472370 7.0067962 002906977 1 345 119025 41063625 18.5741756 7 0135791 .002898551 346 119716 41421736 18.6010752 7.0203490 .002890173 347 120409 41781923 18.6279360 7.0271058 .002881844 348 121104 42144192 18.6547581 7.0338497 .002873563 349 121801 42508549 18.6815417 7.0405806 .002805330 350 122500 42875000 18.7082869 7.0472987 .002857143 351 123201 43243551 18.7349940 7.0540041 002819003 352 123904 43614208 18.7616630 7.0606967 002840909 353 124609 43986977 18.7882942 7.0673767 .002832861 354 125316 44361864 18.8148877 7.0740440 .002824859 355 126025 44738875 18.841437 7.0806988 .002816901 356 126736 45118016 18.8679623 7.0873411 .002808989 357 127449 45499293 18.8941436 7.0939709 .002801120 358 128164 45882712 18.9208879 7.1005485 .002793296 359 128881 46268279 18.9472953 7.1071937 .002785515 360 129600 46656000 18.9736660 7.1137866 .002777778 361 130321 47045881 19.0000000 7.1203674 .002770083 362 131041 47437928 19.0262976 7.1269360 .002762431 363 131769 47832147 19.0525589 7.1334925 .002754821 364 132496 48228544 19.0787840 7.1400370 .002747253 365 133225 48627125 19.1049732 7.1165695 .002739726 366 133956 49027896 19.1311265 7.1530901 .002732240 367 134689 49430863 19.1572441 7.1595988 .002724796 368 135424 49836032 19.1833261 7.1660957 .002717391 369 136161 502-18409 19.2093727 7.1725809 .002710027 370 136900 50653000 19.2353811 7.1790544 .002702703 371 187641 51061811 19.2613603 7.1855162 .002695418 372 138381 51478848 19.2873015 7.1919663 .002688172 270 TABLE VIII.—Continued. No. Squares. Cubes. Square Roots. 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 53582633 19.4164878 7.2240450 .002652520 378 142884 54010152 19.4422221 7.2304268 .002645505 379 143641 54439939 19.4679223 7.2367972 .002638522 380 144400 54872000 19.4935887 7.2431565 .002631579 381 145161 55300341 19.5192213 7.2495045 .002624672 382 143924 55742968 19.5448203 7.2558415 .002617801 383 146689 56181887 19.5703858 7.2021675 .002610966 381 147456 56623104 19.5959179 7.2684824 .002604167 385 148225 57066625 19.6214169 7.2747864 .002597403 386 148996 57512456 19.6468827 7.2810794 002590674 387 149769 57900603 19.6723156 7.2873617 .002583979 388 150544 58411072 19.6977156 7.2936330 .002577820 389 151321 58863869 19.7230829 7.2998936 .002570594 390 152100 59319000 19.7484177 7.3061436 002564103 391 152881 59776471 19.7737199 7.3123828 .0025575-45 392 153664 60236288 19.7989899 7.3186114 002551020 393 154419 60698457 19.8242276 7.3248295 .002544529 394 155236 61162984 19.8494332 7.3310369 .002538071 395 156025 61629875 19.8746069 7.3372339 .002531646 396 156816 62099136 19.8997487 7.3484205 .00:525253 397 157609 62570773 19.9248588 7.3495966 002518892 398 158404 63044792 19.9499373 7.3557624 002512563 399 159201 63521199 19.9749544 7.3619178 002506266 400 160000 64000000 20.0000000 7.3680630 .002500000 401 160801 64481201 20.0249844 7.3741979 .002493766 402 161604 64964808 20.0499377 7.3803227 .002487562 403 162409 65450827 20.0748599 7.3864373 002481390 404 163216 65939264 20.0997512 7.3925418 .002475248 · 405 164025 66430125 20.1246118 406 164836 66923416 20.1494417 7.3986363 7.4047206 002469136 • .002463054 • 407 165649 67419143 20.1742410 7.4107950 .002457002 408 166464 67917312 20.1990099 7.4168595 .002450980 409 167281 68417929 20.2237484 7.4229142 .00244-1988 410 168100 68921000 20.2184567 7.4289589 .002439024 411 168921 69426531 20.2731349 7.4349938 .002433090 412 169744 69934528 20.2977831 7.4410189 .002427184 413 170569 70444997 20.3224014 7.4470342 .002421308 414 171396 70957944 20.3469899 7.4530399 .002415459 415 172225 71473375 20.3715488 7.4590359 .002409639 416 173056 71991296 20.3960781 7.4650223 002403846 417 173889 72511718 20.1205779 7.4709991 .002398082 418 174724 73034632 20.4450483 7.4769664 .002392344 419 175561 78560059 20.4694895 7.4829242 .002386635 420 176-100 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 5912603 7.5125715 002358491 425 180025 76765625 20.6155281 7.5184780 .002352941 426 181476 77308776 20.6397674 7.5243652 .002347418 427 182329 77854483 20.6639783 7.5302482 .002341920 428 183184 78402752 20.6881609 7.5361221 .002336419 429 184041 78953589 20.7123152 7.5419867 .002331002 430 184900 79507000 20.7364414 7.5478423 .002325581 431 185761 80062991 20.7605395 7.5536888 .002320186 432 186624 80621568 20.7846097 7.5595203 .002314815 433 187-189 81182737 20 8086520 75653548 .002309469 434 188356 81746504 20 8326667 7.5711743 .002304147 271 TABLE VIII.-Continued. 1 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769849 002298851 436 190096 82881856 20.8806130 7.5827865 002293578 B7 190969 83453453 20.9045450 7.5885793 .002288330 438 1918H 84027672 20.9281495 7.5943633 002283105 139 192721 84604519 20.9523268 7.6001385 .002277904 440 193600 85184000 20.9761770 7.6059049 002272727 • 141 194481 85766121 21.0000000 7.6116626 .002267574 142 195364 86350888 21.0237960 7.6174116 .002262443 143 196249 86938307 21.0475652 7.6231519 002257336 111 197136 87528384 21.0713075 7.6288837 .002252252 415 198025 88121125 21.0950231 7.6346067 .002247191 446 198916 88716536 21.1187121 7.6403213 .002242152 117 199809 89314623 21.1423745 7.6460272 002237136 • 448 200704 89915392 21.1660105 7.6517247 449 201601 90518849 21.1896201 7.6574133 450 202500 91125000 21.2132034 7.6630943 002232143 .002227171 .002222222 451 203401 91733851 21.2367606 7.6687665 002217295 • 452 204304 92345408. 21.2602916 7.6744303 002212389 453 205209 92959677 21.2837967 7.6800857 .002207506 454 206116 93576664 21.3072758 7.6857328 .002202643 455 207025 94196375 21.3307290 7.6913717 .002197802 456 207936 94818816 21.3541565 7.6970023 .002192982 457 2088-19 95143993 21.3775583 7.7026246 .002188184 458 209764 96071912 21.4009346 7.7082388 .002183406 459 210681 96702579 21.4242853 7.7138148 .002178649 460 211600 97336000 21.4476106 7.719-4426 .002173913 461 212521 97972181 21.4709106 7.7250325 .002169197 462 213444 98611128 21.4941853 7.7306141 .002164502 463 214369 99252847 21.5174348 7.7361877 .002159827 464 215296 99897344 21.5406592 7.7417532 .002155172 465 216225 100544625 21.5638587 7.7473109 .002150538 166 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 7.7804904 .002123142 472 222784 105154048 21.7255610 7.7859928 .002118644 473 223729 105823817 21.7485632 7.7914875 .002114165 474 224676 106496 121 21.7715411 7.7969745 .002109705 475 225625 107171875 21.7941947 7.8024538 .002105263 476 226576 107850176 21.8174242 7.8079254 .002100840 477 227529 108531333 21.8403297 7.8133892 .002096-436 478 228484 109215352 21 8632111 7.8188456 002092050 479 229411 109902239 21.8860686 7.8242942 002087683 480 230400 110592000 21.9089023 7.8297353 .002088333 481 231361 111284641 21.9317122 7.8351688 .002079002 482 484 485 232324 483 233289 112678587 234256 235225 114084125 111980168 21.9544984 7.8405949 .002074689 21.9772610 7.8160134 .002070393 113379904 22.0000000 7.8514211 .002066116 22.0227155 7.8568281 .002061856 486 236196 114791256 22.0454077 7.8622242 .002057613 487 237169 115501303 22.0680765 7.8676130 .002053388 488 238144 116214272 22.0907220 7.8729941 002049180 489 239121 116930169 22.1133444 7.8783681 002044990 490 210100 117649000 22.1359436 7.8837352 .002040816 493 491 241081 118370771 492 242064 243049 22.1585198 7.8890946 002036660 119095488 22.1810730 7.8944468 002032520 119823157 22.2036033 7.8997917 002028398 494 244030 120553784 22.2261108 7.9051294 .002024291 495 245025 121287375 22.2485955 7.9104599 002020202 496 246016 122023936 22.2710575 7.9157832 .002010129 272 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 247009 122763473 22.2934968 7.9210994 .002012072 498 248004 123505992 22.3159136 7.9264085 .002008032 499 219001 124251499 22.3383079 7.9317101 .002004008 500 250000 125000000 22.3606798 7.9370053 .002000000 501 251001 125751501 22.3830293 .9422931 .001996008 502 252304 126506008 22.4053565 7.9475739 .001992032 503 253009 127263527 22.4276615 7.9528177 .001988072 501 254016 128024064 22.1499443 7.9581144 .001984127 505 255025 128787625 22.4722051 7.9633743 .001980198 506 256036 129554216 22.4944438 7.9686271 .001976285 507 257049 130323843 22.5166605 7.9738731 .001972387 508 258064 131096512 22.5388553 7.9791122 .001968504 509 259081 131872229 22.5610283 7.9813114 .001961637 510 260100 132651000 22.5831796 7.9895697 .001960784 511 261121 133432831 22.6053091 7.9947883 .001956947 512 262144 134217728 22.6274170 8.0000000 .001953125 513 263109 135005697 22.6495033 8.0052049 .001949318 514 264196 135796744 22.6715681 8.0104032 .001945525 515 265225 136590875 22.6936114 8.0155946 .001941748 516 266256 137388096 22.7156334 8.0207794 .001937984 517 267289 138188413 22.7376340 8.0259574 .001934236 518 268324 138991832 22.7596134 8.0311287 .001930502 519 269361 139798359 22.7815715 8.0362935 .001926782 520 270100 140608000 22.8035085 8.0414515 .001923077 521 271441 141420761 22.8254244 8.0466030 .001919386 522 272484 142236648 22.8473193 8.0517479 .001915709 523 273529 143055667 22.8691933 8.0568862 .001912046 524 274576 143877824 22.8910463 8.0620180 .001908397 525 275625 144703125 22.9128785 8.0671432 .001904762 526 276676 145531576 22.9346899 8.0722620 .001901141 527 277729 146363183 22.9564806 8.0773743 .001897533 528 278784 147197952 22.9782506 8.0824800 .001893939 5:29 279841 118035889 23.0000000 8.0875794 .001890359 530 280900 118877000 23.0217289 8.0926723 .001886792 531 281961 149721291 23.0434372 8.0977589 .001883239 532 283024 150568768 23.0651252 8.1028390 .001879699 533 284089 151419437 23.0867928 8.1079128 .001876173 534 285156 152273304 23.1084400 8.1129803 .001872659 535 286225 153130375 23.1300670 8.1180414 .001869159 536 287296 153990656 23.1516738 8.1230962 .001865672 537 288369 154854153 23.1732605 8.1281447 .001862197 538 289444 155720872 23.1948270 8.1331870 .001858736 539 290521 156590819 23.2163735 8.1382230 .001855288 540 291600 157464000 23.2379001 8.1432529 001851852 541 $92681 158340421 23.2594067 8.1482765 .001848429 542 293764 159220088 23.2808935 8.1532939 .001845018 543 294849 160103007 23.3023004 8.1583051 .001811621 514 295936 160989184 23.3238076 8.1633102 .001838235 545 297025 161878625 23.3452351 8.1683092 .001834862 516 298116 162771336 23.3666429 8.1733020 .001831502 547 299209 163667323 23.3880311 8.1782888 .001828154 548 300304 164566592 23.1093998 8.1832695 .001824818 5-19 301401 165469149 23.4307490 8.1882441 .001821494 550 302500 166375000 23.4520788 8.1932127 .001818182 551 303601 167284151 23.4733892 8.1981752 .001814882 552 301704 168196608 23.4946802 8.2031319 .001811594 553 305809 169112377 23.5159520 8.2080825 .001808318 551 306916 170031464 23.5372046 8.2130271 .001805054 558 555 308025 170953875 556 309136 557 310249 172808693 311364 173741112 23.5584380 8.2179657 .001801802 171879616 23.5796522 8.2228985 .001798561 23.6008474 8.2278254 .001795332 23.6220236 8.2327403 .001792115 273 TABLE VIII.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 174676879 23.6431808 8.2376614 .001788909 560 313600 175616000 23.6643191 8.2425706 .001785714 561 314721 176558481 23.6854386 8.2474740 .001782531 562 315844 177504328 23.7065392 8.2523715 .001779359 563 316969 178453547 23.7276210 8.2572633 .001776199 564 318096 179406144 23.7486842 8.2621492 .001773050 565 319225 180362125 23.7697286 8.2670294 .001769912 566 320356 181321496 23.7907545 8.2719039 .001766784 567 321489 182284263 23.8117618 8.2767726 001763668 568 322624 183250432 23.8327506 8.2816355 .001760563 569 323761 184220000 25.8537209 8.2864928 .001757469 570 324900 185193000 23.8746728 8.2913444 .001754386 571 326041 186169411 23.8956063 8.2961903 .001751313 572 327184 187149248 23.9165215 8.3010304 .001748252 573 328329 188132517 23.9374184 8.3058651 .001745201 574 329476 189119224 23.9582971 8.3106941 .001742160 575 330625 190109375 23.9791576 8.3155175 .001739130 576 331776 191102976 24.0000000 8.3203353 .001736111 577 332929 192100033 24.0208243 8.3251475 .001733102 578 334084 193100552 24.0416306 8.3299542 .001730104 579 335241 194104529 24.0624188 8.3347553 .001727116 580 336400 195112000 24.0831891 8.3395509 .001724138 581 337561 196122941 24.1039416 8.3443410 .001721170 582 338724 197137368 24.1246762 8.3491256 .001718213 583 339889 198155287 24.1453929 8.3539047 .001715266 581 341056 199176704 24.1660919 8.3586784 .001712329 585 342:225 200201625 24.1867732 8.3634466 .001709402 586 343396 201230056 24.2074369 8.3682095 .001706485 587 344569 202262003 24.2280820 8.3729668 .001703578 588 345744 203297472 21.2487113 8.3777188 .001700680 589 346921 204336469 24.2693222 8.3824653 .001697793 590 348100 205379000 24.2899156 8.3872065 .001694915 591 349281 206425071 24.3104916 8.3919423 .001692047 592 350464 207474688 24.3310501 8.3966729 .001689189 593 351649 208527857 21.3515913 8.4013981 .001686341 594 352836 20958-1584 24.3721152 8.4061180 .001683502 595 354025 210644875 21.3926218 8.4108326 .001680672 596 355216 211708736 24.4131112 8.4155419 .001677852 597 356409 212776173 24.4335834 8.4202460 .001675042 598 357604 213847192 24.4540385 8.4249448 .001672241 599 358801 214921799 24.4744765 8.4296383 .001669449 600 360000 216000000 24.4948974 8.4343267 .00166666 601 361201 217081801 24.5153013 8.4390098 .00166389↓ 603 604 602 362404 218167208 363609 364816 220348864 24.5356883 8.4436877 .001661130 219250227 24.5560583 8.4483605 .001658375 21.5764115 8.4530281 .001C55629 605 366025 221445125 24.5967478 8.4576906 .001652893 606 367236 222545016 24.6170673 8.4623479 .001650165 607 368449 223648543 24.6373/00 8.4670001 001647446 608 369661 224755712 24.6576560 8.4716471 .001644737 609 370881 225866529 21.0779251 8.4762892 .001642036 610 372100 226981000 24.6981781 8.4809261 .001639344 611 373321 228099131 24.7184142 8.4855579 .001636661 614 615 612 613 375769 230346397 376996 378225 232608375 374544 229220928 24.7386338 8.4901848 .001633987 24.7588368 8.4948065 .001631321 231475544 24.7790234 8.4994233 .001628664 21.7991035 8.5040350 .001626016 616 379456 233744896 24.8193473 8.5086417 .001623377 617 380689 234885113 24.8394847 8.5132435 .001620746 618 381924 236029032 24.8596058 8.5178403 .001618123 619 383161 237176659 24.8797106 8.5224321 .001615509 620 381100 238328000 24.8997992 8.5270189 .001612903 : 274 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 621 385641 239483061 24.9198716 8.5310009 .001610306 622 386884 240641848 24.9399278 8.5361780 .001607717 623 388129 241804367 24.9599679 8.54107501 .001605136 624 389376 242970624 21.9799920 8.5453173 .001602564 625 390625 241140625 25.0000000 8.5498797 .001600000 626 391876 245314376 25.0199920 8.5544372 .001597444 627 393129 246491883 25.0399681 8.5589899 .001594896 6:28 394381 247673152 25.0599282 8.5635377 .001592357 629 395641 248858189 25.0798724 8.5680807 .001589825 630 396900 250047000 25.0998008 8.5726189 .001587302 631 398161 251239591 25.1197134 8.5771523 .001584786 632 399124 252435968 25.1396102 8.5816809 633 400689 253636137 25.1594913 8.5862047 63-1 401956 254840104 25.1793566 8.5907238 .001582278 ,001579779 .001577287 H 635 403225 256047875 25.1992063 8.5952380 .001574803 636 404496 257259456 25.2190404 8.5997476 .001572327 637 105769 258474853 25.2388589 8.6042525 .001569859 638 407044 259694072 25.2586619 8.6087526 .001567393 639 408321 260917119 25.2784493 8.6132480 .001564945 640 409600 262144000 25.2982213 8.6177388 .001562500 641 410881 263374721 25.3179778 8.6222248 .001560062 642 412164 264609288 25.3377189 8.6267063 .001557632 643 413449 265847707 25.8574447 8.6311830 .001555210 641 414736 267089984 25.3771551 8.6356551 .001552795 645 416025 268336125 25.3968502 8.6401226 .001550388 646 117316 269586136 25.4165301 8.6445855 001547988 647 418609 270840023 25.4361947 8.0490437 .001545595 649 648 119904 121201 272097792 25.4558441 8.6534974 .001543210 273359149 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 27972626 1 25.5734237 8.6801237 .001529052 655 429025 281011375 25.5929678 8.6845456 .001526718 656 430336 282300416 25.6124969 8.6889630 .001521390 657 431649 283593393 25.6320112 8.6933759 .001522070 658 432964 284890312 25.6515107 8.6977843 001519757 659 434281 286191179 25.6709953 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 25.7487861 8.7197596 .001508296 664 440896 292754944 25.7681973 8.7241414 .001506024 665 412225 294079625 25.7875939 8.7285187 .001503759 666 443556 295-408296 25.8069758 8.7328918 .001501502 667 414889 296740963 25.8263431 8.7372604 .001499250 668 446224 298077632 25.8456960 8.7416246 .001497006 669 447561 299418309 25.8650343 8.7459846 .001494768 670 448900 300763000 25.8843582 8.7503401 .001492537 671 450241 302111711 25.9036677 8.7546913 001490313 672 451584 303404148 25.9229628 8.7590383 .001488095 673 452929 304821217 25.9422435 8.7633809 .001485884 674 454276 306182024 25.9615100 8.7677192 .001-483680 675 455625 307546875 25.9807621 8.7720532 .001481481 676 456976 308915776 26.0000000 8.7763830 .001479290 677 458329 310288733 26.0192237 8.7807084 .001477105 078 459684 311665752 26.0384331 8.7850296 .001474926 679 461041 313046839 26.0576284 8.7893466 .001172754 680 462100 314432000 26.0768096 8.7936593 .001470588 681 463761 315821241 26.0959767 8.7979679 .001468429 682 465124 317214568 26.1151297 8.8022721 .001466276 275 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 318611987 26.1342687 8.8065722 .001464129 681 467856 320013504 26.1533937 8.8108681 .001461988 685 469225 321419125 26.1725047 8.8151598 .001459854 686 470596 322828856 26.1916017 8.8194474 .001457726 087 471969 324242703 26.2106848 8.8237307 .001455604 688 473344 325660672 26.2297541 8.8280099 .001453488 689 474721 327082769 26.2488095 8.8322850 .001451379 690 476100 328509000 26.2678511 8.8365559 .001449275 691 477481 329939371 26.2868789 8.8408227 .001447178 692 478864 331373888 26.3058929 8.8450854 .001445087 693 480249 332812557 26.3248932 8.8493440 .001443001 694 481636 334255384 26.3438797 8.8535985 .001440922 695 483025 335702375 26.3628527 8.8578489 .001438849 696 484416 337153536 26.3818119 8.8620952 .001436782 697 485809 338608873 26.4007576 8.8663375 .001434720 698 487204 340068392 26.4196896 8.8705757 .001432665 699 488601 341532099 26.4386081 8.8748099 .001430615 700 490000 343000000 26.4575131 8.8790400 .001428571 701 491401 344472101 26.4764046 8.8832661 .001426534 702 492804 345948408 26.4952826 8.8874882 .001424501 703 494209 347428927 26.5141472 8.8917063 .001422475 704 495616 348913664 26.5329983 8.8959204 .001420455 705 497025 350402625 26.5518361 8.9001304 .00141840 706 498436 351895816 26.5706605 8.9043366 .001416431 707 499849 353393243 26.5894716 8.9085387 .001414427 708 501264 354894912 26.6082694 8.9127369 .001412429 709 502681 356100829 26 6270539 8.9169311 .001410437 710 501100 357911000 26.6458252 8.9211214 .001408451 711 505521 359425431 26.6645833 8.9253078 .001406470 712 506911 360944128 26.6833281 8.9294902 .001404194 713 508369 362467097 26.7020598 8.9336687 001402525 7:4 509796 363994344 26.7207784 8.9378433 .001400560 715 511225 365525875 26.7394839 8.9420140 001398601 716 512656 367061696 26.7581763 8.9461809 001396648 717 514089 368601813 26.7768557 8.9503438 .001394700 718 515524 370146232 26.7955220 8.9545029 .001392758 719 516961 371694959 26.8141754 8.9586581 .001390821 720 518400 3732-48000 26.8328157 8.9628095 001388889 T21 519841 374805361 26.8514432 8.9669570 .001386963 722 521284 376367048 26.8700577 8.9711007 .001385042 723 522729 377933007 26.8886593 8.9752406 .001383126 724 524176 379503424 26.9072-481 8.9793766 001381215 725 525625 381078125 26.9258240 8.9835089 001379310 726 527076 382657176 26.9443872 8.9876373 001377410 727 528529 384240583 26.9629375 8.9917620 001375516 • 728 529984 385828352 26.9814751 8.9958829 001373626 729 531441 387420489 27.0000000 9.0000000 .001371742 730 532900 389017000 27.0185122 9.0041134 001369863 731 534361 390617891 27.0370117 9.0082229 001367989 732 535824 392223168 27.0554985 9.0123288 .001366120 733 587289 393832837 27.0739727 9.0164309 .001364256 31 538756 395446904 27.0924314 9.0205293 .001362398 735 540225 397065375 27.1108834 9.0246239 .001360544 736 541696 398688256 27.1293199 9.0287149 001358696 737 543169 400315553 27.1477439 9.0328021 .001356852 738 544644 401947272 27.1661554 9.0368857 001355014 739 546121 403583419 27.18455414 9.0409655 .001353180 740 547600 405224000 27.2029410 9.0450419 001351351 741 549081 406869021 27.2213152 9.0491142 001349528 742 550564 408518488 27.2396769 9.0531831 001347709 743 552049 410172407 27.2580263 9.0572482 .001345895 741 553536 411830784 27.2763634 9.0613098 ,001344086 276 TABLE VIII.-Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 746 556516 415160936 27.3130006 9.0694220 .001340483 747 558009 416832723 27.3313007 9.0734726 .001338688 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 .001331558 752 565504 425259008 27.4226181 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 27.5499516 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 414194947 27.62.4546 9.1377971 .001310616 764 583696 445943744 27.6-105499 9.1417874 .001308901 765 585225 447697125 27.6586334 9.1457742 .001307190 766 586756 449455096 27.6767050 9.1497576 .001305483 767 588289 451217663 27.6947648 9.1537375 .001303781 768 589824 452984832 27.7128129 9.1577139 .001302083 769 591361 454756609 27.7308492 9.1616869 .001300390 770 592900 456533000 27.7488739 9.1656565 .001298701 771 594441 458314011 27.7668868 9.1696225 .001297017 772 595984 460099648 27.7848880 9.1735852 .001295337 773 597529 461889917 27.8028775 9.1775-145 .001293661 74 599076 463684824 27.8208555 9.1815003 .001291990 775 600625 465484375 27.8388218 9.1854527 .001290323 776 602176 467288576 27.8567766 9.1894018 .001288660 777 603729 469097433 27.8747197 9.1933474 .001287001 778 605284 470910952 27.8926514 9.1972897 .001285347 779 606811 472729139 27.9105715 9.2012286 .001283697 780 608400 474552000 27.9284801 9.2051641 .001282051 781 609961 4763795-41 27.9463772 9.2090962 .001280410 782 611524 478211768 27.9642629 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 487443403 28.0535203 9.2326189 .001270648 788 620944 489303872 28.0713377 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 614809 517781627 28.3372546 9.2947671 .001245330 804 646416 519718161 28.3548938 9.2986239 001243781 805 648025 521660125 28.3725219 9.3024775 .001242236 806 649636 523606616 28.3901391 9.3063278 .001240695 277- TABLE VIII.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750 .001239157 808 652864 527514112 28.4253408 9.3140190 .001237624 809 654181 529475129 28.4429253 9.3178599 .001236094 810 656100 531441000 28.4604989 9.3216975 .001234568 811 657721 533411731 28.4780617 9.3255320 .001233046 812 659344 535387328 28.4956137 9.3293634 .001231527 813 660969 537367797 28.5131549 9.3331916 .001230012 814 662596 539353144 28.5306852 9.3370167 .001228501 815 664225 541343375 28.5482048 9.3408386 .001226994 816 665856 5-43338496 28.5657137 9.3446575 .001225490 817 667489 545338513 28.5832119 9.3484731 .001223990 818 669124 547343432 28.6006993 9.3522857 .001222494 819 670761 549353259 28.6181760 9.3560052 .001221001 820 672400 551368000 28.6356421 9.3599010 .001219512 821 674041 553387661 28.6530976 9.3637049 .001218027 822 675684 555412248 28.6705424 9.3675051 .0012165-15 823 677329 557441767 28.6879766 9.3713022 .001215067 678976 559476224 28.7054002 9.3750963 .001213592 825 680625 561515625 28.7228132 9.3788873 .001212121 826 682276 563559976 28.7402157 9.3826752 .001210654 827 683929 565609233 28.7576077 9.3864600 .001209190 828 685581 567663552 28.7749891 9.3902419 .001207729 829 687241 569722789 28.7923601 9.3940206 .001206273 830 688900 571787000 28.8097206 9.3977964 .001204819 831 690561 573856191 28.8270706 9.4015691 .001208869 832 692224 575930368 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 001200-480 834 695556 580093704 28.8790582 9.4128690 .001199041 835 697225 582182875 28.8963666 9.4166297 .001197605 836 098896 584277056 28.9136646 9.4203873 .001196172 837 700569 586376253 28.9309523 9.4241420 .001194743 838 702244 588480472 28.9482297 9.4278936 .001193317 839 703921 590589719 28.9654967 9.4316423 .001191895 840 705600 59270-4000 28.9827535 9.4353880 .001190473 8-11 707281 594823321 29.0000000 9.4391307 .001189061 842 708964 596947688 29.0172363 9.4428704 .001187648 843 710649 599077107 29.0344623 9.4466072 .001186240 811 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.0688837 9.4540719 .001183432 846 715716 605495736 29.0860791 9.4577999 .001182033 847 717409 6076-45423 29.1032644 9.4615249 .001180688 818 719101 609800192 29.1204396 9.1652470 .001179245 849 720801 611960049 29.1376046 9.4689661 .001177856 850 722500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 9.4763957 .001175088 852 725904 618470208 29.1890390 9.4801061 .001173709 853 727609 620650477 29.2061637 9.4838136 .001172333 854 729316 622835864 29.2232784 9.4875182 .001170960 855 731025 625026375 29.2403830 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4949188 .001168224 857 734149 629422793 29.2745623 9.4986147 .001166861 858 736164 631628712 29.2916370 9.5023078 .001165501 859 737881 633839779 29.3087018 9.5059980 .001164144 860 739600 636056000 29.3257566 9.5096854 .001162791 861 741321 638277381 29.3428015 9.5133699 .001161440 862 743044 640503928 29.3598365 9.5170515 .001160093 863 744769 642735647 29.3768610 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 VIII.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 869 755161 656234909 29.4788059 9.5427437 .001150748 870 756900 658503000 29.4957624 9.5464027 .001149425 871 758641 660776311 29.5127091 9.5500589 .001148106 872 760384 663054848 29.5296461 9.5537123 .001146789 873 762129 665338617 29.5465734 9.5573630 .001145475 874 763876 667627624 29.5634910 9.5610108 .001144165 875 765625 669921875 29.5803989 9.5646559 .001142857 876 767376 672221376 29.5972972 9.5682982 .001141553 877 769129 674526133 29.6141858 9.5719377 .001140251 878 770884 676836152 29.6310648 9.5755745 .001138952 879 772641 679151439 29.6479342 9.5792085 .001137656 880 774400 681472000 29.6647939 9.5828397 .001136364 881 776161 683797841 29.6816442 9.5864682 .001135074 882- 777924 686128968 29.6984848 9.5900939 .001133787 883 779689 688465387 29.7153159 9.5937169 .001132503 881 781456 690807104 29.7321375 9.5973373 .001131222 885 783225 693154125 29.7489496 9.6009548 .001129944 886 784996 695506456 29.7657521 9.6045696 .001128668 887 786769 697864103 29.7825452 9.6081817 .001127396 888 788544 700227072 29.7993289 9.6117911 .001126126 889 790321 702595369 29.8161030 9.6153977 .001124859 890 792100 704969000 29.8328678 9.6190017 .001123596 891 793881 707347971 29.8496231 9.6226030 .001122334 892 795664 709732288 29.8663690 9.6262016 .001121076 893 797449 712121957 29.8831056 9.6297975 .001119821 894 799236 714516984 29.8998328 9.6333907 .001118568 895 801025 716917375 29.9165506 9.6369812 .001117318 896 802816 719323136 29.9332591 9.6405690 .001116071 897 804609 721734273 29.9499583 9.6411542 .001114827 898 806404 724150792 29.9666481 9.6477367 .001113586 899 808201 726572699 29.9833287 9.6513166 .001112347 900 810000 729000000 30.0000000 9.6548938 .001111111 901 811801 731432701 30.0166620 9.6584684 .001109878 90:2 813604 733870808 30.0333148 9.6620403 .001108647 903 815409 736314327 30.0499581 9.6656096 .001107420 904 817216 738763264 30.0665928 9.6691762 .001166195 905 819025 741217625 30.0832179 9.6727403 .001104972 906 820836 743677416 30.0998339 9.6763017 .001103753 907 8226-49 746142643 30.1164407 9.6798604 .001102536 908 824464 748613312 30.1330383 9.6834166 .001101322 909 826281 751089429 30.1496269 9.6869701 .001100110 910 828100 753571000 30.1662063 9.6905211 .001098901 911 829921 756058031 30.1827765 9.6940694 .001097695 912 831744 758550528 30.1993377 9.6976151 .001096491 913 833569 761048497 30.2158899 9.7011583 .001095290 914 835396 763551944 30.2324329 9.7046989 .001094092 915 837225 766060875 30.2489669 9.7082369 .001092896 916 839056 768575296 30.2654919 9.7117723 .001091703 917 840889 771095213 30.2820079 9.7153051 .001090513 918 842724 773620632 30.2985148 9.7188354 .001089325 919 814561 776151559 30.3150128 9.7223631 .001088139 920 816400 778688000 30.3315018 9.7258883 001086957 921 848241 781229961 30.3479818 9.7294109 .001085776 922 850084 783777448 30.3644529 9.7829309 .001084599 923 851929 786330467 30.3809151 9.7364484 .001083423 924 853776 788889024 30.3973683 9.7399634 .001082251 925 855625 791453125 30.4138127 9.7434758 .001081081 926 857476 794022776 30.4302481 9.7469857 .001079914 927 859329 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.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 931 866761 806954491 30.5122926 9.7644974 .001074114 932 868624 809557568 30.5286750 9.7679922 .001072961 933 870489 812166237 30.5450487 9.7714845 .001071811 934 872356 814780504 30.5614136 9.7749743 .001070664 935 874225 817400375 30.5777697 9.7784616 .001069519 936 876096 820025856 30.5941171 9.7819466 .001068376 937 877969 822656953 30.6104557 9.785-4288 .001067236 938 879844 825293672 30.6267857 9.7889087 .001066098 939 881721 827936019 30.6431069 9.7923861 .001064963 940 883600 830584000 30.6594194 9.7958611 .001063830 941 885481 833237621 30.6757233 9.7993336 .001062699 942 887364 835896888 30.6920185 9.8028036 .001061571 943 889249 838561807 30.7083051 9.8062711 .001060445 944 891136 841232384 80.7245830 9.8097362 .001059322 945 893025 843908625 30.7408523 9.8131989 .001058201 946 894916 846590536 30.7571130 9.8166591 001057082 947 896809 849278123 30.7733651 9.8201169 001055966 948 898704 851971392 30.7896086 9.8235723 .001054852 949 900601 85-4670349 30.8058436 9.8270252 .001053741 950 902500 857375000 30.8220700 9.8304757 .001052632 951 904401 860085351 30.8382879 9.8339238 .001051525 952 906304 862801408 30.8544972 9.8373695 .001050420 953 908209 865523177 30.8706981 9.8408127 .001049318 954 910116 868250664 30.8868904 9.8442536 .001048218 955 912025 870983875 30.9030743 9.8476920 .001047120 956 913936 873722816 30.9192497 9.8511280 .001046025 957 915849 876467493 30.9354166 9.8545617 .001044932 958 917764 879217912 30.9515751 9.8579929 .001043841 959 919681 881974079 30.9677251 9.8614218 .001042753 960 921600 884736000 30.9838668 9.8648183 .001041667 961 923521 887503681 31.0000000 9.8682724 .001040583 962 925444 890277128 31.0161248 9.8716941 .001039501 963 927369 893056347 31.0322413 9.8751135 .001038422 964 929296 895841344 31.0483494 9.8785305 .001037344 965 931225 898632125 31.0644491 9.8819451 .001036269 966 933156 901428696 31.0805405 9.8853574 .001035197 967 935089 904231063 31.0966236 9.8887673 .001034126 968 937024 907039232 31.1126984 9.8921749 .001033058 969 938961 909853209 31.1287648 9.8955801 .001031992 970 940900 912673000 31.1448230 9.8989830 .001030928 971 942841 915498611 31.1608729 9.9023835 .001029866 972 944784 918330048 31.1769145 9.9057817 .001028807 973 946729 921167317 31.1929479 9.9091776 .001027749 974 948676 924010424 31.2089731 9.9125712 .001026694 975 950625 926859375 31.2249900 9.9159624 .001025641 976 952576 929714176 31.2409987 9.9193513 .001024590 977 951529 932574833 31.2569992 9.9227379 .001023541 978 956-484 935441352 31.2729915 9.9261222 .001022495 979 958441 938313739 31.2889757 9.9295042 .001021450 980 960400 911192000 31.3049517 9.9328839 .001020408 981 962361 944076141 31.3209195 9.9362613 .001019368 982 964324 946966168 31.3368792 9.9396363 .001018330 983 966289 949862087 31.3528308 9.9430092 .001017294 984 968256 952763904 31.3687743 9.9463797 .001016260 985 970225 955671625 31.3847097 9.9497479 .001015228 986 972196 958585256 31.4006369 9.9531138 .001014199 987 974169 961504803 31.4165561 9.9564775 .001013171 988 976144 964430272 31.4324673 9.9598389 .001012146 989 978121 967361669 31.4483704 9.9631981 .001011122 990 980100 970299000 31.4642654 9.9665549 .001010101 991 982081 973242271 31.4801525 9.9699095 .001009082 992 984064 976191488 31.4960315 9.9732619 001008065 280 TABLE VIII.—Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 993 986049 979146657 31.5119025 9.9766120 .001007049 994 988036 982107784 31.5277655 9.9799599 .001006036 995 990025 985074875 31.5436206 9.9833055 .001005025 996 992016 988047936 31.5594677 9.9866488 .001004016 997 994009 991026973 31.5753068 9.9899900 .001003009 998 996004 994011992 31.5911380 9.9933289 .001002004 999 998001 997002999 31.6069613 9.9966656 .001001001 1000 1000000 1000000000 31.6227766 10.0000000 .001000000 1001 1002001 1003003001 31.6385840 10.0033322 .0009990010 1002 1004004 1006012008 31.6543836 10.0066622 .0009980040 1003 1006009 1009027027 31.6701752 10.0099899 .0009970090 1004 1008016 101248064 31.6859590 10.0133155 .0009960159 1005 1010025 1015075125 31.7017349 10.0166389 .0009950249 1006 1012036 1018108216 31.7175030 10.0199601 .0009940358 1007 1014049 1021147343 31.7332633 10.0232791 .0009930487 1003 1016064 1024192512 31.7490157 10.0265958 .0009920635 1003 1018081 1027243729 31.7647603 10.0299104 .0009910803 1010 1020100 1030301000 31.7804972 10.0332228 .0009900990 1011 1022121 1033364331 31.7962262 10.0365330 .0.09891197 1012 1021114 1036433728 31.8119474 10.0398410 .0009881423 1013 1026169 1039500197 31.8276609 10.0431469 .0009871668 1014 1028196 1042590744 31.8433666 10.0464506 .0009861933 1015 1030225 1045678375 31.8590646 10.0497521 .0009852217 1016 1032256 1048772096 31.8747549 10.0530514 .0009842520 1017 1034289 1051871913 31.8904374 10 0563485 1018 103032-1 1054977832 31.9061123 10.0596435 .0009832842 0009823183 1019 1038361 1058089859 31.9217794 10.0629364 .0009813543 1020 1040400 1061208000 31.9374388 10.0662271 .0009803922 1021 1042441 1064332261 31.9530906 10.0695156 0009794319 1022 1011184 1067-4626-48 31.9687347 10.0728020 0009784736 1023 1046529 1070599167 31.9843712 10.0760863 .0009775171 1024 1048576 1073741824 32.0000000 10.0793681 .0009765625 1025 1050625 1076890625 32.0156212 10.0826484 .0009756098 1026 1052676 1080045576 32.0312348 10.0859262 .0009746589 1027 1054729 1083206683 32.0468107 10.0892019 .0009737098 1028 1056784 1086373952 32.0624391 10.0924755 .0009727626 1029 1058841 1089547339 32.0780298 10.0957469 .0009718173 1030 1060900 1092727000 32.0936131 10.0990163 .0009708738 1031 1062961 1095912791 32.1091887 10.1022835 .0009699321 1032 1065024 1099104768 32.1247568 10.1055487 .00 9689922 1033 1067089 1102302937 32.1403173 10.1088117 .0009680542 1034 1069156 1105507304 32.1558704 10.1120726 .0009671180 1035 1071225 1108717875 32.1714159 10.1153314 0009661836 1036 1073296 1111934656 32.1869539 10.1185882 .0009652510 1037 1075369 1115157653 32.2024844 10.1218428 .0009643202 1038 1077444 1118386872 32.2180074 10.1250953 .0009633911 1039 1079521 1121622319 32.2335229 10.1283457 .0009624639 1040 1081600 1124864000 32.2490310 10.1315941 .0009015385 1041 1083681 1128111921 32.2645316 10.1348403 .0009606148 1042 1085761 1131366088 32.2800248 10.1380845 .0009596929 1043 1087849 1134626507 32.2955105 10.1413266 .0009587798 1011 1089936 1137893184 32.3109888 10.1445667 .0009578544 1045 1092025 1141166125 32.3264598 10.1478047 0009569378 1046 1094116 1144445336 32.3419233 10.1510406 .0009560229 1047 1096209 1147730823 32.3573794 10.1542744 0009551098 • 1048 1098304 1151022592 32.3728281 1049 1100401 1154320649 32.3882695 10.1575062 10.1607359 .0009541985 .0009532888 1 1050 1102500 1157625000 32.4037035 10.1639636 .0009523810 1051 1104601 1160935651 32.4191301 10.1671893 0009514748 1052 1106704 1164252608 32.4345495 10.1704129 .0009505703 1053 1108809 1167575877 32.1499615 10.1736344 .0009496676 1054 1110916 1170905464 32.4653662 10.1768539 .0009487666 201 TABLE IX.-LOGARITHM OF NUMBERS FROM 0 TO 1000. No. 0 1 2 3 4 5 7 со 8 9 0 10 11 12 13 14 15 16 17 18 19 20 21 22 0 00000 30103 47712 60206 69897 77815 84510 90309 95424 00000 00432 00860 01284 01703 02119 02530 02938 03342 03743 04139 04532 04922 05307 05690 06070 06446 06819 07188 07555 07918 08279 08637 08990 09342 09691 10037 10380 10721 11059 11394 11727 12057 12385 12710| 13033 13354 13672 13988 14301 14613 14922 15229 15533 15836 16137 16435 16732 17026 17319 17609 17898 18184 18469 18752 19033 19312 19590 19866 20140 20412 20683 20952 21219 21484 21748 22011 22272 22531 22789 23045 23300 23553 23805 24055 24304 24551 24797 25042 25285 25527 25768 26007 26245 26482 26717 26951 27184 27416 27646 27875 28103 28330 28556 28780 29003 29226 29447 29667 29885 30103 30320 30535 30749 30963 31175 31386 31597 31806 32015 32222 32428 32633 32838 33041 33244 33445 33646 33846 34044 34242 34439 34635 34830 35025 35218 35411 35603 35793 85984 36173 36361 36549 36736 36922 37107 37291 37475 37658 37840 38021 38202 38382 38561 38739 38916 39094 39270 39445 39619 39794 39967 40140 40312 40483 40654 40824 40993 41162 41330 41497 41664 41830 41996 42160 42325 42488 42651 42813 42975 43136 43297 43157 43616 43775 43933 44091 44248 44404 44560 44716 44871 45025 45179 45332 45484 45637 45788 45939 46090 46240 46389 46538 46687 46835 46982 47129 47276 47422 47567 47712 47857 48001 48144 48287 48430 48572 48714 48855 48996 49136 49276 49415 49554 49693 49831 49969 50106 50243 50379 50515 50651 50785 50920 51055 51189 51322 51455 51587 51720 51851 51983 52114 52244 52375 52504 52634 52763 52892 53020 53148 53275 53403 53529 53656 53782 53908 54033 54158 54283 54407 54531 54654 54777 54900 55022 55145 55267 55388 55509 55630 55751 55871 55991 56110 56229 56348 56467 56585 56703 56820 56937 57054 57171 57287 57403 57519 57634 57749 57863 57978 58093 58206 58320 58433 58546 58659 58771 58883 58995 59106 59218 59328 59439 59550 59660 59770 59879 59989 60097 60206 60314 60423 60531 60638 60745 60853 60959 61066 61172 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47. 48 49 50 69020 69108 69197 69285 69373 69897 69984 70070 70157 70243 69461 69548 69636 69723 69810 70329, 70415 70501 70583 70672 61278 61384 61490 61595 61700 61805 61909 62014 62118 62221 62325 62428 62531 62634 62737 62839 62941 63043 63144 63246 63347 63448: 63548 63649 63749 63849 63949 64048 64147 64246 64345 64444 64542 64640 64738 64836 64933 65031 65128 65225 65896 65992 66087 66181 65321 65418 65514 65609 65706 65801 66276 66370 66464 66558 66652 66745 66839 66932 67025 67117 67210 67302 67394 67486 67578 67669 67761 67852 67943: 68034 68124 68215 68305 68395 68485 68574 68664 68753 68842 68931 282 8 9 TABLE IX-Continued.-LOGARITHM OF NUMBERS FROM 0 TO 1000. No. 0 1 2 3 4 5 6 t- 189 52 53 54 55 56 57 51 70757 70842 70927 71012] 71096 71181 71265 71349 71433 71517 71600 71684 71767 71850| 71933 72016| 72099 72181 72263| 72346 72428 72509 72591 72673 72754 72835 72916 72997 73078 73159 73239 73320| 73399 73480 73560 73639 73719 73799 73878 73957 74036 74115 74194 74273 74351 74429 74507 74586 74663 74741 74819 74896 74974 75051 75128 75205 75282 75358 75435 75511 75587 75664 75740 75815 75891 75967|76042| 76118 75967 76042 76118 76193 76268 76343 76418 76492 76567 76641 76716 76790 76864 76938 77012 77085 77159 77232 77305 77379] 77452 77525 77597 77610 77743 77815 77887 77960 78032 78104 78176 78247 78319 78390 78462 58 59 60 61 62 63 65 66 68 69 71 83383R882 888323F882 CIHAZSZISZ 28828AP382 89; 74 75 76 78 79 80 81 84 85 86 92 94 95 96 98 78533 78604 78675 78746 78817 78888 78958 79029 79099 79169 79239 79309 79379 79449 79518 79588 79657 79727 79796 79865 79934 80003 80072 80140 80209 80277 80346 80414 80482 80550 80618 80686 80754 80821 80889 80956 81023 81090 81158 81224 81291 81358 81425 81491 81558 81624 81690 81757 81823 81889 81954 82020 82086 82151 82217 82282 82347 82413 82478 82543 82607 82672 82737 82802 82866 82930 82995 83059 83123 83187 83251 83315 83378 83442 83506 83569 83632 83696 83759 83822 83885 83948 84011 84073 84136 84198 84261 84323 84386 84448 84510 84572 84634 84696 84757 84819 84880 84942 85003 85065 85126 85187 85248 85309 85370 85431 85491 85552 85612 85673 85733 85794 85854 85914| 85974| 86034| 86094| 86153 86213 86273 86332 86392 86451| 86510| 86570 86629 86688 86747 86806 86864 86923 86982 87040 87099 87157 87216 87274 87332 87390 87448 87506 87564 87622 87680 87737 87795 87852 87910 87967 88024 88081 88138 88196 88252 88309 88366 88423 88480 88536 88593 88649 88705 88762 88818 88874 88930| 88986 89042 89098 89154 89209 89265 89321 89376 89432 89487 89542 89597 89653 89708 89763 89818 89873 89927 89982 90037 90091 90146 90200 90255 90309 90363 90417 90472 90526 90580 90634 90687 90741 90795 90848 90902 90956 91009 91062 91116 91169 91222 91275 91328 91381 91434 91487 91540 91593 91645 91698 91751 91803 91855 91908 91960 92012| 92065 92117 92169 92221 92273 92324 92376 92428 92480 92531 92583 92634 92686 92737 92789 92840 92891 92942 92993 93044 93095 93146 93197 93247 93298 93349 93399 93450 93500 93551 93601 93651 93702 93752 93802 93852 93902 93952 94002 94052 94101 94151 94201 94151 94201 94250 94300 94250 94300 94349 94398 94448 94498 94547 94596 94645 94694 94743 94792 94841| 94890 94939 94988 95036 95085 95134 95182 95231 95279 95328 95376 95424 95472 95521 95569 95617 95665 95713 95761 95809 95856 95904 95952 95999 96047 96095 96142 96190] 96237 96284 96332 96379 96426 96473 96520 96567 96614 96661 96708 96755 96802 96848 96895 96942 96988 97035 97081 97128 97174 97220) 97267 97313 97359 97405 97451 97497 97543 97589 97635 97681 97727 97772 97818 97864 97909 97955 98000 98046 98091 98137 98182 98227 98272 98318 98363 98408 98453 98498 98543 98588 98632 98677 96722 98767 98811 98856 98900 98945 98989 99034 99078 99123 99167 99211 99255 99300 99344 99388 99432 99476 99520 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 Secs. 231 and 232, 1 sin A 1 cosec A' cos A sec A tan A = sec A' cosec A' 1 vers A = 1 sec A exsec A sec A-1; cosec A cot A = sec A 284 TABLE X.-SINES AND COSINES. 0° Sine Cosin 0.00000 One. 1 .00029 One. 2.00058 One. 3 |.00087 One. 4.00116 One. 5.00145 One. 1° 2° 3° 4° Sine Cosin Sine Cosin Sine Cosin Sine Cosin .01803.99984||.03548.99937.05292.99860|| .07034.99752 .01832.99983||.03577.99936||.05321.99858|| .07063.99750|57 .01745.99985.03490.99939 .01774.99984.03519.99938 .05234.99863.06976.99756 60 .05263.99861 .07005.99754 59 .01862.99983.03606.99935 .05350.99857.07092.99748 56 .05408.99854 .07150.99744 54 .01891.99982.03635.99934.05379.99855.07121.99746 55 6.00175 One. || .01920.99982|| .03664.99933||.05408.99854 7.00204 One. 8.00233 One. 9.00262 One. .01949.99981.03693.99932.05437.99852.0179.99742 53 .03723.99931||.05466.99851 .07208.99740 52 . .01978.99980.03723.99931 · .02007.99980||.03752.99930 .05495.99849 .07237.99738 51 10.00291 One. .02036.99979 .03781.99929.05524.99847.07266.99736 50 11.00320.99999||.02065.99979||.03810.99927||.05553.99846|| .07295.99734 49 .00349.99999.02094.99978.03839.99926.05582.99844 .07324.99731 48 13.00378.99999.02123.99977 02123.99977||.03868.99925|| .05611.99842.07353.99729 47 14.00407.99999||.02152.99977.03897 .02152.99977.03897.99924 .05640.99841 .99924.05640|.99841||.07382.99727| 46 15.00436.99999||.02181.99976||.03926.99923|| .05669.99839||.07411.99725 45 16.00465.99999 02211.99976.03955.99922||.05698|.99838|| .07440.99723| 44 17.00495.99999.02240.99975||.03984.99921||.05727.99836||.07469|.99721| 43 18.00524.99999 02269.99974||.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|| .02356.99972 ||.04100.99916.05844.99829.07585.99712 39 22 .00640.99998||.02385.99972||.04129.99915|| .05873.99827.07614|.99710| 38 .00669.99998||.02414.99971||.04159.99913|| .05902.99826.7643.99708 24.00698.99998 02443.99970.04188.99912|| .05931.99824.07672.9970530 25.00727.99997 02472|.99969 || .04217|.99911.03960.99822.07701|.99703 35 26.00756.99997|| .02501|.99969||.04246.99910||.05989.99821 05989.99821.07730.99701 34 27.00785.99997||.02530.99968||.04275.99909||.06018.99819.07759.99699 33 28.00814.99997.02560.99967.04304.99907.06047.99817 .07788.99696 32 29.00844.99996.02589.99966||.04333.99906||.06076|.99815.07817.99694 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 |.07904.99687 33.00960.99995|| .02705.99963.04449.99901|| .06192|.99808.07933.99685 28 34.00989.99995.02734.99963|| .04478.99900.06221.99806.07962.99683|26 35.01018.99995|| .02763.99962||.04507.99898.06250.99804.07991.99680|25 22 36.01047.99995||.02792.99961||.04536.99897|| .06279.99803.08020.99678 37.01076.99994.02821.99960.04565.99896.06308.99801.08049.99676 23 38.01105.99991 .02850 |.99959||.04594.99894 .04594.99894.06337.99799 .06337.99799.08078.99673 39.01134.99994 .02879.99959||.04623.99893||.06366.99797.08107.99671 40.01164.99993 .02908.99958||.04653.99892||.06395.99795.08136|.99668| 20 41.01193.99993|| .02938.99957.04682].99890 .04682.99890.06424.99793.08165.99666 19 42.01222.99993|| .02967.99956||.04711.99889||.06453|.99792.08194.99664 18 43.01251.99992 .02996.99955.04740.99888.06482.99790.08223.99661 17 44.01280.99992|| .03025|.99954||.04769.99886||.06511.99788.08252.99659 16 45.01309.99991|| .03054.99953||.04798.99885||.06540.99786.08281.99657 15 46.01338.99991|| .03083|.99952||.04827.99883.06569.99784 |.08310.99654 47.01367.99991|| .03112|.99952||.04856.99882||.06598.99782.08339.99652|13 14 48.01396.99990.03141.99951.04885.99881 .04885.99881||.06627:99780.08368.99649| 12 .01425.99990.03170.99950.04914.99879.06656.99778 50.01454.99989 04914.99879||.06656.99778.08397.99647|11 .03199.99949.04943.99878||.06685.99776.08426.99644| 10 51.01483.99989| .03228.99948.04972.99876.06714.99774.08455.99642 52.01513.99989 53.01542.99988 54 01571.99988 55 01600.99987 .03257.99947||.05001|.99875||.06743.99772.08484.99639 .03286|.99946||.05030|.99873 .06773.99770 |.08513.99637 .03316.99945.05059.99872.06802 .99768 .08542.99635 .03345.99944||.05088|.99870||.06831.99766.08571|.99632 .05117.99869||.06860.99764 .03374.99943.05117.99869 .06860.99764.08600.99630 .01658.99986.03403.99942.05146.99867 06889.99762.08629.99627 .03432.99941.05175.99866 • |.05175.99866||.06918.99760.08658.99625 .03461.99940.05205.99864 06947.99758 56.01629.99987 57 58.01687.99986 59 .01716.99985 60 | .01745.99985 Cosin Sine Cosin Sine 89° 88° .03490.99939||.05234.99863|| .06976'.99756 06976.99756.08716.99619 Cosin Sine Cosin Sine 87° 86° .08687.99622 Cosin Sine 85° 9 8 QƆ76 LO HAD 285 TABLE X.-SINES AND COSINES. 0 5° 6° 7༠ 8° .13917.99027 9° Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin 08716.99619.10453.9945212187.99255 .15643 15643.98769, 60 1.08745.99617||.10482.99449.12216.99251.13946.99023.15672.98764 59 2 .08774.99614.10511.99446 3.08803.99612||.10540|.99443 4 5 6 .12245.99248.13975.99019.15701.98760 58 .12274.99244.14004.99015.15730.98755 57 08831.99609 .10569.99410.12302.99240 .12302.99240 |.14033.990011 .15758 .98751 56 | 08860.99607||.10597.99437|| .12331|.99237.14061|,99006 .08889|.99604||,10626|.99434 | .12360´.99233 .14090.99002 .08918 99602||.10655|.99431|| .12389.99230´ .14119.98998 8.08947.99599||.10684,99428||.12418¦,99226 7 .15787|.98746 55 .15816.98741' 54 .15845.98737 53 .14148.98994 .15873.98782 52 .14177.98990.15902.98728 51 .14205.98986.15931.98723 50 .14234.98982.15959.98718 49 14263.98978 .15988 .15988.98714 48 .12562.99208.14292.98973 .16017.98709 47 9.08976'.99596||.10713.99424 .12447.99222 10.09005.99594||.10742.99421 .12476.99219 11.09034.99591|| .10771.99418 | .12504.99215 12.09063.99588||.10800.99415|| .12533.99211 13.09092.99586||.10829.99412|| .12562). 99208, 14.09121.99583||.10858.99409|| .12591.99204 .09150.99580||.10887.99406 .12620.99200 16.09179.99578.10916.99402 .12649.99197 15 18.09237.99572|| .10973¦.99396 17.00208.99575||,10945|.99399||.12678.99193 .14320 .98969| .16046.98704 46 .14349.98965.16074.98700 45 .14378.98961 98961 .16103.98695 44 .14407.98957.16132.98690 43 .12706.99189 |.14436.98953| .16160.98686|42 19.09266.99570|| .11002].99393||,12735.99186, .14464.98948.16189.98681 41 20.09295.99567|| .11031.99390 14493.98944.16218.98676 40 21 24 .12764.99182, .11608.98927.16333.9865736 1 .09324.9956-1 .11060.99386 .12793.99178.14522.98940 .16246.98671 39 22.09353.99562||.11039.99383|| .12822.99175 | .14551.98936|| .16275 16275.98667 38 23.00382.99559 11118.90330 | .12851.99171 .12851.99171 .14580.98931 .16304.98662 37 09411.99556.11147.99377 .11147.99377|| .12880.99167 25.094401.99553 .99553|| .11176.99374|| .12908.99163 .14637.98923 .16361.98652 35 .09469.99551||.11205.99370||.12937.99160.14666.98919.16390.98648 34 .09498.99548|| .11234.99367.12966.99156.14695 .12966.99156.14695.98914.16419 .98643 33 28 09527.99545||.11263.99364|.12995.99152| .14723.98910.16447.98638 32 29.09556.99542.11291.99360 .13024.99148 14752.98906 .16476.98633' 31 30.09585.99540.11320.99357 .13053.99144 .14781.98902.16505 .98629: 30 .16505.98629; 31.09614.99537|| .11349.99354 | .13081.99141|| .14810.98897 16533.98624 29 09642|,99534|| .11378|.99351 .11378.99351.13110.99137 .14838.98893 16562.98619, 28 .16591.98614 27 34.09700.99528.11436.99344.13168.99129 .14896.98884.16620.98609 26 32 .09671.99531||.11407.99347 .13139.99133.14867.98889 .14925.98880 · 16648.98604 25 09729.99526||.11465.99341 .11465.99341).13197.99125|| .14925 36.09758.99523||.11494.99337 |.13226.99122||.14954|,98876| .16677.98600 37.09787.99520||.11523|.99334|| .13254.99118.14982.98871.16706.98595 38.09316.99517.11552.99331.13283.99114.15011.98867 .16734.98590 22 39.09345.99514.11580.90327 .13312.99110.15040.98863.16763.98585 21 .09874.99511.11609.99324.13341.99106 .15069.98858 .16792.98580 20 09903.99508.11638.99320 .13370.99102 .15097.9885-1 42.09932.99506.11667.99317.13399.99098.15126.98849.16849.98570 .13370.99102.15097.98851 .16820.98575 19 18 43.09961.99503||.11696.99314.13427.99094 .15155.98845.16878 .98565 17 44.09990.99500||.11725.99310 .18456.99091 .15184.98841 98841 .16906 .98561| 16 45.10019.99497.11754 99307.13485.99087 13485.99087.15212 .15212.98836||.16935.98536 15 10048.99494.11783.99303 .13514.99083 .15241.98832.16964 .98551| 14 47.10077.99491.11812.99300.13543.99079 .15270.98827 .98827|, .16992.98546 13 48.10106.99488|| .11840.99297.13572.99075|| .15299 .15299.98823 .17021 .98541 12 49.10135.99485.11869.99293.13600.99071 .15327.98818.17050.98536 11 50.10164.99482.11898.99290.13629.99067 .15356.98814 .17078.98531 10 51.10192.99479|| .11927.99286||.13658.99063|| .15385 .15385.98809 52.10221.99476||.11956.99283.13687.99059|| 15414.98805 53.10250.99473.11985.99279.13716.99055 .15442.98800 54.10279.99470.12014.99276.13744.99051 55.10308.99467||.12043.99272.13773,99047|| .15500 12071.99269.13802.99043 .99043|| .15529 .15529 .98787 .12100.99265.138311.99039 .15557.98782 56.10337.99464 57 10366.99461 .15471.98796 .15500.98791 .17107.98526 .17136 .98521 .17164.98516 .17193.98511 6 9 8 7 5 .17222 .98506 .17250 .98501 4 .17279 .98496 3 2 58.10395.99458 .12129.99262.13860.99035 .15586.98778.17308 .98491 59.10424.99455] .12158.99258.13889.99031 | .15615 13889.99031.15615 .98773.17336 .98486 1 10453.99452.12187.99255.13917.99027|| .15643 .98769 .15643 .98769.17365.98481 Cosin Sine Cosin Sine Cosin | Sine Cosin Sine Cosin Sine 60 0 84° 83° 82° 81° 80° 286 TABLE X.-SINES AND COSINES. 10° 11° 12° 13° 14° Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin 0.17365 .98481||.19081|.98163 .19081.98163.20791.97815 .20791.97815.22495.97437 .24192.97030| 60 1.17393.98476||.19109.98157|.20820.97809 ‚22523|.97430|| .24220|.97023 59 2.17422.98471 2.17422 .98471.19138.98152.20848.97803 19138.98152 |.20848.97803|.22552.97424 3.17451.98466 3.17451 .98466.19167.98146 19167.98146 |.208771.97797 4.17479 .98461||.19195.98140.20905.97791 .17508 .98455.19224.98135 6.17537.98450 .19252.98129 |.20962.97778 |.22665 07398 7.17565 .98445 19281.93124||.20990.97772|.22693|.97391 .22552.97424.24249.97015. 58 .22580.97417 .24277.97008 57 .22608.97411 20933.97784 .22637.97404 8.17594.98440||.19309.98118.21019].97766 22722.97384 24305.97001 56 .24333.96994 55 .24362.96987 54 .24390.96980 53 .24418.96973 52 9.17623.98435||.19338|.98112||.21047|.97760||.22750|.97378 .24446.96966 51 10.17651.98430|| .19366.98107||.21076|.97754 11.17680.98425 .19395.98101.21104.97748 12.17708.98420 .19423.98096 1.17737.98414.19452.98090.21161.97735 14.17766.98409.19481.98084.21189.97729 15.17794,98404||.19509.98079.21218.97723 16.17823.98399 18.17880.98389 22778.97371 .24474.96959 50 22807.97365 24503.96952 49 .21132.97742 22835.97358 .24531.96945 48 22863.97351 .24559.96937: 47 22892.97345|| .24587|.9693046 22020.97338|| .24615|,9692345 22948.97331 .24644.96916 44 22977.97325 .24672.96909 43 .23005.97318 .24700.96902 42 23033.97311||.24728.96894 41 19538.98073 .21246.97717 17.17852.98394||.19566.98067|.21275.97711 · .19595.98061||.21303.97705 19.17909.98383.19623.98056.21331.97698 • 20.17937.98378.19652.98050 .19652|.98050|| .21360.97692 23062.97304 .24756.96887 40 21.17966.98373.19680.98044 21388.97686.23090.97298 .23090.97298||.24784|.9688039 23 22 .17995.98368 .19709.98039 .21417.97680 23118.97291 24813.96873 38 .18023.98362.197371.93033 .21445.97673 .23146.97284 .24841.96866 37 24.18052.98357.19766).98027 .19766].98027|| .21474.97667 .21474.97667.23175.97278 25.18081.98352 19794.98021 21502.97661 26.18109.98347.19823.98016 .21530.97655 27.18138.98341 .19851.98010|| .21559.97648 28.18166.98336.19880.98004.21587.97642 .23175.97278|| .24869.96858|36 23:203.97271 .24897.96851 35 .23231).97264 .21925.96844 34 .23260.97257|| .24954.96837 33 23288.97251 .24982.96829 32 29.18195.98331.19908.97998 .21616.97636 23316.97244|| .25010|.96822: 31 30.18224.98325.19937|.97992 .21644.97630.23345.97237.25038.96815 30 31.18252.98320 .19965.97987 21672.97623 .23373.97230 25066.96807 29 18281.98315.19994.97981.21701.97617 .23401.97223 .25094.96800 28 33.18309.98310 20022.97975.21729.97611 .23429|.97217 23429.97217|| .25122.96793 27 34.18338.98304.20051.97969.21758.97604 .23458.97210 .25151.96786 26 .23486.97203 25179.96778 25 35.18367.98299||.20079.97963.21786.97598.23486.97203 .18395.98294.20108.97958 .21814.97592 .23514.97196.25207|.96771 24 37.18424.98288.20136.97952.21843.97585 23542.97189 .25235.96764! 23 38.18452.98283.20165.97946 .21871.97579 .23571.97182.25263.96756 22 .18481.98277||.20193.97940.21899.97573 .23599.97176 .23599.97176||.25291|.96749 21 40.18509.98272 20222.97934 21928.97566 .23627.97169.25820.96742| 20 41.18538.98267 20250.97928 .21956.97560 .23656.97162 25348.96734 19 42.18567.98261||.20279|,97922 .21985.97553 23684.97155 .25376.96727 18 43.18595.98256.20307.97916 22013.975-47 .23712.97148||.25404|.96719 17 22041.97541 23740.97141 .25432.96712; 16 45.18652.9824520336|.97910. 22070.97534 .23769|.97134||.25460|.96705 15 46.18681.98240.20393.97899 22098.97528 .23797|,97127|| .25488.96697 14 44.18624.98250 47.18710.98234 .20421.97893.22126.97521 .18738.98229||.20450.97887 .20450.97887|.22155.97515 49 .18767.98223 20478.97881 .22183.97508 23825.97120 |.97120||| .25516.96690 13 .23853.97113 .25545.96682 12 23882.97106.25573.96675 11 50.18795.98218.20507.97875 22212 .97502 .23910.97100 .25601.96667, 10 51.18824.98212.20535.97869 52.18852.98207 20563.97863 53.18881.98201 20592.97857 • 54.18910.98196 55.18938.98190 20620.97851| .20649.97845 56.18967.98185 .20677.97839 57.18995.98179.20706.97833 58.19024.98174 .20734.97827 59.19052.98168|| .20763.97821 60.19081.98163 63|| 201 .20791.97815 Cosin Sine Cosin Sine 79° 78° 22240.97496|| 22268.97489 .23938.97093 .25629.96660 9 .23966.97086.25657 .96653 22297.97483 23995|.97079 || .25685.96645 22325.97476 .24023.97072 .25713 .96638 6 22353.97470.24051.97065 .22382.97463 .24079.97058 .22410¦.97457|| .24108.97051 22438.97450|| .24136.97044 8 7 5 4 3 .25741 .96630- .25769.96623 .25798 .96615 .25826 .966082 22467.97444.24164.97037|.25854 .96600 1 22495 Cosin Sine 237 7170 24192 .97030 Cosin Sine .25882 .96593 0 Cosin Sine 76° 75° 287 TABLE X.-SINES AND COSINES. 15° 16° 17° 18° 19° Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin .25882.96593.27564.96126 29237.95630.30902.95106 .32557.94552 60 .29265.95622.30929.95007 25910.96585.27592 .96118 .30929.95027.32584.94542 59 2.25938.96578 27620.96110.29293.95613.30957.95088 3.25966.96570||.27648.96102.29321.95605.30985.95079 4 5 .25994.96562||,27676|,96094 29348.95596.31012.95070 26022.96555 .27704.96086.29376.95588 .31040.95061 6.26050.96547|| .27731.96078 29404.95579 .31068.95052 777 26079.96540||.27759·96070 8.26107.96532 27787.96062 .32612.94533 58 .32639.94523 57 .32667.94514 56 .32694.94504 55 .32722.94495 54 32749.94485 53 • .29432.95571.31095.95043 9.26135.96524 .27815.96054.29487.95554.31151.95024.32804 10.26163.96517||.27843.96046.29515.95545 11.26191.96509||.27871.96037.29543.95536 .32777.94476| 52 .32804.94466 51 .95015.32832.94457| 50 12.26219.96502.27899.96029.29571.95528 .29571.95528.31233.94997 32887.94438 48 13.26247.96494|.27927.96021.29599.95519 14.26275.96486.27955.96013.29626.95511 .29460.95562.31123.95033 .29515.95545.31178.95015 .31206.95006.32859.94417 49 .31261.94988 .32914.94428 47 .31289.94979 .32942.94418 46 15.26303 26303.96479 .27983.96005 95502.31316 .29654.95502 .31316.94970 .32969.94409 45 16.26331.96471 28011.95997.29682.95493 .29682.95493 .31344 .94961 .32997.94399|44 17.26359.96463 .28039.95989 .29710.95485|| .31372.94952| .33024|,94390|43 18.26387.96456||.28067.95981|| .29737|.95476 29737.95476.31399.94943 .33051.94380 42 19.26415.96448.28095.95972| 29765.95-467 .31427.94933 .33079.94370 41 20.26443.96440.28123.95964 29793.95459 21.26471.96433.28150.95956 24 26 .26500|.96425 .29821.95450 .28178.95948|| .29849 29849.95441 .31510.94906 .33161.94342 38 .28206.95940 29876.95433 .31537.94897.33189.94332 37 .28234.95931 .29904.95424.31565.94888.33216.94322 36 .28262.95923 29932.95415 .31593.94878 .26612|,96394.28290.95915 .29960.95407.31620.94869 33244.94313 35 .33271.94303 31 .33298.94293 33 33326.94284 32 .31454.94924 33106.94361 40 31482.94915 33134.94351 39 23.26528.96417 .26556.96410 25 26584.96402 .26640.96386||.28318.95907 28.26668.96379 .29987.95398 28346.95898 .30015.95389 29 .26696.96371 28374.95890 .30043.95380 30.26724.96363 .284021.95882 .30071.95372 .31648.94860 .31675.94851 .31703.94842 .33353.94274 31 .31730.94832.33381.94264 30 .30098.95363.31758.94823.33408.94254 29 23457.95865.30126.95354.31786.94814.33436.94245 28 33.26808.96340.28485.95857.30154.95345.31813.94805.33463.94235 27 28513.95849.30182.95337.31841.94795 31 26752.96355.28429.95874 32.26780.96347 31.26836.96332 35 .33490.94225 26 26864.96324 28541.95841 .30209.95328.31868.94786.33518.94215 25 36.26892.96316.28569.95832.30237.95319.31896.94777 .33545.94206 24 37.269201.96308 .28507.95824 .30265.95310.31923.94768.33573.94196 23 38.26948.96301 .28625.95816.30292.95301.31951.94758.33600.94186 22 39.26976.96293 .28652.95807.30320.95293.31979.94749.33627.94176 21 .30348.95284 32006.94740.33655.91167 20 40.27004.96285 .28680.95799 41.27032.96277 .28708.95791 .30376.95275.32034.94730.33682.94157 19 42.27060.96269 43.27088.96261 44.27116.96253 45.27144.962461 46.27172.96238 47.27200.96230 48.27228.96222 .28736.95782||.30403|.95266||.32001.94721||.33710.94147| 13 .28764.95774||.30431.95257|| .32089|.94712||.38737|.94137|17 28792.95766.30459.95248.32116.94702.33764.94127 16 28820.95757.30486.95240 28847.95749.30514.95231 .28875.95740||.30542.95222||.32199.94674||.33846.94098 13 .28903.95732||.30570.95213|.32227.94665 49.27256.96214 .28931|.95724|| .30597.95204||.32254.94656 50 27284.96206.28959.95715.30625.95195.32282.94646|.33929|.94068| 10 .32309.94637 51.27312.96198 .28987.95707 .30653.95186 52.27340.96190.29015.95698.30680.95177|| .32337.94627 53.27368.96182|.29042.95690.30708.95168 54.27396.96174.29070.95681 9 .32144.94693 .33792.94118 15 .32171.94684 .338191.94108 14 33874.94088 12 33901.94078 11 33956.94058 .33983.94049 8 D .30736.95159 32364.9461834011.94039 .32392.94609.34038.94029 56 57 .27424.96166|.29098.95673 .30763|.95150 |,32419|.94599 27452.96158.29126.95664|| .30791.95142 .34065.94019 .32447.94590||.34093.94009 27480.96150 .29154.95656| .30819.95133 .30819.95133 .32474.93580.34120.93999 59 27536.96134 + 58.27508.96142 GO.27564.96126 29237.95630 .29182.95647 .30846.95124 32502.94571 .34147.93989 2 .29209.95639 .30874.95115 .32529.94561|| .34175.93979 .30902.95106 .32557.94552.34202.93969 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine 74° 73° 72° 71° 70° 288 TABLE X.-SINES AND COSINES. 20° 21° 22° 23° Sine Cosin Sine Cosin Sine Cosin Sine Cosin .34202.93969.35837.93358.37461.92718.39073.92050 1.34229.93959||.35864.93348||.37488.92707.39100.92039 2.34257.93949||.35891.93337||.37515.92697 5.34339.93919||.35973.93306 · 39127.92028 • 24° Sine Cosin .40674.91355 60 .40700.91343 59 .40727.91331 58 57 3.34284.93939.35918.93327.37542.92686||.39153.92016|| .40753.91319 4.34311.93929||.35945.93316||.37569.92675||.39180.92005|| .40780.91307 56 .35973.93306.37595.92664 37595.92664||.39207|.91994|| .40806.91295 55 6.34366.93909.36000.93295.37622.92653 .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 .37676.92631|| .39287.91959||.40886.91260 52 9.34448.93879||.36081.93264||.37703.92620.39314.91948 .39314.91948.40913.91248 51 10.34475.93869.36108.93253.37730.92609 37730.92609||.39341.91936|| .40939.91236 50 11.34503.93859.36135.93243.37757.92598 36135.93243|| .37757.92598.39367.91925.40966,91224 40 12.34530.93849.36162.93232.37784.92587 .39394.91914.40992 .91212 43 13.34557.93839.36190.93222.37811.92576.39421.91902.41019 .91200 47 14.34584.93829||.36217.93211||.37838|.92565||.39448.91891|| .41045.9118846 15.34612.93819||.36244.93201|| .37865.92554 .39474.91879|| .41072.91176 45 34639.93809 .36271.93190.87892.92543.39501.91868 37892.92543||.39501.91868.41098.91164 | 44 17.34666.93799 .36298.93180.37919.92532.39528.91856 .41125.91152 43 18.34694.93789.36325.93169.37946.92521 .89555.91845 .41151.91140 42 19.34721.93779|| .36352.93159 37973.92510|| .39581.91833|| .41178.91128 41 20.34748.93769 .36379.93148.37999.92499 .39608.91822.41204.91116 40 16 • 21.34775.93750 5.93750.36406.93137|| .38026.92488||.39635.91810.41231.91104 39 22.34803.93748|| .36434.93127||.38053.92477|| .39661.91799.41257.91092 38 .34830.93738|| .36461.93116|| .38080.92466||.39688.91787 .41284.91080 37 24.34857.93728 36488.93106||.38107.92155||.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 .34939 |.93698||.36569.93074.38188.92421||.39795.91741 34966.93688||.36596.93063.38215.92410.39822.91729 .39822.91729.41416.91020 32 29.34993.93677||.36623.93052.8241.92399||.39848.91718||.41443.91008 27.34939.93698 · 34 .41390.91032 33 31 30.35021.93667|| .36650.93042.38268.92388.39875.91706||.41469.90996 30 31.35048.93657 .36677.93031.38295.92377 39902.91694.41496.90984 29 32.35075.93647 .36704.93020 38322.92366 .39928.91688.41522.90972 28 33.35102.93637.36731.93010 38349.92355.39955.91671|| .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 36.35184.93606.36812.92978.38430.92321.40035.91636.41628.90924 24 37.35211.93596.36839.92967 .36839.92967.38456.92310.40062.91625||.41655.90911 38.35239.93585 36867.92956.38483.92299.40088.91613.41681.90899 22 39.35266.93575 36894.92945 38510.92287.40115.91601.41707.90887 21 35293.93565.36921.92935 .36921.92935|| .38537.92276.40141.91590 41734.90875| 20 41.35320.93555.36948 .92924 .36948.92924¦¦.38564.92265 38564.92265.40168.91578 .41760.90863 19 42.35347.93544.36975.92913 38591.92254.40195.91566.41787.90851 18 38617.92243.40221.91555 .41813.90839 17 .38644.92231||.40248.91543 .41840.90826 16 .38671.92220.40275.91531 .40275.91531||.41866.90814 15 .38698.92209.40301.91519 .40301.91519||.41892.90802 14 .38725.92198 .40328.91508||.41919.90790 13 .38752.92186 .40355.91496||.41945.90778 12 43.35375.93534.37002.92902 44.35402.93524 .37029.92892 45.35429.93514.37056.92881 46.35456.93503 .370831.92870 47.35484.93493.37110.92859 48.35511.93483 .37137.92849 51 98 588. 49.35538.93472.37164.92838 · 38778.92175 .40381.91484||.41972.90766 11 50.35565.93462 .37191.92827 .38805.92164 • .35647.93431|| .37272 37272.92794 38886.92130 .40408.91472.41998.90753 10 35592.93452 .37218.92816.38832.92152.40434.91461 .42024.90741 52.35619.93441 .37245.92805 38859.92141 .40461.91449 54.35674.93420.37299.92784 55.35701.93410.37326.92773 56.35728.93400.37353.92762 57.35755.93389.37380.92751 58 38912.92119.40514.91425.42104.90704 38939.92107.40541 .91414.42130.90692 35782.93379 37407.92740 39020 .92073 59.35810.93368 60.35837.93358 9886O LO HO .40488 .91437 .42051.90729 .42077.90717 5 38966.92096.40567.91402 38993.92085 .40594 .91390 .42156.90680 .42183.90668 4 3 · 40621 .91378.42209.90655 2 .37434.92729 .37461.92718 Cosin Sine Cosin Sine 39046 .92062 .92062.40647 .91366 .89073 .92050 92050 |.40674.91355 Cosin Sine Cosin Sine 42235.90643 42262.90631 0 Cosin Sine 69° 68° 67° 66° 65° 289 TABLE X.-SINES AND COSINES. 25° 26° 27° 28° Sine Cosin Sine Cosin Sine Cosin Sine Cosin 29° Sine Cosin 0.42262.90631 1 .43837.89879.45399.89101 .42288.90618.43863.89867 .45425.89087.46973.88281 2.42315.90606.43889.89854.45451.89074.46999.88267 .45399.89101|| .46947.88295 48481.87462 60 .48506.87448 59 .48532.87434 53 3 42341.90594.43916.89841 .45477.89061 .47024.88254 .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 9.42499.90520 10.42525.90507 11.42552.90495 .44124.89739.45684.88955 .47178.88172.48710.87335 51 .47229.88144.48761.8730649 1.88117.48811.87278 47 .44046.89777 .45606|.82995 .47153.88185.48684.87349 52 .44072.80764.45632.88981 .44098.89752.45658.88968.47204.88158.48735.87321 50 12.42578.90483 .44151.89726.45710.88942.47255.88130.48786.87292 48 13.42604.90470 .44177.89713.45786.88928.47281.88117 14.42631.90458.44203.89700.45762.88915 .45762.88915|| .47306.88103|| .48837.87264| 46 15.42657.90446.44229.89687.457871.88902 .45787.88902|| .47332.88089 .48862.87250 45 .48888.87235 44 17.42709.90421.44281.89662.45839.88875.47383.88062 .88875||.47383.88062||.48913.87221 43 18.42736,90-403 44307.89649 .45865.88862|| .47409.88048||.48938.87207| 42 .42762.90396 .44333.89636 .45891.88848.47431.88034|| .48964.87193 41 .42788.90383.44359.89623.45917.88835 .45917.88835||.47460.88020|| .48989.8717840 16.42683.90433.44255.89674.45813.88888.47358.88075 19 20 21 42815.90371 231.42867.90346 .44385.89610.45942.88822.47486.88006 .49014.87164 39 22.42841.90338 .44411.89597||.45968.88808.47511|.87993|| .49040.87150 38 .44437.89584|| .45994.88795|| .47537.87979|| .49065.87136 | 37 .44464.89571 .46020.88782 .47562.87965.49090.87121 36 .44490.89558|| .46046.88768 24.42894.90334 .42920.90321 25 26 1.42946.90309 28 .46123.88728.47665.87909 .47588.87951.49116.87107 35 .44516.89545.46072.88755.47614.87937.49141.87093 34 27 .42972.90296 .44542 .89532|| .46097.88741|| .47039.87923||.49166.87079 33 .42999.90284 .44568 .89519|| .46123 .47665.87909||.49192.8706432 29.43025.9 271.44594.89506 .46149.88715.47090.87896.49217.87050 31 30.43051.90259 .44620.89493 46175.88701 .47716.87882.49242.87036 30 .44646.89480|| .46201.88688.47741.87868.49268.8702129 31.43077.90246 32.431041.90233 33.43130.90221 .44672.89467.46226.88674.47767.87854 .44698.89451 .46252.88661 .47767.87854.49293.87007 28 .47793.87840||.49318.86993 27 34.43156.90208 .44724 .89441|| .46278.88647.47818.87826.49344.86978 26 35.43182.90196.44750.89428.46304.8863-1 .44750.89428|| .46304|.88631|| .47844.87812|| .49369.86964 36.43209.90183.44776.89415.46330.88620.47869.87798.49394.86949 24 23 43261.90158.44828.89389 .46381.88593|| .47920|.87770|| .49445.86921 .43287|.90146|| .44854.89376||,46407.88580|| .47946.87756|| .49470.86906| 21 40.43313.90133|| .44880.89363|| .46433|.88566|| .47971.87743||.49495.86892 20 41.43340.90120 37.43235.90171 .44802.89402||.46355.88607|| .47895.87784.49419.86935 39 42 43 • .44906.89350 .46458.88553 .47997.87729 .48048.87701.49571.86849 .49521.86878 19 43366.90103.44932.89337|| .46484.88539||.48022.87715.49546.86863| 18 17 .43392|.90095|| .44958.89324||.46510|.88526||.48048.87701 44.43418.90082||.44984.89311|| .46536.88512||.48073.87087 .46536.88512.48073.87687.49596.86834 16 .46561.88499.48099.87673 45.43445.90070.45010 .89298 48099.87673.49622.86820 15 48124.87659.49647.86805 14 46.43471.90057|| .45036.89285|| .46587|.88485 47.43497.90045 .45062.89272|| .46613.88472.48150.87645||.49672.86791| 13 48.43523.90032 .45088.89259.46639.88458.48175.87631||.49697.86777 12 .48201 87617 49.43549.90019 || .45114.89245.46664|.88445 .49723.86762 11 .482261.87603.49748.86748 10 50.43575.90007|| .45140.89232 .46690.88431|| .48226.87603 51.43602.89994|| .45166.89219 .46716.88417 .48252.87589||.49773.86733 52.43628.89981 53 .45192.89206.46742.88401 .46742.88404|| .48277.87575 .46767|.88390|| .48303|.87561 .43654.89968|| .45218.89193 .45218.89193.467671.88390 54.43680.89956||.45243.89180.46793.88377 .46793.88377.48328.87546|| .49849|.86690 .49874.86675 .48354.87532 .45269.89167 .46819.88363 .49899.86661 .48379.87518 45295.89153.46844.88349 57.43759.89918||.45321.89140||.46870|.88336||.48405.87504.49924.86646 55.43706.89943 56.43733.89930.45295.89153 58.43785.89905|| .45347.89127.46896.88322 9 .49798.86719 8 .49824.86704 7 6 5 4 3 .48430.87490 .49950.86632 2 .49975.86617 1 60.43837.89879 .45399.89101 .46947.88295 48481.87462 .50000.86603 0 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine 64° 63° 62° 61° 60° .48456.87476 .45373.89114.46921.88308 59.43811.89892.45373.89114 290 TABLE X.-SINES AND COSINES. 30° 31° 32° 33° Sine Cosin Sine Cosin Sine Cosin Sine Cosin 0.50000.86603 51504.85717 52992.84805|| .54464.83867 34° Sine Cosin .55919.82904 60 1.50025.86588||.51529|.85702 .53017.84789 .54488.83851 .55943.82887: 59 2.50050.86573||,51554|.85687||,53041.84774|| .54513.83835 .53041.84774.54513.83835.55968.82871 58 3.50076.86559||,51579|,85672||.53066.84759|| .54537.83819||.55992|.82855 | 57 4.50101.86544|| .51604.85057|| .53091.84743 | .54561.83804|| .56016|.82839 50 5.50126.86530||.51628.85642|| .53115.84728|| .54586.83788 .54586.83788.56040.82822' 55 6.50151.86515||,51653|.85627|| .53140.84712|| .54610.83 .53140.84712.54610.83772.56064.82806 54 7.50176.86501||.51678|.85612||.53164|.84697||.54635.83756||.56088|.82790 .53189.81681 .54059.83740 8.50201.86486.51703.85597 9.50227.86471|| .51728.85582 10.50252.86457|| .51753.85567 11.50277.86442|| .51778.85551 12.50302.86427.51803.85536 13.50327.86413|| .51828.85521 14.50352.86398.51852.85506 15.50377.86384|| .51877.85491|| .53361.84573.54829.83629 53 .56112.82773 52 .53214.84666|| .54683.83724||.56136.82757 51 .53238.84650.54708.83708 .56160.82741 50 .53263.84635.54732.83692|| .56184.82724 49 .53288.84619.54756.83676.56208.82708 48 .53312.84604.54781.83660||.56232.82692 47 .53337.84588.54805.83645.56256.82675 46 .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|.8-4526 .53435.8-4526.54902.83581 .54902.83581.56353.82610' 12 19.50478.86325|| .51977.85431 53460.84511|| .54927|.83565|| .56377.82593 41 20.50503.86310|| .52002.85416 .52002.85416.53484.84495.54951.83549 53484.81495|| .54951|.83549|| .56401.82577. 40 21.50528.86295||.52026.85401||.53509.84480.54975.83533.56425 82561 39 50553.86281||.52051.85385 .53531.84464 .51999.83517.56449.82544 33 23.50578.86266 52076|,85370||.53558|.84448.55024.83501 |.56473.82528 37 24 .50603.86251 .52101.85355 .53583.84433,55048.83485.56497.82511 36 25.50628.86237 .52126.85340 .53607.84417.55072.83469.56521.82495 22 • 28.50704.86192 29.50729.86178 30.50754.86163 .56593 24 35 26.50654.86222.52151.85325||.53632.84402.55097.83453.56545.82478 34 27.50079.86207 .52175.85310 .53656.81386.55121.83437.56569.82462 33 52200.85294.53681.84370.55145.83421 .56593.82446 32 52225.85279|| .53705.84355 .53705.84355.55169.83405 .55169.83405|| .56617 .56617.82429 31 .52250.85261.53730.84339.55194.83389 .55194.83389|| .56641|.82413 | 30 31.50779.86148 52275.85249 53754.84324 | .55218.83373|| .56665|.82396 29 32.50804.86133 .52299.85234.53779.84308 .53779.84308.55242.83356 55242.83356|| .56689.82380 28 33.50829.86119 52324.85218|| .53804.84292 .53804.84292.55266.83340 .56713.82363 27 34.50854.86104|| .52349.85203|| .53828.84277.55291.83324|| .56736|,82347 26 35.50879.86089|| .52374.85188.53853.84261|,55315.83308||.56760 .56760.82330 25 36.50904.86074.52399.85173|| .53877.84245 .55339.83292|| .56784.82314 37.50929.86059||.52423.85157 .53902.84230.55363.83276.56808.82297 23 38.50954.86045|| .52448.85142 .52448.85142.53926.84214.55388.83200.56832.82281 22 39.50979.86030.52473.85127.53951.84198.55412.83244 53951.84198.55412.83244 | .56856|.82264_21 40.51004 86015.52498.85112.53975.84182.55436.83228.56880.82248 20 41.51029.86000|| .52522.85096 .54000.81167|| .55460.83212|| .56904 55460.83212.56904.82231 19 42.51054.85985 .52547.85081|| .54024.84151 .55484.83195|| .56928|.82214| 18 52572.85066|| .54049.84135 55509.83179 .5695282198 17 .56976.82181 16 .52621.85035.54097.84104 .54097.84104|| .55557.83147||.57000.82165' 15 .52646.85020.54122.84088 54122.84088||.55581.83131||.57024,82148 14 .52671.85005 .54146.84072|| .55605|.83115||.57047.82132 13 48.51204.85896 .52696.S.389 .54171.84057|| .55630|.83098|| .57071.82115 12 49 51229.85881|| .52720.84974|| .54195.84041.55654.83082|| .57095 50.51254.85866.52745.84959.54220.84025 .52745.84959||.54220.84025|| .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 .51329.85821|| .52819.84913.54293.83978.55750.83017||.57191.82032 54.51354.85806 52844.84897.54317.83962.55775.83001 43.510791.85970 44.51104.85956 .52597.85051 .51073.84120 .55533.83163 45.51129.85941 46.51154.85926 47.51179.85911 + * 55654.83082.57095.82098, 11 55.51379.85792.52869.84882.54342.83946.55799.82985 .52893.84866.54366.83930 56.51404.85777 .57215.82015 .57238.81999 .55823.82969||.57262|.81982 57.51429.85762||.52918.84851||.54391.83915||,55847.82953||.57286.81965 58.51454.85747.52943.84836 .52943.84836||.54415.83899|| .55871|.82936||.57310|.81949 59.51479.85732 52967.84820||.544401.83883|| .55895.82920.57334.81932 60.51504.85717 52992.84805 .54464.83867 55919.82904.57358.81915 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine 59° • 58° 57° 56° 55° 8 291 TABLE X.-SINES AND COSINES. 35° 36° 37° 38° 39° Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin 0.57358.81915 .58779.80902.60182.79864||.61566.78801 .62932.77715| 60 1 6 .57381.81899||.58802|.80885|| .60205|.79846||.61589.78783||.62955|.77696|59 .57405.81882|| .58826.80867|| .60228.79829||.61612.78765|| .62977.77678 58 3.57429.81865|| .58849.80850||.60251.79811.61635.78747||.63000|.77660| 57 .57453.81848.58873.80833|| .60274.79793.61658.78729||.63022|.77641| 56 .57477.81832||.58896.80816|| .60298.79776.61681.78711|| .63045.77623| 55 .57501.81815.58920.80799 .60321.79758|| .61704|.78694||.63068.77605 54 7.57524.81798.58943.80782|| .60344.79741 .60344.79741||.61726.78676||.63090.77586 53 8.57548.81782|| .58967.80765|| .60367.79723||.61749.78658.63113.77568 52 9.57572.81765|| .58990.80748 .58990.80748.60390.79706|| .61772.78640.63135.77550 51 10.57596.81748|| .59014.80730|| .60414].79688 .59014.80730.60414.79688.61795.78622 .61795.78622|| .63158.77531 50 11.57619.81731|| .59037.80713|| .60437.79671||.61818.78604|| .63180.77513] 40 .57643.81714|| .59061.80696|| .60460|,79653||.61841.78586||.63203.77494 48 12 13.57667.81698.59084.80679|| .60483.79635 .60483.79635||.C1864.78568|| .63225.77476 47 14.57691.81681 59108.80662|| .60506.79018||.61887|.78550||,63248.77458 46 15.57715.81664|| .59131.80644 .59131.80044|| .60529.79600||.61909.78532.63271.77439|45 16 57738|.81647|| .59154.80627|| .60553.79583||.61932.78514||.63293.77421 44 17.57762.81631 .59178.80610.60576.79505|| .61955.78496.63316.77402 43 18.57786.81614|| .59201.80593|| .60599.79547|| .61978.78478.63338.77384| 42 19.57810.81597.59225.80576 .60622.79530 62001.78460 20.57833.81580.59248.80558 59248.80558|| .60645.79512.62024.78142 21 .63361.77366 41 .63383.77347 40 .60668.79494.62046.78424.63406.77329 39 57857.81563.59272.80541.60668.79494 | .63451.77292 37 .63473.77273 36 .63496.77255 35 22 .57881.81546|| .59295.80524.60691.79477|| .62069.78405||.63428.77310 38 57904|.81530|| .59318.80507|| .60714.79459.62092.78387 24.57928.81513.59342.80489.60738.79441 59342.80489 | .60738.79441|| .62115.78369 25.57952.81496|| .59365.80472 .59365.80472 | .60761|.79424|| .62138.78351 26.57976.81479.59389.80455 .60784.79406 .59389.80455 .60784.79406.62160.78333.63518.77236 34 27.57999.81462 .59412.80438.60807.79358.62183.78315|| .63540.77218 33 28.58023.81445|| .59436.80420|| .60830.79371||.62206.78297.63563.77199 32 29.58047.81428.59459.80403 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 58141.81361|| .59552.80334|| .60945.79282||.62320|.78206||.63675|,77107| 27 .58165.81344|| .59576.80316.60968.79264.62342.78188||.63698.77088 26 35.58189.81327|| .59599.80299|| .60991.79247 36.58212.81310.59622|.80282|| .61015.79229 37.58236.81293|| .59646|.80264 .59646.80264|| .61038.79211 38 .62365.78170.63720.77070 25 62388.78152.63742.77051 24 .62411.78134.63765.77033 23 · .62456.78098||.63810.7699621 58260.81276|| .59669.80247.61061.79193||.62433.78116 63787.77014 22 39.58283.81259|| .59693.80230|| .61084.79176||.62456|.78098 40.58307.81242.59716.80212|| .61107.79158||.62479.78079 .63832.76977| 20 45.58425.81157.59832.80125 63854.76959 19 .63877.76940| 18 41.58330.81225.59739.80195.61130.79140.62502.78061 42.58354.81208|| .59763.80178|| .61153.79122|| .62524.78043 43.58378.81191|| .59786.80160 .61176.79105|| .62547.78025||.63899.76921| 17 41.58401.81174|| .59809.80143.61199.79087.62570.78007||.63922.76903 16 .61222.79069.62592.77988||.63944.76884| 15 46.58449.81140.59856.80108.61245.79051.62615.77970.63966.76866 14 58172.81123.59879.80091 .61268.79033 62638.77952.63989.76847 13 48.58496.81106|| .59902.80073|| .61291.79016 62660.77934.64011.76828 12 49.58519.81089||.59926.80056||.61314.78998||.62683.77916 62683.77916.64033.76810 11 | 50 58543.81072||.59949.80038.61337.78980 62706.77897.64056.76791 10 51 .58567.81055|| .59972.80021|| .61360.78962|| .62728.77879 62728.77879.64078.76772 58590.81038||.59995|.80003.61383.78944|| .62751 | .77861 47 52 9 .64100.76754 8 .58614.81021|| .60019.79986 .60019.79986.61406.78926 61406.78926|| .62774|,77843|| .64123|.76735 7 54.58637.81004 .60042.79968.61429.78908 .61429.78908||,62796|,77824 .64145.70717 6 55.58661.80987.60065.79951 '|| .60065.79951.61451.78891.62819.77806.64167).76698 56.58684.80970||.60089.79934.61474.78873||.62842.77788.64190.76679 57 5 4 3 58708.80953.60112.79916.61497.78855 .62864.77769||.64212.76661 58.58731.80936||.60135.79899.61520.78837||.62887.77751 .64234.76642 2 59.58755.80919||.60158.79881.61543.78819||.62909|.77733 60 58779.80902||.60182.79864 .61566.78801 TO 이 ​.64256.76623 1 .62932.77715||.64279.76604 Cosin Sine 50° Cosin Sine Cosin Sine Cosin Sine Cosin Sine 54° 53° 52° 51° 292 TABLE X.-SINES AND COSINES. 40° Sine Cosin 41° 42° 43° Sine Cosin Sine Cosin Sine Cosin 65606.75471|| .66913|.74314|| .68200.73135 0.64279.76604.65606.75471 1.64301.76586||.65628.75452 44° Sine Cosin .69466.71934 60 .65628.75452||.66935|.74295||.68221|.73116||.69487.71911 50 2.64323.76567|| .65650|.75433||.66956.74276||.68242.73096|| .69508.71894| 58 3.64346.76548||.65672|.75414||.66978].74256||.68264|.73076|| .69529.71873 57 50 4.64368.76530.65694.75395||,66999.74237||.68285|.73056||.69549.71853 .64390.76511|| .65716|.75375||.67021|.74217|| .68306|.73036||.69570.71833 7 6 .64412.76492|| .65738.75356||.67043.74198|| .68327|.73016|| .69591.71813 54 64435,76473||.65759.75337||.67064.74178|| .68349.72996|| .69612.71792 5. 8.64457.76455|| .65781.75318 .65781.75318.67086.74159.68370.72976||.69633.71772 52 9.61479.76436 .65803.75299 67107|.74139|| .68391|.72957|| .69054.71752 51 10.64501.76417 .65825.75280|| .67129|.74120|| .68412|.72937||.69675|.71732| 50 .65847.75261 .75261.67151.74100 68434.72917.69696.71711 49 11.64524.76398 12.64546.76380 13 16 .65869.75241 67172.74080.68455.72897.69717.71691 48 .64568.76361 .65891.75222|| .67194.74061 .67194.74061|| .68476.72877|| .69737.71671 47 11.64590.76342.65913.75203 .65913.75203|| .67215].74041 68497.72857.69758.71650 46 15.64612.76323|| .65935.75184 .65935.75184|| .67237.74022 .68518.72837 .69779.71630 45 .64635.76304.65956.75165 .65956.75165|| .67258.74002|| .68539.72817 69800.71610 44 17.64657.76286||.65978.75146 .65978.75146||.67280.73983|| .68561|.72797||.69821|.71590 43 18.64679.76267.66000).75126 .66000|.75126||.67301|.73963|| .68582|,72777|| .69842,71569| 42 19 .64701.76248.66022.75107 .66022.75107|| .67323.73944|| .68603.72757|| .69862.71549| 41 20.64723.76229 .66011|,75088 || .67341|.73924|| .68624|.72737||.69883.71529| 40 .64746.76210 .66066.75069|| .67366.73904|| .68645|,72717|| .69904.71508 39 22.64768.76192.66038.75050 .66038|.75050 || .67387|.73885|| .68666|.72697 |.69925;.71488| 38 23.64790.76173.66109.75030 .66109.75030 || .67409|.73805|| .68688.72677.69946 .71468] 37 24.64812.76154.66131.75011 .67430.73846|| .68709|.72657 .68709.72657.69966.71447 36 .67452.73826 .73826.68730|.72637 69987.71427 35 .67473.73806|| .68751|.72617| .68751.72617.70008.71407 34 .67495.73787|| .68772|.72597 .70029.71386| 33 .67516|.73767|| .68793|.72577 70049.71366) 32 .67538.73747 .68814.72557 | .70070.71345 31 .67559.73728|| .68835.72537|| .70091.71325| 30 21 25.64831.76135.66153.74992 26.64856.76116.66175.74973 27.64878.76097|| .66197).74953 28.64901.76078.66218 74934.67516.73767 .66218.74934 29.64923.76059 .66240.74915 30.61915.76041 .66262.74896 • 31.64967.76022.66284.74876 .67580.73708 .68857.72517.70112.71305] 29 .66306.74857|| .67602|.73683|| .68878|.72497 70132.71284 28 .66327.74838.67623.73669 .67623.73669|| .68899|.72477|| .70153|.71261| 27 .66349.74818.67615 .67645.73649 .68920.72457|| .70174.71243 26 32.64989.76003 33.65011.75984 34.65033.75965 35.65055.75946 .66371.74799 .67666.73623 .68941.72437 | .70195|.71223| 35 .67688.73610|| .68962].72417 |,70215],.71203′ 24 .67709|,73590|| .68983|.72397||.70236.71182 23 67730 73570|| .69004.72377|| .70257,71162 22 .67752.73551 .69025.72357.70277.71141 21 .67773.73531 .69046.72337|| .70298.71121| 20 .69067.72317 72317.70319.71100 19 .69088.72297.70339.71080 18 36.65077.75927.66393 .66393.74780 37.651001.75908.66414.74760 38.65122.75889.66436.74741 39.65144.75870.66458.74722 40.65166.75851.66480.74703 41.65188.75832.66501 | .74683 .67795.73511 42.65210.75813.66523|.74664 43.65232.75794.66545.74644 .67816.73491 44.65254.75775 .66566.74625 45.65276.75756 46.65298.75738 47.65320.75719 .66545 | .74644|| .67837|.73472|| .69109|.72277|| .70360.71059 17 .66588.74606 .66610.74586 .66632.74567 49.65364.75680||.66675.74528 67859.73452.69130.72257 .69130|.72257|| .70381|.71039' 16 .67880.73432|| .69151|.72236|| .70401.71019 15 .67901.73413|| .69172.72216|| .70422.70998 14 .67923.73393|| .69193|.72196|| .70443.70978 13 48.65342.75700 .66653.74548 .67944.73373 .69214.72176|| .70463.70957 12 .67965.73353||.69235|.72156|| .70484).70937. 11 50.65386.75661|| .66697.74509 .66697.74509.67987.73333.69256.72136 .69256.72136|| .70505.70916 10 51.65408.75642 .66718.74489 .6800873314| .69277.72116.70525.70896 9 52.65430.75623||.66740|.74470 .68029.73294.69298.72095|| .70546,70875' 8 53.65452.75604 .66762.74451.68051.73274|| .69319|.72075|| .70567.70855 7 54.65474.75585 .66783.74131||.68072.73254|| .69340|.72055|| .70587|.70834 55.65496.75566 .69361|.72035|| .70608.70813 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 59.65581.75490 60.65606.75471 Cosin Sine 49° 66805.74412.68093.73234.69361.72035 66870.74352 66891.74334 .66913|.74314 .68157.73175|| .69424.71974|| .70670¸.70752 .68179.73155 69415.71954||,70690,70781 .68200.73135 Cosin Sine Cosin Sine 48° 47° 69466.71934 .70711 .70711 Cosin Sine Cosin Sine 46° 45° 5 293 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 0° 1° 2° 3° 4° 5° OHN CO TH LO 0 2 1.0000000 1.0001523 1.0006095 1.0013723 1·0000000 1.0001574 1.0006198 1·0013877 1·0000002 1.0001627 1.0006300 1.0000004 1'0001679 1.0006404 1·0014185 1'0024419 1-0038198 60 1.0024623 1.0038454 59 1·0014030 1·0024829 1.0038711 58 1·0025035 1.0038969 5 4 1.0000007 1·0001733 1.0006509 1·0014341 1'0025241 1.0039227 56 5 1'0000011 1.0001783 1.0006614 1·0014497 1.0025449 1.0039486 55 6 7 1·0000015 1.0001843 1.0006721 1.0000021 1·0001900 1·0006828 1.0014655 1.0025658 1.0039747 54 1·0014813 1.0025867 1.0040008 53 8 9 10 1.0000027 1.0001957 1.0006936 1·0000034 1.0002015 1·0000042 1.0014972 1.0026078 1·0040270 52 1.0007045 1.0015132 1.0026289 1.0040533 51 1.0002073 1.0007154 1·0015293 10026501 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 17 18 20 21 23 24 25 30 31 34 36 39 PEARL 22*** *7.28 ~~*** 85000 **** O***8 AND*6 85888 1.0000108 1.0000137 19 1.0000153 1.0002641 1.0008180 1.0000169 1.0002708 1.0008298 1.0002444 1.0007830 1.0000122 1.0002509 1.0007946 1.0002575 1.0008063 1.0016275 1-0027791 1.0042396 44 1.0016442 1'0028009 1.0042666 43 1·0016609 1.0028228 1.0042937 42 1.0016778 1.0028448 1.0043208 41 1.0016947 1.0028669 1.0043480 40 1.0000187 1'0002776 1·0008417 1·0017117 1·0028890 1.0043753 39 1.0000205 1.0002845 1.0008537 1.0017288 1.0029112 1.0044028 38 1.0000224 1.0002915 1.0008658 1·0017460 1'0029336 1.0044302 37 1.0000244 1'0002986 1.0008779 1.0000264 1.0003058 1.0008902 1.0017633 1.0029560 1.0044578 36 1·0017806 1.0029785 1.0044855 35 26 1.0000286 1.0003130 1.0009025 1.0000308 1'0003203 1.0009149 1.0000332 1.0003277 1.0009274 1·0017981 1.0030010 1.0045132 34 1.0018156 1.0030237 1.0045411 33 1·0018332 1.0030464 1.0045690 32 29 1.0000356 1.0003352 1.0000381 1'0003428 1.0009400 1.0018509 1.0030693 1.0045970 31 1.0009527 1·0018687 1'0030922 1.0046251 30 1.0000407 1.0003505 1.0009654 32 1.0000433 1'00035S2 1.0009783 33 1.0000461 1'0003660 1.0009912 1·0018866 1.0031152 1.0046533 29 1.0019045 1.0019225 1.0000489 1.0003739 1·0010042 1·0019107 35 1.0000518 1.0003820 1·0010173 1·0019589 1.0031383 1.0046815 28 1.0031615 1·0047099 27 1.0031847 1.0047383 26 1.0032081 1.0047669 25 1.0000548 1.0003000 1·0010305 1·0019772 1.0032315 1.0047955 24 37 1.0000579 1.0003982 1.0010438 1·0019956 1.0032551 1·0018242 23 38 1.0000611 1.0004065 1·0010571 1'0000644 1.0004148 1·0010705 40 1.0000677 1.0004232 1.0010841 1·0020140 1.0032787 1.0048530 22 1.0020326 1·0033024 1'0048819 21 1.0020512 1.0033261 1.0049108 20 41 42 1.0000711 1.0004317 1·0010977 1.0000746 1.0004403 1'0020699 1·0011114 1.0020887 43 1·0000782 1.0004490 44 1.0000819 1.0004578 1·0011251 1.0021076 1·0011390 1.0021206 1.0033500 1.0049399 19 1·0033740 1.0049690 18 1.0033980 1.0049982 17 1-0034221 1.0050275 15 45 1.0000857 1.0004666 1.0011529 46 1.0000895 1.0004756 1·0011670 1.0021457 1.0034463 1·0050569 15 : 1.0021648 1.0034706 1.0050864 14 47 1.0000935 48 1.0000975 1.0004846 1.0011811 1.0001937 1·0011953 1.0021841 1.0034950 1.0051160 13 1.0022034 49 50 1.0001016 1.0005020 1.0012096 1.0022228 1.0001058 1.0005121 1·0012239 1.0022423 1.0035195 1.0035440 1·0051754 11 1·0035087 1'0052052 10 1·0051456 12 51 52 53 54 1.0001234 1.0005501 55 1.0001280 1'0005598 1·0012971 56 1.0001327 1.0005696 1·0013120 57 1.0001375 1.0005794 1·0013260 1.0001423 1'0005894 1·0013420 59 1.0001473 1.0005994 1·0013571 1·0001101 1'0005215 1·0012384 1.0001144 1.0005309 1·0012529 1'0022815 1.0001189 1'0005405 1·0012670 1.0023013 1.0012823 1'0023211 1·0023410 1'0022619 1.0035934 1.0052351 9 1.0036182 1.0052651 1.0036431 1.0052952 1.0036681 1·0036932 1·0053254 10053557 ༦ ༦༠ ༢。 7 6 5 60 1.0001523 1.0000095 f 89° 88° 1.0023610 1.0023811 1.0024013 1'0024216 1·0013723 1.0024419 87° 1.0037183 1.0053860 1-0037436 1.0054164 1·0037689 1.0054470 1.0037943 1.0054776 1.0038198 1.0055083 4 9 63 FO 3 86° 85° 84° COSECANTS. 904 TABLE XI.—SECANTS AND COSECANTS. SECANTS. 6° 7° 8° 9° 10° 11° 0 1.0075098 1-0098276 1.0124651 1.0154266 1-0187167 60 1.0055083 1 10075159 1.0098689 1.0125118 1-0154787 1.0187743 59 1'0055391 2 1.0075820 1.0099103 1.0125586 1.0155310 1.0188321 58 1.0055099 3 1-0076182 1.0099518 1-0126055 1.0155833 1.0188899 57 1.0056009 4 1-0076545 1.0099931 1.0126521 1.0156357 1.0189478 56 1.0056319 5 1:0076908 1·0100351 1-0126995 1-0156882 1.0190059 55 1.0056631 6780 1 0077273 1.0100769 1-0127466 1.0157408 1.0190640 54 1.0056943 1.0077639 1-0101187 1.0127939 1.0157931 1.0191222 53 1.0057256 1-0078005 1.0101607 1-0128412 1.0158462 1.0191805 52 1·0057570 9 1.0078372 1·0102027 1-0128886 1.0158991 1-0192389 51 1.0057885 10 1.0078741 1·0102449 1.0129361 1-0159520 1.0192973 50 1.0058200 11 12 13 14 15 16 17 18 19 21 22 23 24 27 HARED U7-02 2**** ***≈≈ ~***3 85880 =**** ****8 EN883 95388 · 1-0079110 1·0102871 1.0129837 1.0160050 1.0193559 49 1·0058517 1-0079480 1·0103294 1·0130314 1.0160582 1.0194146 48 1.0058834 1.0079851 1-0103718 1-0130791 1-0161114 1-0194731 47 1.0059153 1·0080222 1.0104113 1-0131270 1.0161647 1.0195322 46 1.00.59472 1.0080595 1.0104568 1.0131750 1-0162181 1.0195912 45 1.0059792 1-0080968 1·0000113 1-0104995 1.0132230 1-0162716 1.0196502 44 1.0000435 1.0081343 1.0105422 1.0132711 1-0163252 1-0197093 43 1.0000757 1-0081718 1-0105851 1.0133194 1-0163789 1.0197686 42 1.0001081 1·0082091 1-0106280 1-0133677 1.0164327 1.0198279 41 1.0061405 1-0082471 1·0106710 1.0134161 10161865 1.0198873 40 1.0061731 1.0082849 10062057 1'0062384 1.0062712 1-0107141 1.0083228 1.0107573 1.0135132 1-0083607 1.0108006 1.0083988 1.0108440 1.0134646 1.0165405 1-0199468 39 1.0165946 1·0200061 38 1.0135618 1.0160187 1-0200661 37 1.0136106 1-0167029 1-0201259 36 25 1.0063040 1.0084369 1.0108875 1.0136595 1.0167573 1.0201858 35 26 1'0063370 1·0081752 1-0109310 1.0137081 1.0168117 1·0202457 31 1.0003701 1.0085135 1.01097+7 1.0137574 1.0168662 1.0203058 33 28 1.0064032 1.0085519 1-0110184 1·0138066 1·0169208 10203660 32 29 1.0001304 1.0085904 1.0110622 1.0138558 1.0169755 1·0204262 31 30 1'0061697 1'0086290 1-0111061 1.0139051 1'0170303 1.0204866 30 1.0005031 1.0086676 1-0111501 1.0139545 1'0170851 1.0205470 29 32 1.0065366 1-0087061 1.0111942 1.0140010 1-0171401 1.0206075 33 1.0065702 1.0087452 1.0112884 10140536 1-0171952 10206682 27 34 1.0066039 1-0087842 1-0112827 1.0141032 1-0172503 1·0207289 26 35 1.0060376 1.0088232 1.0113270 1-0141530 10173056 1·0207897 25 36 1.0066714 1.0088623 1-0113715 1.0142029 1-0173609 1.0208506 24 37 1.0067054 1.00 9015 1.0114160 1.0142528 1.0174163 1.0209116 23 38 1.0067394 1-0089408 1-0111606 1.0143028 1-0174719 1·0209727 23 1.0007735 39 1.0089802 1.0115054 1-0143530 1-0175275 1·0210339 21 1.0068077 40 1-0090196 1-0115502 1.0144032 1-0175832 1.0210952 20 1.0068419 41 1.0090592 1.0115951 1.0144535 1.0176390 1-0211566 19 1.0068763 42 1.0090988 1.0116100 1.1145039 1-0176949 1·0212180 18 1.0069108 43 1-0001386 1-0116851 1.0145544 10177509 10212796 17 1.0069453 41 1.0091781 1-0117303 1.0146050 1.0178069 1.0213113 16 1.0009799 45 1.0092183 1.0117755 1.0146556 1-0178681 1.0214030 15 1'0070146 46 1.0092583 1·0118209 1-0147064 1.0179191 1·0214649 14 1.0070494 47 1-0092981 1.0118663 1.0147572 1.0179757 1·0215269 13 1.0070813 48 1-0093386 1.0119118 1.0148082 1.0180321 1·0215888 12 1.0071193 1.0071544 1.0093788 1-0119575 1.0148592 1-0180887 1.0216510 11 50 1.0091192 1-0120032 1.0149103 1-0181453 1.0217132 10 1.0071895 51 1·0091596 1.0120489 1-0149616 1.0182020 1:0217755 9 1·0072248 52 1.0072601 1·0095601 1.0120948 1.0150129 1.0182588 1.0218379 8 53 1.0072955 1.0095408 1·0121108 1.0150613 51 1.0073310 1.0095815 1.0121869 1-0151158 55 1·0096223 1·0122330 1.0151673 1-0183158 1.0183723 1·0219630 1.0184298 1·0219004 7 6 1.0220257 5 56 57 1.0073666 1.0074023 1.0096631 1.0122793 1.0074380 1·0097041 10123256 1-0152190 1.0184870 1·0220885 1·0152708 1.0074739 1·0097152 1.0123720 1.0075098 1-0097863 1·0124185 1.0153226 1-0153746 GO 1-0098276 1:0124651 1·0154266 1-0185443 1·0221514 10186017 1.0186591 1·0222774 1.0187167 1·0222144 20 33 14 4 3 1 1-0223106 0 83° 82° 81° 80° 79° 78° COSECANTS. 295 TABLE XI.–ŠECANTS AND COSECANTS. SECANTS. 12° 13° 14° 15° 16° 17° 0 1 1.0223406 1.0263041 1.0306136 1.0224039 1.0263731 1-0224672 1.0264421 1.0352762 1.0306881 1·0307633 1.0402994 1.0353569 1.0403863 1·0354378 1.0404732 1.0456918 60 1.0457848 59 1.0458780 58 1.0225307 1.0265113 1.0308383 1·0355187 1.0105602 10159712 57 1'0225942 1.0265806 1.0309134 1.0355998 1.0406473 1.0460616 56 1-0226578 1-0266499 1-0309886 1.0356809 1.0407346 1.0461581 55 5 1.0227216 1-0267191 1-0310639 1.0357621 1·0108219 1·0462516 51 6 1.0227854 1·0267889 1.0311393 1.0358435 10409094 1.0463453 53 7 1'0228493 1-0268586 1·0312147 1.0359249 1-0109969 1.0464391 52 1-0229133 1-0269283 1·0312903 1.0360065 1.0410845 1.0465330 51 9 1-0229774 1.0269902 1.0313660 1·0360881 1.0111723 1.0466270 50 10 1·0230416 1·0270631 1.0314418 1·0361699 11 1·0412601 1.0467211 49 12 1·0231059 1-0271381 1.0315177 1.0362517 1-0413481 1.0468153 48 13 10231703 1-0272082 1.0315936 1.0363337 1.0414362 1.046 096 47 14 1-0232348 10272785 1.0316697 1-0364157 1 0415243 1.0470040 46 15 1-0232991 1.0273188 1.0317159 1-0361979 1.0416126 1-0470986 45 16 17 18 19 20 23 25 26 28 30 31 36 1 PARA 22*** ***** M**** 8*88* ***** ***98 ***** 35.83 1.0233641 1.0274192 1.0318222 1.0365801 1-0417009 1-0471932 44 1.0234288 1·0271897 1-0318985 1.0366625 1.0417894 1.0472879 43 1-0234937 1.0275603 1.0319750 1.0367449 1-0118780 1.0473828 42 1-0235587 1-0276310 1-0320516 1-0368275 1.0419667 1.0474777 41 1·0236237 1-0277018 1-0321282 1.0369101 1.0420554 1.0475728 40 1-0236889 1·0277727 1.0322050 1.0369929 1-0421443 1.0176679 39 1-0237541 1-0278137 1.0322818 1-0370757 1-0122333 1.0477632 38 1.0238195 1·0279148 1·0323588 1.0371587 1.0423224 1.0178586 37 24 1·0238849 1·0279860 1·0239501 1-0280573 1.0325130 1.0240161 1.0281287 1·0240818 1.0282002 1.0326676 1-0241476 1·0282717 1-0324359 1-0372417 1-0424116 1·0479540 36 1-0373249 ! 1.0125009 1.0480496 35 1 1.0325903 1.0374082 1-0425903 1.0481453 34 1.0374915 1-0126798 1·0482411 33 1.0327451 1.0375750 1.0427694 1-0483370 32 29 1-0242135 1.0283431 1.0328227 1-0376585 1.0428591 1-0484330 31 1.0242795 1.0284152 1.0329003 1.0377422 1.0429489 1.0485291 30 } 1-0243456 1-0284871 1.0329781 1·0378260 1.0430388 1.0486253 29 32 1·0244118 1.0285590 1-0244781 1.0286311 34 1·0215115 1.0287033 1.0216110 1-0287755 1.0330559 1.0379098 1.0431289 1.0487217 28 1-0331339 1.0379938 1.0432190 1.0488181 27 1.0332119 1.0380779 1-0433092 1.0489146 26 10332901 1.0381621 1.0433995 1-0490113 25 1-0246776 1-0288179 1·0247442 1.0289203 1-0333683 1-0382463 1.0434900 1-0491080 24 1.0334467 1-0383307 1.0435805 1.0492049 23 38 1.0248110 1·0289929 1.0335251 1.0384152 1.0436712 1-0193019 22 1.0248779 1·0290655 1.0336037 1-0384998 1·0437619 1·0493989 21 1.0249418 1.0250119 1-0291383 1.0336823 1.0385844 1.0438528 1.0194961 20 1-0292111 1.0337611 1.0386692 1.0439437 1.0495931 19 1-0250790 1·0292840 1.0338399 1.0387541 1.0440348 1.0496908 18 1.0251463 44 1-0293571 1.0339188 1.0388391 1-0252136 1.0294302 1-0339979 1.0389242 1·0295031 1-0252811 1·0340770 1.0390094 1.0253486 1-0295768 1.0341563 1.0390947 1.0251162 1·0296502 1.0342356 1.0391800 1.0254839 1.0297237 1:0343151 1-0255518 1.0297973 1.0343946 1-0256197 1-0208711 1.0441259 1.0497883 17 1·0442172 1·0498859 16 1'0143086 1.0499836 15 1.0444001 1.0500815 14 1·0444917 1.0501794 13 1-0392655 1.0145833 1.0502774 12 1.0393511 1-0446751 1.0503756 11 1.0344743 1.0394368 1'0417670 1·0501738 10 1.0256877 1-0299449 1-0345540 1.0395226 1.0448590 1.0505722 9 1-0257558 1.0300188 1.0346338 1.0396085 1-0449511 1.0306706 8 1.0258240 1.0300928 54 1-0347138 1.0396945 1-0258923 1·0301669 1.0347938 1*0259607 1-0302411 1.0348740 1.0450433 1.0507692 7 1.0397806 1.0451357 1·0508679 6 1-0398669 1.0452281 1·0509667 5 57 1.0260292 59 60 1.0260978 1-0303898 1.0350346 1.0400396 1.0261665 1.0304643 1.0262352 1.0305389 1:0351955 1.0263041 1.0306136 1-0352762 77° 76° 75° 1·0303154 1.0349542 1-0399532 1.0453206 1.0510656 4 1.0454132 1.0511646 3 1-0351150 1.0401261 1.0455060 1.0512637 8 1·0402127 1.0455988 1-0513629 1 1.0402994 1.0456918 1.0514622 74° 73° 72° COSECANTS. 296 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 18° 19° 20* 21° 22° 23° 0 1.0514622 1.0576207 1.0641778 1-0711450 1 1-0515617 1-0577267 1'0642905 2 CO - LO 3 4 1.0516612 1·0517608 1·0579390 1-0518606 1*0580453 1-0578329 1.0644033 5 1.0519605 1.0581517 6 1·0520601 1.0582583 1.0712647 1-0713844 1.0645163 1-0715013 1·0646294 1.0716241 1.0617425 1-0717445 1.0648558 1-0785347 1.0786616 1.0864946 1.0787885 1.0866289 1·0863604 1.0718647 7 1.0521605 1.0583649 1.0649693 1-0719351 1.0792975 1.0794250 1.0873021 1.0789156 1·0867631 1-0790427 1.0868979 1-0791700 1.0870326 1.0871675 8 1-0522607 1-0584717 1·0650828 1.0721056 1·0795527 1.0874375 9 1·0523610 1-0585786 1.0651964 10 1·0524614 1.0586855 1.0653102 11 1.0525619 1.0587926 1.0654240 1-0721678 12 1.0526625 1-0588999 1.0655380 13 1.0527633 1·0590072 1.0656521 1-0725887 1.0727098 1-0801928 1.0722262 1.0796805 1-0875727 1·0723469 1·0798081 1·0877080 1-0799361 1.0800646 1.0878435 1-0879791 883588 88858 G3 60 59 57 50 55 54 53 52 51 50 49 48 1.0881148 47 14 1.0528641 1.0591146 1:0657663 1-0728310 1.0803212 1·0882506 46 15 1-0529651 1.0592221 1.0658807 1-0729523 1·0804497 1.0883866 45 16 1.0530661 1.0593298 1.0659951 1-0730737 1.0805781 1.0885226 41 17 1·0531673 1-0594376 1.0661097 1.0731953 1.0807071 1-0886589 43 18 1.0532686 1.0595154 1.0662243 1.0733170 '1-0808360 1 0887952 42 19 1.0333699 1.0596534 1.0663391 1-0734388 1.0809650 1'0859317 41 20 1-0534714 1.0597615 1.0661510 1-0735607 1.0810942 1 0890682 40 21 1·0535730 1·0598697 1.0665690 1.0736827 1.0812234 22 1-0536747 1-0599781 1.0666842 1.0738048 1-0813528 1.0892050 1.0893418 23 1.0537765 1.0600865 1.0667994 1-0739271 1.0814823 1-0894788 24 1-0538785 1.0601951 1.0669148 1.0740495 1.0816119 1.0896159 25 1.0539805 1.0603037 1.0670302 1-0741720 1-0817417 1.0897531 26 1.0510826 1.0604125 1.0671458 1.0742946 1·0818715 1-0898901 27 1-0541849 1.0605214 1.0672615 1.0744173 1.0820015 1·0900279 28 1.0542873 1.0606304 1.0673774 1.0745402 1.0821316 1.0901655 29 1-0543897 1.0607395 1'0674933 1-0746631 1.0822618 1·0903032 31 30 1.0511923 1.0608187 1-0676094 1-0747862 1.0823922 1.0904411 31 1.0545950 1.0609580 1-0677255 1.0749095 1.0825227 32 1-0546978 1.0610675 1.0678418 1.0750328 1.0826533 1.0905791 1.0907172 33 1-0518007 1.0611770 1.0679582 1.0751562 34 2·0549037 1.0612867 1-0680747 1.0752798 35 1-0550068 1.0613965 1-0681914 1·0754035 36 1.0551101 1.0615061 1.0683081 1-0755273 37 1.0552134 1.0616164 1-0684250 1.0756512 38 1-0553169 1.0617265 1.0685420 1.0757753 39 1.0554204 1·0618367 1.0686591 1.0758995 40 1-0555241 1-0619471 1.0687763 1.0760237 41 1.0556279 1.0620575 1.0688936 1.0761481 42 1.0557318 1.0621681 1.0690110 1-0762727 1-0839661 1.0827840 1.0829149 1·0909938 1.0830458 1.0911323 1.0831769 1·0912709 1.0833081 1-0914097 1.0834395 1.0315485 1.0835709 1.0916876 1-0837025 1.0918267 1.0838342 1-0919659 1-0921053 1.0908551 26 21 23 22 21 20 2358* ***** ***** **NNA 39 33 37 36 34 33 32 30 29 28 27 25 19 18 43 1.0558358 1·0622788 1.0691286 1.0763973 1-0840980 1.0922418 17 44 1.0559399 1.0623896 1.0692463 1-0765221 1.0842301 1.0923815 16 45 1.0560441 1.0625005 1.0693641 1-0766170 1.0843623 1.0925243 15 46 1.0561485 1-0626115 1.0691820 1.0767720 1.0844947 1.0926642 14 47 1.0562529. 1.0627227 1.0696000 1.0768971 1.0846271 1·0928012 13 48 1.0563575 1.0628339 1.0697182 1-0770224 1·0847597 1·0929144 12 49 1.0564621 1.0629453 1.0698364 10771477 1.0848921 1.0930846 11 50 1.0565669 1-0630568 1.0699548 1·0772732 1·0850252 1·0932251 10 51 1.0566718 1.0631684 1·0700733 1.0773988 1.0851582 1.0933656 9 52 1.0567768 1-0632801 1.0701919 1-0775246 1.0852913 1.0935063 8 53 1.0568819 1.0633919 1.0703105 1.0776504 1.0854245 1·0936471 7 54 1-0569871 1.0635038 1.0701295 1-0777764 1-0855578 1.0937880 6 55 1.0570924 1.0636158 1·0705484 1.0779025 1.0856912 1-0939291 56 57 58 59. 60 1·0571978 1.0637280 1.0706675 1.0780287 1-0573034 1-0638403 1-0707867 1.0781550 1.0574090 1-0639527 1.0709060 1-0782815 1.0860924 1-0943530 1.0575148 1-0640652 1.0710254 1.0784080 1·0862263 1.0944946 1·0576207 1.0641778 1.0711450 1-0785347 1'0863604 1·0946363 1.0858248 1.0940702 4 1-0859585 1.0942116 3 71° 70° 69° 68° 67° 66° COSECANTS. 297 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 24° 25° 26° 27° 28° 29° 0 1 1·0919201 1.0946363 1·1033779 1-0917781 1-1035277 1.1036775 3 1-0950622 1·1038275 1.1130761 1·1126019 1.1223262 1.1325701 1.1433541 1-1127599 1-1224927 1.1327453 1.1435385 1.1129179 1.1226592 1·1329207 1.1437231 1-1228259 1.1330962 60 59 58 1.1439078 57 1.0952044 1.1039777 1.1132345 1*1229928 1.1332719 1.1440927 56 1.0953167 1.1041279 5 1·1133929 1.1231598 1-1334478 1-1442778 55 24 26 28 29 80 31 32 33 34 35 36 37 39 40 41 42 44 45 ora 12212 07-22 22*** ***22 7**** 83882 4+**** 1.0954892 1.1042783 1.1135516 1.1233269 1.1336238 1.1444630 54 7 1.0956318 1.1044289 1'1137103 1-1231942 1.1337999 1-1446184 53 8 1.0957746 1.1045795 1-1138692 1.1236616 1-1339762 1.1448339 52 1.0959174 1-1017303 1.1140282 1.1238292 1-1341527 1-1450196 51 1.0960601 10 1.1048813 1-1141874 1·1239969 1.1313293 1.1452055 50 1.0962036 1.1050321 1-1143467 1.1211618 1.1345060 1.1453915 49 1.0963468 13 14 1.0961902 1·0966337 1.0967774 1-1051836 1.1053319 1-1146658 1.1145062 1.1213328 1.1316829 1-1455776 48 1.1215010 1.1348600 1.1457639 47 1.1054864 1.1148255 1.1246693 1 1350372 1.1459504 46 15 1.1056380 1-1149854 1.1248377 1-1352146 1-1461371 45 16 1.0969212 1.1057898 1.1151454 1·1250063 1.1353921 1.1463238 44 17 1-0970651 1·1059417 18 19 21 23 1.1153056 1.1251750 1.0972091 1.1060937 1.1154659 1·1253139 1-0973533 1.1062458 1.1156263 1.1255130 1.0974976 1.1063981 1.1157869 1.1256821 1.0976420 1.1065506 1.1159176 1.1258514 1·0977866 1·1067031 1-1161084 1.1260209 1.0979313 1-1068558 1.1162691 1·1261905 1.1355697 1.1357476 1.1466979 1.1359255 1·1468852 1-1361036 1.1470726 1.1465108 43 42 41 40 1-1362819 1.1472602 39 1.1364603 1.1474479 38 1.1366389 1.1476358 37 1.0980761 1·1070087 1.1161306 1-1263603 1.1368176 1.1478239 36 10982211 1-1071616 25 1.1165919 1.1265302 1.1369965 1-1480121 35 1·0983662 1·1073147 1.1167533 1.1267003 1-1371755 1.1482005 34 1·0985114 27 1·1074680 1.1169148 1·1268705 1.1373547 1.1483890 33 1.0986568 1-1076211 1.1170706 1-1270408 1-1375341 1·1485777 32 1-0988023 1.1077749 1.1172384 1.0989479 1-1079285 1·1174004 1.1272113 1-1377135 1-1487665 31 1-1273819 1·1378932 1.1489555 30 1.0990936 1·0992395 1·1080823 1.1175625 1.1082363 1-1177248 1.1275527 1·1491447 1.1380730 29 1-1277237 1'0993855 1.1083903 1:1178872 1.1278948 1.1382529 1.1384330 1-1493340 1.1495235 23 27 1.0995317 1·1085145 1.1180198 1·1280660 1.1386133 1-1497132 26 1.0996779 1·1086989 1·1182124 1.1282371 1-1387937 1.1499030 25 1.0998243 1.1088533 1-1183753 1.1284089 1.1389742 1·1500930 21 1.0999709 1·1090079 1.1185303 1-1235806 1.1391550 1.1502831 23 1.1001175 1.1091627 1.1187014 1.1287524 1.1002644 1.1093176 1.1188617 1·1289244 1.1004113 1-1094726 1.1190281 1.1290965 1.1393358 1·1501731 1.1395169 1.1500638 1.1396980 1.1508541 22 21 20 1.1005584 1.1096277 1.1191916 1.1292687 1.1398791 1.1510152 19 1·1007056 1.1097830 1.1193553 1·1294412 1.1400608 1-1512361 18 1.1008529 1.1099385 1.1195191 1.1206137 1-1102425 1·1514272 17 1.1010001 1·1100940 1.1196831 1.1297861 1-1404213 1-1516185 16 1·1011480 1-1102198 1.1198472 1.1299593 1.1406062 1-1518099 15 46 1-1012957 1-1101056 1.1200115 1.1301323 1·1407883 1-1520015 11 1.1014436 1'1105616 1.1201759 47 1.1303035 1·1409706 1-1521932 13 1.1015916 1-1107177 48 1.1203405 1.1304788 1.1411530 1-1523851 12 1-1017397 1-1108740 49 1.1205051 1.1306522 1.1413356 1-1525772 11 1 60 3 68318 5888 - 1·1018879 50 1.111034 1·1206700 1.1308258 1.1415183 1-1527691 10 1.1020363 51 1·1111869 1-1208350 1.1309996 1.1417012 1.1529618 9 1.1021849 52 1-1113436 1.1210001 1.1311735 1.1418842 1.1531543 53 1.1023335 1·1115001 1.1211653 1-1313475 1.1420674 1.1533470 1.1024823 1.1116573 1.1213308 1-1315217 1·1422507 1.1535399 65 1·1026313 1-1118144 1.1214963 1.1316961 1.1424342 1.1537329 776 ∞ ∞ 8 6 1.1027803 1-1119716 1-1216620 1.1318706 11126179 1.153926) 4 1.1029295 1.1121290 1.1218278 1.1320452 1.1428017 1.1541195 3 1·1030789 1.1032283 1·1033779 1.1122865 1.1219938 1.1322200 1.1124442 1.1221600 1.1323950 1.1126019 1-1429857 1-1543130 2 1·1431698 1-1545067 1 1-1223262 1.1325701 1-1433541 1.1547005 0 65° 64° 63° 62° 61° 60° COSECANTS. 298 TABLE XI. -SECANTS AND COSECANTS. SECANTS. 30° 31° 32° 33° 34° 35° 0 1 2 10 678Q O 9 14547005 1.1666334 1.1518945 1·1550887 1.1670416 1-1552830 1·1672459 1.1798222 1.1554775 1.1674504 1.1800372 1.1556722 1·1558670 1.1678599 1.1804676 1.1560620 1.1680649 1.1806831 1.1562572 1.1564525 1-1684755 1-1791784 1.1923633 1.2062179 60 1.2207746 1-1668374 1·1793928 1.1925886 1-2064547 1-2210233 59 1.1796074 1-1928142 1.2066917 58 1.2212723 1·1930399 1.2069288 57 1.2215215 1.1932658 1.2071662 56 1.2217708 1.1676551 1.1802523 1·1934918 1.2074037 55 1*2220204 1.1937181 1.2076415 1'2222702 1-1939446 1.2078794 53 1*2225202 1.1682701 1.1808988 1.1941712 1.2081175 1.2227703 1.1811146 1.1943980 1.2083559 52 51 1·2230207 1.1566480 1.1686810 1.1813307 1.1916251 1.2085911 50 1.2232713 11 1.1568136 1.1688867 1.1815469 1.1948523 1-2088331 49 1.2235222 13 1.1570394 1.1690926 1.1817633 1-1950796 1.2090720 48 1.2237732 13 1.1572354 1.1692986 1.1819798 1.1953072 1.2093112 47 1·2210244 14 15 16 17 18 19 21 22 23 24 25 26 27 28 29 30 31 32 33 31 35 19 2022 ***** ANXA. 728*9 93380 **** ***?A AND** 85988 1.1574315 1.1695048 1.1821966 1.1955350 1.2095505 46 1.2242758 1.1576278 1.1697112 1.1824135 1·1957629 1.2097900 45 1-2215274 1.1578243 1'1699178 1.1826306 1-1959911 1·2100297 44 1.2247793 1.1580209 1.1701245 1.1828179 1-1962194 1-2102696 1.2250313 43 1-1582177 1-1703314 1.1830654 1.1964479 1-2103097 42 1.2252836 1.1584146 1·1705385 1-1832830 1·1966767 1.2107500 41 1-2255361 11586118 1·1707457 1.1833008 1.1969056 1-2109905 40 1.2257887 1·1588091 1·1709531 1.1837188 1.1971346 1-2112312 39 1.2260416 1.1590065 1.1711607 1.1839370 1-1973639 1.2114721 1.2262947 38 1.1592041 1-1713685 1.1841554 1·1975934 1·2117132 1.2265480 37 1.1501019 1-1715764 1.1843739 1-1978230 1.2119545 1.2268015 36 1.1595999 1-1717845 1.1845927 1.1980529 1-2121960 1.2270552 35 1-1597980 1.1719928 1.1848116 1-1982829 1.2124377 1.2273091 31 1-1599963 1.1722013 1.1850307 1.1985131 1.2126795 1-2275633 33 1.1601917 1-1724099 1.1852500 1.1987435 1.2129216 1.2278176 32 1.1603933 1.1726187 1-1854694 1·1989741 1-1605921 1-1728277 1-1856890 1-1992049 1-1607911 1·1730368 1.1859089 1·1609902 1·1732462 1.1861289 1.1611894 1·1731557 1.1863190 1.1998985 1·1613389 1.1736653 1-1865694 1.2001300 1.2131639 1.2134064 1.1994359 1-2136191 1.1996671 1.2138920 1.2141351 1.2143784 1-2280722 31 1.2283269 30 1-2285819 1.2258371 1.2290921 27 1.2293180 26 1-1615885 1.1738752 1.1867900 1-2003618 1-2146218 1.2296039 25 36 1-1617883 1·1710852 1.1870107 1.2005937 1.2148655 1.2298599 24 37 1.1619882 1.1742954 1.1872316 1.2008258 1.2151094 1.2301161 40 1.1621883 1.1745058 1.1874527 1:1623886 1.1747163 1*1876740 1.1625891 1-1749270 1-2010582 1.2153535 1.2303725 22 1*2012907 1.2155978 1.2306292 1-1878951 1-2015234 1.2158423 1-2308861 20 1·1627897 1-1751379 1.1881171 1-2017563 1.2160870 1.2311432 42 1.1629905 1-1753490 1.1883389 1-2019891 12163319 1.2314004 43 1.1631914 1.1755603 1.1885609 1.2022226 1.2165770 1.2316579 41 1.1633925 1.1757717 1·1857831 1.2024561 1.2165223 45 1-1635938 1-1759833 1.1890055 1.2020898 1-2319156 1-2170678 1-2321736 ***** ***** ***=- 2878* ***** ***** ***** 22fe9 29 28 19 18 46 1·1637953 1.1761951 1·1892280 47 1.1639969 1.1764070 43 1·1611987 49 1·1641007 1.1894508 1'1766191 1-1896737 1-1768314 1-1898968 1.2029236 1.2173135 1·2324317 1.2031577 1-2175594 1.2326200 1.2033919 1.2178055 1-2329486 1-2036204 1.2180518 14 13 12 1.2332074 11 50 1-1616028 1.1770439 1.1901201 1.2038610 1.2182983 1.2331661 10 51 1-1648051 1·1772506 1.1903436 1.2040958 1.2185450 1.2337256 9 52 1-1650076 1.1774694 1.1905673 1.2043308 1.21879:9 1-2339850 8 53 1·1652102 1-1776821 1.1907911 1.2045660 1.2190390 1.2342446 7 51 1.165-1130 1·1778956 1-1910152 1-2048014 1.2192864 1.2345044 6 55 1·1656160 1.1781089 1·1912391 1.2050370 1-2195339 1-2347645 5 57 53 59° 1-1658191 1.1783225 1-1914638 1·1660221 1.1755362 1.1916834 1·1662259 1.1787501 1-1919132 1-1004296 1.1789042 1-1921331 1.1666331 1-1791784 1-1923633 58° 57° 1.2052728 1-2197816 1.2350213 4 1*2055088 1.2200296 1-2352852 3 1.2057450 1-2202777 1.2355459 2 1.2039814 12205260 1.2062179 1·2338069 1 1.2207746 1.2360680 56° 55° 54° COSECANTS. 299 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 36° 37° 38° 39° 40° 41° 1.2360680 1-2521357 1.2690182 1.2367596 1.3054073 1.3250130 60 ✪ 1-2363293 1.2524102 1-2693067 1.2870628 1.3057261 1.6253482 69 1 1.2365909 1-2526850 1.2695955 1.2873663 1.3060151 1.3256837 58 2 1.2368526 1.2529601 1.2698845 1.2876700 1.3063614 1.3260191 57 3 1.2371146 1.2532353 1-2701737 1.2879740 1.3066839 1.3263551 56 4 1-2373768 1-2535108 1.2704632 1.2882782 1.3070038 1.3266918 55 5 1-2376393 1.2537865 1-2707529 1.2885827 1-3073239 1·3270281 51 878 6 9 1.2379019 1-2510625 1-2381647 1.2543387 1.2713331 1-2384278 1-2546151 1.2386911 1.2548917 1.2710429 1.2888875 1.3076142 1.3273653 53 1.2891925 1.3079649 1.3277021 52 1.2716235 1.289 1977 1.3082858 1.3280399 51 1-2719142 1.2898032 1.3086069 1.3283776 50 10 1.2389516 1-2551685 1.2722052 1.2901090 1.3089281 1.3287156 49 11 1.2392183 1.2554456 1.2724963 1.2904150 1.3092501 1.3290539 48 12 1.2394823 1-2557229 1.2727877 1.2907213 1.3095720 1.3293925 47 13 1.2397461 1-2560005 1.2730794 1.2910278 1.3098943 1-3297314 46 14 1-2100108 1.2562782 1.2733712 1.2913316 1.3102168 1.3300706 45 15 1 2402751 1.2565562 1.2736634 1.2918416 1.3105396 1.3304100 44 16 1.2405102 1.2568345 1.2739557 1.2919489 1.3108626 1.3307497 43 17 1.2108053 1-2571129 1.2712184 1-2922561 1.3111860 1-3310897 42 18 1-2410704 1-2573916 1.2745112 1.2925612 1.3115095 1-3314301 41 19 1.2413359 1.2576705 1.2748313 1.2928723 1.3118334 1.3317707 40 20 1.2410016 1-2579497 1.2751276 1.2931806 1.3121575 1.3321115 39 21 1.2418675 1-2582291 1.2751212 1.2934892 1.3124820 1.3324527 22 1.2421336 1.2585087 1-2757151 1.2937980 1.3128066 1-3327942 37 23 1.2423999 1.2587885 1-2760091 1.2941071 1.3131316 1.3331359 36 24 1.2126665 1-2590686 1.2763034 1.2914164 1:3134568 1:3334779 25 1.2429333 1.2593489 1.2765980 1.2947260 1.3137823 1.3338203 31 26 1.2432003 1.2596294 1.2768928 1.2950359 1.3141081 1.3341629 27 1.2481675 1.2599102 1.2771878 1-1953160 1-3144341 1.3345058 32 28 1.2437349 1.2601912 1.2774831 1.2956561 1-3147604 1.3348489 29 1.2140026 1.2604724 1.2777787 1.2959670 1.3150870 1.3351924 30 1.2442704 1.2607539 1.2780744 1.2962779 1.3154139 1.3355362 31 1.2415385 1.2610356 1.2783705 1.2965890 1-3157410 1.3358802 32 1.2148069 1.2613175 1.2786667 1-2969004 1-3160684 1.3362246 33 1.2450754 1.2615997 1-2789032 1.2972121 1-3163961 1.3365692 34 12453112 1.9618820 1.279 2600 1.2975240 1.3167210 1-3369141 183533 18828 2**** 33 31 30 29 28 27 26 25 35 1.2456131 1-2621647 1.2795570 1.2978362 1.3170523 1.3372594 21. 36 1.2458823 1.2621475 1.2798543 1.2981487 1.3173808 1.8376049 23 37 1.2461518 1.2627306 1.2801518 1.2984614 1-3177096 1.3379507 22 38 1-2464214 1.2630140 1.2801495 12987743 1.3180386 13382968 21 39 1-2466913 1-2632975 1.2807175 1.2990876 1-3183680 1 3386132 20 40 1.2169614 1.2635813 1.2810457 1.2991011 1.3186976 1-3389898 19 41 1-2472317 1-2635653 1.2813442 1.2997148 1.3190274 1.3393368 18 42 1.2475022 1-26-11196 1.2816130 1-3000288 1.3193576 1.3396841 17 43 1.2477730 12644311 1.2819119 1.3003431 1.3196881 1.3400316 16 41 1-2180440 1.2047188 1.2822412 1.3006576 1.3200188 1.3103795 15 45 1.2483152 1.2650038 1.2625407 1-3009721 1-3203498 1.3407276 14 46 1.2485866 1-2652890 1'2828401 1.3012875 1.3206810 1-3410761 13 47 1.2488583 1.2655745 1.2831404 1.3010028 1.3210126 1.3414248 12 48 1-2491302 1.2658601 1.2831106 1.3019184 1-3213444 1-3417738 11 49 1.2491023 1.2661460 1.2837411 1.3022313 1.3216765 1.3421232 10 50 1.2496746 1.2664322 1.2810418 1'3025504 1.3220089 1-3424728 9 51 1-2499471 1-2667186 1.2843123 1.3028667 1.3223416 1.3428227 8 62 1.2502199 1.2670052 1.2846140 1·3031831 1.3226745 1 3131729 7 53 1.2501929 1.2672921 1.2849455 1·3035003 1.3230078 1-3435234 6 51 1.2507661 1.2675792 1.2852472 1-3038175 1.3233413 1.3438742 5 55 1.2510396 1-2678665 1.2855492 1-3041349 1.3236750 1.3442253 4 56 1-2513133 1-2681541 1-2858514 1.3044526 1.3240091 1-3445767 57 1.2515872 1.2681419 1.2861539 1.3047706 1.3213135 1.3149284 2 68 1.2518613 1.2687299 1.2861566 1-3050888 1.3216781 1.3152801 69 1-2521357 1.2690182 1-2867596 1-3054073 1-3250130 1.3456327 0 60 53° 52° 51° 50° 49° 48° COSECANTS. 300 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 42° 0 43° 1.3456327 1.3673275 1.3901636 1.4142136 1.3159853 1.3670985 1.3905543 1-4146251 1.3163382 1.3680699 44° 45° 46° 47° 1.4395565 1.4662792 60 1.4399904 1.4667368 59 1.3909453 1:4150370 1·4404216 1.4671918 58 3 1.3466914 1.3684416 1.3913366 14154493 14408592 1.4676532 57 4 1·3170149 1.3688136 1.3917283 14158619 1.4412941 1-4681120 56 5 1-3173987 1-3691859 1·3921203 1.4162749 1-4417295 1.4685713 55 6 1.3477528 1-3695586 1.3925127 1-4166883 1.4421652 1-4690309 54 7 1.3481072 1.3699315 1.3929054 14171020 1.4426013 1.4694910 53 8 1.3181619 1.3703018 1.3932985 9 1-3488168 1.3706784 1.3936918 1.4179306 10 1·3491721 13710523 "1.3940856 1-4183454 1.4175161 1:4430379 1-4434748 1:4439120 1:1708736 1.4699511 52 1.4704123 51 50 11 1.3495277 1.3714266 1-3914796 1.4187605 1-4443497 1.4713354 49 12 1.3498836 1.3718011 1-3918740 1-1191761 1.4447878 1-4717975 48 13 1-3502398 1-3721760 1.3952688 1.4195920 1:4462962 1.4722600 47 14 1-3505963 1.3725512 1.3956639 1-4200082 11186651 1:4727230 46 15 1-3509531 1.3729268 1-3960593 1.4204248 1.4461043 1-4731864 45 16 1.3513102 1.3733026 1.3964551 14208418 1-4165439 1.4736502 44 17 1.3516677 1.3736788 1.3968512 1.4212592 1-4469839 1-4741144 43 18 1.3520254 1.3740553 3972477 1-4216769 1-4474243 1.4745790 42 19 1.3523834 1.3711321 1.3976445 1.4220950 1.4178651 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 1.3534593 1.3755645 1.3988369 1-4233514 1-4491898 1.4764417 38 23 1.3538185 1-3759426 1.3992351 1-4237710 1.4496322 1-4769081 37 21 1.3541780 1.3763210 1.3996336 1.4211909 1:4500749 1.4773755 36 25 1.3515379 1.3766998 1·4000325 1.4246112 1-1505181 1-4778131 35 26 1.3548980 1.3770789 1·4004317 1.4250319 1.4509616 1-4788111 34 27 1.3552585 1-3771583 1-1008313 1.4254529 1.4514055 1-4787795 33 28 1.3556193 1.3778380 1-4012312 1.4258743 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 1.3793602 1-4028343 1.4275636 1-4536311 1-4811278 28 33 1.3571277 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 1.1040403 1.4288345 1-4549712 1:4825120 25 36 1.3585164 1.3808877 1.4044430 1.4292588 1-4551187 1-4830142 24 37 1.3588800 1.3812701 1.1048461 1-4296836 1-4558666 1.4834868 23 38 1-3592138 1.3816534 1·1052194 1-4301087 1.4563149 1.4839599 22 39 1.3596080 1.3820367 1.4056532 1'4305342 1-4567636 1-4814334 21 40 1.3599725 1.3821201 1.1060573 1.4309600 1-4572127 1.4849073 20 41 1.3603372 1.3828014 1.4064617 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 1:1322399 1.4585623 1-4863317 17 44 1.3614334 45 1:3617995 1.3839584 1.4076772 1.3843437 1-4326672 1-4080831 1.4330950 1-4590130 1.4868073 16 1.4594641 1-4872834 15 46 1.3621658 1.3847294 1-4084893 1-4335231 47 1.3625324 1-3851153 1.4088958 1-4339516 1.4599156 1*4877599 14 1.4603675 1.4882369 13 48 1-3628994 1.3855017 1.4093028 1.4313805 1-4608198 14887142 12 49 1.3632667 1.3858883 1.4097100 1.4348097 14612726 1.4891920 11 50 1.3636313 1-3862753 1-1101177 14352393 1-4617257 1.4896703 10 51 1.3610022 1-3866626 1'4105257 1.4356693 1.4621792 1·4901489 9 52 1.3613704 1.3870503 1·4109340 1.4360997 1.4626331 1.4906280 8 53 1·3647389 1.3871383 1-4113427 1.4365305 1.1630875 1.4911076 7 54 1.3651078 1-3878266 1-4117517 1-4369616 55 1.3651770 1-3882153 1-4121612 56 1.3658461 1.3886043 57 58 59 60 47° 1.3673275 1.3901636 46° 1-4125709 1.3662162 1.3889936 1-4129810 1-3665863 1.3893832 1-4133915 1.3669567 1.3897733 1-4138024 1*4142136 45° 1.4373932 1.4378251 1.4644529 1-4925488 1.4382574 1.4649089 1.4930301 1.4386900 1.4653652 1-4935118 1.4391231 1.4658220 1.4939940 1.4395565 1-4662792 1.4944765 44° 43° 1.4635422 1.4915876 1.4639973 1.4920680 6 5 4 ลลล 3 2 1 0 42° COSECANTS. 301 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 48° 49° 50° + 51° 52° 53° 1:49 14765 1.5242531 1:4949596 1.5247634 1.5557238 1.5562634 1.5890157 1.6242692 1-6616401 60 1.5895868 1-6248743 1.6622819 59 14951131 3 1.4959270 1.5252741 1.5568035 1'5257854 1.5573441 1.5901584 1.6254799 1-6629243 58 1.5907306 1.6260861 1.6635673 57 & 1.1964113 1-5262971 1.5578852 1.5913033 1.6266929 1-6642110 56 5 1-4968961 1.5268093 1.5584268 1.5918766 1-6273003 1-6648553 55 6 1-4973813 1-5273219 1.5589689 1.5924501 1.6279083 1.6655002 54 7 1-4978670 1-5278351 1.5595115 1.5930247 1.6285169 1.6661458 53 8 1-4983531 1-5283487 1.5600546 1.5935996 1.6291261 1-6667920 52 9 1-4988397 1-5288627 1.5605982 1-5941751 1-6297359 1-6671389 51 110 10 14993267 1-5293773 1.5611424 .1.5917511 1·6303462 1.6680864 50 -4 11 1.4998141 1.5298923 1-5616871 1.5953276 1.6309572 1-6687345 49 12 1.5003020 1.5301078 1-5622322 1.5959048 1.6315688 1.6693833 48 13 1.5007903 1-5309238 1-5627779 1.5964824 1-6321809 1-6700328 47 14 1.5012791 1.5314403 1.5633241 1.5970606 1.6327937 1.6706828 46 15 1.5017683 1.5319572 1.5638708 1-5976391 1·6331070 1-6713336 45 16 1·5022580 1.5324746 1.5644181 1-5982187 1.6340210 1.6719850 44 17 15027481 1-5329925 1.5619658 1.5987986 1.6316355 1.6726370 43 18 1.5032387 1.5335109 1.5655141 1.5993790 16352507 1-6732897 42 19 1-5037297 1.5340297 1.5660628 1.5999600 1-6358661 1.6739430 41 20 1.5042211 1.5345491 1-5666121 1-6005416 1-6364828 1-6745970 40 21 1-5047131 1-5350689 1.5671619 1.6011237 1-6370997 1-6752517 22 1-5052051 1-5355892 1.5677123 1.6017061 1.6377173 1-6759070 23 1·5056982 1.5361100 1.5682631 1.6022896 1-6383355 1-6765629 24 1-5061915 1.5366313 1.5688145 1.6028734 1.6389542 1.6772195 25 1.5066852 1-5371530 1·5693664 1.6034577 1.6395736 1-6778768 26 1:5071793 1.5376752 1.5699188 1.6040126 1-6401936 1.6785317 27 1.5076739 1.5381980 1.5704717 1.6046281 1-6108142 1.6791933 28 1.5081690 1-5387212 1-5710252 1-6052142 1-6414354 1.6798525 29 1.5086645 1.5392449 1.5715792 1.6058008 1-6420572 1-6805124 30 1-5091605 1.5397690 1-5721337 1.6063879 1.6426796 1-6811730 31 1-5096569 1.5402937 1-5726887 1-6069757 1-6133027 1-6818312 32 1.5101538 1.5408189 1.5732443 1.6075640 1.6439263 1-6824961 33 1.5106511 1-5413445 1.5738004 1.6081528 1-6415506 1-6831586 34 1-5111489 1.5118706 1-5743570 1.6087423 1-6451754 35 1.5116472 1-5423973 1-5749141 1.6093323 1·6158009 36 1.5121459 1-5429244 1-5751718 1.6099228 1.6464270 1.6838219 1-6844857 1-6851503 25 24 37 1-5126150 1.5134520 1-5760300 1.6105140 1-6470537 1.6858155 38 1.5131446 1-5139801 1.5765887 1.6111057 1-6476811 1·6864814 22 39 1.5136147 1.5445087 1-5771479 16116980 40 1.5141452 1.5150378 1-5777077 1.6122908 1-6489376 41 1.5146462 1.5155673 1.5782680 1-6128843 1'6 193668 42 1.5151477 1·5160974 1.5788289 1.6134788 1.6501966 1-6483090 1.6871479 1.6878151 1-6884830 1.6891516 21 20 19 18 43 1.5156196 1.5463280 1.5793902 1-6140728 1-6508270 1-6898208 17 44 1-5161520 1-5471590 1-5799521 1.6146680 1-6514581 45 1.5166548 1-5176906 1.5805146 1.6152637 1.6520898 1.6904907 1-6911613 16 15 2**** ***** ***** ***** 22-23 39 38 37 36 35 34 33 32 31 30 29 28 27 26 23 46 1.5171581 1.5482226 1-5810776 1.6158600 1.6527221 1.6918326 14 47 1.5176619 1-5487552 1.5816411 1.6161569 1-6533550 1.6925045 13 48 1.5181661 1.5492882 1-5822051 1.6170544 1.6539885 1-6931771 12 49 1.5186708 1-5498218 1.5827097 1-6176524 1.6546227 1.6938501 11 50 1-5191759 1:5503558 1.5833318 1.6182510 1·6552575 1.6945244 10 51 15196815 1.5508901 1.5839005 1-5841667 52 53 51 55 56 1.5222166 1.5535706 57 1-5227250 1.5541081 1.5873058 58 1-5232339 1.5546462 1-5878752 59 60 1-5237433 1.5551818 1.5881452 1-5242531 1.5557238 1-5890157 1.6188502 1.5201876 1.5514254 1-6194500 1.6565290 1.5206012 1.5519610 1·5850334 1-6200504 1.5212012 1.5524970 1.5356007 1.6206513 1.5217087 1.5530335 1.5861685 1-6212529 1.5867369 1.6218549 1.6224576 1-6230609 1.6236648 1.6212692 1-6558929 1.6951990 1.6958744 1.6571657 1-6965504 1-6578030 1.6972271 1-6581409 1-6979044 1.6590795 1.6985825 1.6597187 1-6992612 1.6603586 1-6999107 16609990 1.7006208 1-6616401 1-7013016 1876 LO 9 5 • TO $7 pm – 4 3 41° 40° 39° 38° 37° 36° COSECANTS. 302 TABLE XI.-SECANTS AND COSECANTS. SECANTS. ке 1 54° 1.7013016 1.7019831 55° 56° 57° 58° 59° 2 1.7431468 1.7882916 1-7441715 1.7890633 1.7026653 1.7448969 1.7898357 1-7033482 1-7456230 1.7906090 1.7040318 1.7463499 1-7913831 1-7047160 1-7470776 1.8360785 1.8369013 1-8377251 1·8383198 1.8393753 1-8906016 1-7921580 1-8102018 1-8914845 1.8870799 1-9110010 1*8879589 1-9425445 1-8888388 1.9434861 1-8897197 1.9444238 1-9453725 1.9463173 6 1.7054010 1.7478060 1-7929337 1-8410292 1.892368+ 1.9472632 7 1.7060867 1.7485352 1.7937102 1.8418574 1.8932532 1.9482102 8 1.7067730 1.7492651 1.7941876 1-8426866 1.8941391 1 9491583 9 1.7074601 1-7499958 1.7952658 1-8135166 1·8950259 10 1-7081478 1.7507273 1·7960119 1-8143176 1-8959138 11 1.7088362 1.7514595 1.7968247 1-8451795 1.8968026 12 1.7095254 1.7521924 13 1.7102152 1.7529262 1.7976054 1.8460123 1.7983869 1-8408460 1-9501075 1.9510577 1.9520091 1-8976924 1-9529615 1-8985832 1.9539150 14 1.7109058 1·7536607 1.7991693 1.8476806 1.8994750 15 1.7115970 1-7513959 1-7999524 1-8185161 1.9003678 16 17 1.7122890 1.7551320 1.7129817 18 1.7136750 1-8007365 1-8493525 1.7558687 1.8015213 1-8501898 1.7566063 1.8023070 1-8510281 1.9518697 1-9558254 1.9012616 1.9567822 1.9021564 1-9577402 1.9030522 44 43 1.9586992 19 J.7143691 1.7573446 1-8030935 1-8518672 19039491 1-9596593 41 20 1.7150639 1.7580837 1-8038809 1.8527073 1.9018169 1.9606206 40 21 1'7157591 1.7588236 1.8046691 1-8535483 1.9057457 1.9615829 39 22 1.7161556 1.7595642 1.8051582 1.8543903 1.9066156 1-9625464 38 23 1.7171525 1.7603057 1-8062481 1-8552331 1.9075464 1.9635110 37 24 1.7178501 1.7610478 1.8070388 1-8560769 1-9081483 1.9644767 36 25 1.7185184 1.7617908 1-8078304 1.8569216 1-9093512 1.9654435 35 26 1.7192475 1.7625345 1-8086228 1-8577672 1.9102551 1.9664114 34 27 1-7199472 1.7632791 1.8094161 1.8586138 1-9111600 1.9673805 33 28 1.7206477 1.7640211 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 1.76626 19 1.8125977 1 8620093 1.9147899 1-9712680 29 32 1.7234568 1.7670133 1.8133953 1.8628605 33 1.7241609 1.7677625 1-8141937 34 1.7248657 1.7685125 1-8149929 1.9156999 1.8637126 19166110 1.8645657 1-9722427 28 1.9732185 35 1.7255712 1.7692633 1-8157930 1.8654197 1-9175230 1-9741954 1.9181362 .1.9751735 26 25 36 1.7262774 1.7700149 1-8165940 1.8662747 1.9193503 1-9761527 24 37 1.7269814 1.7707672 1-8173958 1.8671306 1.9202655 1.9771331 23 38 39 1.7284005 1.7276921 1.7715204 1-8181985 1-7722743 1-8679875 40 1.7291096 1.8190021 1.7730290 1.8198065 1-8688453 1.8699010 41 1.7298195 1.7737815 1-8206118 1-8705637 1-9211817 1-9781116 1.9220990 1.9790972 1-9230173 1.9800810 1.9239366 1-9810659 21 20 19 42 1.7305301 1.7745409 1-8214179 1.8711244 1-9248570 1.9820520 43 1-7312414 1.7752980 1-8222249 1.8722859 1.9257781 1.9830393 44 1-7319535 1-7760559 1-8230328 1-8731485 45 1.7326663 1-7768146 1-8238416 1-8740120 1.9267009 1.9840276 1.9276211 1·9850172 16 888583 HANJA 2*7** ***** 88-88 ***78 2*7** ***22 2**** 60 59 57 56 55 54 53 52 51 50 49 48 47 46 45 42 18 17 15 46 47 48 49 1.7333798 1.7775741 1-8246512 1.7340941 1.7783311 1-7818091 1-7790955 1-7355248 1-7798574 1-8748764 1-8254617 50 1.7362413 1-7806201 1-8757119 1-8262731 1.8766082 1.8270851 1-8774755 1-8278985 1-8783438 1-9285480 1.9860080 1.9294746 1.9869997 14 13 1-9301013 1.9879927 1.9313290 12 1.9889869 11 1-9322578 1.9899822 10 51 1.7369585 1-7813836 1-8287125 1-8792131 1-9331876 1.9909787 9 52 1.7376761 1.7821479 1.8295274 1.8800833 1-9341185 1.9919764 53 1.7383951 1.7829131 1-8303432 1*8809545 1-9350505 1.9929752 7 54 1.7391145 1.7836790 1.8311599 1.8818266 1.9359835 1.9939733 6 55 1.7398347 1.7844457 1-8319774 1.8826998 1-9369176 -1.9949764 5 56 1.7405556 1.7852133 1.8327959 1-8835738 1.9378527 1.9959788 4 57 1.7412773 I-7859817 1-8336152 1-8844189 1.9387889 1.9969823 3 58 59 60 1-7419997 1.7867508 1-8344354 1.8853219 1.7427229 1.7875208 1-8352565 1-8862019 1-7431468 1.7882916 1-8360785 1.8870799 35° 34° 33° 1-9397262 1.9979870 1.9106616 1.9989929 32° 31° 1.9416010 2·0000000 30° 0 COSECANTS. 303 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 60° 61° 62° 63° 64° 65° 2.0000000 9-0010083 2.0626653 2.1300545 60 2.2026893 2-2811720 2-3662016 2-0637484 59 2-1312205 1 2*2039176 2-2825335 2-3676787 0020177 20030298 2-0648328 2.0659186 2-1335570 58 2-1323830 2-2052075 2.2838967 2.3691578 57 3 2.2064691 2-2852618 2-3706390 2-0010402 56 2-0670056 2.1347274 4 2.2077323 2.2566286 2-3721222 2.0050533 2-0680940 2.1358993 55 5 2-2089972 2-2879974 2.3736075 2-006067 54 2.0591836 6 2·1370726 2:2102637 2.2893679 2.3750949 53 7 2-0070828 2.0702716 2.1382475 2-2115318 2.2907493 2-3765843 52 8 2.0080991 2-0713670 2-1394238 2-2128016 2-2921115 2.3780758 51 2.0091172 2.0724606 9 2.1406015 2-2140730 2-2934906 2-3795694 50 10 2.0101362 2-0735556 2-1417808 2.2153460 2-2948685 2.3810650 49 2-0111561 11 2-0746519 2.1429615 2.2166208 2.2962483 2.3825627 48 2:0121779 12 2-0757496 2-1441438 2.2178971 2.2976299 2.3840625 47 2-0132005 2.0768486 13 2-1453275 2-2191752 2.2990131 2.3855645 46 2-0142243 2.0779489 2.1465127 14 2.2201548 2.3003988 2-3870685 45 2-0152491 2-0790506 15 2.1476993 2-2217362 2.3017860 2.3885746 16 2.0162756 2.0801536 2-1488875 44 2.2230192 2.3031751 2:3900828 43 2-0173031 2.0812580 17 2-1500772 2.2243039 2.3045660 2.3915931 42 18 2-0183318 2-0823637 2.1512684 2-2255903 2.3059588 2.3931055 41 2.0193618 2.0834708 2.1524611 19 2.2268783 2.3073536 2.3946201 40 2.0203929 2.0815792 2.1536553 20 2-2281681 2.3087501 2.3961367 2.0214253 2.0856890 21 2-1548510 2.2294595 2.3101486 2.3976555 22 2-0224589 2-0868002 2-1560182 2.2307526 2.3115490 2.3991764 23 2.0231937 2.0879127 2.1572469 2.2320474 2.3129513 2.4006995 2-0215297 2-0890265 2.1584471 21 2-2333138 2-3143554 25 2.0255670 2.0901418 2.1596489 2.4022247 2.2316420 2.3157615 2.4037520 26 2.0266056 2-0912584 2.1608522 2.2359419 2-3171695 2.4052815 2.0276+53 2.0923761 2.1620570 27 2-2372435 2.3185794 2-1068132 2.0286863 2-0934957 2.1632633 28 2-2385468 2.3199912 2.4083469 2.0297286 2-0946164 2-1644712 29 2-2398517 2-3214049 2-1098829 30 2-0307720 2-0957385 2.1656806 2-2411585 2.3228205 2.4114210 2-0318168 2.0968620 2.1668915 31 2.2424669 2.3242381 2.4129613 2-0328628 2.0979869 2.1681040 32 2-2437770 2.3256575 2:4145038 2-0339100 2.0991131 2.1693180 33 8.2450889 2.3270790 2.4160184 34 2-0349585 2.1002408 2.1705335 2.2464025 2-3285023 2.4175952 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 2.0381114 2.1036320 2:1741895 37 2-2503536 2.33278-10 2:4222488 • 2.0391649 2.1047652 38 2-1754113 2.2516741 2.3342152 2-4238044 2:0402197 2.1058998 39 2-1766346 2-2529964 2.3356482 2.4253622 តតតតត ឥឥឥឥ ឥឥ ឥឥឥឥន 30, 27 24 23 20 40 2-0412757 2.1070359 2-1778595 2-2513201 2·3370833 2.4269222 19 2.0423330 2-1081733 2.1790859 41 2.2556461 2-3385203 2-4284844 18 2-0433916 2.1093121 2.1803139 42 2-2569736 2-3399593 2.4300489 17 43 2.0444515 2-1104523 2.1815435 2-2583029 2.3414002 2-4316155 16 2.0455126 2.1115940 2-1827746 44 2.2596339 2.3428432 2-4331844 15 2.0465750 2-1127371 2.1840074 45 2.2609667 2.3442881 2.4347555 14 46 2.0476386 2-1138815 2-1852117 2-2623012 2-3457349 2.4363289 13 2.0487036 2-1150274 2.1864775 47 2.2636376 2.3471838 2.4379045 12 2:0497698 2.1161748 2.1877150 48 2:0508373 2-1173235 2.1889541 49 2.1901947 50 59 FA**6 85688 51 52 53 54 55 2.0519061 2:1184737 2.0529762 2.1196253 2.1914370 2.0510476 2-1207783 2.1926808 2.0551203 2-1219328 2-1939262 2.0561942 2.1230887 2-1951733 2:0572695 2.1242460 2-1964219 2.2649756 2.2663155 2.3500875 2-2676571 2.3515124 2-2690005 9.2529992 2-4442294 2-2703457 2.3544581 2.4458163 2-2716927 2-3559189 2-4474054 2.3573818 2-2730415 2.4189968 2.2743921 2-3588467 2.4505905 2.3486347 2-1394823 11 2.4410624 10 2.4426448 9 98744 6 2-0583460 2.1254048 2-1976721 2.2757445 2.3603136 2.4521865 57 60 2.0594239 2-1265651 2.1989240 2.0605031 2-1277267 2.2001775 2-0615836 2-1288899 2.2014326 2.0626653 2.1300545 2.2026893 29° 28° 27° 2-2770987 2.3617826 2-4537848 2-2784546 2.3632535 2.4553853 9.2798124 2-3647265 2.4569882 2-2311720 23662016 2.4585933 26° 25° 24° COSECANTS. 304 TABLE XI-SECANTS AND COSECANTS. SECANTS. 66° 67° 68° 69* 70° 71° C HP CONHO 0 1 2 3 4 2.4585933 2.5593047 2-6694672 2.7904281 2.9238041 3:0715335 GO 2.460 2008 2.5610599 2 6713906 2-7925114 2-9201431 3:0741507 2 4618106 2.5628176 2.6733171 2.7916611 2.9291858 3-0767525 59 58 2:1634227 2-5645781 2.6752465 2.7967873 2.9308326 3-0793590 57 2-1030371 2.5663412 2.6771790 2.7989140 2.9331833 3.0819702 6 2-4600538 2.5681069 2-6791145 2-8010141 2.9355380 3.0815560 56 55 6 2:4682729 2.5698752 2.6810530 2·8031777 2.9378968 3.0872066 51 7 24693913 2.5716162 2-6829915 2-8053148 2-9402597 3.0898319 53 8 2.4715181 2.5734199 2-6819391 2.8074554 2-9126265 3.0921620 52 9 2-1731442 2.5751963 2.6868867 2.8095995 2-9119975 3.093)267 51 10 2:4747726 2.5769753 2.6888371 2·8117471 2.9473725 3-0977363 50 11 2.4764034 2.5787570 2'6907912 2.8138982 2.9197516 3.1003805 49 12 2:1780366 2.5805114 2.69 27480 2.8160529 2.9521348 3.1030296 48 13 2.4796721 2.5823284 2.6947079 2.8182111 2.9545221 3-1056835 47 14 2.1813100 2.5841182 2.6966709 2-8203729 2.9569135 15 2:1829503 2.5859107 2.6986370 2-8225382 2 959 3090 16 2 1815929 2:5877058 2.7006061 2-8247071 17 2 4862380 2-5895037 2-7025784 2.8268796 2.9617087 2-9611125 3.1136740 3.1163172 43 3.1063122 3.1110057 45 44 46 18 2.4878854 2.5913043 2.7015538 2-8290556 2-9665205 3.1190252 42 19 2.4895352 2-5931077 20 2:4911874 2.5919137 2.7065323 2-7085139 2.8334185 2-8312353 2-9689327 3.1217081 41 2:9713190 3.1243959 20 21 2-4928421 2.5967225 22 2.194499 ! 2.5985341 23 2:4961586 2.0003484 24 2-1978201 2.6021654 2·7104987 2.8356054 2.7124866 2-8377938 2.7141777 2.8399899 2-7164719 2-8421877 2.9737695 3:1270886 39 2.9761942 2-9786231 25 2.4994818 2.6039852 2.7181693 2.8443891 3.1297862 3.1321887 2.9810563 3.1351962 36 2-9831936 3.1379086 35 38 37 26 2.5011515 26058078 2.7204698 2.8165941 2-9859352 3.1406259 34 27 2.5028207 2.6076332 2-7224735 2-8188028 2-9883811 3.1433483 33 28 2.5011923 2.6091613 2.7244804 2.8510152 2.9908312 3.1160756 32 29 2.5061663 2-6112922 2-7261905 2.8532312 2.9932856 3.1488079 31 30 2-5078428 2.6131259 2.7285038 2.8554510 2.9957143 3.1515153 30 31 2.5095218 2.6149624 2-7305203 2-8576744 2.9982073 3.1542877 29 32 2.5112032 2.6168018 2.7325400 2.8599015 3.0006746 3.1570351 28 33 2-5128871 2-6186439 2.7345630 2-8621324 3-0031462 3.1597876 27 34 2.5145735 2.6204888 2-7363892 2.8613670 3-0056221 3.1625152 26 35 2.5162624 2.6223366 2-7386186 2-8666053 3-0081021 3.1653078 25 36 2.5179537 2.6241872 2.7106512- 2.8688474 3.0105870 3.1680756 24 37 2-5196175 2.6260106 2-7426871 2-8710932 3.0130760 3-1708484 23 38 2-5213438 2-6278969 2.7447263 2-8733428 3-0155691 3.1736264 22 39 40 2-5230126 2.6297560 2.5247440 2-6316180 2.7167687 2-8755961 3-0180672 3-1764095 21 2-7488144 28778532 3-0205693 3.1791978 20 41 2-5204478 2-6334828 2-7508634 2.8801142 3.0230759 3.1819913 19 42 2.5281541 2.6353506 2.7529157 2.8523789 3.0255868 3.1847899 18 43 2.5298630 2-6372211 2-7549712 2.8846474 3.C281023 3.1875937 17 44 2-5315744 2.6390946 2.7570301 2.8869198 3-0306221 3·1901028 16 45 2-5332883 2-6409710 2-7590923 2.8891960 3:0331461 3.1932170 15 46 2-5350048 2.6428502 2-7611578 2.8911760 3.0356752 3.1960365 14 47 2.5367238 2-6447323 2.7632267 2.8937598 3.0382081 3.1988613 13 48 2-5381453 2.6166174 2-7652988 2.8960175 3-0107462 3.2016913 12 49 2.5401691 2.6485054 2.7673744 2.8983391 3.0132884 3.2045266 11 50 2.5118961 2.6503962 2-7694532 2.9006316 3.0-158352 3.2073673 10 51 2-5436253 2-6522901 2.7715355 2.9029339 3.0183861 3.2102132 9 52 53 54 2-5453571 2.6541868 2.7736211 2.5170915 2.6560865 2-7757100 2.5488284 2-6579891 2.7778024 2.9052372 3.0509123 3.2130644 2.9075443 3-0535026 3.2159210 2.9098553 55 2:5505680 2-6598947 2.7798982 2.9121703 3-0560675 3.2187830 3.0586370 3.2216503 876∞ 5 56 2.5523101 2-6618033 2-7819973 2.9144892 57 2-55105-18 2.6637148 2-7840999 2.9168121 58 2.5558022 2.6656292 2.7862059 2-9191389 3-0612111 3.0637898 3 0663731 3.2215230 4 69 2.5575521 2.6675467 2.7882153 2-9211697 3 0089610 3-2274011 3.2302546 3-2331736 3 2 1 60 2-5593017 2.6694672 2.7901281 2-9238044 3 0715535 3.2360680 23° 22° 21° 20° 19° 18° COSECANTS. 305 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 72° 73* 74° 75° 76° 77° 3.2360680 3.1203036 3.6279553 1 3.2389678 3.4235611 3.6316395 3.8679025 2 3.2418732 3.1268251 3.6353316 3-8721112 3 3.2117810 3:1300956 3.6390315 3-8763293 3.8637033 4.1335655 4-4454115 60 4-4510198 4.1383939 4.1132339 4-4566428 4.1480856 59 58 4.4622803 57 4 3.2477003 3.1333727 3.6427392 3-8805570 4.1529491 4-4679324 56 5 10 67899 3.2506222 3.2535196 3.2564825 3.4366563 3-1399165 3.6161518 3-8817913 4.1578243 4.1735993 55 3.6501783 3.8890411 4.1627114 4-4792810 54 3.4432433 3.6539097 3-8932976 4.1676102 4-4819775 53 3.2594211 3.4165167 3.6576191 3.8975637 4.1725210 4.1906889 52 3.2623652 3.1498568 3.6613964 3.9018395 4.1774138 4-4961152 51 3.2653149 3:1531735 3.6651518 3.9061250 4.1823785 4*5021565 50 11 3*2682702 3.4561969 3.6689151 3.9104203 4.1873252 4-5079129 49 12 3.2712311 3.1598269 3.6726865 3.9117251 4-1922840 4.5136811 48 13 3.2741977 3.4631637 3.6764660 3.9190103 4.1972519 4-5191711 47 14 3.2771700 3:1665073 3.6802536 3-9233651 4-2022380 4-5252730 46 15 3.2801179 3.1698576 3:6810193 3.9276997 4.2072333 4.5310903 45 16 3.2831316 3.4732146 3.6878532 3-9320443 4-2122408 4-5369229 44 17 3.2861209 3.4765785 3.6916652 3-9363988 4.2172606 4-5427709 43 18 3.2891160 3-1799492 3.6951854 3.9107633 4-2292928 4.5486311 42 19 3.2921168 3.1833267 3.6993139 3.9151379 4.2275 73 4.5515134 41 20 3.2951231 3.4867110 3.7031506 3.9195224 4.2323943 4.5604030 40 21 3.2981357 3.490r023 3.7069956 3.9539171 4.2374637 4.5663183 39 22 3-3011539 3-4935001 3.7108189 3-9583219 4.2125457 4.5722414 38 23 3.3041778 3.1969055 3.7147105 3-9627369 4.2476402 4.5731862 37 21 3.3072076 3.5003175 3.7185805 3-9671621 4-2527174 4.5841430 36 25 3.3102432 3.5037365 3.7221589 3.9715975 4.2578671 4.5904 35 26 3.3132817 3-5071625 3-7263157 3.9760131 4.2629996 4.5961070 34 27 3.3163320 3-5105954 3.7302109 3.9804991 4.2681449 46021126 33 28 3-3193853 3-5140354 3.7311146 3.9819651 4.2733029 4.6081313 32 29 3.3221444 3-5174824 3.7380563 3.9894121 4.2784733 4-6141722 31 30 3.3255095 3-5209365 3.7419775 3.9939292 4.2836576 4.6202263 30 31 3-3285805 3-5243977 3-7459068 3.9984267 4.2888513 4-6262967 29 32 3-3316575 3.5278660 3.7198117 4:0029317 4.2940610 4.6323835 28 33 3-3317105 3-5313411 3.7537911 4-0074532 4.2992867 4.6381867 27 34 3-3378294 3-5348210 3-7577462 4-0119823 4.3045225 4-6146061 26 35 3.3409244 3-5383138 3.7617100 4.0165219 4-3097715 4.6507127 25 36 3.3140254 3.5418107 3.7656821 4.0210722 4.3150336 4.6568956 24 37 3-3471324 3.5453149 3.7696636 4-0256332 4.3203090 4.6630652 23 38 3-3502455 3.5488263 3.7736535 4.0302018 4.3255977 4.6692516 22 39 3.3533617 3-5523150 3.7776522 4.0347872 4-3308996 4.6754518 21 40 3.3561900 3.5558710 3.7816596 4-0393801 4.3362150 4-6816718 20 41 3.3596214 3.5591042 3.7856760 4-0139844 4.3415438 4.6879119 19 42 3-3627589 3.5629418 3.7897011 4.0485992 4.3468861 4.6941660 18 43 3-3659026 3.5661928 3.7937352 4-0532249 4.3522419 4.7004372 17 41 3-3690524 3-5700481 3.7977782 45 3.3722081 3.5736108 3-8018301 4.0578615 4.3576113 4.0625091 4.7067256 16 4-3629913 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 4.7256945 13 48 3.3817138 3-5843437 3.8140399 4.0765181 4.3792257 4-7320521 12 49 3-3848948 3.5879362 3-8181280 4-0812100 4.3816638 4·7381277 11 50 3.3880820 3.5915363 3-8222251 4.0859130 4.3901158 4·7418206 10 51 3.3912755 3.5951439 3.8263313 4.0906272 4.3955817 4.7512312 9 52 3.3944754 3.5987590 3.8304467 4-0953526 53 3-3976816 3.6023818 3-8345713 54 3.4008941 3.6060121 3.8387052 55 3:1011130 3.6096501 3-8128182 4.1000893 4.1048371 4-4120637 4-1095967 4-4010616 4-4065556 4-7576596 8 4.7641058 7 4.7705699 6 4.4175859 4.7770519 5 56 3:1073382 3.6132957 3.8470006 4-1143675 4-4231224 4.7835520 4 57 3-4105699 3.6169490 3.8511622 4.1191498 4-4286731 4-7900702 3 58 59 60 3.6206101 3.4138080- 3.8563332 4.1239435 3.6212708 3.4170526 3-8595135 4-1287487 3.4203036 3.6279553 3-8637033 4.1335655 4.4342382 4.7966066 4.4398176 4.8031613 4-4154115 4.8097313 • 17° 16° 15° 14° 13° 12° COSECANTS. 306 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 78° 79° 80° 81° 82° €.3° 4-8097313 1 4.8163258 2 4-8229357 5.2408431 5.7587705 5.2486979 5.7682867 5.2565768 5.7778350 6.3924532 7.1852965 8.2055039 60 6.4042154 7.2001996 8-2210052 50 6*4160216 7-2151653 8.2415719 53 3 4.8295643 5.2614798 5.7874153 6.4278719 7-9301940 8.2642405 57 4-8362114 5 2724070 5.7970280 6.4397666 7-2152859 8-2810171 56 4-8128774 5.2803587 5.8066732 6.4517059 7-2604417 8-3038812 55 6 4.8495621 5.2883347 5.8163510 6.4636901 7.2756616 8.3238415 54 7 4-8562657 5.2963354 5.8260617 6.4757195 7-2909160 8.34389S6 53 8 4-8629883 5.3013608 5.8358053 6.4877914 7.3062951 8.3610531 52 9 4-8697299 5.3121109 5.8455820 6.4999148 7.3217102 8.3813065 51 10 4-8761907 5.3201860 5.8553921 6.5120812 7.3371909 8-4016586 50 11 4.8832707 5.3285861 5.8652356 6.5242938 7.3527377 8.4251105 49 12 4.8900700 5.3367114 5.8751128 6.5365523 7.3683512 8*1456629 48 13 4.8968886 5.3448620 5.8850238 6.5188586 7.3840318 8-4663165 47 14 4.9037267 5.3530379 5.8949688 6.5612113 7.3997798 8-4870721 46 15 4.9105841 5.3612393 5.9019179 6.5736112 7.4155959 8.5079301 45 16 4-9174616 5.3094664 5.9149614 6.5860587 7:1314803 8-5288923 44 17 4.9243586 5.3777192 5.9250095 6.5985540 7-4474335 8-5499581 43 18 4-9312754 5.3859979 5.9350922 6.611 973 7:1631560 8-5711295 42 19 4.9382120 5.3913026 5.9152098 6.6236S90 7.4795182 8.5924065 41 20 4.9151687 5.4026333 5.9553625 6-6363293 7:4957106 8.6137901 40 21 4-9521453 5.4109903 5-9655504 6.6190181 7.5119437 8.6352812 39 22 4.9591421 5.4193737 5.9757737 6.6617568 7.5282478 8-6568805 38 23 4.9661591 5.4277835 5.9860326 6.6745446 7-5446236 8.6785889 37 21 4.9731961 5-1362199 5.9963274 6.6873822 7.5610713 8.9004071 36 25 4.9802541 5:1446831 6.0006381 6.7002699 7.5775916 8.7223361 35 26 4 9873323 5-4531731 6-0170250 6.7132079 7.5941849 8.7443766 31 27 4.9914311 5.4616901 6.0271282 6-7261965 7.6108516 8-7665295 33 28 5.0015505 5.4702342 6.0378680 $7392360 7.6275923 8-7887957 32 29 5.0086907 5.4788056 6·0183110 *7723268 7.6441075 8-8111761 31 30 5.0158517 5.4874043 6.0588580 354691 76612976 8.8336715 30 ! 31 '5·0230337 54960305 6.0691085 32 5.0302367 5-5046843 6.0799964 6733632 67919095 7-6782631 8.8562828 29 7.6953017 88790109 28 33 5:0374607 5-5133659 6·0906219 6.8052082 7.7124227 8-9018567 27 34 5.0447060 5.5220754 6·1012850 6.8185597 7.7296176 8.9248211 26 35 5-0519726 5.5308129 6.1119861 6.8319612 7-7468901 8.9179051 25 36 5-0592606 5.5395786 6.1227253 6.8451222 7-7642406 8.9711095 24 37 5.0665701 5.5183726 6.1335028 6-8589338 7-7816697 8.9944354 23 38 5.0739012 5.5571951 6.1443189 6.8721995 7.7991778 9-0178837 22 39 5.0812539 $5.5660160 6.1551736 6.8861195 7.8167656 9.0411553 21 40 5'0886284 5-5749258 6.1660674 6.8997912 7-8341335 9.0651512 20 41 5-0960248 5.5838343 6.1770003 6.9135239 7-8521821 9-0889725 19 42 5.1034431 5.5927719 6.1879725 6.9273089 7.8700120 9.1129200 18 43 5:1108835 5.6017386 6.1989843 6.9111496 7.8879238 9.1369949 17 44 5.1183461 5.6107345 6.2100359 6.9550161 7.9059179 9.1611980 16 45 5-1258309 5.6197599 6-2211275 6*9689991 7.9239950 9.1855305 15 46 5.1333381 5.6288148 6.2322594 6.9830092 47 5.1408677 5-6378995 6.2431316 6.9970760 7.9421556 9-2099934 14 7.9601003 9.2345877 13 48 5.1484199 5.6470140 6.2546446 7-0112001 7.9787298 9-2593115 12 49 5.1559948 5.6561584 6.2658984 7-0253820 7-9971445 9.2811749 11 50 5.1635924 5-6653331 6.2771933 7.0396220 8.0156150 9.3091699 10 51 5-1712128 5-6745380 6.2885295 7-0539205 8.0312321 9.3343006 9 52 5.1788563 53 5.1865228 51 5.1912125 55 5.6837734 6.2999073 5.6930393 6-3113269 5.7023360 6.3227884 7.0971700 5.2019254 E·7116636 6.3342923 7·1117059 7-0682777 8.0529062 9-3595682 8 7.0826941 · 56 5.2096018 57 5.2174216 5.7230223 6.3458386 5.7304121 6.3574276 7.1263019 7.1409587 8.0716681 9-3849738 8·0905182 9:4105184 8.1094573 9:4362033 8.1284860 8.1476048 9-4879981 7 6 5 9.4620296 4 3 58 5.2252050 5.7398333 6.3690595 71556764 8.1668145 9.5141110 2 69 5.2330121 5.7492861 6.3807317 7.1704556 60 5-2108131 5.7587705 6.3021532 7.1852965 8.1861157 9.5103686 8.2053090 9.5667722 11° 10° 9° 8° 7° 6° COSECANTS. 307 TABLE XI.-SECANTS AND COSECANTS. SECANTS. 84° 85° 86° 87° 88° 89° 0123GD 9.5667722 11.473713 14.335587 19.107323 28-653708 57.298688 60 9*5933233 11.511990 14.395471 19-213970 28.891398 58.269755 59 9-6200229 11.550523 14.455859 19.321816 29.139169 59.274308 59 9-6168721 11.589316 14.516757 19.430882 29.388 24 60-314110 57 4 9.6738730 11.628372 14.578172 19-541187 29-641373 61.391050 56 9.7010260 11.667693 14.610109 19-652754 29.899026 62.507153 55 10 11 12 13 14 15 16 17 18 20 21 OPAGO ERRER DE°2* ***** ****~ ~***3 8588&= 9-7283327 11.707282 14-702576 19.765601 30.161201 63.664595 54 9-7557941 11-747141 14.765580 19.879758 30.428017 61.865716 53 B 9.7834124 11.787274 14.829128 19.993241 30.699598 66.113036 52 9.8111880 11.827683 14-893226 20-112075 30.976074 67-409272 51 9.8391227 11.868370 14.957882 20-230281 31-257577 68.757360 50 9.8672176 11.909340 15.023103 20.319893 31.544246 70·160174 49 9.8954744 11.950595 15.088896 20-470926 31-836225 71.622052 49 9-9238943 11.992137 15.155270 20-593409 32.133663 73.145827 47 9-9524787 12.033970 15.222231 20-717368 32.436713 74-735856 46 9-9812291 12.076098 15-289788 20-842830 32-715537 76-396554 45 10-010147 12-118522 15.357919 20-969821 33-060300 78.132742 44 10.039234 12-161246 15.426721 21.098376 33.381176 79.949684 43 10-068491 12.204274 15.496114 21.228515 33.708345 81.853150 42 19 10-097920 12-247608 15-566135 21.360272 31-041994 83-849470 41 10.127522 12.291252 15.636793 21.493676 34.382316 85.915609 40 10-157300 12.335210 15.708096 21.628759 31.729515 88-149244 39 22 10-187254 12.379184 15.780054 21-765553 35.083800 90.468863 23 10.217386 12-424078 15.852676 21.904090 35 145391 92.913869 21 10-247697 12.468995 .15.925971 22.011103 35.814517 95-494711 36 25 10.278190 12-511240 15.999948 22.186528 36-191414 98-223033 35 26 10.308866 12.559815 16.074617 22.330499 36-576332 101-11185 34 27 10.339726 12.605721 16 149987 22.476353 36.969528 104.17574 28 10.370772 12.651971 16.226069 22.624126 37.371273 107.43114 29 10-402007 12.698560 16.302873 22-773857 37.781849 110.89656 30 10.433431 12.715195 16.380408 22-925586 38.201550 114.59301 31 10.465046 12-792779 16.458686 23.079351 38.630683 118.54440 32 10-196851 12.840416 16.537717 23.235196 39-069571 122.77803 33 10-528857 12-888410 16.617512 23-393161 39-518549 127.32526 31 10-561057 12-936765 16.698082 23.553291 39.977969 132.22229 35 10.593155 12-985486 16.779139 23.715630 40-448201 137-51108 36 10.626054 13:031576 16.861594 23.880224 40-929630 143-24061 37 10.658854 13.081010 16-944559 24.047121 41.122660 149-46837 38 10-691859 13.133882 17.028346 21.216370 41-927717 39 10-725070 13-184106 17.112966 24-388020 42 445245 156.26228 163.70325 40 10.758488 13-234717 17.198434 24.562123 42.975713 171.88831 20 88783 *~*~* **N** **272 33 32 31 30 29 28 27 26 25 24 23 21 20 41 10.792117 13.285719 17.284761 24.738731 43.519612 180 93496 19 42 10-825957 13-337116 17.371960 21-917900 44 077458 190.98680 18 43 10.860011 13.388914 17-460016 25.099685 44.619795 202.22122 17 44 10.894281 13.441118 17.519030 25.284144 45.237195 214-85995 16 45 10 928768 13-493731 17-638928 25-471337 45.840260 229-18385 15 46 47 48 49 60 63 ANGAS O*8*3 85888 - 10.963476 13.546758 17.729753 25.661324 46.459625 245.55402 14 10.998406 13.600205 17-821520 25.851169 47.095961 264-41269 13 11.033560 13.651077 17.914243 20-049937 47-749974 286-47918 12 11.068910 13.708379 18.007937 26.248691 48-422111 312-52297 11 11-101549 13-763115 18.102619 26.450510 49.114062 313-77516 10 11.140389 13-818291 18-198303 26-655455 49-825762 52 11.176462 13.873913 18.295005 26.863603 50-558396 381-97230 429-71873 11-212770 13-929985 18.392742 27.075030 51.312902 491 10702 51 11-249316 13.986514 18-491530 27-289814 52.090272 572-95809 55 11-286101 14-043504 18.591387 27-508035 52-891564 687.54960 8768 ∞ 9 £6 11.323129 14.100963 18.692330 27.729777 53.717896 859-43689 4 57 11.360402 14.158894 18-794377 27.955125 54.570461 1145 9157 3 11.397922 14.217301 18.897515 28.184168 55-450534 1718-8735 2 59 11·435693 14.276200 19.001854 28-416997 56.359462 3437-7468 1 60 11-473713 14-335587 19.107323 28.653708 57.298688 Infinite. 5° 4° 3° 2º 1° 0° COSECANTS. 308 TABLE XII.—TANGENTS AND COTANGENTS. 0° 1° 2° 3° Tang 10TH Q 0.00000 1.00029 2.00058 3437.75 1718.87 3 .00087 1145.92 Cotang Tang Cotang Infinite. .01746 57.2900 .01775 56.3506 .01804 55.4415 .01833 54.5613 Tang Cotang || Tang .03492 28.6263 .03521 28.3994 .03550 28.1664 Tang Cotang .03241 19.0811 60 .05270 18.9755 59 .05299 18.8711 58 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.665656 .00145 €87.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 .00204 491.106 .01949 51.3032 .03696 27.0566 .05115 18.3655 53 8.00233 9.00262 429.718 .01978 50.5485 .03725 26.8450 .05174 18.2677 52 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.075050 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.980249 12 .00349 286.478 13 .00378 264.441 .02095 .02124 47.0353 47.7395 .03842 26.0307 .05591 17.8803 48 .03871 25.8348 .05620 17.793447 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05049 17.7015 46 AAAA CO0 51 15 .00436 16.00465 17.00495 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 214.838 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 19 .00553 180.932 20 .00582 171.885 21 .00611 163.700 02269 44.0661 .02298 .02328 42.9641 .04016 21.8978 .05766 17.3432 42 43.5081 .04016 24.7185 .05795 17.2558 41 .01075 21.5118 .05824 17.1693 40 02357 42.4335 .04104 24.3675 .05854 17.0837 39 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.999038 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.831936 25.00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 26 .00756 132.219 ,02502 39.9655 .04250 23.5321 .05999 16.6681 34 7890 723 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 29 .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .00116 16.3499 30 31 .00902 110.892 32 .00931 107.426 .02648 37.7680 .04395 .02677 37.3579 33 .00960 104.171 .02706 22.7519 .06145 16.272229 .04424 22.6020 .00175 16.195223 36.9560 .04454 22.4541 .06204 16.1190 31 .00989 35 .01018 36.01047 101.107 .02735 98.2179 36.5627 .04483 22.3081 .06233 16.0435 26 .02764 30.1776 .04512 22.1640 .06262 15.9687 25 95.4895 .02793 35.8006 .04511 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.821123 38 .01105 39 .01135 40 .01164 90.4633 .02851 35.0695 88.1436 .02881 34.7151 85.9398 .02910 .04599 21.7426 .06350 15.7483 22 .04628 21.6056 .06379 15.6762 21 31.3678 .04658 21.4704 .06408 15.6048 20 42 .01222 43 .01251 79.9-434 44 .01280 78.1263 41 .01193 83.8435 .02939 34.0273 81.8470 .02968 33.6935 .02997 .03026 33.0452 .04687 21.3369 .06437 15.5340 19 .04716 21.2049 .06467 15.4638 18 33.3662 .01745 21.0747 .06496 15.3943 17 .04774 20 9460 .06525 15.325416 45 .01309 76.8900 .03055 46 .C1338 74.7292 32.7303 .04803 20.8188 .06554 15.257115 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47.01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.122213 48 .01396. 49 .01425 53 .01542 54 .01571 71.6151 .03143 70.1533 .03172 31.5284 50 .01455 68.7501 .03201 31.2416 51 .01484 67.4019 .03230 52 .01513 66.1055 .03259 64.8580 .03288 63.6567 .03317 31.8205 .04891 20.4465 .06642 15.055712 .04920 .04949 20.3253 .06671 14.989811 20.2056 .06700 14.9244 10 30.9599 30.6833 30.4116 .04978 20.0872 .05007 | 19.9702 .05037 19.8546 .06730 14.8596 9 .06759 14.7954 30.1446 55.01600 62.4992 .03346 56 .01629 61.3829 .03376 57.01658 60.3058 .03405 58 .01687 59.2659 .03434 59 .01716 58.2612 57.2900 60 .01746 Cotang Tang Cotang Tang Cotang Tang .06993 14.3007 0 Cotang Tang .05066 19.7403 29.8323 .05095 19.6273 29.6245 .05124 19.5156 29.3711 .05153 19.4051 29.1220 .05182 19.2959 .03463 28.8771 .05212 19.1879 .03492 28.6363 .05241 19.0811 .06788 14.7817 .06817 .06847 .06876 14.5438 14.6685 14.6059 .06905 14.4823 .06934 3 14.4212 .06963 14.3607 OHWWA UTO -700∞ 8 4 1 89° 88° 87° 86° 309 TABLE XII.—TANGENTS AND COTANGENTS. 4° 5° 6° що Tang Cotang Tang Cotang Tang Cotang Tang Cotang 0 1 .06993 14.3007 .07022 2.07051 14.2411 14.1821 .08749 11.4301 .08778 11.3919 .08807 11.3540 3 .07080 14.1235 .08837 11.3163 9.43515 4 .07110 14.0655 .08866 11.2789 9.40904 .10510 9.51436 .12278 .10540 9.48781 .10569 9.46141 .10599 .10629 8.14-3560 .12308 8.12481 59 .12338 8.10536 58 .12367 8.08600 57 .12397 8.06674 56 5 .07139 14.0079 6 .07168 13.9507 7.07197 .08895 11.2417 .08925 13.8940 .08954 .10657 9.38307 12426 8.04756 55 11.2048 .10687 9.35724 .12456 8.02848 54 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.80599 .12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 7.93438 49 12 .07344 13.6174 .09101 10.9882 10863 9.20516 .12633 7.91582 48 • 13 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12662 7.8973447 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12692 7.8789546 15 .07431 13.4566 .09189 10.8829 16 .07461 13.4039 .09218 • 17.07490 13.3515 18 .07519 13.2996 19 .07548 13.2480 20.07578 13.1969 .10952 .10981 10.8483 09247 10.8139 .11011 9.08211 10.7797 .09277 .09306 10.7457 .09335 10.7119 9.13093 .12722 7.86064 45 9.10646 .12751 7.84242 44 .12781 7.82428 43 .11040 9.05789 .12810 7.80622 42 .11070 9.03379 .12840 7.78825 41 .11099 9.00983 .12869 7.77035 40 21.07607 23.07665 24.07695 25 .07724 12.9469 13.1461 .09365 10.6783 .09394 22.07636 13.0958 10.6450 .09423 13.0458 10.6118 12.9962 .09453 10.5789 .09482 .11128 8.98598 .12899 7.75254 39 .11158 8.96227 12929 7.73480 38 .11187 8.93807 .12958 7.71715 37 .11217 8.91520 .12988 7.69957 36 10.5462 .11246 8.89185 .13017 7.68208 35 26.07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.66466 34 • 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 7.64732 83 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.6300532 • 29.07841 12.7536 .09600 10.4172 .1136-1 8.79964 .13136 7.6128731 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.5957530 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 29 32 .07929 12.6124 .09688 10.3224 .11452 8.73172 13224 7.5617628 33 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 26 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.4946524 37.08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 28.08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 39.08134 12.2946 09893 10.1080 .11 ) 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 42 .08221 12.1632 10.0183 .09981 10.0187 .11747 .11718 8.53402 .13491 7.41240 19 8.51259 .13521 7.39616 18 * 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.3799917 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 7.3478615 46.08339 11.9923 .10099 9.90211 .11865 8.42795 .12639 7.33190 14 47.08368 11.9504 .10128 9.87338 .11895 8.40703 .13669 7.31600 13 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 7.2844211 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 53 54 .08573 .08544 11.7045 11.6645 .10305 9.70441 .12072 8.28376 55 .08602 11.6248 56.08632 11.5853 57.08661 11.5461 58.08690 .10334 9.67680 .10363 9.64935 .10393 9.62205 .10422 9.59490 .10452 11.5072 .12101 .12131 8.24345 .12160 8.26355 8 7.23754 .13817 7 7.22204 .13846 7.20661 6 .13876 .13906 7.19125 8.22344 .13935 7.17594 .12190 | 8.20352 .13965 7.16071 9.56791 .12219 8.18370 .13995 7.14553 59.08720 60.08749 .10481 11.4685 11.4301 .10510 9.54106 9.51436 Cotang Tang Cotang Tang 12249 12278 8.14435 Cotang Tang .14024 8.16398 7.13042 .14054 Cotang 7.11537 0 Tang 85° 84° 83° 82° 310 TABLE XII.—TANGENTS AND COTANGENTS. · 8° 9° 10° 11° Tang 0.14054 1.14084 2 .14113 Cotang Tang Cotang 7.11537 .15838 6.31375 7.10038 .15868 6.30189 7.08516 Tang Cotang Tang Cotang .15898 6.29007 .17633 5.67128 .17663 5.66165 .17693 5.65205 .19438 5.14455 60 .19468 5.13658 59 .19498 5.12862 58 3 .14143 4.14173 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 56 5.14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 6.14232 7.14262 8 .14291 9.14321 10.14351 7.02637 .16017 6.24321 6.91174 .16047 6.99718 .16077 6.98268 6.96823 .17813 5.61397 .19619 5.09704 51 6.23160 .17843 5.60452 .19649 5.08921 53 6.22003 .17873 5.59511 .19680 5.0813952 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 .16137 6.19703 .17933 5.57638 .19740 5.0658450 11.14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 12.14410 6.93952 .16196 6.17419 .17993 5.55777 13 .14440 6.92525 .16226 6.16283 .18023 5.51851 .19801 5.05037 48 .19831 5.04207 47 14 .14470 15 .14199 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.0273445 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17.14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .199825.00451 22 FRX) 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 20012 4.9969541 20.14648 .14648 6.82694 .16435 6.08144 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 23.14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 36 995993379 8858 42 38 25 .14796 6.75838 .16585 6.02962 .18384 5.43966 .20191 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.00797 .18414 5.42192 .20254 4.93721 33 28 .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .11915 6.70450 30 .14945 6.69116 .16704 5.98646 .16734 5.97576 .18504 5.40429 .20315 4.92249 31 .18531 5.39552 .203-15 4.91516 30 31 .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 .15061 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 35.15094 36.15124 37.15153 6.62523 .16884 5.92283 6.61219 .16914 5.91236 6.59921 .16944 5.90191 .18684 5.35206 .18714 5.34345 .20497 4.87882 25 .20527 4.87162 .18745 5.33187 .20557 4.8644423 40 42.15302 43.15332 44 .15362 45.15391 46.15421 47.15451 48.15481 49 .15511 50 .15540 38 .15183 6.58627 .16974 5.89151 39 .15213 6.57339 .17004 5.88114 .15243 6.56055 41 .15272 6.54777 .17063 5.86051 6.53503 .17093 5.85024 6.52234 .17123 5.84001 6.50970 .17153 5.82982 6.49710 .17183 5.81966 6.48456 .17213 5.80953 6.47206 .17243 5.79944 6.45961 .17273 5.78938 6.44720 .17303 5.77936 6.43484 .18775 5.32631 .20588 4.85727 22 .18805 5.31778 .20618 4.85013 21 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 .18865 5.30080 .20679 4.83590 19 .18895 5.29235 .20709 4.82882 18 .18925 .18955 5.27553 5.28393 .20739 4.8217517 .20770 4.81471 16 .18986 5.26715 .19016 .20800 4.80769 15 5.25880 .20830 4.80068 14 .19046 5.25048 .20861 4.79370 13 .19076 5.24218 .20891 4.7867312 .19106 5.23391 .20921 4.77978 11 .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 53.15630 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 · 54 .15660 6.38587 .17453 5.72974 55 15689 6.37374 .17483 5.71992 56 .15719 6.36165 .17513 5.71013 57.15749 6.34961 .17543 5.70037 58 .15779 6.33761 .17573 5.69064 59.15809 6.32566 .17603 5.68094 60 .15838 6.31375 .17633 5.67128 Cotang Tang Cotang Tang .19257 5.19293 .21073 4.74534 6 .19287 5.18480 .21104 4.73851 5 .19317 5.17671 .19347 5.16863 .21134 4.73170 4 .21164 4.72490 3 .19378 5.16058 .19408 5.15256 .19438 5.14455 Cotang Tang .21195 4.71813 2 .21225 4.71137 .21256 4.70463 0 Cotang Tang 81° 80° 79° 78° 311 TABLE XII-TANGENTS AND COTANGENTS. 12° 13° 14° 15° Tang 0.21256 1 .21286 4.69791 2.21316 4.69121 Cotang Tang Cotang 4.70463 .23087 4.33148 .23117 .23148 4.32001 Tang Cotang Tang Cotang 4.32573 24933 4.01078 26795 .24964 4.00582 .24995 4.00086 3.73205 60 .26826 3.72771 59 .26857 3.72338 58 3.21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 8.71907 57 4 .21377 4.67786 .23209 4.30860 25056 3.99099 .26920 3.71476 56 5 .21408 4.67121 23240 4.30291 25087 3.98607 .26951 3.71046 55 6.21438 4.66458 23271 4.20724 .25118 3.98117 .26982 3.70616 54 7.21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8.21499 4.65138 23332 4.28595 .25180 3.97139 .27011 3.6976152 9 .21529 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.6933551 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 3.6806148 13 .21651 4.61868 .23-485 4.25795 25335 3.94713 27201 3.67038 47 14 .21682 4.61219 23516 4.25239 .25366 3.94232 .27232 3.6721746 15 .21712 .23547 4.60572 4.24685 .25397 3.93751 .27263 3.66796 45 16.21743 4.59927 23578 4.24132 .25428 3.93271 .27294 3.66376 41 17 .21773 4.59283 .92608 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 .23070 4.22481 .25521 3.91839 .27388 3.65121 41 20.21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21.21895 4.56726 .23731 4.21387 25583 3.90890 .27451 3.64289 39 22.21925 4.56091 .23762 4.20342 25614 3.90417 .27482 3.63874 38 23 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27513 3.63461 37 24 .21986 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 36 .22017 .23854 4.54196 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 28.22108 4.52316 .23946 4.17600 .25800 29 .22139 4.51693 .23977 4.17064 .25831 30 .22169 4.51071 .24008 4.16530 .25862 .27638 3.88068 3.87601 .27670 | 3.61405 .27701 3.87136 .27732 3.86671 3.61814 33 32 3.6099631 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 27764 3.6018129 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 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 3.81824 .27858 3.58966 26 35 22322 4.47986 .21162 4.13877 .26017 3.84364 .27889 3.58562 25 36.22353 4.47374 . 24193 4.13350 .26048 3.839061 .27921 3.5816024 37 .22383 4.46764 39 38 .22414 .22444 40.22475 4.46155 41 .22505 12.22536 13.22567 14 .22597 45 .22628 46 22658 4.41340 47.22689 4.40745 48.22719 4.40152 .24223 4.12825 .2425-4 4.45548 .24285 4.44942 .24316 .24347 4.10736 4.44338 4.43735 .24377 4.10216 .24-108 4.43134 .24439 4.42534 4.09182 .24470 4.41936 4.08666 .24501 4.08152 .24532 4.07639 .24562 4.07127 26079 3.83449 27952 3.57758 23 4.12301 4.11778 4.11256 .26110 3.82992 .27983 3.57357 22 .26141 3.82537 .28015 3.5695721 .20172 3.82083 .28016 3.56557 20 .26203 3.81630 .28077 3.56159 19 .26235 3.81177 .28109 3.5576118 4.09699 26266 3.80726 .28110 3.55364 17 49 .22750 4.39560 .24593 4.06616 26297 .26328 3.79827 .20359 3.79378 .26390 3.78931 .26421 .20152 3.80276 .28172 3.54968 16 .28203 3.54573 15 .28234 3.54179 14 .28266 3.5378513 .28297 3.78485 3.53393 12 3.78040 .28329 3.53001 11 50 .22781 4.38969 .246244.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 28391 3.52219 9 52 22842 4.37793 53 .22872 4.37207 54 .22903 4.36623 .24686 4.05002 .26546 3.76709 28423 3.51829 8 24717 4.04586 .26577 3.76268 .28151 3.51441 .24747 4.04081 .26608 3.75828 .28486 3.51053 6 55 .22934 56 .22964 4.36040 4.35459 .24778 4.03578 .26639 57 22995 4.31879 188859 58.23026 .23056 GO .23087 Cotang .21809 4.03076 .24840 4.02574 4.34300 .24871 4.02074 .24902 4.33723 4.01576 .21933 4.01078 4.33148 Tang Cotang Tang 3.75388 .26670 .28549 3.50279 3.74950 3.74512 .28580 .26701 3.49894 .26733 3.74075 .28612 3.49509 .28043 .26764 3.49125 3.73640 .28675 .26795 3.73205 3.48741 Cotang Tang Cotang Tang .28517 3.50666 5 3 0 1170 76° 75° 74° ゾ ​312 TABLE XII.—TANGENTS AND COTANGENTS. 19° 16° 170 18° Tang 0.28675 Cotang Tang Cotang Tang Cotang Tang Cotang 3.48741 .30573 3.27085 1 .28706 3.48359 .30605 3.20745 2 28738 3.47977 .30637 3.26406 3 .28769 3.47596 .30669 .28800 3.47216 .28832 3.46837 6.28864 3.46458 .30700 3.26067 3.25729 .30732 3.25392 .32492 3.07768 32524 3.07464 .32556 3.07160 32588 3.06857 .34433 2.90421 60 .34465 2.90147 59 .34498 2.89873 58 .34530 2.89600 57 • 32621 3.06554 .34563 2.89327 56 .32653 3.06252 .34596 2.89055 55 .30764 3.25055 .82685 3.05950 .34628 2.88783 54 7.28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.88511 53 8.28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.8824052 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 2.8743049 12.29053 13 .29084 14 .29116 15.29147 3.43084 3.44202 .30955 3.23048 3.43829 .30987 3.22715 3.43456 .31019 3.22384 .31051 3.22053 32878 3.04152 .34824 2.87161 48 .32911 3.03854 .34856 2.86892 47 .32943 3.03556 .34889 2.8662446 .32975 3.03260 .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.8608944 17 .29210 3.42343 .31115 3.21392 .33010 3.02667 .34987 2.85822 43 19 29242 19.29274 20.29305 3.41973 .31147 3.21063 .33072 3.02372 35020 2.85555 42 3.41604 .31178 3.20734 3.41236 .31210 3.20406 .33104 3.02077 .35052 2.8528941 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22.29368 23 .29400 24 25 27 .29526 28 .29558 29 .29590 3.40502 .31274 3.19752 3.40136 .31306 3.19426 .29432 3.39771 .31338 3.19100 .29463 3.39406 .31370 3.18775 23 .29495 3.39042 3.38679 3.38317 .31466 3.37955 .33201 3.01196 .35150 2.84494 38 .33233 3.00903 .35183 2.84229 37 • #3379 885 .33266 3.00611 .35216 2.83965 36 .33298 3.00319 .35248 2.83702 35 .31402 3.18451 .33330 .31434 3.18127 3.00028 .33363 2.99738 .35281 2.83439 341 .35314 30 .29621 3.37594 3.17804 .31498 3.17481 .31530 3.17159 .33395 2.99447 .35346 .33427 2.99158 .35379 2.83176 33 2.8291432 2.8265331 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 32 .29685 3.36875 33 .29716 3.36516 34 .29748 3.36158 35 .29780 3.35800 .31594 .31626 .31562 3.16838 3.16517 3.16197 .33492 2.98580 .35445 2.82130 29 .33524 2.98292 .35477 2.81870 28 .33557 2.98004 .35510 2.81610 27 .31658 3.15877 .33589 2.97717 .35543 2.81350 26 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29811 37.29843 38 29875 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 23 3.34732 .31786 3.14605 .33718 2.96573 .35671 2.80316 22 39 29906 3.34377 40 .29938 3.34023 41.29970 .31818 3.14288 .31850 3.13972 3.33670 .31882 42 .30001 3.33317 .31914 .33751 2.96288 .33783 2.96004 .35707 2.80059 21 .35740 2.79802 20 1859 8ANAKANAZA 3.13656 .33816 2.95721 .35772 2.79545 19 3.13341 .33848 2.95437 .35805 2.7928918 43 .30033 44 .50065 45 .30097 46.30128 47.30160 48 .30192 49.30224 50 .30255 3.32965 .31946 3.32614 .31978 3.12713 3.32264 .32010 3.12400 3.31914 .32042 3.12087 3.31565 .32074 3.11775 3.31216 .32106 3.11464 3.30868 .32139 3.11153 3.30521 .32171 3.10842 3.13027 .33881 2.95155 .35838 2.79033 17 .33913 2.91872 35871 2.78778 16 .33945 2.94591 .33978 .34010 2.94028 .35904 2.78523 15 2.94309 35937 2.78269 14 .35969 2.7801413 .34043 2.93748 .36002 2.7776112 51.30287 53.30351 3.29483 3.30174 .32203 3.10532 .34140 52 .30319 3.29829 .32235 .32267 54 .30382 3.29139 .32299 55 .30414 3.28795 .32331 56 .30446 3.28452 .32363 57.30478 3.28109 .32396 58 .30509 3.27767 .32428 59.30541 3.27426 60 .30573 3.27085 Cotang Tang 73° 3.09298 .34075 2.93468 .34108 2.93189 2.92910 3.10223 .34173 2.92632 3.09914 .34205 2.92354 3.09606 .34238 2.92076 .34270 2.91799 3.08991 .34303 2.91523 3.08685 .34335 2.91246 3.08379 .34368 2.90971 .32460 3.08073 .34400 2.90696 32492 3.07768 .34433 2.90421 Cotang Tang Cotang Tang .36035 2.7750711 .36068 2.7725410 .36101 2.77002 9 .36134 2.76750 8 .36167 2.76498 .36199 2.76247 6 .36232 2.75996 5 .36265 2.75746 4 .36298 2.75496 3 72° 71° .36331 2.75246 2 .36364 2.74997 1 .36397 2.74748 0 Cotang Tang 70° HO 313 TABLE XII.—TANGENTS AND COTANGENTS. 20° 21° 22° 23° Tang 0 .36397 Cotang Tang Cotang 2.74748 .38386 2.60509 Tang Cotang Tang Cotang .40403 2.47509 .42447 2.35585 60 1 .33430 2.36463 2.74199 2.74251 .38420 2.60283 .38453 2.60057 3.36496 2.74004 .38487 2.59831 .40436 2.47302 .40470 2.47095 .40504 2.46888 42482 · 2.35395 59 .42516 2.3520558 .42551 2.3501557 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 53 8.36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 52 9 .36694 2.72526 .38687 2.58184 .40707 2.45655 .42757 2.33881 51 10 .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 50 12 11 .36760 .36760 .36793 2.71792 13 .36826 2.71548 14 .36859 2.71305 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.3350549 .38787 2.57815 .40809 2.45043 .42860 2.33317 48 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 .38854 2.57371 .40877 2.44636 42929 2.32943 46 15 .36892 16 .36925 17.36958 18 .36991 19.37024 20.37057 22 21 .37090 .37123 23 37157 2.69131 2.71062 2.70819 .38921 2.56928 2.70577 .38955 2.56707 2.70335 38988 2.56487 2.70094 .39022 2.56266 2.69853 .39055 2.56046 2.69612 .39089 2.55827 2.69371 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 .40945 2.44230 .42998 2.32570 44 .40979 2.44027 .43032 2.32383 43 .41013 2.43825 .43067 2.32197 42 .41047 2.43623 .41081 .43101 2.32012 41 2.43422 .43136 2.31826 140 .41115 2.43220 43170 2.31641 39 .39122 2.55608 .41149 2.43019 .43205 2.3145638 .39156 2.55389 .41183 2.42819 .43239 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 30 25 37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.3090235 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 28 .37322 29 .37355 30.37388 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 2 67937 .39324 2.54299 .41353 2.41819 .43412 2.3035132 2.67700 31.37422 32 .37455 33 .37488 34 .37521 35.37554 36 .37588 2.66046 .39458 2.53432 2.66752 .39492 2.53217 2.66516 .39526 2.53001 2.66281 .39559 2.52786 .39593 2.52571 .39357 2.54082 2.67462 .39391 2.53865 2.67225 .39425 2.53648 2.66989 .41387 2.41620 .43447 2.30167 31 .41421 2.41421 .43481 2.29984 30 .41455 2.41223 .43516 2.29801 20 .41490 2.41025 .43550 2.29619 28 .41524 2.40827 .43585 2.29437 27 .41558 2.40629 .43620 2.29254 26 .41592 2.40432 43654 2.2907325 • .41626 2.40235 .43689 2.28891 24 37.37621 2.65811 .39626 2.52357 .41660 2.40038 .43724 2.28710 23 2.05576 41 .37754 42.377787 43.37820 44.37853 38 .37654 39 .37687 2.65342 .39694 2.51929 40.37720 2.65109 .39727 2.51715 2.64875 .39761 2.51502 2.64642 .39795 2.51289 2.64410 .39829 2.51076 2.64177 .39862 2.50864 .39660 2.52142 .41694 2.39841 .43758 2.28528 22 .41728 2.39645 .43793 2.28348 21 .41763 2.39449 .41797 .41831 .43828 2.28167 20 2.39253 2.39058 .43862 2.27987 19 .43897 2.27806 18 .41865 2.38863 .43932 2.2762617 41899 2.38668 .43966 2.27447 16 45 .37887 2.63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 15 46.37920 2.03714 39930 2.50440 .41968 2.38279 .44036 2.27088 14 4737953 2.63483 .39963 2.50229 .42002 2.38081 .44071 2.26909 13 48 .37986 2.63252 .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 .38020 50 .38053 2.63021 .40031 2.49807 2.62791 .40065 2.49597 .42070 2.37697 .44140 2.26552 11 .42105 2.37504 .44175 | 2.26374 |10 51 52.38120 53 .38153 54.38186 55.38220 2.61646 .38086 2.62561 .40098 2.49386 2.62332 .40132 2.49177 2.62103 2.48967 .40166 2.61874 .40200 2.48758 .40234 2.48549 .42139 2.37311 .44210 2.26196 9 .42173 2.37118 .42207 2.36925 .44244 .42242 2.36733 .42276 2.36541 8 2.26018 .44279 17 2.25840 .44314 2.25663 6 .44349 2.25486 5 56 .38253 2.61418 .40267 2 48340 .42310 2.36349 .44384 2.25309 4 57.38286 2.61190 40301 2.48132 .42345 2.36158 .44418 2.25132 3 • 58 .38320 2.60963 59 .38353 2.60736 .40335 2.47924 .40369 2.47716 60 .38386 2.60509 .40403 2.47509 Cotang Tang Cotang Tang .42379 2.35967 .42413 2.35776 42447 2.35585 Cotang Tang .44453 2.24956 2 .44488 2.24780 .44523 | 2.24604 Cotang Tang HO 1 0 69° 68° 67° 66° 314 TABLE XII.-TANGENTS AND COTANGENTS. 27° 24° 25° 26° Tang 0.44523 Cotang Tang Cotang 2.24604 .46631 2.14451 .44558 2.24428 .46666 2.14288 2 .44593 2.24252 .46702 2.14125 3.44627 2.24077 .46737 2.13963 Tang Cotang .48773 2.05030 .48809 2.04879 .48845 Tang Cotang .50953 1.96261 60 .50989 1.96120 59 .48881 4.44662 2.23902 .46772 2.13801 .48917 2.04728 2.04577 .51063 2.04426 .51099 .51026 1.95979 58 1.95838 57 1.9569856 5.44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.95557 55 6 .44732 2.23553 .46843 2.13477 .48989 2.04125 .51173 1.95417 54 7.44767 2.23378 8.44802 2.23204 9.44837 2.23030 10 .44872 2.22857 .46879 2.13316 .46914 .46950 2.12993 .46985 2.12832 .49026 2.03975 .51209 1.95277 53 2.13154 .49062 2.03825 .51246 1.95137 52 .49098 2.03675 .51283 1.94997 51 .49134 2.03526 .51319 1.94858 50 11 .44907 2.22683 12 .44942 2.22510 13.44977 2.22337 .47021 2.12671 .47056 2.12511 .47092 2.12350 .49170 2.03376 51356 1.94718 49 .49206 2.03227 .51393 1.94579 48 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15.45047 16.45082 17.45117 18 .45152 2.21992 .47163 2.12030 2.21819 .47199 2.11871 2.91647 .47234 2.11711 2.21475 .47270 2.11552 .49315 2.02780 .51503 1.94162 45 .49351 2.02631 .51540 1.94023 44 .49387 2.02483 .51577 1.93885 43 .49423 2.02335 .51614 1.93746 42 19 .45187 2.21304 20.45222 21.45257 2.21132 .47305 2.11392 .47341 2.11233 .49459 2.02187 .51651 1.93608 41 .49495 2.02039 .51688 1.93470 40 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 1.93195 38 23.45327 2.20619 .47418 2.10758 .49604 2.01596 .51798 1.9305737 24.45362 2.20419 25.45397 26.45432 .45467 28 .45502 29.45538 30.45573 31 .45608 32.45643 33 .45678 34.45713 35.45748 .47483 2.10600 2.20278 .47519 2.10442 2.20108 .47555 2.10284 2.19938 2.19769 .47626 2.09969 2.19599 .47662 2.09811 2.19430 .47698 2.09654 2.19261 .47733 2.09498 2.19092 .47769 2.09341 2.18923 .47805 2.09184 2.18755 .47840 2.09028 2.18587 .47876 2.08872 36.45784 2.18419 .47912 2.08716 .49640 2.01449 .51835 1.9292036 .49677 2.01302 .51872 1.92782 35 .49713 2.01155 .51909 1.92645 34 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 .49786 2.00862 .51983 1.92371 32 .49822 2.00715 .52020 1.92235 31 .49858 2.00569 .52057 1.92098 30 .49894 2.00423 .52094 1.91962 29 .49931 2.00277 .52131 1.91826 28 .49967 2.00131 .52168 1.91690 27 .50004 1.99986 .52205 1.91554 26 .50040 1.99841 .52242 1.91418 25 .50076 1.99695 .52279 1.9128224 37.45819 38.45854 39 .45889 40 .45924 41.45960 42.45995 43.46030 44 .46065 2.18251 .47948 2.08560 2.18084 .47984 2.08405 2.17916 .48019 2.08250 2.17749 .48055 2.08094 2.17582 .48091 2.07939 2.17416 .48127 2.07785 2.17249 .48163 2.07630 2.17083 .50113 1.99550 .52316 1.91147 23 .50149 .50185 1.99261 1.99406 .52353 1.91012 22 .52390 1.90876 21 .50222 1.99116 52427 1.90741 20 .50258 1.98972 .52464 1.90607 19 .50295 1.98828 52501 1.90472 18 • .50331 1.98684 .52538 1.9033717 .48198 2.07476 .50368 1.98540 .52575 1.9020316 45 .46101 2.16917 .48231 2.07321 .50404 1.98396 .52613 1.9006915 46.46136 2.16751 .48270 2.07167 .50441 1.98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 1.98110 .52687 1.89801 13 48 .46206 2.16420 .48342 49 .462-42 2.16255 .48378 50 .46277 2.16090 .48414 2.06860 .50514 1.97966 .52724 1.89667 12 2.06706 .50550 1.97823 .52761 1.89533 11 2.06553 .50587 1.97681 .52798 1.8940010 8881 51.46312 2.15925 .48450 2.06400 .50623 1.97538 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 53 .46383 2.15596 .48521 2.06094 .50696 1.97253 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 57 .46525 2.14940 .48665 2.05485 .50843 1.96685 58 .46560 2.14777 .48701 2.05333 .50879 1.96544 59 .46595 2.14614 .48737 .52836 1.89266 9 .52873 1.89133 .52910 1.89000 .52947 1.88867 .52985 1.88734 .53022 1.88602 4 .53059 1.88469 .53096 1.88337 2.05182 .50916 1.96402 .53134 1.88205 60 .16631 2.14451 .48773 2.05030 .50953 1.96261 Cotang 65° Tang Cotang Tang Cotang Tang 64° 63° Cotang Tang .53171 1.88073 62° 315 TABLE XII.—TANGENTS AND COTANGENTS. 28° 29° 30° 31° Targ 2 1.87941 1.87809 3 .53283 .53320 .53358 .55469 .55507 1.87677 .55545 1.87546 55583 1.87415 1.80281 1.80158 Cotang Tang Cotang Tang Cotang Tang Cotang 01 .53171 1.88073 .55431 1.80405 .57735 1.73205 .60086 1.66428 60 1 .53208 .53246 .57774 1.73089 .60126 1.66318 59 1.80034 1.79911 6 .53395 1.87283 .55621 1.79788 .55659 1.79665 .57813 1.72973 .57851 1.72857 .57890 1.72741 .60245 .57929 1.72625 .60284 .57968 1.72509 .60165 1.6620958 .60205 1.66099 57 1.6599056 1.6588155 .60324 1.65772 54 7 .53432 1.87152 .55697 1.79542 .58007 1.72393 .60364 1.65663 53 8.53470 1.87021 .55736 1.79419 .58046 1.72278 .60403 1.65554 52 9 .53507 1.86891 .55774 1.79296 .58085 1.72163 .60443 1.65445 51 10 .53545 1.86760 11.53582 1.86630 .55812 .55850 12.53620 1.86499 13.53657 1.86369 14 .53694 1.86239 15 .53732 1.86109 1.79174 1.79051 .55888 1.78929 .55926 .55964 .56003 1.78563 .58124 1.72047 .60483 1.65337 50 .58162 1.71932 .60522 1.05228 49 .58201 1.71817 .60562 1.6512048 1.78807 .58240 1.71702 .60602 1.65011 47 1.78685 .58279 1.71588 .60642 1.64903 46 .58318 1.71473 .60681 1.64795 45 16.53769 1.85979 17 .53807 1.85850 .56079 .56041 1.78441 1.78319 .58357 1.71358 .60721 1.64687 44 .58396 1.71244 .60761 1.64579 43 18 .53844 1.85720 .56117 1.78198 .58435 1.71129 .60801 1.64471 42 19 .53882 1.85591 .56156 1.78077 .58174 1.71015 .60841 1.64363 41 20.53920 1.85462 .56194 1.77955 .58513 1.70901 .60881 1.64256 40 21 .53957 22.53995 23.54032 26.54145 1.84689 1.85333 .56232 1.77834 1.85204 .56270 1.77713 1.85075 24.54070 1.84946 .56347 1.77471 25.54107 1.84818 .56385 1.77351 .56424 1.77230 .58552 1.70787 .60921 1.64148 39 58591 1.70673 .60960 1.6404138 .56309 1.77592 .58631 1.70560 .61000 1.63934 37 .58070 1.70446 .61040 1.6382636 .58709 1.70332 61080 1.63719 35 .58748 1.70219 .61120 1.6361234 27.54183 1.84561 .56462 1.77110 .58787 1.70106 .61160 1.6350533 28.51220 29 .54258 30.54296 35.54484 1.84433 .56501 1.76990 1.84305 .50539 1.76869 1.84177 .56577 1.76749 31.54333 1.84049 .56616 1.76629 32.54371 1.83922 .56654 1.76510 33.54409 1.83794 .56693 1.76390 34.54446 1.83667 .56731 1.76271 1.83540 .56769 1.76151 .58826 1.69992 .61200 1.6339832 .58865 1.69879 .58905 .612-10 1.63292 31 1.69766 .61280 1.63185 30 58944 1.69653 .61320 1.63079 29 .58983 1.69541 .59022 1.69428 .59061 1.69316 .61360 1.62972 28 .61400 1.62866 27 .61440 1.6276026 .59101 1.69203 .61480 1.62654 25 36.54522 37 .54560 38 .54597 39 .54635 40 .54073 1.83413 1.83286 .56816 1.75913 1.83159 .56885 1.83033 .56808 1.76032 .59149 1.69091 .61520 1.62548 24 .59179 1.68979 .61561 1.62442 23 1.82906 1.75794 .56923 1.75075 .56962 1.75556 .59218 1.68866 .61601 1.62336 22 .59258 1.68754 .61641 1.62230 21 .59297 1.68643 .61681 1.62125 20 41 .54711 42.54748 43.54786 44 .54824 45 .54862 46.54900 47 5-1938 48 .54975 1.81025 .57541 1.73788 1.82780 .57000 1.75437 1.82654 .57039 1.75319 1.82528 .57078 1.75200 1.82402 .57116 1.75082 1.82276 .57155 1.74964 1.82150 .57193 1.74846 .59533 1.67974 1.82025 .57232 1.74728 .59573 1.67863 1.81899 .57271 1.74610 .59612 1.67752 .62003 49 .55013 1.81774 .57309 1.74492 .59651 1.67641 .62043 50 .55051 1.81649 .57348 1.74375 .59691 1.67530 .62083 51.55089 1.81524 .57386 1.74257 .59730 1.67419 .62124 52 .55127 1.81399 .57425 1.74140 .59770 1.67309 .6216-4 1.60865 53.55165 1.81274 .57464 1.74022 .59809 1.67198 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 55 .55241 .59888 1.66978 56.55279 1.80901 .57580 1.73671 59928 1.66867 57.55317 1.80777 .57619 1.73555 58.55355 1.80653 .57657 1.73438 59.55393 1.80529 .57696 1.73321 60 .55431 1.80405 .57735 1.73205 Cotang Tang Cotang Tang .59336 1.68531 .61721 1.62019 19 .59376 1.68419 .61761 1.61914 18 .59415 1.68308 .61801 1.61808 17 .59454 1.68196 .61842 1.6170316 .59494 1.68085 .61882 1.61598 15 .61922 1.61493 14 .61962 1.6138813 1.61283 12 1.6117911 1.6107410 1.60970 .62245 62204 1.60761 1.60657 9876 .62285 1.60553 5 62325 1.60449 4 .59967 1.66757 .62366 1.60345 3 61° 60° 59° .60007 1.66647 .60046 1.66538 .60086 1.66428 Cotang Tang .62446 1.60137 1 .62487 1.60033 0 Cotang 58° .62406 1.60241 2 Tang 316 TABLE XII.-TANGENTS AND COTANGENTS. 33° 32° 34° 35° Tang Cotang Tang Cotang Tang Cotang Tang Cotang 0 62487 1.60033 .64941 1.53986 .67451 1.48256 70021 1.42815 60 1 .62527 1.59930 .64982 1.53888 .67493 1.48163 .70064 1.42726 59 2.62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3.62608 1.59723 .65065 1.53693 .67578 1.47977 .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7.62770 1.59311 .65231 1.53302 .67748 1.47607 70325 1.4219853 .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 .65255 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 14.03055 1.58593 .65521 1.52622 .68045 1.46962 .70629 15.63095 1.58490 .65563 1.52525 1.41584 46 .68088 1.46870 .70673 1.41497 45 16.63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1.4140944 17.63177 1.58286 .65616 1.52332 .68173 1.46686 .70760 1.11322 43 AAR FARHANNARA M38 18 19 .63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 63258 1.58083 .65729 1.52139 .68258 1.40503 .70848 1.4114841 20.63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.4106140 21 .63340 1.57879 .65813 1.51946 .68343 1.46320 .70935 22 1.40974 39 .63380 1.57778 .65854 1.51850 .68386 1.46229 .70979 1.40887 38 23 .63421 1.57676 .65896 1.51754 68429 1.46137 • .71023 1.40800 37 24 .63462 1.57575 .65938 1.51658 .68471 1.46046 .71066 1.40714 36 25 .63503 1.57474 .65980 1.51562 .68514 1.45955 .71110 26.63544 1.40627 35 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 201.63666 30 .63707 1.57170 .66105 1.51275 .68642 1.45682 .71242 1.40367 32 1.57069 .66147 1.51179 .68685 1.45592 .71285 1.40281 31 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31.63748 32 .63789 1.56868 .66230 1.50988 .68771 1.45410 .71373 1.40109 29 1.56767 .66272 1.50893 .68814 1.45320 .71417 33.63830 3-1.63871 35.63912 1.40022 28 1.56667 .66314 1.50797 .68857 1.45229 .71461 1.3993627 1.56566 1.56466 36.63953 1.56366 .66356 1.50702 .66398 1.50607 .66410 1.50512 .68900 1.45139 .71505 1.39850 26 .68942 1.45049 .71549 1.39764 25 .68985 1.44958 71593 37 .63994 1.56265 1.3967924 .66482 1.50417 .69028 1.44868 .71637 38 .64035 1.39593 23 1.56165 .66521 1.50322 .69071 1.44778 .71681 39 .64076 1.56065 1.39507 22 .66566 1.50228 .69114 1.44688 .71725 40 .64117 1.55966 .66608 1.50133 1.39421 21 .69157 1.44598 .71769 1.3933620 41 .61158 1.55866 .66650 1.50038 .69200 1.44508 .71813 42.64199 1.55766 1.39250 19 .66692 1.49944 .69243 1.44418 43.64240 .71857 1.3916518 1.55666 .66734 1.49819 .69.286 1.44329 44.64281 .71901 1.55567 .66776 1.39079 17 1.49755 .69329 1.44239 .71946 45.64322 1.55467 1.38994 16 .66818 1.49661 .69372 1.44149 46.64363 .71990 1.55368 1.38909 15 .66860 1.49566 .69416 1.44060 .72034 47 .64104 1.55269 1.38824 14 .66902 1.49472 .69459 1.43970 18 .61116 .72078 1.55170 1.38738 13 .66941 1.49378 .69502 1.43881 49 .64487 72122 1.55071 1.38653 12 .66986 1.49284 .69545 1.43792 .72167 50 .64528 1.54972 1.3856811 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 52 .64610 72255 1.54774 1.38399 9 .67113 1.49003 .69675 1.43525 53 .64652 72299 1.54675 1.38314 8 .67155 1.48909 .69718 1.43436 54 .64693 72344 1.54576 • 1.38229 7 .67197 1.48816 .69761 55 .64734 1.54478 1.43347 72388 • 1.38145 6 .67239 1.48722 .69804 1.43258 56 .64775 1.54379 .72432 1.38060 5 .67282 1.48629 .69847 1.43169 57 .64817 .72477 1.54281 1.37976 4 .67324 1.48536 .69891 1.43080 1888 58.64858 1.54183 .72521 1.37891 3 .67366 1.48442 .69934 1.42992 59 .72565 .64899 1.54085 1.37807 2 .67409 1.48349 .69977 1.42903 60 64941 .72610 1.53986 1.37722 .67451 Cotang Tang 1.48256 Cotang Tang Cotang Tang .70021 1.42815 .72654 1.37638 Cotang Tang 57° 56° 55° 54° 317► TABLE XII.—TANGENTS AND COTANGENTS. 39° 36° 37° 38° Tang 1 .72699 1.37554 Cotang Tang Cotang 0 .72654 1.37638 .75355 1.32704 .75401 1.32624 Tang Cotang Tang Cotang 2 .72743 1.37470 .72788 1.37386 .72832 1.37302 72877 1.37218 .72921 1.37134 7 .72966 1.37050 9 8.73010 .73055 10 .73100 .75447 1.32544 .75492 1.32464 .75538 1.32384 .75584 1.32304 .75629 1.32224 .75675 1.32144 1.36967 .75721 1.36883 .75767 1.36800 .78129 1.27994 .78175 1.27917 78222 1.27841 .78269 1.27764 .78316 1.27688 .78363 1.27611 .78110 1.27535 .80978 1.23490 60 .81027 1.23416 59 .81075 1.23343 58 .81123 1.2327057 .81171 1.23196 56 .81220 1.23123 55 .81268 1.23050 54 .78457 1.27458 .81316 1.22977 53 1.32064 .78504 1.27382 .81364 1.2290452 1.31984 .75812 1.31904 .78551 1.27306 .81413 1.22831 51 .78598 1.27230 .81461 1.2275850 11 73144 1.36716 .75858 1.31825 .78645 1.27153 .81510 1.2268549 12.73189 1.36633 .75901 1.31745 .78692 13 73234 1.36549 .75950 1.31666 .78739 1.27077 1.27001 .81558 1.22612 48 .81606 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 1.2239445 16 .73368 1.36300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 17 73413 1.36217 18.73457 1.36134 .76134 .76180 1.31269 1.31348 .78928 1.26698 .81800 1.22249 43 .78975 1.26622 .81849 1.2217642 19 .73502 1.36051 20 .73547 1.35968 21 .73592 22 73637 23.73681 .76226 1.31190 .76272 1.31110 1.35885 .76318 1.31031 1.35802 .76364 1.30952 .79022 1.26546 .81898 1.22104 41 .79070 1.26471 .81946 1.22031 40 .79117 1.26395 .81995 1.21959 39 .79164 1.26319 .82044 1.21886 38 1.35719 .76410 1.30873 .79212 1.26244 .82092 1.2181437 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 35 888 29 30 1.34896 26 .73816 1.35472 27 .73861 1.35389 .76594 1.30558 28 .73906 1.35307 .76640 1.30480 .73951 1.35224 .76686 1.30401 .73996 1.35142 .76733 1.30323 31 .74041 1.35060 76779 1.30244 32 .74086 33.74131 .76548 1.30637 .79354 1.26018 .79401 1.25943 .79149 1.25867 .79496 .82238 • 1.21598 34 .82287 1.21526 33 .82336 1.21454 32 1.25792 .82385 1.21382 31 .79544 1.25717 .82434 1.21310 30 .79591 1.25642 .82483 1.21238 29 1.34978 .76825 1.30166 .79639 1.25567 .82531 1.21166 28 .76871 1.30087 .79686 1.25492 .82580 1.21094 27 34 .74176 35 74221 1.34814 .76918 1.30009 .79734 1.25417 .82629 1.2102326 36.74267 37 .74312 1.34732 .76964 1.29931 1.34650 .77010 1.29853 1.34568 .77057 1.29775 79781 1.25343 .82678 1.2095125 .79829 1.25268 .82727 1.20879 24 .79877 1.25193 82776 1.2080823 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 40 .74402 1.34405 .77149 1.29618 .74447 1.34323 .79972 1.25044 .82874 1.20665 21 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 56.75173 57.75219 Cotang Tang 42 .74538 1.34160 .77289 1.29385 43 .74583 1.34079 .77335 1.29307 44 .74628 1.33998 .77382 45 .74674 1.33916 .77428 46 .74719 1.33835 .77475 1.29074 47 .74764 1.33754 .77521 1.28997 48 .74810 1.33673 .77568 1.28919 49.74855 1.33592 .77615 1.28842 50 .74900 1.33511 .77661 1.28764 51 .74946 1.33430 .77708 1.28687 52.74991 1.33349 .77754 1.28610 53 .75037 1.33268 .77801 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 55 .75128 1.33107 .77895 1.28379 .80738 1.23858 1.33026 .77941 1.28302 .80786 1.23784 1.32946 .77988 1.28225 80834 1.23710 1.32865 .78035 1.28148 .80882 1.23637 59.75310 1.32785 .78082 1.28071 80930 1.23563 60 .75355 1.32704 .78129 1.27994 .80978 1.23490 Cotang Tang Cotang Tang 58.75264 .80115 1.24820 .83022 1.20451 18 .80163 1.24746 .83071 1.20379 17 1.29229 .80211 1.24672 .83120 1.2030816 1.29152 .80258 1.24597 .83169 1.20237 15 .80306 1.24523 .83218 1.20166 14 80354 1.24419 .83268 1.20095 13 .80402 1.24375 .83317 1.20024 12 .80450 1.24301 .83366 1.1995311 .80498 1.24227 .83415 1.19882 10 .80546 1.24153 .83465 1.19811 9 .80594 1.24079 ,83514 1.19740 8 1.28533 .80642 1.24005 .83564 1.19669 77 .83613 .83662 1.19528 5 .83712 1.19457 4 .83761 1.19387 3 1.19599 6 .83811 1.19316 2 .83860 1.19246 1 .83910 1.19175 0 Cotang Tang 53° 52° 51° 50° 318 TABLE XII.-TANGENTS AND COTANGENTS. 40° 41° Tang Cotang Tang 0 1 2.84009 .83910 1.19175 .86929 .83960 1.19105 .86980 1.19035 .87031 3.84059 1.18964 .87082 • 4.84108 5.84158 1.18894 .87133 1.18824 Cotang Tang Cotang 1.15037 .90040 1.11061 1.14969 .90093 1.10996 1.14902 .90146 1.10931 1.14834 .90199 1.10867 1.14767 .90251 1.10802 .87184 1.14699 43° Tang Cotang 42° .93252 1.07237 60 .93306 1.07174 59 .93360 1.07112 58 .93415 1.0704957 .93469 1.06987 56 188858 .90304 1.10737 .93524 1.06925 55 6.84208 1.18754 .87236 1.14632 .90357 1.10672 .93578 1.0686254 7 .84258 1.18684 87287 1.14565 .90410 1.10607 .93633 1.06800 53 8.84307 1.18614 9 .84357 1.18544 10 .84407 1.18474 .87338 1.14498 87389 1.14430 87441 1.14363 .90463 1.10543 11.84457 12.84507 13 .84556 1.18404 .87492 1.14296 1.18334 .87543 1.14229 1.18264 14 .84606 15.81656 .87595 1.14162 1.18194 .87646 1.18125 .87698 1.14095 1.10156 .93688 .90516 1.10478 .93742 .90569 1.10414 .93797 90621 1.10349 .93852 .90674 1.10285 .93906 .90727 1.10220 .93961 .90781 1.06738 52 1.0667651 1.06613 50 1.0655149 1.0648948 1.0642747 .94016 1.06365 46 1.14028 90834 1.10091 .94071 1.0630345 16.84706 17.84756 008 288 18 1.18055 1.17986 .84306 | 1.17916 19 .84856 1.17846 .87749 1.13961 .87801 1.13894 .87852 .87904 1.13761 .90887 1.10027 .94125 1.0624141 .90940 1.09963 94180 1.06179 43 1.13828 .90993 1.09899 .94235 1.06117 42 .91046 1.09834 .94290 1.0605641 20 .84906 1.17777 .87955 1.13694 .91099 1.09770 .94345 1.0599440 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 94400 1.0593239 22 85006 1.17638 .88059 1.13561 .91206 1.09642 .94455 1.05870 38 23.85057 1.17569 .88110 1.13494 24.85107 1.17500 .88162 1.13428 25.85157 1.17430 .88214 1.13361 26.85207 1.17361 .88265 1.13295 27.85257 1.17292 .88317 28.85308 1.17223 .88369 .91259 1.09578 .94510 1.05809 37 .91313 1.09514 .94565 1.0574736 .91366 1.09450 .94620 1.05685 35 .91419 1.09386 .94676 1.05624 34 1.13228 .91473 1.09322 .94731 1.0556233 1.13162 .91526 1.09258 .94786 1.0550132 29.85358 1.17154 .88421 1.13096 .91580 1.09195 94841 1.05439 31 + 30.85408 1.17085 .88473 1.13029 .91633 1.09131 .94896 1.05378 30 31.85458 1.17016 88524 32.85509 1.16947 .88576 33.85559 1.16878 34 .85609 1.16809 35 .85660 1.16741 1.12963 .91687 1.09067 .94952 1.0531729 1.12897 .91740 1.09003 .95007 1.05255 28 • .88628 1.12831 .91794 1.08940 .95062 1.05194 27 .88680 1.12765 .91847 1.08876 .95118 1.05133 26 .88732 1.12699 .91901 1.08813 .95173 1.05072 25 36.85710 1.16672 .88784 1.12633 .91955 1.08749 .95229 1.05010 24 37.85761 1.16603 88836 1.12567 .92008 1.08686 .95284 1.0494923 1.16535 41 40 .85912 .85963 1.16329 .89045 1.12303 42.86014 1.16261 38 .85811 39 .85862 1.16466 .88940 1.12435 1.16398 .88992 1.12369 88888 1.12501 .92062 1.08622 .95340 1.0488822 .92116 1.08559 .95395 1.04827 21 .92170 1.08496 .95151 1.04766 20 .92224 1.08432 .95506 1.04705 19 .89097 1.12238 .92277 1.08369 .95562 1.0464418 43 .86064 1.16192 44.86115 1.16124 .89149 .89201 1.12106 1.12172 .92331 1.08306 .95618 1.0458317 .92385 1.08243 .95673 1.0452216 45.86166 1.16056 .89253 1.12041 .92439 1.08179 .95729 1.04461 15 46 .86216 1.15987 .89306 47 .86207 1.15919 .89358 1.11975 .92493 1.08116 1.11909 .92547 1.08053 .95785 1.04401 14 .95841 1.0434013 48 .86318 1.15851 .89410 1.11844 .92601 1.07990 .95897 1.04279 12 1888 59.86878 1.15104 60 .86929 1.15037 49 .86368 1.15783 50 .86419 1.15715 51 .86470 1.15647 52 .86521 1.15579 53.86572 1.15511 51 .86623 1.15443 .89725 1.11452 55 .86674 1.15375 .89777 1.11387 56.86725 1.15308 .89830 1.11321 57.86776 1.15240 .89883 1.11256 58 86827 1.15172 .89935 1.11191 .89988 .90040 .89463 1 11778 .92655 1.07927 .95952 1.04218 11 .89515 .89567 1.11648 .89620 1.11582 .89672 1.11517 1.11713 .92709 1.07864 .96008 1.0415810 .92763 1.07801 .96064 1.04097 .96120 92872 1.07676 .96176 .92926 1.07613 .96232 92980 1.07550 .96288 1.03855 5 .93034 1.07487 .96344 1.03794 4 .93088 1.07425 .96400 1.03734 3 .93143 1.07362 .96457 1.03674 1.11126 .93197 1.07299 .96513 1.03613 1.11061.93252 1.07237 .96589 1.03553 Cotang Tang Cotang Tang Cotang Tang Cotang Tang .92817 1.07738 1.04036 1.03976 1.03915 6 OIL-GLO TOO ON HO 9 7 49° 48° 47° 46° 319 TABLE XII. TANGENTS AND COTANGENTS. 44° Tang Cotang 44° 44° Tang Cotang Tang Cotang 0 1 .96569 1.03553 60 20 .96625 1.03493 59 21 .97756 .97700 1.02355 40 40 1.02295 39 | 41 .98843 1.01170 20 .98901 1.01112 19 2 .96681 1.03433 58 22 .97813 1.02236 38 38 42 .98958 1.01053 18 6.96907 .96738 1.03372 57 23 .96791 1.03312 56 24 .96850 1.03252 55 25 1.03192 54 26 .97870 1.02176 37 43 .99016 1.00994 17 .97927 1.02117 36 44 .99073 1.00935 16 .97984 1.02057 35 45 .99131 1.00876 15 .98041 1.01998 34 34 46 .99189 1.00818 14 .96963 1.03132 53 27 .98098 1.01939 33 33 47 .99247 1.00759 13 8 .97020 1.03072 52 28.98155 1.01879 32 32 || 48 .99301 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 49 .99362 1.00642 11 10 .97133 1.02952 50 30 .98270 1.01761 30 30 50 .99120 1.00583 10 11 .97189 1.02892 49 31 .98327 1.01702 29 29 51 .99478 1.00525 9 12 .97246 1.02832 48 32 .98381 1.01642 28 28 52 .99536 1.00467 8 13 .97302 1.02772 47 33 .98441 1.01583 27 53 .99594 1.00408 7 14 .97359 1.02713 46 34 .98499 1.01524 26 54 .99652 1.00350 6 15 .97416 1.02653 45 35 .98556 1.01465 25 55 .99710 1.00291 5 16 .97472 17 .97529 1.02593 44 1.02533 43 37 36 .98613 1.01406 24 56 .99768 1.00233 4 .98671 1.01347 23 57 .99826 1.00175 3 18 .97586 1.02474 42 38 .98728 1.01288 22 58 .99884 1.00116 2 19 .97643 1.02414 41 39 .98786 20 .97700 1.02355 40 40.98843 1.01229 21 59 1.01170 20 20 60 .99942 1 1.00000 1.00000 0 1.00058 Cotang Tang 45° Cotang 45° Tang Cotang Tang 45° 320 TABLE XIII.—VERSINES AND EXSECANTS. 0° 1° 2° 3° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0123 GLON∞✪ .00000 .00000 .00015 .00000 .00000 .00016 .00000 .00000 .00016 .00015 .00061 .00061 .00137 .00137 0 .00016 .00062 .00062 .00139 00139 1 .00016 .00063 .00063 .00140 .00140 2 .00000 .00000 .00017 .00017 .00064 .00064 .00142 .00142 3 4 .00000 .00000 .00017 .00017 .00065 .00065 .00143 .00143 5 .00000 .00000 .00018 .00018 .00066 .00066 .00145 .00145 6 .00000 .00000 .00018 .00018 00067 .00067 .00146 .00147 .00000 .00000 .00019 .00019 00068 .00068 .00148 .00148 8 .00000 .00000 00020 .000:20 00069 .00069 .00150 .00150 8 9 .00000 .00000 10 .00000 .00000 11 .00001 12 .00001 .00020 .00020 00021 .00021 .00001 .00021 .00021 .00001 .00022 .00070 .00070 .00151 .00151 9 .00071 .00072 .00153 .00153 10 .00073 .00073 .00154 .00155 11 .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 21 30 31 33 36 39 41 46 50 DEP02 78*****82. 788I3O588; =***99*998 H2OHBADO.: 17 16 .00001 .00001 .00024 .00001 .00001 .00025 .00024 .00078 .00078 .00162 .00163 16 .00025 .00079 .00079 .00164 .00164 17 .00001 .00001 .00026 .00026 .00081 .00081 .00166 .00166 18 .00002 .00002 .00026 .00026 .00082 .00082 .00168 .00168 19 .00002 .00002 .000:27 .00027 .00083 .00083 .00169 .00169 20 .00002 .00002 .00028 .00028 .00084 .00084 .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 26 .00003 .00003 .00031 .00003 .00003 .00032 .00003 .00003 .00033 .00033 .00031 .00031 .00089 .00089 .00178 .00178 25 .00031 .00032 00090 .00090 .00179 ► .00180 26 .00091 .00091 .00181 .00182 27 .00093 .00093 .00183 .00183 28 29 .00004 .00004 .00034 .00034 .00094 .00091 .00185 .00185 29 .00004 .00004 .00034 .00034 .00095 .00095 .00187 .00187 30 .00004 .00004 .00035 .00035 00096 .00097 .00188 .00189 31 32 .00004 .00001 34 .00036 .00036 .00005 .00005 .00037 .00037 .00005 .00005 .00037 .00037 .00005 .00005 .00038 .00038 .00005 .00005 .00039 .00039 .00006 .00000 .00040 .00040 .00006 .00006 .000-11 .00041 .00098 .00098 .00099 .00100 .00100 .00190 .00190 32 .00099 .00192 .00192 33 .00194 .00194 34 .00102 .00102 .00196 .00196 35 .00103 .00103 .00197 .00198 36 .00104 .00104 .00199 .00200 37 .00106 .00106 .00201 .00201 38 .00006 .00006 40 .00007 .00007 42 .00007 .00007 .00041 .00041 .00042 .00042 .00007 .00007 .00043 .00043 .00014 .00044 .00107 .00107 .00203 .00203 39 .00108 .00108 .00205 .00205 40 .00110 .00110 .00207 .00207 41 .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 .00009 .00009 .00018 .00048 .00117 .00117 .00216 .00216 46 .00009 .00009 .00048 .00048 .00118 .00118 .00218 .00010 00010 .000 19 .00049 .00119 .00120 .00218 47 .00220 .00220 48 49 .00010 .00010 .00050 .00050 .00121 .00121 .00011 .00011 .00051 .00051 .00122 .00122 .00222 .00222 49 .00224 .00224 50 51 .00011 .00011 .00052 .00052 52 .00011 .00011 .00053 .00053 53 .00012 .00012 .00054 54 .00012 .00012 .00055 · 55 .00013 .00013 .00056 56 .00013 .00013 .00057 57 .00014 .00014 58 .00014 .00014 59 .00015 .00015 .00124 .00124 .00125 .00125 .00054 .00127 .00127 00055 .00128 .00128 00056 .00130 .00130 .00057 .00131 .00058 .00058 .00133 .00059 .00059 .00134 .00060 .00060 .00226 00226 51 .00228 .00228 52 .00230 .00230 53 .00232 .00232 54 .00234 .00234 55 .00131 .00236 .00236 56 .00133 .00238 .00238 57 .00134 .00240 .00210 58 .00136 .00136 .00242 .00242 59 60 .00015 .00015 .00061 00061 .00137 .00137 .00244 00244 60 321 TABLE XIII.—VERSINES AND EXSECANTS. 4º 5° 6° що Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .00244 .00244 .00381 00382 .00548 .00551 .00745 .00751 1 .00246 .00246 .00383 00385 .00551 .00554 .00749 .00755 10 14 20 21 23 31 33 36 39 43 45 47 49 53 54 56 57 RATNO-∞∞O ERRERLEBAR ZRRHKRKARA ☎88483588* =83I99EGGA NX8XBOXARO .00248 .00248 .00386 .00387 .00554 .00557 .00752 .00758 .00250 .00250 .00388 .00390 .00557 .00560 .00756 .00762 OHQ∞ 0 2 .00252 .00252 .00391 .00392 00560 .00563 .00760 .00765 .00254 .00254 .00393 .00395 .00563 .00566 .00763 .00769 .00256 .00257 .00396 .00397 .00566 00569 .00767 .00773 .00258 .00259 .00398 .00400 .00569 .00573 .00770 .00776 7 8 .00260 .00262 .00261 .00401 .00403 .00572 .00576 .00774 .00780 8 .00263 .00404 .00405 .00576 .00579 .00778 .00784 9 .00264 .00265 .00106 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 .00273 .00274 .00414 .00116 .00588 .00592 .00792 .00799 13 .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 18 • .00279 .00280 .00425 .00427 .00601 .00604 00807 .00813 17 .00281 .00282 .00428 .00429 .00604 .00608 .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 .00814 00821 19 .00286 .00287 .00433 .00435 .00610 .00614 .00818 00825 20 .00288 00289 .00436 .00438 .00614 .00617 .00822 00828 21 .00290 .00291 .00438 .00440 .00617 .00621 .00825 .00832 22 .00293 .00293 .00441 .00443 .00620 00624 .00829 .00836 23 24 .00295 .00296 .00297 .00298 .00299 .00300 .00301 .00304 .00444 .00446 .00623 .00627 .00833 .00840 24 .00447 .00449 .00626 .00630 .00837 .00844 25 .00449 .00451 .00630 00840 .00634 .00848 26 .00302 .00452 .00454 .00633 .00637 00844 .00851 27 .00305 .00455 .00457 .00636 .00640 00848 .00855 28 .00306 .00458 .00307 00460 00640 .00644 .00852 .00859 29 .00308 .00309 .00460 .00463 .00643 .00647 .00856 .00863 30 .00311 .00312 .00463 .00313 .00646 .00465 .00650 .00859 .00867 31 .00466 .00314 .00315 .00316 .00468 .00649 .00654 00863 .00871 32 .00469 .00471 .00653 .00657 .00867 .00875 33 .00317 .00318 .00472 .00474 .00656 .00660 .00871 00878 34 .00320 .00321 .00474 .00477 .00659 .00664 .00875 00882 35 .00322 .00323 .00477 .00480 .00663 .00667 00878 .00886 36 .00324 .00326 .00480 .00482 .00666 .00671 .00882 .00890 37 .003:27 .00328 .00483 00485 .00669 .00674 .00886 .00894 38 .00329 .00330 .00486 .00488 .00673 .00677 00890 .00898 39 40 .00332 .00333 .00489 .00491 .00676 .00681 00894 .00902 40 .00334 .00339 .00335 .00492 .00336 00337 .00340 .00494 .00680 .00684 .00898 .00906 41 .00494 .00497 .00683 .00688 .00902 .00910 42 .00497 .00500 .00686 .00691 .00906 .00914 43 .00341 .00500 .00342 .00503 .00690 .00695 .00909 .00918 44 .00343 .00503 .00345 .00506 .00693 .00698 .00913 .00922 45 .00346 .00347 .00506 .00509 .00697 .00701 .00917 .00926 46 .00351 .00348 .00350 .00352 .00509 .00512 .00700 .00705 .00512 .00515 .00703 .00925 .00708 .00353 .00354 .00515 .00518 .00707 • .00356 .00357 .00518 .00521 .00710 47 00934 48 00938 .00929 00712 .00715 00930 .00921 49 .00933 00942 50 .00358 .00359 .00521 .00524 .00714 .00361 .00362 00524 .00527 .00717 • .00363 .00364 .00527 .00721 .00530 .00365 .00367 .00530 .00533 .00724 .00368 .00369 .00533 00536 00728 .00937 .00719 .00911 00722 .00954 .00945 .00726 .00958 .00949 .00730 .00953 .00733 .00946 51 .00950 52 53 54 .00962 55 00370 .00372 .00536 .00539 .00731 .00957 .00737 .00966 56 .00373 00374 .00539 00542 .00735 .00740 .00961 .00970 57 .00375 · 60 | .00381 .00542 00377 .00742 .00747 .00379 .00378 .00545 .00548 00382 .00551 .00745 .00548 .00751 .00545 .00738 .00744 00965 .00975 58 .00969 .00979 59 .00973 .00983 | 60 322 TABLE XIII.-VERSINES AND EXSECANTS. 8° 9° 10° 11° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 THOR BOTH LO CO .00973 .00983 .01231 .01247 .00977 .00987 .01236 .00981 .00991 .01240 .01251 .01524 .01519 .01543 .01548 .01837 .01872 0 .01813 .01877 1 .01256 .01529 .01553 .01818 .01883 .00985 .00995 .01245 .01261 .01534 .01558 .01854 .01889 .00989 .00999 .00994 .01004 .00998 .01008 .01249 .01265 .01254 .01270 .01259 .01275 .01540 .01564 .01860 .01895 .01545 .01569 .01865 .01901 .01550 .01574 .01871 .01906 .01002 .01012 .01263 .01279 .01555 .01579 .01876 .01912 .01006 .01016 .01268 .01284 .01560 S .01010 .01020 .01272 .01289 .01565 10 .01014 .01024 .01277 .01294 .01570 .01585 .01882 .01918 8 .01590 .01888 .01924 9 .01595 .01893 .01930 10 11 .01018 .01029 .01282 .01298 .01575 .01601 .01899 .01936 11 12 .01022 .01033 .01286 .01303 .01580 .01606 .01904 .01941 12 13 .01027 .01037 .01291 .01308 .01586 .01611 .01910 .01947 13 14 .01031 .01041 .01296 .01313 .01591 .01616 .01916 .01953 11 15 01035 .01046 .01300 1 .01318 .01596 .01622 .01921 .01959 15 31 51 AHAAR ZANIHAKKRA H38HHO6889 =88*995998 528H 16 .01039 .01050 .01305 .01322 .01601 .01627 .01927 .01965 16 17 .01043 .01054 .01310 .01327 .01605 .01633 .01933 .01971 17 18 .01047 .01059 .01314 01332 .01612 .01638 .01939 .01977 18 19 .01052 .01063 .01319 .01337 .01617 .01643 .01944 .01983 19 20 .01056 .01067 .01324 .01342 .01622 .01649 .01950 .01989 20 .01080 .01338 .01050 .01071 .01329 .01346 01084 .01076 .01333 .01351 .01069 .01073 .01084 .01343 .01027 .01654 .01632 .01659 .01956 .01995 21 .01961 .02001 .01356 .01638 .01665 .01967 .02007 23 .01361 .01643 .01670 .01973 .02013 24 .01077 .01089 .01348 .01366 .01618 .01676 .01081 .01093 .01352 .01371 .01653 .01681 .01984 .01979 .02019 25 .02025 26 .01086 .01097 .01357 .01376 .01659 .01687 .01990 .02031 27 28 29 30 .01098 .01111 .01103 .01115 .01090 .01102 .01362 .01381 .01664 .01094 .01106 .01367 .01386 .01371 .01391 .01376 .01395 .01692 .01996 .02037 28 32 .01107 .01119 .01381 .01400 33 .01111 .01124 .01386 .01405 31 .01116 .01128 .01391 .01410 35 .01120 .01133 .01396 .01124 .01137 .01400 .01415 .01420 37 .01129 .01142 .01405 .01425 .01133 .01146 .01410 .01430 .01137 .01151 .01415 .01435 40 .01142 .01155 .01420 .01440 .01728 .01669 .01698 .01675 .01703 .01680 .01709 .01685 .01714 .02019 .01690 01720 .02025 .01696 .01725 .02031 .01701 .01731 .02037 .01706 .01736 .02042 .01712 .01742 .02048 .01717 .01747 .0:2054 .01723 .01753 .01758 .02002 .02008 .02043 .02049 30 .02013 .02055 31 .02061 32 .02067 33 ER FARANRKRAS E88 22 .02073 31 .02079 35 .02085 36 .02091 37 .02097 33 .02060 .02103 39 .02066 .02110 40 41 .01146 .01160 .01425 .01445 42 .01151 .01164 .01430 .01450 43 .01155 .01169 .01435 .01455 41 .01159 .01173 .01439 .01161 45 46 48 .01177 .01191 .01191 .01195.01209 53 .01200 .01214 54 .01204 .01219 .01164 .01178 .01444 .01466 .01168 .01182 .01419 .01471 .01173 .01187 .01454 *01476 .01459 .01481 49 .01182 .01196 .01464 .01486 .01777 .01809 .01186 .01200 .01469 .01491 .01782 .01815 .02125 .02171 50 .01205 .01474 .01496 .01788 .01820 .02131 .02178 51 .01479 .01501 .01793 .01826 .02137 .02184 52 .01484 .01506 .01799 .01832 .02143 .02190 53 .01489 .01512 .01804 .01837 .02149 .02196 5-1 01733 .01764 .01739 .01769 .017-14 .01775 01750 .01781 .01755 .01786 .01760 .01792 .01766 .01798 .01771 .01803 .02072 .02116 41 .02078 .02122 43 .02084 .02128 43 .02090 .02134 44 .02095 .02140 45 .02101 .02146 46 .02107 .02113 .02119 .02153 47 .02159 48 .02165 49 89 =393935998 55 .01209 .01223 .01494 .01517 .01810 .01843 .02155 .02203 55 56 .01213 .01228 .01499 .01522 .01815 57 .01218 .01233 ..01504 .01527 58 59 60 8888 .01222 .01237 .01509 .01532 .01849 .01821 .01854 .01826 .01860 .02161 02209 56 .02167 .02215 57 .02173 .02221 1 58 .01227 .01242 .01514 .01537 .01832 .01866 .02179 .02228 59 .01231 .01247 .01519 .01543 01837 .01872 .02185 02234 60 323 TABLE XIII.-VERSINES AND EXSECANTS. 12° 13° 14° 15° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .02185 1 .02191 3 .02234 .02570 .02240 .02197 .02247 .02203. .02583 .02253 02210 .02259 .02589 .02563 .02630 02637 .03528 .02970 .03407 .03061 .02977 .03069 .03415 .03536 0 .02576.02644 .02985 .03076 .03422 .03544 .02651 .02992 .03084 .03430 .03552 .02658 .02999 .03091 .03438 .03560 02216 .02266 .02596 .02665 .03006 .03099 .03445 .03568 .02222 .02272 .02602 .02672 .03013 .03106 .03453 .03576 | .02228 .02279 .02609 .02679 .03020 .03114 .03.160 .03584 7 .02234 02:285 02616 .C2686 .03027 .03121 .03-468 .03592 8 • 9 .02240 .02291 .02622 .02693 .03034 .03129 .03476 .03601 9 10 .02246 .02298 .02629 .02700 .03041 .03137 .03483 .03609 10 11 .02252 ..02301 02635 .02707 .03048 .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 .02662.02735 .03077 .03175 .03521 .03650 15 16 .02283 .02336 .02669 .02742 .03084 .03182 .03529 .03658 16 17 20 21 26 29 31 FEER ZAREKENARO 783 .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 · .02308 02362 .02696 .02770 .03113 .03213 .03560 .03691 20 .02314 .02369 .02702 .0277 .03120 .03221 .03567 .03699 21 22 .02320 .02375 .02709 .02784 .03228 .03127 .03575 .03708 23 02382 .02327 .02716 .02791 .03134 .03236 .03583 .03716 • 24 02333 .02388 .02722 .02799 .03142 .03244 .03590 .03724 - 25 .02339 .02395 .02729 .02806 .03149 .03251 .03598 .03732 25 .02345 .02402 02736 .02813 .03156 .03259 .03606 .037-11 26 27 02352 .02408 02743 .02820 .03163 .03267 .03749 .03614 28 .02415 .02358 .02749 02827 .03171 .03275 .03621 .03758 .02364 .02421 .02756 02834 .03178 .03282 .03629 .03766 30 .02370 .02428 .02763 02842 .03185 .03290 .03637 .03774 30 222 783*HRAX23 22 24 29 .02377 .02435 .02770 02849 .03193 .03298 .03645 .03783 31 .02396 32 02283 .02441 33 .02589 34 .02777 .02448 .02782/ .02856 .03200 .03306 .03653 .03791 32 02863 .03207 .03313 .03660 .03799 33 .02454 .02790 .02870 .03214 .03321 .03668.03808 3-4 35 .02402 .02797 .02461 .02878 .03222 .03329 .03676 .03816 35 36 .02108 .02468 .02804 .02885 03229 .03337 .03684 .03825 36 37 .02415.02474 .02811 .02892 03236 .03345 .03692 .03833 37 41 43 47 48 49 50 53 55 57 59 60 ********** C**RAAD88A 6800 * 38 .02421 .02181 .02818 .02899 03244 .03353 .03699 .03842 38 39 .02427 .02488 .02824 .02907 .03251 .03360 .03707 .03850 39 40 .02434 .02194 02831 .02914 .03258 03368 .03715 .03858 + .02440 .02501 .02838 .02921 03266 .03376 .03723 .03867 41 42 .02417 .02508 .02845 .02928 .03273 .02453 .02515 .02852 .02936 .03281 .03731 .03384 .03739 03392 44 .02459 02521 .02859 .02943 .03288 .03400 .03747 .03875 42 .03884 .03892 43 • 45 .02466 .02528 .02866 .02950 .03295 .03408 .03754 .03901 46 .02472 .02535 .02873 .02958 .03303 .03416 .02880 .02542 .02479 .02965 .03310 .03424 .03762 .03909 .03770 .03918 | 47 .02485 .02548 .02894 .02555 .02492 .02-198 .02562 02900 .02887 .02972 .03318 .03432 .03778 .03927 48 .02980 .03325 .03439 .02987 .03333 .02569 .02504 .02907 .02511 .02576 .02914 .02517 02582 .02921 .02994 .02524 .02589 02928 .02530 .02596 02935 .03024 03370 .02537 .02603 .02942 .03032 .03377 .03495 .02543 .02610 .02949 .03039 .03385 .03503 .03447 .03802 .03952 51 03340 .03455 .03810 03347 .03463 .03002 .03471 .03818 .03355 .03009 .03826 .03479 .03362 .03017 .03987 .03487 .03834 .03995 .03842 .03850 .04004 57 .03786 .03935 49 .03794 .03914 50 .03961 .03969 .03978 54 333 333995*38 5888 40 44 45 46 52 53 55 56 58 .02550 .02617 .02956 .03392 .03046 .02550 02624 .02963 .03054 02563 .02630 .02970 03061 .03512 .03400 .03520 .03528 03407 .03858 .04013 58 .03866 .04021 59 .03874 04030 60 324 TABLE XIII.-VERSINES AND EXSECANTS. 16° 17° 18° 19° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. હર્ષ સ 0 .03874 .04030 .04370 .04569 .03882 .04039 .04378 .04578 .03890 .04047 .03898 .04056 .04894 .05146 .04903 .05156 .05418 .05762 0 .05458 .05778 1 .04387 .04588 .04395 .04912 .05166 .05467 .05783 .04597 .04921 .05176 .05477 .05794 .03906 .04065 .04404 .04606 .04930 .05186 .05486 .05805 .03914 .04073 .04412 .04616 .04939 .05196 .05496 .05815 .03922 .04082 .01421 .04625 .04948 .05206 .05505 .05826 7 .03930 .04091 .04429 .04635 .04957 .05216 .05515 .05836 8 .03938 .01100 .01438 .04644 9 .03946 .04108 04446 .04653 .04967 .04976 .05236 05226 05524 .05847 .05534 .05858 9 10 .03954 .04117 .04455 .04663 .04985 .05246 .05543 .05869 10 11 .03963 .04126 .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 .04498 .04710 .05030 .05297 05591 .05922 15 16 .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 .05067 .05337 .05630 .05965 19 20 .04036 .04206 .04541 .04757 .05076 .05347 .05639 .05976 20 21 .04044 .04214 .04550 .04767 .05085 .05357 .05649 .05987 21 .04052 .04223 .04559 .04776 .05094 .05367 .05658 .05998 22 23 .04060 .04232 .04567 .04786 .05103 .05378 .05668 .06009 23 .04069 .04241 .04576 .04795 .05112 .05388 .05678 .06020 24 25 .04077 .04250 .01585 .04805 .05122 .05398 .05687 .06030 25 .04085 .04259 .04593 .04815 .05131 .05408 .05697 .06041 26 • 27 .04093 .04268 .04602 .04824 .05140 .05418 .05707 .06052 28 .04102 .04277 .04611 .04834 .05149 .05429 .05716 · .06063 28 29 37 .04110 .04286 30 .04118 .04295 31 .04126 .04304 32 .04135 .04313 33 .04143 .04322 34 .04151 .04331 35 .04159 .04340 36 .04168 .04349 .04176 .04358 .04620 .041843 .05158 .05439 .05726 .06074 29 .04628 .04853 .05168 .05449 .05736 .00085 30 .04637 .04863 .05177 .05460 .05746 .06096 .04646 .04872 .05186 .05470 05755 .06107 32 .04655 .04882 .05195 .05480 .05765 .06118 33 .04663 .04891 05205 .05490 * .05775 .06129 34 22 7N3TRONKA. HABI 27 .04672 .04901 05214 .05501 .05785 .06140 35 .04681 .04911 05223 .05511 .05794 .06151 36 .04690 .04920 .05232 .05521 .05804 .06162 37 38 .01184 .04367 .04699 .04930 .05242 .05532 .05814 .06173 38 39 .04193 .04376 .04707 .04940 .05251 05542 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 .04752 .04989 .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 50 .04285 .04477 .04796 .05038 .05344 .05646 .05922 .06295 49 .04805 .05047 .05354 .05657 .05932 .06306 50 51 .04293 .04486 52 .04302 .04495 53 .04310 .04504 54 .04319 .04514 .01814 .05057 .05363 .05667 .05942 .06317 51 .04823 .05067 .05373 .05678 .05951 .06328 52 .04832 .05077 .05382 .05688 .05961 .06339 53 .04841 .05087 .05391 .05699 .05971 .06350 54 55 .04327 .04523 .04850 .05097 .05401 .05709 .05981 .06362 55 56 .04336 .04532 57 .04344 .04541 58 .04353 .04551 .04858 .05107 .05410 .04867 .05116 .05720 .05991 .06373 56 .05420 .05730 .06001 .06384 57 .04876 .05126 .05429 .05741 .06011 .06395 58 59 .04361 .04560 .04885 .05136 .05439 .05751 .06021 .06407 59 60 .04370 .04569 .04894 .05146 .05448 .05762 .06031 .06418 60 325 TABLE XIII.-VERSINES AND EXSECANTS. 20° 21° 22° 23° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 1 .06031 .06418 .06041 .06429 2 .06051 .06140 3 .06061 .06452 4 .06071 .06463 41 45 46 50 51 53 DOTTO FRATROLPER F384H2KARS HBOSBOD389 #333994998 5884685828 5 .00081 .06474 6 .06091 .06-186 ༩ .06101 .06497 91 8.06111 .06121 .06508 .06642 .07115 .06652 .07126 .06663 .07138 .06673 .07150 .06684 .07162 .06694 .07174 .06705 .07186 .06715 .07199 .06726 .07211 07282 .07853 .07293 .07866 .07303 .07879 .07314 .07892 .07325 .07904 .07336 .07917 .07347 .07930 .07358 .07369 .07950 .08636 0 .07961 .08649 1 .07972 .08663 2 .07984 .08676 3 .07995 .08690 .08006 .08703 5 .08018 .08717 6 .07943 .08029 .08730 7 .07955 .08041 .08744 8 .06520 .06736 .07223 .07380 .07968 .08052 .08757 9 10 .06131 .06531 .06747 .07235 .07391 .07981 .08064 .08771 10 11 .06141 .06542 .06757 .07247 .07402 .07994 .08075 .08784 11 12 .06151 .06554 .06768 .07259 .07413 .08006 .08086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 15 .06171 .06577 .06181 .06588 .06789 .07283 .07435 .08032 .08109 .08825 14 .06799 .07295 .07446 .08045 .08121 .08839 15 18 20 16 .06191 .06600 17 .06201 .06611 .06211 .06622 19 06221 .06634 .06231 .06810.07307 .07457 .08058 .08132 .08852 16 .06820 .07320 .07468 .08071 .08144 .08866 17 .06831 .07332 .07479 .08084 .08155 .08880 18 .06841 .07344 .07490 .08097 .08167 .08893 19 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 92 .06252 .06668 06873 .07880 .07523 .08135 .08201 .08934 29 .06262 .06272 .06691 .06680 .06884 .07393 .07534 .08148 .08213 .08948 23 .06894 .07405 .07545 .08161 .08225 .08962 25 .06282 .06703 .06905 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 .07568 .08187 .08248 .08989 20 .06302 .06726 .06926 .07442 .07579 .08200 .08259 .09003 27 .06312 .06738 .06937 .07454 .07590 .08213 20 .06323 .06749 .06948 .07466 .07601 30 .06333 .06761 .00958 .07479 31 .06343 .06773 32 .06353 .06784 33 34 .06374 .06807 35 .06384 .06969 .06980 .07503 .06363 .06796 .06990 .07516 .07001 .06819 .07012 .07491 .07528 .07540 .07668 36 .06394 .06831 .07022 .07553 .07679 .08226 .07612 .08239 .07623 .08252 .07034 .08265 .07645 .08278 .07657 .08291 .08305 .08318 .08271 .09017 28 .08282 .09080 29 .08294 .09014 30 28 FBRAAakaaa 24 37 .06404 .06843 .07033 .07565 .07090 .08331 .08306 .09058 31 .08317.09072 32 .08329 .09086 33 .08340 .09099 34 08352 .09113 35 .08364 .09127 36 .08375 .09141 37 40 .06415 .06851 .07014 .07578 06425 .06866 .07055 .07590 .06435 .06878 .07065 .07602 .07701 .08344 .08387 .09155 38 .07713 .08357 .08399 .09169 39 .07724 .08370 .08410 .09183 40 .06445 .06889 .07076 .07615 .07735 .08383 .08422 .09197 41 47 06507 .06960 42 .06456 .06901 .07087 .07627 .07746 .08397 .06466 .06913 .07098 .07640 .07757 .08410 .06476 .06925 .07108 .07652 .07769 .08423 .06486 .06936 .07119 .07665 .07780 .08436 .06497 .06918 .07130 .07677 .07791 .08449 .07141 .07690 .07802 .08463 .06517 .06972 .07151 .07702 .07814 .08476 .06528 .06984 .07162 .07715 .06538 .06995 .07173 .07727 .08431 .09211 42 .08445 .09224 43 .08457 .09238 44 .08469 .09252 45 .08481 .09266 46 .08492 .09280 47 .08504 .09204 43 07825 .08489 .08516 .09308 49 .07836 .08503 08528 .09323 50 • .06548 07007 .07184 .07740 .07818 .08516 .08539 .09337 | 51 .06559 .07019 .07195 .07752 .07859 .08529 .08551 .09351 52 .06569 .07031 .07206 .06580 .07043 .07216 .07765 .07870 .08542 .08563 .09365 53 .07778 .07881 .08556 .08575 .09379 54 .06590 .07055 .07227 .07790 .06600 .07067 .07238 .07803 .07893 .08569 .08586 .09393 55 .07904 .08582 .08598 .09407 56 .06611 .07079 .06621 .07091 .07260 .07828 1 07927 .07249 .07816 .07915 .08596 .08610 .09421 57 .08609 .08622 .09-435 58 59 .06632 .07103 .07271 .07841 .07938 .08623 .08634 .09449 59 60 .06642 .07115 .07282 .07853 .07950 .08636 .08645 .09464 60 326 TABLE XIII.-VERSINES AND EXSECANTS. 24° 25° 26° 27° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 16 20 21 23 26 29 30 OHRDTOLAR FARERADAAR ZORANAKARA HA .08645 .09464 .09369 .10338 .08657 .09478 .09382 .10353 .08669 .09492 .0939-1 .10368 .10121 .11260 .10133 .11276 .10146 .11292 .10899 .12233 0 .10913 .12249 1 .10026 .12266 .08681 .09506 .09406 .10383 .10159 .11308 .10939 .12283 .08693 .09520 .09418 .10398 .10172 .11323 .08705 .09535 .08717 .09549 .09443 .09431 .10413 .10184 .11339 .10428 .10197 .08728 .09563 .09455 .10443 .10210 .11355 .11371 .10952 .12299 .10965 .10979 .12333 .12316 .10992 .12349 .08740 .09577 .09468 .10458 .10223 .11387 .11005 .12366 .08752 .09592 .09-480 .10473 .10236 .11403 .11019 .12383 9 10 .08764 .09606 .09493 .10488 .10248 .11419 .11032 .12400 10 11 .08776 .09620 .09505 .10503 .10261 .11435 .11045 .12416 11 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .12433 12 13 .08800 .09649 .09530 .10533 14 .08812 .09663 .09542 15 .08824 .09678 .09554 .10287 .11467 .11072 .12450 13 .10549 .10300 .11483 .11085 .12467 14 .10564 .10313 .11199 .11098 .12484 15 .08836 .09692 .09567 .10579 .10326 .11515 .11112 .12501 16 17 .08848 .09707 .09579 .10594 .10338 .11531 .11125 .12518 17 18 08860 .09721 .09592 .10609 .10351 .11547 .11138 .12531 18 .03872 .09735 .09604 .10625 .10364 .11563 .03884 .09750 .09617 .10640 .10377 .11579 .11165 .11152 .12551 19 .12568 20 .08896 .09764 .09629 .10655 22 .08908 .09779 .09642 .08920 .09793 .09654 .10670 .10103 .10686 .08932 .09808 .09666 .10701 .10390 .11595 .11611 .10416 .11627 .10429 .11643 .11178 .12585 21 .11192 .12602 22 .11205 .12619 23 .11218 .12636 24 .08911 .09822 .09679 .10716 .10442 .11659 .11232 .12653 25 .08956 .09837 .09691 .10731 .10455 .11675 .11245 .12670 26 .08968 .09851 .09704 .10747 .10468 .11691 .11259 .12687 27 .08980 .09866 .09716 .10762 .10481 .11708 .11272 .12704 28 .08992 .09880 .09729 .10777 .10494 .11724 .11285 .12721 29 .09004 .09895 .09741 .10793 .10507 .11740 .11299 .12738 30 .09016 .09909 .09754 .10808 .10520 .11756 .11312 .12755 31 32 .09028 .09924 .09767 .10824 33 .09040 .09939 .09779 .10839 34 .09052 35 .09064 .09953 .09968 .09792 .10851 .09801 .10870 36 47 51 52 5889 ====994988 HOBHRONORS .09076 .09982 37 .09089 .09997 38 .09101 .10012 39 .09113 .10026 40 .09125 .10041 .09817 .10885 .09829 .10901 41 42 .09137 .10055 .09149 .10071 43 .09161 .10085 .09842 .10916 .0985-1 .10932 .09867 .10947 .09880 .10963 .09892 .10978 .09905 .10994 .10533 .11772 .10546 .11789 .10559 .11805 .10572 .11821 .10585 .11838 .10598 .11854 .10611 .11870 .10624 .11886 .10637 .11903 11326 .12772 32 .11339 .11353 .11366 .11380 .12789 33 .12807 34 .12824 35 SPER ARAJARNARD HOCHB .12841 36 .11393 .12858 37 .11407 .12875 .10650 .10663 .09174 .10100 45 .09186 .10115 46 .09198 .10130 .09210 .10144 .09918 .11009 .09930 .11025 .09943 .11041 .09955 .11056 09222 .10159 .09968 .11072 .11919 .11936 .10676 .11952 .10689 .11968 .10702 .11985 .10715 .12001 .10728 .12018 .10741 .12034 .11420 .12892 .11434 .11447 .11461 .12910 40 .12927 .12944 42 41 .11474 .12961 43 .11488 .12979 44 .11501 .12996 45 .11515 .13013 46 .11528 5389 =9939 .13031 47 .11542 .13048 48 .09234 .10174 .09981 .11087 .10755 :12051 .11555 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11569 .13083 50 09259 .10204 .10006 .11119 .10781 .09271 .10218 .10019 .12084 .11583 .13100 51 .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 .11705 .13:257 60 327 TABLE XIII.—VERSINES AND EXSECANTS. 28° 29° 30° 31° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 29 31 35 37 39 40 42 43 OTHRRIDONOOD HRREDOIRAR 728**27X2. H88*885889 #33#*95898 HABIBONACO .11705 .13257 .12538 .14335 .13397 .15470 .14283 .16663 0 .11719 .13275 .12552 .11733 .13292 .12566 .11746 .14354 .13412 .15489 .14298 .16684 1 .14372 .13427 .15509 .14313 .16704 2 .13310 .12580 .14391 .13411 .15528 .14328 .16725 3 .11760 .13327 .12595 .14409 .13456 .15548 .14343 .16745 .11774 .13345 .12609 .14428 .13470 .15567 .14358 .16766 .11787 .13362 .12623 .14446 .13485 .15587 .14373 .16786 .11801 .13380 .12637 .14465 .13499 .15606 .14388 .16806 .11815 .13398 .12651 .14483 .13514 .15626 .14403 .16827 .11828 .13415 .12665 .14502 .13529 .15645 .14418 .16848 9 10 .11842 .13433 .12679 .14521 .13543 .15665 .14433 .16868 10 11 .11856 12 .11870 .13468 13 .11883 .13486 .13451 .12694 .14539 .13558 .15684 .14449 .16889 11 .12708 .14558 .13573 .15704 .14464 .16909 12 .12722 .14576 .13587 15724 .14479 .16930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .14494 .16950 14 15 .11911 .13521 .12750 .14614 .13616 .15763 .14509 .16971 15 16 .11925 .13539 .12765 .14632 .13631 .15782 .14524 .16992 16 17 .11938 .13557 .12779 .14651 .13646 .15802 .14539 .17012 17 18 .11952 .13575 .12793 .14670 .13660 .15822 .14554 .17033 18 19 .11966 .13593 .12807 .14689 .13675 .15841 .14569 .17054 19 20 .11980 .13610 .12822 .14707 .13690 .15861 .14584 .17075 20 21 .11994 .13628 .12007 .13646 23 .12021 .13664 .12864 .12035 .13682 .12879 25 .12049 .13700 26 .12063 .13718 .12907 .14820 .12836 .14726 .13705 .15881 .14599 .17095 21 .12850 .14745 .13719 .15901 .14615 .17116 22 .14764 .13734 .15920 .14630 .17137 23 .14782 .13749 .15940 .14645 .17158 24 .12893 .14801 .13763 .15960 .14660 .17178 25 .13778 .15980 .14675 .17199 26 30 .12077 .13735 .12921 .14839 .12091 .13753 .12936 .14858 .12104 .13771 .12950 .14877 .12118 .13789 .13793 .16000 .14690 .17220 27 .13838 .16019 .14706 .17241 28 .13822 .16039 .14721 .17262 29 .12964 .14896 .13837 .16059 .14736 .17283 30 31 .12132 32 .12146 .13825 13807 .12979 .14914 .13852 .16079 .14751 .17304 31 .12993 .14933 .13867 .16099 .14766 .17325 32 33 .12160 .13843 .12174 .13861 .12188 .13879 .18007 .14952 .13881 .16119 .14782 .17346 33 .13022 .14971 .13896 .16139 .14797 .17367 34 .13036 .14990 .13911 .16159 .14812 .17388 35 36 .12202 .13897 .13051 .15009 .13926 .16179 .14827 .17409 36 .12216 .13916 .13065 .15028 .13941 .16199 38 .12230 .13934 .13079 .15047 .13955 .16219 .14843 .17430 37 .14858 .17451 38 .12244 .13952 .13094 .15066 .13970 .16239 .12257 .13970 .13108 .15085 .13985 .16259 41 .12271 .13988 .13122 .15105 .14000 .16279 .14873 .17472 .14888 .14904 .17493 40 .17514 41 .12285 .14006 .13137 .15124 .12299 .14024 .13151 .15143 .14015 .14030 .16299 .14919 .17535 42 .16319 .14934 .17556 43 45 .12327 .14061 .13180 .15181 46 .12341 .14079 .13195 .15200 .14059 .16359 .14074 .16380 47 .12355 .14097 .13209 .15219 .14089 .16400 48 .12369 .14115 .13223 .15239 .14104 .16420 49 .12383 .14134 .13238 .15258 .11119 .16440 50 .12397 .14152 .13252 .15277 .14134 .16460 51 .12411 .14170 .13267 .15296 55 52 .12425 .14188 53 .12439 .14207 54 .12454 .14225 .12468 .14243 56 .12482 .14262 57 .12496 .14280 58 .12510 .14299 59 .12524 .14317 .13383 60 12538 .14335 .14149 .16481 .13281 .15315 .14164 .16501 .15072 .17747 52 .13296 .15335 .14179 .16521 .15087 .17768 .13310 .15354 .14194 .16541 .13325 .15373 .14208 .16562 .13339 .15393 .14223 .16582 .13354 .15412 .14238 .16602 .13368 15431 .14253 .16623 .15451 .14268 .16643 .13397 .15470 .14283 .16663 .15057 .17726 51 53 .15103 .17790 54 .15118 .17811 55 .15134 .17832 56 .15149 .17854 57 15164 .17875 58 .15180 .17896 59 .15195 .17918 | 60 44 .12313 .14042 .13166 .15162 .14044 .16339 .14949 .17577 44 .14965 .17598 45 .14980 .17620 46 .14995 .17641 47 .15011 .17662 48 .15026 .17683 49 .15041 .17704 50 889 7333995998 588ER 39 3.8 TABLE XIII.-VERSINES AND EXSECANTS. 32° 33° 34° 35° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0123) .15195 .17918 .16133 .19236 .15211 .17939 .161.19 .15226 .17961 .17096 .20622 .19259 .17113 .206-45 .18085 .22077 0 .18101 .22102 1 .16165 .19281 .17129 .20669 .18118 .22127 2 .15241 .17982 .16181 .19304. .17145 .20693 .18135 .22152 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 5 .15272 .18025 .162.12 .19349 .17178 .20740 .18168 22202 6 .15288 .18047 .16228 .19372 .17194 .20764 .18185 22227 6 • 7 .15303 .18068 .16244 .19394 .17210 .20788 .18202 .22252 8 .15319 .18090 .16260 .19417 .17227 .20812 .18218 .22277 9 .15334 .18111 10 .15350 .18133 .16276 .19440 .16292 .19463 .17243 .20836 .18235 22302 9 • .17259 .20859 .18252 .22327 10 11 .15365 .18155 .16308 .19485 17276 .20883 .18269 .22352 11 12 15381 .18176 .16324 .19508 .17292 .20907 .18286 .22377 12 13 .15396 .18198 .16340 .19531 .17308 .20921 • .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 .20955 .183.9 22428 14 15 .15427 .18241 .16371 .19576 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18353 22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .18369 .22503 17 18 .15474 .18307 .16419 .19645 .17390 .21051 .18386 .22528 18 19 .15189 .18328 .16435 .19668 .17407 .21075 .18403 .22551 20 .15505 .18350 .16451 .19691 .17423 .21099 .18420 .22579 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 .17472 .21171 .18470 .22655 23 .15567 .18437 .16515 .19782 .17489 .21195 .18487 .22680 .15583 .18459 .16531 .19805 .17505 .21220 .18504 22706 25 .15598 .18481 .16547 .19828 .17522 .21244 .18521 .22731 26 27 .15614 .18503 .16563 .19851 .17538 .21268 .18538 .22756 28 .15630 .18525 .16579 .19874 .17551 .21292 .18555 22782 28 29 .156-15 .18547 .16595 .19897 .17571 .21316 .18572 22807 29 30 .15661 .18569 .16611 .19920 .17587 .21341 .18588 22833 30 31 .15676 .18591 32 .15692 33 .15708 .18635 .16627 .19944 .17604 .21365 .18605 22858 .18613 .16644 .19967 .17620 .21389 .18622 22884 .16660 .19990 .17637 .21414 .18639 .22909 34 .15723 .18657 .16676 .20013 .17653 .21438 .18656 22935 34 35 .15739 .18679 .16692 .20036 .17670 .21462 .18673 .22960 36 .15755 .18701 .16708 .20059 .17686 .21487 .18690 .22986 37 .15770 .18723 38 .15786 .18745 39 .15802 .18767 40 .15818 .18790 .16724 .20083 .17703 .21511 .18707 .23012 37 .16740 .20106 .17719 .21535 .18724 .23037 .16756 .20129 .17736 .21560 .18741 23063 .16772 .20152 .17752 .21584 .18758 .23089 40 41 .15833 .18812 .16788 .20176 42 .15849 .18834 .16805 .20199 43 .15865 .18856 .16821 .202:22 .18878 52 888888 44 .15880 45 .15896 .18901 46 .15912 .18923 .16869 .20292 47 .15923 .18945 .16885 .20316 48 .15943 .18967 .16902 20339 49 .15959 .18990 .16918 .20363 50 .15975 .19012 51 .15991 .19034 .16950 .16006 .19057 .16966 .20433 53 .16022 .19079 .16983 .20457 54 .16038 .19102 .16999 .20480 55 .16054 .19124 .17015 .20504 56 .16070 .19146 .17031 20527 57 .16085 .19169 .17047 .20551 .16837 20246 .16853 .20269 .17769 .21609 .17786 .21633 .17802 .21658 .17819 .21682 .17835 .21707 .17852 .21731 .17868 .21756 .17885 .21781 .18775 .23114 41 .18792 .23140 42 .18809 .23166 43 .18826 .23192 .18843 .23217 45 .17902 .16934 .20386 .17918 .21805 .21830 .18860 .23243 46 .18877 .23269 .18894 .23295 .18911 .23321 47 .18928 .23347 50 .20410 .17935 .21855 .17952 .21879 .17968 .21904 .17985 .18001 .21953 .18018 .21978 .18945 .23373 51 .18962 .23399 .18979 .23424 53 52 FAQ HARIRAKKA. H831386889 =99‡494898 ☎0B 19 20 32 35 .21929 .18996 .23450 54 .19013 .23476 55 .19030 .23502 56 .18035 .22003 .19047 .23529 57 58 60 .16133 .19236 .16101 .19191 .17064 .20575 .18051 .22028 59 .16117 .19214 .17080 20598 .18068 .22053 .17096 .20622 .18085 .22077 .19064 .23555 58 .19081 23581 59 .19098 23607 60 329 TABLE XIII.-VERSINES AND EXSECANTS. 36° 37° 38° 39° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 016100 10 .19098 .23607 .20136 .25214 .19115 .23633 .20154 .25241 .21199 .26902 22285 ,28676 0 .19133 .23659 .20171 .25269 .21217 .26931 .21235 22304 .28706 1 .26960 22322 .28737 2 .19150 .23685 .20189 .25296 .21253 .26988 22340 .28767 3 .19167 .23711 20207 .25324 .21271 .27017 .22359 .28797 5 .1918 .23738 .20224 .25351 .21289 .27046 .22377 .28828 .19201 .23764 .20242 .25379 .21307 .27075 .22395 .28858 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 .21860 .27162 .22450 .28950 9 10 .19270 23869 .20312 .25489 .21378 .27191 .22469 .28980 10 11 .19287 .23895 20329 .25517 .21396 .27221 .22487 .29011 11 12 .19304 .23922 .20347 .255-15 .21414 .27250 22506 .29042 12 13 .19321 .23948 .20365 .25572 11 .19338 .23975 20382 .25600 15 .19356 .24001 20400 .25628 .21432 .27279 .21450 .27308 .21468 .27337 22524 .29072 13 .22542 .29103 14 22561 .29133 15 16 .19873 .24028 .20417 .25656 .21486 .27366 22579 29164 16 • 17 .19390 .24054 .20435 .25683 .21504 .27396 18 .19407 .24081 .20453 .25711 .21522 .27425 21 29 30 31 22 ZARTA.NAR. H 19 .19424 .24107 .20470 .25739 .21540 .27454 20 .19442 .24134 .20488 .25767 .21558 .27483 22598 .29195 17 .22616 .29226 18 .22634 .29256 19 .22653 .29287 20 .19459 .24160 .20506 .25795 .21576 .27513 .19476 .24187 .205.23 .25823 .21595 .27542 .22671 .29318 21 .22690 .29349 22 .19493 .24213 .20541 .25851 .21613 .27572 .22708 .29380 23 24 .19511 24240 .20559 .25879 .21631 25 26 .19528 .19545 .24293 .20594 .24267 .20576 .25907 .27601 .22727 .21649 .27630 .29411 24 22745 .29442 25 .25925 .21667 .27660 .22764 .29473 26 27 .19562 .24320 .20612 .25963 .21685 .27689 .22782 .29504 27 28 .19580 .24347 .20629 .25991 .21703 .27719 .19597 .24373 .20647 .26019 .21721 .27748 • .19614 .24400 .20665 .26047 .21739 27778 • .19632 .24427 .20682 .26075 .21757 .27807 .22801 .29535 28 22819 22838 .29597 30 .22856 29628 31 .29566 29 32 .19649 .24454 .20700 .26104 .21775 .27837 22875 .29659 32 33 .19666 .24481 .20718 .26132 .21794 .27867 22893 .29690 33 22 FRRIHQNAR. H83 34 .19684 .24508 .20736 .26160 .21812 .27896 22912 .29721 3-4 35 .19701 .24534 .20753 .26188 .21830 .27926 22930 .29752 35 36 .19718 24561 20771 .26216 .21848 .27956 22949 .29784 36 37 .19736 .24588 20789 .26245 .21866 .27985 .22967 .29815 37 38 .19753 .24615 .20807 .26273 .21884 .28015 22986 .29846 38 89 588#995998 ERBE 39 .19770 .21642 20824 .26301 40 .19788 .24669 .20842 .21902 .28045 .23004 .29877 39 41 .19805 .24696 .20860 42 .19822 .24723 .26330 .26358 21921 .28075 .23023 .29909 40 • .21939 .28105 .23041 .29940 41 43 .19840 .24750 .20878 .26387 20395 .21957 .28134 .23060 .29971 42 .26115 .21975 .28164 .23079 .30003 43 44 .19857 .24777 .20913 .26143 .21993 .28194 .23097 .30034 44 45 .19875 .24804 .20931 .26472 .22012 .28224 .23116 .30066 45 46 .19892 .24832 .20949 .26500 47 20967 .26529 49 .19909 .24859 48 .19927 .24886 .20985 26557 .19944 .24913 .21002 .26586 50 .19962 .24940 .21020 .26615 .19979 51 .24967 .21038 .266-13 52 .19997 .24995 :21056 .26672 53 .20014 .25022 .21074 .26701 .20032 .25049 .21092 .26729 .20049 .25077 .21109 .26758 .21127 .30097 46 .23153 .30129 47 .23172 .30160 48 .23190 .30192 49 23209 .30223 50 .23228 .30255 51 .23246 .30287 52 .22157 .28464 23265 .30318 53 .22176 .28495 23283 .30350 54 .22194 .28525 .23302 .30382 55 22212 .28555 .23321 .22231 .28585 .23339 22249 .28615 23358 22267 .28646 .23377 .30509 59 .22285 .28676 23396 .30541 60 54 55 56 .20066 .25104 57 .20084 .25131 .21145 60 .20136 .25214 .26787 .26815 .21163 .26844 .26873 .21199 .26902 • 58 .20101 25159 59 .20119 .25186 .21181 888 .30413 56 .30445 57 .30477 58 699 .22030 .28254 .23134 .22048 .28284 · .22066 .28314 .22084 .28344 22103 .28374 22121 .28404 .22139 28434 **** HO*X 330 TABLE XIII.—VERSINES AND EXSECANTS. 40° 41° 42° 43° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .23396 .30541 .24529 .32501 1 .23414 .30573 .24548 .32535 .25686 .34563 .25705 .34599 .26865 .36733 0 .26884 .36770 1 2 .23433 .30605 .24567 .32568 .25724 .34634 .26904 .36807 2 3 .23452 .30036 .24586 .32602 .25744 .34669 .26924 .36844 3 4 .23470 .30668 .24605 .32636 .25703 .34704 .26944 .36881 5 .23489 .30700 .24625 .32009 .25783 .34740 .26964 .36919 .23508 .30732 24644 .32703 .25802 .34775 .26984 .36956 6 77 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 32770 .25841 .34846 .27024 .37030 8 9 .23564 .30829 .24701 32804 .25861 .34882 .27043 .37068 9 10 .23583 .30861 .24720 .32838 .25880 .34917 .27063 .37105 10 11 .23602 30893 .24739 .32872 .25900 .34953 .27083 .37143 11 12 .23620 .30925 .24759 .32905 .25920 .34988 .27103 .37180 12 13 .23639 .30957 .24778 .32939 .25939 .35024 .27123 .37218 13 14 .23658 .30989 .24797 .32973 .25959 35060 .27143 .37255 14 15 .23677 .31022 .24816 .33007 .25978 .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 .37-406 18 19 .23752 .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 .27283 .37519 21 22 .23808 .31248 .2-1950 .33245 .26115 .35346 .27303 .37556 22 23 .23827 .31281 .24970 .33279 .26135 .35382 .27323 .37594 23 24 .23816 .31313 .24989 .33314 .26154 .35418 .27343 .37632 2-1 25 .23865 .31346 .25008 .33348 .26174 .35454 .27363 .37670 26 23884 .31378 .250.27 .33382 .26194 .35490 27 .23903 .31411 .25047 .33416 .26213 .27383 .37708 26 .35526 .27403 .37746 28 .23922 .31443 .25066 .33451 .26233 .35562 .27423 .37784 29 .23941 .31476 .25085 .33485 .26253 .35598 .27443 .37822 20 30 .23959 .31509 .25104 .33519 .26272 .35634 27463 .37860 30 31 .23978 .31541 32 .23997 .81574 .25124 .33554 .26292 35670 .27483 .37898 31 22 728ANRARQA H 25 .25143 .33588 .26312 .85707 .27503 .37936 32 33 .24016 .31607 .25162 .33622 .26331 .35743 .27523 .37974 33 3-1 .24035 .31610 .25182 .33057 .26351 .35779 .27543 .38012 3-4 35 24054 .31672 .25201 .33691 .26371 .35815 .27563 .38051 35 36 .24073 .31705 .25220 .33726 .26390 .35852 .27583 .38089 36 37 .24092 .31738 .25240 .33760 .26410 .35888 .27603 .38127 37 41 389 ES33995988 N231285888 .24111 .81771 .25259 .33795 .26430 .35924 .27623 .38165 33 40 .24130 .31804 .25278 .24149 .31837 .25297 .33830 .26449 .35961 .27643 .38204 39 .33864 .26469 .35997 .27663 .38242 40 .24168 .31870 .25317 33899 .26-189 ,36034 .27683 .38280 41 43 .24206 .31936 44 42 .24187 .31903 .25336 .33934 .25356 .33968 .24225 .31969 25375 .34003 45 .24244 32002 .25394 .34038 .26509 .36070 .27703 .38319 42 .26528 .36107 .27723 .38357 43 .26548 .30143 .27713 .38396 44 .26568 .36180 .27764 .38434 45 46 .24262 .32035 .25414 .34073 .26588 .36217 .27784 .38473 46 47 .24281 .32008 .25433 .34108 .26607 .36253 .27804 .38512 47 49 50 48 .24300 .32101 .25452 .34142 .24320 .32134 .25-472 .34177 .24339 .32168 .25491 .34212 .24858 .32201 .25511 .34247 .24377 .32234 25530 34282 .24396 .32207 .255-19 .34317 .26627 .36290 .27824 .38550 48 .26647 .36327 .27844 .38589 49 .26667 .36363 .27864 .38628 50 .26686 .36400 .26706 .36437 .27884 .38660 51 .27905 .38705 52 .26726 .36474 .27925 .38744 53 54 .24415 .32301 .25569 55 .24434 .82334 .25588 .34387 56 .24153 .32368 .25608 .34423 57 .24472 .32401 25627 .31458 .24491 .32434 .25647 .34493 59 .24510 .32468 .25666 .3-1528 .26845 .34352 .26746 .36511 .27945 .38783 5-1 .26766 .36548 .27965 .38822 55 .26785 .36585 .27985 .38860 56 .26805 .36622 .28005 .38899 57 .26825 .36659 28026 .38938 58 .36696 .28046 .38977 59 60 .24529 32501 .25686 .3-1563 .26865 .36733 .28066 .39016 60 331 TABLE XIII.—VERSİNES AND EXSECANTS. 44° 45° 46° 47° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .28066 .39016 .29289 .41421 1 .28086 .39055 .29310 .41463 2 .28106 .39095 .29330 .41504 .30534 .43956 .30555 .43999 .30576 .44042 .31800 .46628 0 .31821 .46674 1 .31843 .46719 2 3 .28127 .39134 .29351 .41545 .30597 .44086 .31864 .46765 .28147 .39173 29372 .41586 .30618 .44129 .31885 .46811 .28167 .39212 .29392 .41627 .30639 .41173 .31907 .46857 .28187 39251 .29413 .41669 .30660 .44217 .31928 .46903 .28208 .39291 .29433 .41710 .30681 .44260 .31949 .46949 28228 .39330 .29-154 .41752 .30702 .44304 .31971 .46995 9 .28248 .39369 .29475 .41793 .30723 .448347. .31992 .47041 9 10 .28268 .39409 .29495 .41835 .30744 .44391 .32013 .47087 10 11 28289 .39448 .29516 .41876 .30765 .44435 .32035 .47134 11 12 .28309 .39487 .29537 .41918 .30786 .44179 .32056 .47180 12 13 283:29 .39527 .29557 .41959 .30807 .44523 .32077 .47220 13 14 .28350 .39566 .29578 .42001 .30828 .44567 .32099 .47272 14 15 .28370 .39606 .29599 .42042 .30849 44610 .32120 .47319 15 B 16 .28390 .39646 .29619 .42084 .30870 .44654 .32141 .47365 16 17 .28410 .39685 .296-10 .42126 .30891 .41698 .32163 .47411 17 18 .28431 .39725 .29661 .42168 .30912 .44742 .32184 .47458 18 19 .28-451 .39764 .29681 .42210 .30933 .44787 .32205 .4750-1 19 20 .28471 .39804 .29702 .42251 .30954 .44831 .32227 .47551 20 21 .28492 .39844 .29723 42293 .30975 .44875 .32248 .47598 21 22 .28512 .39384 .29743 42335 .30996 .44919 .32270 .47644 22 23 .28532 .39921 .29764 .42377 .31017 .44963 .32291 .47691 23 24 .28553 .39963 .29785 .42419 .31038 .45007 .32312 .47788 24 25 26 .28593 .40043 .28573 .40003 .29805 .42461 .31059 .45052 .32334 .47781 25 .29826 .42503 .31080 .45096 .32355 .47831 26 27 .28614 .40083 .29847 42545 .31101 .45141 .32377 .47878 27 .28634 .40123 .29868 42587 .31122 .45185 .32398 .47925 28 • .28655 .40163 .29888 .42630 .31143 .45229 .32420 .47972 29 30 .28675 .40203 .29909 .42672 .31165 .45274 .32441 .48019 30 31 .28695 .40243 .29930 .42714 .31186 .45319 .32462 .48066 31 32 .28716 .40283 33 .28736 .40324 34 .28757 .40364 35 .28777 .40404 .29951 .42756 ,31207 .45363 .32-184 .48113 32 .29971 .42799 .31228 .45408 .32505 .48160 33 22 FROHA❤NAR. 788. .29992 .42841 .31249 .45452 .32527 .48207 34 .30013 .42883 .81270 .45-497 .32548 .48254 35 36 .28797 .40414 .30034 .42926 .31291 .45542 .32570 .48301 36 37 .28818 .40485 .30054 .42968 .31312 .45587 .32591 .48349 37 889 ERRI#999988 EBRERAKOAS 38 .28838 .40525 .30075 .43011 .31334 .45631 39 .28359 .40565 .30006 .43053 40 .28879 .40606 .30117 .43096 41 .28900 .40646 .30138 .43139 42 .28920 .40687 .30158 .43181 44 .28961 .40768 45 .28981 .40808 46 .29002 43 .28941 .40727 .30179 .43224 .30200 .43267 .30221 .43310 .40849 .30242 .43352 47 .29022 .40890 .30263 .43395 48 .29043 .40930 .30283 .43438 .31355 .45676 .31376 .45721 .31397 .45766 .31418 .45811 .31439 .45856 .31461 .45901 .31482 .45946 .31503 .45992 .31524 .46037 .31545 .46082 .32613 .48396 38 .32634 .32656 .48491 40 .48443 39 49 .29063 50 • .40971 .29084 .41012 .30325 .43524 .30304 .43481 51 .20104 57 .41053 .30346 .43567 52 .29125 .41093 .30367 .43610 53 .29145 .41134 .30388 .43653 54 .29166 .41175 .30-109 .43696 55 .29187 .41216 .30430 .43739 56 29207 .41257 .30451 .43783 .29228 .41298 .30471 .43826 58 .29248 .41339 .30-192 .43869 59 .29269 .41380 .30513 .43912 .31779 60 .29289 .41421 .30534 .43956 .31567 .46127 .31588 .46173 .31609 .46218 .32893 .49015 51 .31630 .46263 .32914 .49063 .31651 .46309 .32936 .49111 53 .31673 .40354 32957 .49159 54 .31694 .46400 .32979 .31715 .46445 .32677 .48538 41 .32699 .48586 42 .32720 .48633 .32742 .48681 44 .32763 .48728 .32785 .48776 46 .32806 .48824 47 .32828 .48871 .32849 .48919 .32871 .48967 88% #38I995438 H2S) 50 52 .49207 55 .33001 .49255 56 .31736 .46491 33022 .49303 57 .31758 .46537 33044 .49351 58 • .46582 .31800 .46628 33065 .49399 59 .33087 .49448 60 332 TABLE XIII.-VERSINES AND EXSECANTS. 48° 49° 50° 51° · Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. OHQ 0 .33087 .49448 34394 .52425 .35721 .55572 .37068 .58902 0 1 .33109 .49496 .34416 .52476 .35744 .55626 .37091 .58959 1 .33130 .49544 .34438 .52527 .35766 .55680 .37113 .59016 2 .33152 .49593 .34460 .52579 .35788 .55734 .37136 .59078 3 .33173 .49641 .34482 .52630 .35810 55789 .37158 .59130 + .49835 9 .33195 .49690 .33217 .49738 .34526 .33238 .49787 .33260 .33282 .34504 .52681 35833 .55843 .37181 .59188 • .52732 35855 .55897 .37204 .59245 .34548 .52784 .35877 .55951 .37226 .59302 .34570 .52835 .35900 .56005 .37249 .59360 8 19884 .34592 .52886 .35922 .56060 .37272 .59418 9 10 .33303 .49933 .34614 .52938 .35941 .56114 .37294 .59-475 10 11 .33325 .49981 .34636 .52989 .35967 .56169 .37317 .59533 11 12 .33347 .50030 .34658 .53041 .35989 .56223 .37340 .59590 12 13 .33368 .50079 .34680 .58092 .36011 .56278 .37362 .59648 13 14 .33390 .50128 .34702 .53144 .36031 .56332 .37385 .59706 14 15 .33-412 .50177 .34724 .53196 .36056 .56387 .37408 .59764 15 16 .33434 .50226 .34746 .53247 .36078 .56412 .37430 .59822 16 17 .33455 .50275 .34768 .53299 .36101 .56497 .37453 .59880 17 20 PAR ENRI 18 .33477 .50324 .34790 .53351 .36123 .56551 .37476 .59938 18. 19 .33499 .50373 .34812 .53403 .36146 .56606 .37498 .59996 19 .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 .37507 .60171 22 .33586 .50570 .34900 .33607 .50619 25 .33629 .50669 .34945 .53611 .36235 .56826 .37589 .60229 23 .34923 .53663 .36258 .56881 .37612 .60287 24 .53715 .36280 .56937 .37635 .60346 25 26 .33651 .50718 .31967 .53768 .36302 .56992 .37658 .60404 26 31 INARA HAA7 27 .33673 .50767 .34989 .53820 .36325 .57047 .37680 .60463 27 28 .33694 .50817 .35011 .53872 .36347 .57103 .37703 .60521 28 29 .33716 50866 .35083 .53924 .36370 .57158 .37726 .60580 29 30 .33738 .50916 .35055 .53977 .36392 .57213 .37749 .60639 30 .33760 .50966 .35077 .54029 .36415 .57269 .37771 .60698 31 32 .33782 .51015 .35099 .54082 .36437 .57324 37794 • 33 .33803 .51065 .35122 .54134 .36460 .57380 .60756 32 .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 37 .33891 .51265 .35188 .54292 .36527 .57547 .37885 .60992 36 .35210 .51345 .365-19 .57603 .37908 .61051 37 38 .33912 .51314 .35232 .54398 .36572 .57659 .37931 .61111 38 43 46 8G =333995998 N88) 39 .33934 .51364 .35254 .51151 .36594 .57715 .37954 .61170 39 40 .33956 .51415 .35277 .51504 .36617 .57771 .37976 .61229 40 41 .33978 .51465 .35299 .54557 .36639 .57827 .37999 .61288 41 42 .34000 .51515 .35321 .54610 .34022 .51565 .36662 .57883 .38022 .61348 42 .35343 .54663 .36684 .57939 .38045 .61107 43 44 .34044 .51615 .35365 .54716 .36707 .57995 .38068 .61467 44 45 .31065 .51665 .35388 47 49 .34153 .51867 50 .34175 .51918 .54769 .341087 .51716 .35410 .54822 .34109 .51766 .35432 .54876 48 .34131 .51817 .35-154 .54929 .54982 .55036 .35476 .36729 .58051 .36752 .58108 .36775 .58164 .36797 .58221 .36820 .58277 .38091 .61526 45 .38113 .61586 46 .38136 .61646 47 .38159 .61705 48 .38182 .61765 49 .35499 .36842 .58333 .38205 .61825 50 51 .34197 .51968 .35521 .55089 .36865 .58390 38228 .61885 51 52 .34219 .52019 .35543 .55143 .36887 .58417 .38251 .61945 52 53 .34241 .52069 .35565 54 .34262 .52120 69 59 .34372 60 34394 .55196 .35588 .55250 55 .34284 .52171 .35610 .55303 56 .34306 52222 .35632 .55357 57 .34328 .52273 .35654 .55411 58 .34350 .52323 .35677 .55465 .37023 .52374 .35699 .55518 .370-15 .52425 .35721 .55572 .37068 .36910 .58503 38274 .62005 53 .36932 .58560 38296 .62065 54 .36955 .58617 38319 .62125 55 • .36978 .58674 38342 .62185 56 .37000 .58731 .38365 .62246 57 .58788 .38388 .62306 58 .58845 38411 .62266 59 .58902 38434 .62427 60 333 TABLE XIII.-VERSINES AND EXSECANTS. 52° 53° 54° 55° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. CHOO JLO CO 0 .38434 .62427 .39819 .66164 .38457 .62487 .39842 .66228 .41221 .70130 42642 .41245 .70198 74345 0 42666 .74417 1 .38480 62548 .39865 .66292 .41269 .70267 .42690 .74490 3 .38503 .62609 .39888 .66357 .41292 .70335 .42714 .74562 4 .38526 .62669 .39911 .66421 .41316 .70403 .42738 .74635 5 .38549 .62730 .39935 .66486 .41339 .70472 .42762 .74708 6 .38571 .62791 .39958 .66550 · .41363 .70540 77 .38594 .62852 .39981 .66615 .41386 .70609 8 .38617 .62913 .40005 66679 .41410 .70677 9 .38640 .62974 .40028 .66744 .41433 .70746 10 .38663 .63035 .40051 .66809 .41457 .70815 .42785 .74781 .42809 .74854 .42833 .74927 8 .42857 .75000 42881 .75073 10 11 .38686 63096 .40074 .66873 .41481 .7088-1 .42905 75146 11 12 .38709 .63157 .40098 .66938 .41504 .70953 .42929 .75219 12 13 .38732 .63218 .40121 .67003 .41528 .71022 .42953 .75293 14 .38755 .63279 .40144 .67068 .41551 .71091 .42976 .75366 14 15 .38778 .63341 .40168 .67133 .41575 .71160 .43000 75440 15 16 .38801 .63402 .40191 .67199 .41599 .71229 .43624 .75513 45 48 51 55 56 CARR FARAHUNAQA H****808* =33#994990 788758 17 .38824 .63464 .40214 .67264 .38847 .63525 .67329 .38870 63587 .40261 .67394 .40237 20 .38893 .63648 .40284 .67460 .41622 .41646 .71368 .41670 .71437 .41693 .71506 .71298 .43048 75587 .43072 .75661 .43096 .75734 22 21 .38916 .38939 .38962 63710 .40307 .67525 .41717 .71576 .43120 .43144 .75808 20 .75882 21 .63772 .40331 .67591 .41740 .71646 .43168 .75956 22 63834 .40354 .67656 • .41764 .71715 .43192 .76031 23 24 .38985 63895 .40378 .67722 .41788 .71785 .43216 .76105 .39009 .63957 .40401 .67788 .41811 .71855 .43240 .76179 25 26 .39032 .64019 .40424 .67853 .41835 .71925 .43264 .76253 27 28 .39078 .64144 29 .39101 .64206 .39055 .64081 .40448 .67919 .40471 .40494 .68051 .41859 .71995 .43287 .76328 27 .67985 .41882 72065 .43311 .76402 28 .41906 .72135 43335 .76477 29 30 .39124 .64268 .39147 .64330 .40541 32 .39170 .64393 .40565 .40518 .68117 .41930 ..72205 .43359 .76552 30 .68183 .41953 .72275 .43383 .76626 .68250 .41977 72346 .43407 .76701 32 33 .39193 .64455 .40588 .68316 34 .39216 .64518 .40611 .68382 .42001 .72416 .42024 .43431 .76776 33 .72487 .43455 .76851 35 .39239 .64580 .40635 .68449 .42048 .72557 .43479 .76926 36 .39262 .64643 .40658 .68515 .42072 .72628 .43503 .77001 36 .39286 .64705 .40682 68582 .42096 .72698 .43527 .77077 38 .39309 .64768 .40705 .68648 .42119 72769 .43551 .77152 38 39 .39332 .64831 .40728 .68715 .42143 .72840 .43575 .772 39 .39355 .64894 .39378 .64957 .40775 .39401 .65020 .39424 .65083 .40752 .68782 .42167 .72911 .43599 .77303 .68848 .42191 72982 .43623 .77378 .40799 .68915 .40822 .42214 .73053 .43647 .77454 42 .68982 .42238 41 .39447 .65146 .408-16 .69049 .42262 39-471 .65209 .40869 .69116 .42285 73124 .43671 .73195 .43695 73267 .77530 43 .77606 .43720 .77681 .39494 .65272 .40893 .69183 .42309 73338 .43744 .77757 46 .39517 .65336 .40916 .69250 .42333 73409 .43768 .77833 47 .39540 .65399 .40939 .69318 .42357 73481 43792 .77910 48 .39563 .65462 .40963 .69385 .42381 .73552 .43816 .77986 49 .39586 .65526 .40986 .69452 .42404 .73624 .43840 78062 .39610 .65589 .41010 .69520 .42-128 .73696 .43864 .78138 51 52 .39633 .65653 .41033 .69587 .42452 .73768 .43888 .78215 52 53 .39656 .65717 .41057 .69655 .42476 .73840 .43912 .78291 53 ROTONDOO FRATERERAR FRKEARNAR. HABIASH889 =333999998 F8% 54 .39679 .65780 .41080 .69723 .42499 73911 .43936 .78368 5-1 57 .39702 .65844 .41101 .69790 .42523 39726 .65908 .41127 .69858 .42547 .39749 .65972 .41151 .69926 .42571 .74128 58 .39772 .66036 .41174 .69994 .42595 .74200 59 .39795 .66100 .41198 .70062 42619 60 .39819 .66164 41221 .73983 .43960 .78445 55 74056 .43984 .78521 56 .44008 78598 57 .44032 78675 58 • .74272 .44057 .78752 59 .70130 .42642 74345 .41081 78829 60 334 TABLE XIII.—VERSINES AND EXSECANTS. 56° 57° 58° 59° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .44081 .78829 .45536 .83608 .47008 88708 .48496 .94160 0 1 .44105 .78906 .45560 .83690 .47033 88796 .48521 .94254 2 .44129 .78981 .45585 83773 .47057 88881 • .48546 .94349 3 .44153 .79061 .45609 .83855 .47082 88972 .48571 .94143 3 .44177 .79138 .45634 .83938 .47107 89060 48596 .94537 .44201 .79216 .45658 .84020 .47131 .89148 .48621 .94632 .44225 .79293 .45683 .84103 .47156 .89:237 48646 .94726 7 .44250 79371 .45707 .84186 .47181 .89325 .48671 .94821 • 8 .44274 .79449 .45731 .84269 .47206 .89414 .48696 .94916 8 9 .44298 .79527 .45756 .84352 .47230 .89503 .48721 .95011 9 10 .44322 .79604 .45780 .81135 .47255 .89591 .48746 .95106 10 11 .44346 79682 .45805 .84518 .47280 .89680 .48771 .95201 11 12 .41370 79761 .45829 .84601 .47304 .89769 .48796 .95296 12 13 .44395 .79839 .45854 .84685 .47329 .89858 .48821 .95392 13 14 .44113 .79917 .45878 .84768 .47354 .89948 .48846 .95487 14 15 .4448 .79995 .45903 .84852 .47379 .90037 .48871 .95583 15 16 .44167 .80074 .45927 .84935 .47403 .90126 .48896 .95678 16 17 .44491 .80152 .45951 .85019 .47428 .90216 .48921 .95774 17 41 42 44 47 52 :982 233HRANARA 2831986889 #8**99***8 NNN 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 .90185 .48996 .96062 20 21 .44588 .80467 .46049 .85355 .47527 .90575 .49021 .96158 21 .44612 .80546 .46074 .85439 .47552 .90665 .49046 .96255 26 28 30 .44806 .81180 .44637 .80625 .46098 .85523 .44661 .80704 .46123 .85608 .44685 .80783 .46147 .85692 .44709 .80862 .46172 .85777 27 .44734 .80942 .46196 .85861 .44758 .81021 .46221 .85946 29 .44782 .81101 .46246 86031 .46270 .47577 .90755 .49071 .96351 23 .47601 .47626 .90845 .90935 .47651 .91026 .47676 .91116 .47701 .91207 .47725 .91297 .49096 .96418 24 .49121 .96544 25 .49146 .96641 .49171 .96738 27 .49196 .96835 .49221 .96932 29 .86116 .47750 .91388 .49246 .97029 30 35 .44831 .81260 .44855 .81340 .46319 .86286 33 .44879 .81419 .46344 .86371 .41903 .81499 .46368 .86457 .44928 .81579 .46393 .86542 .44952 .81659 .46417 .46295 .86201 .47775 .91479 .49271 .97127 31. .47800 .91570 .49296 .97224 32 .47825 .91661 .49321 .97322 33 .86627 .47849 .91752 .47874 .91844 .47899 .91935 .49346 .97420 34 .49372 .97517 35 CHAR 288*Hakak. 78871 26 .49397 .97615 36 37 .44976 .81740 .46442 .86713 .47924 92027 .49422 .97713 37 .45001 .81820 .46466 .86799 .47949 .92118 .49447 .97811 38 .45025 .81900 .46491 .86885 .47974 92210 .49472 .97910 40 .45049 .81981 .46516 .86990 .47998 .92302 .49497 .98008 .45073 .82061 .46540 .45098 .82142 .87056 .48023 92394 .49522 .98107 .46565 .87142 .48048 92486 43 45122 .82222 • .46589 .87229 .48073 .92578 .45146 .82303 .46614 .87315 .48098 48 .45244 .82627 .46712 49 .45268 .82709 50 .45292 .82790 .46737 .46762 45 .45171 .82384 .46039 .87401 46 .45195 .82465 .46663 .87488 .45219 .82546 .46688 .87574 .48172 .92947 .87661 .48197 .93040 .87748 .48222 93133 .87834 .48247 .93226 .48123 .48148 .92670 92762 .49623 .98502 .92855 .49618 .98601 .49517 .98205 .49572 .98304 .49597 .98403 .49673 .98700 47 .49698 .98799 .49723 .98899 49 .49748 .98998 50 51 .45317 .82871 .45341 .82953 .46786 .87921 .48272 .93319 .46811 88008 .48297 .93412 53 .45365 .83034 .46836 .88095 .48322 + 54 .45390 .83116 .46860 .88183 .483-47 52 93505 .49824 .99298 53 .93598 .49773 .99098 .49799 .99198 JUO OAARAL FEEL A COC 39 40 41 42 43 44 45 46 48 .49849 99398 54 55 .45414 .83198 .46885 56 .45439 .83280 .46909 57 .45463 .83362 88270 .48372 93692 .49874 • .99498 55 .88357 .48396 .93785 .49899 .99598 56 .46934 88145 .48421 93879 .49924 .99698 57 58 .45487 .83414 .46959 88532 .48416 .93973 .49950 .99799 58 59 .45512 .83526 60 .45536 46983 88620 .48471 .94066 .49975 .99899 83608 47008 88708 .48496 .94160 .50000 1.00000 383 59 335. TABLE XIII.-VERSINES AND EXSECANTS. 60° 61° 62° 63° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0.50000 1.00000 .51519 1.06267 .53053 1.13005 .54601 1.20269 0 1 .50025 1.00101 .51511 1.06375 .53079 1.13122 .54627 1.20395 1 21.50050 1.00202 .51570 1.06483 .53101 1.13239 .54653 1.20521 2 3.50076 1.00303 .51595 1.06592 .53130 1.13356 .54679 1.20647 3 .50101 1.00404 .51621 1.06701 .53156 1.13473 .54705 1.20773 4 .50126 1.00505 .51646 1.06809 53181 1.13590 .54731 1.20900 5 .50151 1.00607 .51672 1.06918 .53207 1.13707 .54757 1.21026 6 .50176 1.00708 .51697 1.07027 53233 1.13825 .54782 1.21153 77 • 8 .50202 1.00810 .51723 1.07137 53258 1.13942 .54808 1.21280 8 9.50227 1.00912 .51748 1.07246 53284 1.14060 .54834 1.21407 9 10 .50252 1.01014 .51774 1.07356 .53310 1.14178 .54860 1.2153510 11 .50277 1.01116 .51799 1.07465 53336 1.14296 .54886 1.21662 11 12.50303 1.01218 .51825 1.07575 .53361 1.14414 .54912 1.2179012 13 .50328 1.01320 .51850 1.07685 .53387 1.14533 .54938 1.21918 13 14 .50353 1.01422 .51876 1.07795 .53413 1.14651 .54964 1.22045 14 15.50378 1.01525 .51901 1.07905 .53439 1.14770 .51990 1.22174 15 16 .50404 1.01628 .51927 1.08015 .53464 1.14889 .55016 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 1.2255918 19 .50479 1.01936 .52003 1.08347 .53542 1.15246 .55094 1.2268819 20 .50505 1.02039 .52029 1.08458 .53567 1.15366 .55120 1.2281720 21 .50530 1.02143 .52054 1.08569 .53593 1.15485 .55146 1.22946 21 22 .50555 1.02246 .52080 1.08680 .53619 1.15605 .55172 1.2307522 23 .50581 1.02349 .52105 1.08791 .53645 1.15725 .55198 1.23205 23 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 1.23464 25 26 .50656 1.02661 .52182 1.09126 53722 1.16085 .55276 1.23594 26 27 .50682 1.02765 .52207 1.09238 .53748 1.16206 .55302 1.23724 27 28.50707 1.02869 .52233 1.09350 .53774 1.16326 .55328 1.23855 28 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 1.2437832 33.50834 1.03391 .52361 1.09911 .53903 1.16932 .55458 1.2450933 34 50859 1.03496 .52386 1.10024 .53928 1.17053 .55484 1.24640 34 35 .50884 1.03601 .52412 1.10137 .53954 1.17175 .55510 1.24772 35 36 .50910 1.03706 .52438 1.10250 .53980 1.17297 .55536 1.24903 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 38 140 39 .50986 .51011 41 51036 1.04022 .52514 1.10590 .54058 1.17663 .55615 1.2530039 1.04128 .52540 1.10704 .54083 1.17786 .55641 1.25432 40 1.04233 42 .51062 44 43 .51087 .51113 45 .51138 46 .51163 1.04764 47 .51189 1.04870 18 .51214 1.04977 49 51 993 721 .52566 1.10817 1.04339 .52591 1.10931 1.04445 .52617 1.11045 1.04551 1.11159 .52642 1.04658 .52668 1.11274 .52694 1.11388 .52719 1.11503 .52745 1.11617 .54109 1.17909 55667 1.25565 41 54135 1.18031 .55693 1.25697 42 .54161 1.18154 55719 1.2583043 .54187 1.18277 .55745 1.25963 44 .54213 1.18401 .54238 1.18524 .54264 .55771 1.26097 45 55797 1.26230 46 .55823 1.18648 1.26364 47 .54290 1.18772 .55849 1.26498 48 .51239 1.05084 .52771 1.11732 .54316 1.18895 .55876 1.26632 49 50 .51265 1.05191 .52796 1.11847 .54342 1.19019 .55902 1.26766 50 52.51316 53 54 56 58 .51417 57 .51443 .51468 .51290 1.05298 .52822 1.11963 1.05405 .52848 1.05512 .52873 .51341 .51366 1.05619 .52924 1.05727 55 .51392 1.05835 .52950 1.05942 .52976 1.12657 .54368 1.19144 .55928 1.26900 51 1.12078 .54394 1.19268 .55954 1.27035 52 1.12193 54420 1.19393 .55980 1.27169 53 .52899 1.12309 .54446 1.19517 .56006 1.27304 54 1.12425 .54471 1.19642 .56032 1.27439 55 1.12540 .54497 1.19767 .56058 1.27574 56 .54523 1.19892 .56084 1.27710 57 1.06050 .53001 1.12773 .54549 1.20018 .56111 1.27845 58 59 .51494 1.06158 .53027 1.12889 .54575 1.20143 .56137 1.27981 59 60.51519 | 1.06267 .53053 1.13003 54601 1.20269 .56163 1.28117 160 336 TABLE XIII.-VERSINES AND EXSECANTS. 64° 65° 66° 67° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 1.56189 1.28253 2.56215 1.28390 3.56241 1.28526 4 .56267 1.28663 5.56294 1.28800 6.56320 1.28937 7.56346 1.29074 0.56163 1.28117 .57738 1.36620 .59326 .57765 1.36768 .59353 -.57791 1.36916 | .59379 .57817 1.37064 | .59406 .57814 1.37212 .59433 .57870 1.37361 .59459 1.46665 .57896 1.37509 .57923 1.37658 1.45859 60927 1.55930 0 1.46020 '60954 1.56106 1.46181 .60980 1.56282 2 1.46342 .61007 1.56458 3 1.46504 .61034 1.56634 4 61061 1.56811 .59486 1.46827 .61088 1.56988 .59512 1.46989 .61114 1.57165 7 • 8.56372 9.56398 10 .56425 1.29211 .57949 1.37808 .59539 1.47152 .61141 1.57342 8 1.29349 .57976 1.37957 .59566 1.47314 .61168 1.57520 9 1.29487 .58002 1.38107 .59592 1.47477 .61195 1.57698 10 11 .56451 1.29625 .58028 1.38256 .59619 1.47640 .61222 1.5787611 12.56477 1.29763 .58055 1.38406 .59645 1.47804 .61248 1.5805412 13.56503 1.29901 14.56529 .58081 1.38556 1.30040 .58108 .59672 1.38707 .59699 1.47967 1.48131 .61302 .61275 1.58233 13 1.5841214 15.56555 1.30179 .58134 1.38857 .59725 1.48295 .61329 1.5859115 16.56582 17.56608 18 .56634 1.30318 .58160 1.39008 .59752 1.48459 .61356 1.5877116 1.30457 1.30596 19 .56660 20 .56687 1.30735 1.30875 21 .56713 22 .56739 23.56765 24 .56791 1.31436 1.39766 1.39918 .58345 1.40070 .58372 1.40222 25 .56818 1.31576 .58398 1.40375 26.56844 1.31717 .58425 1.40528 27.56870 1.31858 .58451 1.40681 28 .56896 1.31999 .58478 1.40835 29 .56923 1.32140 .58504 1.40988 30 .56949 1.32282 .58531 1.41142 31.56975 1.32424 32.57001 1.32566 33.57028 1.32708 .58610 1.41605 31.57054 1.32850 .58637 1.41760 35.57080 1.32993 .58663 1.41914 36.57106 1.33135 .58690 1.42070 37.57133 1.33278 .58716 1.42225 1.31015 1.31155 .58319 1.31295 .58187 .58213 1.39311 .58240 1.39462 .58266 .58293 1.39159 .59779 1.48624 .61383 1.5895017 .59805 1.48789 .61409 1.59130 18 1.39614 59832 .59859 1.48954 .61436 1.49119 1.5931119 .61463 1.59491 20 .59885 1.49284 .61490 1.59672 21 .59912 1.49450 .59938 .61517 1.59853 122 1.49616 .61544 1.60035 23 .59965 1.49782 .61570 1.60217 24 .59992 1.49948 .61597 1.60399 25 .60018 1.50115 .61624 1.60581 20 .60045 1.50282 .61651 1.60763 27 .60072 1.50449 .61678 1.60946 28 .60098 1.50617 .61705 1.61129 29 .60125 1.50784 .61732 1.61313 30 .58557 58584 1.41296 .60152 1.50952 .61759 1.61496 31 1.41450 .60178 1.51120 .61785 1.61680 32 .60205 1.51289 .61812 1.61864 33 .60232 1.51457 .61839 1.62049 34 .60259 1.51626 .61866 1.62234 35 .60285 1.51795 .61893 1.6241936 38.57159 39.57185 40 .57212 45.57343 46.57369 47 .57396 1.33422 .58743 1.42380 1.33565 .58769 1.42536 1.33708 .58796 1.42692 41 .57238 1.33852 58822 42 .57264 1.33996 .58849 43 .57291 1.34140 .58875 1.43162 44 .57317 1.34284 .58902 1.43318 1.34429 .58928 1.43476 1.34573 .58955 1.48633 1.34718 .58981 1.43790 48.57422 1.34863 .60312 .60339 1.51965 .61920 1.62604 37 1.52134 .61947 1.62790 38 .60365 1.52304 .61974 1.6297639 .60392 1.52474 .62001 1.6316240 1.42848 .60419 1.52645 .62027 1.63348 41 1.43005 .60445 1.52815 .62054 1.63535 42 .60472 1.52986 .62081 1.6372243 .60499 1.53157 .62108 1.6390944 .60526 1.53329 .62135 1.64097 45 .60552 1.53500 .62162 1.64285 46 .60579 1.53672 .62189 1.64473 47 .59008 1.43948 .60606 1.53845 .62216 1.64662 48 49 .57448 1.35009 .59034 1.44106 60633 1.54017 .62243 1.64851 49 50 .57475 1.35154 .59061 1.44264 .60659 1.54190 .62270 1.05040 50 51 .57501 1.35300 .59087 52.57527 1.35446 .59114 1.44582 53 .57554 1.35592 .59140 1.44741 54 .57580 1.35738 .59167 1.44900 55 .57606 1.35885 .59194 1.45059 56 .57633 1.36031 59220 1.45219 57 .57659 1.36178 58 .57685 1.36325 1.44423 .60686 1.54363 62297 1.65229 51 • 59 .57712 1.36473 60 .57738 1.36620 .59247 1.45378 .59273 1.45539 .60873 1.55580 .59300 1.45699 .60900 59326 1.45859 .60927 .60713 1.54536 .60740 1.54709 .60766 1.54883 .60793 1.55057 60820 1.55231 .60847 1.55405 .62324 1.65419 52 .62351 1.65609 53 .62378 1.65799 54 62405 1.65989 55 • .62431 1.66180 56 .62458 .62485 1.6637157 1.6656358 1.55755 1.55930 .62512 1.66755 59 62539 1.6694760 888! 337 TABLE XIII.—VERSINES AND EXSECANTS. 68. 69° 70° 71° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .62539 1.66947 1 .62566 1.67139 2 62593 1.67332 .62620 1.67525 .64163 1.79043 .65798 .64190 1.79254 .65825 .64218 1.79466 .64245 1.79679 1.92380 67443 2.07155 0 • 1.92614 .67471 2.07415 1 .65S53 1.92849 .67498 2.07675 2 65880 1.93083 .67526 2.07936 3 4 62647 1.67718 .64272 1.79891 .65907 1.93318 .67553 2.08197 5 .62674 1.67911 .64299 1.80104 .65935 1.93554 .67.81 2.08459 Ꮳ .62701 1.68105 .64326 1.80318 .65962 1.93790 7 .62728 1.68299 .64353 1.80531 .65989 1.94026 8 .62755 1.68494 .64381 1.80746 9 .62782 1.68689 .64408 1.80960 10 .62809 1.68884 .64435 1.81175 .66017 1.94263 .66044 1.94500 .66071 1.94737 .67608 2.08721 7 .67636 2.08983 .67663 2.09246 8 9 .67691 2.09510 6 .67718 2.09774 10 11 .62836 1 69079 .64462 1.81390 .66099 1.9-1975 .67746 2.1003811 12 .62863 1.69275 .64489 1.81605 .66126 1.95213 .07773 2.10303 12 13 .62890 1.69471 .64517 1.81821 .66154 1.95452 .67801 2.1056813 14 .62917 1.69667 .64544 1.82037 .66181 1.95691 .67829 2.10834 14 15 .62944 1.69864 .64571 1.82254 .66208 1.95931 .67856 2.11101 15 16 .62971 1.70061 17 .62998 1.70258 18 .63025 1.70455 .64598 .64625 .64653 1.82906 1.82471 .66236 1.96171 .07881 2.1136716 1.82688 .66263 1.96411 .67911 2.11635 17 .66290 1.96652 .67939 2.11903 18 19 .63052 1.70653 .64680 1.83124 .66318 1.96893 .07966 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 22.63133 1.71249 23 .63161 24 .63188 .64761 1.83780 1.71448 .64789 1.83999 .64816 1.71647 1.84219 .66400 1.97619 .68049 2.12979 .66427 1.97862 .68077 2.13249 23 .66455 1.98106 .68104 2.13520 24 25 .63215 1.71847 .64843 1.84439 .66482 1.98349 .68132 2.13791 25 26 .63242 1.72047 .64870 1.84659 .66510 1.98594 .68159 2.1406326 27 .63269 1.72247 .64898 1.84880 .66537 1.98838 .68187 2.14335 27 28 .63296 1.72448 .64925 1.85102 .66564 1.99083 68214 2.14608 28 29 .63323 1.72649 30 .63350 1.72850 .64952 1.85323 .66592 1.99329 .68242 2.14881 29 .64979 1.85545 .66619 1.99574 .68270 2.15155 30 31 32 .63377 1.73052 .63404 33 63431 .65007 1.85767 .66647 1.99821 .68297 2.15429 31 1.73254 .65034 1.85990 .66674 2.00067 68325 2.1570432 ་ 1.73456 .65061 1.86213 .66702 2.00315 .68352 2.15979 33 34 .63458 1.78659 .65088 1.86437 .66729 2.00562 .68380 2.1625534 35 .63485 1.73862 .65116 1.86661 .66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 .65143 1.86885 .66784 2.01059 .68435 2.16808 36 37 .63539 1.74269 .65170 1.87109 .66811 2.01308 .68463 2.1708537 38 .63566 1.74473 .65197 1.87334 .66839 2.01557 .68490 2.1736338 39 .63591 1.74677 .65225 1.87560 .66866 2.01807 .68518 2.1764139 40 .63621 1.74881 .65252 1.87785 .66894 2.02057 .68546 2.1792040 41 63648 1.75086 .65279 1.88011 .66921 2.02308 .68573 2.18199 41 33= 42 63675 1.75292 .65306 1.88238 .66949 2.02559 .68601 2.18479 42 43 63702 1.75497 .65334 1.88465 .66976 2.02810 .68628 2.18759 | 43 44 .63729 1.75703 .65361 1.88692 .67003 2.03062 .68656 2.1904044 45 .63756 1.75909 65388 1.88920 .67031 2.03315 68684 2.1932245 • 46 .63783 1.76116 65416 1.89148 .67058 2.03568 .68711 2.19604 46 47 .63810 1.76323 65143 1.89376 .67086 2.03821 .68739 2.19886 47 48 49 51 52 1.76530 .03338 .63365 1.76737 .65497 50 .63892 .65525 1.76945 .63919 1.77154 .65552 63946 .65-170 1.89605 .67113 2.04075 .68767 2.20169 48 1.89834 .67141 2.04329 .68794 2.20453 49 1.90063 .67168 2.04584 .68822 2.20737 50 1.77362 53 63973 1.77571 54 .64000 1 77780 55 64027 1.77990 56 .64055 1.78200 1.90293 .65579 1.90524 .67223 1.90754 .65607 .65634 1.90986 .65661 1.91217 .65689 1.91449 .67196 2.04839 .68849 2.21021 51 2.05094 .67251 2.05350 .67278 2.05607 .67306 2.05864 .68877 2.21306 52 .68905 2.21592 53 .68932 2.21878 54 .68960 2.22165 55 67333 2.06121 68988 2.22452 56 · 57 64082 1.78410 .65716 1.91681 .67361 2.06379 .69015 2.22740 57 • 60 .64163 58 .64109 1.78621 1.91914 59 64136 1.78832 .65771 1.92147 .65798 1.79043 1.92380 .65743 67388 2.06637 .69043 2.23028 58 = .67416 2.06896 .69071 2.2331759 .67443 2.07155 .69098 2.23607 60 338 TABLE XIII.-VERSINES AND EXSECANTS. 72° 73° 74° 75° Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 0 .69098 2.23607 .70763 2.42030 .72436 1 .69126 2.23897 .70791 2.42356 .72464 2.62796 2.63164 2.69154 2.24187 3 .69181 2.24478 4.69209 .70818 2.42683 .70846 2.43010 2.24770 .70874 2.43337 .72492 2 63533 .74118 2.86370 .74146 2.86790 .74174 2.87211 0 72520 2.63903 .72548 2.64274 .74202 .74231 2.87633 3 2.88056 4 5.69237 2.25062 .70902 2.43666 .72576 2.64645 .74259 2.88479 5 69264 2.25355 .70930 2.43995 · 7 .69292 2.25648 .70958 2.44324 72604 72632 2.65391 2.65018 .74287 2.88901 6 .74315 2.89330 8 .69320 2.25942 .70985 2.44655 THERE BO .69347 2 26237 .71013 2.44986 .72660 2.65765 .72688 .74343 2.89756 8 2.66140 .74371 2.90181 9 10 .69375 2.26531 .71041 2.45317 .72716 2.66515 .74399 2.90613 10 11 .69403 2.26827 .71069 2.45650 .72744 2.66892 .74427 2.9104211 12 .69-430 2.27123 .71097 13 .69458 2.27420 2.45983 72772 2.67269 2.67269.74455 2.91473 12 14 .69486 2.27717 15 .69514 2.28015 .71125 .71153 2.46651 .71180 2.46986 2.46316 .72800 2.67647 .74484 2.91904 13 .72828 2.68025 .74512 2.92337 14 .72856 2.68405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 .72884 2.68785 .74568 2.93204 16 21 22 ZRBZAQNARA H 20 .69652 .69680 22.69708 23 .69735 21.69703 25 .69791 26.69818 27.69846 .69874 2.31939 37.70124 28 29 .69902 30 .69929 2 32244 2.32551 31 .69957 2.32858 32 .69985 2.33166 .71654 2.52787 33 70013 2.33474 .71682 2.53134 31.70010 2.33783 .71710 2.53482 35 .70068 2.34092 .71738 2.53831 36.70096 2.34403 .71766 2.54181 2.34713 .71794 2.54531 38.70151 2.35025 .71822 2.54883 17 .69569 2.28612 .71236 2.47658 18 .69597 2.28912 .71264 2.47995 19 .69624 2.29212 .71292 2.48333 2.29512 .71320 2.48671 2.29814 .71348 2.49010 2.30115 .71375 2.49350 2.30418 .71403 2.49691 2.30721 .71431 2.50032 2.31024 .71459 2.50374 2.31328 .71487 2.50716 2.31633 .71515 2.51060 .71543 2.51404 .71571 2.51748 .72912 2.69167 .74596 2.93640 |17 .72940 2.69549 .74624 2.94076 18 .72968 2.69931 .74652 2.94514 19 .72996 2.70315 .74680 2.94952 20 .73024 2.70700 .74709 2.95392 21 .73052 2.71085 .74787 2.95832 22 .73080 2.71471 74765 2.96274 23 .73108 .73136 2.72246 2.71858 .74793 2.96716 24 74821 2.97160 25 .73164 2.72635 74849 2.9760426 .73192 2.73024 .74878 2.98050 27 73220 2.73414 .74906 2.98497 28 .73248 2.73806 .74934 2.98944 29 .71598 2.52094 .78276 2.74198 .74962 2.99393 30 .71626 2.52440 73304 2.74591 .74990 לי 2.99843 31 73332 2.74984 .75018 3.00293 32 .73360 2.75379 .75047 3.00745 33 73388 2.75775 .75075 3.0119834 • 73416 2.76171 .75103 3.01652 35 73111 2.76568 .75131 3.02107 36 • .73472 2.76966 .75159 3.02563 37 .78500 2.77365 .75187 3.03020 38 39.70179 2.35336 .71850 2.55235 73529 2.77765 .75216 3.03179 39 40.70207 2.35649 .71877 2.55587 .78557 | 2.78166 .75244 3.03938 40 43.70290 41 .70235 2.35962 .71905 42 .70263 2.36276 .71933 2.36590 2.55940 .73585 2.78568 .75272 3.04398 41 2.56294 .73613 2.78970 .75300 3.04860 42 .71961 2.56649 .78641 2.79374 .75328 3.05322 48 44 70318 2.36905 .71989 2.57005 .73669 2.79778 .75356 3.0578644 45.70346 2.37221 .72017 2.57361 .73697 2.80183 .75385 3.06251 45 46.70374 2.37537 72045 2.57718 .73725 2.80589 .75413 3.0671746 47.70401 2.37854 .72073 2.58076 73753 2.80996 .75441 3.0718447 48.70429 2.38171 49.70457 50.70485 .72101 2.58434 .73781 2.81404 .75469 3.0765248 . 2.38489 72129 2.58794 2.38808 .73809 2.81813 .75497 3.0812149 .72157 2.59154 .73837 2.82223 .75526 3.08591 50 56 51.70513 2.39128 52.705-40 53.70568 2.39768 54 .70596 2.40089 55 .70624 2.40411 .70652 2.40734 57 .70679 2.41057 58 70707 2.41381 59.70735 60.70763 .72185 2.59514 .73865 2.82633 .75554 3.0906351 2.39448 .72213 2.59876 .72241 2.60238 .72269 2.60601 72296 2.60965 .72324 2.61330 .73893 2.83045 75582 3.0953552 .73921 2.83457 .75610 3.1000958 .73950 .73978 2.84285 2.83871 .75639 3.10484 54 .75667 3.10960 55 .74006 2.84700 .72352 2.61695 .71034 2.85116 .72380 2.62061 .74062 2.85533 2.41705 .72408 2.62428 .74090 2.85951 2.42030 .72436 2.62796 .74118 2.86370 .75695 3.11437 56 75723 3.11915 57 • 75751 75780 3.1239458 3.12875 + 75808 3.13357 160 808 59 339 TABLE XIII.—VERSINES AND EXSECANTS. 76° 77770 78° 79° Vers. Vers. Exsec. Exsec. Vers. Exsec. Vers. Exsec. 0 .75808 3.13357 1 .75836 3.13839 2 .75864 3.14323 3 .75892 4 .75921 3.14809 3.15295 .75949 3.15782 .75977 3.16271 77505 3.44541 .77533 3.45102 .77562 3.45664 3.46228 .77590 .77618 3.46793 .77647 .77675 .79209 3.80973 .80919 4.24084 0 .79237 3.81633 .80948 4.24870 1 .79266 3.82294 .80976 4.25658 2 79294 3.82956 .81005 4.26448 3 .79323 3.83621 .81033 4.27241 4 3.47360 .79351 3.84288 .81062 4.28036 3.47928 .79380 3.84956 .81090 4.28833 6 .76005 3.16761 .77703 3.48498 .79408 3.85627 .81119 4.29634 ལ 8.76031 3.17252 77732 3.49069 .79437 3.86299 .81148 4.30436 8 9.76062 3.17744 .77760 3.49642 .79465 3.86973 .81176 4.31241 9 10 .76090 3.18238 77788 3.50216 .79493 3.87649 .81205 4.32049 10 11 76118 3.18733 77817 3.50791 .79522 3.88327 .81233 4.32859 11 12 .76147 3.19228 .77845 3.51368 .79550 3.89007 .81262 4.33671 12 13.76175 77874 3.19725 3.51947 .79579 3.89689 .81290 4.3448613 + 14 .76203 3.20224 .77902 3.52527 .79607 3.90373 .81319 4.3530414 15 .76231 3.20723 .77930 3.53109 .79636 3.91058 .81348 4.36124 15 16.76260 3.21224 .77959 3.53692 .79664 3.91746 .81376 4.3694716 17.76288 18.76316 3.21726 .77987 3.54277 3.22229 .78015 3.54863 .79693 3.92436 .81405 4.3772 17 .79721 3.93128 .81433 4.38600 18 19 .76344 20.76373 3.22734 3.23239 .78044 .78072 3.55451 3.56041 79750 3.93821 .81462 4.39430 19 • .79778 3.94517 .81491 4.40263 20 21.76401 3.23746 .78101 3.56632 .79807 3.95215 .81519 4.41099 21 22.76-429 3.24255 .78129 3.57224 .79835 3.95914 .81548 4.41937 22 23.70458 3.24704 .78157 3.57819 .79864 3.96616 .81576 4.42778 23 24.76486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 25 .76514 3.25787 .78214 3.59012 .79921 3.98025 .81633 4.44468 25 26.76542 3.26300 .78242 3.59611 .79949 3.98733 .81662 4.45317 26 27 .76571 3.26814 .78271 3.60211 .79978 3.99443 .81691 4.46169 27 28.76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 28 29.76627 3.27847 .78328 3.61417 .80035 4.00869 81748 4.47881 29 30 .76655 3.28366 .78356 3.62023 .80063 4.01585 .81776 4.48740 30 31 76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 32 33 .76712 3.29406 .76740 3.29929 .78413 3.63238 .80120 4.03024 .81834 4.50468 32 .78441 3.63849 .80149 4.03746 .81862 4.51337 33 34.76769 3.30452 .78470 3.64461 .80177 4.04471 .81891 4.5220834 35 .76797 3.30977 78498 3.65074 .80206 4.05197 .81919 4.53081 35 36.76825 3.31503 37 .76854 3.32031 3.66307 38.76882 3.32560 39.76910 3.33090 40 .76938 3.33622 .78526 3.65690 .78555 .78583 3.66925 .78612 3.67545 .80320 .78640 3.68167 .803-18 80234 4.05926 .81948 4.53958 36 .80263 4.06657 .81977 4.54837 37 .80291 4.07390 82005 4.55720 38 4.08125 .82034 4.56605 39 4.08863 .82063 4.57493 40 41.76967 3.34154 42.76995 3.34689 78669 3.68791 .78697 3.69417 .80377 4.09602 .82091 4.58383 41 .80405 4.10344 .82120 4.5927742 43 .77023 3.35224 .78725 3.70044 .80434 4.11088 .82148 4.60174 43 44.77052 3.35761 .78754 3.70673 .80462 4.11835 .82177 4.61073 44 45.77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61976 45 46.77108 .78811 3.36839 3.71935 .80520 4.13334 .82234 4.62881 46 47.77137 3.37380 .78839 3.72569 .80548 4.14087 .82263 4.63790 47 48.77165 3.37923 78868 3.73205 .80577 4.14842 82292 4.64701 48 49.77193 3.38466 .78896 3.73843 .80605 4.15599 .82320 4.65616 49 50 .77222 3.39012 .78924 3.74482 .80634 4.16359 .82349 4.66533 50 51.77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 51 52.77278 3.40106 .78981 3.75766 .80691 4.17886 .82406 4.68377 52 53 .77307 3.40656 .79010 3.76411 .80719 4.18652 .82435 4.69304 53 54.77335 3.41206 .79038 3.77057 .80748 4.19421 .82463 4.70234 54 55 .77863 3.41759 .79067 3.77705 .80776 4.20193 .82492 4.71166 55 56 .77392 3.42312 .79095 3.78355 .80805 4.20966 .82521 4.72102 56 57 .77420 3.42867 .79123 58 .77448 3.43424 79152 889998 59 .77477 3.43982 79180 601 .77505 3.44541 79209 3.79007 .80862 3.79661 3.80316 3.80973 .80833 4.21742 82549 4.73041 57 4.22521 .82578 4.73983 58 .80891 4.23301 .82607 4.74929 59 .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. ACTA CONTO 0.82635 4.75877 .84357 5.39245 .86083 6.18530 1.82664 4.76829 2.82692 4.77784 3.82721 .84385 5.40422 .86112 6.20020 .82750 .82778 7.82836 8.82864 4.83581 .84414 5.41602 4.78742 .81143 5.42787 4.79703 .84471 5.43977 4.80667 .84500 5.45171 .82307 4.81635 .84529 4.82606 .84558 .84586 5.48779 .86140 6.21517 .86169 6.23019 .87900 .87813 7.20551 0 • 87842 7.22500 1 .87871 7.24457 2 7.26425 .86198 6.24529 .87929 7.28402 86227 6.26044 .87957 | 7.30388 OHRM CHŁO 3 5.46369 .86256 6.27566 .87986 7.32384 5.47572 .86284 6.29095 .88015 7.34390 7 .86313 6.30030 .88044 7.36405 8 9 82593 4.84558 10 .82922 4.85539 .84615 5.49991 .81611 5.51208 .86342 | 6.32171 .88073 7.38431 ទ .86371 6.33719 .88102 7.40466 |10 11.82950 4.86524 .84673 5.52429 .86400 6.35274 .88131 7.42511 11 12 .82979 4.87511 .84701 5.53655 .86428 6.36835 .88160 7.44566 12 13.83003 4.88502 .84730 5.54886 .86457 6.38403 .88188 7.46632 13 4.89497 222 FAXJKANKRA 24 5.03787 14 .83036 15 .83065 4.90495 .84788 16 .83094 4.91496 .84816 5.58606 17 .83122 4.92501 .84815 18 .83151 4.93509 .84874 19 .83180 4.94521 .84903 5.62369 20 .83208 4.95536 .84931 5.63633 21.83237 4.96555 .84960 5.64902 22 .83266 4.97577 .84989 5.66176 23 .83294 4.98603 .85018 5.67454 83323 4.99033 .85046 5.68738 25.83352 5.00666 .85075 5.70027 26.83380 5.01703 .85104 5.71321 27.83109 5.02743 .85133 5.72620 28 .83438 .85162 .84759 5.56121 .86486 6.39978 .88217 7.4870714 5.57361 .86515 6.41560 .88246 7.50793 15 .86544 6.43148 .88275 7.5288916 5.59855 .86573 6.44743 .88304 7.5499617 5.61110 .86001 6.46346 .88333 7.57113 18 .86630 6.47955 .88362 7.59241 19 .86659 6.49571 .88391 7.61379 20 .86688 6.51194 88420 7.63528 21 .86717 6.52825 .88118 7.65688 22 .86746 6.54462 .88477 7.67859 23 .86774 6.56107 .88506 7.70041 24 .86803 6.57759 .88535 7.72234 125 .86832 6.59418 .88564 7.74438 26 86861 6.61085 .88593 7.76653 27 5.73924 29 .83467 83407 | 5.04834 30.83495 .85190 5.75233 31 83524 3283553 33.83581 34 .83610 35.83639 36.83667 37.83696 38 5.05880 .85219 5.06941 .85248 5.77866 5.08000 .85277 5.79191 5.09062 .85305 5.80521 5.10129 .85334 5.81856 5.11199 .85363 5.83196 5.12273 .85392 5.84542 5.13350 .85420 5.85893 .83725 5.14432 .85449 39 .83754 5.15517 85478 40.83782 5.16607 .85507 5.76547 .86890 .86919 6.64141 .86947 6.66130 6.62759 .88622 7.78880 28 .88651 7.81118290 88680 7.83367 30 • .86976 6.67826 .88709 7.8562831 .87005 6.69530 .88737 7.87901 32 .87034 6.71242 .88766 7.9018633 .87063 6.72962 .88795 7.92482 34 .87092 6.74689 .88824 7.94791 35 .87120 6.70424 .88853 7.97111 36 .87149 6.78167 .88882 7.99414 37 5.87250 .87178 6.79918 .88911 8.01788 38 5.88612 .87207 6.81677 .88940 8.04116 39 5.89979 .87236 6.83443 .88969 8.06515 40 41.83811 42 .83840 43.83868 44.83897 45.83926 46.83954 47 .83983 5.17700 .85586 5.91352 5.18797 .85564 5.92731 5.19898 85593 5.94115 5.21004 .85622 5.95505 5.22113 .85051 5.96900 5.23226 .85680 5.98301 5.24343 .87265 6.85218 .88998 8.08897 | 41 .87294 6.87001 .87322 6.88792 .87351 6.90592 .89027 8.11292 42 .87880 6.92400 .89055 8.1369943 .89084 | 8.16120 44 .89113 8.18553 45 .85708 5.99708 .87409 .87438 6.94216 .89142 8.2099946 6.96040 .89171 8.23459 47 48.84012 49 .84041 5.26590 50 .84069 5.27719 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 .85766 6.02538 .87496 | 6.99714 .89229 8.28417 49 .85795 6.03962 .87524 7.01565 .89258 8.30917 50 FOOTBOKARO 51.84098 52 .84127 5.29991 53.84155 54 .81184 5.28853 85823 0.05392 87553 7.03423 .89287 8.33430 | 51 85852 6.06828 87582 7.05291 .89316 8.35957 52 5.31133 .85881 6.08269 87611 7.07167 .89345 8.38497 53 5.32279 .85910 6.09717 .87640 7.09032 .89374 8.410525-4 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 .84270 5.35743 58.84299 5.36906 .85996 6.14096 .87720 7.14760 .89460 8.46800 57 .86025 6.15568 50 .84328 5.38073 601 .84357 5.39245 86054 6.17046 .86083 6.18530 .87755 7.16681 .87784 7.18612 .878137,20551 .89489 8.5141158 .89518 8.54037 59 .89547 8.56677 160 341 TABLE XIII.-VERSINES AND EXSECANTS. 84° 85° 86° Vers. Exsec. Vers. Exsec. Vers. Exsec. OHRM TH LO 0 .89547 8.56677 .91281 10.47371 .93024 13.33559 0 .89576 8.59332 .91313 10.51199 93053 13.39547 1 .89605 8.62002 .91342 10.55052 .93082 13.45586 .89634 8.64687 .91371 10.58932 .93111 13.51676 .89663 8.67387 .91400 10.62837 .93140 13.57817 .89692 8.70103 .91429 10.66769 .93169 13.61011 .89721 8.72833 .91458 10.70728 93198 13.70258 .89750 8.75579 .91487 10.74714 .93227 13.76558 8 .89779 8.78341 .91516 10.78727 .93257 13.82913 8 9 .89808 8.81119 .91545 10.82768 .93286 13.89323 9 10 .89836 8.83912 .91574 10.86837 .93315 13.95788 10 11 .89865 8.86722 .91603 10.90934 .93344 14.02310 11 12 .89894 8.89547 .91632 10.95060 .93373 14.08890 12 13 .89923 8.92389 .91661 10.99214 .93402 14.15527 13 14 89952 8.95248 .91690 11.03397 .93431 14.22223 14 • 15 .89981 8.98123 .91719 11.07610 .93-460 14.28979 15 16 .90010 9.01015 .91718 11.11852 .93489 14.33795 16 17 .90039 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90068 9.06849 .91806 11.20427 93547 43 45 46 47 52 53 59 22 788JA24*2. ™****0.88% #38I995928 5081885688 19 .90097 9.09792 .91835 11.24761 .93576 14.49611 18 14.56614 20 .90126 9.12752 .91864 11.29125 .93605 14.63679 .90155 9.15730 .91893 11.33521 .93634 14.70810 21 .90184 9.18725 .91922 11.37948 .93663 14.78005 22 .90213 9.21739 .91951 11.42408 .93692 14.85268 23 .90242 9.24770 .91980 11.46900 .93721 14.92597 21 .90271 9.27819 .92009 11.51424 .93750 14.99995 25 26 .90300 9.30887 .92038 11.55982 .93779 15.07462 26 90329 9.33973 .92067 11.60572 .93808 15.14999 27 • .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 80 31 .90444 9.46505 .92183 11.79278 93924 15.45869 31 32 .90473 9.49685 .92212 11.81042 .93953 15.53772 33 .90502 9.52886 .92241 11.88841 .93982 15.61751 33 222 7337NAKHA. H88 19 20 32 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 .9-4098 15.94456 37 38 .90647 9.69186 .92386 12.13388 .94127 16.02835 38 39 .90676 9.72507 .92415 12.18411 .94156 16.11297 39 40 .90705 9.75819 .92114 12.23472 .91186 16.19843 40 41 .90734 9.79212 .92473 12.28572 .94215 16.28476 41 42 .90763 9.82596 .92502 12.33712 .91244 16.37196 42 .90792 9.86001 .92531 12.38891 .94273 16.46005 43 44 .90821 9.89428 .92560 12.41112 .90850 9.92877 .92589 12.49373 - .90879 9.96348 .92618 12.54676 .90908 9.99841 .92647 12.60021 48 .90937 10.03356 .92676 12.65408 49 .90966 10.06894 .92705 12.70838 50 .90995 10.10-155 .92731 12.76312 .91024 10.14039 .92763 12.81829 .91053 10.17646 .92792 12.87391 .91082 10.21277 .92821 12.92999 .94302 .94331 .94360 16.72975 46 .94389 16.82152 47 .91418 16.91424 48 .94147 17.00794 49 .94476 17.10262 50 .94505 17.19830 .94534 17.29501 .9-1563 17.39274 53 16.54903 41 16.63893 45 51 =333995998 HRS 52 .91111 10.24932 .92850 12.98651 .94592 17.49153 54 .91140 10.28610 92879 13 04350 .9-1621 17.59139 55 .91169 10.32313 .92908 13.10096 .94650 17.69233 56 57 +91197 10.36040 .92937 13.15889 .94679 17.79138 57 .91226 10.39792 .92966 13.21730 .91255 10.43569 .92995 13.27620 .91284 10.47371 .93024 13.33559 .94708 17.89755 58 .94737 .94766 18.00185 59 18.10732 60 342 TABLE XIII.—VERSINES AND EXSECANTS. 87° 88° 89° Vers. Exsec. Vers. Exsec. Vers. Exsec. 2 3 O1QMOT LO 0 .94766 18.10732 .96510 27.65371 .98255 56.29869 0 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 .94825 18.32182 .96568 28.13917 .98313 58.27431 2 .94851 18.43088 .96597 28.38812 .98342 59.31411 3 .94883 18.54119 .96626 28.64137 .98371 60.39105 .94912 18.65275 .96655 28.89903 .98400 61.50715 .94941 18.76560 96684 29.16120 .98429 62.66460 6 .94970 18.87976 .96714 29.42802 .98458 63.86572 8 .94999 18.99524 .96743 29.69960 .98487 65.11304 9 .95028 19.11208 .96772 29.97607 .98517 66.40927 9 10 .95057 19.23028 .96801 30.25758 .98546 67.75736 10 11 .95086 19.34989 .96830 30.54425 .98575 69.16047 11 12 .95115 19.47093 .96859 30.83623 .98604 70.62285 12 13 .95144 19.59341 .96888 31.13366 .98633 72.14583 13 14 .95173 19.71737 .96917 31.43671 .98662 73.73586 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 22 78314QNARA H 19 .95318 20.36027 .97062 33.04199 .98807 82.84947 19 20 .95347 20.49368 .97092 33.38232 .98836 84.94561 20 21 .95377 20.62876 22 .95406 20.76555 .97121 .97150 24.08380 33.72952 .98866 87.14924 21 .98895 89.46886 22 .95435 20.90409 .97179 34.44539 .98924 91.91387 23 .95-464 21.04410 .97208 34.81452 .98953 94.49471 .95493 21.18653 .97237 35.19141 .98982 97.22303 26 .95522 21.33050 .97266 35.57033 .99011 100.1119 26 27 .95551 21.47635 .97295 35.96953 .99040 103.1757 27 28 .95580 21.62413 .97324 36.37127 .99069 106.4311 28 29 .95609 21.77386 .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 31 32 .95696 22.23520 .97440 38.06957 .99186 121.7780 32 33 .95725 22.39316 .97470 38.51855 .99215 126.3253 33 34 .95754 22.55329 .97499 38.97797 .99244 131.2223 34 35 ≈≈= =33995988 588H) 40 .95783 36 .95812 22.88022 37 .95842 23.04712 38 .95871 23.21637 39 .95900 23.38802 .95929 22.71563 .97528 39.44820 .99273 136.5111 35 97557 39.92963 .99302 142.2406 36 .97586 40.42266 .99331 148.4684 37 .97615 40.92772 .99360 155.2623 38 .97644 41.44525 .99389 23.56212 .97673 41.97571 .99418 162.7033 39 170.8883 40 41 .95958 23.73873 .97702 42.51961 .99447 42 .95987 23.91790 .97731 43.07746 .99476 179.9350 41 189.9868 42 43 .96016 24.09969 .97760 43.64980 .99505 44 .96045 24.28414 .97789 44.23720 .99535 201.2212 43 213.8600 44 45 .96074 24.47134 .97819 44.84026 .99564 46 .96103 24.66132 .97848 45.45963 47 .96132 24.85417 97877 46.09596 48 .96161 25.04994 .97906 46.74997 .99593 .99622 .99651 228.1839 45 244.5540 263.4427 47 285.4795 49 .96190 25.24869 .97935 47.42241 50 .96219 25.45051 .97964 48.11406 .99680 311.5230 49 99709 342.7752 50 51 .96248 25.65546 .97993 48.82576 52 .96277 25.86360 .98022 49.55840 .99738 .99767 380.9723 51 53 .96307 26.07503 .98051 50.31290 .99796 54 .96336 26.28981 .98080 51.09027 55 .96365 26.50804 .98109 51.89156 .99825 .99855 428.7187 52 490.1070 53 571.9581 54 686.5496 55 56 .96394 26.72978 .98138 52.71790 57 59 .96423 58 .96452 27.18417 .96481 60 .96510 26.95513 .98168 53.57046 .98197 54.45053 27.41700 .98226 55.35946 27.65371 .98255 56.29869 .99881 858.4369 56 .99913 .99942 99971 1.00000 1144.916 57 1717.874 58 3436.747 59 Infinite 60 888 333995988 F8BHBA80 46 48 343 TABLE XIV.-CUBIC YARDS PER 100 FEET. SLOPES : 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 22 24 26 2 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 4 193 222 252 281 341 370 400 430 245 282 319 356 431 468 505 542 6 300 344 389 433 522 567 611 656 7 356 408 460 512 616 668 719 771 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 11 601 682 761 845 1008 1090 1171 1253 12 667 756 844 933 1111 1200 1289 1378 13 731 831 926 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 1511 1659 1778 1896 17 · 1023 1149 1274 1401 1653 1779 1905 2031 24 26 27 28 31 33 35 36 38 39 40 41 42 43 44 45 46 47 48 002 FR*TRONKAA NACIKOHAO, #33I985098 728X 1100 1233 1366 1500 1767 1900 2033 2167 1179 1319 1460 1601 1882 2023 2164 2305 1259 1407 1555 1704 2000 2148 2296 2444 1342 1497 1653 1808 2119 2275 2431 2586 1426 1589 1752 1915 2241 2404 2567 2730 1512 1682 1853 2023 2364 2534 2705 2875 1600 1778 1955 2133 2489 2667 2844 3022 J 1690 1875 2060 2245 2616 2801 2986 3171 1781 1974 2166 2359 2744 2937 3130 3322 1875 2075 2274 2475 2875 3075 3275 3475 1970 2178 2384 2593 3007 3215 3422 3630 2068 2282 2496 2712 3142 3350 3571 3786 2167 2389 2610 2833 3278 3500 3722 3944 2268 2497 2726 2956 3416 3645 3875 4105 2370 2607 28-14 3081 3556 3793 4030 4267 2475 2719 2964 3208 3697 3942 4186 4431 2581 2833 3085 3337 3841 4093 4344 4596 2690 2949 3208 3468 3986 4245 4505 4764 2800 3067 3333 3600 4133 4400 4667 4933 2912 3186 3460 3734 4282 4556 4831 5105 3026 3307 3589 3870 4433 4715 4996 5278 3142 3431 3719 4008 4586 4875 5164 5453 3259 3556 3852 4148 4741 5037 5333 5630 3379 3682 3986 4290 4897 5201 5505 5808 3500 3811 4122 4433 5056 5367 5678 5989 3623 3942 4260 4579 5216 5534 5853 6171 3748 4074 4400 4726 5378 5704 6030 6356 3875 4208 4541 4875 5542 5875 6208 6542 4004 4314 4684 5026 5707 6048 6389 6730 4134 4482 4830 5179 5875 6223 6571 6919 4267 4622 4978 5333 6044 6400 6756 7111 4401 4764 5127 5490 6216 6579 6942 7305 4537 4907 5278 5648 6389 6759 7130 7500 4675 5053 5430 5808 6564 6942 7319 7697 52 4815 5200 5584 5970 6741 7126 7511 7896 4956 5741 5349 6134 6919 7312 7705 8097 5100 5500 5900 6300 7100 7500 7900 8300 55 5245 5053 6060 6468 282 7690 8097 8505 56 5393 5807 6222 6637 7467 7881 8296 8711 57 55-12 5961 6386 6808 7653 8075 8497 8919 58 5693 6122 6552 6981 7841 8270 8700 9130 39 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 : 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 22 24 26 28 1 46 5-1 61 69 83 91 98 106 2 96 111 126 141 170 185 200 215 3 150 172 194 217 261 283 306 328 4 207 237 267 296 356 385 415 411 5 269 306 343 380 454 491 528 565 333 378 422 467 556 600 611 6.99 402 451 506 557 661 713 765 817 474 533 593 652 770 830 889 948 9 550 617 683 750 883 950 1017 1083 10 630 704 778 852 1000 1074 1148 1222 11 713 794 876 957 1120 1202 1283 1365 12. 800 889 978 1067 1244 1333 1422 1511 13 891 987 1083 1180 1372 1469 1565 1661 11 985 1089 1193 1296 1501 1607 1711 1815 15 1083 1194 1306 1417 1639 1750 1861 1972 16 1185 1304 1422 1541 1779 1896 2015 2133 36 41 43 41 45 46 47 49 50 51 52 53 54 55 56 57 58 1082 7837HQNARA H28**26*8= =8BE420 HAD1R8LX88 17 1291 1417 1543 1669 1920 2046 2172 2298 1400 1533 1667 1800 2067 2200 2333 2467 1513 1654 1794 1935 2217 2357 2498 2639 1630 1778 1926 2074 2370 2519 2667 2815 21 1750 1906 2061 2217 2528 2683 2839 2994 1874 2037 2200 2363 2689 2852 3015 3178 2002 2172 2343 2513 2854 3024 3194 3305 21 2133 2311 2489 2667 3022 3200 3378 3556 2269 2454 2639 2824 3194 3380 3505 3750 26 2407 2600 2793 2985 3370 35C3 3756 3918 27 2550 2750 2950 3150 3550 3750 3950 4151 28 2696 2901 3111 3319 3733 3941 4148 4356 2846 3061 3276 3491 3920 4135 4350 4565 3000 3222 3414 3667 4111 4333 4556 4778 3157 3387 3617 3846 4306 4535 4765 4994 3319 3556 3793 4030 4504 4741 4978 5215 33 3483 3728 3972 4217 4706 4950 5194 5139 3-1 3652 3904 4156 4407 4911 5163 5415 5667 35 3824 4083 4343 4602 5120 5380 5639 5898 4000 4267 4533 4800 5333 5600 5867 6133 4180 4454 4728 5002 5550 5824 6098 6372 4363 4644 4926 5207 5770 6052 6333 6615 4550 4839 5128 5417 599-4 6283 6572 68C1 4741 5037 5333 5630 6222 6519 6815 7111 4935 5239 5543 5846 6454 6757 7061 7365 5133 5414 5756 6067 6689 7000 7311 7622 5335 5654 5972 6291 6928 7246 7565 7883 5541 5867 6193 6519 7170 7496 7822 8148 5750 6083 6417 6750 7417 7750 8083 8417 5963 0304 6644 6985 7667 8007 8348 8689 6180 6528 6876 7224 7920 8269 8617 8965 6400 6756 7111 7467 8178 8533 8889 9214 6624 6987 7350 7713 8439 8802 9165 9528 6852 7222 7593 7963 8764 9074 9444 9815 7083 7461 7839 8217 8972 9350 9728 10106 7319 7704 8089 8474 9244 9630 10015 10400 7557 7950 8343 8735 9520 9913 10306 10698 7800 $200 8600 9000 9800 10200 10600 11000 8046 8-154 8861 9269 10083 10491 10898 11206 8296 8711 9126 9541 10370 10785 11200 11615 8550 8972 9394 9817 10661 11083 11506 11928 8807 9237 9667 10096 10956 11385 11815 12244 9069 9506 9943 10380 11254 11691 12128 12565 9333 9778 10222 10667 11556 12000 12444 12889 345 TABLE XIV. —CUBIC YARDS PER 100 FEET. SLOPES 1 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 20 28 30 32 1 48 56 63 ΤΟ 78 107 115 122 104 119 133 148 163 222 237 252 167 189 211 233 256 344 367 389 4 237 267 296 326 356 474 504 533 5 315 352 389 426 463 611 648 685 G 400 414 489 533 578 756 800 844 7 493 514 596 648 700 907 959 1011 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 1333 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 1914 2389 2500 2611 16 1659 1778 1896 2015 2133 2607 2726 2811 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 31 33 40 41 42 43 & FINHARKAA. H8358688 ** 2370 2519 2667 2815 2963 3556 3704 3852 2567 2722 2878 3033 3189 3811 3967 4122 2770 2933 3096 3259 3122 4074 4237 4144 2981 3152 3322 3493 3663 4344 4515 4685 3200 3378 3556 3733 3911 4622 4800 4978 3426 3611 3796 3981 4167 4907 5093 5278 26 3659 3852 4041 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 5018 5263 6122 6337 6552 30 4667 4889 5111 5333 5556 6144 6667 6889 4937 5167 5396 5626 5856 6774 7004 7233 5215 5452 5689 59.26 6163 7111 7348 7585 5500 5744 5989 6233 6478 7456 7700 7914 31 5793 6011 6296 65-48 6800 7807 8059 8311 6093 6352 6611 6870 7130 8167 8426 8C85 6-400 6667 6933 7200 7467 8533 8800 9007 37 6715 6989 7263 7537 7811 8907 9181 9456 7037 7319 7600 7881 8163 9289 9570 9832 7367 7656 7944 8233 8522 9678 9967 10256 7701 8000 8296 8593 8889 10074 10370 10607 8048 8352 8656 8959 9263 10478 10781 11085 8400 8711 9022 9333 9644 10889 11200 11511 8759 9078 9396 9715 10033 11307 11626 11944 41 9126 9452 9778 10104 10430 11733 12059 12385 45 9500 9833 10167 10500 10833 12167 12500 12833 46 9881 10222 10563 10904 11244 12607 12948 13289 47 10270 10619 10967 11315 11663 13056 13404 13752 UO JAN! 48 10667 11022 11378 11733 12089 13511 13867 14222 49 11070 11433 11796 12159 12522 13974 14337 14700 50 11481 11852 12222 12593 12963 14444 14815 15185 51 11900 12278 12656 13033 13411 14922 15300 15678 52 12326 12711 13096 13481 13867 15407 15793 16178 53 12759 13152 13544 13937 14330 15900 16293 16685 54 13200 13600 14000 14100 14800 16400 16800 17200 55 13648 14056 14463 1-1870 15278 16907 17315 17722 56 14104 1-4519 14933 15348 15763 17422 17837 18252 57 14567 14989 15411 15833 16256 17944 18367 18789 58 15037 15-167 15890 16326 16756 18474 1890-4 19333 59 15515 15952 16389 16826 17263 19011 19448 19885 60 16000 16114 16889 17333 17778 19556 20000 20444 346 TABLE XIV. -CUBIC YARDS PER 100 FEET. SLOPES 1½: 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 20 28 30 32 1 50 57 65 72 80 109 117 124 2 111 126 141 156 170 230 211 259 3 183 206 228 250 272 361 383 406 4 267 296 326 356 385 504 533 563 361 398 435 472 509 657 694 731 467 511 556 600 644 822 867 911 583 635 687 739 791 998 1030 1102 8 711 770 830 889 948 1185 1214 1304 9 850 917 983 1050 1116 1383 1450 1517 10 1000 1074 1148 1222 1296 1593 1667 1741 11 1161 1243 1324 1406 1487 1813 1894 1976 12 1333 1422 1511 1600 1689 2044 2133 2222 13 1517 1613 1709 1806 1902 2287 2383 2480 11 1711 1815 1919 2022 2126 2541 2641 2748 15 1917 2028 2139 2250 2361 2806 2917 3028 16 2133 2252 2370 2489 2607 3081 3200 3319 17 2361 2187 2613 2739 2865 3369 3491 3620 18 2600 2733 2867 3000 3133 3667 3800 3933 42 44 46 48 49 128 FRAKK***** 588-33588 =33I995BD 2850 2991 3131 3272 3413 3976 4117 4257 20 3111 3259 3-107 3556 3704 4296 4444 4592 3383 3539 3694 3850 4005 4628 4783 4939 3667 3830 3993 4156 4318 4970 5133 5296 3961 4131 4302 4472 4642 5324 5494 5665 4267 4444 4622 4800 4978 5689 5867 6044 25 4583 4769 4951 5139 5324 6065 6250 6435 26 4911 5104 5296 5489 5681 6452 6644 6837 27 5250 5-150 5650 5850 6050 6850 7050 7250 28 5600 5807 6015 6222 6430 7259 7467 7674 5961 6176 6391 6606 6820 7680 7894 8109 30 0333 6556 6778 7000 7222 8111 8333 8555 6717 6946 7176 7406 7635 8554 8783 9013 32 7111 7348 7585 7822 8059 9007 92-H 9-182 7517 7761 8006 8250 8494 9472 9717 9962 34 7933 8185 8437 8689 8941 9948 10200 10452 8361 8620 8880 9139 9398 10435 10694 10954 36 8800 9067 9333 9600 9867 10933 11200 11467 37 9250 9524 9798 10072 10316 11443 11717 11991 9711 9993 10274 10556 10337 11963 12244 12526 39 10183 10472 10761 11050 11339 12494 12783 13072 40 10667 10963 11259 11556 11852 13037 13333 13630 11161 11165 11769 12072 12376 13591 13894 14198 11667 11978 12289 12600 12911 14156 14467 14778 12183 12502 12820 13139 13457 14731 15050 15369 12711 13037 13363 13689 14015 15319 15644 15970 13250 13583 13917 14250 14583 15917 16250 16583 13800 14141 14481 14822 15163 16526 16867 17207 14361 14709 15057 15106 15754 17146 17494 17843 14933 15289 15644 16000 16356 17778 18133 18489 15517 15880 16243 16606 16968 18420 16111 16481 16852 17222 17592 19074 51 16717 17094 17472 17850 52 17333 17719 18104 53 17961 18354 18746 54 55 56 19911 58 59 60 24000 18228 19739 18874 20415 19531 21102 18800 19000 19400 19300 20200 21800 22200 19250 19657 20065 20472 20880 22509 22917 20326 20741 21156 21570 20583 21006 21428 21850 21267 21696 22126 22556 21961 22398 22835 22667 23111 23556 18489 18783 19146 19444 20117 20494 20800 21185 19815 19139 21494 21887 22600 23324 23230 23644 24059 22272 23961 24383 24805 22985 24701 25133 25563 23272 23709 25457 25894 26332 24444 26222 26667 27111 347 TABLE XIV.-CUBIC YARDS PER 100 FEET. SLOPES 2 : 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 20 28 30 32 10 11 12 TORGOT 10 CO 20000 HR 1 52 59 67 74 81 111 119 126 119 133 143 163 178 237 252 267 200 222 241 267 289 378 400 422 4 296 326 356 385 415 533 563 593 407 444 481 519 556 704 741 778 533 578 622 667 711 889 933 978 674 726 778 830 881 1089 1141 1193 8 830 889 948 1007 1067 1304 1363 1422 9 1000 1067 1133 1200 1267 1533 1600 1667 1185 1259 1333 1407 1481 1778 1852 1926 1385 1467 1548 1630 1711 2037 2119 2200 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 3111 15 16 17 18 19 21 23 24 25 26 28 29 30 31 32 33 34 35 36 37 38 40 41 42 43 46 48 292002 2****27H20 7800005089 #33‡=====8 PROHO85888 2333 2444 2556 2667 2778 3222 3333 3444 2607 2726 2844 2963 3081 3556 3674 3793 2896 3022 3148 3274 3400 3904 4030 4156 3200 $333 3167 3600 3733 4267 4400 4533 3519 3659 3800 3941 4081 4644 4785 4926 [ 3852 4000 4148 4296 4444 5037 5185 5333 4200 4356 4511 4667 4822 5444 5600 5756 4563 4730 4889 5052 5215 5867 6030 6193 4911 5111 5281 5452 5622 6304 6474 6644 5333 5511 5689 5867 6044 6756 6933 7111 5741 5926 6111 6296 6-481 7222 7407 7593 6163 6356 6548 6741 6933 7704 7896 8089 6600 6900 7000 7200 7400 $200 8400 8600 7052 7259 7467 7074 7831 8711 8919 9126 7519 77:3 7948 8163 8378 9237 9452 9667 8000 8222 8444 8667 8889 9778 10000 10222 8496 8726 8956 9185 9415 10333 10563 10793 9007 9244 9481 9719 9956 10901 11141 11378 9533 9778 10022 10267 10511 11489 11733 11978 10074 10326 10578 10:30 11081 12089 12341 12593 10630 10889 11148 11407 11667 12704 12963 13222 11200 11467 11733 12000 12267 13333 13600 13867 11785 12059 12333 12607 12381 13978 14252 14526 12385 12667 12948 13230 13511 14637 14919 15200 39 13000 13289 13578 13867 14156 15311 15600 15889 13630 13926 14222 14519 14815 16000 16296 16593 14274 14578 14981 15185 15189 16704 17007 17311 14933 15244 15526 15867 16178 17-422 17733 18044 15607 15926 16224 16563 16881 18156 18474 18793 44 16296 16622 16948 17274 17600 18901 19230 19556 45 17000 17333 17667 18000 18333 19667 20000 20333 17719 18059 18400 18741 19081 20444 20785 21126 47 18 152 18800 19148 19496 19844 21237 21585 21933 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 21:33 21911 22289 22667 23044 24556 24933 25311 52 53 22341 22726 23111 23496 23163 23556 23948 24341 23881 25422 25807 26193 24733 26304 26696 27089 54 24000 24400 21800 25200 25600 27200 27600 28000 2-1852 25259 25667 26074 26481 28111 28519 28926 56 25719 26133 26548 26963 27378 29037 29452 29867 57 26600 27022 27444 27867 28289 29978 30400 30822. 27496 27926 28356 28785 29215 30933 31363 31793 59 28407 28844 29281 29719 30156 31904 32341 32778 29333 29778 30222 30667 31111 32889 33333 33778 348 TABLE XIV.-CUBIC YARDS PER 100 FEET. SLOPES 3: 1. Depth Base Base Base Base Base Base Base Base 12 214 16 18 20 28 30 32 1 56 63 ΤΟ 78 85 115 122 130 2 133 148 163 178 193 252 267 281 233 256 278 300 322 411 433 456 356 385 415 444 474 593 622 652 500 537 574 611 648 796 833 870 667 711 756 800 814 1022 1067 1111 856 907 959 1011 1063 1270 1322 1374 1067 1126 1185 1244 1301 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 2618 12 2133 2222 2311 2400 2489 2814 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 PAR ZABIHANH.. 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 6814 7007 7659 7822 7985 23 6900 7070 7241 7411 7581 8263 8433 8504 24 7467 7641 7822 8000 8178 8889 9067 9144 25 8056 8241 8426 8611 8796 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 14122 14352 32 12800 13037 13274 13511 13748 14696 14933 15170 23 13567 13811 14056 14300 14544 15522 15767 16011 34 14356 14607 14859 15111 15363 16370 16622 16874 35 15167 15426 15685 15911 16204 17241 17500 17759 36 16000 16267 16533 16800 17067 18133 18400 18667 50 51 52 57 5887 7333994990 NOOJROADRO 16856 17130 17404 17678 17952 19048 19322 19596 17733 18015 18296 18578 18859 19985 20267 20548 18633 18922 19211 19500 19789 20941 21233 21522 19556 19852 20148 20111 20741 21926 22222 22516 20500 20804. 21107 21411 21715 22930 23233 23537 21467 21778 22089 22400 22711 23956 24267 24578 22456 22774 23093 23411 23730 25004 25322 25641 23467 23793 24119 24444 24770 26074 26400 26726 24500 24833 25167 25500 25833 27167 27500 27833 25556 25896 26237 26578 26919 28281 28622 28963 26633 26981 27330 27678 28026 29419 29767 30115 27733 28089 28144 28800 29156 30578 30933 31289 28856 29219 29581 29944 30307 31759 32122 32485 30000 30370 30741 31111 31481 32963 33333 33704 31167 31544 31922 32300 32678 34189 34567 34944 32356 32741 33126 33511 33896 35437 35822 56207 33567 33959 34352 34744 35137 36707 37100 37493 34800 35200 35600 36000 36400 38000 38400 38800 36056 36463 36870 37278 37685 39315 39722 40130 37333 37748 38163 38578 38993 40652 41067 41481 38633 39056 39478 39900 40322 42011 42433 42856 39956 40385 40815 41244 41674 43393 43822 44252 41300 41737 42174 42611 43048 60* 42667 43111 43556 44000 41444 46222 44796 45233 45670 46667 47111 349 TABLE XV.-CUBIC YARDS IN 100 FEET LENGTH. Area. Sq. Ft. Cubic Yards. Area. Cubic Area. Sq. Ft. Yards. Sq. Cubic Area. Cubic Area. Yards. Sq. Cubic Yards. Sq. Ft. Yards. Ft. Ft. 12345 3.7 51 188.9 101 374.1 151 559.3 201 744.4 7.4 52 192,6 102 377.8 152 563.0 202 748.2 11.1 53 196.3 103 381.5. 153 566.7 203 751.9 14.8 54 200.0 104 385.2 ་ 154 570.4 204 755.6 18.5 55 203.7- 105 388.9 155 574.1 205 759.3 22.2 56 207.4 106 392.6. 156 577.8 206 763.0 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' 222,2 110 407.4 160 592.6 210 777.8 11 40.7 61 225.9 111 411.1 161 596.3 211.781.5 12 44.4 62 229.6 112 414.8 162 600.0 212 785.2 13 48.1. 63 233.3 113 418.5 163 603.7 213 788.9 14 51.9 64 23740 114 422.2 164 607.4 214 792.6 15 55.6 65 240.7 115 425.9 165 611.1 215 796.3 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. 63 251.9 118 437.0 168 622.2 218 807.4 19 -70,4 69 255.6 119 440.7 169 625.9 219 811.1 20 7431 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 8135 72 266.7 122 451.9 172 637.0 222 822.2 85.2. 78 270.4 123 455.6. 173 640.7 223 825.9 88.9 74 274.1 124 459:3 174 644.4 224 829.6 92.6. 5 277.8 125 463.0 175 648.2 225 833.3 96.3 76 281.5 126 466.7 176 651.9 226 837.0 100:0. 285.2 127 470.4 177 655.6 227 -840.7 28 103.7 288.9 128: .474.1 178 659.3 228 844.4 29 107.4 79 292.6. 129 477.8 .179 668.0 220. 848.2 30 1111/ *80 296.3 130 481.5 180 666.7 280 851,9 31 114.8 .81 ·300.0 131 485.2 181 670 4 231 855.6 32 118.5. 82 303.7 132 488.9 182 674.1 859.3 33 122.2 83 307.4- 133 492.6 18% 6078 863.0 34: 125.9 84 : 311.1 134 496.3 184 681:5 234 866.7 35 129 6 85 314.8 135 500.0 185 688.2 235 870.4 36 193.3 ,86 $18.5 136 503.7 186 688.9. 236 874.1 37 1370 87 322 2 137 --507.4 187 692.6 237 877.8 38 140.7 -88 325.9€ 1138 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 4 3407 142 525.9. 199 711.1 242 896.3* 43 159.3 93- 844.4 143 529:6 193 714.8 243 900.0˜ 44 163.0 94 818.2 144 533.3. 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 1147 544.4 197 729.6 247 914.8 48 177:8 98 363.0 148 548.2 198 733.3 248 918.5 497 181.5 99 366.7 149 551.9 199 737.0 249 922.2- 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. Area. Sq. Cubic Yards. Cubic Sq. Yards. Sq. Area. Cubic Area. Cubic Yards. Sq. Yails. Sq. Area. Cubic Yards. Ft. Ft. Ft. Ft. Ft. -251 929.6 301 1114.8 351 1300.0 401 1485. 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 1692.6 258 955.6 308 1140.7 358 1325.9 408 1511.1 458 1696.3 259 959.3 309 1144.4 359 1329.6 409 1514.8 459 1700.0 260 963.0 310 1148.2 360 1333.3 410 1518.5 460 1703.7 261 966.7 311 1151.9 361 1337.0 411 1522.2 461 1707.4 $ 262 970.4 312 1155.6 362 1340.7 412 1525.9 462 1711.1 263 974.1 313 1159.3 363 1344.4 413 1529.6 463 1714.8 261 977.8- 1163.0 314 364 1348.2 411 1533.3 464 1718.5 265 981.5 315 1166.7 365 1351.9 415 1537,0 465 1722.2 $266 985.2 316 1170.4 366 1355.6 416 1540.7 466 1725.9 267 988.9 317 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 1366.7 419 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 1196.3 323 373 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 1389.9 425 1574.1 475 1759.3 276 1022.2 326 1207.4 376 1392.6 426 1577.8 476 1763.0 27 1025.9 327 1211.1 377 1396.3 427 1581.5 477 1766.7 218 1029.6 328 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 1225.9 381 1411.1 431 1596.3 481 1781.5 282 044.4 332 1229.6 382 1414.8 432 1600.0 482 1785.2 283 1048.2 333 1233.3 383 1418.5 433 1603 7 483 1788.9 284 1051.9 334 1237.0 384 1422.2 434 1607.4 484 1792.6 €285 1055.6 335 1240.7 395 1425.9 435 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 8 311 1263.0 391 1448.2 441 1633.3 491 1818.5 1081.5 342 293 1085.2 $8 294 -2088.0 1266.7 1270.4 344 1274.1 392 1451.9 442 1637.0 492 1822.2 393 1455.6 443 1640.7 493 1825.9 394 1459.3 444 1644.4 494 1829.6 295 1092.6 345 $277.8 395 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 1235.2 397 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 1848.2 300 1111.1 350 1296.3 400 1481.5 450 1666.7 500 1851.9. 351 TABLE XV.-CUBIC YARDS IN 100 FEET LENGTH. Area. Cubic Aree Cubic Sq. Yards. Sq. Ft. Ft Area. Yards. Sq. Area. Cubic Yards. Sq. Cubic Yards. Area. Cubic Sq. Yards. Ft. Ft. Ft. 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 2233.3 653 504 1866.7 554 2051.9 604 2237.0 654 505 1870.4 555 2055.6 605 506 1874.1 556 2059.3 606 507 1877.8 508 1881.5 509 1885.2 513 514 557 2063.0 558. 2066.7 559 2010.4 510 1888.9 560 2074.1 511 1892.6 561 2077.8 512 1896.3 562 2081.5 1900.0 563 1903.7 2240.7 655 2244.4 607 2248.2 608 2251.9 609 2255.6 610 2259.3 611 2263.0 612 2266.7 2418.5 703 2603.7 2422.2 704 2425.9 705 656 2429.6 657 2433.3 658 2437.0 2607.4 2611.1 706. 2614.8 707 2618.5 708 2622.2 659 2440.7 709 2625.9 660 2444.4 710 2629.6 661 2448.2 711 2633.3 662 2451.9 712 2637.0 2085.2 613 2270.4 663 2455.6 713 2640.7 564 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 2111.1 521 1929 6 571 2114.8 522 1933.3 572 2118.5 620 2296.3 621 2300.0 622 2303.7 670 2481.5 720 2666.7 523 1937.0 573 2122.2 623 2307.4 524 1940.7 574 2125.9 624 2311.1 671 2485.2 672 2488.9 673 2492.6 674 721 722 2674.1 723 2677.8 2670.4 2496.3 724 2681.5 525 1944.4 575 2129.6 625 2314.8 675 2500.0 725 2685.2 526 1948.2 576 2133.3 626 2318.5 676 2503.7 726 2688.9 527 1951.9 577 2137.0 627 2322.2 677 2507.4 727 2692.6 528 1955.6 578 2140.7 628 2325.9 678 2511.1 728 2696.3 529 1959.3 579 2144.4 629 2329.6 679 2514.8 729 2700.0 530 1963.0 580 2148 2 630 2333.3 680 2518.5 730 2703.7 531 1966.7 581 2151.9 631 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 2363.0 688 2548.2 738 2733.3. 539 1996.3 589 2181.5 639 2366.7 540 2000.0 590 541 2003.7 591 542 2007.4 543 2011.1 2185.2 610 2370.4 2188.9 641 2374.1 544 2014.8 592 2192.6 593 2196.3 594 2200.0 644 642 2377.8 689 2551.9 690 2555.6 691 2559.3 692 2563.0 739 2737.0 643 2381.5 693 2566.7 2385.2 694 2570.4 599 545 2018.5 595 2203.7 645 2388.9 546 2022.2 596 2207.4 646 2392.6 547 2025.9 597 2211.1 647 2396.3 548 2029.6 598 2214.8 648 2400.0 549 2033.3 2218.5 649 2403.7 695 274.1 696 2577.8 697 2581.5 740 741 2744.4 742 2748.2 713925.9. 744 2755.6 745 2759.3 746-2763.0 747 2766.7 2740.7 698 2585.2 748 2770.4 699 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. Area. Area. Area. Cubic Sq. Yards. Sq. Cubic Yards. Cubic Cubic Area. Cubic Sq. Yards. Sq. Yards. Sq. Yards. Ft. Ft. Ft. Ft. Ft. 751 2781.5 801 2966.7 851 3151.9 901 3337.0 951 3522.2 752 2785.2 802 2970.4 852 3155.6 902 952 3340.7 3525.9 753 2788.9 803 2974.1 852 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 2992.6 858 908 3177.8 3363.0 958 3548.2 759 2811.1 809 2996.3 859 3181.5 909 3366.7 959 3551.9 760 2814.8 810 3000.0 860 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 3007.4 862 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 8570.4 765 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 3122.2 974 3607.4 775 2870.4 825 3055.6 875 3240.7 925 3425.9 975 3611.1 776 826 2874.1 3059.3 876 3244.4 926 3429.6 976 3614.8 777 2877.8 827 3063.0 877 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 3263.0 931 3448.2 981 3633.3 782 2896.3 832 3081.5 882 3266.7 932 3451.9 982 3637.0 783 2900.0 833 3085.2 883 3270.4 933 3455.6 983 3640 7 794 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 3096.3 886 3281.5 936 3466.7 986 3651.9 787 2914.8 788 2918.5 837 3100.0 838 3103.7 789 2922.2 839 3107.4 790 2925.9 840 3111.1 887 3285.2 937 3470.4 987 3655.6 888 3288.9 938 3474.1 988 3659.3 889 3292.6 939 3477.8 989 3663.0 890 3296.3 940 3481.5 990 3666.7 791 2929.6 841 3114.8 891 3300.0 941 3485.2 991 3670.4 } 792 2933.3 793 842 3118.5 2937.0 843 3122.2 892 3303.7 942 3488.9 992 3674.1 794 2940.7 795 2944.4 796 2948.2 893 3307.4 844 3125.9 894 3311.1 845 3129.6 895 3314.8 846 3133.3 943 3492.6 993 3677.8 914 3496.3 991 3681.5 945 3500.0 995 3685.2 896 3318.5 946 3503.7 996 3688.9 797 2951.9 847 3137.0 897 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 3144.4 899 3329.6 949 3514.8 999 3700.0 800 2963.0 850 3148.2 900 3333.3 950 3518.5 1000 3703 T 353 TABLE XVI. CONVERSION OF ENGLISH INCHES INTO CENTIMETRES. Ius. 0 1 2 3 4 10 5 6 CO 7 8 9 Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. 0 0.000 2.540 5.080 7.620 10 25.40 27.94 30.48 33.02 20 50.80 30 40 60 ΤΟ 80 90 100 12.70 15.24 17.78 20.32 22.86 38.10 40.64 43.18 45.72 48.26 53.34 55.88 58.42 60.96 63.50 66.04 68.58 71.12 73.66 76.20 78.74 81.28 83.82 86.36 88.90 91.44 93.98 96.52 99.06 101.60 104.14 106.68 109.22 111.76 114.30 116.84 119.38 121.92 124.46 50 127.00 129.54 132.08 134.62 137.16 139.70 142.24 144.78 147.32 149.86 152.40 154.94 157.48 160.02 162.56 165.10 167.64 170.18 172.72 175.26 177.80 180.34 182.88 185.42 187.96 190.50 193.04 195.58 198 12 200.96 203.20 205.74 208.28 210.82 213.36 215.90 218.44 220.98 223.52 226.06 228.60 231.14 233.68 236.22 238.76 241.30 243.84 246.38 248.92 251.46 254.00 256.54 259.08 261.62 264.16 266.70 269 24271.78274.32 276.86 CONVERSION OF CENTIMETRES INTO ENGLISH INCHES. 10.16 35.56 Cm. 0 1 2 3 4 5 6 7 8 9 Ins. Ins. Ins. Ins. Ins. 0 0.000 0.394 0.787 1.181 1.575 10 5.512 20 30 40 50 60 9.449 Ins. Ins. Ins. Ins. Ins. 1.969 2.362' 2.756 3.150 3.543 3.937 4.331 4.742 5.118 5.906 6.299 6.693 7.087 7.480 7.874 8.268 8.662 9.055 9.843 10.236 10.630 11.024 11.418 11.811 12.205 12.599 12.992 13.386 13.780 14.173 14 567 14.961 15.355 15.748 16.142 16.530 16.929 17.323 17.71718.111 18.504 18.898 19.292 19.685 20.079 20.473 20.867 21.260 21.654 22.048 22 441 22.835 23.229 23.622 24.016 24.410 24.804 25.197 25.591 25.985 26.378 26.772 27.166 70 27.560 27.953 28.347 28.741 29.134 29.528 29.922 30.316 30.709 31.103 31.497 31.890 32.284 32.678 33.071 33.465 33.859 34.253 34.646 35.040 35.434 35.827 36.221 36.615 37.009] 37.40237.796 38.190 38.583 38.977 39.370 39.764 40.158 40.552 40.945 41.339 41.733 42.126 42.520 42.914 CONVERSION OF ENGLISH FEET INTO METRES. 80 90 100 Feet. 0 10 20 30 40 3 4 5 0 1 2 Met. Met. Met. Met. Met. 0.000 0.3048 0.6096 0.9144 1.2192 3.0479 3.3527 3.6575 3.9623 4.2671 6.0359 6.4006 6.7055 7.0102 7.3150 9.1438 9.4486 9.7534 10.058 10.363 12.192 12.496 12.801 13.106 13.411 50 15.239 15.544 15.849 16.154 16.459 18.287 18.592 18.897 19 202 19.507 21.335 21.640 21.945 22.250 22.555 24.383 24.688 24.993 25.298 25.602 27.431 27.736 28.041 28.346 28.651 100 30.479 30.784 31.089 31.394 31.698 60 70 2888 Met. | 6 | 7 8 9 Met. Met. Met. Met. Met. 1.5239 1.8287 2.1335 2.4383 2.7431 4.5719 4.8767 5.1815 5.4863 5.7911 7.6198 7.9246 8.2294 8.5342 8.8390 10.668 10.972 11.277 11.582 11.887 13.716 14.020 14.325 14.630 14.935 16.763 17.068 17.373 17.678 17.983 19.811 20.116 20.421 20.726 21.031 22.859 23.164 23.469 23.774 24.079 25.907 26.212 26.517 26.822 27.126 28.955 29.260 29.565 29.870 30.174 32.003 32.308 32.613 32.918 33.222 1 CONVERSION OF METRES INTO ENGLISH FEET. 1 5 | 6 | 7 | 8 | 9 0 1 2 3 4 Feet. Feet. Feet. Feet. Feet. Feet. | Feet. Feet. Feet. Feet. 0 0.000 3.2809 6.5618 9.8427 13.123 16.404 19.685 22.966 26.247 29.528 10 32.809 36.090 39.371 42.651 45.932 49.213 52.494 55.775 59.056 62.337 20 65.618 68.899 72.179 75.461 78.741 82.022 85.303 88.584 91.865 95.146 30 98.427 101.71 104.99 108.27 111.55 114 83 118.11 121.39 124.67 127.96 40 131.24 134.52 137.80 141.08 144.36 147.64 150 92 154.20 157.48 160.76 50 164.04 167.33 170 61 173.89 177.17 180.45 183.73 187.01 190.29 193.57 60 196.85 200.13 203.42 206.70 209.98 213.26 216.54 219.82 223.10 226.38 ΤΟ 229.66 232.94 236.22 239 51 242.79 246.07 249.35 252.63 255.91 259.19 80 262.47 265.75 269.03 272.31 275.60 278.88282.16 285.44 288.72 292.00 90 295.28 298.56 391.84 305.12 308.40 311.69 314.97 318.25 321.53 324.81 328.09 331.37 334.65 337.93| 341.21| 344.49|347.78 351.06 354.34357.62 100 354 TABLE XVII. CONVERSION OF ENGLISH STATUTE-MILES INTO KILOMETRES. Miles. 0 1 2 3 4 5 6 17 J 8 9 Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. 0.0000 1.6093 3.2186 4.8279 6.4372, 8.0465 9.6558 11.2652 12.8745 14.4818 16.093 17.70219.312 20.921 22.530 24.139 25 749 27.358 28.967 30.577 32.186 33.795 35.405 37.014 38.623 40.232 41.842 43.451 45.060 46.670 48.279 49.888 51.498 53.107 54.716 56.325 57.935 59.544 61.153 62.763 64.372 65.981 67.591 69.200 70.809 72.418 74.028 75.637 77.246 78.856 80.465 82.074 83.684 85.293 86.902 88.511 90.121 91 780 93.339 94.949 96.558 98.167 99 777 101.39 102.99 104.60 106.21 107.82 109.43 111.04 112.65 114.26 115.87 117.48 119.08 120.69 122.30 123.91 125.52 127.13 128.74 130.35 131.96 133.57 135.17 136.78 138.39 140.00 141.61 143.22 144.85 146.44 148.05 149.66 151.26 152.87 154.48 156.09 157.70 159.31 160.93 162.53 164 14 165 75 167.35 168.96 170.57 172.18 173.79 175.40 CONVERSION OF KILOMETRES INTO ENGLISH STATUTE-MILES. 0 10 20 30 40 50 60 70 80 90 100 'Kilom. 0 1 2 3 4 10 5 6 M 8 9 0 10 20 30 40 50 60 70 80 Miles. Miles. Miles. Miles. Miles. Miles. Miles. Miles. Miles. Miles. 0.0000 0.6214 1.2427 1.8641 2.4855 3.1069 3.7282 4.3497 4.9711 5.5924 6.2138 6.8352 7.45658.0780 8.69941 9.3208 9.9421 10.562 11.185 11.805 12.427 13.049 13 670 14.292 14.913 15.531 16.156 16.776 17.399 18.019 18.641 19.263 19.884 20.506 21.127 21.748 23 370 22.990 23.613 24.233 24.855 25.477 26.098 26.720 27.341 27.962 28.584 29.204 29.827 30.447 31.069 31.690 32.311 32.933 33.554 34.175 34.797 35.417 36.040 36.660 32.311|32.933 37.282 37.904 38.525 39.14739 768 40.389 41.011 41.631 42.254 42.874 43.497 44.118 44.739 45.361 45.982 46.603 47.225 47.845 48.468 49.088 49.711 50.332 50.953 51.575 52.196 52.817 53.439 54.059 54.682 55.302 90 55.924 56.545 57.166 57.788 58.409 59.030 59.652 60.272 60.895] 61.515 100 62.138 62.759 63.380 64.002|64.623 65.244 65.866 66.486 67.109 67.729 LENGTH IN FEET OF 1' ARCS OF LATITUDE AND LONGITUDE. TABLE XVIII. Lat. 1' Lat. 1' Long. Lat. 1' Lat. 1' Long. 1° 6045 6085 31 6061 5222 2° 6045 6083 32° 6062 5166 ३० 6045 6078 33° 6063 5109 4° 6045 6071 34° 6064 5051 50 6045 6063 35° 6065 4991 6° 6045 6053 36° 6066 4930 6046 6041 37° 6067 4867 80 6046 6027 38° 6068 4802 до 6046 6012 39° 6070 4736 10° 6047 5994 40° 6071 4669 11° 6047 5975 41° 6072 4600 12° 6048 5954 420 6073 4530 13° 6048 5931 43° 6074 4458 14° 6049 5907 44° 6075 4385 15° 6049 5880 45° 6076 4311 16° 6050 5852 46° 6077 4235 170 6050 5822 47° 6078 4158 18° 6051 5790 48° 6079 4080 19° 6052 5757 49° 6080 4001 20° 6052 5721 50° 6081 3920 210 6053 5684 51° 6082 3838 22° 6054 5646 52° 6084 3755 23- 6054 5605 53° 6085 3671 24° 6055 5563 54° 6086 3586 25° 6056 5519 55° 6087 3499 26° 6057 5474 56° 6088 3413 27° 6058 5427 57° 6089 3323 28° 6059 5378 58° 6090 3233 29° 6060 5327 59° 6091 3142 30° 6061 5275 60° 6092 3051 TABLE XIX.-TO REDUCE MEAN TO SIDEREAL TIME. Solar Add Hours. Min. Sec. Min. Sec. Solar Add Solar Add Solar Add Solar Add Min. Sec. Sec. Sec. Sec. Sec. 162 CO TŁO TO E- 0 9.86 0 19.71 1 0 29.57 4 0 39.43 5 0 49.28 0.59.14 1 9.00 8 1 18.85 -QDJDO7000 0.16 31 5.09 1 0.00 31 0.08 0.33 32 5 26 2 0.01 32 0.09 0.49 33 5.42 0.01 33 0.09 0.66 34 5.59 0.01 34 0.09 5 0.82 35 5.75 0 01 35 0.10 0.99 36 5.92 0.02 36 0.10 1.15 37 6.08 0.02 37 0.10 8 1.31 38 6.24 0.02 38 0.10 9 1 28.71 9 1.48 39 6.41 9 0.03 39 0.11 10 1 38.57 10 1.64 40 6.57 10 0.03 40 0.11 11 1 48.42 11 1.81 41 6.74 11 0.03 41 0.11 12 1 58.28 12 1.97 42 6.90 12 0.03 42 0.12 13 2 8.13 13 2.14 43 7.07 13 0.01 43 0.12 14 2 17 99 14 2.30 44 7.23 14 0.01 44 0.12 15 2 27.85 15 2.46 45 7.39 15 0.04 45 0.12 16 2 37.70 16 2.63 46 7.56 16 0.04 46 0.13 17 2 47.56 17 2.79 47 7.72 17 0.05 47 0.13 18 2.57.42 18 2.96 48 7.89 18 0.05 48 0.13 19 3 7.27 19 3.12 49 8.05 19 0.05 49 0.13 20 3 17.13 20 3.29 50 8.22 20 0.06 50 0.14 21 3 26.99 21 3.45 51 8.38 21 0.06 51 0.14 22 3 36.84 22 3 61 52 8.54 22 0.06 52 0.14 23 346.70 23 3.78 53 8.71 23 0.06 53 0.15 24 3 56.56 24 3.91 54 8.87 24 0.07 54 0.15 25 4 6.40 25 26 4 16.26 26 27 4 26.13 27 28 4 36.00 29 4 45.86 30 4 55.71 C❤XNNX 4.11 55 9.04 25 0.07 55 0.15 4.27 56 9.20 26 0.07 56 0.15 4.44 28 4.60 29 4.76 30 4.93 5888888 57 9.37 9.53 59 9.69 NOR 27 0.08 57 0.16 28 0.08 58 0.16 29 0.08 59 0.16 60 9.86 30 0.08 60 0.16 356 TABLE XIX-Continued.-TO REDUCE SIDEREAL TO MEAN TIME. Sid. Subtract ubt Sub- Sub- Sub- Sub- Sid. Sid. Sid. Sid. tract tract tract tract Hours. Min. Sec. Min. Min. Sec. Sec. Sec. Sec. Sec. Sec. 1 0 9.83 1 0.16 31 5.08 1 0.00 2 0 19.66 2 0.33 32 5.24 2 0.01 223 31 0.08 32 0.09 3 0 29.49 3 0.49 33 5.41 3 0.01 33 0.09 4 0 39.32 4 0.66 34 5.57 0.01 34 0.09 5 0 49.15 5 0.82 35 5.73 0.01 35 0.10 6 0 58.98 6 0.98 36 5.90 0.02 36 0.10 7 1 8.81 ༩ 1.15 37 6.06 0.02 37 0.10 8 1 18.64 8 1.31 38 6.23 8 0.02 38 0.10 9 1 28.47 9 1.47 39 6.39 9 0.03 39 0.11 10 1 38.30 10 1.64 40 6.55 10 0.03 40 0.11 11 1 48.12 11 1.80 41 6.72 11 0.03 41 0.11 12 1 57.95 12 1.97 42 6.88 12 0.03 42 0.11 13 2 7.78 18 2.13 43 7.04 13 0 04 43 0.12 14 2 17.61 14 2.29 44 7.21 14 0.04 44 0.12 15 2 27.44 15 2.46 45 7.37 15 0.04 45 0.12 16 2 37.27 16 2.62 46 7.54 16 0.04 46 0.13 17 2 47.10 17 2.79 47 7.70 17 0 05 47 0.13 18 2 56.93 18 2.95 48 7.86 18 0.05 48 0.13 19 3 6.76 19 3.11 49 8.03 19 0.05 49 0.13 20 3 16.59 20 3.28 50 8.19 21 3 26.42 21 3.44 51 836 22 3 36.25 22 3.60 52 8.52 23 3 46.08 23 3.77 53 8.68 24 3 55.91 24 3.93 54 8.85 25 4 5.74 25 4.10 55. 9.01 27 29 30 186088 26 4 15.57 26 4.26 56 9.17 4 25.41 27 4.42 57 9.34 28 4 35.24 28 4.59 58 9.50 4 45.07 29 4.75 59 9.67 4 54.90 30 4.92 60 9.83 27RAJAQnka. 20 0.06 50 0.14 21 0.06 51 0.14 0.06 52 0.14 0.06 53 0.14 0.07 54 0.15 25 0.07 55 0.15 26 0.07 56 0.15 0.07 57 0.16 28 0.08 58 0.16 29 0.08 59 0.16 30 0.08 €0 0.16 357 TABLE XX. TO REDUCE TIME TO DEGREES. TO REDUCE DEGREES TO TIME. H. M. O H. M. M. S. M. S. S. T. S. T. Hours. Minutes. Hours. CAWN- 0 4 8 0 12 0 16 0 20 38 39 42 43 46 47 48 49 50 =>x^2 =22#2 9=229 723** 2ARRA 58818 8nage #33IR OL*** 0 24 11 12 13 16 0 28 0 32 0 36 0 40 60 0 44 0 48 0 52 14 0 56 64 15 1 0 1 4 1 8 18 1 12 19 1 16 20 1 20 21 1. 24 1 28 1 32 24 1 36 25 1 40 26 1 44 28 29 1 56 27 1 48 1 52 30 2 0 32 33 2 12 35 2 16 2 20 31 4 2 8 36 2 24 37 2 28 2 32 2 36 2 40 2 44 2 48 2 52 2 56 3 0 3 4 3 8 3 12 3 16 3 20 100 ENKER ADAAO 58845 86882 PRPCP PEP22 538** ***** ***** ****g 3 24 3 101 6 44 1 28 102 6 48 11 3 32 103 6 52 3 36 104 6 56 3 40 105 7 0 3 44 106 7 4 31 3 48 107 7 8 4 3 52 108 7 12 67 3 56 109 7 16 5 4 0 110 7 20 51 4 4 115 7 40 62 4 8 120 8 0 4 12 125 8 20 4 16 130 8.40 4 20 135 9 0 4 24 140 9 20 4 28 145 9 40 4 32 150 10 0 4 36 155 10 20 10 ΤΟ 4 40 160 10 40 10 oSìto Fafog 105 858 1858 Degrees. M. O M. S. T. S. "/ // T. // 123 J 0 15 51 12 45 0 30 52 13 0 0 45 53 13 15 4 1 0 54 13 30 5 1 15 55 13 45 1 30 56 14 O 1 45 57 14 15 2 0 58 14 30 9 2 15 59 14 45 10 2 30 60 15 0 11 2.45 61 15 15 12 3 0 62 15 30 13 3 15 63 15 45 1124 14 3 30 64 16 O 120 15 3 45 65 16 15 127 16 4 0 66 16 30 135 17 4 15 67 16.45 1424 18 4 30 68 17 0 150 19 4 45 69 17 15 157 20 5 0 ΤΟ 17 30 4 44 165 11 0 11 105 21 5 15 71 17 45 ^2 4 48 170 11 20 114 172 22 5 30 72 18 0 4 52 175 11 40 12 180 23 5 45 73 18 15 74 4 56 180 12 0 124 1871 24 6 0 74 18 30 5 0 135 12 20 13 195 25 6 15 75 18 45 76 4 190 12 40 13 2021 26 6 30 76 19 0 S 195 13 0 14 210 27 6 45 77 19 15 78 5 12 200 13 20 144 217½ 28 7 0 78 19 30 5 16 205 13 40 15 225 29 7 15 79 19 45 5 20 210 14 0 15 232 30 730 80 20 0 5 24 215 14 20 16 240 31 7 45 81 20 15 5 28 220 14.40 16 2471 32 8 0 82 20 30 5 32 225 15 0 17 255 33 8 15 83 20 45 84 5 36 230 15 20 17 2621 34 8 30 84 21 0 5 40 235 15 40 18 270 25 8 45 85 21 15 86 5 44 240 16 0 183 2771 36 9 0 86 21 30 87 5 48 245 16 20 19 285 37 9 15 87 21 45 88 5 52 250 16 40 193 2921 38 9 30 88 22 0 5 56 255 17 0 20 300 39 9 45 89 22 15 6 0 260 17 20 20 3071 40 10 0 90 22 30 6 4 270 18 0 21 315 41 10 15 91 22 45 6 8 280 18 40 21 322/ 42 10 30 92 23 0 6 12 290 19 20 22 330 43 10 45 93 23 15 6 16 300 20 0 337 44 11 0 94 23 30 6 20 310 20 40 23 345 45 11 15 95 23 45 6 24 320 21 20 23 3521 46 11 30 96 24 0 6 28 330 22 0 24 360 47 11 45 97 24 15 6 32 340 22 40 48 12 0 98 24 30 6 36 350 6 40 360 23 20 49 12 15 99 24 45 24 0 50 12 30 100 25 0 358 UNIVERSITY OF MICHIGAN 3 9015 06438 6157 1