remote storage THE UNIVERSITY OF ILLINOIS LIBRARY From the library of Harry Harkness Stoek Professor of Mining Engineering 1909-1923 Purchased 1923. 526.9 yeztA REMOTE STORAG THE UNIVERSITY OF ILLINOIS LIBRARY From the library of Harry Harkness Stoek Professor of Mining Engineering 1909-1923 Purchased 1923. 526.9 J63t4- £.AL...s.. .1 . Call Number Overdue books are subject to a fine of 2 cents a day. Name Identification number . ' 1 /' .uyziL Address. , / / Univ. of III Lib. Call Slip Digitized by the Internet Archive in 2017 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/theorypracticeof00john_0 THE THEORY AND PRACTICE OF SURVEYING. DESIGNED FOR THE USE OF SURVEYORS AND ENGINEERS GENERALLY, BUT ESPECIALLY FOR THE USE OF Students in Engineering, BY J. B. JOHNSON, C.E., Professor of Civil Engineering in Washington University, St. Louis, Mo. Formerly Civil Engineer on the U. S. Lake and Mississippi River ’Purveys ; Member of the American Society of Civil Etigineers. FOURTH EDITION. NEW YORK : JOHN WILEY & SONS, 15 Astor Place. 1888. Copyright, 1880, By J. B. JOHNSON. JLsti- PREFACE TO THE FIRST EDITION. No apology is necessary for the appearance of a new book on Surveying. The needs of surveyors have long been far be- yond the accessible literature on this subject, to say nothing of that which has heretofore been formulated in text-books. The author’s object has been to supply this want so far as he was able to do it. The subject of surveying, both in the books and in the schools, has been too largely confined to Land Surveying. The engineering graduates of our technical schools are probably called upon to do more in any one of the departments of Railroad, City, Topographical, Hydrographical, Mining, or Geodetic Surveying than in that of Land Surveying. Some of these subjects, as for example City, Geodetic, and Hy- drographical Surveying, have not been formulated hitherto, in any adequate sense, in either English or any other languasre, to the author’s knowledge. In the case of Geodetic Surveying there has been a wide hiatus between the matter given in text-books and the treatment of the subject in works on Geodesy and in special reports of geodetic operations. The latter were too technical, prolix, and difficult to give to stu- dents, while the former were entirely inadequate to any rea- sonable preparation for this kind of work on even a small scale. The subjects of City and Hydrographical Surveying as here presented are absolutely new. Part I. treats of the adjustment, use, and care of all kinds IV PREFACE. of instruments used in surveying, eitlier in field or office.* In describing the adjustments of instruments the object has been to present to the mind of the reader the geometrical relations from which a rule or method of adjustment would naturally follow. The author has no sympathy with descriptions of ad- justments as mechanical processes simply to be committed to memory, any more than he has with that method of teaching geometry wherein the student is allowed to memorize the demonstration. Many surveying instruments not usually described in books on surveying are fully treated, such as planimeters, panto- graphs, barometers, protractors, etc. The several sets of prob- lems given to be worked out by the aid of the corresponding instruments are designed to teach the capacity and limitations of such instruments, as well as the more important sources of error in their use. This work is such as can be performed about the college campus, or in the near vicinity, and is sup- posed to be assigned for afternoon or Saturday practice while the subject is under consideration by the class. More ex- tended surveys require a special field-season for their success- ful prosecution. t The methods of the differential and integral calculus have been sparingly used, as in the derivation of the barometric for- mula for elevations, and of the L M Z formulae in Appendix D. Such demonstrations may have to be postponed to a later period of the course. % * Certain special appliances, as for example heliotropes, filar micrometers, current-meters, etc., are treated in the subsequent chapters. f At Washington University all the engineering Sophomores go into the field for four weeks at the end of the college year, and make a general land and topographical survey, such as shown in Plate II. At the end of the Junior year the civil-engineering students go again for four weeks, making then a geodetic and railroad survey. Some distant region is selected where the ground, boarding facilities, etc., are suitable. PREFACE. V Part II. includes descriptions of the theory and practice of Surveying Methods in the several departments of Land, Topo- graphical, Railroad, Hydrographical, Mining, City, and Geo- detic Surveying; Surveys for the Measurement of Volumes; and the Projection of Maps, Map Lettering, and Topographi- cal Signs. The author has tried to treat these subjects in a concise, scientific, and practical way, giving only the latest and most approved methods, and omitting all problems whose only claim for attention is that of geometrical interest. In treating the trite subject of Land Surveying many prob- lems which are more curious than useful have been omitted, and several new features introduced. The subjects of com- puting areas from the rectangular co-ordinates, and the supply^ ing of missing data, are made problems in analytical geometry, as they should be. A logarithmic Traverse Table for every minute of arc from zero to 90°, arranged for all azimuths from zero to 360°, to be used in connection with a four-place loga- rithmic table, serves to compute the co-ordinates of lines when the transit is the instrument used. A traverse table com- puted for every 15 minutes of arc is no longer of much value. The isogonic declination-curves shown on Plate I. will be found to embody all the accessible data up to 1885, reduced from the U. S. Coast Survey chart. Appendix A will be found of great value as outlining the Judicial Functions of the Surveyor by the best possible authority. The chapter on Mining Surveying was written by Mr. C, A. Russell, C.E., U. S. Deputy Mineral Surveyor of Boulder, Colorado. He has had an extended experience in Hydro- graphic Surveying, in addition to many years’ practice in sur- veying mines and mining claims. The chapter on City Surveying was written by Mr. Wm. Bouton, C.E., City Surveyor of St. Louis, Mo. Mr. Bouton has done a large proportion of the city surveying in St. Louis vi PREFACE. for the last twenty years, and has gained an enviable reputa- tion as a reliable, scientific, and expert surveyor. It is believed that the ripe experience of these gentlemen which has been embodied in their respective chapters will ma- terially enhance the value of the book. The author also desires to acknowledge his indebtedness to his friend H. S. Pritchett, Professor of Astronomy in Wash- ington University, for valuable assistance in the preparation of the matter on Time in Chapter XIV. Although the theorems and the notation of the method of least squares are not used in this work, yet two problems are solved by what is called the method of the arithmetic mean (which, when properly defined, is the same as the method of least squares), which will serve as a good introduction to the study of the method of least squares. These problems are the Rating of a Current-meter, in Chapter X., and the Adjustment of a Quadrilateral, in Chapter XIV. The author has found that such solutions as these serve to make clear to the mind of the student exactly what is accomplished by the least- square methods of adjusting observations. The chapter on Measurement of Volumes is not intended to be an exhaustive treatment of the subject of earthwork, but certain fundamental theorems and relations are established which will enable the student to treat rationally all ordinary problems. The particular relation between the Henck pris- moid and the warped-surface prismoid is an important one, but one which the author had nowhere found. An earthwork table (Table XL) has also been prepared which gives volumes directly, without correction, for the warped-surface prismoid. The author has no knowledge that sucli a table has ever been prepared before. A former work by the author on Topographical Surveying oy the Transit and Stadia is substantially included in this oook. PREFACE, Vll The methods recommended for measuring base-lines with steel-tapes are new ; but they have been thoroughly tested, and are likely to work a material change in geodetic methods. The author wishes to acknowledge his obligations to many instrument-manufacturers for the privilege they have very kindly accorded to him of having electrotype copies made from the original plates, for many of the cuts of instruments given throughout the book ; persons familiar with the valuable cata- logues published by these firms will recognize the makers among the following: W. & L. E. Gurley, Troy, N. Y. ; Buff & Berger, Boston, Mass.; Fauth & Co., Washington, D. C. ; Queen & Co. and Young & Sons, Philadelphia, Pa.; Keuffel & Esser, New York ; and A. S. Aloe and Blattner & Adam of St. Louis, Mo. Also to Mr. W. H. Searles for the privilege of using copies of plates from his Field-book for Tables L, VI., and VII. Hoping this work will assist in lifting the business of sur- veying to a higher professional plane, as well as to enlarge its boundaries, the author submits it to surveyors and engineers generally, but especially to the instructors and students in our polytechnic schools, for such crucial tests as the class-room only can give. J, B. J. St. Louis, Sept. 23, 1886. ^ Sc* TABLE OF CONTENTS. PAGE Introduction j BOOK I. SURVEYING INSTRUMENTS. CHAPTER I. INSTRUMENTS FOR MEASURING DISTANCES. The Chain ; 1. The Engineer’s Chain 5 2. Gunter’s Chain 5 3. Testing the Chain 6 4. The Use of the Chain 8 The Steel Tape : 5. Varieties 9 6. The Use of Steel Tapes 10 Exercises with the Chain : 7-17. Practical Problems ii, 12 CHAPTER II. INSTRUMENTS FOR DETERMINING DIRECTIONS. The Compass : 18. The Surveyor’s Compass described 13 19. The General Principle of Reversion. 15 20. To make the Plate perpendicular to the Axis of the Socket 16 21. To make the Plane of the Bubbles perpendicular to the Axis of the Socket 16 22. To adjust the Pivot to the Centre of the Graduated Circle 16 X cox TRACTS. 23. To straighten the Needle 24. To make the Plane of the Sights normal to the Plane of the Hubbles. 25. To make the Diameter through the Zero-graduations lie in the Plane of the Sights 26. To remagnetize the Needle 27. The Construction and Use of Verniers The Declination of the Needle: 28. The Declination defined 29. The Daily Variation 30. The Secular Variation. . 31. Isogonic Lines 32. Other Variations of the Declination 33. To find the Declination of the Needle Use of the Needle Compass : 34. The Use of the Compass 35. To set off the Declination 36. Local Attractions 37. To establish a Line of a Given Bearing 38. To find the True Bearing of a Line.- 39. To retrace an Old Line The Prismatic Compass : 40. The Prismatic Compass described Exercises : 41-44. Exercises for the Needle Compass 38, The Solar Compass : 45. The Burt Solar Compass 46. Adjustment of the Bubbles 47. Adjustment of the Lines of Collimation 48. Adjustment of the Declination Vernier 49. Adjustment of the Vernier on the Latitude Arc 50. Adjustment of Terrestrial Line of Sight to the Plane of the Polar Axis Use of the Solar C»,' mpass : 51. Conditions requiring its Use 52. To find the Declination of the Sun 53. To correct the Declination for Refraction 54. Errors in Azimuth due to Errors in the Declination and Latitude Angles 55. Solar Attachments Exercises with the Solar Compass : 56-59. Practical Problems 53, ■AGR 17 17 17 18 18 20 20 21 23 29 29 34 36 36 37 37 37 38 39 39 41 41 42 43 43 44 44 45 49 52 54 CONTENTS. xi CHAPTER III. INSTRUMENTS FOR DETERMINING HORIZONTAL LINES. PAGE Plumb-line and Bubble : 60. Their Universal Use in Surveying and Astronomical Work 55 61. The Accurate Measurement of small Vertical Angles 58 62. The Angular Value of one Division of the Bubble 58 63. General Considerations 59 The Engineer’s Level : 64. The Level described 60 65. Adjustment of Line of Sight and Bubble Axis to Parallel Positions. 63 66. Lateral Adjustment of Bubble 67 67. The Wye Adjustment 67 68. Relative Importance of Adjustments ^ 68 69. Focussing and Parallax 68 70. The Levelling-rod 70 71. The Use of the Level 71 Differential Levelling : 72. Differential Levelling defined 72 73. Length of Sights 73 74. Bench-marks 74 75. The Record 75 76. The Field work 76 Profile Levelling : 77. Profile Levelling defined 77 78. The Record 78 Levelling for Fixing a Grade : 79. The Work described 81 The Hand Level: 80. Locke’s Hand Level 81 Exercises with the Level : 81-85. Practical Problems 82 CHAPTER IV. INSTRUMENTS FOR MEASURING ANGLES. THE TRANSIT. 86. The Engineer’s Transit described 83 87. The Adjustments slated 86 88. Adjustment of Plate Bubbles 86 89. Adjustment of Line of Collimation 87 xn CONTENTS. I'AGK 90. Adjustment of the Horizontal Axis 87 91. Adjustment of the Telescope Hubble 89 92. Adjustment of Vernier on Vertical Circle 89 93. Relative Importance of Adjustments 89 Instrumental Conditions affecting Accurate Measuremen is : 94. Eccentricity of Centres and Verniers 90 95. Inclination of Vertical Axis 91 96. Inclination of Horizontal Axis 92 97. Error in Collimation Adjustment 93 The Use of the Transit : 98. To measure a Horizontal Angle 93 99. To measure a Vertical Angle 94 100. To run out a Straight Line . . 95 101. Traversing 97 The Solar Attachment : 102. Various Forms described 99 103. Adjustments of the Saegmuller Attachment 102 The Gradienter Attachment : 104. The Gradienter described 104 The Care of the Transit : 105. The Care of the Transit 104 Exercises with the Transit : 106-114. Practical Problems 105-107 the sextant. 115. The Sextant described 108 116. The Theory of the Sextant no 117. The Adjustment of the Index Glass in 118. The Adjustment of the Horizon Glass in 119. The Adjustment of the Telescope to the Plane of the Sextant in 120. The Use of the Sextant 112 Exercises with the Sextant : 121. 121^. Practical Problems 112, 113 The Goniograph : 122. The Double-reflecting Goniograph 113 CHAPTER V. THE PLANE TABLE. 123. The Plane Table described 117 124. Adjustment of the Plate Bubbles 119 125. Adjustment of Horizontal Axis 119 CON TEN 7' S. XIU PAGE 126. Adjustment of Vernier and Bubble to Telescopic Line of Sight.. . 119 The Use of the Plane Table: 127. General Description of its Use 120 128. Location by Resection 123 129. Resection on Three Known Points 123 130. Resection on Two Known Points 124 131. The Measurement of the Distances by Stadia 125 Exercises with the Plane Table : 132-135. Practical Problems 126 t CHAPTER VI. ADDITIONAL INSTRUMENTS USED IN SURVEYING AND PLOTTING. The Aneroid Barometer; 136. The Aneroid described 127 137. Derivation of Barometric Formulae 129 138. Use of the Aneroid 136 The Pedometer ; 139. The Pedometer described 137 The Length of Men’s Steps 138 The Odometer : 140. Description and Use 139 The Clinometer : 141. Description and Use 141 The Optical Square : 142. Description and Use 142 The Planimeter : 143. Description 143 144. Theory of the Polar Planimeter 144 145. To find the Length of Arm to use 150 146. The Suspended Planimeter 152 147. The Rolling Planimeter 152 148. Theory of the Rolling Planimeter 154 149. To Test the Accuracy of a Planimeter 157 150. The Use of the Planimeter 158 151. Accuracy of Planimeter Measurements, 160 The Pantograph : 152. Description and Theory 161 Various Styles of Pantographs 163 153. Use of the Pantograph 165 xiv CONTENTS. PAGR Protractors : 154. Various Styles described 166 Parallel Rulers : 155. Description and Use 169 Scales ; 156. Various Kinds described 169 BOOK II. SUR VE YING ME TJ/ODS. CHAPTER VII. LAND-SURVEYING. 157. Land-surveying defined 172 158. Laying out Land 172 The United States System of Laying out the Public Lands: 159. Origin and Region of Application of the System 173 160. The Reference Lines 173 161. The Division into Townships 174 162. The Division into Sections 175 163. The Convergence of the Meridians 176 164. The Corner Monuments 178 Finding the Area of Superficial Contents of Land when the Limiting Boundaries are given : 165. The Area defined 179 By Triangular Subdivision : 166. By the Use of the Chain alone 180 167. By the Use of the Compass or Transit and Chain 180 168. By the Use of the Transit and Stadia 181 From Bearing and Length of the Botmdary Lines : 169. The Common Method 181 170. The Field Notes 182 171. Method of Computation stated 185 172. Latitudes, Departures, and Meridian Distances 1S5 173. Computation of Latitudes and Departures of the Courses 187 174. Balancing the Survey 190 175. The Error of Closure 193 17C. The Form of Reduction 194 177. Area Correction due to Erroneous Length of Chain 197 CONTENTS. XV PAGE Area from the Rectangular Co ordinates of the Corners : 178. Conditions of Application of the Method. 200 179. Theory of the Method 201 180. The Form of Reduction 203 Supplying Missing or Erroneous Data : 181. Equations for Supplying Missing Data — Four Cases 203 Plotting : i^ia. Plotting the Survey 208 Irregular Areas : 182. The Method by Offsets at Irregular Intervals 208 183. The Method by Offsets at Regular Intervals 210 The Subdivision of Land : 184. The Problems of Infinite Variety 213 185. To cut from a Given Tract of Land a Given Area by a Right Line starting from a Given Point in the Boundary 213 186. To cut from a Given Tract of Land a Given Area by a Right Line running in a Given Direction 215 Examples : 187-196^. Practical Problems 220-222 CHAPTER VIIL TOPOGRAPHICAL SURVEYING BY THE TRANSIT AND STADIA. 197. Topographical Survey defined 223 198. Available Methods 223 199. Method by Transit and Stadia stated 224 Theory of Stadia Measurements : 200. Fundamental Relations 224 201. Method Used on the Government Surveys 230 202. Another Method of Graduating Rods 231 203. Adaptation of Formulae to Inclined Sights • 231 204. Description and Use of the Stadia Tables 233 205. Description and Use of the Reduction Diagram 235 The Instruments : 206. The Transit 235 207. Setting the Cross-wires 236 208. Graduating the Stadia Rod 237 General Topographical Surveying : 209. The Topography 241 210. Methods of Field Work 241 XVI CONTENTS. I'ACR 211. Reduction of the Notes 249 212. Plotting the Stadia Line 252 213. Check Readings 253 214. Plotting the Side Readings 254 215. Contour Lines 259 216. The Final Map 2O2 217. Topographical Symbols 263 218. Accuracy of the Stadia Method 263 CHAPTER IX. RAILROAD TOPOGRAPHICAL SURVEYING. 219. Objects of the Survey 265 220. The Field Work 265 221. The Maps 267 222. Plotting the Survey 269 223. Making the Location on the Map 271 224. Another Method 275 CHAPTER X. HYDROGRAPHIC SURVEYING. 225. Hydrographic Surveying defined 277 The Location of Soundings ; 226. Enumeration of Methods 278 227. By Two Angles read on Shore 279 228. By Two Angles read in the Boat — The Three-point Problem 279 229. By one Range and one Angle ' 282 230. Buoys, Buoy-flags, and Range-poles 283 231. By one Range and Time-intervals 284 232. By means of Intersecting Ranges 284 233. By Means of Cords or Wires 284 Making the Soundings : 234. The Lead 285 235. The Line 285 236. Sounding Poles 287 237. Making Soundings in Running Water 2S7 238. The Water-surface Plane of Reference 2S7 239. Lines of Equal Depth 23 • CONTENTS. XVll PAGE 240. Soundings on Fixed Cross-sections in Rivers 288 241. Soundings for the Study of Sand-waves 289 242. Areas of Cross-section 290 Bench-marks, Gauges, Water-levels, and Water-Slope : 243. Bench-marks 291 244. Water Gauges 291 245. Water-levels 292 246. River-slope 293 The Discharge of Streams : 247. Measuring Mean Velocities of Water-currents 294 248. Use of Sub-surface Floats 295 249. Use of Current Meters 300 250. Rating the Meter 301 251. Use of Rod Floats 307 I 252. Comparison of Methods 308 253. The Relative Rates of Flow in Different Parts of the Cross section 309 254. To find the Mean Velocity on the Cross-section 312 255. Sub-currents 316 256. The Flow over Weirs 316 257. Weir Formulae and Corrections 319 258. The Miner’s Inch 322 259. Formulae for the Flow of Water in Open Channels — Kutter’s For- mula 323 260. Cross sections of Least Resistance 328 Sediment Observations : 261. Methods and Objects 329 262. Collecting the Specimens of Water 331 263. Measuring out the Samples 331 264. Siphoning off, Filtering, and Weighing the Sediment 332 CHAPTER XL MINING SURVEYING. 265. Definitions 333 266. Stations 335 267. Instruments 335 268. Mining Claims 339 Underground Surveys : 269. Mining Surveying proper 343 270. To determine the Position of the End or Breast of a Tunnel and its Depth below the Surface at that Point 343 XVlll CONTENTS. PAC.n 271. Required, the Distance that a Tunnel will have to be driven to cut a Vein with a Certain Dip. — Two Cases 346 272. Required the Direction and Distance from the Dreast of a Tunnel to a Shaft, and the Depth at which it will cut the Shaft 34S 273. To Survey a ]\Iine with its Shafts and Drifts 351 274. Conclusion 354 CHAPTER XII. CITY SURVEYING. 275. Land-surveying Methods inadequate in City Work 356 276. The Transit 357 277. The Steel Tape 357 Laying Out a Town Site : 278. Provision for Growth 359 279. Contour Maps / 360 280. The Use of Angular Measurements in .Subdivisions 360 281. Laying out the Ground 361 2S2. The Plat to be Geometrically consistent 363 283. Monuments 363 284. Surveys for Subdivision 365 285. The Datum-plane 369 286. The Location of Streets 369 287. Sewer Systems 370 2S8. Water supply 370 289. The Contour Map 371 Methods of Measurement : 290. The Retracing of Lines 371 291. Erroneous Standards 372 292. True Standards 373 293. The Use of the Tape 374 294. Determination of the “Normal Tension” 376 295. The Working Tension 380 296. The Effect of Wind 381 297. The Effect of Slope. 382 298. The Temperature Correction 382 299. Checks 383 Miscellaneous Prohlems : 300. The Improvement of Streets 3^4 301. Permanent Bench-marks 384 CONTENTS. XIX 302. The Value of an Existing Monument 303. The Significance of Possession 304. Disturbed Corners and Inconsistent Plats 305. Treatment of Surplus and Deficiency 306. The Investigation and Interpretation of Deeds 307. Office Records 308. Preservation of Lines 309. The Want of Agreement between Surveyors.. . PAGE 385 3S7 38S 3S9 391 391 392 393 CHAPTER XIII. THE MEASUREMENT OF VOLUMES. 310. Proposition 394 311. Grading over Extended Surfaces 396 312. Approximate Estimates by Means of Contours 399 313. The Prismoid 402 314. The Prismoidal Formula 402 315. Areas of Cross-section 404 316. The Centre and Side Heights 405 317. The Area of a Three-level Section 405 318. Cross-sectioning 406 319. Three-level Sections, the Upper Surface consisting of two Warped Surfaces 408 320. Construction of Tables for Prismoidal Computation • 410 321. Three-level Sections, the Surface divided into Four Planes by Diagonals 413 322. Comparison of Volumes by Diagonals and by Warped Surfaces. , . 415 323. Preliminary Estimates from the Profiles... 417 324. Borrow-pits 420 325. Shrinkage of Earthwork 420 326. Excavations under Water 421 CHAPTER XIV. GEODETIC SURVEYING. 527. Objects of a Geodetic Survey 424 328. Triangulation Systems 425 329. The Base-line and its Connections 427 330. The Reconnaissance 429 XX CONTENTS. pAore 331. Instrumental Outfit for Reconnaissance 431 332. The Direction of Invisible Stations 432 333. The Heights of Stations 432 334. Construction of Stations 437 335. Targets 438 336. Heliotropes 442 337. Station Marks 444 Measuremi-:nt of the Base Line: 338. Methods 447 The Steel Tape 449 339. Method of Mounting and Stretching the Tape 450 340. M. Jaderin’s Method 453 341. The Absolute Length of Tape 455 342. The Coefficient of Expansion 456 343. The Modulus of Elasticity 457 344. Effect of the Sag 457 345. Temperature Correction 459 346. Temperature Correction when a Metallic Thermometer is used... 460 347. Correction for Alignment 461 348. Correction for Sag 463 349. Correction for Pull 465 350. Elimination of Corrections for Sag and Pull 465 351. To reduce a Broken Base to a Straight Line 468 352. To reduce the Length of the Base to Sea-level 468 353. Summary of Corrections 469 354. To compute any Portion of a Broken Base which cannot be directly measured 472 355. Accuracy attainable by Steel tape and Metallic-wire Measure- ments 473 Measurement of the Angles : 356. The Instruments 477 357. The Filar Micrometer 480 358. The Programme of Observations 483 359. The Repeating Method. . . 484 360. Method by Continuous Reading around the Horizon 485 361. Atmospheric Conditions 487 362. Geodetic Night Signals 488 363. Reduction to the Centre 488 Adjustment of the Measured Angles : 364. Equations of Condition 49I 365. Adjustment of a Triangle 493 CONTENTS. xxi PAGE Adjustment of a Quadrilateral : 366. The Geometrical Conditions 494 367. The Angle-equation Adjustment 495 368. The Side-equation Adjustment 497 369. Rigorous Adjustment for Angle- and Side-equations 501 Example of Quadrilateral Adjustment..^ 504 Adjustment of Larger Systems : 370. Used only in Primary Triangulation 506 371. Computing the Sides of the Triangles 506 Latitude and Azimuth : 372. Conditions 508 373. Latitude and Azimuth by Observations on Circumpolar Stars at Culmination and Elongation 508 374. The Observation for Latitude 512 375. First Method 513 376. Second Method 513 377. Correction for Observations not on the Meridian 514 378. The Observation for Azimuth 515 379. Corrections for Observations near Elongation 517 380. The Target 518 381. The Illumination of Cross-wires 518 Time and Longitude : 382. Fundamental Relations 519 383. Time 520 384. Conversion of a Sidereal into a Mean Solar Time Interval, and vice versa 522 385. To change Mean Time into Sidereal Time 524 386. To change from Sidereal to Mean Time 525 387. The Observation for Time 526 388. Selection of Stars, with List of Southern Time-Stars for each Month. 526 389. Finding the Mean Time by Transit 530 390. Finding the Altitude 531 391. Making the Observations 532 392. Longitude 534 393. Computing the Geodetic Positions 535 394. Example of ZAf Z Computation 539 Geodetic Levelling : 395. Of Two Kinds 340 ( A ) Trigonometrical Levelling: 396. Refraction 540 397. Formulae for Reciprocal Observations 541 XXll CONTENTS. PAGB 398. Formulae for Observations at One Station only 543 399. Formulae for an Observed Angle of Depression to a Sea Horizon. . 545 400. To find the Value of the Coefficient of Refraction 546 { B ) Precise Spirit-Levelling; 401. Precise Levelling defined 547 402. The Instruments 548 403. The Instrumental Constants 550 404. The Daily Adjustments 553 405. Field Methods 555 406. Limits of Error 558 407. Adjustment of Polygonal Systems 559 408. Determination of the Elevation of Mean Tide 563 CHAPTER XV. PROJECTION OF MAPS, MAP-LETTERING, AND TOPOGRAPHICAL SYMBOLS. Projection of Maps: 409. Purpose of the Map 564 410. Rectangular Projection 564 41 1. Trapezoidal Projection 565 412. The Simple Conic Projection 566 413. De ITsle’s Conic Projection 567 414. Bonne’s Projection 567 415. The Polyconic Projection 568 416. Formulae used in the Projection of Maps. 568 417. Meridian Distances in Table VIII 571 418. Summary 572 419. The Angle of Convergence of Meridians 574 Map-Lettering and Topographical Symbols: 420. Map Lettering 575 421. Topographical Symbols 576 CONTENTS. xxiii PAGE APPENDIX A. The Judicial Functions of Surveyors 579 APPENDIX B. Instructions to U. S. Deputy Mineral Surveyors 589 APPENDIX C. Finite Differences 605 APPENDIX D. Derivation of Geodetic Formula 611 TABLES. I. — Trigonometrical Formula 625 11 . — For Converting Metres, Feet, and Chains 629 III. — Logarithms of Numbers to Four Places 630 IV. — Logarithmic Traverse Table 632 V. — Stadia Reductions for Horizontal Distance and for Eleva- tion 640 VI. — Natural Sines and Cosines 648 ^VII. — Natural Tangents and Cotangents 657 VIII. — Coordinates for Polyconic Projection 669 IX.— Vai.ues of Coefficient in Kutter’s P^ormula 670 X. — Diameters of Circular Conduits, by Kutter’s Formula 671 XL — Earthwork Table — Volumes by the Prismoidal Formula 672 SURVEYING INTRODUCTION. Surveying is the art of making such field observations and measurements as are necessary to determine positions, areas, volumes, or movements on the earth’s surface. The field opera- tions employed to accomplish any of these ends constitute a survey. Accompanying such survey there is usually the field record, the computation, and the final maps, plats, profiles, areas, or volumes. The art of making all these belongs, therefore, to the subject of surveying. Inasmuch as all fixed engineering structures or works involve a knowledge of that portion of the earth’s surface on which they are placed, together with the necessary or resulting changes in the same, so the execution of such works is usually accompa- nied by the surveys necessary to obtain the required informa- tion. Thus surveying is seen to be intimately related to en- gineering, but it should not be confounded with it. All engineers should have a thorough knowledge of surveying, but a surveyor may or may not have much knowledge of engineer- ing. The subject of Surveying naturally divides itself into — I. The Adjustment, Use, and Care of Instruments. II. Methods of Field Work, III. The Records, Computations, and Final Products. All the ordinary instruments that a surveyor may be called upon to use in any of the departments of the work will be dis- cussed in the following pages. The most approved methods 2 INTRODUCTION. only will be given for obtaining the desired information, and many problems that are more curious than useful will not be mentioned. The student is assumed to possess a knowledge of geometry, and of plane and spherical trigonometry. He is also supposed to be guided by an instructor, and have access to most of the instruments here mentioned, with the privilege of using them in the field. The'field work of surveying consists wholly of measuring dis- tances, angles, and time, and it is well to remember that no meas- urement can ever be made exactly. The first thing the young sur- veyor needs to learn, therefore, is the proportionate error in the special work assigned him to perform. It is of the utmost importance to his success that he shall thoroughly study this subject. He should know what all the sources of error are, and their relative importance; also the relative cost of diminishing the size of such errors. Then, with a given standard of accuracy, he will know how to make the survey of the required standard with the least expenditure of time and labor. He must not do all parts of the work as accurately as possible, or even with the same care. For, if the expense is proportioned to the accuracy of results, then he is the most successful surveyor who does his work just good enough for the purpose. The relative size of the various sources of error is of the utmost importance. One should not expend considerable time and labor to reduce the error of measurement of a line to i in 10,000 when the unknown error in the length of the measuring unit may be as high as i in 1000. The surveyor must carefully discriminate, also, between com- pensating errors and cumulative errors. A compensating error is one which is as likely to be plus as minus, and it is therefore largely compensated in, or eliminated from, the result. A cumulative error is one which always enters with the same sign, and therefore it accumulates in the result. Thus, in chaining, the error in setting the pin is a compensating error, wliile the error from erroneous length of chain is a cumulative error. If a mile is cliained with a 66-foot chain, tiiere are 80 measurements INTRODUCTION. 3 taken. Suppose the error of setting the pin be 0.5 inch, and the error in the length of the chain be o.i inch. Now the theory of probabilities shows us that in the case of compensating errors the square root of tlie number of errors probably'^ remains un- compensated. The probable error from setting the pins is therefore 9 X 0.5 inch = 4.5 inches. The error from erroneous length of chain is 80 X o. i inch = 8 inches. Thus we see that although the error from setting the pins was five times as great as that from erroneous length of chain, yet in running one mile, the resulting error from the latter cause was nearly twice that from the former. A careful study of the various sources of error affecting a given kind of work will usually enable tlie surveyor either to add greatly to its accuracy without increasing its cost, or to greatly diminish its cost without diminishing its accuracy. The surveyor should have no desire except to arrive at the truth. This is the true scientific spirit. He should be most severely honest with himself. He should not allow himself to change or “fudge” his notes without sufficient warrant, and then a full explanation should be made in his note-book. Neither should he make his results appear more accurate than they really are. He should always know what was about the relative accuracy witli which his field work was done, and carry his results only so far as the accuracy of the work would war- rant. He is either foolish or dishonest who, having made a survey of an area, for instance, with an error of closure of i in 300, should carry his results to six significant figures, thus giv- ing the area to i in 500,000. It is usual to carry the computa- tions one place farther than the results are known, in order that no additional error may come in from the computation. It is not unusual, however, to see results given in published docu- ments to two, three, or even four places farther than the observa- tions would warrant. *The meaning of this statement is that on the average this will occur oftener than any other combination, and that any single result will, on the average, be nearer to this result than to any other. 4 INTRODUCTION. The student should make himself familiar witli the structure and use of every part of every instrument put into his hands. The best way of doing this is to take the instrument all apart and put it together again. This, of course, is not practicable for each student in college, but when he is given an instrument in real practice, he should then make himself thoroughly familiar with it before attempting to use it. The adjustments of instruments should be studied as problems in descriptive geometry and not as mechanical manipulations, learned in a mechanical way; and when adjusting an instrument the geometry of the problem should be in the mind rather than the rule in the memory. Students of engineering in polytechnic schools are urged to make themselves familiar with every kind of instrument in the outfit of the institution, and to do in the field every kind of work herein described if pos-sible. Otherwise he may be called upon to do, or to direct others to do, what he has never done himself, and he will then find that his studies prove of little avail with- out the real knowledge that comes only from experience. BOOK 1. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. CHAPTER I. INSTRUMENTS FOR MEASURING DISTANCES. THE CHAIN. 1. The Engineer’s Chain is 50 or 100 feet long, and should be made of No. 12 steel wire. The links are one foot long, including the connecting rings, and links should be brazed to prevent giving. The connections are designed so as to admit of as little stretch as possible. Every tenth foot is marked by a special form of brass tag. If the chain is adjustable in length, it should be made of standard length by meas- uring from the inside of the handle at one end to the outside of the handle at the other. If it is not adjustable, measure from the outside of the handle at the rear end to the standard mark at the forward end. 2. Gunter’s Chain is 66 feet long, and is divided into 100 links, each link being 7.92 inches in length. This chain is mostly used in land-surveying, where the acre is the unit of measure. It was invented by Edmund Gunter, an English All joints in rings Fig. I. 6 SURVEYING. astronomer, about 1620, and is very convenient for obtaining areas in acres or distances in miles. Thus, One mile = 80 chains ; also, One acre = 160 square rods, = 10 square chains, = 100,000 square links. If, therefore, the unit of measure be chains and hundredths (links), the area is obtained in square chains and decimals, and by pointing off one more place the result is obtained in acres. This is the length of chain used on all the U. S. land surveys. In all deeds of conveyance and other documents, when the word chain is used it is Gunter’s chain that is meant. 3. Testing the Chain. — No chain, of whatever material or manufacture, will remain of constant length. The length changes from temperature, wear, and various kinds of distor- tion. A change of temperature of 70° F. in a lOO-foot chain will change its length by 0.05 foot, or a change of i in 2000. If the links of a chain are joined by three rings, then there are eight wearing surfaces for each link, or eight hundred wearing surfaces for a 66- or loo-foot chain. If each surface should wear o.oi inch, the chain is lengthened by eight inches. It is not uncommon for a railroad survey of, say, 300 miles to be run with a single chain. If such a chain were of exactly the right length at the beginning of the survey, it might be six inches too long at the end of it. The change of length from distortion may come from a flattening out of the connecting rings, from bending the links, or from stretching the chain beyond its elastic limit, thus giv- ing it a permanent set. Both the wear and the distortion are likely to be less for a steel chain than for an iron one. When a bent link is straightened it is permanently lengthened. When we remember that all unknowm changes in the length of the chain produce cumulative errors in the meas- ured lines, we see how important it is that the true length of ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, / the chain should be always known, or better, that the standard length (50, 66, or 100 feet) should be properly measured from one end of the chain and marked at the other. This chain test is most readily accomplished by the aid of a standard steel tape, which is at least as long as the chain. By the aid of such a tape a standard length may be laid off on the floor of a large room, or two stones may be firmly set in the ground at the proper distance apart and marks cut upon their upper sur- faces. If stones are used they should reach below the frost- line. Or a short tape, or other standard measuring unit, may be used for laying off such a base-line. By whatever means it is accomplished, some ready means should at all times be available for testing the chain. Since a chain always grows longer with use, the forward end of the chain will move farther and farther from the standard mark. A small file- mark may be made on the handle or elsewhere, and then re- moved when a new test gives a new position. Care must be exercised to see that there are no kinks in the chain either in testing or in use. In laying out the standard base the temperature at which the unit of measure is standard should be known (this tempera- ture is stamped on the better class of steel tapes), and if the base is not laid out at this temperature, a correction should be made before the marks are set. The coefficient of expansion of iron and steel is very nearly 0.0000065 for 1° F. If T'o be the temperature at which the tape is standard, T the tem- perature at which the base is measured, and L the length of the base, then 0.0000065 {T^— T)L is the correction to be applied to the measured length to give the true length. When the chain is tested by this standard base the tem- perature should be again noted, and if this is about the mean temperature for the field measurements no correction need be made to the field work. If it is known, at the time the chain is tested, that the temperature is very different from the prob- 8 S UR VE YING. able mean of the field work, then the standard mark can be so placed on the chain as to make it standard when in use. 4. The Use of the Chain. — The chain is folded by taking it by the middle joint and folding the two ends simultaneous- ly. It is opened by taking the two handles in one hand and throwing the chain out with the other. Since horizontal distances are always desired in surveying, the chain should be held horizontally in measuring. Points vertically below the ends of the chain are marked by iron pins, the head chainman placing them and the rear chainman remov- ing them after the next pin is set. The chain is lined in either by the head or rear chainman, or by the observer at the instru- ment, according as the range-pole is in the rear, or in front, or not visible by either chainman. When chaining on level ground, the rear chainman brings the outside of the handle against the pin, and the head chainman sets the forward side of his pin even with the standard mark on the chain. By this means the centres of the pins are the true distance apart. On uneven ground both chainmen cannot hold to the pin ; one end being elevated in order to bring the chain to a horizontal position. In this case there are three difficulties to be over- come. The chain should be drawn so taut that the stretch from the pull would balance the shortening from the sag; the chain should be made horizontal ; the elevated end-mark must be transferred vertically to the ground. It is practically im- possible to do any of these exactly. The first could be deter- mined by trial. Stretch the chain between two points at the same elevation, having it supported its entire length. Then remove the supports, and see how strong a pull is required to bring it to the marks again. This should be done by the chain- men themselves, thus enabling them to judge how hard to pull it when it is off the ground. To hold the chain horizontal on sloping ground is very difficult, on account of the judgment being usually very much in error as to the position of a hori- ADJUSTMENT, USE, AND CADE OF INSTRUMENTS. 9 zontal line. In all such cases the apparently horizontal line is much too nearly parallel with the ground. Sometimes a level has been attached to one end of the chain, in which case it should be adjusted to indicate horizontal end-positions for a certain pull, this being the pull necessary to overcome the shortening from sag. To hold a plumb-line at the proper mark, with the chain at the right elevation, and stretched the proper amount, requires a steady hand in order that the plumb-bob may hang stationary. This should be near the ground, and when all is ready, it is dropped by the chainman letting go the string. The pin is then stuck and the work proceeds. It is common in this country for the rear chainman to call “ stick” when he is ready, and for the head chainman to answer “ stuck” when he has set the pin. The rear chainman then pulls his pin and walks on. There should be eleven pins, marked with strips of colored flannel tied in the rings to assist in finding them in grass or brush. In starting, the rear chainman takes a pin for the initial point, leaving the head chainman with ten pins. When the last pin is stuck, the head chainman calls out,” and waits by this station until the rear chainman comes up and delivers over the ten pins now in his possession. The eleventh pin is in the ground, and serves as the initial point for the second score. Thus only every ten chains need be scored. Good chaining, therefore, consists in knowing the length of the chain, in true alignment, horizontal and vertical, and in proper stretching, marking, and scoring. THE STEEL TAPE.- 5. Varieties. — Steel tapes are now made from one yard to 1000 feet in length, graduated metrically, or in feet and tenths. A pocket steel tape from three to ten feet long should always be carried by the surveyor. A 50-foot tape is best fitted to city surveying where there are appreciable grades. For cities lO SU/^ VE YING. without grades a loo-foot tape might be found more useful. For measuring base-lines, or for some kinds of mining surveying, a 300 or 500 foot tape is best. These are of small cross-section, being about o.i inch wide and 0.02 inch thick. A tape about Fig. 2. 0.5 inch wide and 0.02 fneh thick (Fig. 2) is perhaps best suited to general surveying. 6. The Use of Steel Tapes. — Steel tape-measures are used just as chains are. They are provided with handles, but the end graduation-marks are usually on the tape itself and not on the handle. They are graduated to order, the graduations being either etched or made on brass sleeves which are fastened on the tape. Their advantages are many. They do not kink, stretch, or wear so as to change their length, so that, with careful handling, they remain of constant length except for temperature. They are used almost exclusively in city and bridge work, and in the measurement of secondary base-lines. The same precautions must be taken in regard to alignment, pull, and marking with the tape, as was described for the chain.'^* * For methods of using the steel tape in accurate measurements, see Chap- ter XIV., Base-Line Measurements. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II EXERCISES. To be worked out on the ground by the use of the chain or tape alone. 7. To chain a line over a hill between two given points, not visible from each other. Range-poles are set at the given points. Then the two chainmen, each with a range-pole, range themselves in between the two fixed points, near the top of the hill, by successive approximations. The line can then be chained. 8. To chain a line across a valley between two fixed points. Establish other range-poles by means of a plumb-line held on range between the points. 9. To chain a line between two fixed points when woods intervene, and the true line is not to be cleared out. Range out a trial line by poles, leaving fixed points. Find the resulting error at the terminus, and move all the points over their proportionate amount. The true line may then be chained. 10. To set a stake in a line perpendicular to a given line at a given point. All multiples of 3, 4, and 5 are the sides of a right-angled triangle; also any angle in a semicircumference is a right angle. 11. To find where a perpendicular from a given point without a line will meet that line. Run an inclined line from the given point to the given line. Erect a per- pendicular from the given line near the required point, extend it till it intersects the inclined line, and solve by similar triangles. 12. To establish a second point that shall make with a given point a line parallel to a given line. Diagonals of a parallelogram bisect each other. 13. To determine the horizontal distance from a given point to a visible but inaccessible object. Use two similar right-angled triangles. 14. To prolong a line beyond an obstacle? in azimuth* and distance. First Solution : By an equilateral triangle. Second Solution : By two rectangular offsets on each side of the obstacle. Third Solution : By similar triangles, as in Fig. 3. From any point as A run the line AB, fixing the half and three quarter points at X and j. From any other point as C, run CxD, making xD = Cx. From D *The azimuth of a line is the angle it forms with the meridian, and is meas- ured from the south point in the direction S.W. N.E. to 360 degrees. It thus becomes a definite direction when the angle alone is given. Thus the azimuth of 220'’ corresponds to the compass-bearing of N. 40° E. 12 SUR VE YING. run DyE making DE = AB= 4Z?/, fixing the middle point z. From B run BzII, making zH = Bz. Then is HE parallel and equal to DB, A C, and CH. D B Fig. 3. Stakes should be set at all the points lettered in the figure. Check: Measure HE and AC. If they are equal the work is correct. 15. To measure a given angle. Lay off equal distances, b, from the vertex on the two lines, and measure the a third side a of the triangle. Then tan ^ A= -- ■ -r— V4b'^ — a* 16. To lay out a given angle on the ground. Reverse the above operation. .<4 is known; assume ^ and compute a. Then from A measure oft AB = b. From B and A strike arcs with radii equal to a and b respectively, giving an intersection at C. Then CAB is the required angle. If b is assumed not greater than 0.6 the length of the chain, angles may be laid out up to go®. 17. Other Instruments for measuring distances with great accuracy will be discussed under the head of Base-Line Measurements, Chapter XIV. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 3 CHAPTER II. INSTRUMENTS FOR DETERMINING DIRECTIONS. THE COMPASS. l8. The Surveyor’s Compass consists essentially of a line of sight attached to a horizontal graduated circle, at the centre of which is suspended a magnetic needle free to move, the whole conveniently supported with devices for levelling. Fig. 4 shows a very good form of such an instrument. In ad- dition to the above essential features, the instrument here shown has a tangent-screw and vernier-scale at e for setting off the declination of the needle; a tangent-scale on the edge of the vertical sight for reading vertical angles, the eye being placed at the sight-disk shown on the opposite standard ; and an H SUR VE YTNG. auxiliary graduated circle, with vernier, shown on the front part of the plate, for reading angles closer than could be done with the needle. The compass is mounted either on a tripod or on a single support called a Jacob’s-staff. It is connected to its support by a ball-and-socket joint, which furnishes a con- venient means of levelling. Although the needle-compass does not give very accurate results, it is one of the most useful of surveying -instruments. Its great utility lies in the fact that the needle always points in a known direction, and therefore the direction of any line of sight may be determined by referring it to the needle-bear- ing. The needle points north in only a few localities; but its declination from the north point is readily determined for any region, and then the true azimuth, or bearing of a line, may be found. It has grown to be the universal custom, in finding the direction of a line by the compass, to refer it to cither the north or the south point, according to which one gives an acute angle. Thus, if the bearing is ioo° from the south point it is but 8o° from the north point, and the direction would be defined as north, 8o° east or west, as the case might be: thus no line can have a numerical bearing of more than 90°. In accordance with this custom, all needle- compasses are graduated from both north and south points each way to the east and west points, the north and south points being marked zero, and the east and west points 90°. When the direction of a line is given by this system it is called the bearing of the line. When it is simply referred to the position of the needle it is called the magnetic bearing. When it is corrected for the declination of the needle, either by setting off the declination on the declination-arc or by correcting the observed reading, it is called the true bear- ing, being then referred to the true meridian. Becau.se the graduated circle is attached to the line of sight and moves with it, while the needle remains stationary, E and ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 5 W are placed on the compass-circle in reversed position. Thus when the line of sight is north-east, the north end of the needle points to the left of the north point on the circle, and hence E must be put on this side of the meridian line. In reading the compass, always keep the north end of the circle pointing forward alo7tg the line, and read the north end of the needle. The north end of the needle is usually shaped to a special design, or, if not, it may be distinguished by knowing that the south end is weighted by having a small adjustable brass wire slipped upon it to overcome the tendency the north end has to dip. ADJUSTMENTS OF THE COMPASS. 19. The General Principle of almost all instrumental ad- justments is the Principle of Reversion, whereby the error is doubled and at the same time made apparent. A thorough mas- tery of this principle will nearly always enable one to deter- mine the proper method of adjusting all parts of any survey- ing instrument. It should be a recognized principle in sur- veying, that no one is competent to handle any instrument who is not able to determine when it is in exact adjustment, to locate the source of the error if not in adjustment, to dis- cuss the effect of any error of adjustment on the work in hand, and to properly adjust all the movable parts. The methods of adjustment should not be committed to memory — any more than should the demonstration of a proposition in geometry. The student in reading the methods of adjust- ment should see that they are correct, just as he sees the cor- rectness of a geometrical demonstration. Having thus had the method and the reason therefor clearly in the mind, he should trust his ability to evolve it again whenever called upon. He thus relies upon the accuracy of his reasoning, rather than on the distinctness of his recollection. i6 S UR VE YING. 20. To make the Plate perpendicular to the Axis of the Socket. — This must be done by the maker. It is here men- tioned because the axis is so likely to get accidentally bent. Instruments made of soft brass must be handled very care- fully to prevent such an accident. If this adjustment is found to be very much out, it should be sent to the makers. If much out, it will be shown by the needle, and also by the plate-bubbles. 21 . To make the Plane of the Bubbles perpendicular to the Axis of the Socket. — Level it in one position, turn i8o°, and correct one half the movement of each bubble by the adjusting-screw at the end of the bubble-case. Now level up again, and revolve i8o°, and the bubbles should remain at the centre. If not, adjust for one half the movement again, and so continue until the bubbles remain in the centre for all positions of the plate. The student should construct a figure to illustrate this and almost all other adjustments. Thus, in this case, let the figure consist of two lines, one repre- senting the axis of the socket, and the other the axis of the bubble, crossing it. Now if these two lines are not at right angles to each other, when the one is horizontal (as the bubble-axis is when the bubble rests at the centre of its tube) the other is inclined from the vertical. Now with this latter fixed, let the figure be revolved i8o° about it (or construct another figure representing such a movement), and it will be seen that the bubble-axis now deviates from the horizontal by twice the difference between the angle of the lines and 90°, By now correcting o}ie half of this change of direction on the part of the bubble- axis, it will be made perpendicular to the socket-axis. Then by relevelling the instrument, which consists of moving the socket-axis until the bubbles return to the middle of the tubes, the instrument should now revolve in a horizontal plane. 22. To adjust the Pivot to the Centre of the Graduated Circle. — When the two ends of the needle do not read exactly alike it may be due to one or more of three causes: The circle may not be uniformly graduated ; the pivot may be bent out of its central position ; or the needle may be bent. All ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 1 7 our modern instruments are graduated by machinery, so that they have no errors of graduation that could be detected by eye. One or both of the other two causes must therefore ex- ist. If the difference between the two end-readings is con- stant for all positions of the needle, then the pivot is in the centre of the circle, but the needle is bent. If the difference between the two end-readings is variable for different parts of the circle, then the pivot is bent, and the needle may or may not be straight. To adjust the pivot, therefore, find the posi- tion of the needle which gives the maximum difference of end- readings, remove the needle, and bend the pivot at right angles to this position by one half the difference in the extreme variation of end-readings. Repeat the test, etc. Since the glass cover is removed from the compass-box in making this adjustment, it should be made indoors, to prevent any disturbance from wind. 23. To straighten the Needle, set the north end exactly at some graduation-mark, and read the south end. If not 180° apart, bend the needle until they are. This implies that the preceding adjustment has been made, or examined and found correct. 24. To make the Plane of the Sights normal to the Plane of the Bubbles. — Carefully lewel the instrument and bring the plane of the sights upon a suspended plumb-line. If this seems to traverse the farther slit, then that sight is in adjustment. Reverse the compass, and test the other sight in like manner. If either be in error, its base must be re- shaped to make it vertical. 25. To make the Diameter through the Zero-gradua- tions lie in the Plane of the Sights. — This should be done by the maker, but it can be tested by stretching two fine hairs vertically in the centres of the slits. The two hairs and the two zero-graduations should then be seen to lie in the same plane. The declination-arc must be set to read zero. i8 SURVEYING. 26. To remagnetize the Needle. — Needles sometimes lose their magnetic properties. They must then be remagnetized. To do this take a simple bar-magnet and rub each end of the needle, from centre towards the ends, with the end of the magnet which attracts in each case. In returning the magnet for the next stroke lift it up a foot or so to remove it from the immediate magnetic field, otherwise it would tend to mil- lify its own action. The needle should be removed from the pivot in this operation, and the work continued until it shows due activity when suspended. An apparently sluggish needle may be due to a blunt pivot. If so, this should be removed, and ground down on an oil-stone. THE VERNIER. 27. The Vernier is an auxiliary scale used for reading frac- tional parts of the divisions on the main graduated scale or limb. If we wish to read to tenths of one division on the limb, we make 10 divisions on the vernier correspond to either 9 or 1 1 divisions on the limb. Then each division on the vernier is one tenth less or greater than a division on the limb. If we wish to read to twentieths or thirtieths of one division on the limb, there must be twenty or thirty divisions on the vernier corresponding to one n^ore or less on the limb. The zero of the vernier-scale marks the point on the limh whose reading is desired. Suppose this zero-point corresponds exactly with a division on the limb. The reading is then made wholly on the limb. If a division on the vernier is less than a division on the limb, then, by moving the forward a trifle, the next fo 7 'ward division on the vernier corresponds with a division on the limb. (The particular division on the limb that may be in coincidence is of no consequence.) On the other hand, if a division on the vernier greater than a division on the limb, then by moving the vernier forward a trifle, the next backward division on the ADJUSTMENT, USE, AND CAEE OF INSTRUMENTS. 1 9 vernier comes into coincidence. Thus we have two kinds of verniers, direct and retrograde according as they are read forward or backward from the zero-point. Most verniers in use are of the direct kind, but those commonly found on sur- veyors’ compasses for setting off the declination are generally of the retrograde order. In Fig. 5 are shown two direct verniers, such as are used on transits with double graduations. Thus in reading to the right the reading is 138° 45', but in reading to the left it is 221° 15'. In each case we look along the vernier in the direction of the graduation for the coincident lines. In Fig. 6 is shown a special form of retrograde vernier in which the same set of graduation-lines on the vernier serve for 20 SURVEYING. either right- or left-hand angles. Here a division of the vernier is larger than a division on the limb, and it must therefore be read backwards. Thus, we see that the zero of the vernier is to the left of the zero of the limb, the angle being 30' and something more. Starting now toward the right (backwards) on the vernier scale, we reach the end or 15-minute mark, without finding coincident lines ; we then skip to the left-hand side of the vernier scale and proceed iozvards the right again until we find coincident lines at the twenty-sixth mark. The reading is therefore ^o-\-26=^6 minutes. This is the form of vernier usually found on surveyors’ compasses for setting off the declination. We have therefore the following Rules. First. To find the ‘‘ smallest reading" of the vernier., divide the value of a division on the limb by the number of divisions in the vcriiier. Second. Read forward along the limb to the last graduation precedmg the zero of the vernier ; then read forward along the vernier if direct^ or backward if retrograde, until coincident lines are found. The number of this line on the vernier from the zero- graduation is the number of smallest-reading" units to be added to the reading made on the hmb. These rules apply to all verniers, whether linear or circular. THE DECLINATION OF THE NEEDLE. 28. The Declination^ of the Needle is the horizontal angle it makes with the true meridian. At no place on the earth is this angle a constant. The change in this angle is called the variation of the declination. 29. The D aily Variation in the Declination consists in a Formerly called variation of the needle, and still so called by navigators and by many surveyors. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2 1 swinging of the needle through an arc of about eight, minutes daily, the north end having its extreme easterly variation about 8 A.M. and its extreme westerly position about p.30 P.M. It has its mean or triLe declination about 10.30 A.M. and 8 P.M. It varies with the latitude and with the season, but the foITow- ing table gives a fair average for the United States. A more extended table may be found in the Report of the U. S. Coast and Geodetic Survey for 1881, Appendix 8. TABLE OF CORRECTIONS TO REDUCE OBSERVED BEARINGS TO THE DAILY MEAN. Month. Add to N.E. and S.W. bearings. Subtract from N.W. and S.E. bearings. Add to N.W. and S.E. bearings. Subtract from N.E. and S.W. bearings. 6 7 8 9 10 II 12 I 2 3 4 5 6 A.M. A.M. A.M. A.M. A.M. A.M. M. P.M. P.M. P.M. P.M, P.M, P.M. January l' 2' 2' l' 0' 2' 3' 3' 2' l' l' o' April 3 4 4 3 I I 4 5 5 4 3 2 I July 4 5 5 4 I I 4 5 5 4 3 2 I October 2 2 2 I I 3 3 3 2 I 0 0 This table is correct to the nearest minute for Philadelphia, where the observations were made. 30. The Secular Variation of the magnetic declination is probably of a periodic character, requiring two or three cen- turies to complete a cycle. The most extensive set of obser- vations bearing on this subject have been made at Paris, where records of the magnetic declination have been kept for about three and a half centuries. The secular variation for Paris is shown in Fig. 7, and that for Baltimore, Md., in Fig. 8.* Whether or not either of these curves will return in time to the same extreme limits here given is unknown, as is also the cause of these remarkable changes. The extraordinary varia- tion in the declination at Paris of some 32°, and that at * These taken from the Coast Survey Report of 1882. 22 SURVEYING. Baltimore of some 5 °, show the necessity of paying careful attention to this matter. No reliance should be placed on 1540 CO 80 ICOO 20 40 GO 80 1700 20 40 CO 80 1800 20 40 60 '80 1900. MM _ i n~n M 1 1 1 1 1 1 Secular Vdriationl 1 df,ilie Magnetic Declination at Pay is Fra Tice.. \ Observed declinations arc shou'n by dots. V Computed] {declinations by first t jyeriodic , 7 icr rn'iula,by cu rve. r 1 1 1 M J ■ 4 r Fig, 7. old determinations of the declination unless the rate of change be known, and even then this rate is not likely to be constant ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 23 a great many years. They also show the necessity of record- ing the date and the declination of the needle on all plats and records of surveys, with a note stating whether the bearings given were the true or magnetic bearings at the time they were taken. 31. Isogonic Lines are imaginary lines on the earth’s sur- face joining points whose declinations are equal at any given time. The isogonic line joining points having no declination is called the agonic line. There is such a line crossing the United States passing just east of Charleston, S. C., and just west of Detroit, Mich. All points east of this line have a western declination, and all points west of it have an eastern declination. The isogonic lines for 1885 for the whole of the United States are shown on Plate It will be noted that where the observations are most thickly distributed, as in Missouri for instance, there the isogonic lines are most crooked ; showing that if the declinations were accurately known for all points of this map the isogonic lines would be much more irregular, and would be changed very much in position in many places. The isogonic lines given on this chart are all moving west- ward, so that all western declinations are increasing and all eastern declinations are decreasing. They are not all moving at the same rate, however, those in New Brunswick and also those near the eastern boundaries of California and Oregon being about stationary. P'or many points in the United States and Canada the rate of change in the declination has been observed, and formulae determined for computing the declination for each point, which formulae will probably remain good for the next twenty years. The following tables t give this information. In these tables t is the time in calendar years. Thus for July i, 1885, ^==1885.5. In the first table all the formulae have been re- * Reduced from charts in the U. S. Coast and Geodetic Survey Report for 1882. f Taken from the above report. 24 SURVEYING. ferred to one date — Jan. i, 1850. Here m is used to represent the time in years after 1850, or ;;/=/— 1850. Thus, for July i, 1885, m = 35.5. The annual value of this secular change in the declination is marked at various points on the isogonic chart given in Plate I., but from the small number of the observa- tions, both in time and space, it is evident that no great reli- ance can be placed on any such chart for exact information. It will be seen that the change in the declination over the Northern States will average about one minute to the mile in an east and west direction. A value of the declination found in one end of a county may be some forty minutes in error in the other end of the same county. This shows that the declina- tion must be known for the exact locality of the survey. In fact, the surveyor can never be sure of his declination until he has observed it for himself for the given time and place. This is best done by means of a transit instrument, and such a method is given in the chapter on Geodetic Surveying. If, however, no transit is at hand, a result sufficiently accurate for compass surveying may be obtained by the compass itself. FORMULA EXPRESSING THE MAGNETIC DECLINATION AT VARIOUS PLACES. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 25 FORMULA EXPRESSING THE MAGNETIC DECLINATION AT VARIOUS PLACES- Continued. 26 SURVEYING. Approximate expression. FORMULAE EXPRESSING THE MAGNETIC DECLINATION AT VARIOUS PLACES— Continued. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2/ * Approximate expression. 28 SURVEYING. * There are pfiven in the U. S. Coast and Geodetic Survey Report for 1882, and issued as a separate pamphlet, the declinations of the magnetic needle at some 2500 points, mostly in the United States, all reduced to the epoch Jan. i, 1885. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2g 32. Other Variations of the Declination. — In addition to the daily and secular changes in the declination, there are others worthy of mention. T/ie annual variation is very small, being only about a half- minute of arc from the mean position for the year. It may therefore be neglected. The lunar ineqiialities are still smaller, being only about fif- teen seconds of arc from the mean position. Magnetic disturbances are due to what are called magnetic storms. They may occur at any time, and cannot be predicted. They may last a few hours, or even several days. “ The fol- lowing table of the observed disturbances, in a bi-hourly series, at Philadelphia, in the years 1840 to 1845, will give an idea of their relative frequency and magnitude : Deviations from nor- mal direction. Number of disturbances. 3'. 6 to 10'. 8 2189 10'. 8 to 18'. I 147 18'. I to 25'. 3 18 25'. 3 to 32'. 6 3 Beyond 0 At Madison, Wis., where the horizontal magnetic intensity is considerably less, very much larger deflections have been noticed. Thus, on October 12, 1877, 4 ^^ ^^id on Ma^y 28, 1877, one of 1° 24', were observed.” * The geometric axis of a needle may not coincide with its magnetic axis, and hence the readings of two instruments at the same station may differ slightly when both are in adjust- ment. In this case the declination should be found for each instrument independently. 33. To Find the Declination of the Needle. — The - From Report of the U. S. Coast and Geodetic Survey-for 1882. 30 SURVEYING, method here given is by means of the compass and a plumb- line, and is sufficiently accurate for compass-work. The com- pass-sights are brought into line with the plumb-line and the pole-star (Polaris), when this is at either eastern or western elongation. This star appears to revolve in an orbit of i° i8' radius. Its upper and lower positions are called its upper and lower culminations, and its extreme east and west positions are called its eastern and western elongations, respectively. When it is at elongation it ceases to have a lateral component of motion, and moves vertically upward at eastern and downward at western elongation. If the star be observed at elongation, therefore, the observer’s watch may be as much as ten or fifteen minutes in error, without its making any appreciable error in the result. The method of making the observation is as follows : Suspend a fine plumb-line, such as an ordinary fishing-line, by a heavy weight swinging freely in a vessel of water. The line should be suspended from a rigid point some fifteen or twenty feet from the ground. Care must be taken to see that the line does not stretch so as to allow the weight to touch the bottom of the vessel. Just south of this line set two stakes in the ground in an east and west direction, leaving their tops at an elevation of four or five feet. Nail to these stakes a board on which the compass is to rest. The top of this board should be smooth and level. This compass-support should be as far south of the plumb-line as possible, to enable the pole-star to be seen below the line-support. A sort of wooden box may be provided, in which the compass is rigidly fitted and levelled. Several hundred feet of nearly level ground should be open to the northward, for setting the azimuth-stake. Prepare two stakes, tacks, and lanterns. Find from the table given on page 32 the time of elongation of the star. About twenty minutes before this time, set the compass upon the board, bringing both sights in the plane defined by the plumb-line and star. The ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 3 1 line must be illuminated. The star will be found to move slowly east or west, according as it is approaching its eastern or western elongation. When it ceases to move laterally, the compass is carefully levelled, the rear compass-sight brought into the plane of the line and star, and then the forward com- pass-sight made to coincide with the rear sight and plumb-line. (If the forward sight were tall enough, we could at once bring both slits into coincidence with line and star.) Continue to ex- amine rear sight, line, and star, and rear sight, forward sight, and line alternately, until all are found to be in perfect coinci- dence, the instrument still being level. If this is completed within fifteen minutes of the true local time of elongation, the observation may be considered good ; and if it is completed within thirty minutes of the time of elongation, the resulting error in azimuth will be less than one minute of arc. Having completed these observations, remove the plumb-line and set a stake in the line of sight as given by the compass, several hun- dred feet away. In the top of this stake a tack is to be set exactly on line. For setting this tack, a board may be used, having a vertical slit about J inch wide, covered with white cloth or paper, behind which a lamp is held. This slit can then be accurately aligned and the tack set. A small stake with tack is now set just under the compass (or plumb-line), and the work is complete for the night. Great care must be taken not to disturb the compass after its final setting on the line and star. At about ten o’clock on the following day, mount the com- pass over the south stake. From the north stake lay off a line at right angles to the line joining the two stakes (by compass, optical square, or otherwise) towards the west if eastern elongation, or towards the east if western elongation had been observed. Carefully measure the distance between the two stakes by some standardized unit. From the table of azimuths on page 33 find the azimuth of the star at elongation for the 32 SURVEYING. given time and latitude. Multiply the tangent of this angle by the measured distances between the stakes, and care- fully lay it off from the north tack, setting a stake and tack. This is now in the meridian with the south point. With the compass in good adjustment, especially as to the bubbles and the verticality of the sights, the observation for declination may now be made. If this be done at about 10.30 A.M., it will give the mean daily declination. Many readings should be made, allowing the needle to settle independently each time. The fractional part of a division on the graduated limb should be read by the declination-vernier, thus enabling the needle to be set exactly at a graduation-mark. If all parts of this work be well done, it will give the declination as accurately as the flag can be set by means of the open sights. MEAN LOCAL TIME (ASTRONOMICAL, COUNTING FROM NOON) OF THE ELONGATIONS OF POLARIS. [The table answers directly for the year 1885, and for latitude -|- 40°.] Date. Eastern Elongation. Western Elongation. Date. Eastern Elongation. Western Elongation. Jan. I 0'' 35“-3 12^ 24” .6 July I 12^ 39 ° ^6 32™. 8 “ 15 23 36 .1 II 29 .3 i ( 15 II 44 •7 23 34 -o Feb. I 22 29 .0 10 22 .2 Aug. I 10 38 .2 22 27 .5 “ 15 21 33 .7 9 27 .0 “ 15 9 43 •3 21 32 .6 Mar. I 20 38 .5 8 31 .8 Sept. I 8 36 •7 20 26 .0 i i 15 19 43 -4 7 36 .6 i “ 15 7 41 .7 19 31 -I Apr. I 18 36 .4 6 29 •7 Oct. I 6 38 •9 18 28 .2 i ( 15 17 41 .4 5 34 •7 1 “ 15 5 43 •9 17 33 .2 May I 16 38 .6 4 31 .8 Nov. I 4 37 .0 16 26 .4 ( < 15 15 43 .7 3 36 •9 “ 15 3 41 •9 15 31 -3 June I 14 37 -I 2 30 •3 Dec. I 2 38 •9 14 28 .2 * * 15 13 42 .2 I 35 •4 15 I 43 .6 13 33 -o ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 33 AZIMUTH (FROM THE NORTH) OF POLARIS, WHEN AT ELONGA- TION, BETWEEN THE YEARS 1886 AND 1895, FOR DIFFERENT LATITUDES BETWEEN + 25° AND + 50". Lat. 1886.0 00 00 b i 838 .o 1889.0 1890.0 1891.0 1892.0 1893.0 1894.0 1895.0 0 0 / c / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / + 25 I 26.0 I 25.7 I 25.3 I 25.0 I 24.6 I 24.3 I 23.9 I 23.6 I 23.2 I 22.9 26 26.7 26.4 26.0 25.7 25-3 25.0 24.6 24.3 23.9 23.6 27 27-5 27.1 26.8 26.4 26.0 25-7 25.4 25.1 24.7 24-3 28 28.3 27.9 27.6 27.2 26.8 26.5 26.2 25.8 25.4 25 -ij 29 29.1 28.8 28.4 28.0 27.6 27-3 27.0 26.6 26.3 25.9 30 30.0 29.6 29-3 28.9 28.5 28.2 27.8 27-5 27.1 26.8 31 30.9 30.5 30.2 29.8 29.4 29. 1 28.8 28.4 28.0 27.6 32 31-9 31-5 31.2 30.8 30.4 30.1 29.7 29-3 29.0 28.6 33 33-0 32.6 32.2 31.8 31.4 31*1 30.7 30.3 30.0 29.6 34 34.0 33-6 33*3 32.9 32.5 32.1 31.8 31-4 31.0 30.6 35 35.2 34 8 34-4 34.0 33.6 33-2 32.9 32.5 32.1 31-7 36 36.4 36.0 35-6 35-2 34-8 34-4 34.0 33.6 33.2 32.9 37 37-6 37.2 36.8 36.4 36.0 35-6 35-2 34-8 34.5 34.1 38 38.9 38.5 00 37.7 37.3 36.9 36.5 36.1 35.7 35.3 39 40.3 39-9 39-5 39-1 38.7 38.3 37-9 37.5 37.1 36.7 40 41.8 41.4 41.0 40.5 40. 1 39*7 39-3 38.9 38.5 38.1 41 43-3 42.9 42.5 42.0 41.6 41.2 40.8 40.4 40.0 39-6 42 44.9 44-5 44.1 43-6 43-2 42.8 42.4 42.0 41.5 41. 1 43 46.6 46.1 45*7 45.3 44.9 44.4 44.0 43-6 43.2 42.7 44 48.4 47-9 47-5 47.1 46.6 46.2 45-8 45-3 44.9 44.4 45 50.3 49.8 49.4 48.9 48.5 48.1 47.6 47.1 46.7 46.2 46 52.2 51.8 51.3 50.9 50.4 50.0 49-5 49.0 48.6 48.2 47 54.3 53.8 53-4 52.9 52.5 52.0 51.5 51.0 50.6 50.2 48 56.5 56.0 55-6 55.1 54-6 54.2 53-7 53-2 52.8 52.3 49 I 58.8 I 58.3 I 57.9 57-4 56.9 56.5 56.0 55-5 55-0 54-5 + 50 2 01.3 2 00.8|2 00.3 I 59.8 I 59-3 I 58.8 I 58.4 I 57.9 I 57-4 I 56.9 3 34 SUR VE YING. If the elongation of Polaris does not come at a suitable time for observing for declination, the upper culmination, which occurs 5*' 54™*^ after the eastern, or the lower culmination, & 03"\4 after the western elongation, may be chosen. The objection to this is that the star is then moving at its most rapid rate in azimuth. It is so near the pole, however, that if the observation can be obtained within two minutes of the time of its culmination the resulting error will be less than T of arc. This will then give the true meridian without having to make offsets. It must be remembered that the time of elongation given in the table is the local time at the place of observation. In- asmuch as hourly meridian time is now carried at most points in this country to the complete exclusion of local time, it will be necessary to find the local time from the known meridian or watch time. Thus, all points in the United States east of Pitt.s- burgh use the fifth-hour meridian time (75° w. of Greenwich); from Pittsburgh to Denver, the sixth-hour meridian time (90° w. of Greenwich), etc. To find local time, therefore, the longi- tude east or west of the given meridian must be found. This can be determined with sufficient accuracy from a map. Thus, if the longitude of the place is 80° w. from Greenwich, it is 5° w. of the fifth-hour meridian, or local time is twenty min- utes slower than meridian time at that place If meridian time is used at such a place, the elongation will occur twenty min- utes later than given by the table. If the longitude from Washington is given on the map consulted, add it to 77° if west of Washington, and subtract it from 77° if east of Wash- ington, to get longitude from Greenwich. USE OF THE NEEDLE-COMPASS. 34. The Use of the Needle-compass is confined almost exclusively to land-surveying, where an error of one in three hundred could be allowed. As the land enhances in value. T ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 35 however, there is an increasing demand for more accurate means of determining areas than the compass and chain af- ford. The original U. S. land-surveys were all made with the needle, or with the solar, compass and Gunter’s chain. Hence all land boundaries in this country have their directions given in compass-bearings, and their lengths in chains of sixty-six feet each. The compass is used, therefore, — 1. To establish a line of a given bearing. 2. To determine the bearing of an established line. 3. To retrace old lines. If the true bearing is to be used, the declination of the needle from the meridian must be determined and set off by the vernier. If the magnetic bearing is used, the declination of the needle at the time the survey was made should be recorded on the plat. If old lines are to be retraced, the declinations at the times of both surveys must be known. The needle should be read to the nearest five minutes. This requires reading to sixths of the half-degree spaces, but this can be done with a little practice. Always lift the needle from the pivot before moving the in- strument. If the needle is sluggish in its movements and settles quickly it has either lost its magnetic force or it has a blunt pivot. In either case it is likely to settle considerably out of its true posi- tion. The longer a needle is in settling the more accurate will be its final position. It can be quickly brought very near its true position by checking its motion by means of the lifting screw. In its final settlement, however, it must be left free. Careful attention to the instrumental adjustments, to local disturbances, and close reading of the needle are all essential to good results with the compass. 36 SURVEYING. < 35* "I o set off the Declination, we liave only to remem- ber that the declination arc is attached to the line of sight and that the vernier is attached to the graduated circle. If the declination is west, then when the line of sight is north the north end of the needle points to the left of the zero of the graduated circle. In order that it may read zero, or north, the circle must be moved towards the left, or opposite to the hands of a watch. On the other hand, if the declination is east, the circle to which the vernier is attached should be moved with the hands of a watch. This at once enables the observer to set the vernier so that the needle-readings will be the true bearings of the line of sight. 36. Local Attractions may disturb the needle by large or small amounts, and these often come from unknown causes. The observer should have them constantly in mind, and keep all iron bodies at a distance from the instrument when the needle is being read. The glass cover may become electrified from friction, and attract the needle. This can be discharged by touching it with a wet finger, or by breathing upon it. Read- ing-glasses should not have gutta-percha frames, as these be- come highly electrified by wiping the lens, and will attract the needle. Such glasses should have brass or German-silver frames. No nickel coverings or ornaments should be near, as this metal has magnetic properties. A steel band in a hat- brim, or buttons containing iron, have been known to cause great disturbance. In cities and towns it is practically impos- sible to get away from the influence of some local attraction, such as iron or gas pipes in the ground, iron lamp-posts, fences, building-fronts, etc. For this reason the needle should never be used in such places. In many regions, also, there are large magnetic iron-ore de- posits in the ground, which give special values for the declina- tion at each consecutive station occupied. It is practically impossible to use magnetic bearings in such localities. 77/^’ test for local attraction in the field-work is to read the ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 3/ bearing of every line from both ends of it. If these are not the same, and no error has been made, there is some local dis- turbance at one station not found at the other. If there is known to be mineral deposits in the region it may perhaps be laid to that. If not, the preceding station should be occupied again, and the cause of the discrepancy inquired into. If the forward and reverse bearings of all lines agree except the bear- ings taken from a single station, then it may be assumed there is local attraction at that station. ELIMINATION OF LOCAL ATTRACTIONS. 37. To establish a Line of a Given Bearing, set the com- pass up at a point on tiie line, turn off the declination on the declination-arc, and bring the north end of the needle to the given bearing. The line of sight now coincides with the re- quired line, and other points can be set. 38. To find the True Bearing of a Line, set the compass up on the line, turn off the declination by the vernier, bring the line of sight to coincide with the line with the south part of the graduated circle towards the observer, and read the north end of the needle. This gives the forward bearing of the line. 39. To retrace an Old Line, set the compass over one well-determined point in the line and turn the line of sight upon another such point. Read the north end of the needle. If this reading is not the bearing as given for the line, move the vernier until the north end of the needle comes to the given bearing, when the sights are on line. The reading of the declination-arc will now give. the declination to be used in retracing all the other lines of the same survey. If a second well-determined point cannot be seen from the instrument-sta- tion, a trial-line will have to be run on an assumed value for the declination, and then the value of the declination used on the first survey computed. Thus, if the trial-line, of length /, comes out a distance x to the right of the known point on 38 SURVEYING. the line, the vernier is to be moved in the direction of the hands of a watch an angular amount whose tangent is j. If the trial-line comes out to the /eyt of the point, move the vernier in a direction opposite to the hands of a watch. PRISMATIC COMPASS. 40. The Prismatic Compass is a hand-instrument pro- vided with a glass prism so adjusted that the needle can be read while taking the sight. A convenient form is shown in Fig. 9 , which is carried in the pocket as a watch. The line of sight is established by means of the etched line on the glass cover 5. It is used in preliminary and reconnoissance work, in clearing out lines, etc. EXERCISES FOR COMPASS ALONE OR FOR COMPASS AND CHAIN. 41. Run out a line of about a mile in length, on somewhat uneven ground, establishing several stations upon it, using a constant compass-bearing. Then ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 39 run back by the reve7'se bearings and note how nearly the points coincide with the former ones. The chain need not be used. 42 . Select some half dozen points that enclose an area of about forty acres (one quarter mile square) on uneven ground. Let one party make a compass- and-chain survey of it, obtaining bearing and length of each side. Then let other parties take these field-notes and, all starting from a common point, run out the lines as given by the Jield-note'i, setting other stakes at all the remaining corners, each party leaving special marks on their own stakes. Let each party plot their own survey and compare errors of closure. 43 . Select five points, three of which are free from local attraction, while two consecutive ones are known to be subject to such disturbance. Make the sur- vey, finding length and forward and reverse bearings of every side. Determine what the true bearing of each course is, and plot to obtain the error of closure. 44 . Let a number of parties observe for the declination of the needle, using a common point of support for the plumb-line. Let each party set an inde- pendent meridian stake in line with the common point. Note the distance of each stake fro7n the 77tea7t positio7i, and compute the corresponding angular dis- crepancies. (March and September are favorable months for making these observations, for then Polaris comes to elongation in the early evening.) The above problems are intended to impress upon the student the relative errors to which his work is subject. THE SOLAR COMPASS. 45. The Burt Solar Compass essentially consists first, of a polar axis rigidly attached in the same vertical plane with a terrestrial line of sight, the whole turning about a vertical axis. When this plane coincides with the meridian plane, the polar axis is parallel with the axis of the earth. Second, attached to the polar axis, and revolving about it, is a line of collimation making an angle with the polar axis equal to the angular dis- tance of the sun for the given day and hour from the pole. This latter angle is 90° plus or minus the sun’s declination, according as the sun is south or north of the equator. The polar axis must therefore make an angle with the horizon equal to the latitude of the place, and the line of collimation must deviate from a perpendicular to this axis by an angular amount equal to, and in the direction of, the sun’s declination. With these angles properly set, and the line of collimation 40 SUR VE YING. turned upon the sun, the vertical plane through the terrestrial line of sight, and the polar axis must lie in the meridian, for otherwise any motion of the line of collimation about its axis would not bring it upon the sun. In Fig. lo is shown a cut of this instrument as manufac- ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 4 1 tured by Young & Sons, Philadelphia. The polar axis is shown at /, and the terrestrial line of sight is defined by the slits in the vertical sights, the same as in the needle-compass. The line of collimation is defined by a lens at the upper end of the arm and a silver plate at the lower end, containing gradua- tions with which the image of the sun, as formed by the lens, is made to coincide. The polar axis is given the proper incli- nation by means of the latitude-arc /, and the line of collima- tion is inclined from a perpendicular to this axis by an amount equal to the sun’s declination by means of the declination-arc d. When these arcs are properly set, the arm a is revolved about the polar axis, and the whole instrument about its verti- cal axis, until the image of the sun is properly fixed on the lines of the silver plate, when the terrestrial line of sight, as defined by the vertical slits, lies in the true meridian. Any desired bearing may now be turned off by means of the hori- zontal circle and vernier, shown at v. The accuracy with which the meridian is obtained with this instrument depends on the time of day, and on the accuracy with which the lati- tude- and declination-angles are set off. It is necessary to at- tend carefully, therefore, to the ADJUSTMENTS OF THE SOLAR COMPASS. 46. To make the Plane of the Bubbles perpendicular to the Vertical Axis. — This is done by reversals about the verti- cal axis, the same as with the needle-compass. 47. To adjust the Lines of Collimation. — The declination- arm a has two lines of collimation that should be made paral- lel. As it is shown in the figure, it is set for a south declina- tion. This is the position it will occupy from Sept. 20 to March 20. When the sun has a north declination, as from March 20 to Sept. 20, the declination-arm is revolved 180° about the polar axis, and a line of collimation established by 42 SUR VE Y/NG. a lens and a graduated disk on opposite ends from those pre- viously used. Each end of this arm, therefore, has both a lens and a disk, each set of which establishes a line of collimation. The second adjustment consists in making these tivo lines of col- limation parallel to each other. They are made parallel to each other by making both parallel to the faces of the blocks con- taining the lenses and disks. To effect this, the arm must be detached and laid upon an auxiliary frame which is attached in the place of the arm, and which is called an adjuster. With the latitude- and declination-arc set approximately for the given time and place, lay the declination-arm upon the adjuster, and bring the sun’s image upon the disk. Now turn the arm care- fully bottom side up (not end for end) and see if the sun’s image comes between the equatorial lines on the disk.* If not, adjust the disk for one half the displacement, and reverse again for a check. When this disk is adjusted, turn the arm end for end, and adjust the other disk in a similar manner. Having now made both lines of collimation parallel to the edges of the blocks, they are parallel to each other. 48. To make the Declination-arc read Zero when the Line of Collimation is at Right Angles to the Polar Axis. — Set the vernier on the declination-arc to read zero. By any means bring the line of collimation upon the sun. When carefully centred on the disk, revolve the arm 180° quickly about the polar axis, and see if the image now falls exactly on the other disk. If not, move the declination-arm by means of the tangent-screw until the image falls exactly on the disk. Read the declination-arc, loosen the screws in the vernier-plate, and move it back over one half its distance from the zero-reading. Centre the image again, reverse 180°, and test. This adjustment depends on the parallelism of the two lines of collimation. If the vernier-scale is not adjustable, * It would not be expected to fall between the hour-lines on the disk, since some time has elapsed. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 43 one half the total movement is the index error of the declina- tion-arc, and must be taken into account in all settings on this arc. The two preceding adjustments should be made near the middle of the day. 49. To adjust the Vernier of the Latitude-arc. — Find the latitude of the place, either from a good map or by a transit- observation. Set up the compass a few minutes before noon, with the true declination set off for the given day and hour. Bring the line of collimation upon the sun, having it clamped in the plane of the sights, or at the twelve-hour angle, and follow it by moving the latitude-arc by means of the tangent- screw, and by turning the instrument on its vertical axis. When the sun has attained its highest altitude, read the lati- tude-arc? Compare this with the known latitude. Move the vernier on this arc until it reads the true latitude ; or, if this cannot be done, the difference is the index error of the latitude- arc. If, however, the latitude used with the instrument be that obtained by it, as above described, then no attention need be paid to this error. This error is only important when the true latitude is used with the instrument in finding the meridian, or where the true latitude of the place is to be found by the in- strument. In using the solar compass, therefore, ahvays use the latitude as given by that instrument by a meridian observa- tion on the sun."^ 50. To make the Terrestrial Line of Sight and the Polar Axis lie in the same Vertical Plane. — This should be done by the maker. The vertical plane that is really brought into the meridian by a solar observation is that containing the polar axis, and by as much as tJie plane of the sights deviates from * Since the sun may cross the meridian as much as 15 minutes or more before or after mean noon, this observation may have to be taken that much before or after 12 o’clock mean time. It is, however, in all cases, an observation on the sun ai culmination. 44 SUR VE YING. this plane, by so much will all bearings be in error. TIic best test of this adjustment is to establish a true meridian by the transit by observations on a circumpolar star ; and then by making many observations on this line, in both forenoon and afternoon, one may determine whether or not the horizontal bearings should have an index-correction applied. USE OF THE SOLAR COMPASS. 51. The Solar Compass is used on land and other surveys where the needle-compass is either too inaccurate, or where, from local attraction, the declination of the needle is too vari- able to be accurately determined for all points in the survey. Where there is no local attraction, however, and the declination of the needle is well known, the advantages of the solar com- pass in accuracy are fairly offset by several disadvantages in its use which do not obtain with the needle-compass. Thus, the solar compass should never be used when the sun is less than one hour above the horizon, or less than one hour from noon. Of course it cannot be used in cloudy weather. For such times as these bearings may be obtained by a needle which is always attached, but then the instrument becomes a needle-com- pass simply. It is also much more trouble, and consumes more time in the field than the needle-compass. But more significant than any of these is the fact that if the adjustments are not carefully attended to, the error in the bearing of a line may be much greater by the solar compass than is likely to be made by the needle-compass, when there is no local attrac- tion. It is possible, however, to do much better work with the solar compass than can be done with the needle-com- pass. 52. To find the Declination of the Sun. — On account of the inclination of the earth’s axis to the plane of its orbit, the sun is seen north of the celestial equator in summer, and south ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 45 of it in winter. This deviation, north or south of the equator, is called north or south declination, and is measured from any point on the earth’s surface in degrees of arc. On about the 2ist of June the sun reaches its most northern declination, and begins slowly to return. Its most southern point is reached about December 2ist. In June and Decem- ber, therefore, the sun is changing its declination most slowly, while at the intervening quadrant-points of the earth’s orbit, March and September, it is changing its declination most rapidly, being as much as one minute in arc for one hour in time. It is evident, therefore, that we must regard the decli- nation of the sun as a constantly changing quantity, and, for any given day’s work, a table of declinations must be made out for each hour of the day. The American Ephemeris and Nautical Almanac gives the declination of the sun for noon of each day of the year for both Greenwich and Washington. Since the time universally used in this country is so many hours from Greenwich, it is best to use the Greenwich declina- tions. Since, also, we are five, six, seven, or eight hours west of Greenwich, the declination given in the almanac for Green- wich noon of any day will correspond to the declination here 7, 6, 5, or 4 o’clock A.M. of the same date, according as East- ern, Central, Mountain, or Western time is used. As this standard time is seldom more than 30 minutes different from local time, and as this could never affect the declination by more than 30 seconds of arc, it will here be considered sufficient to correct the Greenwich declination by the change, as found for the standard time used. Thus, if Central (90th meridian) time is used, the declination given in the almanac is the declination at 6 o’clock A.M. at the place of observation. To this must be added (algebraically) the hourly change in declination, which is also given in the almanac. A table may thus be prepared, giv- ing the declination for each hour of the day. 53. T o correct the Declination for Refraction. — All rays 46 SURVEYING. of light coming to the earth from exterior bodies are refracted downward, thus causing such bodies to appear higher than they really are. This refraction is zero for normal (vertical) lines, and increases towards the horizon. It varies largely, also, with the special temperature, pressure, and hydrometrical condition of the atmosphere. Tables of refraction give only the mean values, and these may differ largely from the values found for any given time, especially for lines near the horizon. It is for this reason that all astronomical observations made near the horizon are very uncertain. There is but one setting on the solar compass that has reference to the position of the sun in the heavens, and that is the declination. Now, the re- fraction changes the apparent altitude of the body ; and by so much as a change in the altitude changes the declination, by so much does the apparent declination differ from the true dec- lination. Evidently it is the apparent declination that should be set off. When the sun is on the meridian, the change in altitude has its full effect in changing the declination, but at other times the change in declination is less than the change in altitude. The correction to the declination due to refraction is found from the following final equations : * tan N — cot cp cos tan q = sin N cos (d N) tan z cot (d + JV) cos q ‘ dS = ■— dz cos * See Chauvenet’s “ Spherical Astronomy,” vol. i., p. 171, and Doolittle’s “ Practical Astronomy,” p. 159. ADJUSTMENT, USE, AND CATE OF INSTRUMENTS. 4/ where q) = latitude ; t = hour angle from the meridian ; S = declination of sun ; ^ — zenith distance of sun; TV and q being auxiliary angles to facilitate the computation. From these equations we may compute the auxiliary angle q, and the zenith distance z, for each hour from noon, for every day of the year. Then from a table of mean refractions, giving the refraction for given altitudes, or zenith distances, which is dz, we may find the corresponding d8, which is the correction to be applied to the declination for refraction. In this manner the following table has been computed for the latitude of 40°. For any other latitude the correction is found by multiplying the correction given in the table by the corresponding coefficient, as given in the table “ Latitude Co- efficients.” These coefficients were obtained by plotting the ratios of the actual refraction at the different latitudes to that at latitude 40°, for each hour from 7 A.M. to 5 P.M. and for the various declinations. It was found that this ratio was almost a constant, except for very low altitudes, where the inherent uncertainties of an observation are very large, from the actual refraction varying so largely from the mean, as given in the tables. A mean value of this ratio was chosen, therefore, which enables the corrections at other latitudes to be found in terms of those in latitude 40° without material error. These ratios are given in the Table of Latitude Coefficients. EXAMPLE. Let it be required to prepare a table of declination settings for a point whose latitude is 38° 30', which lies in the “ Central Time Belt,” and for April 5, 1890. Since the time is 6 h. earlier than that at Greenwich, the declination given in the Ephemeris for Greenwich mean noon (6° 9' 57") is the declination for the given place at 6 A.M. If the point were in the “ Eastern Time Belt ” it would be the declination at 7 A.M., etc. Suppose it is desired to prepare declination settings from 7 A.M. to 5 P.M. From the table of 48 SUR VE YING. TABLE OF REFRACTION CORRECTION TO BE APPLIED TO THE DECLINATIONS. Refraction Refraction 1 Refraction Refraction Date. Correction. Date. Correction. Date. Correction. Date. Correction. Latitude 40°. Latitude 40°. Latitude 40“. Latitude 40“. Jan. Feb. Mar. May. *i h. l' S8" 13 I h. i' 16" 30 h. 42" »4 I h. 23" 2 2 16 14 2 25 31 15 2 27 3 3 04 15 3 I 48 April. 47 57 18 16 3 34 3 4 6 23 16 17 4 5 2 8 47 39 2 3 4 5 i* 2 *7 i3 4 5 i' U 4 5 2 2 54 II 18 19 2 I I 12 20 4 C 1 2 39 44 54 14 08 »9 20 1 2 22 26 3 2 59 20 3 I 40 0 6 3 21 3 33 1 4 6 01 21 4 2 31 7 A j 22 4 47 22 5 6 49 8 5 2 23 5 I 15 9 I 51 23 T 07 9 I 36 24 I 21 2 2 07 24 2 I 15 10 2 41 25 2 25 3 2 51 25 3 1 33 II 3 51 26 3 32 12 4 5 40 26 4 2 18 12 4 I 10 27 4 46 13 27 5 5 29 13 5 58 28 5 I 13 14 15 16 17 18 1 2 3 4 1 2 2 5 46 01 40 00 28 Mar. 2 3 4 1 2 3 4 5 I 1 1 2 4 03 10 27 06 39 14 ll *7 18 1 2 3 4 5 I 1 34 38 48 06 49 29 30 June. 1 2 1 2 3 4 5 I 20 24 31 44 II 19 20 I I 42 56 5 I 0 59 19 20 1 2 32 36 3 I 19 21 2 I 6 2 I 06 21 3 45 4 2 23 22 3 2 31 7 3 I 21 22 4 I 02 5 3 30 23 4 4 35 8 4 I 56 23 5 I 42 6 4 43 9 5 4 04 7 5 I 10 24 37 24 I 30 25 I I 10 I 55 25 2 34 8 I t8 26 50 II 2 I 02 26 3 42 9 2 22 27 3 2 22 12 3 I 15 27 4 58 10 3 29 28 4 4 07 13 4 I 47 28 5 I 36 II 4 43 14 5 3 34 12 5 I 09 29 29 j 28 32 39 18 30 I I 32 15 I 52 2 13 I 2 I 44 16 2 58 May. ■a 14 2 22 Feb. 3 2 13 17 3 I 10 I IS 3 29 1 2 4 3 41 18 19 4 5 1 3 39 08 2 3 4 5 I 55 30 16 17 4 5 I 42 08 3 4 I T 26 20 21 2 48 54 4 5 1 2 26 30 18 19 1 2 18 22 5 2 37 22 3 I OS 6 3 37 20 3 29 6 3 04 23 4 I 32 7 4 53 21 4 42 7 4 3 21 24 5 2 51 8 5 26 22 5 08 8 I I 21 25 I 45 9 I 25 23 I 18 9 2 I 3 * 26 2 50 10 2 29 24 2 22 10 3 1 56 27 3 I 01 II 3 36 25 3 29 11 4 3 04 28 4 I 25 12 4 51 26 4 42 12 29 5 2 34 13 5 I 22 27 5 08 * The hours are counted each way from noon. Thus 9 a.m. and 3 p.m. would correspond to the 3d hour in the table. SURVEYING. 48^ Refraction Refraction Refraction Date. Correction. Date. Correction. Date. Correction. Date. Latitude 40 ®. Latitude 40 ®. Latitude 40 ®. June. Aug. Oct. Nov. 28 I h. 18 " 17 I h. 32 " 6 I h. 1 ' 03 " 20 29 18 2 36 7 2 I 10 21 jSy. 3 29 43 09 19 20 3 4 1 ^ . 45 02 8 9 3 4 1 2 27 06 22 23 1 2 5 i' 21 5 I 42 10 5 4 39 24 22 I 34 3 I 19 23 2 38 II I I 07 25 26 4 2 23 24 3 48 12 2 I 15 5 3 30 25 4 I 06 13 3 I 33 27 28 6 4 43 26 5 I 49 14 4 2 18 7 5 I 10 27 I 36 15 5 5 29 29 8 I 20 28 2 41 9 2 24 29 3 51 16 I I 12 10 3 31 30 4 I 10 17 2 1 20 II 4 44 31 5 I 58 18 3 I 40 Dec. 12 5 I II 19 4 2 31 I Sept. 20 5 6 49 2 »3 I 21 I I 39 3 14 2 25 2 2 44 16 4 15 3 32 3 3 54 21 I I 16 4 46 4 4 I 14 22 2 I 25 17 5 I 13 5 5 2 08 23 3 I 48 e 18 24 4 2 47 6 I 22 6 I 42 25 5 8 39 19 2 26 7 2 47 7 8 20 3 33 8 3 57 2l 4 47 9 4 I 19 26 I I 21 9 22 5 I 15 10 5 2 18 27 2 I 31 48 28 3 I 56 23 I 23 II I 29 4 3 04 10 24 2 27 12 2 50 30 5 II 01 II 25 3 34 13 3 I 01 12 26 4 14 4 I 25 N 26 13 27 5 I 18 15 5 2 34 Nov. 2 I 37 14 28 25 29 36 51 22 16 I 48 2 3 2 04 29 30 1 2 3 17 18 2 3 I 54 05 3 4 4 5 3 13 21 57 15 16 19 4 I 32 17 Aug. 5 I 20 5 2 51 18 * 21 1 52 6 I I 32 19 2 I 26 22 2 58 7 2 I 44 3 2 30 23 3 I 10 8 3 2 13 20 4 3 37 24 4 I 39 9 4 3 41 21 5 4 53 25 5 3 08 22 6 5 I 26 23 26 I 55 10 37 24 '7 I 28 27 2 I 02 It I I . 8 9 2 3 32 39 28 29 3 4 I I 15 47 12 13 2 3 1 2 50 22 25 10 4 55 30 5 3 34 14 4 4 07 26 II 5 I 30 Oct. 27 28 12 I 30 I X 59 15 42 56 29 13 2 34 2 2 I 06 16 14 3 42 3 3 I 21 17 15 4 58 4 4 I 56 18 3 31 30 16 5 I 36 5 5 4 04 19 4 4 35 3- Refraction Correction. Latitude 40°. h. 46" 2 01 2 40 4 59 1 50 2 06 2 49 5 33 1 54 2 n 2 59 6 oi 1 5S 2 16 3 04 6 23 2 00 2, 19 3 09 6 38 2 20 I ” 6 47 I " 6 49< 2 00 2 19 3 09 6 13 ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. TARLE OF LATITUDE COEEFICIENTS. Latitude. Coefficient. Latitude. Coefficient. I.atitude. Coefficient. 15“ •30 30° •65 45 ” 1.20 16 •32 31 .68 46 1.24 17 •34 32 •71 1 47 1.29 18 •36 33 •75 1 48 r 33 19 .38 34 •78 1 49 1.38 20 .40 35 .82 50 1.42 21 .42 36 .85 51 1.47 22 .44 37 .89 52 1-53 23 .46 38 .92 53 1.58 24 .48 39 .96 54 1.64 25 •50 40 1 .00 55 1.70 26 •53 41 1.04 56 1.76 27 .56 42 1.08 57 1.82 28 •59 43 1 . 12 58 1.88 29 .62 44 1 . 16 59 1.94 Note. — F or any other latitude than 40° the refraction corrections given in the preceding table are to be multiplied by the coefficients given in this table to obtain the true refraction corrections for that latitude. latitude corrections we find that the refraction corrections will be .94 of those given in the table. The following table of declination settings may now be made out : Hour. Declination. Refr. Cor. Setting. Hour. Declination. Refr. Cor. Setting. 7 + 6° 10' 54" + 2' 00" + 6° 12' 54" I + 6° 16' 35" + yi " 4 - 6 ° 17' 12" 8 6 II 51 4- I 10 , 1 0 13 01 2 6 17 31 + 41 6 18 12 9 6 12 47 + 51 6 13 38 3 6 18 28 + 51 6 19 19 10 6 13 44 + 41 6 14 25 4 6 19 25 4- i' 10" 6 20 35 II 6 14 41 + 37 6 15 18 5 6 20 22 4 - 2 00 6 22 22 From March 20th to September 20th the declination is positive, while from September 20th to March 20th it is nega- tive. From December 20th to June 20th the hourly correction is positive, while from June 20th to December 20th it is nega- tive. The refraction correction is always positive. Particular attention mu.st be given to all these signs in making out the table of declination settings. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 49 54. Errors in Azimuth due to Errors in the Declina- tion and Latitude Angles. — The spherical triangle involved in an observation by the solar compass is shown in Fig. II, where Pis the pole, Z the zenith, and 5 the sun. Then the angle at P=t, the hour-angle from the meridian ; “ “ .2'= the azimuth from the north point ; ‘‘ S = q, the variable or parallactic angle. Also, the arc PZ =: the co-latitude = 90° — (p ; “ PS — the co-declination = 90° — 6 ; “ ZS = the co-altitude, or zenith dis- tance = 90° — h. Taking the parenthetical notation of the figure, we have, from spherical trigonometry, cos (^) = cos [c] cos {U) sin {c) sin (^) cos (^). But in terms of d', 0, and A, this becomes sin d = sin 0 sin h — cos 0 cos h cos A. . (i) In a similar manner, from two other fundamental equations of the spherical triangle, we may write cos d cos t — cos 0 sin h -f- sin 0 cos h cos A ; (2) cos d sin / = cos ^ sin (3) If we differentiate equation (i) with reference to A and d, 4 50 SUR VK YING. and then with reference to A and 0, we obtain, after some reductions by the aid of equations (2) and (3) and = dA^ — d d cos 0 sin /’ dcf) cos 0 tan { (4) (5) Now, if the change (or error) in S and 0 be taken as i minute of arc, or, in other words, if the settings for declination or lati- tude be erroneous by that amount, either from errors in the instrumental adjustments or othemise, then equations (4) and (5) show what is the error due to this cause in the azimuth, or in the direction of the meridian, as found. In the following table, values of dA^ and dA^ are given for various values of 0 and t (latitude and hour-angle). In this table no attention is paid to signs, as it is intended mainly to show the size of the errors to which the work is liable from inaccurate settings for declination and latitude ; the values may, however, be used as corrections to the observed azimuths from such inaccuracies by observing the instructions in the appended note. * dAh signifies the change in A due to a small change, d 8 , in d, the other functions involved in equation (i) remaining constant. Similarly for dA^, when

*« ^ Also, from the vertical angles taken at B, we have : Elevation of A below B = AB tan Va\ “ “ above tan Vp'. We now have a check on both the relative elevations and on the distances AP and BP. Assuming the elevation of A to be zero, we have: Elevation of P above A — AP tan Vp = AB tan Vb + BP tan Vp'. This equality will not result unless the observations were well taken, the computations accurately made, and the instrument carefully adjusted. The ad- justments mainly involved here are the plate-bubbles and the vernier on the vertical circle. If the points are a considerable distance apart, as over a half- mile, the elevations obtained by reading the vertical angles are appreciably too great, on account of the earth’s curvature. This may be taken as eight inches for one mile and proportional to the square of the distance. Or, we may write: Elevation correction on long sights, in inches,* = — 8 (distance in miles'!®. If the distances are all less than about half a mile, no attention need be paid to this correction in this problem. For a full discussion of this subject see chap. XIV. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I07 109. Find the height of a tree or house above the ground, on a distant hill, without going to the immediate locality. no. Find the horizontal length and bearing of a line joining two visible but inaccessible objects. Use the magnetic bearing if the true bearing of the base- line is not known. ^ 11^ Find the horizontal length and bearing of a line joining two inaccessi- ble points both of which cannot be seen from any one position. Let A and B be the inaccessible points. Measure a base CD such that A is seen from C, and B from D. Auxiliary bases and triangles may be used to find the lengths of ^ C and BD. Knowing A C and CD and the included angle, compute AD in bearing and distance. The angle ADB may now be found, which, with the adjacent sides AD and BD known, enables the side AB to be found in bearing and distance. 1 12. With the transit badly out of level, or with horizontal axis of the tele- scope thrown considerably out of the horizontal, measure the horizontal angle between two objects having very different angular elevations. Do this with both telescope normal and telescope reversed, and note the difference in the values of the angle obtained in the two cases. 1 13. Select a series of points on uneven ground, enclosing an area, and occupy them successively with the transit, obtaining the traverse angles. That is, knowing or assuming the azimuth of the first line, obtain the azimuths of the other connecting lines, or courses, with reference to this one, returning to the first point and obtaining the azimuth of the first course as carried around by the traversed line. This should agree with the original azimuth of this course. The distances need not be measured for this check. 1 14. Lay out a straight line on uneven ground by the method given in Art. 100, occupying from six to ten stations. Return over the same line and estab- lish a second series of points, paying no attention to the first series, and then note the discrepancies on the several stakes. In returning, the two final points of the first line become the initial points of the second, this return line being a prolongation of the line joining these two points. If these deviate ever so little, therefore, from the true line, the discrepancy will increase towards the initial point. Similar exercises to those given for the solar compass may be assigned for the solar attachment. io8 SUR VE YING. on board ship. It is exclusively used in observations at sea, and is always used in surveying where angles are to be meas- ured from a boat, as in locating soundings, buoys, etc., as well as in reconnoissance work, explorations, and preliminary sur- veys. It has been in use since about 1730. The accompanying cut shows a common form of this in- strument as manufactured by Fauth & Co., Washington. The limb has a 7|-inch radius, and reads to 10 seconds of arc. THE SEXTANT. I15. The Sextant is the most convenient and accurate hand-instrument yet devised for measuring angles, whether horizontal, vertical, or inclined. It is called a sextant because its limb includes but a 60° arc of the circle. It will measure angles, however, to 120°. It is held in the hand, measures an angle by a single observation, and will give very accurate re- sults even when the observer has a very unstable support, as Fig. 21. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. IO9 There is a mirror 7 ^/(Fig. 22), called the Index Glass, rigidly- attached to the movable arm MA, which carries a vernier reading on the graduated limb CD. There is another mirror, I, called the Horizon Glass, rigidly attached to the frame of the instrument, and a telescope pointing into this mirror, also rigidly attached. This mirror is silvered on its lower half, but clear on its upper half. A ray of light coming from H passes through the clear portion of the mirror / on through the tele- scope to the eye at E. Also, a ray from an object at O strikes the m\xxox M, is reflected to m, and then through the telescope to E. Through one half of the objective come the rays from H, and through the other half the rays from O, each of which sets of rays forms a perfect image. By moving the arm MA it is evident these images will appear to move over each other, no SUR VE YING. and for one position only will they appear to coincide. The bringing of the two images into exact coincidence is what the observation consists in, and however unsteady the motion of the observer may be, he can occasionally see both images at once, and so by a series of approximations he may finally put the arm in its true position for exact superposed images. The angle subtended by the two objects is then read off on the limb. Ii6. The Theory of the Sextant rests on the optical principle that “ if a ray of light suffers two successive reflec- tions in the same plane by two plane mirrors, the angle be- tween the first and last directions of the ray is twice the angle of the mirrors.” To prove this, let ( 9 J/and mEhe the first and last posi- tions of the ray, the latter making with the former produced the angle E. The angle of the mirrors is the angle A. The angles of incidence and reflection at the two mirrors are the angles i and PM, and p 7 n being the normals. We may now write : Angle E = OMm — MmE, — lit ^ j I angle A = ImM — mMA = (9o°-O-(9O°-0 = i— t\ Therefore E = 2A. Q. E. D. When the mirrors are brought into parallel planes, the angle A becomes zero, whence E also is zero, or the rays OM and Hin are parallel. This gives the position of the arm for the zero-reading of the vernier. The limb is graduated from this point towards the left in such a way that a 60° arc of the circle will read to 120°. That is, a movement of 1° on the arc really measures an angle of 2° in the incident rays, so it must ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. Ill be graduated as two degrees instead of one. The very large radius enables this to be done without difficulty. ADJUSTMENTS OF THE SEXTANT. 117. To make the Index Glass perpendicular to the Plane of the Sextant. — Bring the vernier to read about 30° and examine the arc and its image in the index glass to see if they form a continuous curve. If the glass is not perpendi- cular to the plane of the arc, the image will appear above or below the arc, according as the mirror leans forward or back- ward. It is adjusted by slips of thin paper under the project- ing points and corners of the frame. 118. To make the Horizon Glass Parallel to the Index Glass for a Zero-reading of the Vernier. — Set the vernier to read zero and see if the direct and reflected images of a well-defined distant object, as a star, come into exact coinci- dence. If not, adjust the horizon glass until they do. If this adjustment cannot be made, bring the objects into coincidence, or even with each other so far as the motion of the arm is con- cerned, and read the vernier. This is the index error of the instrument and is to be applied to all angles read. The better class of instruments all allow the horizon glass to be adjusted. This adjustment is generally given as two, but it is best con- sidered as one. If made parallel to the index glass after that has been adjusted, it must be perpendicular to the plane of the instrument. 119. To make the Line of Sight of the Telescope parallel to the Plane of the Sextant. — The reticule in the sextant carries four wires forming a square in the centre of the field. The centre of this square is in the line of collima- tion of the instrument. Rest the sextant on a plane surface, pointing the telescope upon a well-defined point some twenty feet distant. Place two objects of equal height upon the extremities of the limb that II2 SUI^ VE YING. will serve to establish a line of sight parallel to the limb. Two lead-pencils of same diameter will serve, but they had best be of such height as to make this line of sight even with that of the telescope. If both lines of sight come upon the same point to within a half-inch or so at a distance of 20 feet, the resulting maximum error in the measurement of an angle will be only about i". THE USE OF THE SEXTANT. 120. To measure an Angle with the sextant, bring its plane into the plane of the two objects. Turn the direct line of sight upon the fainter object, which may require the instru- ment to be held face downwards, and bring the two images into coincidence. The reading of the limb is the angle re- quired. It must be remembered that the angles measured by the sextant are the true angles subtended by the two objects at the point of observation, and not the vertical or horizontal projection of these angles, as is the case with the transit. The true vertex of the measured angle is at E, Fig. 21. It is evident the position of E is dependent on the size of the angle, being at a great distance back of the instrument for a very small angle. The instrument should therefore not be used for meas- uring very small angles except as between objects a very great distance off. The sextant is seldom or never used for measur- ing angles where the position of the instrument (or the vertex of the angle) needs to be known with great accuracy. EXERCISES FOR THE SEXTANT. 121. Measure the altitude of the sun or a star at its culmination by bringing the direct image, reflected from the surface of mercury held in a flat dish on the ground, into coincidence with the image reflected from the index glass. Half the observed angle is the altitude of the body. The altitude of a terres- trial object may be obtained in the same manner, in which case the vessel of mercury should rest on an elevated stand ; the sextant could then be brought near to it and the angular divergence of the two incident rays to the mercury surface and index glass reduced to an inappreciable quantity. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II3 If the observation of a heavenly body be made on the meridian and the declination of the body at the time of observation be known, the latitude of the place is readily found. I2ia. Measure the angle subtended by two moving bodies, as of two men walking the street in the same direction, or of two boats on the water. (This is to illustrate the capacity of the sextant, for none but a reflecting instrument bringing two converging lines of sight into coincidence is competent to do this.) The exercises given in Arts. 106, 108, 109, and no for the transit may also serve for the sextant. Further applications of the sextant in locating soundings are given in chap. X. 122. The Double-reflecting Gpniograph is a kind of dou- ble sextant and three-arm protractor* combined. It enables the two variable angles of the “ three-point problem” f to be measured at once, and then provides for the immediate plot- ting of these angles upon the sheet, without reading off the values of the angles unless they are to be put on record. The angles may be read, however, and plotted afterwards if de- sired. This very ingenious and convenient instrument is the invention of Lieutenant Constantin Pott, of the English Navy. The construction and principles of the instrument are shown in Figs. 23, 24, and 25. To the graduated circle whose centre is D, Fig. 24, there are attached one fixed and two movable arms, each having one radial fiducial edge. The main frame- work of the instrument lies on the prolongation of the fixed arm A. Immediately back of the centre of the circle is a cylindrical frame containing two fixed mirrors, s s, one above the other, and also a free opening, W, Fig. 23. These corre- spond to the fixed mirror and clear glass on the sextant. Im- mediately back of these mirrors is the telescope, P, and on each side of this is a movable mirror, 55 , attached to the slide bars //. These bars are fastened to the mirrors and slide freely through the studs Z set upon the movable arms B P,. * For a description of the three arm protractor, see chapter VI., p. 167. f See chapter X., p. 280, for a discussion of this problem. 8 IT4 SURVEYING. The distance of these studs from the centre of the graduated circle is the same as that of the axes of the movable mirrors 5 S. Therefore a circle whose centre coincides with the centre of the graduated circle may pass through these four axes. The theory "of the instrument is shown in Fig. 25. The ray of light R is reflected from c to ^ and thence down the tele- scope to A. The object in the prolongation of AB casts the ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II5 ray Be directly down the telescope. The angle formed by the incident and final reflected ray, Rfe^ is twice the angle subtended by the planes of the mirrors C g e, as was shown in the case of the sextant. When the rays R and B coincide the mirrors S S and s s, Fig. 24, are parallel. The slide-bar then has the position Ca. When the arm has come into the posi- tion dB'f making the angle 0 with the fixed arm dB^ the slide- bar has come into the position Ca\ making an angle |-0 with its former position ca, since this is an angle in the circumfer- ence. The mirror has also turned through an angle and since it was parallel to the mirror ss in its first position it now makes an angle y — \ with it. The angle which is the angle subtended by the incident ray Rc and by the direct ray BA, is therefore equal to the angle 0, which is the angle ada' read on the graduated circle. ii6 SUR VE YING. Both movable arms are provided with clamp-screws, K Ky and tangent screws, M M. The instrument is held, while observing, by the handle Hy Fig. 23 ; but when used for plot- ting the point of observation this handle is unshipped and the instrument manipulated by the two milled heads F and 6*. The centre at dy Fig. 24, is open, so that when the instrument is adjusted to the plotted positions of the three known stations, the point of observation is marked by a pencil through the open centre. It is therefore a double sextant for observing and a three-arm protractor for plotting. ADJUSTMENT, USE, AND CADE OF INSTRUMENTS, llj CHAPTER V. THE PLANE TABLE. 123. The Plane Table consists of a drawing-board properly mounted on which rests an alidade carrying a line of sight rigidly attached to a plain ruler with a fiducial edge. The line of sight is usually determined by a telescope, as in Fig. 26. This telescope has no lateral motion with respect to the ruler, but both may be moved at pleasure on the table. The telescope has a vertical motion on a transverse axis, as in the transit. It is also provided with a level tube, either detachable or permanently fixed. The table is levelled by means of one round or two cross bubbles on the ruler of the alidade. The line of sight of the telescope is usually parallel to the fiducial edge of the ruler, though this is not essential. It is only necessary that they should make a fixed horizontal angle with each other. The table itself must have a free hori- zontal angular movement and the ordinary clamp and slow- motion screw. The table corresponds to the graduated limb in the transit, the alidades in the two instruments performing similar duties. Instead, however, of reading off certain hori- zontal angles, as is done with the transit, and afterwards plotting them on paper, the directions of the various pointings are at once drawn on the paper which is mounted on the top of the table, no angles being read. The true relative positions of certain points in the landscape are thus transferred directly to the drawing-paper to any desired scale. The magnetic bearing of any line may be determined by means of the decli- nator, which is a small box carrying a needle which can swing some ten degrees either side of the zero-line. The zero-line ii8 SUR VE YING. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I IQ being parallel to one edge of the box, the magnetic meridian may be at once marked down on any portion of the map, and the bearing of any intersecting line determined by means of a protractor. The instrument has been long and extensively used for mapping purposes, and is still the only instrument used for the “ fillingdn” of the topographical charts of the U. S. Coast and Geodetic Survey. An extended account of the instrument and the field methods in use on that service may be found in Appendix 13 of the Report of the U. S. Coast and Geodetic Survey for 1880. The following discussion is partly from that source. ADJUSTMENTS OF THE ALIDADE. 124. To make the Axes of the Plate-bubbles parallel to the Plane of the Table. — Level the table with the alidade in any position, noting the readings of the bubbles. Mark the exact position of the alidade on the table, take it up carefully, and, reversing it end for end, replace it by the same marks. If the bubbles now have the same readings as before, with refer- ence to the table they are parallel to the plane of the table. If not, adjust the bubbles for one half the movement and try again. 125. To cause the Line of Sight to revolve in a Vertical Plane. — This adjustment is the same as in the transit. It need not be made with such extreme accuracy, however, and the plumb-line test is sufficient. With the instrument carefully levelled, cause the line of sight to follow a plumb-line through as great an arc as convenient. If the line of sight deviates from the plumb-line raise or lower one end of the transverse axis of the telescope, until it will follow it with sufficient exact- ness. 126. To cause the Telescope-bubble and the Vernier on the Vertical Arc to read Zero when the Line of Sight is Horizontal. — This adjustment is also the same as in the 120 SURVEYING. transit. The methods given for the transit may be used with the plane table, or a sea horizon may be used as establishing a horizontal line, or a levelling-instrument may be set up beside the plane table having the telescopes at the same elevation, and both lines of sight turned upon the same point in the horizontal plane as determined by the level. The bubble and vernier are then both adjusted to this position of telescope. This adjustment is important if elevations are to be deter- mined either by vertical angles or by horizontal lines of sight. If only geographical position is sought this adjustment may be neglected. THE USE OF THE PLANE TABLE. 127. In using the Plane Table at least two points on the ground, over which the table may be set, must be plotted on the paper to the scale of the map before the work of locating other points can begin. This requires that the distance between these points shall be known, which distance becomes the base- line for all locations on that sheet. Any error in the measure- ment or plotting of this line produces a like proportional error in all other lines on the map. The plane table is set over one of these plotted points, the fiducial edge of the ruler brought into coincidence with the two points, and the table revolved until the line of sight comes on the distant point. The table is now clamped and carefully set by the slow-motion screw in this position, when it is said to be oriented, or in position. In Figs. 27 and 28, let T, T,' T," T^" represent the plane- table sheet and the points a and p the original plotted points. The corresponding points on the ground are A and P, the latter being covered by p in Fig. 27, and the former by in Fig. 28. In Fig. 27, the plotted point p is centred over the point P, the ruler made to coincide with ap, and the telescope made to read on A by shifting the table. For plotting the directions of I2I Fig. 28. lO 122 SUR VE YING. other objects on the ground, the alidade is made to revolve about p just as the transit revolves about its centre. A needle is sometimes stuck at this point, and the ruler caused to press against it in all pointings, but this defaces the sheet. Other pointings are now made to B, C, and D, which may be used as stations, and also to a chimney (c/l), a tree (/.), a cupola {cup.), a spire (sp.), and a windmill {w.ml). Short lines are drawn at the estimated distance from p, and these marked with letters, as in the figure, or by numbers, and a key to the numbers kept in the sketch- or note-book. The table is now removed to A, the other known point, and set with the point a on the plot over the point A on the ground, when the table is approximately oriented. The ruler is now set as shown in Fig. 28, coinciding with a and p, but pointing towards /. The table is then swung in azimuth until the line of sight falls on P, when it is clamped. It is now oriented for this station, and pointings are taken on all the objects sighted from P, and on such others as may be sighted from subsequent stations, the alidade now revolving about the point a on the paper. The intersections on the plot of the two pointings taken to the same object from^ and P will evi- dently be the true position on the plot for those points with reference to, and to the scale of, the line ap. These intersec- tions are shown in Fig. 28. It is evident that if other points, as D or C, be now occu- pied, the table oriented on either A or P, and pointings taken on any of the objects sighted from both A and P, the third or fourth line drawn to the several objects should intersect the first two in a common point. This furnishes a check on the work, and should be taken for all important points. It is pref- erable also to have more than two points on the sheet pre- * It will be noted that this process of orienting the plane table is practically identical with that by which the limb of the transit is oriented in traversing (art. loi). ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 23 viously determined. Thus, if B were also known and plotted at b, when the table had been oriented on any other station, and a pointing taken to the fiducial edge of the ruler should have passed through b. As fast as intersections are obtained and points located the accompanying details should be drawn in on the map to the proper scale. If distances are read by means of stadia wires on a rod held at the various points (see chap. VIII), then a single pointing may locate an object, the distance being taken off from a scale of equal parts, and the point at once plotted on the proper direction-line. It is now common to do this in all plane-table surveying. 128. Location by Resection. — This consists in locating the points occupied by pointings to known and plotted points. The simple case is where a single pointing has been taken to this point from some known point, and a line drawn through it on the sheet. It is not known what point on this line represents the plotted position of this station. The setting of the instrument can therefore be but approximate, but near enough for all purposes. The table can be oriented as before, there being one pointing and corresponding line from a known point. A station is then selected, a pointing to which is as nearly 90 degrees from the orienting line as possible, and the alidade so placed that while the telescope sights the object the fiducial edge of the ruler passes through the plot of the same on the sheet. The intersection of this edge with the former line to this station gives the station’s true position on the sheet. This latter operation is called resection. Another re- section from any other determined point may be made for a check. 129. To find the Position of an Unknown Point by Re- section on Three Known Points. — This is known as the Three-point Problem, and occurs also in the use of the sextant in locating soundings. It is fully discussed in that connection 124 SUJ^VEVING. (see chap. X.), so that only a mechanical solution suitable for the problem in hand will be given here. It is under- stood there are three known points, A, B, and C, plotted in a, b, and c on the map. The table is set up over any given point (not in the circumference of a circle through A, B, and C\ Fasten a piece of tracing-paper, or linen, on the board, and mark on it a point p for the station /^occupied. Level the table, but of course it cannot be oriented. Take pointings to A, B, and C, and draw lines on the tracing-paper from p towards a, b, and c, long enough to cover these distances when drawn to scale. Remove the alidade and shift the tracing- paper until the three lines drawn may be made to coincide exactly with the three plotted points a, b, and c. The point p is then the true position of this point on the sheet. This being pricked through, the table may now be oriented and the work proceed as usual. 130. To find the Position of an Unknown Point by Re- section on Two Known Points. — This is called the Two- point Problem, and but one of several solutions will be given. It is evident that if the table could be properly oriented over the required point, its position on the sheet could be at once found by resection on the two known points. The table may be oriented in the following manner: Let A and B be the known points plotted in a and b on the sheet. Let C be the unknown point whose position c on the sheet is desired. Select a fourth point D, which may be occupied, and so placed that intersections from C and D on A and B will give angles between 30 and 120 degrees. Fasten a piece of tracing linen or paper on the board, marking a point at random. Set up over D, orienting the table as nearly as may be by the needle or otherwise. Draw lines from d! towards A, B, and C. Mark off on the latter the estimated distance to C, to scale, calling this point C . Set up over C, with d over the station, orienting on D by the line c'd\ This brings the table ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 12 $ parallel to its former position at D, From c' draw lines to A and B, intersecting the corresponding lines drawn from d in a' and b' . We now have a quadrilateral db'c'd similar to the quadrilateral formed by the true positions of the plotted points abed, but it differs in size, since the distance c'd was assumed, and also in position (azimuth), since the table was not properly oriented at either station. Remove the alidade, and shift the tracing until the line a'b' coincides with a and b on the sheet. Replace the alidade on the tracing, bringing it into coincidence with dd , c'b\ or c' d , and revolve the table on its axis until the line of sight comes upon A, B, or D, as the case may be. The table is now oriented, when the true posi- tion of c may be readily found by resecting from a and b, which, when pricked through, gives its position on the sheet. The student may show how the same result could have been obtained with- out the aid of tracing-paper. If the fourth point D may be taken in range A and B, the table may be properly oriented on this range, and a line drawn towards C from any point on this range line on the plot. Then C is occupied, and the table again properly oriented by this line just drawn, when the true position of c may be found by resecting from a and b, as before. In general, if the table can be properly oriented over any unknown point from which sights may be taken to two or more known (plotted) points, the position of this unknown point is at once found by resection from the known points. The student would do well to look upon the table and the attached plot as analogous to the graduated horizontal limb in the transit. The principles and methods of orienting are pre- cisely similar, the pointings differing only in this, that with the transit the horizontal angle, referred to the meridian, is read off, recorded, and afterwards plotted, while with the plane table this bearing is immediately drawn upon the sheet. 131. The Measurement of Distances by Stadia. — This 126 SURVEYING. method of determining short distances is now generally used in connection with the plane table. It is fully discussed in chap- ter VIIL, where the principles of its action and its use with the transit are given at length. The same principles, field methods, and tables apply to its use with the plane table, with such modifications as one accustomed to the use of the plane table would readily introduce. When used in this way it enables a point to be plotted from a single pointing, it being located by polar coordinates (azimuth and distance), in- stead of by intersections. EXERCISES WITH THE PLANE TABLE. 132. Make a plane table survey of the college campus, measuring the length of one side for a base. 133. Having located several points on the sheet by intersections, occupy them and check their location by resection. 134. Locate a point (not plotted) by resection on three known points (art. 129). 135. Locate a point (not plotted) by resection on two known points, first taking the auxiliary point D not in line with AB, and then by taking it in line with AB. This gives a check on the position of the point, and shows the ad- vantages of the second method when it is feasible. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 12 / CHAPTER VI. ADDITIONAL INSTRUMENTS USED IN SURVEYING AND PLOTTING. THE ANEROID BAROMETER. 136. The Aneroid Barometer consists of a circular me- tallic box, hermetically .sealed, one side being covered by a corrugated plate. The air is mostly removed, enough only being left in to compensate the diminished stiffness of the cor- Fig. 29. rugated cover at higher temperatures. This cover rises or falls as the outer pressure is less or greater, and this slight motion is greatly multiplied and transmitted to an index pointer moving over a scale on the outer face. The motion of the index is compared with a standard mercurial barom- eter and the scale graduated accordingly. Inasmuch as all 128 SUR VE YING. barometric tables are prepared for mercurial barometers, wherein the atmospheric pressure is recorded in inches of mercury, the aneroid barometer is graduated so that its read- ings are identical with those of the mercurial column. Figure 29 shows a form of the aneroid designed for eleva- tions to 4000 feet above or to 2000 feet below sea-level. It has a vernier attachment and is read with a magnifying-glass to single feet of elevation. It must not be supposed, how- ever, that elevations can be determined with anything like this degree of accuracy by any kind of barometer. The barometer simply indicates the pressure at the given time and place, but for the same place the pressure varies greatly from various causes. All barometric changes, therefore, cannot be attrib- uted to a change in elevation, when the barometer is carried about from place to place. If two barometers are used simultaneously, which have been duly compared with each other, one at a fixed point of known elevation and the other carried about from point to point in the same locality, as on a reconnoissance, then the two sets of readings will give very close approximations to the differences of elevation. If the difference of elevation be- tween distant points is desired, then long series of readings should be taken to eliminate local changes of pressure. The aneroid barometer is better adapted to surveys than the mer- curial, since it may be transported and handled with greater ease and less danger. It is not so absolute a test of pressure, however, and is only used by exploring or reconnoissance parties. For fixed stations the mercurial barometer is to be preferred. It has been found from experience that the small aneroids of i| to 2 \ inches diameter give as accurate results as the larger ones. 137. Barometric Formulse. — In the following derivation of the fundamental barometric formula the calculus is used, so that the student will have to take portions of it on trust until ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 29 he has studied that branch of mathematics. All that follows Eq. (4) he can read. Let H — height of the “ homogeneous atmosphere”* in lat. 45 °- h — corresponding height of the mercurial column, d r= the relative density of the “ homogeneous atmos- phere” with reference to mercury. z = difference of elevation between two points, with barometric readings of h' and at the higher and at the lower point respectively. Then from the equilibrium between the pressures of the mercurial column and atmosphere we have : h = dH (i) Also, for a small change in elevation, dz, the corresponding change in the height of the mercurial column would be dh =. 6dz (2) Substituting in (2) the value of as given by (i), we have : dh = -j^dz'y or. dz = H- h Integrating (3) between the limits h' and we have: , K log. y (3) ( 4 ) * “ Homogeneous atmosphere” signifies a purely imaginary condition wherein the atmosphere is supposed to be of uniform density from sea-level to such upper limit as may be necessary to give the observed pressure at the ob- served temperature. 9 130 SUR VE YING. where the logarithm is in the Napierian system. Dividing by the modulus of the common system to adapt it to computation by the ordinary tables, we have : 2=2.30258// iog„ A (5) If Ho be the height of the homogeneous atmosphere at a temperature of 32° F., and if ho be the height of the mercurial column at sea-level at same temperature, and if and be the specific gravities of mercury and air respectively, then, evidently, or, Ho = — — (6) From experiment we have : ho — 29.92 inches, .fm= 13-596 ga - 0.001239 whence Ho = 26,220 feet. This is on the assumption that gravity is constant to this height above sea-level. When this is corrected for variable gravity we have : Ho = 26,284 feet (7) Equation (7) gives the height of the homogeneous atmos- phere at a temperature of 32° F. But since the volume of a gas under constant pressure varies directly as the temperature, and since the coefficient of expansion of air is 0.002034 for 1° F., we have for the height of the homogeneous atmosphere at any temperature : //=//„ [i +0.002034 (/— 32°)] ... (8) ADJUSTMENT, USE, AND CADE OF INSTRUMENTS. I3I If the temperature chosen be the mean of the temperatures at the two points of observation, as t’ and tx for the upper and lower points respectively, then we should have: //= j^i + 0.002034 (^--32)] = 26,284 [l +O.OOIOI7 64)] . . (9) Substituting this value of Hm Eq. (5) we obtain : h z — 60,520 [i + 0.001017 (/'+ — 64)] log . (10) If we wish to refer this equation to approximate sea-level (height of mercurial column of 30 inches) and to a mean tem- perature of the two stations of 50° F., we may write : 3 ? , K , h' , 30 , 30 K Also, when t' lOO®, we have /'-f /,-64=36". Substituting these equivalents in eq. (10), we obtain z = 60520 (i -|- 0.001017X (n) 8 = 62737 log ^ - 62737 log 132 SUK VE YING. In this equation, the two terms of the second member rep- resent the elevations of the upper and lower points respec- tively, above a plane corresponding to a barometric pressure of 30 inches and for a mean temperature of the two positions of 50° F. Table I. is computed from this equation, the arguments be- ing the readings of the barometer, Ji! and /;, at the upper and lower stations respectively, the tabular results being elevations above an approximate sea-level. The difference between the two tabular results gives the difference of elevation of the two points, for a mean temperature of 50° and no allowance made for the amount of aqueous vapor in the air. For other tem- peratures, and for the effect of the humidity (which is not ob- served, but the average conditions assumed to exist), a certain correction needs to be applied, which correction is not an abso- lute amount, but is always a certain proportion of the difference of elevation as obtained from eq. (ii) or table I. If the two elevations taken from the table be called A' and A^y and the correction for temperature and humidity be C, we would have ^ = {A'-A,)(i + C) (12) It is seen, therefore, that ^7 is a coefficient which, when mul- tiplied into the result obtained from table L, gives the correc- tion to be applied to that result. The values of C are given in table II. for various values of t' + The following example will illustrate the use of the tables : ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 133 TABLE I. BAROMETRIC ELEVATION.* 30 Containing A = 62737 log T • Argument, h. h h. A. Dif. for .01. k. A. Dif. for .CI, h. A. Dif. for .OI« Inches. Feet. Feet. Inches. Feet. Feet. Inches. Feet. Feet. II. 0 27,336 —24.6 14.0 20,765 -19.5 17.0 — 16.0 II. I 27,090 24.4 14. 1 20,570 19.3 17. I 15,316 15.9 II .2 20,846 24.2 14.2 20,377 19. 1 17.2 15,157 15.8 II . 3 26,604 24.0 14.3 20,186 18.9 17.3 14,999 15.7 II . 4 26,364 23.8 14.4 19,997 18.8 17.4 14,842 15.6 II . 5 26,126 23.6 14.5 19,809 18.6 17.5 14,686 15.5 II. 6 25,890 23.4 14.6 19,623 18.6 17.6 14,531 15.4 II. 7 25,656 23.2 14.7 19,437 18.5 17.7 14,377 15.4 II. 8 25,424 23.0 14.8 19,252 18.4 17.8 14,223 15.3 II. 9 25,194 22.8 14.9 19,068 18.2 17.9 14,070 15.2 12.0 24,966 22.6 15.0 18,886 18. 1 18.0 13,918 15.1 12. 1 24,740 22.4 15. 1 18,705 18.0 18. 1 13,767 15.0 12.2 24,516 22.2 15.2 18,525 17.9 18.2 13,617 14.9 12.3 24,294 22.1 15.3 18,346 17.8 18.3 13,468 14.9 12.4 24,073 21.9 15.4 18,168 17.6 18,4 13,319 14.7 12.5 23,854 21.7 15.5 17,992 17.5 18.5 13,172 14.7 12.6 23,637 21.6 15.6 17,817 17.4 18.6 13,025 14.6 12.7 23,421 21.4 15.7 17,643 17.3 18.7 12,879 14.6 12.8 23,207 21.2 15.8 17,470 17.2 18.8 12,733 14.4 12.9 22,995 21.0 15.9 17,298 17. 1 18.9 12,589 14.4 13.0 22,785 20.9 16.0 17,127 16.9 19.0 12,445 14.3 13. 1 22,576 20.8 16. 1 16,958 16.9 19. 1 12,302 14.2 13.2 22,368 20.6 16.2 16,789 16.8 19.2 12,160 14.2 13.3 22,162 20.4 16.3 16,621 16.7 19-3 12,018 14. I 13.4 21,958 20. 1 16.4 16,454 16.6 19.4 11,877 14.0 13.5 21,757 20.0 16.5 16,288 16.4 19.5 11,737 13.9 13.6 21,557 19.9 16.6 16,124 16.3 19.6 11,598 13.9 13.7 21,358 19.8 16.7 15 961 16.3 19.7 11,459 M CO 13.8 21,160 19.8 16.8 15,798 16.2 19.8 11,321 13.7 13.9 20,962 -19.7 16.9 15,636 — 16.0 19.9 11,184 -13.7 14.0 20.765 17.0 15,476 20.0 11,047 * This table taken from Appendix 10 , Report U. S. Coast and Geodetic Survey, i88i. 134 SURVEYING. TABLE I. Barometric Elevation. — Continued. 30 Containing A = 62737 log — . Argument, h. h. A. Dif. for .01. h. A. Dif. for .01, h. A. Dif. for .01. Inches. Feet. Feet. Inches. Feet. Feet. Inches. Feet. Feet. 20.0 11,047 — 13.6 23.0 7,239 -II. 8 26.0 3,899 — 10.5 20. 1 10,911 13-5 23. I 7,121 II. 7 26. I 3.794 10.4 20.2 10,776 13.4 23.2 7,004 II. 7 26.2 3,690 10.4 20.3 10,642 13.4 23.3 6,887 II. 7 26.3 3,586 10.3 20.4 10,508 13-3 23-4 6,770 II .6 26.4 3,483 10.3 20.5 10,375 13-3 23-5 6,654 II .6 26.5 3.380 10.3 20.6 10,242 13-2 23.6 6,538 II -5 26.6 3,277 10.2 20.7 10,110 I 3 -I 23.7 6,423 II . 5 26.7 3,175 10.2 20.8 9.979 I 3 -I 23.8 6,308 II. 4 26.8 3,073 10. I 20.9 9,848 13.0 23-9 6,194 II. 4 26.9 2.972 10 . I j 21.0 9,718 12.9 24.0 6,080 11-3 27.0 2,871 10. 1 21. I 9.589 12.9 24.1 5,967 II . 3 27.1 2,770 10. 0 21.2 9,460 12.8 24.2 5.854 1 II . 3 27.2 2,670 10. 0 21.3 9.332 12.8 24.3 5,741 II. 2 27.3 2,570 10. 0 21.4 9,204 12.7 24.4 5.629 ii. I 27.4 2,470 9.9 21.5 9.077 12.6 24.5 5,518 II. I 27.5 9.9 21.6 8,951 12.6 24.6 5,407 II . I 27.6 2,272 9.9 21.7 8,825 12.5 24.7 5.296 II. 0 27.7 2,173 9.8 21.8 8,700 12.5 24.8 5,186 10.9 27.8 2,075 CO O' 21.9 8,575 12.4 24.9 5,077 10.9 27.9 1,977 9-7 22.0 8,451 12.4 25.0 4,968 10.9 28.0 1,880 9.7 22.1 8,327 12.3 25.1 4,859 10.8 28.1 1,783 9.7 22.2 8,204 12.2 25.2 4,751 10.8 28.2 1,686 9.7 22.3 8,082 12.2 25.3 4,643 10.8 28.3 1,589 9.6 22.4 7,960 12.2 25-4 4,535 10.7 28.4 1,493 9.6 22.5 7,838 12. 1 25.5 4,428 10.7 28.5 1,397 9.5 22.6 7,717 12.0 25.6 4,321 10.6 28.6 1,302 9.5 22.7 7,597 12.0 25.7 4,215 10.6 28.7 1,207 9.5 22.8 7,477 II. 9 25.8 4,109 10.5 28.8 1,112 9.4 22.9 7,358 -II. 9 25-9 4,004 -10.5 28.9 1,018 -9.4 23.0 7.239 26.0 3,899 29.0 924 ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 135 TABLE I. Barometric Elevations. — Continued. 30 Containing A = 62737 log ~j^- Argument, h. \h. A. Dif. for .01. h. A. Dif. for • OI, h. A. Dif. for .01. Inches. 29.0 29.1 29.2 29*3 29.4 29-5 29.6 29.7 Feet. 924 830 736 643 550 458 366 274 Feet. -9.4 9.4 9-3 9-3 9.2 9.2 -9.2 Inches. 29.7 29.8 29.9 30.0 30.1 30.2 30.3 30.4 Feet. 274 182 91 00 -91 181 271 361 Feet. -9.2 9.1 9.1 9.1 9.0 9.0 -9.0 Inches. 30.4 30.5 30.6 30-1 30.8 30.9 31.0 Feet. 361 451 540 629 717 805 — 893 Feet. —9.0 8.9 8.9 8.8 ' 8.8 - 8.8 TABLE 11 . CORRECTION COEFFICIENTS TO BAROMETRIC ELEVATIONS FOR TEMPERATURE AND HUMIDITY.* tx + 1>. c. tx + t'. c. tx + t'. c. 0° —0. 1025 60 — C >.0380 120 -I-0.0262 5 — .0970 65 - .0326 125 + -0315 10 — .0915 70 — .0273 130 + .0368 15 — .0860 75 - .0220 135 + .0420 20 — .0806 80 - .0166 140 + -0472 25 - .0752 85 - .0112 145 + -0524 30 — .0698 90 - .0058 150 + -0575 35 - .0645 95 - .0004 155 -{- .0626 40 - .0592 100 + .0049 160 + -0677 45 - -0539 105 + .0102 165 + .0728 50 — .0486 ITO + .0156 170 + .0779 55 - .0433 II5 + .0209 175 -f- .0829 60 — .0380 120 + .0262 180 + .0879 *This table compiled from tables I. and IV. of Appendix 10 of Report of the U. S. Coast and Geodetic Survey for 1881. 136 SU/^ VE YING. Example, From observations made at Sacramento, Cal., and at vSum- mit on the top of the Sierra Nevada Mountains, the annual means were : // = 23.288 in. /' = 42.i/^ — 30.014 in. — 59.9. From table I. we have A' = 6901.0 feet. A^= — 12.7 “ ^'-^, = 6913.7 “ From table II. we find for = i02°.o, C = .0070. . • . .S' = 6913.7 (i + .0070) = 6962 feet. 138. Use of the Aneroid. — The aneroid barometer should be carried in a leather case, and it should not be removed from it. It should be protected from sudden changes of tempera- ture, and when observations are made it should have the temperature of the surrounding outer aif It should not be carried so as to be affected by the heat of the body, and should be read out of doors, or at least away from all artificially warmed rooms. Always read it in a horizontal position. The index should be adjusted by means of a screw at its back, to agree with a standard mercurial barometer, and then this ad- justment left untouched. When but a single instrument is used it is advisable to pass between stations as rapidly as possible, but to stop at a number of stations during the day for a half-hour or so, reading the barometer on arrival and on leaving. The difference of these two readings shows the rate of change of barometric readings due to changing atmospheric conditions, and from these iso- lated rates of change a continuous correction-ctirve can be con- ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 13 / structed on profile or cross-section paper from which the instrumental corrections can be taken for any hour of the day.* The observations should be repeated the same day in reverse order, the corrections applied as obtained from this correction curve, and the means taken. Observations should be made when the humidity of the air is as nearly constant as possible, and never in times of changeable or snowy weather. Let the student measure the heights of buildings, hills, etc., and then test his results by level or transit. THE PEDOMETER. 139. The Pedometer is a pocket-instrument for register- ing the number of paces taken when walking. It is generally Fig. 30,— Front View. Fig. 31.— Back View. made in the form of a watch, the front and back views being shown in Figs. 30 and 31. * Mr. Chas. A. Ashburner, Geologist of the Penn. Geol. Survey, has used this method with good results. 138 SURVEYING. When the instrument is attached to the belt in an upright position, as here shown, the jar given it at each step causes tlie weighted lever shown in Fig. 31 to drop upon the adjustable screw vS. The lever recovers its position by the aid of a spring, and in so doing turns a ratchet-wheel by an amount propor- tional to the amplitude of the lever’s motion. T.his may be adjusted to any length of pace by means of the screw 5 , which is turned by a key. The face is graduated like that of a watch, and gives the distance travelled in miles. This instrument may also be used on a horse, and when adjusted to the length of a horse’s step will give equally good results. The accuracy of the result is in proportion to the uniformity of the steps, after having been adjusted properly for a given individual. The instrument is only used on explorations, preliminary sur- veys, and reconnoissance-work. The Length of Mens Steps has been investigated by Prof. Jordan,* of the Hanover Polytechnic School. From 256 step-measurements by as many different individuals, of lines from 650 to 1000 feet in* length, carefully measured by rods and steel tapes, he concludes that the average length of step is 2.648 feet, ranging from 2.066 to 3.182 feet. The mean deviation from this amount for a single measurement was ± 0.147 feet, or 5^ per cent. The average age of the persons making these step-measurements was 20 years. The length of step decreases with the age of the individual after the age of 25 to 30 years. It is also proportional to the height of the person. The results for 18 different-sized persons gave the following averages : Height of person 5'.o8 5'.25 5 ' 4 i 5'.58 5'-74 5 '- 9 f> 6'.07 6'. 23 6 '.40 6'.56 Length of step. . 2 .46 2 .53 2 .56 2 .59 2 .62 2 .69 2 .72 2 .76 2.79 2.85 * See translation in Engineering News and American Contract Journal for July 25, 1885. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 39 On slopes the step is always shorter than on level ground, whether one goes up or down. The following averages from the step-measurement of 136 lines on mountain-slopes along trails were found : Slope 0° 5 ° 10° 15° 20° 25° 30° Length of step in ascending 2'.53 2'. 30 2'.03 i '.84 i '.64 i '48 l'.25 Length of step in descending 2'.53 2'.43 2'. 36 2'.30 2'. 20 T.97 I '.64 The length of the step is also found to increase with the length of the foot. One steps farther when fresh than when tired. The increase in the length of the step is also in nearly direct proportion to the increase of speed in walking. When the proper personal constants are determined, and when walking at a constant rate, distances can be determined by pedometer, or by counting the paces, to within about two per cent of the truth. One should always take his 7 iatural step, and not an artificial one which is supposed to have a known value, as three feet, for instance. Let a base be measured off and each student determine the length of his natural step when walking at his usual rate, or, what is the same thing, find how many paces he makes in icx) feet. He then has always a ready means of determining distances to an approximation, which in many kinds of work is abundantly sufficient. THE ODOMETER. 140. The Odometer is an instrument to be attached to the wheel of a vehicle to record the number of revolutions made by it. One form of such an instrument is shown in Fig. 32 attached to the spokes of a wheel. Each revolution is recorded by means of the revolution of an axis with reference to the instrument, this axis really being 140 SURVEYING. held stationary by means of an attached pendulum which does not revolve. The instrument really revolves about this fixed axis at each revolution of the wheel, and the number of times Fig. 32. it does this is properly recorded and indicated by appropriate gearing and dials. This method of measuring distances is more accurate than by pacing, as the length of the circumference of the wheel is a constant. This length multiplied by the number of revolu- tions is the distance travelled. It is mostly used by exploring parties and in military movements in new countries which have not been surveyed and mapped. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I4I THE CLINOMETER. 141. The Clinometer is a hand-instrument for determin- ing the slope of ground or the angle it makes with the horizon. It consists essentially of a level bubble, a graduated arc, and a line of sight, so joined that when the line of sight is at any angle to the horizon the bubble may be brought to a central position and the slope read off on the graduated arc. Such a combina- tion is shown in Fig. 33. It is called the Abney level and clinometer, being really a hand-level when the vernier is set to read zero. The position of the bubble is visible when looking through the telescope, the same as with the Locke hand-level, shown in Fig. 16, p. 82. The body of the tube is made square, so that it may be used to find vertical angles of any surface by placing the tube upon it and bringing the bubble to the centre. The graduations on the inner edge of the limb give the slope in terms of the relative horizontal and vertical components of any portion of the line; thus, a slope of 2 to i signifies that the horizontal component is twice the vertical. In reading this scale the edge of the vernier-arm is brought into coincidence with the graduation. This instrument is very useful in giving approximate slopes in preliminary surveys, the instrument being pointed to a posi- 142 SUJiVEYING. tion as high above the ground as its own elevation when held to the eye. THE OPTICAL SQUARE. 142. The Optical Square is a small hand-instrument used to set off a right angle. It is shown in Fig. 34, the method of its use being evident from the figure. Thus, while the rod at 0 is seen directly through the opening, the rod at p is seen in ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I43 the glass as the prolongation downwards of that of Oy it being reflected from the mirrors /and ^ in succession, they having an angle of 45° with each other. By this means a line may be located at right angles to a given line at a given point, or a point in a given line may be found in the perpendicular to this line from a given point. THE PLANIMETER. 143. The Planimeter is an ingenious instrument used for measuring irregular areas. It is a marked example of high mathematical analysis embodied in a very simple and useful mechanical appliance. Many forms of it are now in use, three of the best of which will be described. The instrument has come to be a necessity in all kinds of surveying and engineer- ing work where irregular areas have to be evaluated. It is important that the student should thoroughly understand its principles, that he may use it with the greatest efficiency. The demonstration of its competency to measure areas is necessarily somewhat involved, and requires a little patient consideration. The demonstrations here given, though fol- lowing the methods of the calculus, are free from the peculiar notation there used. The form of the instrument shown in Fig. 35 is known as Amsler’s Polar Planimeter. The point e is fixed by means of a needle-point puncturing the paper. The point d is made to pass over the perimeter of the area to be measured, and the record given by the rolling-wheel c and the 144 SUR VE YING. record-disk / is the area in the unit for which the length of the arm h was set. The rolling-wheel is mounted on an axis which is parallel to the arm h, and moves with a minimum amount of friction. It is evident that any motion of the wheel c in the direction of its axis would not cause it to re- volve, while any motion at right angles to this axis is fully recorded by the wheel. The arm ei is of fixed length, while the length of the arm h is adjustable. 144. Theory of the Polar Planimeter.* — In Fig. 36 let C represent the point where the instrument is fastened to the paper, and ClPthe arm, of fixed length m, whose only motion is that of revolution on ^7 as a centre, causing P to move in a circular arc. RT the other arm, revolving on P as a centre, and carrying at the fixed distance RP n) from Pa rolling- wheel whose periphery touches the paper at R and whose axis is parallel to PP. RT also carries at a distance TP 1) from P the tracing-point, P ; / is a constant while the instrument is in use, though capable in the best instruments of having dif- ferent values given to it for different purposes. * The demonstration here given was published by Mr. Fred. Brooks, in the Journal of the Association of Engineering Societies, vol. iii., p. 294, and is rep- resented as a joint production by himself and Mr. Frank S. Hart. A few slight changes and additions are here made. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I45 T and R can move nearer to or further from C only by the motion of the arm TR on Z’ as a centre varying the angle X. The distance CT = + / cos X)' + (/ sin XJ = ^ R 2rnl cos as may be seen by dropping a perpendicular Tq from T on CP produced. To every particular value of X correspond particular values of CT, CR, angle CRP, etc. ; and successive small variations in X are accompanied by successive small variations in these quantities. When T, starting at any given distance from C, is moved through any path to the same or another place equally distant from C, the usefulness of the instrument depends upon W’s coming back to its first value by passing in reverse order through the changes it has once made. This is secured by the usual construction of the instrument, which prevents T and R from crossing the line of CP', in other words, X, ex- pressed as arc to radius unity, is never less than 0 nor more than 7t (a half-circumference). The only motion possible besides those above described is the turning of the rolling-wheel on its axis, which is produced by the component of the motion of R perpendicular to RP, that is, tangential to its periphery ; but the wheel does not turn for the component of the motion of R in the direction RP, which is parallel to the axis. Suppose, for simplicity, that the periphery of the wheel has a length of one unit and that the number of turns and fractions of a turn it makes upon any trial is recorded ; for, whatever the size and graduations may be, a simple calculation would reduce the results to the re- quired equivalents. To illustrate, let the arm RT turn on P as a centre, while CP remains fixed, from the position of the full line to that of the dotted line sk ; R moves to s, describing 10 146 SUR VE YING. an arc which is everywhere at right angles to its radius RP ; hence the record of the wheel is the length of the arc Rs. On the other hand, supposing that CsP is a right angle -j and 11 cos CPs ~ — , let both arms revolve around C with X fixed m equal to CPs ; the wheel is at every point moving parallel to its own axis, and its record is zero. The distance Ck of the tracing-point from C in this case may be found by substituting n the value — for the cos X in the general expression for CT, which gives V nf P 2 nl. The circumference described by the tracing-point with this radius may be called the zero-cir- cumference. If both arms similarly revolved around C with X fixed at any other value between 0 and n, the axis of the wheel would make an oblique angle with the direction of i?’s path, and the wheel would partly roll and partly slip. The further CRP 7t varied from — , the less in proportion would be the slipping component, and the greater the rolling component and the 7t record of the wheel. With CRP'> T would describe an arc outside the zero-circumference and the wheel would make n what we will call a positive record. With CRP < T would describe an arc inside the zero-circumference and the wheel would turn in the contrary direction, which we will call nega- tive ; provided that T revolved in the same direction in both cases. Motion of T through any arc in the direction of the hands of a watch may be considered positive ; then motion of T in the opposite direction over the same arc back to its start- ing-place must be considered negative, and would obviously be ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I47 attended by a turning of the wheel equal in amount to that attending the positive movement, but with its direction re- versed. In the practical use of the instrument T may move over any path, near enough to the zero-circumference to be reached, whose beginning and end are equally distant from C. Hence X is the same at the end as at the beginning. The record thus made on the wheel is proportional to the area included between the zero-circumference and T\ path and the radial lines through its beginning and end from the centre as will now be explained. 7"’s path may be resolved into an infinite number of parts, consisting of infinitesimal arcs (/) described from Pas a centre by changes in X, CP being fixed, and of infinitesimal arcs (y) described from (7 as a centre with X fixed. This is illustrated by large arcs of the two classes on the diagram. The area in question may be correspondingly divided into elementary por- tions (illustrated by the large divisions made on the diagram by fine radial lines) each of which may be described as plus the area included between one of these infinitesimal arcs and radial lines through its extremities from (7, minus the sector included by the same radii and an arc of the zero-circumfer- ence. Hence the area is a minus quantity if T moves inside the zero-circumference, positive if outside ; provided that T moves around C in the direction of the hands of a watch. If T moves around (7 in a contrary direction, both the signs in the above expression are to be changed ; for as the area of a sector is equal to its arc multiplied by half its radius, the area becomes negative when the arc becomes negative. If this be borne in mind it will be seen that the algebraic sum of all the elements corresponding to the second term in the above ex- pression is the sector of the zero-circumference included by radii passing through the points of beginning and ending of 148 SURVEYING. T's path ; and that the algebraic sum of all the elements corresponding to the first term is the area inclosed by T's path and lines from C to its beginning and end, however irreg- ular T's path may be. We will first consider that class of infinitesimal arcs (/) and corresponding elements of area, due to changes in X alone. Their accumulated effect upon both the area and up 07 i the record of the rolling-wheel is zero. As to the wheel, from the condition that X passes again in reverse order through the changes it has once made, it follows that for every infinitesi- mal motion, like Rs, of R, recorded by the wheel for the infinitesimal change (/) between two consecutive values of W, there must be in some other place a motion in the opposite direction of the same magnitude for the infinitesimal change back again between two consecutive values of X equal to the former pair. As to the area, each infinitesimal arc / (like Tk) has, as previously stated, its corresponding element of area ; and the equally large arc with the contrary sign, just now referred to, in another place where X has the same values, must also have its corresponding element of area, exactly as large as the former, but with its algebraic sign reversed. The effect of the first class of elements into which T's path was resolved is thus eliminated. Hence the total record of the wheel for T's whole path is the record due to the second class of its elements, the infini- tesimal arcs (y) described from C with X fixed for each ; and the total area included between the zero-circumference, T's path, and the terminal radii is the sum of all the elements of area corresponding to this second class of arcs which we have now to consider. J expresses in terms of arc to radius unity any infinitesimal angle ydy between radial lines passing from C through the extremities of an infinitesimal arc Tf. The corresponding element of area is the difference between the sector TfC and the sector included by the zero-circumfer- ADJUSTMENT, USE, AND CADE OF INSTRUMENTS. I49 ence and the same radii. Making use of the algebraic expres- sions given above, from \J ( 11 ^ + 2^/ cos. X) subtract J + 2?^/) and the difference J I {in cos. X — n) is the required element of area. The corresponding record of the wheel is made by the motion of R through the path Re = / X CR, This path may be resolved into two components, Rh, which has no effect upon the record, and he, which is the record — J CR X cos {n — CRP). By dropping the perpendicular Cg upon PR produced it will be seen that CR cos {rt -- CRP) = Rg = m cos X — 11 . Hence record of wheel is y X {m cos X — n). Therefore the element of area corresponding to an infinitesimal arc,J, is just I times the record due to the same arc ; hence the sum of the elements of area for all the arcs {f') is / times the total record corresponding, which is the essential thing that was to be proved. In the application of the instrument to get the area of a closed figure, 7"’s path ends in the same point where it began, and we have two cases according as this is accomplished by 67^’s making a complete revolution around C, or by its mov- ing backward as much as it has once moved forward. In the first case, C is within the figure ; in the second, outside. In both cases the area between 7”s path and the terminal radii is the area of the closed figure. The sector within the zero-circumference, which we have been deducting, is in the first case the whole circle n {ml + + 2nl) ; in the second, nothing. Hence add ir {m^ + + 2nl) to / times the record in the first case, and add nothing to it in the second, in order to get the required area of the closed figure. 150 SUR VE YING. To show that the proper summation is made on the wheel for the areas outside and inside the zero-circle, let Af, — area generated by the line CT when the point T is outside of circle ; “ Ai = area generated by the line CT when the point T is inside of circle ; “ 5 = area of sector between radii to points where the perimeter crosses the zero-circle ; “ A — area of the figure. Then Ao — S — outer area, and S — Ai = inner area. The sum of these is A = (A,-S) + {S-A,) = A„-yj,. But since Ai is recorded negatively on the wheel, while Ao is recorded positively, the wheel record is Ao— { — Ai) = Ao-^ Ai = A. 145. To find Length of Arm to give area in any desired unit. In the previous article it was shown that the area was always / times the wheel record, where / was the length of the arm carrying the tracing-point, or the distance PT in Fig. 36. The wheel record is evidently its net circumferential move- ment, or nc, where n = number of revolutions of wheel shown by the differ- ence between the initial and final readings, and c = circumference of wheel. We may then write for the area of the figure A = Inc. ADJUSTMENT, USE, AND CARE OE INSTRUMENTS. 15I If I and c are given in inches A will be found in square inches, and the same for any other unit. To cause an area of i square inch to register i revolution of the wheel, we will have I = Ic, If c were 2 inches, this would give l—\ inch, which would be too short for practical purposes. Let us assume, then, that I square inch shall be registered as o.i revolution of the wheel. Then we have I = O.I Ic, On an instrument the author has used c — 2.347 inches, whence for 0.1 revolution to correspond to i square inch area we have / = = 4.26 inches. 2.347 When this length of arm is carefully set off by the appro- priate clamp- and slow-motion screw, the area is given in square inches by multiplying the number of revolutions of the wheel by 10. A vernier is provided for reading the revolu- tions of the wheel to thousandths ; hence if it be read to thousandths, and two figures pointed off, the result is the area of the diagram moved over in square inches. It is evident that c can be evaluated in centimetres, and the corresponding metrical length of / found for giving the result in the metric notation. The exact circumference of the wheel is determined by the makers, and remains a constant for that 152 SUR VE YING. individual instrument, giving a corresponding set of values of /. Since no two instruments are likely to have exactly the same wheel-circumference, so the settings for one instrument cannot be used for another. It must be kept in mind that the result is given in absolute units of area of the diagrani, and this result must then be evaluated according to the significance of such unit on the diagram. Thus, if a sectional area has been plotted with a* vertical scale of lo feet to the inch and a horizontal scale of 100 feet to the inch, then one square inch on this diagram rep- resents looo square feet of actual sectional area. The number of square inches in the figure as given by the planimeter must then be multiplied by lOOO to give the area of the section in square feet. 146. The Suspended Planimeter.— This is shown in Fig. 37. It is essentially a polar planimeter, the pole being at C, Fig. 37. It has the advantage of allowing the wheel to move over the smooth surface of the plate 5 , instead of over the paper, thus giving an error about one sixth as great as that of the ordina- ry polar instrument. The theory of its action is essentially the same as the other. 147. The Rolling Planimeter is the most accurate instru- ment of its kind yet devised. Its compass is also indefinitely increased, since it may be rolled bodily over the sheet for any ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 53 distance, on a right line, and an area determined within certain limits on either side. It is therefore especially adapted to the measuring of cross-sections, profiles, or any long and narrow surface. Fig. 38 shows one form of this instrument as de- signed by Herr Corradi of Zurich. It is a suspended planim- eter, inasmuch as the wheel rolls on a flat disk which is a part of the instrument, but it could not be called a polar pla- nimeter, the theory of its action being very different from that instrument. The frame B is supported by the shaft carrying Fig. 38. the two rollers To this frame are fitted the disk A and the axis of the tracing-arm F. The whole apparatus may thus move to and fro indefinitely in a straight line on the two rollers while the tracing-point traverses the perimeter of the area to be measured. The shaft carries a bevel-gear wheel, which moves the pinion R^. This pinion is fixed to the axis of the disk, and turns with it, so that any motion of the rollers 7 ?, causes the disk to revolve a proportional amount, and the component of this motion at right angles to the axis of the wheel E is recorded on that wheel. If the instrument remains 154 SUR VE YING. stationary on the paper (the rollers R not turning) and the tracing-point moved laterally, it will cause no motion of the wheel, since its axis is parallel to the arm F, and turns about the same axis with F, but 90° from it ; the wheel E, therefore, moves parallel with its axis and does not turn. 148. Theory of the Rolling Planimeter. — This will be developed by a system of rectangular coordinates, the path of the fulcrum of the tracing-arm being taken as the axis of Fig. 39. abscissae. The path of the tracing-point will be considered as made up of two motions, one parallel to the axis of abscis- sae and the other at right angles to it. The elementary area considered will be that included between the axis of abscissae and two ordinates drawn to the extremities of an elementary portion of the path. But since this element of the perimeter is supposed to be made up of two right lines, one perpendicu- lar to the axis of abscissae and the other parallel to it, our ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I55 elementary area must also be divided in a similar manner. It will at once be seen that one part of this area is zero, since the two ordinates bounding it form one and the same line. This is the part generated by the motion at right angles to the axis of abscissae. Now, we have just shown in the previous article that the wheel-record for this part of the path is also zero. We are brought therefore to this important conclusion : that all components of motion of the tracing-pomt at right angles to the axis of abscisses have no influence upon the re suit. We will therefore only discuss a differential motion of the tracing- point in the direction of the axis of abscissae. In Fig. 39, which is a linear sketch of the instrument shown in Fig. 38, with the corresponding parts similarly lettered, it is to be shown that the motion of the wheel E caused by the movement of the tracing-point over the path dx is equal to the corresponding area_y<^.r multiplied by some constant which is a function of the dimensions of the instrument. It is evident that a motion of the tracing-point in the di- rection of the axis of abscissae can only be obtained by moving the entire instrument on the rollers by the same amount, and therefore when the point moves over the path dx the circum- ferences of the rollers have moved the same amount. This causes a movement of the pitch circle of of dx This motion is conveyed to the disk through R^, so that any point on this disk, as a, distant ad from the axis, moves through a R ad distance equal to dx Let aby Fig. 39, be this distance, Then we have ab = dx^ ad (I) The motion of that portion of the disk on which the roller rests, equal to ab, causes the circumference of the wheel E to 156 SUR VE YING. revolve by an amount equal to the component of the distance ab perpendicular to the axis of the wheel. Tliis component part of the disk’s motion is bc^ and this is the measure of the wheel’s motion. It therefore remains to show that be ■=^ ydx multiplied by an instrumental constant. Now, be — ab sin bae (2) But bae = a since gae and bad are both right angles. Also, bae — supplement of dag= a -|- ft- Also, from the triangle dag, we have or sin dag : sin agd :: D : ad, D a sm (« + /?) = ( 3 ) Since Fga is also a right angle, we have the angle formed y by Fg and the axis of abscissae equal to a, whence sin a = We may now write : bc = ab^,x.{a^^) = ab--^-^ = ab^^. . ( 4 ) Now, substituting the value of ab from (i), we have eb — ydx D . R, F^R,-R, is) Since D, R^, F, R,, and R^ are all constants for any one instrument, we see that the wheel-record is a function of the area generated by the tracing-point and the instrumental con- stants, which was to be shown. It now follows that the sum- mation of all these elementary areas included between the path of the tracing-point, the limiting ordinates, and the axis ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 157 of abscissae, is represented by the total wheel-movement , or the difference between its initial and final readings. If, therefore, the area to be measured is of this character, being bounded by one right line and limiting ordinates, it would not be necessary to move the point over the entire perimeter, but only along the irregular boundary, provided the instrument could be ad- justed with the point ^ exactly over the base of the figure, and with the axis B at right angles to it, so that in rolling the in- strument along, the point g would remain over the base-line. In other words, the axis of abscissa of the instrument would have to coincide exactly with this base-line. Then for motion of the tracing-point over this line, as well as for its motion over the end-ordinates, the wheel would not revolve, neither would there be any area generated between these lines and the axis. In general this cannot be done, and it is only mentioned here in order to more clearly illustrate the working of the instru- ment. As in the case of the polar instrument, the proper length of arm F, to be used with the rolling-planimeter to give results in any desired unit, depends on the other instrumental con- stants. These being known, the value of A may be computed in the same manner as with the polar planimeter. 149. To test the Accuracy of the Planimeter, there is usually provided a brass scale perforated with small holes. A needle-point is inserted in one of these and made fast to the paper or board, while the tracing-point rests in another. The latter may now be moved over a fixed path with accuracy. Make a certain number of even revolutions forward, or in the direction of the hands of a watch, noting the initial and final readings. Reverse the motion the same number of revolutions, and see if it comes back to the first reading. If not, the dis- crepancy is the combined instrumental error from two meas- urements due to slip, lost motion, unevenness of paper, etc. If this test be repeated with the areas on opposite sides of 158 SURVEYING. the zero-circle in the case of the polar-planimeter, or on oppo- site sides of the axis of abscissae in case of the rolling-planimc- ter, with the same score in both cases, it proves that tlie pivot- points a, b, k, and the tracing-point d (Fig. 35), arc in the same straight line, in case of the polar instrument, and that the cor- responding points in the suspended and rolling planimctcrs form parallel lines; in other words, that the axis of the meas- uring-wheel is parallel to the tracing-arm. If the results differ when the areas lie on opposite sides of the axis or zero-circle, these lines are not parallel and must be adjusted to a parallel position. 150. Use of the Planimeter. — The paper upon which the diap:ram is drawn should be stretched smooth on a level sur- face. It should be large enough to allow the rolling-wheel to remain on the sheet. The instrument should be so adjusted and oiled that the parts move with the utmost freedom but without any lost mo- tion. This requires that all the pivot-joints shall be adjustable to take up the wear. The rim of the measuring-wheel must be kept bright and free from rust. The instrument must be han- dled with the greatest care. Having set the length of the tracing-arm for the given scale and unit, it is well to test it upon an area of known dimensions before using. If it be found to give a result in error by ^ of the total area, the length of the tracing-arm must be changed by an amount equal to this same ratio of its former length. If the record made on the wheel was too small then the length of the tracing-arm must be di- minished, and vice versa. If the paper has shrunk or stretched, find the proportional change, and change the length of the tracing-arm from its true length as Just founds by this same ratio, making the arm longer for stretch and shorter for shrink- age. Or the true length of arm may be used, and the results corrected for change in paper. ADTUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 59 To measure an area, first determine whether the fixed point, or pole, shall be inside or outside the figure. It is preferable to have it outside when practicable, since then the area is ob- tained without correction. If, however, the diagram is too large for this (in case of the polar planimeter) the pole may be set inside. In either case inspection, and perhaps trial, is nec- essary to fix upon the most favorable position of the pole, so that the tracing-point may most readily reach all parts of the perimeter. If the area is too large for a single measurement, divide it by right lines and measure the parts separately. Having fixed the pole, set the tracing-point on a well-defined portion of the perimeter, and read and record the score on the rolling-wheel and disk. This is generally read to four places. Move the tracing-point carefully and slowly over the outline of the figure, in the direction of the hands of a watch, around to the initial point. Read the score again. If the pole is outside the figure, this result is always positive when the motion has been in the direction here indicated. If the pole is inside the figure, the result will be negative when the area is less than that of the zero-circle, positive if greater. With the pole inside the figure, however, the area of the zero- circle must always be added to the result as given by the score, and when this is done the sum is always positive, the motion being in the direction indicated. The area of this zero-circle is found in art. 144, to be tt + 2 nl). The value of /, which is the length of the tracing-arm, is known. The values of in and n should be furnished by the maker. If these are unknown, the area of the zero-circle can be found for any length of arm /, by measuring a given area with pole outside and inside, the difference in the two scores being the area of this circle. By doing this with two very different values of / we may obtain two equations with two unknown quantities, m and n, from which the absolute values of these quantities may be found. Thus we would have: i6o SURVEYING. A = 71 {m^ +r + 2nl ) ; ^' = ;r (;«’ + P + 2;//') ; whence 7t wherein /, A, and A' are known. The values of in and n are then readily found. In using the rolling-planimeter, it is advisable to take the initial point in the perimeter on the axis of abscissae, as in this position any small motion of the tracing-point has no effect on the wheel, and so there is no error due to the initial and final positions not being exactly identical. The planimeter may be used to great advantage in the solution of many problems not pertaining to surveying. In all cases where the result can be represented as a function of the product of two variables and one or more constants, the corresponding values of the variables may be plotted on cross- section paper by rectangular coordinates, thus forming with the axis and end-ordinates an area which can be evaluated for any scale and for any value of the constant-functions by setting off the proper length of tracing-arm. Thus, from a steam- indicator card the horse-power of the engine may be read off, and from a properly constructed profile the amount of earth- work in cubic yards in a railway cut or fill. Some of these special applications are further explained in Part II. of this work. 151. Accuracy of Planimeter-measurements. — Professor Lorber, of Loeben, Austria, has thoroughly investigated the relative accuracy of different kinds of planimeters, and the re- sults of his investigations are given in the following table. It ADJUSTMENT, USE, AND CAEE OF INSTRUMENTS. l6l will be seen that the relative error is less as the area measured is larger. The absolute error is nearly constant for all areas, in the polar planimeter. The remarkable accuracy of the rolling- planimeter is such as to cause it to be ranked as an instrument of precision. TABLE OF RELATIVE ERRORS IN PLANIMETER-MEASUREMENT3. Area in — The error in one passage of the tracer amounts on an average to the following fraction of the area meas- ured by — The ordinary po- lar planimeter- Unit of vernier: 10 sq. mm. = .015 sq. in. Suspended plani- meter -Unit ol vernier: I sq. mm. = .001 sq. in. Rolling planime- ter-Unit of ver- nier: I sq. mm. = .001 sq. in. Square cm. Square inches. 10 1.55 TZ 20 3.10 TTTT zIwS: 50 7.75 TZZ zluz W&T 100 15.50 zrkz itVt TcVlF 200 31.00 ±27 f ttVj ITTS 300 46.50 .... Wr? TGZZIS THE PANTOGRAPH. 152. The Pantograph is a kind of parallel link-motion apparatus whereby, with one point fixed, two other points are made to move in a plane on parallel lines in any direction. The device is used for copying drawings, or other diagrams to the same, a larger, or a smaller scale. The theory of the instru- ment rests on the following: Proposition : If the sides of a parallelogram, jointed at the corners A, B, C, and D, and indefinitely extended, be cut by a right line in four points, as E, F, G, and H, then these latter pomts will lie in a straight line for all values of each of the parallelogram angles from zero to 180°, and the ratio of the dis- tances EF, EG, and GH will remain unchanged. II SURVEYING. 162 In Fig. 40, let A, B, C, Z>be the parallelogram, whose sides (extended) are cut by a right line in F, G, E, and //. It is evident that one point in the figure may remain fixed while the angles of the parallelogram change. Let this point be G, Since GC and GH, radiating from G, cut the parallel lines DE and CH^ we have GD : DE :: GC \ CH. Also, for similar reasons, % ED : DG :: EA : AF. Now since the sides of the parallelogram, as well as all the intercepts, AF, GD, DE, and CH, remain constant as the angles of the figure change, when the figure has taken the position shown by the dotted lines, we still have GU : UE wGC \ CH'-, also, E’U : D'G :: EA ' : A’F. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 163 From the first of these proportions we know that G, E\ and H' are in the same straight line, and the same for G, E' , and F ; therefore, they are all four in the same straight line. To show that they are the same relative distance apart as before we have, FG\GE\ EH :: BC \ DE \ CH^DE\ also, FG : GE : EH :: EC : UE : CH-UE. But BC = EC, DE = UE, and CH^DE^ CH - UE\ therefore we may write, FG\GE\ EH :: FG : GE : EH, Q. E. D. It is evident that two of the points E, F, G, and -^T may become one by the transversal passing through the point of intersection of two of the sides of the parallelogram. The above proposition would then hold for the three remaining points. In the Pantograph only three of the four points E, F, G, and H (Fig. 40) are used. One of these may therefore be taken at the intersection of two sides of the parallelogram, but it is not necessarily so taken. These three points are: the fixed point, the tracing-point, and the copying-point. In Fig. 41, i^is the fixed point, held by the weight P; B is the tracing-point, and D is the copying-point, or vice versa as to B and D. The parallelogram is E, G, B, H. The points 164 SURVEYING. /% and D must lie in a straight line, B being at the inter- section of two of the sides of the parallelogram. The points Ay Ey and C are supported on rollers. In Fig. 42, the fixed- Fig. 41. point is the point of intersection of two of the sides of the parallelogram. The upper left-hand member of the frame is not essential to its construction, serving simply to stiffen the copying-arm, the fourth side of the parallelogram being the side holding the tracing-point. In Fig. 43, neither of the three points is at the intersection of two of the sides of the parallelogram, and hence there is a ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 165 fourth point unused, having the same properties as the fixed, tracing, and copying points, it being at the intersection of the line joining these three points with the fourth side of the par- allelogram. From the theoretical discussion, and from the figures shown, it becomes evident that there may be an indefinite variety of only essential conditions are that the fixed, the tracing, and the copying points shall lie in a straight line on at least three sides of a jointed parallelogram, either point serving any one of the three purposes. 153. Use of the Pantograph. — The use of the instrument is easily acquired. Since both the tracing and copying points should touch the paper at all times, such a combination as that shown in Fig. 41 is preferable to those shown in Figs. 42 and 43, since in these latter the tracing point is surrounded by sup- ported points, and so would not touch the paper at all times unless the paper rested on a true plane. In most instruments where the scale is adjustable, the two corresponding changes in position of tracing and copying points for different scales are indicated. To test these marks, see that the adjustable points are in a straight line with the fixed point, and to test the FD scale see that the ratio (Fig. 41) is that of the reduction desired. Thus, if the diagram is to be enlarged to twice the original size, make FD = 2FB ; or make DE_F^ DG ~ ~BG = scale of enlargement. If the drawing is to be reduced in size, make B the copying- point and D the tracing-point. If the drawing is to be copied to the same scale, make BF — BD and make B the fixed point. The figure is then copied to same scale, but in an inverted position. In the best instruments the arms are made of brass, but very good work may be done with wooden arms. PROTRACTORS. 154. A Protractor is a graduated circle or arc, with its cen- tre fixed, to be used in plotting angles. They are of various designs and materials. Semicircular Protractors, such as shown in Fig. 44, are usually made of horn, brass, or german-silver. They are grad- uated to degrees or half-degrees, and the angle is laid off by holding the centre at the vertex of the angle, with the plain edge, or the o and 180 degree line on the given line from which the angle is to be laid off. In the full circle protractor, shown in Fig. 45, there is a movable arm with a vernier reading to from i to 3 minutes. The horn centre is set over the given point, the protractor oriented with the zero of the circle on the given line, and the arm set to the given reading when the other line may be drawn. ADJUSTMENT, USE, AND CATE OF INSTRUMENTS. 1 6 / The three-arm protractor, Fig. 46, has one fixed and two movable arms by which two angles may be set off simulta- neously. It is used in plotting observations by sextant of two Fig. 45. angles to three known points for the location of the point of observation. This is known as the three-point problem and is discussed in Chap. X. Fig. 46. Paper protractors are usually full circled, from 8 to 14 inches in diameter, graduated to half or quarter degrees. They are printed from engraved plates on drawing- or tracing- 5-6^/’ VE Y I NG. 1 68 paper or bristol-board, and are very convenient for plotting topographical surveys. The map is drawn directly on the protractor sheet, the bearing of any line being taken at once from the graduated circle printed on the paper. These “ pro- tractor sheets” can now be obtained of all large dealers. The coordinate protractor"^ is a quadrant, or square, with angular graduations on its circumference, or sides, and divided over its face by horizontal and vertical lines, like cross-section paper. A movable arm can be set by means of a vernier to read minutes of arc, this arm being also graduated to read distances from the centre outward. Having set this arm to read the proper angle, the latitude is at once read off on the * Called a Trigonometer by Keuffel & Esser, the makers. ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 1 69 vertical scale and the departure on the horizontal scale for the given distance as taken on the graduated arm. A quadrant protractor giving latitudes and departures for all distances under 2500 feet to the nearest foot, or under 250 feet to the nearest tenth of a foot, has been used. The radius of the cir- cle is i8| inches. Both the protractor and the arm are on heavy bristol-board, so that any change due to moisture will affect both alike and so eliminate errors due to this cause. The instrument was designed to facilitate the plotting of the U. S. survey of the Missouri River.* It has proved very efficient and satisfactory. A similar one on metal, shown in Fig. 47, is now manufactured, and serves the same purpose. PARALLEL RULERS. 155* The Parallel Ruler of greatest efficiency in plotting is that on rollers, as shown in Fig. 48. The rollers are made of exactly the same circumfer- ence, both being rigidly attached to the same axis. It should be made of metal so as to add to its weight and prevent slipping. It is of especial value in connec- tion with the paper protractors, for the parallel ruler is set on any given bearing and then this transferred to any part of the sheet by simply running the ruler to place. Two triangles may be made to serve the same purpose, but they are not so rapid or convenient, and are more liable to slip. The parallel ruler is also very valuable in the solution of problems in graphical statics. SCALES. 156. Scales are used for obtaining the distance on the drawing or plot which corresponds to given distances on the Fig. 48. For sale by A. S. Aloe & Co., St. Louis, Mo. 170 SURVEYmC. object or in the field. There is such a variety of units for both field and office work, and a corresponding variety of scales, that the choice of the particular kind of scale for any given kind of work needs to be carefully made. Architects usually make the scale of their drawings so many feet to the inch, giving rise to a duodecimal scale, or some multiple of A surveyor who uses a Gunter’s chain 66 feet in length plots his work to so many chains to the inch, making a scale of some multiple of engineer usually uses a lOO-foot chain and a level rod divided to decimal parts of a foot ; so he finds it convenient to use a decimal scale for his maps and drawings, reduced to the inch-unit however. Here the field- unit is feet and the office-unit is inches, both divided deci- mally. This gives rise to a sort of decimal-duodecimal system, the scale being some multiple of Various combinations of all these systems are found. Figure 49 shows one form of an ivory scale of equal parts for the general draughtsman. The lower half of the scale is designed to give distances on the drawing for 4, 40, or 400 units to the inch when the left oblique lines and bottom figures are used, and for 2, 20, or 200 units to the inch when the right oblique lines and top figures are used. Thus, if we are plotting to a scale of 400 feet to the inch, and the dis- tance is 564 feet, set one point of the dividers on the vertical line marked 5, and on the fourth horizontal line from the bot- tom. Set the other leg at the intersection of the sixth inclined ADJUSl'MENT, USE, AND CARE OF INSTRUMENTS. IJl line with this same horizontal line, and the space subtended by the points of the dividers is 564 feet to a scale of Figure 50 is a cut of an engineer’s triangular boxwood scale, 12 inches long, being divided into decimal inches. There are six scales on this rule, a tenth of an inch being sub- divided into I, 2 , 3, 4, 5, and 6 parts, making the smallest Fig, 50. graduations of an inch respectively. This is called an engineer’s or decimal-inch scale The architect’s triangular scale is divided to give J, f, f, I, ij, 2, 3, and 4 inches to the foot. Such a scale is of less service to the civil engineer. BOOK II. SURVEYING METHODS. CHAPTER VII. LAND-SURVEYING. 157. Land-surveying includes laying-out, subdividing, and finding the area of, given tracts of land. In all cases the boundary- and dividing-lines are the traces of vertical planes on the surface of the ground, and the area is the area of the horizontal plane included between the bounding vertical planes. In other words, the area sought is the area of the horizontal projection of the real surface. 158. In laying out Land the work consists in running the bounding- and dividing-lines over all the irregularities of the surface, leaving such temporary and permanent marks as the work may demand. These lines to lie in vertical planes, and their bearings and horizontal distances to be found. The bearing of a line is the horizontal angle it makes with a merid- ian plane through one extremity, and its length is the length of its horizontal projection. This reduces the plot of the work to what it would be if the ground were perfectly level. If all the straight lines of a land-survey lie in vertical planes, and if their bearings and horizontal lengths are accurately deter, mined, then as a land survey it is theoretically perfect, what- ever the purpose of the survey may be. 4 ZAJVD SUJ^FEYING. 173 THE UNITED STATES SYSTEM OF LAYING OUT THE PUBLIC LANDS. 159* The Public Lands of the United States have included all of that portion of our territory north of the Ohio River and west of the Mississippi River not owned by indi- viduals previous to the dates of cession to the United States Government. All of this territory, except the private claims, has been subdivided, or laid out, in rectangular tracts bounded by north and south and east and west lines, each tract having a particular designation, such that it is impossible for the pat- ents or titles, as obtained from the Government, to conflict. This has saved millions of dollars to the land-owners in these regions by preventing the litigations that are common in the old colonial States, and is one of the greatest boons of our national Government. The system was probably devised by Gen. Rufus Putnam,* an American officer in the Revolution- ary War. It was first used in laying out the eastern portion of the State of Ohio, in 1786-7, then called the Northwest Territory. This was the first land owned and sold by the national Government. The details of the system have been modified from time to time, but it remains substantially un- changed. The following is a synopsis of the method which is given in detail in the Instructions to Survey or s-General, issued by the Commissioner of the General Land Office, at Washing- ton, D. C., and obtained on application. 160. The Reference-lines consist of Principal Meridians and Standard Parallels. The principal meridians may be a hundred miles or more apart, but the standard parallels are 24 miles apart north of 35° north latitude, and 30 miles apart south of that line. These lines should be run with great care, * See article by Col. H. C. Moore in Journal of the Association of En- gineering Societies, vol. ii., p. 282. 174 SURVEYING. using the solar compass or solar attachment. The magnetic needle cannot be relied on for this work, for two reasons : there may be local attraction from magnetic deposits, and the dec- lination changes rapidly (about a minute to the mile) on east and west lines. The transit alone might be used to run out the meridians, as this consists simply of extending a line in a given direction. The transit could not boused for running the parallels, however, for these are ever changing their direction, since they are at all points perpendicular to the meridian at that point. This change in direction is due to the convergence of the meridians. The solar compass is the only surveying in- strument that can be used for running a true east and west line an indefinite distance. The needle-compass would do if there were no local attraction and if the true declination were known and allowed for at all points. The solar compass (or solar attachment) is the instrument recommended for this work. In running these reference-lines, every 8o chains (every mile) is marked by a stone, tree, mound, or other device, and is called a “ section corner.” Every sixth mile has a different mark, and is called a “ township corner.” i6i. The Division into Townships. — From each “town- ship corner” on any standard parallel auxiliary meridians are run north to the next standard parallel. Since these meridians converge somewhat towards the principal meridian, they will not be quite a mile apart when they reach the next standard parallel. But the full six-mile distances have been marked off on this parallel from the principal meridian, and it is from these township corners that the next auxiliary meridians will start and run north to the next standard parallel, etc. Thus each standard parallel becomes a “ correction-line” for the merid- ians. The territory has now been divided into “ranges” which are six miles wide and twenty-four miles long, each range being numbered east and west from the principal meridian. LAND SURVEYING. 175 These ranges are then cut by east and west lines joining the corresponding township corners on the meridians, thus dividing the territory into “ townships,'’ each six miles square, neglect- ing the narrowing effect of the convergence of the meridians. The townships are numbered north and south of a chosen parallel, which thus becomes the “ Principal Base-line.” The fifth township north of this base-line, lying in the third range west of the principal meridian, would be designated as “ town five north, range three west.” Each township contains thirty- six square miles, or 23,040 acres. 162. The Division into Sections. — The township is di- vided into thirty-six sections, each one mile square and contain- ing 640 acres. This is done by beginning on the south side of each township and running meridian lines north from the sec- tion corners” already set, marking every mile or “ section corner,” and also every half-mile or “ quarter-section corner.” When the fifth section corner is reached, a straight line is run to the corresponding section corner on the next township line. This will cause this bearing to be west of north on the west, and east of north on the east, of the principal meridian. When this northern township boundary is a standard or correction-line, then the sectional meridians are run straight out to it, and thus this line becomes a correction-line for the section-lines as well as for the township-lines. The east and west division-lines are then run, connecting the corresponding section corners on the meridian section lines, always marking the middle, or quarter- section points. Evidently, to run a straight line between two points not visible from each other, it is necessary first to run a random or trial line, and to note the discrepancy at the second point. From this the true bearing can be computed and the course rerun, or the points on the first course can be set over the proper distance. The sections are numbered as shown in Figs. 51 and 52. When account is taken of the convergence of meridians, the SUR VE YING. 176 sections in the northern tiers of each township will not be quite one mile wide, east and west ; but as the section corners are set at the full mile distances on the township-lines, the southern sections in the next town north begin again a full mile in width. In setting the section and quarter-section corners on the cast and west town lines the full distances are given from the east towards the west across each township, leaving the deficiency on the last quarter-section, or 40-chain distance, until the next correction-line is reached, when the town meridians are again adjusted to the full six-mile distances. 163. The Convergence of the Meridians is, in angular amount,* c = m sin ^ {L L ) ; where in — meridian distance in degrees, or difference of longi- tude, and L and L are the latitudes of the two positions. In other words, the angular convergence of the meridians is the difference in longitude into the sine of the mean latitude. The convergence in chains of two township-lines six miles apart, from one correction-line to another twenty-four miles apart, in lat. 40°, is C = 24 X 80 X sin iT ; where in degrees, = -g^ sin 40°, since one degree of longitude in lat. 40° = 53 miles. Thus ^ = 4^.37 for each six-mile dis- tance, east or west, in lat. 40°. Whence C = 2.42 chains, which is what the northern tier of sections in the north range between correction-lines lacks of being six miles east and west. In a similar manner, we may find that the north sections in a town are about six feet narrower, east and west, than the corresponding southern sections in the same town. * From Eq. (G), p. 621, when cos \Ah\s taken as unity. LAND SURVEYING. 177 Figures 51 and 52 show the resulting dimensions of sections in chiains when no errors are made in the field-work. The north and south distances are all full miles. , Fig. 51. 79.40 80 80 80 80 80 6 5 4 3 2 I 79.92 79.92 79-92 79.92 79-92 79-92 7 8 9 10 II 12 79-94 79-94 18 17 16 15 14 13 79-95 79-95 19 20 21 22 23 24 79-97 79-97 30 29 28 27 26 25 ’ 79-98 79-98 31 32 34 34 35 36 80 80 80 80 80 80 CORRECTION-LINE. In Fig. 51 it will be observed that in the northern tier of sections the meridians must bear westerly somewhat so as to meet the full-mile distance, laid off on the township-line. In Fig. 52 they continue straight north to the town-line, which is in this case a correction-line. If the distances on this correction-line be summed they will be found to be 2.42 chains short of six miles as above computed. The law provides that all excesses or deficiencies, either 12 178 SU/! VE YING. CORRF-CTION-I.INF.. 78.08 6 78.10 79.90 5 79,90 4 79.90 3 79.90 2 79.90 79.92 7 78.12 8 9 10 II 12 79-94 18 78.13 17 16 15 14 13 79-95 19 78.14 20 21 22 23 24 79-97 30 78.16 29 28 27 26 25 79.98 31 32 33 34 35 36 78.18 80 80 80 80 80 Fig. 52. from erroneous measurements or bearings or from the conver- gence of meridians, shall, so far as possible, be thrown into the northern and western quarter-sections of the township. 164. Corner Monuments have been established on all United States land surveys at the corners of townships, sec- tions, and quarter-sections, except at the quarter-section corner at the Centre of each section. These corners have consisted of stones, trees, posts, and mounds of earth. Witness- or bear- ing-trees have always been blazed and lettered for the given town, range, and section, one tree in each section or town meeting at that corner, whenever such trees were available. The bearings and distances to such trees, and a description of LAND SURVEYING. 179 the same, are given in the field-notes. All such corners and witness points, except those made of stone, are subject to de- struction and decay, and when these are lost there is no means of relocating the boundary-lines. They were designed to serve only until the land should be sold off to individuals, when it was expected the owner would replace them with marks of a more permanent character. This has seldom been done, so that in many instances the sectional boundaries can now only be redetermined by personal testimony, line fences and other circumstantial evidence.* FINDING THE AREA OR SUPERFICIAL CONTENTS OF LAND WHEN THE LIMITING BOUNDARIES ARE GIVEN. 165. The Area of a Piece of Land is the area of the level surface included within the vertical planes through the bound- ary-lines. This area is found in acres, roods, and perches, or, better, in acres only, the fractional part being expressed decimally. Evidently the finding of such an area involves two distinct operations, viz. : the Field-work, to determine the positions, directions, and lengths of the boundary-lines ; and the Computation, to find the area from the field-notes. There are several methods of making the field observations, giving rise to corresponding methods of computation. Thus, the area maybe divided into triangles, and the lengths of the sides, or the angles and one side, or the bases and altitudes measured, and the several partial areas computed. Or the bearings and distances of the outside boundary-lines maybe determined and the included area computed directly. This is the common method employed. Again, the rectangular coordinates of each of the corners of the tract may be found in any manner with reference to a chosen point which may or may not be a point in the boundary, and the area computed from these coordi- nates. These three methods will be described in detail. * See Appendix A. i8o SU/^VEVING. I. Area by Triangular Subdivision. i66. By the Use of the Chain Alone. — In Fig. 53 let ABCDEF be the corner bound- aries of a tract of land, the sides being straight lines. Measure all the sides and also the diag- onals AC, AD, AE, and FB. The area required is then the sum of the areas of the four tri- angles ABC, A CD, ADE, and AEF. These partial areas are computed by the formula Area = Vs{s — a){s — b){s — c), where s is the half sum of the three sides a, b, c in each case. For a Check, plot the work from the field-notes. Thus, take any point as A and draw arcs of circles, with A as the com- mon centre, with the radii AB, AC, AD, AE, and ^ A' taken to the scale of the plot. From any point on the first arc, as B, and with a radius equal to BC to scale, cut the next arc, whose radius was AC, giving the point C. From C find D with the measured distance CD, etc., until F is reached. Measure FB on the plot, and if this is equal to the measured length of this line, taken to the scale of the drawing, the field-work and plot are correct. It is evident the point A might have been taken anywhere inside the boundary-lines without changing the method. 167. By the Use of the Compass, or Transit, and Chain. — If the compass had been set up at A the outer boundaries could have been dispensed with, except the lines AB and AF. All that would be necessary in this case would be the bear- ings and distances to the several corners. We then have two LA//D SURVEYING. i8i sides and the included angle of each triangle given when the area of each triangle is found by the formula: Area = \ab sin C. In this case there is no check on the chaining or bearings. I’he taking-out of the angles from the given bearings could be checked by summing them. This sum should be 360° when A is inside the boundary-line, and 360° minus the exterior angle FAB when A is on the boundary. If the boundary- lines be measured also, then the area of each triangle can be computed by both the above methods and a check obtained. 168. By the Use of the Transit and Stadia.*— Set up at A, or at any interior or boundary point from which all the corners can be seen, and read the distances to these corners and the horizontal angles subtended by them. The area is then computed by the formula given in the previous article. The distances may be checked by several independent read- ings, and the angles by closing the horizon (sum = 360°). The above methods do not establish boundary-lines, which is usually an essential requirement of every survey. II. Area from Bearing a 7 id Length of the Boundary-lines. 169. The Common Method of finding land areas is by means of a compass and chain. The bearings and lengths of the boundary-lines are found by following around the tract to the point of beginning. If the boundary-lines are unobstructed by fences, hedges, or the like, then the compass is set at the corners, and the chaining done on line. If these lines are ob- structed, then equal rectangular offsets are measured and the * The stadia methods are described in Chapter VIII. i 82 SURVEYING. bearings and lengths of parallel lines are determined. In this case the compass positions at any corner for the two courses meeting at that corner are not coincident, neither are the final point of one course and the initial point of the next course, the perpendicular offsets from the true corner overlapping on angles less than i8o° and separating on angles over i8o°. The chaining is to be done as described in art. 4, p. 8, the 66-foot or Gunter’s chain being used. Both the direct and the reverse bearing of each course should be obtained for a check as well as to determine the existence of any local attraction. For the methods of handling and using the compass see Chapter II. 170. The Field-notes should be put on the left-hand page and a sketch of the line and objects crossing it on the right- hand page of the note-book. The following is a convenient form for keeping the notes. They are the field-notes of the survey which is plotted on p. 184. It will be seen that the “tree” was sighted from each corner of the survey and its bearing recorded. If these lines were plotted on the map they would be found to intersect at one point. If the plot had not closed, then these bearings would have been plotted and they would not have intersected at one point, the first line which deviated from the common point indicating that the preceding course had been erroneously measured, either in bearing or distance, or else plotted wrongly. In general such bearings, taken to a common point, enable us to locate an error either in the field-notes or in the plot. The bearings of all division-fences were taken, as well as their point of inter- section with the course, so that these interior lines could be plotted and a map of the f^rm obtained. The “old mill” is located by bearings taken from corners .5 and G. The reverse- bearings are given in parenthesis. LAATD SURVEYING. 183 FIELD NOTES— COMPASS SURVEY. Oct. 23, 1885. No. of Course. Point. Bearing. Distance along the Course. Remarks. S. 76° 50' E.. . . West Ch. True bearings given. Variation of needle 5° 50' east. Henry Flagg, Compassman. PeterLong, } . John Short, 7 . 20 Yard “ 9-75 11.54 13.90 25.42 I 4 4 Orchard “ i< Corner B South Wt.= I (North) B. T N. 54° 15' E... N. 58° E... North Courses i and 2 are along the centres of the highway. Old Mill 2 Fence 12.50 24.10 34.68 Corner C S. 89° 55' E.... (West) Wt.= I B. T N. 22° 20' W . . Old Mill N. 26° 45' W.. . Fence N. 61° 45' W... 9.90 10.70 12.45 24.00 3 Mill Creek Fence N. 64° W Corner D N. 27° 40' E. . . (S. 27“ 45 ' VV.) - 1 B. T S. 85° W N. 19° 10' W. . (S. 19° 15' E.).. 4 Corner E 7.40 Wt.= 2 B. T S. 62° 30' W... South C Fence 15.80 D Corner F N. 86° 50' W. . 25.58 Wt.= 2 (S. 86° 45’ E.).. B. T S. 40° 15' E N. bank Mill Creek. 0.30 0. 80 6 S. “ Corner G S. 47 ° 30' W. . . (N. 47° 30' E.). 1.50 Wt.= 5 Fence S. 32° E 0.00 Offset, 0.40 0.00 “ .60 3.00 “ .80 6.00 7 “ .70 9.00 12.00 • * •••••• * ‘ , “ .20 13.60 13.60 Corner H S. 77 ° 45' W... (N. 77 ° 45 ' E.). Wt.= 3 8 Corner A S 89° W 3.53 Wt.= I (N. 89° E.) SUR VE YING. 184 -Highway- LAND SURVEYING. 185 COMPUTING THE AREA. Fig. 55 . 171. The Method stated. — In Fig. 55 ,* let ABODE be the tract whose area is desired. Let us suppose the bearings and lengths of the several courses have been observed. Pass a meridian through the most westerly corner, which in this case is the corner A. Let fall perpendiculars upon this meridian from the several corners, and to those lines drop other perpendicu- lars from the adjacent corners, as shown in the figure. Then we have: Area ABODE = bBODfb - bBAEDfb = bBOe + eODf - (bBA -f- AEa + aEDf). (i) Hence twice the area ABODE is 2A ^ (bB + eO)Bc + {eO + fD)Dd — {bB)Ab — (aE)Aa — (^5* -f- fD)Eg. ... ( 2 ) We will now proceed to show that these distances are all readily obtained from the lengths and bearings of the courses. 172. Latitudes, Departures, and Meridian Distances. — The latitude of a course is the length of the orthographic pro- jection of that course on the meridian, or it is the length of the course into the cosine of its bearing. If the forward bearing of the course is northward its latitude is called its 7wrthing. and is reckoned positively ; while if the course bears southward its latitude is called its southing, and is reckoned negatively. * The lines OD and OX in this figure are used in art. 185. i86 SURVEYING. The departure of a course is the length of its orthographic projection on an east and west line, or it is the length of the course into the sine of its bearing. If the forward bearing of the course is eastward its departure is called its easting, and is reckoned positively ; while if its forward bearing is westward its departure is called its westing, and is reckoned negatively. The meridian distance of a point is its perpendicular dis- tance from the reference meridian, which is here taken through the most westerly point of the survey. The meridian distance of a course is the meridian distance of the middle point of that course ; therefore The double meridian distance of a eourse is equal to the sum of the meridian distances to the extremities of that course. The D. M. D.’s of the two courses adjacent to the reference meridian are evidently equal to their respective departures. The D. M. D. of any other course is equal to the D. M. D. of the preceding course plus the departure of that course plus the departure of the course itself, easterly departures being counted positively and westerly departures negatively. This is evident from Fig. 55. Thus in Fig. 55 Dd is the latitude and dC is the departure of the course DC. If the survey was made with the tract on the left hand, then the latitude of this course is positive and the departure negative ; while the reverse holds true if the survey was made with the tract on the right hand. In this discussion it will be assumed that the survey is made by going around to the left, or by keeping the tract on the left hand, although this is not essential. The D. M. D. of this course CD is fD -f eC', or it is the D. M. D. of BC f- cC — dC). In equation (2), art. 171, the quantities enclosed in brack- ets are the double meridian distances of the several courses, all of which are positive, while the distances into which these are multiplied are the latitudes of the corresponding courses. If we go around towards the left the latitudes of the courses LAND SURVEYING. 187 AB^ DE, and EA are negative, and therefore the correspond- ing products are negative, while the latitudes of the courses BC and CD being positive, their products are positive. We may therefore say that twice the area of the figure is equal to the algebraic sum of the products of the double ineridia^i distances of the several courses into the corresponding latitudes, north latitudes being reckoned positively and south latitudes negatively, and the tract being kept on the left in making the survey. If the tract be kept on the right in the survey, then the numerical value of the result is the same, but it comes out with a negative sign. 173. Computing the Latitudes and Departures of the Courses. — Since the departure of a course is its length into the sine, and its latitude its length into the cosine, of its bear- ing, these may be computed at once from a table of natural or logarithmic sines and cosines. When bearings were (formerly) read only to the nearest 15 minutes of arc, tables were used giving the latitude and departure for all bearings expressed in degrees and quarters for all distances from i to 100. Such tables are called traverse tables. It is customary now, how- ever, to read even the needle-compass closer than the nearest 15 minutes; and if forward and back readings are taken on all courses, and the mean used, these means will seldom be given in even quarters of a degree. If the transit or solar compass is used, the bearing is read to the nearest minute. The old style of traverse table is therefore of little use in modern survey- ing. The ordinary five- or six-place logarithmic tables of sines and cosines are computed for each minute of arc, and these may be used, but they are unnecessarily accurate for or- dinary land-surveying. For this purpose a four-place table is sufficient. If the average error of the field-work is as much as I in 1000 (and it is usually more than this), then an accuracy of I in 5000 in the reduction is evidently all-sufficient, and this is about the average maximum error in a four-place table; that r88 SURVEYING. is, the average of the maximum errors that can be made in the different parts of the table. Table III. is a four-place table of logarithms of numbers from I to 10,000, and Table IV. is a similar table of logarithms of sines and cosines, from o to 360 degrees. If a transit is used in making the survey, and if it is graduated continu- ously from o to 360 degrees, then the azimuths of the several sides are found, all referred to the true meridian or to the first side. If it is desired now to take out the latitudes and de- partures, the same as for a compass-survey, where the bearings N o 190 of the sides are given directly referred to the north and south points, it may be done by Table IV. Since the log sine changes very fast near zero and the log cosine very fast near 90°, the table is made out for every min- ute for the first three degrees from these points ; for the rest of the quadrant it gives values 10 minutes apart, but with a tabular difference for each minute. It is very desirable to make the table cover as few pages as possible for convenience and rapidity in computation. In this table the zero-point is LAND SURVEYING. 189 south and angles increase in the direction SWNE, so that in the first quadrant both latitudes and departures are negative. In the second quadrant latitude is positive and departure nega- tive, in the third both are positive, and in the fourth latitude is negative and departure positive. These relations are shown in Fig. 56. For any angle, falling in any quadrant, if reckoned from the south point in the direction here shown, the log sin (for departure) and log cosine (for latitude) may be at once found from Table IV. If these logarithms are both taken out at the same time and then the logarithms of the distance from Table III., this can be applied to both log sin and log cos, thus giving the log departure and log latitude, when from Table III. again we may obtain the lat. and dep. of this course, giving these their signs according to the quadrant in which the azi- muth of the line falls. If Table IV. is to be used for bearings of lines as given by a needle-compass, then enter the table lor the given bearing, in the first set of angles, beginning at o and ending at 90°. Example: Compute the latitudes and departures of the survey plotted in Fig. 55, p. 185, by Tables III. and IV. The following are ihe field-notes as they would appear, first, as read by a transit and referred to the true meridian; and, second, as read by a needle-compass: Station. Azimuth referred to the South Point. Compass bearing. Distance. A 290° 45' S. 69° 15' E. 7.06 B 217° 15' N. 37° 15' E. 5-93 C 140° 30' N. 39*" 30' W. 6.00 D 57" 45' S. 57° 45' W. 4-65 E 30° 00' S. 30° 00' w. 4.98 SURVEYING. 190 The following is a convenient form for computing the lati- tudes and departures: Course AH 4th Q. Course BC 34 0. Course Cl) ad g. Course DK isi g. Course KA isi g. log sin (dep.) = 9.9708 9.7820 9-8035 ^.C)2-I2 9,6990 log dist. = .8488 .7731 .7782 .6675 .6972 log dep. = ,8196 .5551 .5817 •5947 .3962 Departure = 4-6.60 + 3.59 — 3.82 - 3-93 - 2.49 log cos (lat.) = 9-5494 9.9009 9.8874 9-7272 9-9375 log dist. = .8488 •7731 .7782 .6675 .6972 log lat. = .3982 .6740 .6656 •3947 •6347 Latitude = — 2.50 4-4.72 4-4-63 — 2.48 - 4-31 It is seen that Table IV. answers equally well for either set of bearings, and also that Table III. would have given the lati- tudes and departures to the fourth significant figure as well as to the third. If the proper quadrant is given for each course in the heading as shown above, then the signs may be at once given to the corresponding latitudes and departures. 174. Balancing the Survey. — If the bearings and lengths of all the courses had been accurately* determined, the survey would “ close that is, when the courses are plotted succes- sively to any scale the end of the last course would coincide on the plot with the beginning of the first one. Furthermore, the sum of the northings (plus latitudes) would exactly equal the sum of the southings (minus latitudes), and the sum of the * The error of closure simply shows a want of uniformity of measurement, for if all the sides were in error by the same relative amount, the survey would close just the same. For instance, if an erroneous length of chain were used, the survey might close but the area be considerably in error. See arts. 175 and 177. LAND SURVEYING. I9I eastings (plus departures) would exactly equal the sum of the westings (minus departures). It is evident that such exactness is not attainable in practice, and that neither the north and south latitudes nor the east and west departures will exactly balance, there always being a small residual in each case. These residuals are called the errors of latitude and departure respectively. The distribution of these errors is called bal- ancing the survey. In the form for reduction of the field-notes given below, wherein this example is solved, it is seen that the error of lati- tude is 6 links and the error of departure is 5 links. The dis- tribution of these errors is made by one of the following: FORM FOR COMPUTING AREAS FROM BEARINGS AND DISTANCES OF THE SIDES. Sta- Courses. Dif. Lat. Departure. Balanced. Q + tions. Bearings. 1 Dist. N. •f s. E. + W. Lat. Dep. s' d Area. Area. A S. 69° 15' E. Ch. 7.06 2.50 6.60 — 2.52 -I-6.61 6.61 16.66 B N. 37° 15' E. 5-93 4.72 3-59 + 4-71 + 3-6 o 16.82 79.22 C N. 39° 30' W. 6.00 4-63 3-82 -(-4.62 - 3-8 i 16.61 76.74 D S. 57° 45' W. 4-65 2.48 3-93 - 2.49 — 3-92 8.88 22.11 E S. 30® 00' W. 4.98 4-31 2.49 - 4-32 — 2.48 2.48 10.71 28.62 9-35 9.29 10.19 10.24 155-96 49.48 9.29 10,19 49.48 Error in lat. = .06 Error in dep. = .05 106.48 Error of closure = 0.0027 Area = 53-24 sq. ch. = 5. 324^ erf s = I ia 366. 2862 192 SUR VE YING. RULES FOR BALANCING A SURVEY. Rule i. As the sum of all the distauces is to each particidar distance, so is the whole error in latitude {or departure) to the cor- rection of the corresponding latitude {or departure), each correc- tion being so applied as to diminish the whole error in each case. Rule 2. Determine the relative difficulties to accurate measurement and alignment of the several courses, selecting one course as the standard of reference. Thus, if the standard course would probably give rise to an error of i, determine what the errors for a7i equal distance on the other courses would probably be, as 1^,2, 1,0.5 Multiply the length of each course by its number, or weight, as thus obtained. Then we would have : As the sum of all the multiplied lengths is to each multiplied length, so is the whole error in latitude {or departure) to the cor- rection of the corresponding latitude {or departure), each correc- tion being so applied as to diminish the whole error in each case. These two rules are based on the assumption that the error of closure is as much due to erroneous bearings as to erroneous chaining,* which experience shows to be true in needle-compass work. If, however, the bearings are all taken from a solar compass (or attachment) in good adjustment, or if the exterior lines are run as a traverse with a transit, so that the angles of the pe- rimeter are accurately measured, then the above assumption does not hold, as it is highly probable that the error of closure is almost wholly due to erroneous chaining. Especially would this be highly probable if the azimuth is checked by occupying * Let the student prove the correctness of rules i and 3 for the assumed sources of error. LAATD SURVEYING. 193 the first station on closing and redetermining the azimuth of the first course, as found from the traverse, and comparing it with the initial (true or assumed) azimuth of this course. If it thus appears that the traverse is practically correct as to angular measurements, it may be fairly assumed that the error of closure is almost wholly due to erroneous chaining. In this case use Rule 3. As the arithmetical sum of all the latitudes is to any one latitude, so is the zvhole error in latitude to the correction to the corresponding latitude, each correction being so applied as to diminish the whole error in each case. Proceed similarly with the departures.* In the solution given on p. 191 the first rule is applied. In ordinary farm-surveying it is not common to give the lengths of the courses nearer than the nearest even link or hundredth of a chain. In balancing, therefore, the same rule may be observed. 175. The Error of Closure is the ratio to the whole pe- rimeter of the length of the line joining the initial and final points, as found from the field-notes. The length of this line is the hypotenuse of a right triangle of which the errors in latitude and departure are the two sides. Its length is there- fore equal to the square root of the sum of the squares of these two errors. This divided by the whole perimeter gives the error of closure, which ratio is usually expressed by a vulgar fraction whose numerator is one, being in the above example. The error of closure for ordinary rolling country should not * It is evident that the courses could here be weighted for different degrees of difficulty in the chaining ; but instead of multiplying the lengths of the courses by their weights, multiply the latitudes and departures by the weights of the corresponding courses, and then distribute the errors in latitude and departure by these multiplied latitudes and departures. 13 194 SUR VE YING. be more than i in 300. In city work it sliould be less than i in 1000, and should average less than i in 5 CXX). 176. The Form of Reduction. — On p. 191, the ordinary form of reduction is shown. Here the courses are not weight, ed for different degrees of difficulty in chaining; and since it was a compass-survey the effect of erroneous bearings is sup- posed to equal that from erroneous chaining, and so the first rule for balancing is used. The balanced latitudes and de- partures having been found, the double meridian distances are next taken out. In taking out these it is preferable to begin with the most westerly coriier^ whether this be the first course recorded or not. In the example solved on p. 19 1, it is the first corner occupied, but in that given on p. 198 it is not the first course. By beginning with the most westerly corner (which is equivalent to passing the reference meridian through that corner), all the double meridian distances will be positive; otherwise some of them may be negative. If attention be paid to signs we may begin at any corner to compute the double meridian distances. A check on the computation of the D. M. D.’s is that, when computed continuously in either direction and from any cor- ner, the numerical value of the D. M. D. of the last course must equal its departure. This is a very important check and must not be neglected, as it proves the accuracy of all the D. M. D.’s. We are now able to compute the double-areas according to equation (2), art. 171, since the terms entering in that equation have their numerical values determined. The several products, being the partial double-areas, are written in the last two col- umns, careful attention being paid to the signs of these prod- ucts. Thus, when the reference meridian is taken through the most westerly corner, then all the D. M. D.’s are positive and the results take the sign of the corresponding latitude. If some of the D. M. D.'s are negative, then the signs of these par- LAATD SURVEYING. 195 tial areas are opposite to those of the corresponding latitudes. The algebraic sum of the partial double-areas is twice the area of the figure, as shown in eq. (2), art. 171. If the dis- tances are given in chains, then the area is given in sq. chains, and dividing by ten gives the area in acres. If the dis- tances were given in feet, as it often is, being measured by a loo-foot chain or tape, then the area is in sq. feet, and this must be divided by 43560, the number of sq. feet in one acre, to give the area in acres. This is best done by logarithms, as shown in the example solved on p. 198. It is preferable to ex- press areas in acres and decimals rather than in roods and perches, as was formerly the custom. On the following page is the reduction of the field-notes given on p. 183. Here the several courses have been weighted for various degrees of difficulty in the chaining. Thus, the first and second courses were along the public highway and on even ground. These are taken as the standard and given the weight unity. The third course is on very uneven ground and is judged to give rise to about three times the error of courses one and two per unit’s distance. It is therefore weighted pthree. The proper weight to give to the several courses is thus seen to depend on the character of the obstructions to ac- curate work, and represents simply the judgment of the sur- veyor as to the probable relation of these sources of error. The short course FG was very difficult to measure, as there were precipitous bluffs, and the course GH was also on very uneven ground. Following the column of weights in the tabular reduction are the multiplied distances ; the errors of latitude and depart- ure are distributed according to the results in this column by Rule Two, p. 192. This survey was also made with a needle- compass. In the following example the transit was used, and the 96 SU/^ VE Y TNG. ^ ' tji+ ;z:+ ui iS ^5.2 SQ m o *-• O i-i M I I + + + I I I c<^ xrt M CO '8 i; > boiifl f2'5'5.S CO w w W W ^ ^ ^ ^ ^ Sd i-i m o O o O VO c in ir> CO u rt o O O' O O' W U) CO 00 'I- CO C/J cn C/5 c/5’ <:muQwf^oiu a, V -o u O u W O N O -I- O' O' + *o- c^ ZAN’D SURVEYIVG. 197 survey began at A. The azimuth of the line AB (Fig. 57) was found by a solar attachment, and then the other courses ran as a traverse, the horizontal limb of the transit being oriented by the back azimuth of the last course. The azimuths of the courses are all referred to the south point as zero, and increase in the direction SWNE. After the last course FA was run, the instrument was carried to A and oriented by a back sight on i^and the azimuth of AB again determined. This agreed so well with the original azimuth of this course that the azimuths of all the courses were proved to be correct.f The error of closure is therefore due to the chaining alone. A hundred-foot chain was used so that the distances are all given in feet. The obstructions to chaining were about uni- form, so the courses are all given equal weight. In balancing. Rule Three must be used, since the errors are supposed to come only from the chaining. If the errors in latitude and departure had been distributed by Rule One, or in proportion to the lengths of the courses, the resulting area would have been 56.41 acres, a difference of 0.07 acres, or about one eight-hundredth of the total area. 177. Area Correction due to Erroneous Length of *The lines MB and 00 ' in this figure are used in art. 186. f From the azimuth check here obtained, as compared to the errors in lat- itude and departure, decide whether the latter are due mostly to the chaining or whether the errors in azimuth have had an equal influence, and so determine whether to use rule i or rule 3 in balancing. 198 SURVEYING. Areas. • • • 00 0 • * * CO GO * • • • \n 0 • I r ; CO CO r . . . CT' . CO CO 0 vO CO + Areas. •t" 0 N • . vn CO CO . .CO O- O- o' c 4 cT t * »A -t- 00 0 , • 0 0 - : ; ci ci • • • . • . M CO O' vO 0 vO vo" o‘ CO C 7 ' CO CO VO j D. M. D. O' 0 CO -t N 10 CO - 1 - CO »0 CO VO r> CO r'. CO N CX CX fi Balanced. d VO 0 O' 0 0 *H VO 0 M « M M A (check) 164° 05' | Station. pq u Q W CO O o' •ct a. o *0 o II o U 4/6‘J 4 - 172 ( 2,470,012 sq. ft. Error of closure = • = i in 360. (435^0 sq. ft. = lA.) Area = 6484 tor 56.70 acres. ZAArn SURVEYING. 199 Chain. — If the measuring unit has not the length assigned to it in the computation, then the computed area will be errone- ous. Such an error will not show in the balancing of the work or elsewhere, and hence an independent correction must be ap- plied for this error. If the chain was too long by one one- thousandth part of its length, for instance, then all the courses are too short in the same ratio. And since similar plane fig- ures are to each other as the squares of their like parts, we would have true area : computed area :: (1001)“ : (1000)*, or true area = computed area (nearly) ;* or, in general, if / = length of chain and Al = error in length, being positive for chain long and negative for chain short, and if Al is small as compared with /, as it always is in this case, then if we let A = true area. A' = computed area Ca — correction to computed area, and A = relative error of chain, IA-2AI ^ we have A = — A' — {i 2 A')A' whence, A — A' = Cj_= 2 A A'. That is to say, the relative area correction due to erroneous length of chain is twice the relative error of the chain^ being positive for chain long, and negative for chain short. * The error in this approximation is one one-millionth in this case, and would always be inconsiderable in this class of problems. 200 S UR VE YING. FINDING TH?: AREA OR SUPERFICIAL CONTENTS OF LAND WHEN THE RECTANGULAR COORDINATES OF THE COR- NERS ARE GIVEN WITH RESPECT TO ANY POINT AS AN ORIGIN. 178. Conditions of Application of this Method. — Where many tracts of land, all bounded by straight lines, are somewhat confusedly intermingled, as is the case in many of the older States, and where the area of each tract over an ex- tended territory is to be found, this method is greatly to be preferred to that by means of the boundary-lines. In this case it is only necessary to make a general coordinate survey of the whole territory, as described in Chapter VIII., on Topographi- cal Surveying, using the stadia for obtaining distances, and be- ing careful to locate every corner of each tract. If areas alone are required, no attention need be paid to the obtaining of elevations for contour lines, and so the work is greatly facilitated. A transit and two or three stadia rods would be the instru- ments used. The survey would then be carefully plotted and the coordinates measured on the sheet, or they could be com- puted from the field-notes. If the plotting is carefully done the former method is preferable. It is best to choose the origin of coordinates entirely outside the tract and so that the whole area falls in one quadrant, thus making all the coor- dinates of one sign. Large tracts of mineral land are sometimes acquired by large companies, including perhaps hundreds of individual es- tates. In such cases a topographical map of the region is necessary ; and when this survey is rnade, a little extra care to obtain all the “ corners” of private claims will enable the areas of all suclrlots to be determined with great accuracy and at small additional cost. The method probably has no advan- tages when the area of but a single tract is desired. LAATD SUJ^VEVING. 201 179. The Method of Finding the Area from the Rec- tangular Coordinates of the Corners is as follows : Let Fig. 58 be the same tract as that given in Fig. 55, and Fig. 58. let the origin be one chain west of A and three chains south of B. Then, from the balanced latitudes and departures for this case, given on p. 191, we find the following coordinates of the corners fb, etc., denoting the latitudes of the corners A, B, etc., and similarly with Xb, etc., for departures : = 5 - 52 , n = 3-00, = 77L n = 12.33, 7 , = 9.84. .^^=1.00, Xb = 7.6i, Xc= 11.21, xa = 7.40, = 3.48. The area of the figure ABODE is equal to the areas ybBCy^ -^-y^CDy^ — {y^EDy^ y^AEy, + y,,BAyJi ; 202 SURVEYING. or ^ i [(jj'c-jz,) (^6+^c) + (^d-7c) (^0 + x^-{ya -y,) (xa + x,) - {ye — ya) (^e + -^a) “ ( Ja “ /b) {^a + ^b)]- ( I ) . By developing equation (i) we obtain A = i [y^Xc — yaXt, +ybXa — JbXc + “ /c^'6)^ while its bearing is , , . ^c — the arc whose tan is . yc -yb 181. Supplying Missing or Erroneous Data. — In any closed survey there are two geometric conditions that must be fulfilled, viz. : 1. The sum of all the latitudes must be zero. 2. The sum of all the departures must be zero. 204 SU/^ VE Y/NG. These two conditions give rise to two corresponding equa- tions. If /j, 4 , /g, etc., be the lengths of the several courses, and if etc., be their compass-bearings, then our two geo- metric conditions give /, sin 6 ^ -(- 4 sin -f- 4 sin 0 ^ -j- etc., =0. . . (i) /, cos + 4 cos + ^3 cos <^3 + etc., = o. . . (2) Since we have two independent equations, we can solve for two unknown quantities. These two unknowns may be any two of the functions entering in the above equations. Thus, if any two distances, any two bearings, or any one distance and any one bearing are missing, they may be found from these equations. Or, if but one bearing or distance is missing, it may be found from one of these equations and the other equation used for balancing either the latitudes or departures. When all bearings and distances are given, these equations are really used in balancing ; but if they are both used to deter- mine missing quantities, there can be no balancing of errors, for when the missing quantities are computed by these equa- tions, both latitudes and departures will exactly balance. In other words, all the errors of the survey are thus thrown into these two quantities. This artifice should therefore never be resorted to except where it is impracticable to actually measure the quantities themselves in the field. There are four cases to be solved : I. Where the bearing and length of one course are un- known. II. Where the bearing of one course and length of another o O are unknown. III. Where two bearings are unknown. IV. Where two lengths are unknown. LAND SURVEYING, 205 The bearings will be reckoned from both north and south points around to the east and west points, as is common in compass surveying. Then the length of a course into the sin of its bearing gives its departure, and into the cos of its bear- ing gives its latitude. North latitude is plus and south latitude minus; east departure plus and west departure minus. In every case let the sum of the departures of all known courses, taken with the opposite sign, be D, and the sum of their latitudes, taken with the opposite sign, be L. Then D and L are the departure and latitude necessary to close the survey. Case I . — Bearing and le^igth of one course unknown. The two condition equations here become (3) Whence tan ( 4 ) Having found the bearing, find 4i from either of equations ( 3 ). Particular attention must always be paid to the signs of D and L. Evidently sin 6^ (dep.) and cos (lat.) have the same signs as D and L respectively, whence the quadrant which includes the bearing may be determined and the proper letters applied. For this purpose Fig. 56 may be consulted. Case 1 1. — The bearing of one course and the length of another unknoivn. In this case let a be the known bearing of the course whose length is unknown, and let / be the known length of the course whose bearing is unknown. Then we have s\n a-\- 1 sin 6^ = L cos a-\- 1 cos 6^ = L 2o6 SUR VE Y I NG. If we let sin a — s^ and cos a = Cy we have L = sD+cL± Vr - {D^ + U) + {sD + cL)\ . ( 6 ) Here there are two values of which will satisfy the equa- tion, and so there arc two solutions to the problem. If the surveyor has no knowledge whatever of either the unknown length or bearing, the problem is indeterminate. If he has seen the tract he could usually tell which length or which resulting bearing was the correct one, when the problem would become determinate. When 4^ is found, substitute in one of equations ( 5 ) and find 6^. Pay careful attention to the signs of the trigonometrical functions of all bearings. Case III. — When two bearuigs are unknown. Let I' and 1" be the known lengths of the courses whose bearings are unknown. Then the equations become I' sin -|- /" sin 6^ = D;) I cos -[■ cos 6n = L. \ ‘ • • • ( 7 ) Whence Where cos 6^ = KL ± DVD - D^ + D • (8) K = ^ p D' D 2l" This case is also indeterminate unless one is able to tell which of the two sets of bearings is the correct one. Case IV. — When the lengths of two courses are unknown. Let a and b be the known bearing of the courses whose lengths are unknown. ZAJVn SURVEYING. 207 Our equations here become whence sin a-{- Insm b = D\ cos a-\- In cos b = L. Ls,m a — D cos a ” sin {a — b) • • (9) (10) This case is determinate. In case there is but one unknown, then either one of equa- tions (3) will solve. In taking out either the sine or the cosine from the tables, however, two angles will always be found equidistant from the east or west point if the sine, and equi- distant from either the north or south point if the cosine, either of which may be chosen. In such case both sine and cosine must be found, when the signs alone of these two func- tions will determine the quadrant in which the bearing is found. Hence, if the single unknown is a bearing; both of the equa- tions (3) must be used in order to determine which of the two bearings given by the table is the correct one, but one alone is sufficient to obtain the numerical value of the bearing. Thus, if the sine equation is used to compute the bearing, then the latitude may be taken out for the given length and bearing; and these will then not balance, but will have to be balanced in the usual way, while the departures will, of course, balance, since the residual departure D necessary to close the survey as to departures was used to compute the corresponding bearing. The reverse of this would be true if the cosine equation were used to compute the bearing. 2o8 S UR VE YING. PIXDTTING THE FIELD-NOTES. iSia. To plot a Compass Survey select a point for the initial station, and pass a meridian through it in pencil, l^y means of a semicircular protractor, sucli as is shown in Fig. 44, mark the bearing and draw an indefinite line from the sta- tion point. On this line lay off to scale the length of the course, thus establishing the next corner. Througli this draw another pencil meridian, and proceed as before. If the plot- ting is perfect the length of the line joining the final with the initial point, taken to scale, is the error of closure of the sur- vey; and the horizontal and vertical components of this line, taken to scale, should be the errors in departure and latitude respectively as obtained by the computation. If preferred, the bearings of the successive courses may be so combined as to give the deflection-angle at each station, and these laid off from the preceding course as already drawn. Errors are more likely to accumulate in the plot by this method, however, than by that first given. Again, the rectangular coordinates of the several corners may be computed and these plotted from a pair of rectangular axes, but this is not a common practice. For the plotting of transit surveys, especially where the stadia is used, see Chapter VIII. THE AREAS OF FIGURES BOUNDED BY CURVED OR IRREGULAR LINES. 182. The Method by Offsets at Irregular Intervals. — Where a tract of land is bounded by a body of water, as a stream or lake, it is customary to run straight lines as near the boundary as practicable and then to take rectangular offsets at selected intervals from these bordering-lines to the irregular boundary. These small areas are then computed as trapezoids, LAND SURVEYING. 209 the distance along the base-line being the altitude and the half- sum of the adjacent offsets being the mean width. The offsets should therefore be run at such intervals as to make this method of computation sufficiently accurate. Such offsets were taken from the course GH in Fig. 54, the notes for which are given on p. 183. The work of computation may be shortened by using a modified form of the method of areas from the rectangular coordinates of the corners, b which, in this case, are the ends ” of the offset lines. Let Fig. 59 be an area to be determined from the offsets from the line AK. The position and lengh of the offsets are given. Take the origin at A and let the distances along AK be the abscissae, and the lengths of the offsets be the ordinates. Using the second of equations (3), p. 202, we have D F i 1 Vp 1 ! 1 1 ^ K Si S! o5,i 1 1 j Oil i 1 • • 1 "1 1 ^.1 1 1.00 V 0 1.21 2JJ3 3.56 5 . 04 . 5.75 7.00 Fig. A — *k\Xa{yk — J'i.) + —yc) + ^e{yb —yd) + ^d{y<,- y.) + - yf) + ^Ay.-yg) + Xg {yf -y^ + (0 But here Xi,, ya, and y^g are all zero ; also x^ = Xjg^ hence this equation becomes A=i[Xg{yi- ya) -f-x^iy, — y.) + ^e {yd- yf) +Mye-yg)+-Vg {y/-yi,)+ Myg+yh)l(2) From eq. (2) we have the * The plus sign is here used, since we have gone around the figure in a direc- tion opposite to that followed in the general case 210 SURVEYING. RULE FOR FINDING AREAS FROM RECTANGULAR OFFSETS AT IRREGULAR INTERVALS. Multiply the distajice along the course of each intermediate offset from the first by the difference between the two adjace7it offsets^ always subtractuig the following from the preceding. Also midtiply the distance of the last offset from the first by the smn of the last two offsets. Divide the sum of these products by two. The following is the nunnerical reduction for finding the area of the irregular tract shown in Fig. 59. Offset. Distance from A. Length of Offset. Differences. Products. ch. ch. ch. sq. ch. B 0.00 1-53 C I . 21 1.76 - 0.47 — 0.57 D 2.23 2.00 - 0.56 - 1.25 E 3-56 2.32 + .09 + .32 F 5.04 I. 91 + .87 + 4.38 G 5.75 1-45 .91 + 5-23 H 7.00 1. 00 + 2.45 +17.15 2 ) 25.26 Area = 12.63 sq. chs. = 1 . 263 acres. It is evident that an area bounded on all sides by irregular or curved lines could have a base-line run through it, and off- sets taken from this line to both boundaries and the area com- puted by this method. Example 196, p. 221, should be so computed. 183. The Method by Offsets at Regular Intervals. — If the intervals between the offsets, or ordinates, are all equal the computation is much simplified. On the'assumption that the area is a series of trapezoids, we have the lAATB SURVEYING. 2II RULE FOR FINDING THE AREA FROM RECTANGULAR OFFSETS AT REGULAR INTERVALS. Add together all the intermediate offsets and one half the end offsets^ and multiply the sum by the constant interval between them. The following rules for finding areas are found from the suc- cessive orders of differences in each case and may all be derived by a rigid development.* They assume that the bounding-line is curved and that rectangular ordinates have been measured at uniform intervals from a base-line traversing the figure. Let the common interval between ordinates be d\ let the lengths of the ordinates be h^, h^, h^ . , , , hn \ and let the number of intervals be N, 1. N = 1, A = ^{h^-{- hi), Trapezoidal Rule. N = 2, A = ^ {h^ + Simpson’s ^ Rule. 'id III. iV = 3 , ^ — -g- {K+ ZK + iK + ^0’ Simpson’s f Rule. IV. = 4, ^ ^ [7 (/^o + h) + 32 {K + h) + I2/.J. V. iv = 6, A IK ■VK + K+ K+ 5 {K+ K+ K)+Kl This is called Weddel’s Rule. If a quadrant be computed by this rule, the result is o.yjgA instead of o. 785 r^ the true value. When an area, bounded by a base line and two end ordi- *See appendix C. 212 SURVEYING. nates, be divided by imaginary lines parallel to the end ordi- nates and equally spaced, as in Fig. 6o, and if the middle ordi- Fig. 6o. nates of these partial areas be measured, then if ^ common width of the partial areas and /i^, etc., their middle ordi- nates, a the first end ordinate and b the last one, we have, approximately. I. ^ = d:2h, where signifies the summation of all the h's. The following rules are, however, more accurate : II. A = d:2h + - ^ Poncelet’s Rule ; or, [Rule. III. A = + Francke’s /2 The various rules above given are often used to determine areas of irregular figures such as steam diagrams, cross-sections of structural forms, streams, excavations, etc. The most ready and accurate means of determining all such areas, how- ever, is by means of the planimeter. LAJVB SURVEYING. 213 THE SUBDIVISION OF LAND. 184. The Problems arising in the subdivision of land are of almost infinite variety. All such problems are solved by the application of the fundamental principles and relations of geometry and trigonometry with which the student is supposed to be familiar. There are, however, two classes of problems of such frequent application that they will be given in detail. 185. To cut off from a Given Tract of Land a Given Area by a Right Line, starting from a Given Point in the Boundary. — In Fig. 55, p. 185, let O be the middle point on the line AB, from which a line is to be run in such a manner as to cut off three acres from the western portion of the tract. We may at once assume that the dividing-line will cut the side DC in some point X, whose distance from D is to be found. First compute the area OAED, using the balanced latitudes and departures given on p. 191, we have the following: Course. Lat. Dep. D. M. D. Double Areas. + - AO ch. — 1.26 ch. + 3.30 3.30 4.16 OD (+ 8.07) (+ 3 - 10 ) 9.70 78.28 DE - 2.49 - 3.92 8.88 22.11 EA - 4.32 — 2.48 2.48 10.71 (— 8.07) (— 3.10) Sums + 78.28 36.98. — 36.98 2)41.30 » Area = 20.65 sq. chs. = 2.065 acres. Here the latitude and departure of the course OD are such as to make the latitudes and departures balance. The area is 214 SURVEYING. found to be 2.065 acres, leaving 0.935 acres to be laid off from OD by the line OX. It remains now to find the point X. First compute the length and bearing of the line OD from Case I., p. 205. Thus we have D +3.10 Whence 6 — 21° from the table of natural tangents. From the table of natural sines, we find sin 21° = 0.358. Hence from eq. (3), p. 205, we have / sin ^ = Z?, or 0.358/ = 3.10. Whence I — 8.66 chains. The bearing is evidently N. 21° E. We now have to find the distance DX such that the area ODX shall be 9.35 sq. chains. Since the area of any triangle is one half the product of two sides into the sine of the in- cluded angle (another way of saying it is equal to half the base into the altitude), we have 9.35 =1(8.66 sin . . . . (i) From the bearings of OD and DX we find the angle ODX to be bo"" 30', hence sin ODX = 0.870, from which we find DX = 2.48 chains. The length and bearing of the line OX may be computed from its latitude and departure, the same as was done for the line OD above, or we may compute the angle DOX and length LJATB SURVEYING. 215 OX by solving the triangle DOX. The bearing of OX may then be found, and the line run from O. There will then be two checks on the work, viz. : the measured lengths of OX and DX must be equal to their computed values. To find the angle DOX, let the three angles of the triangle be D, O, and X, and the sides opposite these angles be d, 0 , and X, respectively. Then we have tanHX-£?) = ^tani(^+ 0- This equation gives the angle (X — O'), whence 0 = 1 (X+ £?)- J {X- 6'),and ^ (X + O) + i (X - O). Also, d = OX = OB Sin X and 0 BX = OB sm X We therefore have the following RULE FOR CUTTING OFF A GIVEN AREA BY A LINE START ING FROM A GIVEN POINT IN THE BOUNDARY Having first surveyed the tract and plotted the same, join the given point on the plot with the corner which will give the nearest approximation to the desired area. Compute the length and bearing of this line, and of the area thus cut off. Subtract this area from the desired area, and the remainder is the area to be cut off in the form of a triangle, one side of which has bearing and distance given, and another side has its bearing alone given. From these data compute the lengths and bearings of the other sides, one of which is the line sought. This line may then be run, and its length measured, as well as the length of the portion of the opposite boundary cut off, for a check on the accuracy of the work. i86. To cut off from a Given Tract of Land a Given Area by a Right Line running in a Given Direction. 2i6 SUR VE YING. — Let the problem be to cut off 30 acres from the northern portion of the tract shown in Fig. 57, p. 197, by a line whose bearing is N. 80° E., or whose azimuth is 260°.* Pass a line parallel to the required line through the corner nearest to the probable position of the desired line. Let MB, Fig- 57 > such a line. Compute the lengths of the lines EM and MB by Case IV., p. 206. From the computation, p. 198, we have the following: Courses Azimuth. Lengths. Ralanccd Latitudes. Balanced Departures. D. M. D.’s Double Areas. BC 205° 39' 1004 ft. -j- 906 ft. + 432 ft. 2738 -f- 2,480,628 CD II 2 12 896 + 339 - 834 2336 + 791,804 DE 55 00 912 — 522 - 750 752 — 392,544 EM 0 04 (926) — 926 — I I - 926 MB 260 00 (1171) + 203 + 1153 1153 + 234.059 (4- 723) (-1152) 2)3,113,021 Therefore to close requires Z = — 723 and Z> = -}- 1152. Area = 1,556,510 sq. ft. , ■ = 35.73 ac’s. From equation (10), p. 207, we have EM^ D cos 260° — L sin 260° sin 259° 56' ‘ (+ 1152) (+ -1736) - (- 723) (+ -9848) + .9846 200 + 712 .9846 = 926 ft. * In this problem it would have shortened the operation somewhat if the meridian of the survey had been taken parallel to the dividing-line. The bear- ings could have all been changed to give angles from this meridian, and original computation made from these new bearings. ZAJVn SURVEYING. 217 Whence from eq (9), we have MB = D — EM sin 4' sin 260"^ + 1152 - (926) (- .0011) = i = II7I ft. + .9848. ^ Inserting these values of the lengths of the courses EM and MB., we can compute the area BCDEM. This is found to he 35.73 acres, or 5.73 acres too much. The problem now is to pass a line north of MB and parallel to it, so that the area included between the parallel lines and the intercepted por- tions of £'/^and BC shall be 5.73 acres, or 249,710 sq. ft. Let 00 ' be such a line. This line can be run when either MO or BO' is known. It is best, however, to compute both these distances, using one for a check. To find these distances. Let X = perpendicular distance between the parallel lines MB and 00 ' . Let angle EMB^EOO’ and angle 00 ' B =0. Then we have Area MOO' B — MB . x — ^x"^ cot 0 cot 0 = MB . X + (cot 0 — cot B). . . (i) Since ^and 0 are known angles, their cotangents are known quantities in any case. So, for simplicity, let (cot 0 — cot B)~K\ 2I8 SURVEYING. also, let the distance MB = D, and area MOO'B = A. Then the equation becomes Dx^\Kx^', ( 2 ) , 2Z? 2 A ^ ^ - K' D jzA , D' K^\J K' = - J±^4/2^Ar+Z>“; = i(± ^2AK^D‘-D) (3) That sign of the radical is to be used which will give a positive value to x. The other sign would give the value of X to be used in laying off the given area on the opposite side of MB, provided the sides OM dind O' B were continuous in that direction. Using equation (3) for the problem in hand, we have e = 79° 56' ; 0= 54° 21'; A — 249,710 sq. ft. ; D — 1 1 71 ft. ; K = 0.7172 - 0.1775 = o 5397: LAJVI? SURVEYING. 2ig whence ;ir = — (± /269, 537 + 1,371,241 - 1171) = 203.6 feet. We can now find MO and BO' from MO = -I—a, and BO' = ; Sin u sin 0 whence MO — 206.8 feet, and BO' = 250.6 feet. The length of the line 00 ' is 00 ' — MB + X (cot 0 — cot We may therefore write the following RULE FOR CUTTING OFF A GIVEN AREA BY A LINE PASSING IN A GIVEN DIRECTION. Having first surveyed the tract and plotted the same, pass a line on the plot in the required direction through the corner which will give the nearest approximation to the desired area. Compute the lengths of the two unknown courses bounding this area, and then the area itself. Subtract this from the given area, and the remainder is the area which is to be cut off by a line parallel to the first trial line. This auxiliary area will always be a trapezoid, whose area, the length and bearing of one of the parallel sides, and the bearings of tlie remaining sides are known. The lengths of these sides may then be computed, one of the end lengths laid off, and the dividing- line run. Measure the length of this line and also of the other end line for checks. 220 SU/iVEYING. EXAMPLES. 187. Compute the area, plot the survey, and determine error of closure from the following field-notes : Station. Bearing. Distance. A S. 46^ E. 20.00 ch. B S. 74^ E. 30.95 C N. 33 i E. 18.80 D N. 56 W. 27.60 E W. 21.25 F S. 51I w. 13.80 ( Error of closure = i in 201. This being a compass-survey, the errors in latitude and departure must be distributed in proportion to the lengths of the courses, regardless of their bear- ings, or according to Rule i, p, 192. If the errors in the bearings (or deflection angles) had been very small as compared with the errors in measuring the dis- tances, as is the case when the deflection angles are measured with a transit, then Rule 3, p. 193, should have been used. This would have changed the result by 0.08 acres, the result then being 104.35 acres. 188. Find the area and error of closure from the following field-notes : Station. Bearing. Distance. A E. 130 rods. B 00 137 C N. 81 W. 186 D S. 54 E S. 36 w. 125 F S. 45 E. 89 G N. 40 E. 70 LAArn SURVEYING. 221 What would be the resulting difference in area from the use of Rules i and 3 ? 189. In the example, art. 187, suppose the length and bearing of the first course were unknown. Let these be found as in Case I., art. 180. 190. Suppose the length of course A and bearing of B are unknown in same example. Compute by Case II. 191. Let the first two bearings be unknown. Compute them by Case III. 192. Let the lengths of the first two courses be unknown. Find them by Case IV. 193. Let it be required to cutoff twenty-five acres from the west end of the tract given in art. 187 by a line passing through a point on the course EC at a distance of ten chains from B. Find the length and bearing of the division- line, and the other intersecting point on the boundary. 194. Let it be required to divide the tract given in art. 187 into three equal portions by north and south lines. Find the lengths and points of intersection of such lines with the boundary-lines. 195. Compute the coordinates of the corners of the tract given in art. 187, taken with reference to a point 35 chains directly south of A, and then com- pute the area of the tract from these coordinates by the formula given in art. 179. This area should, of course, be the same as that obtained by any other method where the same balanced latitudes and departures are used. 196. An irregular tract of land has a straight line run through it and rec- tangular offsets taken to the boundary. Find the area of the tract from the following notes : Distance. Width. ch. ch. 0 2.35 10 8.42 14 12.60 , 20 11.38 25 10.75 28 6.15 30.50 0.00 Is it significant whether or not this tract lies on both sides or wholly on one side of the base-line? I96ct. Compute the area of the tract of which the following are the field- 222 SUR VE YING. notes. The rectangular offsets are taken on both sides of a straight axial line R signifying right and L left. Distances. Side. Width or Lenfjlh of OlTset. Distances. Side. Width or Lcneth of Offset, ch. ch. I ch. ch. O R 4.23 18 R 15.80 O L 0.00 20 L 5-00 5 R 7.16 25 R 12.20 7.50 L 3-45 30 L 2.62 10 R 12.68 30 R 6.48 10 L 6.00 30 L 0.00 12 R 10.75 Note. — For a valuable paper on the Judicial Functions of the Surveyor, by fudge Cooley of the Michigan Supreme Court, see Appendix A. CHAPTER VIIL TOPOGRAPHICAL SURVEYING BY THE TRANSIT AND STADIA.* 197. A Topographical Survey is such a one as gives not only the geographical positions of points and objects on the surface of the ground, but also furnishes the data from which the character of the surface may be delineated with respect to the relative elevations or depressions. 198. There are three general methods of making such a survey. First, with a compass (or transit) and chain, to determine geographical position, and with a level for obtaining relative elevations. Second, with' a plane-table, either with or without stadia- rods. Third, with a transit instrument and stadia rods. The first method is very laborious, slow, and expensive. It is therefore not adapted to large areas. The second method has been more extensively used for this purpose than any other. The use of the plane-table is fully described in Chap- ter V. This method is giving place, however, to the third, which has been in use in America since about 1864, when it was officially adopted on the United States Lake Survey. The system was first used in Italy about 1820. In what fol- lows, the third method will alone be described. *The word “stadia” is Italian and was originally used to designate the rod used by the inventor of the method. It is now too firmly established to be changed. On the U. S, Coast and Geodetic Survey the word “telemeter” is used in place of “stadia,” but this, which very properly means distance-meas- urer, has been appropriated for other appliances used for measuring at a dis- tance, as temperature, for example. It would therefore seem that “stadia” is the better word to use. 224 SUR VE YING. 199. The Principle of the location of points by the transit and stadia, both horizontally and vertically, is that of polar coordinates. That is, the location of the point geographically is by obtaining its angular direction from the meridian through the instrument, which is read on the limb of tlie transit, and its distance from the instrument, which is read through the telescope on the stadia-rod which is held at the point. This distance is found by observing what portion of the image of the graduated rod is included between certain cross-hairs in the telescope. The farther the rod is from the instrument, the greater is the portion of the rod’s image which falls between the cross-wires. For elevation, the vertical angle is read on the vertical circle of the transit, when the telescope is directed towards a point of the stadia-rod as far from the ground as the telescope is above the stake over which it is set. The tangent of this angle of elevation, or depression, into the given horizontal dis- tance is the amount by which the point is above or below the instrument station. In this way, both the chain and levelling-instrument are dis- pensed with, and the slow and laborious processes of chaining over bad ground, and levelling up and down hill, are avoided. The horizontal distances are obtained as well, in general, as by the chain ; and the levelling may be done within a few tenths of a foot to the mile which is amply sufficient for topo- graphical purposes. THEORY OF STADIA MEASUREMENTS. 200. Fundamental Relations. — In Fig. 61 let LS be any lens, or combination of lenses, used for the object-glass of a telescope. Let be a portion of the object (in this case the stadia- rod), and let be its image. The point of the object A^ has its image formed at A^, and so with ^2 and B^, TOPOGRAPHICAL SUR VE YING. 225 Let the position of the image for parallel rays, or for an object an infinite distance away ; and let C be the centre of the instrument, or the intersection of the plumb-line, extended, with the axis of the telescope. Let and be the “ principal points,” * and let the distance FE^ — f (focal length), ^ 'Z/ ] (conjugate foci), — t (for image, intercepted portion), — s (for stadia, intercepted portion). Then, since A^E^ is parallel to A^E^, and B^E^ is parallel to B^E^, we have A,B, : A,B, :: IE,: 0E„ or, (i) Also, from the law of lenses we have * As optics is generally taught in the English text-books, Ei and E^ are made to coincide in a point at or near the centre of the lens; and this is called the “optical centre.” The “principal points” of the ordinary objective fall inside the surfaces of the lens, but they never coincide. The ordinary theory is sufficiently approximate for the development of stadia formulae but it saves confusion to make the conditions rigid, and it is equally simple. 15 226 SUJ^VEV/NG. ( 2 ) On these two equations rests the whole theory of stadia measurements. Since the distance FE^ = f = focal distance, is a constant for any lens or fixed combination of lenses, we see from equa- tion (2) that if the object P approaches the lens the distance /, is diminished, and therefore /, must be increased ; that is, the image recedes farther from the lens as the object ap- proaches it, and vice versa. If the extreme wires in the reticule of the telescope be sup- posed to be placed at and in the figure, then is the visual angle which is equal to A^E^B.^. But as the image changes its distance from the objective as the object is nearer to or farther from the instrument, so the reticule is moved back and forth,* for it must always be in the plane of the image. Therefore lE^ = is a variable quantity, while A^B^ is constant for fixed wir^s. Therefore the visual angles at E^ and E^ are variable. If these angles were constant, the space intercepted on the rod, and the distance of the rod from the objective, would be in constant ratio. Since this is not true, we must find the rela- tion that does exist between the distance Efi and the space intercepted on the rod, From equation (i) we have I s /. if. I I I but from equation (2) frf /; * If the objective is moved in focusing it does not appreciably affect these relations. TOPOGRAPHICAL SUR VE YING, 22J S I I ~ifr f f: or ( 3 ) that is, the distance of the rod from the objective is equal to the intercepted space in the rod multiplied by the constant objective, and i is the distance between extreme wires. If the distance between the extreme wires be made o.oi of the focal length of the objective, then the distance of the stadia- rod from the objective (rigidly from is a hundred times the intercepted space on the rod, plus the focal length of the ob- jective. Again, if a base be measured in front of the instrument, with its initial point a distance f in front of the object-glass of the telescope, then the rod may be held at any point on this base-line, and its distance from the initial point, and the space intercepted by the extreme wires, will be in constant ratio. The lines Aft' and in Fig. 6i show this relation, for they are the lines defining the space on the rod which is inter- cepted by the extreme wires as the rod moves back and forth. Evidently the rod cannot approach so near as F , for then the image would be at an infinite distance behind the lens. Usu- ally the extreme position of reticule does not correspond to a position of rod nearer than ten to fifteen feet. It must be remembered that any motion of the eye-piece, with reference to the image and wires, is only made to accom- 228 S UR VE YING. modate different eyes, and has no effect in changing the rela- tion of wire interval and image. The eye-piece is simply a magnifier with which to view the image and wires, but in all erecting-instruments it also reinverts the image so as to make it appear upright. The effect of the eye-piece has no place in the discussion of stadia formulae. If the distance of the stadia is to be reckoned from the centre of the instrument, which it usually is, and if this dis- tance = and the distance from the centre of the instrument to the objective {CE^ in Fig. 6i) =^:, then we have, from (3), ^ = /a + ^ = +/+ ^ (4) Since /, /, and c are constant for any instrument, we may measure f and c directly, and then find the value of i by a single observation. Proceed as follows: 1st. Measure the distance from the centre of the instru- ment (intersection of plumb-line with telescope) to the objec- tive, and call this c. 2d. Focus the instrument on a distant point, preferably the moon or a star, and measure the distance from the plane of the cross-wire to the objective, and call this f. 3d. Set up the instrument, and measure the distance /+ ^ forward from the plumb-line, and set a mark. From this mark as an initial point, measure off any convenient base, as 400 feet. 4th. Hold the rod at the end of this base, and measure the space intercepted by the extreme wires. If we call the length of this base b, and the distance intercepted Sy then we have, from equation (3), or (5) TOPOGRAPHICAL SURVEYING. 229 Here we have the value of i in terms of known quantities. If it is desirable to set the wires at such a distance apart s that-^ will be a given ratio, as^-J-g-, then t must equal o.oif. It is possible to set the wires by this means to any scale, so that a rod of given length may read any desired maximum distance. If it is desired that ^ should be determined with great ac- curacy for a given instrument, with wires already set, so as to have a coefficient of reduction for distance, for readings on a rod graduated to feet and tenths, for instance, proceed as fol- lows : Make two sets of observations for distance and intercepted interval. The distances should differ widely, as 50 feet and 500 feet, or 100 feet and 1000 feet, according to the length of rod used. The shorter distance should not be less than 50 feet, and the longer one not more than 1000 feet with the most favorable conditions of the atmosphere. The distances are to be measured from the centre of the instrument. Make several careful determinations of the wire interval at each position of the rod, and take the mean of all the results at each distance, and call that the wire interval, s, for that distance, d. We then have two equations and two unknown quantities, these latter being and in the formula, equation (4), z Here the d and s are observed, and ^ and (/ -f" found. Knowing these, a table could be prepared giving values of d for any tabular value of s for that instrument. This applies to the reading of distances from levelling-rods. 230 PURVEYING. Some engineers prefer, in this case, to observe the wire interval for various measured distances, from the sliortcst to the longest, to be read in practice, and prepare a table by inter- polation. If the observed positions are sufficiently numerous, this method should give identical results with those obtained by the use of the formula. The two methods may be used to check each other. From equation (4) we see that the distance of the rod from the centre of the instrument is a constant ratio times the intercepted space on the rod, plus a constant c). If diagrams or designs be drawn on the stadia-rod to the i i scale or so that \o j. yards on the rod would correspond to 10 yards in distance, and if the rod were decorated with symbols of this size, then the distance of the rod from the instrument could be read at once by noting how many symbols were intercepted between the wires. To this distance must then be added the small distance (/+ c), which is from 10 to 16 inches in ordinary field-transits. On all side-readings, taken only to locate points on a map, this correction need not be added, as one foot is far within the possibilites of plotting. 201. On the Government Surveys the base is usually measured from the centre of the instrument^ and its length is taken as about a mean of those which the stadia is intended to measure, and the symbols scaled by this reading. Then, of course, the distance read is always in error by a small amount, except when it is the same as the base for which it was gradu- ated. For all shorter distances the reading is too small, and for all greater distances the reading is too large. Sometimes several different lengths of base are taken, as 400, 600, and 800 feet, all from centre of instrument, and a mean value of wire interval used for giving the scale for the diagrams. This is practically the same as the other, for in either case the scale is correct for but a single distance. TOPOGRAPHICAL SURVEYING. 231 The correction to any reading on a stadia so graduated, in order to give the distance from the centre of the instrument, is where K = correction, in feet ; B — distance read on stadia, in feet ; B' = length of base, in feet, for which the stadia was graduated. If B' = 1000 feet, B = 100 feet, and c -\-f= 1.5 feet, then K - 1.5 (I- iVA) = + 1-35 feet. If B had been 2000 feet, then K = 1.5 (i - =•- 1.5 feet. These corrections are not usually applied. 202. Another Method of determining the scale for gradu- ating the rod is to measure the base from the plumb-line, as above, and then, from a fixed point on the lower part of the rod, find the intervals that correspond to various distances, as 100 feet, 200 feet, 300 feet, etc., and mark these on the board, always keeping the lower wire on the fixed, initial point of the rod. Then each lOO-foot space is subdivided into ten equal parts, or symbols ; so that, in reading the rod afterwards, if the lower wire is always set on the initial point, the reading always gives the correct distance from the centre of the instrument. The objection to this method is that the initial point on the rod cannot always be seen, on account of obstructions. 203. Adaptation of Formulae to Inclined Sights. — The previous discussion is applicable to horizontal sights only. 232 SUJ^ VE YING. If the rod be held on the top of a hill, and the telescope pointed towards it, the reading on the rod will give the linear distance from instrument to rod, provided iJic rod be held per- pcndicidar to the line of sight. As it would be inconvenient to do this, let the rod be held vertical in all cases. When the line of sight is inclined to the rod, the space intercepted is increased in the ratio of i to the cos of the angle with the horizon. Thus, the space A' B' (Fig. 62) for the rod perpendicular to the line of sight becomes AB for the rod vertical. But A'B' = AB cos v.^ Let A'B' = r', the reading on the stadia for perpendicular position ; and Let AB = r, the actual reading obtained for a vertical position. Then r' — r cos v. But in equation (4) we have*^, s = r', and therefore r' c tr * This assumes that A' B' is perpendicular to CB and CA, which it is practi- cally, since the angle ACO' is so very small, usually about 15'. TOPOGRAPHICAL SURVEYING. 233 -[“/"is the distance C 0 '\ whereas the distance on the horizon- tal, CO., is generally desired, and for this we have CO = d— CO' cos V — c f) cos v — r cos’ cos V, (7) This is the equation for reducing all readings on the stadia to the corresponding horizontal distances. The vertical distance of O' above O is equal to CO' sin F. But CO' = / + /+ ^ = r cos 2/ + /+ c, hence 00 ' = k = r cos vsmv (/+ c) sin v = sm2v c)smv. ( 8 ) Equation (8) is used for finding the elevation of the point on which the stadia is held above or below the instrument sta- tion. 204. Table V. gives the values d and h computed from these formulae for a stadia reading of 100 feet (or metres, or yards), with varying angles up to 30°. It will be noted that the second term in the right member of equations (7) and (8) is always small, and its value depends on the instrument used. The values of this term are taken out separately in the table ; and three sets of values are given oi(c-\-/), — viz., 0.75 feet, i.oo feet, and 1.25 feet. If the work does not require great accuracy, these small corrections may be omitted. The use of the table directly involves a multiplication for 234 SUR VE YING. every result obtained. Thus, if the stadia reads 460 feet, the angle of inclination 6° 20', and we have f c = i foot, then d = 4.60 X 9878 + 0.99 = 455.4 feet, and h = 4.60 X 10.96 + o.ii = 50.53 feet. The table is not generally used for reductions iox d when the angle of elevation is less than 3 to 5 degrees. When = 5° 44', this reduction amounts to just one per cent. When an error of I in lOO can be allowed, then the reduction to the horizontal would not be used under 6°. If the second term in e-^-pbe also neglected, these two errors tend to compensate ; and if ^ 4“ / the instrument used is i foot, and both these corrections be omitted, they do exactly compensate when the stadia reading is 100 feet, vertical angle 5° 44'. a << 200 it <( 4° 04'. ti u 300 <( << 3° 20'. u u 400 2° 52'. u it 500 (( 2° 32'. u 1000 << 1° 46'. « n 2000 << << 1° 18'. Therefore the reduction to the horizontal need never be made when v is less than 2°, and it generally may be neglected when V is less than 6°. In obtaining the difference of elevation, h, the term in c -j- y may be omitted for all angles under 6° if errors of o.l foot are not important. For elevations on the main line, how- ever, this term should always be included. In practice, therefore, the tables are mostly used to obtain the difference of elevation from the given stadia reading and angle of elevation. TOP OCR A PHICA L SURVE YING. 235 205. Reduction Diagram. — Since the use of these tables involves a multiplication each time, and since a table for vary- ing distances and angles would be very voluminous, it is prefer- able to take out the elevations from a diagram. Such a diagram has been prepared, to be used in place of the table. It is ar- ranged with both coordinates in feet, but can be used for both coordinates in metres, since the same unit is used for both. It will only be neccessary to re-number the divisions, to adapt it to the new scale. This diagram has been prepared with great care, and is arranged to give distances to 500 yards or metres, or 1500 feet, with elevations to 50 feet. For longer distances or higher elevations for a single pointing, the results may be obtained from the table. Elevations are taken off from the diagram to the nearest tenth of a foot, with great readiness ; as the smallest spaces are 2 millimetres square, and these correspond to two-tenths of a foot in elevation. It is of more convenient use than extended tables, and is just as accurate ; the nearest tenth of a foot being quite as exact as one is warranted in writing elevations when obtained in this manner. Corrections to the distances read are also obtained from this diagram for large vertical angles.* THE INSTRUMENTS. 206. The Transit. — That the transit may be best adapted to this work, there are certain features it should possess, though all of them are by no means essential. They will be named in the order of their importance. 1st. The horizontal limb should be graduated from zero to 360°, preferably in the direction of the movement of the hands of a watch. * The diagram is printed on heavy lithographic paper 20 by 24 inches, from an engraved plate, and can be had from the publishers of this volume. Price 50 cents, post paid. 236 SURVEYING. 2d. The instrument should have a vertical circle rigidly at- tached to the telescope axis, and not simply an arm that is fastened by a clamp-screw, and which reads on a fixed arc be- low. So much depends on the vertical circle holding its adjust ment that its arrangement should be the best possible. Since the telescope is not transited, the vertical circle need not be complete. 3d. The telescope should be inverting, for two reasons : first, in order to dispense with two of the lenses, and so obtain a better definition of image ; and, second, that the objective may have a longer focal length, thus giving a flatter image and a less distorted field. 4th. The stadia wires should be fixed instead of adjustable, as in the latter case they are not stable enough to be reliable. 5th. The bubbles on the plate of the instrument should be rather delicate, so that a slight change in level may become apparent. They should also hold their adjustments well. This is very important, in order that the readings of the vertical angles may be reliable. It is also of great importance in carrying azimuth where the stations are not on the same level. 6th. The horizontal circle should read to thirty seconds ; and there should be no eccentricity, so that one vernier-read- ing shall be practically as good as two. yth. The instrument (or tripod) should have an adjustable centre, for convenience of setting over points. 8th. A solar attachment to the telescope will be found very convenient. In most regions the azimuth can be checked up by the reading of the needle, but in many places this is not reliable. 207. Setting the Cross-wires. — The engineer should al- ways have at hand a spider’s cocoon of good wires, and a small bottle of thick shellac varnish. If the dry shellac is carried it may be dissolved in alcohol. If no such cocoon is at hand a spider may be caught and made to spin a web. The small, TOPOGRAPHICAL SUR VE YING. 237 black, out-door spider makes a good web for stadia purposes. A new wire should be allowed to dry for a few minutes, and an old one should be steamed to make it more elastic. The wires for stadia-work should be small, round, and opaque. Some wires are translucent, and some are flat and twisted like an auger-shank. Scratches must be made across the face of the reticule where the wires are to lie. These must be made with great care, so as to have them equally spaced from the middle wire, parallel to each other, and perpendicular to the vertical wire. The distance apart of the extreme wires is to be computed by equation (5) for any desired scale on the rod. Take a piece of web on the points of a pair of dividers, by wrapping the ends several times about the points, which should be separated by about an inch ; stretch the wire, by spreading the dividers, as much as it will bear; and lay the dividers across the reticule in such a way that the web comes in place. The dividers must be supported underneath, so that the points will drop just a trifle below the top of the reticule ; otherwise they would break the web. Move the dividers until the web is seen, by the aid of a magnifying-glass (the eye-piece will do), to be in exact position. Then take a little shellac on the end of a small stick or brush, and touch the reticule over the web, being careful to have no lateral motion in the movement. The shellac will harden in a few minutes, when the dividers may be removed. Shellac is not soluble in water. 208. Graduating the Stadia-rod. — The stadia-rod is usually a board one inch thick, four or five inches wide, and twelve to fourteen feet long. Sometimes this is stiffened by a piece on the back. To graduate the rod, it is necessary to know what space on the rod corresponds to a hundred feet (or yards, or metres) in distance. Either of the three methods cited on pp. 230-1 may be used for doing this, but the first is recommended. Thus, measure off c /in front of the plumb- 238 SUI^VILYING. line, and set a point. From this point measure off any con- venient base, as 200 yards, on level ground, and hold the blank rod (which has had at least two coats of white paint), at the end of this base-line. Have a fixed mark or target on the upper part of the rod, on which the upper wire is set. Have an assist- ant record the position of the lower wire as he is directed by the observer. Some sort of an open target is good for this pur- pose, but any scheme is sufficient that will enable the observer to fix the position of the extreme wires at the same moment with exactness. This work should be done when there is no wind, and when the atmosphere is very steady : a calm, cloudy day is best. Repeat the operation until the number of results, or their accordance, shows that the mean will give a good result. If the base was 200 yards long, divide this space into two equal parts, then each of these parts into ten smaller parts, and finally each small space into five equal parts; and one of these last divisions represents two yards in distance. Dia- grams are then to be constructed on this scale, in such a way that the number of symbols can be readily estimated at the greatest distance at which the rod is to be read. The individ- ual symbols should be at least three inches across ; so that, if one of these is to represent teit ufiits^ as yards or metres, then 100 units will cover 2^ feet, and a rod 14 feet long will read a distance of 560 units (yards or metres). If it is desired to read distances of a quarter of a mile or more, the rod should be graduated to read to yards (or five-foot units, or metres) ; but if it is not to be used for distances over 500 to looo feet, it might be graduated to read to feet. This question must be decided before the wires are set, and then they must be spaced accordingly. In measuring the base, care should be taken to test the chain or tape carefully by some standard. If the rod is to be graduated to read to feet, of course the base should be some even hundreds of feet, as 600. TOPOGRAPHICAL SUR VE YING. 239 In Fig. 63 are shown four designs for stadia-rods which have been long in use, and are found to work well. They are intended to be all in black on a white ground.^* It will be noticed that the shortest lines in these diagrams all cover a space of two units on the rod. In diagrams 2 and 3 the units are either yards or metres, while in i they are units of five feet each. In diagram 4 the units are of two feet each. The is 4-1' Fig. 63. successive units are found at the middles and limits of these lines and spaces. Wherever the wire falls, there should be a white ground on some part of the cross-section ; and the more white ground the better, provided the figures are distinct. The black paint may be put on heavy, so that one coat will be sufficient. The 50- and lOO-unit marks should be distinguished by special designs. There should usually be at least two boards with each instrument, and sometimes three and four are needed. Of course, these are all duplicates. After the unit scale is obtained, or the space on the rod corresponding to a hundred * Some engineers prefer red on the loo-unit figures. 240 SUR VE YING. units in distance, these lOO-unit spaces should be so distributed as to be symmetrical with reference to the C7ids of the rod. The reason of this will appear later. Having determined how many lOO-unit spaces there will be on the rod, fix the position of the two end lOO-unit symbols with reference to this symmetry, and then the rod is subdivided from these points. Special pains should be taken to have the angular points of the diagrams well defined and in position. These points are on the lines of subdivision of the rod. After one rod is subdivided, the others of that set may be laid alongside, and all fastened rigidly together ; and then, by means of a try-square or T -square, the remaining rods may be marked off. The wire interval should be tested every few months by remeasuring a base, as was done for graduation, and reading the rod on it, to see if this shows the true measured distance. This is to provide against a possible change in the value of the wire interval. If the wires are stretched reasonably tight when they are put in, they seldom change, If they are too loose, they swell in wet weather, and may sag some. The reticule should be so firm that the variable strain on the adjusting- screws will not distort it appreciably. If the wire interval is found to have changed, either the rods must be regraduated, or else a correction must be made to all readings of importance. What are called the “ side shots/* which make up a large proportion of the readings taken, would not need to be corrected. If the wires are adjustable, any unit scale may be chosen at pleasure, and the wires adjusted to this scale. Then, if the intervals change, the matter is corrected by adjusting the wires. The adjustable wires are generally used to obtain dis- tances from levelling-rods, where it is desirable that each foot on the rod shall correspond to a hundred feet in distance. For the ordinary stadia-rods, fixed wires are preferable. TOPOGRAPHICAL S UR FRYING. 241 GENERAL TOPOGRAPHICAL SURVEYING. 209. The Topography of a region includes not only the character and geographical distribution of the surface-cover- ing, but also the exact configuration of that surface with reference to its elevations and depressions. Thus any point is geographically located when its position with reference to any chosen point and a meridian through it is found, but to be topographically located its elevation above a chosen level surface must also be known. A topographical survey consists in locating by means of three coordinates a sufficient number of points to enable the intervening surface to be known or inferred from these. Evidently the points chosen should be such as would give the greatest amount of information. As for geographical outline, the corners, turns, or other critical points are chosen, so for configuration the points of changes in slope, as the tops of ridges and bottoms of ravines, or the brow and foot of a hill, are chosen as giving the greatest information. 210. Field-work. — Let it be required to make a topo- graphical survey of either a small tract, a continuous shore- line, or of a large area, for the purpose of making a contour map. In case of the small tract, any point may be taken as a point of reference, and the survey referred to it as an origin. In case of an extended region, a series of points should be determined with reference to each other, both in geographical position and in elevation. These determined points should not be more than about three miles apart. The points of ele- vation or bench-marks need not be identical with those fixed in geographical position. These last are best determined by a system of triangulation, and are called “ triangulation stations." In the succeeding discussion, the symbol A will be used for triangulation station, and B.M. for bench-mark, 16 242 SUR VE YING, Firsts a system of triangulation points is established, the angles observed, azimuths and distances computed, and the stations plotted to scale on the sheet which is to contain the map. This plotting is best done, for small areas, by comput- ing the rectangular coordinates (latitudes and departures), and plotting them from fixed lines which have been drawn upon the map, accurately dividing it into squares of lOOO or 5000 units on a side. They may, however, be plotted directly from the polar coordinates (azimuth and distance) as given by the triangulation reduction. For this purpose, the sheet on which the map is first drawn, called the field sheet, should have a protractor circle printed upon it, about twelve mches hi diam- eter, These protractor sheets of drawing-paper can be obtained of most dealers in drawing-materials, or the protractor circle may be printed to order on any given size or quality of paper.* These protractor circles are very accurate,, and are graduated to 15' of arc. Plotting can be done to about the nearest 5'. Second, a line of levels is run, leaving B.M.’s at convenient points whose elevation are computed, all referred to a com- mon datum. If the A’s are not also B.M.’s, then a B.M. should be left in the near vicinity of each A. This is not essential, however. Third, the topographical survey is then made, and referred to, or hung upon, this skeleton system of A’s and B.M.’s. The topographical party should consist of the observer, a recorder, two or three stadia-men, and as many axemen as may be necessary, generally not more than two. The azimuth, preferably referred to the true meridian, is known for every line joining two A’s, as well as the length of such line. Set up the transit over a A, and set the horizontal circle * Messrs. Queen & Co., Philadelphia, or Blattner & Adam of St. Louis, can furnish such sheets. TOPOGRAPHICAL S UR VE YING. 243 (which should be graduated continuously from 0° to 360° in the direction of the hands of a watch) so that vernier A will read the same as the azimuth of the triangulation line by which the instrument is to be oriented. Clamp the plates in this position, and set the telescope to read on the distant A. Now clamp the instrument below, so as to fix the horizontal limb, and unclamp above. The azimuths of the triangulation lines are generally referred to the south point as the zero, and in small systems of this sort the forward and back azimuths are taken to be 180° apart. When the instrument has been set and clamped, all subsequent readings taken at that station are given in azimuth by the readings of vernier A on the horizon- tal limb. For any pointing, therefore, the reading of this vernier gives the azimuth of the point referred to the true meridian, and the rod reading gives the distance of the point from the instrument station. These enable the point to be plotted on the map. To draw the contour lines, elevations must also be known. If the elevation of the A is known, measure the height of instrument (centre of telescope) above the A on the stadia,* as soon as the instrument is levelled up over that station. Sup- pose this comes to the 212-unit mark. Write in the note-book, as a part of the general heading for that station, “ Ht. of Inst. = 212.” Then, for all readings from that station for eleva- tions, bring the middle horizontal wire to the 212-unit mark on the rod, and read the vertical angle. From this inclination and distance, the height of the point above or below the instrument station is found- If the rod be graduated sym- metrically with reference to the two ends, one need not be careful always to keep the same end down, and so errors from this cause are avoided. * Or, if preferred, a light staff, about five feet long, may be carried with the instrument for this purpose, it being graduated the same as the stadia rods for this instrument. 244 SUJ^ VE YJNG. The record in the note-book consists of — 1st. A Description of the Point, as, “ N.E. cor. of house,” “intersec. of roads,” “top of bank,” “ C.P.” for “contour point,” which is taken only to assist in drawing the contours, “ □ i6 ” for “ stadia station i6,” etc. 2d. Reading of Ver. A. 3d. Distance. 4th. Vert. Angle. These four columns are all that are used in the field. There should be two additional columns on the left-hand page, for reductions, viz. : 5th. Difference of elevation, corresponding to the given vertical angle and distance, and which is taken from a table or diagram. 6th. Elevatioji. which is the true elevation of each point referred to the common datum. The right-hand page should be reserved for sketching. It will be found most convenient to let the sketching pro- ceed from the bottom to the top of the page ; as in this case the recorder can have his book properly oriented as he holds it open before him, and looks forward along the line. The notes may advance from top to bottom, or vice versa, as de- sired. If there are many “ side shots” from each instrument station, one page will not usually contain the notes for more than two stations, and sometimes not even for one. The sketch is simply to aid the engineer when he comes to plot the work, and may often be omitted altogether. One soon becomes accustomed to impressing the characteristics of a landscape on his memory so as to be able to interpret his notes almost as well as though he had made elaborate sketches. For beginners the sketches should be made with care. The observer should usually make his own sketches and plot his own work. After the instrument is oriented over a station, and its TOPOGRAPHICAL SUR VE YING. 245 height taken on the stadia, the stadia-men go about holding the rods at all points which are to be plotted on the map, either in position or in elevation, or both. The choice of points depends altogether on the character of the survey ; but since a single holding of the rod gives the three coordinates of any point within a radius of a quarter of a mile, it is evident the method is complete, and that all necessary information can thus be obtained. For very long sights, the partial wire inter- vals (intervals between an extreme and the middle wire) may be read separately on the stadia, and in this way twice as great a distance read as the rod was designed for. The limit of good reading is, however, usually determined by the state of the atmosphere, rather than by the length of the rod. When the air is very tremulous, good readings cannot be made over distances greater than 500 feet ; while, when the atmosphere is very steady, a half-mile may be read with equal facility. Before the instrument is removed from the first station, the forward stadia-man selects a suitable site for the next instrument station (generally called stadia station, and marked □, to distinguish it from a triangulation station. A), and drives a peg or hub at this point. This peg is to be marked in red chalk, with its proper number, and should have a taller mark- ing-stake driven by the side of it. The peg for the □ should be large enough to be stable ; for it must serve as a reference point, both in position and elevation, during the period of the survey. It is often desirable to start a branch line, or to duplicate some portion of the work, with one of these stations as the starting-point ; and, since each □ is determined, in position and elevation, with reference to all the others, one can start a branch line from one of these as readily as from a A. It is not usually necessary to put a tack in the top, but the centre may be taken as the point of reference. The stadia- man first holds his stadia carefully over the centre of this □, with its edge towards the instrument, so as to enable the 246 SURVEYING. observer to get a more accurate setting for azimuth. The observer could just as well bisect the face of the rod ; but, if held in this position, the centre of the rod may not be so nearly over the centre of the peg as when held edgewise. This holding of the rod edgewise for azimuth checks the care- lessness of the stadia-man, and is done only for readings on instrument stations. At a signal from the observer, the stadia is turned with its face to the instrument, and the observer reads the distance and vertical angle. It is advisable, in good work, to re-orient and relevel the instrument just before reading to the forward □. The transit is very apt to get out of level after being used for some time, with more or less stepping around it, and the limb may have shifted slightly on the axis, both of which might be so slight as to make no material difference for the side readings, but which would be important in the continued line itself. It is best, therefore, to level up again, and reset on the back station, before reading to the forward one. If it is inconvenient for the rear rodman to go back to this station to give a reading, a visible mark should be left there, to enable the observer to reset upon it for azimuth, as it is not necessary to read distance and vertical angle again. When the instrument is moved, it is set up over the new station, and the new height of instrument determined and recorded. The rear stadia-man is now holding his rod, edge- wise, on the station just left ; and by this the observer orients his instrument, making vernier A read 180° different from its previous reading on this line. Clamping the plates at this reading, the telescope is turned upon the rod on the back sta- tion, and the lower plate clamped for this position. The circle is now oriented, so that, for a zero-reading of vernier A, the telescope points south. It will be noted that the telescope is never reversed in this work. TOPOGRAPHICAL SURVEYING. 247 The distance and vertical angle should both be reread, on this back reading, for a check. If the vertical circle is not in exact adjustment, this second reading of the vertical angle will show it, for the numerical value of the angle should be the same, with the opposite sign. If they are not the same, then the numerical mean of the two is the true angle of elevation, and the difference between this and the real readings is the index error of the vertical circle. This error may be corrected in the reduction, or the vernier on the vertical circle may be adjusted. The second reading of the vertical angle on the stadia- stakes is thus seen to furnish a constant check on the adjust- ment of the vertical circle, and should therefore never be neglected. If the circle is out of adjustment by a small amount, as one minute or less, in ordinary work it would not be necessary either to adjust it or to correct the readings on side-shots, for the elevations of contour points are not required with such extreme accuracy. The mean of the two readings on stadia-stakes would still give the true difference of elevation between them, so that there would be no continued error in the work. The work proceeds in this manner until the next A is reached. In coming to this station, it is treated exactly as though it were a newH; and the forward reading to it, and the back reading from it, are identical with those of any two consecutive IZl's. Having thus occupied the second A, and having oriented the instrument by the last □. turn the tele- scope upon some other A whose azimuth from this one is known. The reading of vernier A for this pointing should be this azimuth, and the difference between this reading and the known azimuth of the line is the accumulated error in azimuth due to carrying it over the stadia line. This error should not exceed five minutes in the course of two or three miles in good work. 248 SUR VE YTNG. The check in distance is to be found from plotting the line, or from computing the coordinates of the single triangulation line, and also of the meandered line, and comparing the re- sults. The elevations are checked by computing the elevation of the new A from the stadia line, and comparing this with the known elevation from the line of levels. In case the elevations of the A’s are not given, but only certain B.M.’s in their vicinity, then the check can be made on these just the same. Thus, in starting, read the stadia on the neighboring B.M., and from this vertical angle compute the elevation of the A over which the instrument sets, and then proceed as before. In a similar manner, the check for eleva- tion at the end of the line may be made on a B.M. as well as on the A. A quick observer will keep two or three stadia-men busy giving him points; so that in flat, open country, with long sights, it may be advisable to have three or even four stadia- men for each instrument. In hilly country more time will be required in making the sketches, and hence fewer stadia-men are required. After the instrument is oriented at each new station, the needle should be read as a check. To make this needle-read- ing agree with the readings of the verniers on the horizontal circle (the north end with vernier A, and the south end with vernier B, for instance), graduate an annular paper disk the size of the needle-circle, and figure it continuously from 0° to 360°, in the reverse direction to that on the horizontal limb of the instrument, and paste it on the graduated needle-circle in such a position that the north end of the needle reads zero when the telescope is pointing south. If the variation is 6° east, this will bring the zero of the paper scale 6° east of south on the needle-circle. This position of the paper circle is then good within the region of this variation of the needle. When TOPOGRAPHICAL SURVEYING. 249 the survey extends into a region where the variation is differ- ent, the scale will have to be reset. With these conditions, when the instrument is oriented for a zero-reading when the telescope is south, the reading of the north end of the needle will always agree with the reading of vernier A, and the south end with vernier B. It is so easy a matter to let the needle down, and examine at each □ to see if this be so, that it well pays the trouble. No record need be made of this reading, as it is only used to check large errors. 2II. Reducing the Notes. — The only reduction necessary on the notes is to find the elevation of all the points taken, with reference to the fixed datum, and sometimes to correct the distance read on the rod for inclined sights. The difference of elevation between the El and any point read to, as well as the correction to the horizontal distance, can be taken from Table V. or from the diagram. The methods of using these have been explained (see pp. 234-5). After the differences of elevation are taken out, the final elevations of the points are to be computed by adding algebraically the difference of eleva- tion to the elevation of □. The following is a sample page with these reductions; 250 SURVEYING. Gazzam, Observer. Baikr, Recorder. Elevation = 24'. 94. April 20, 1883. At □ 4. Ht. of Inst. = 87. Object. Azimuth. Ver. A. Distance. Vert. Angle. Difference of Elevation. Eleva- tion above Datum. yds. □ 3 328“ 10' 199 — 0° 10' — i '.56 — Bridge 127° 40' 70 4-0® 32' + i '-9 26'.8 S.E. cor. of house 142“ 35' 90 4-0*’ 15' + i '-2 26'. I On road 180° 25' 114 + 0® 7' + o'. 7 25'. 6 Water-level, foot of hill. . . . 230° 15' 224 - 0® 57' — 10'. 9 14'. 0 □ 5 128° 33' 30* 216 + 0° 55' -f- io'.38 — C.P 190° 48' 210 fi® 2' -fii '.4 36'. 3 At □ 5. Ht. of Inst. = 78. Mean = + io'.26. 35' .20. 0 4 308° 33' 30’ 215 - 0® 54' — 10'. 13 — S.W. cor. of house 43“ 30' 104 + 3" 3' -f-16'.o 5 i '-2 Edge of bank 332° 10' 98 + i° 57 ' -f-io'. I 45'.3 S.E. cor. of R.R. station. . . 85“ 30' 158 + 1° 2' + 8'.5 43'. 7 Railroad track 43 ° 55' 40 + 2° 53 ' + 6'.o 4 i'.2 “ “ 79 ° 30' 270 + 0® 9' -f- 2 '.I 37'- 3 06 79 ° 30' 200 — 0° 2' — o'. 36 — At 0 6. Ht. of Inst. = 79. Mean = - o'. 54 . 34 '' .66. 0 5 259° 30' 200 + 0° 4' + o'. 72 — Cor. of house 277° 55' II 2 + 3° 26' +i 9'-7 54'-4 Top of hill 87° 25' 198 -f- 4° 48' + 49'- 3 84'. 0 Wagon road 58° 15' 186 + 4 ° 25' + 42'.9 77'. 6 40° Vl' 216—3 4 - 6 ° rr' + 73'.53 H'-' 0 / 213 C.P 41° 45' III -f 4 ° 41' -f-27'.o 61'. 7 0 7 5° 25' 194 -|-o° 12' + 2'. 04 — TOPOGRAPHICAL SURVEYING. 251 It will be noted that the reading on □ 5 from □ 4 has a distance of 216 yards, and a vertical angle of -{-0° 55'; while on the back reading, from □ 5 to □ 4 the distance is 215 yards, and the vertical angle — 0° 54'. The distance was probably between 215 and 216 yards, and the vertical circle was prob- ably slightly out of adjustment. The difference of elevation is taken out for both cases, however, being respectively 10.38 feet and 10.13 feet. The mean of these is 10.26 feet, which stands as a part of the general heading at Q 5. The true elevation of 0 5 is then found by adding 10.26 to 24.94, giving 35.20 feet, which is also set down as part of the general heading. The elevations on the side-readings from this station can now be taken out. These side-elevations are only used for obtaining the contours, and hence are only taken out to tenths of a foot. When the contours are ten feet apart or more, these side-elevations need only be taken out to the nearest foot. The elevations of the stadia stations should, however, always be taken out to hundredths, to prevent an accumula- tion of errors in the line. The reduction for distance may also be taken from that portion of the diagram arranged for this purpose. This is used the same as the other portion ; and the correction is found, which is to be always subtracted from the rod-reading. Thus, in the reading on □ 8 from 0 6, we have a reading of 216 yards, and a vertical angle of 6° 33'. The correction here is 2.16 X 1-3 = 2.8 yards, as found from the table. Calling this 3 yards it is subtracted from the 216, leaving 213 yards as the distance to be plotted. It is only the stadia-line distances that need ever be corrected in this way, the corrections being usually so small that it is not important on the side-shots. It will be noted that two 0 ’s were set from 0 6. This was done because a branch-line was run from 0 6 over the bluffs. In order to make it unnecessary to occupy 0 6 again 252 SURVEYING. when the branch-line came to be run, □ 8 was set while 0 6 was occupied in the main-line work. When the branch-line came to be run, the instrument was taken directly to Ci] 8, and oriented on □ 6 by the readings previously taken from Q 6. The right-hand page of the note-book, opposite the notes given above, is occupied with a sketch of the locality, with the □’s marked on, the general direction of the contour lines, the railroad, stream, houses, etc.* 212. Plotting the Stadia Line. — It is customary to first plot the stadia stations alone, from one El to the next, to find whether or not it checks within reasonable limits. This part of the work should be done with extreme care, so that if it does not check it cannot be attributed to the plotting. In case it does not check within the desired limit, then the line of investigation will be about as follows until the error is found: 1st. Replot the stadia line. 2d. Recompute and replot the triangulation line. 3d. By examining the discrepancy on the plot, try and decide whether the error is in azimuth or distance, and, if possible, where such error occurred, and its amount. 4th. Examine the note-book carefully, and see if there is any evidence of error there. 5th. If there is a large probability that the error is of a certain character, and that it occurred at a certain place, take the instrument to that station, set it up, and redetermine the azimuths or distances which seem to be in error. 6th. If there is no high probability of any certain errors to be examined for in this way, then go back and run the line over, taking readings on \I\'s only. If the elevations had been found to check, the vertical angles may be omitted on this duplicate line ; and, on the other hand, if the plot came out all right, but the elevations could not be made to check, then a duplicate line must be run to determine this alone ; and in this * These notes were taken from a field-book of a topographical survey of Cr^:ve Coeur Lake by the engineering students of Washington University. TOPOGRA PHICAL SUR VE YING. 253 case the vertical angles between H’s are all that need be read. In cases of this kind, it will be found a great help to have the El’s so well marked that they can be readily found. With reasonable care in reading and in the handling of the instrument, it will never be necessary to duplicate a line entire, for all readings between H’s are checked. The vertical angles and distances are checked by reading them forward and back over every stadia line; and the azimuth is checked by the needle readings, and also when the second A is reached. If, in the progress of the work, the readings on the back Q for distance and vertical angle do not fairly agree with these quantities as read from the previous station, the recorder should note the fact : and the observer should then re-examine these readings ; and, if found to be right, the first readings, taken from the other station, should be questioned, and the mean not taken in the reduction. For plotting the stadia lines a parallel ruler (moving on rollers) is very desirable ; otherwise, triangles must be used. The plotting is done by setting the parallel ruler or triangle on the proper azimuth as found from the protractor printed on the sheet, moving it parallel to itself to the station from which the point is to be plotted, and drawing a pencil line in the right direction. Then, with a triangular scale, — or, better, with a pair of dividers and a scale of equal parts, — lay off the correct distance on this line ; and this gives the point. If the instrument was oriented in the field for a zero read- ing for a south pointing, then the protractor on the sheet must have its south point marked zero, and increase around to 360° in the same direction in which the limb of the instrument in- creases, preferably in the direction of the movement of the hands of a watch. 213. Check Readings. — To enable the observer to locate large errors in azimuth or distance, or both, it is a good prac- tice to take azimuth readings to a common object from a series of consecutive stations, if such be possible. If the plot does 254 Sl/J^ VE YIA^G. not close, go back and plot in these azimuths ; and if there has been no error in azimuth or distance between IZl’s, and no error in reading the azimuths for these pointings, then all these lines will meet in a common point on the plot. If all but one in- termediate line meet at a point, then the error probably was in reading the azimuth of this pointing alone. If several of the first pointings intersect in a point, and the remaining point- ings of the set taken to this object intersect in another point, then it is highly probable that the error was in reading the azimuth or distance of the line connecting these two .sets of IZl’s ; and the relative position of the points of intersection will enable the observer to decide whether the error was in azimuth or distance, and about how much. If, in this way, the error be located, the instrument can be taken to this point, and the readings retaken. 214. Plotting the Side-readings. — Having plotted the stadia line and made it check, the next step is to go back and plot in the side-readings. For doing this, a much more rapid method may be used than that described above. Divide the sheet into squares by horizontal and vertical lines spaced uniformly at from 1000 to 5000 units apart, ac- cording to scale. These lines are to be used for orienting the auxiliary protractor, and also to test the paper for stretch or shrinkage. The side-readings are now plotted by the aid of a paper protractor, such as is shown in Fig. 64. This is made frorn a regular field-protractor sheet. The graduated circle printed on the sheet is used ; and this is some 12 inches in diameter, and graduated to 15 minutes. The sheet is trimmed down to near the graduated circle, and the edges divided, as shown in the figure, to any convenient small scale.* This sheet is to be * It is sometimes desirable to make the open space DFE rectangular and graduate the sides of the space ABF instead of the outer edges. The pro- tractor can then be used nearer the edge of the sheet. TOPOGRAPHICAL SURVEYING. 255 laid upon the plot, with its centre, C, coinciding with the □. It is oriented by bringing the corresponding spaces on opposite edges to coincide with any one of the spaced lines on the plot. This circle then has its position parallel to that of the protractor circle printed on the sheet, and an azimuth taken from the one will agree with an azimuth taken from the other. When this auxiliary protractor has been so centred and oriented, let it be held in place by weights. Now the part ADEB folds back, on the line AB, into the position indi- cated by the dotted lines. The portion DEE is cut out en- tirely, so that when the flap is turned back the space AFB is left open. This space is to be large enough to include the longest side-readings when plotted to scale ; that is, the radius, CF, of the circle to the scale of the drawing must exceed the longest readings. We now have a protractor circle about the □, with this station for its centre. Take a triangular scale, select the side to be used in laying off the distances, and paste a piece of strong paper on the lower side at the zero point. Make a needle-hole through this 2S6 SURVEYING. paper close to the edge, at the zero of the scale, h'astcn a needle through this hole into the point which marks the exact position of the □. The scale can now swing freely around the needle, on the auxiliary protractor ; and its zero remains at the centre of the station from which the points are to be plotted. To plot any point, swing the scale around to the proper azimuth, and at the proper distance mark with the pencil the position of the point. If this marks a feature of the land- scape, it should be drawn in at once, before going farther; and if the elevation of the point will be needed in sketching the contours, this should also be written in. For contour points, the elevation is all that is put down. In this manner the points can be plotted very rapidly. A six-inch triangular scale, divided decimally, will be found best for this. If there is very much of this work to be done, it might be found advisable to have a special scale constructed for the { l<]m7T linn II II I'll TTpTTiiinr II III I'll' Til niMliri'iT ■mn — 1 .2 3 4 5 6 Fig. 65, purpose. Fig. 65 is one form of such a scale drawn one-third size, which would be found very convenient and cheap. It should be graduated on a bevel edge, and to such a scale that the units of distance used on the rod may be plotted to the scale of the drawing. The small needle-hole, in line with the graduated edge, should be only large enough to fit the needle-point used, so that there would be no play. The rule then turns on an accurate centre, which will not wear. Such scales, six inches long, could be constructed very cheaply of German silver by any instrument-maker. TOPOGRAPHICAL SUR VE YING. 257 A special form of protractor, shown in Fig. 66, has also been used with great success in France and on the Mississippi River surveys. '' It is essentially a semicircular protractor, provided with 258 SUR VE YING. a needle-pointed pivot at its centre, and having the straight edge graduated so that distances can be measured off each way from the pivot ; the angular deflection is given by the graduated circle, reading from a point marked on the paper. The bottom of the plate is flush with the bottom of the pro- tractor, and the hole F is at the centre, and should be only large enough to admit a fine needle. The screw D has a hole drilled in its axis to admit the needle-point. It is also split, so that when it is screwed down it will clamp the needle firmly. If the latter is broken, it can readily be replaced by a new one. In addition to the scale on the beveled edge, a diagonal scale is also provided as shown. This instrument combines all the requisites for rapid and accurate plotting of points located by polar co-ordinates or by intersections. In using this protractor the needle-point is placed at, say, the first station, and pressed firmly down. A meridian line is then decided upon, and a point is marked on it at the outer edge of the protractor circle. This will be the initial point from which the angles will be read. As azimuth is read from the south around by the west, it is plain that the circle, numbered as shown and revolved about the pivot till the proper reading coincides with the meridian line, will give the direction of the required point along the graduated diameter, while from the latter the distance can be pricked off. A point can be plotted in any direction without lifting the protractor from its position. In going to the second station it is not necessary to draw a meridian line through it. The azimuth between the first and second stakes being known, if the pivot be set at the lat- ter, and the protractor revolved so that the straight edge coin- cides with the line passing through the two stakes, then the point on the circle corresponding to the azimuth of the line will be a point on the meridian line. This point being marked on the paper is the origin for the angles plotted from the TOPOGRAPHICAL SUR VE YING. 259 second station, and it is evident that they will bear the proper relations to the points plotted from the first station. Other methods are employed for plotting the side shots, such as solid half-circle protractors, of paper or horn, weighted in position, with their centres over the station. This is ori- ented on a meridian drawn through the point, and then all the points plotted whose azimuth falls between 0° and 180°, when the protractor is laid over on the other side, and the remaining points plotted. In this case the ruler is laid across the pro- tractor, with some even division at the station. This method is more troublesome, less rapid, and defaces the drawing more, than the other methods given above. The plotter should have an assistant to read off to him from the note-book. When all the elevations have been plotted, the contour lines are sketched in. The plotting should keep pace with the field-work as close- ly as possible, being done at night and at other times when the field-work is prevented or delayed. In difficult ground the map could be carried into the field and the contours sketched in on the ground. At least the stadia lines should be plotted up and checked before the observer leaves the immediate local- ity. Where the elevations are checked on B.M.’s, these checks should be immediately worked out. This much, at least, could be done each evening for that day’s work. 215. Contour Lines. — In engineering drawings the config- uration of the surface is represented by means of contour lines. A contour line is the projection upon the plane of the paper of the intersection of a horizontal, or rather level, plane with the surface of the ground. These cutting level planes are taken, five, ten, twenty, fifty, or one hundred feet apart vertically, beginning with the datum-plane, which is usually taken below any point in the surface of the region. Mean sea-level is the universal world’s datum which should always be used when a reasonably accurate connection with the sea can be ob- 26o SURVEYING. tained.* Such contour lines are shown on Plate 11 . The proper drawing of these contours requires some accurate knowledge of the surface to be depicted, aside from the elevations of isolated points plotted on the map. This knowledge may consist of a vivid mental picture of the ground, derived from personal ob- servation, or it may be gained from sketches made upon the ground. Even with this knowledge the draughtsman must keep vividly in mind the true geometrical significance of the contour line, in order to properly depict the surface by this means. The ability to draw the contour lines accurately on a field-sheet is the severest test of a good topographer. They are first sketched and adjusted in pencil and then may be drawn in ink. A few fundamental principles may be stated that will assist the young engineer in mastering this art. 1. All points in one contour line have the same elevation above the datum-plane. 2. Where ground is uniformly sloping the contours must be equally spaced, and where it is a plane they are also straight and parallel. 3. Contour lines never intersect or cross each other. 4. Every contour line must either close upon itself or ex- tend continuously across the sheet, disappearing at the limits of the drawing. It cannot have an end within these limits (an apparent exception, though not really one, is the following). 5. No contour should ever be drawn directly across a stream or ravine. The contour comes to the bank, turns up stream, and disappears in the outer stream line. If the bed of the stream, or ravine, ever rises above this plane, then the contour crosses it ; but in the case of a stream the crossing is never actually shown. In the case of a ravine the crossing is shown, if points have been established in its bed. 6. Where a contour closes upon itself, the included area * See in Chapter XIV. , Precise or Geodesic Levelling, p. 563. TOPOGRAPHICAL SUR VE YING. 261 is either a hill-top or a depression without outlet. If the latter, it would in general be a pond or lake. In other words, such contours enclose either maximum or minimum points of the surface. 7. If a higher elevation seems to be surrounded by lower ones on the plot, it is probably a summit ; but if a lower eleva- tion seems to be surrounded by higher ones, it is probably a a ravine, or else an error ; otherwise it is a depression without outlet, in which case there would probably be a pool of water shown. 8. Contour lines cut all lines of steepest declivity, as well as all ridge and valley lines, at right angles. 9. Maximum and minimum ridge and valley contours must go in pairs ; that is, no single lower contour line can intervene between two higher ones, and no single higher contour line can intervene between two lower ones. 10. Vertical sections, or profiles, corresponding to any line across the map, straight or curved, can be constructed from a contour map, and conversely a contour map may be drawn from the profiles of a sufficient number of lines. 11. Each contour is designated by its height above the datum-plane, as the fifty-foot contour, the sixty-foot contour, etc. In flat country, where the contour lines are few and wide apart, always put the number of the contour on the higher side, otherwise it sometimes may be impossible to tell on which side is the higher ground. 12. In taking surface-elevations for determining contour lines, points should always be taken on the ridge and valley lines, and at as many intermediate points as may be desirable. There are two general systems of selecting these points. By one system points are chosen approximately in lines or sec- tions cutting the contours about at right angles, the critical points being the tops and bottoms of slopes ; while by the other system points are selected nearly in the same contour line, — that is, on the same horizontal plane, — the critical points 262 SURVEYING. being the ridge and valley points, these being the points of maximum and opposite curvature in the contour lines them- selves. By the second method one or two principal contours maybe followed continuously, the points being taken as nearly as may be on these contour lines. If such principal contours are 50 feet apart, then when these are accurately drawn on the map, any desired number of additional contours may be inter- polated between the principal ones. 216. The Final Map. — The field-sheets are drawn as de- scribed above, in pencil, or partly in pencil and partly in ink, or wholly in ink, according to the use to be made of them. If they are simply to serve as the embodiment of the field-sur- vey, to be used only for the construction of the final maps, they are usually left in pencil, a six-H pencil being used. The field-sheets are usually small, about 18x24 inches. The final sheets maybe of any desired size. Usually several field-sheets are put on one final sheet, which will be worked up wholly in ink, or color, the scale remaining the same. The work on the field-sheet is then simply transferred to the final sheet by the most convenient means available. Tracing-paper (not linen) may be used. This is carefully tacked or weighted down over the field-sheet, and the principal features, such as triangulation stations, stream and contour lines, roads, buildings, fence lines, etc., are traced in ink. The tracing-paper is then removed and laid upon the final sheet, orienting it by making the triangula- tion stations on the tracing coincide with the corresponding stations on the final sheet, where they have been carefully plotted from the triangulation reduction. All the matter on the tracing may now be transferred to the paper beneath by passing over the inked lines with a dull point, bearing down hard enough to leave an impression on the paper below. If preferred, the tracing may have its under surface covered with plumbago (soft pencil-scrapings), after the tracing is made, and then with a very gentle pressure of the tracing-point will leave a light pencil line on the final sheet. In either case, when the TOPOGRAPHICAL SUR VE YING. 263 tracing is removed, these lines may be inked in on the final sheet. If the map is to be photo-lithographed it must be drawn wholly in black, as given in Plates 11 . and III. If not, it is best to use some color in its execution. The water-lines may be drawn in blue, and the contours in brown on arable land, and in black on barren or rocky land. In this way the character of the surface may be partly given. Where the slopes are very steep the contour lines become nearly coincident, but to further em- phasize the uneven character of the ground, cross-hatching, or hachures, may be employed on slopes greater than 45° from the horizontal. All these conventional practices are illustrated on Plate III., except the use of colors, this map having been drawn for the purpose of being photo-lithographed. Plate II. is a photo-lithograph copy of a student’s map of the annual field survey of the engineering students of Washington University. 217. Topographical Symbols are more or less conven- tional, and for that reason given forms should be agreed upon. The forms given in Plate III. were used on all the Mississippi River surveys made under the Commission, and are recom- mended as being elegant and fairly representative or natural. Evidently the rice, cotton, sugar, and wild-cane symbols would find no place in maps of higher latitudes. The cypress-tree symbols may be used for pine to distinguish them from decid- uous growth, and the sugar-cane symbol could be used for corn if desired. It is not important to distinguish between different kinds of cultivated crops, since these are apt to change from year to year, but it is sometimes desirable to do so to give a more varied and pleasing appearance to the map. The grouping of the trees in a large forest is also varied simply for the appearance, to prevent monotony. Colors are sometimes used in place of pen-drawn symbols, but these are necessarily so very conventional as to require a key to interpret them, and besides it makes the map look cheap and unprofessional. 218. Accuracy of the Stadia Method. — In measuring dis- 264 SURVEYING. tances by stadia the errors made in reading the rod are as apt to be plus as minus. They therefore follow the law of com- pensating errors, which is that the square root of the number of errors remains (probably) uncompensated. If the rod was properly graduated, therefore, the only error is that from read- ing the position of the wires. On inclined sights the distance read on the rod is accurately reduced to the horizontal by means of proper tables or diagrams. There is another pecu- liarity of this system, and that is that the accuracy depends very largely on the state of the atmosphere. If this is clear and steady the accuracy attainable for given lengths of sight is much greater than when it is either hazy or very unsteady from the effects of heat. Or, for a given degree of accuracy, the lengths of sight may be taken much longer under favorable atmospheric conditions than under unfavorable ones. It is impossible, therefore, to specify any given degree of precision for given lengths of sight for all atmospheric conditions. The results obtained on the U. S. Lake Survey are perhaps a fair average for various conditions. On that service the errors of closure of 141 meandered lines was computed with a mean result of one in 650. The lengths of sight averaged from 800 to 1000 feet, with a maximum length of about 2000 feet. The official limit of error of closure was one in 300. The average length of the lines run was one and a half miles. If care is taken to shorten up the sights for unsteady atmo- sphere, and to reduce all readings to the horizontal, it would not be difficult to reduce the error of closure on lines aver- aging from one to two miles in length, to one in 1000 or one in 1200. Since the absolute error increases as the square root of the length of the line run, it is evident that the relative error diminishes as the length of line increases. Thus, for a single reading of say 400 feet the error might possibly be two feet, but for 100 such sights the error probably would be but 10 X 2 feet = 20 feet, the distance now being 40,000 feet, giving an error of one in 2000. CHAPTER IX. RAILROAD TOPOGRAPHICAL SURVEYING, WITH THE TRANSIT AND STADIA. 219. Objects of the Survey. — Since the transit and stadia are the best means of making a general topographical survey, so they are the means that are best adapted to make a railroad survey, so far as this is a topographical survey. The map of a railroad survey may serve two purposes : Firsts to enable the engineer to make a better location of the line than could be done in the field. Secondy to give all necessary data relating to right of way, as the drawing of deeds, assessment of damages, etc. In flat or gently undulating country, it is not advisable to locate by a map ; but even here the map is quite as essential for determining questions relating to the right of way. In either case, therefore, a good topographical map of the line is of prime importance, and all the data for this map may be taken on the preliminary survey.* Both these ends may be served by the same map. The method of location by contours (sometimes called “ paper lo- cation”) is often absolutely necessary in rough ground, but is still more often judicious in simpler work, inasmuch as a better location can often be made in this way. 220. The Field-work. — In this case there would be no A’s or B.M.’s to check on; but the errors in distance and ele- vation would be no more, probably, than are now made on * By " preliminary survey” is here meant a survey of a belt of country which it is expected will embrace the final line, and not a mere reconnoisance made to determine the feasibility of a line, or which of several lines is the best. 266 SUR VE YING. preliminary surveys. In fact, the errors in distance would not be nearly so great, unless the chain be tested frequently for length, and the greatest care taken on irregular ground. If a chain ico feet long has 600 wearing-surfaces, which most of them have, and if each of these surfaces be supposed to wear O.Oi inch, which it will do in the course of a 200- or 300-mile survey, then the chain has lengthened by six inches, or the error in distance is now i in 200 from this cause alone. If we add to this the uncertain errors that come from chaining up and down hill, and over obstructed ground, it is certain that the stadia measures will be much the more accurate. In the matter of elevations, since the local change of ele- vation is alone significant, and not the total difference of ele- vation of points at long distances apart, the line of levels carried by the stadia would be amply sufficient for a prelimi- nary survey. The following observations are applicable to the prelimi- nary survey for final location, when it is expected the line will be included in the belt of country surveyed: 1st. All data should be taken that will contribute to the so- lution of all questions of location, such as elevations for con- tour lines ; streams requiring culverts, trestles, or bridges, and the necessary size of each, if possible ; all depressions which cross the line, and will require a water-way, together with the approximate size of the area drained ; highways and private roads or lanes ; buildings of all kinds, fences, and hedges ; character of surface, as rock, clay, sand, etc. ; character of vegetation, as cultivated, forest, prairie, marsh, etc. ; the loca- tion of any natural rock that may be used for structures on the line, such as culverts or abutments ; high-water marks if in a bottom subject to overflow; and, in fact, all information which will probably prove of value in determining the location, or in making up a report with estimates to the board of directors, or in letting contracts for earthwork. RAILROAD TOPOGRAPHY. 267 2d. All data that may be found useful in respect to land titles or right of way, or that may relate to claims for dam- ages, such as section corners, boundaries, fences, buildings, streets, roads, lanes, farm roads, cultivated and uncultivated land, as well as such as may be cultivated, public and private grounds, orchards, forests, together with the value of the forest timber, mineral lands, stone quarries, proximity to villages, etc. Since the bearings and position of all boundary-lines are of great importance in the matter of right of way, every such boundary should have at least two readings upon it in the field ; and these should be as far apart as possible. 221. The Maps. — Before any plotting is done, two ques- tions of importance must be decided. They are — first., whether one set of maps is to serve for both the location and for the further use of the company, or whether a set of contour maps, worked up in pencil, shall serve for the location, and another set for the continuous use of the company; second, what shall be the scale of the maps ? These will be argued separately. Whether one or two sets of maps will be decided on, will de- pend largely on the care that is exercised with the locating- sheets. If these are carefully worked up for the location, and kept clean, they can be utilized for the final maps. If they become too badly soiled by field use, new sheets would prob- ably be substituted for the uses of the company. If it is expected, at the start, to have a different set of sheets for the final maps, then “ protractor sheets” should be used for the location. In this case, plot on these sheets only such of the field-notes as will contribute to the location ; and these need only be plotted in pencil. When the location has been made, such features may be transferred from the locating- sheets to the final maps, as may be desired. These would con- sist mainly in the stadia stations, the contours, and the located line. The rest of the field-notes may then be plotted on the final sheets, and the whole worked up in ink. 268 SURVEYING. If, on the other hand, one set of maps is to serve both pur- poses, then it would, perhaps, be best to use plain sheets, as the protractor circle would somewhat disfigure the final maps. The protractor sheets would, however, furnish a ready means of taking off the bearings of lines from the final charts, which might be thought to compensate for the slight marring of the map’s appearance. If plain sheets are chosen, then they should be divided into squares by lines drawn in ink parallel to the sides of the paper, in the direction of the cardinal points of the compass. Both the stadia stations and the side-readings may then be plotted by means of the auxiliary protractor, this being oriented by the meridian lines on the sheet. Even here, only those readings would at first be plotted that will contrib- ute to the location, and these marked in pencil. After the location has been decided on, and the location notes taken off, as described below, then the stadia stations, contour lines, the located line of road, and such other features as should be pre- served on the final map, are inked in, and the map thoroughly cleaned. The rest of the field-notes may now be plotted, and the map finished up. If the road runs through a settled region, the questions of right of way are among the first things to be settled ; so that preliminary maps showing the relation of the road belt to the property lines are essential to the settlement of damages, and to obtaining the right of way from the property-holders. Coincident, therefore, with the making of maps to determine the location must come the construction of preliminary right- of-way maps or tracings. On these latter need be plotted only the boundary-lines, fences, more important buildings, roads, etc., or just sufficient to enable the right-of-way agent to nego- tiate intelligibly with the property-owners.* Neither the lo- * For an excellent article on the subject of right-of-way maps and permanent railway-property records, by Charles Paine, see The Railroad Gazette of Nov. 14, 1884. Reprinted in book form in “ Elements of Railroading." RAILROAD TOPOGRAPHY. 269 eating nor the final map should be on a continuous roll. The roll requires more room for storage, is more apt to get dusty, and is much more inconvenient for reference. When sheets are used, the survey plot covers a more or less narrow belt across the map. One of the edges of the sheet, either where the plot enters upon it or disappears from it, should be trimmed straight, and the plot extended quite to this edge. This edge is then made to coincide with one of the parallel or meridian lines of the next sheet ; so that when the line is plotted, the sheets may be tacked down in such a way as to show the con- tinuous plot of the survey. The scale of the map will depend on whether or not separate sets of charts are to serve the purposes of location and of the continuous use of the company. For the purpose of location, a scale of 400 feet to one inch does very well ; but for the final detail sheets the scale should be larger. If both purposes are to be served by one set of maps, then the scale should be about 200 feet to one inch,* with 5- or lo-foot contours. The sheets should be about twenty by twenty-four inches. 222. Plotting the Survey. — In case the map is plotted on a protractor sheet, the methods of plotting will be identical with those for general topographical work, except that here there will be no checks, either for distance, azimuth, or eleva- tion, except such as are carried along or independently de- termined. For distance, there is no check, except the dupli- cate readings between instrument stations, unless the survey is through a region which has already been surveyed. In this case the section lines may serve as a check on the distances. The azimuth should be checked at every station by reading the needle, as described on p. 248, and also by independently determining the meridian frequently, either by a solar attach- ment or by a stellar observation. If the line is not nearly * Some engineers prefer a scale of 100 feet to one inch for the final charts of the company. 270 SUR VE YING. nortli and south, or, in other words, if it is extended materially in longitude, then the azimuth must be constantly corrected for convergence of meridians, as is shown in Chap. XIV. The elevations can only be checked by the duplicate read- ings between instrument stations.* All the greater care should be used, therefore, on readings between stations. The first plotting, whether there are to be two sets of maps or one, will consist in representing on the sheet only such data as will assist in deciding on the location. These will be mainly contour points, streams, important buildings near the line, principal highways, other lines of railway, villages with their streets and alleys near the proposed location, the lines of de- markation between cultivated and timbered or wild land, etc. From the plotted elevations, aided by the sketches in the note- book, the contour lines are drawn in ; if necessary, this may be done on the ground. This is sufficient for determining upon a location. When this has been done, then the natural features, the contour lines, the stadia stations, and the located line, may be inked in (or transferred by means of tracing-paper, in case the final maps are to be on separate sheets), and the remainder of the notes plotted. In drawing the contour lines in ink, make those upon bar- ren or rocky land in black, and those on arable land in brown. If they are ten feet apart, make every tenth one very heavy, and every fifth one somewhat heavier than the others. If this be done, only the 50- and lOO-foot contours need be numbered. In case a map does not contain at least two of these numbered contours, then every contour which does appear on the map should be numbered, giving its elevation above the datum of the survey. * It may be observed that the same lack of sufficient checks on the distance, azimuth, and elevation obtains with the ordinary preliminary survey with tran- sit, level, and chain. If preferred, all bearings may be taken from the needle, and then each alternate station only need be occupied by the instrument. See series of articles on this subject by the author in “ The Railroad Gazette ” for Feb. 3d, Mar. 2d, 9th, and 30th, 1888, RAILROAD TOPOGRAPHY, 271 The streams should be water-lined in blue, and an arrow should tell the direction of its flow. The name should also be given when possible. All fences should be shown, and especial pains taken to represent division fences in their true position ; for it is from this map that the deeds for the right of way are to be drawn. Outhouses may be distinguished from dwellings by diago- nal lines intersecting, and extending slightly beyond the out- line. The character of the buildings may be^hown by colors, as red for brick, yellow for frame, pale sepia for stone ; the outlines always being in black. The stadia stations should be left on the finished sheets; as, in case of a disputed boundary, or for other cause, the map may be replotted if the positions of the instrument stations are left on it. The numbers of the stations should, of course, be appended. The magnetic bearings of boundary-lines may be given on the map, or they may be determined, as occasion requires, by means of the auxiliary protractor and the true meridian lines when the variation of the needle is known. For this purpose, the magnetic meridian should be drawn on each map, diverg- ing from one of the meridian lines, and the amount of the variation marked in degrees and minutes. 223. Making the Location. — When a preliminary survey is made, as above described, for the purpose of making what is called a “ paper location,” the location is first made on the map, and then staked out in the field. Every railroad line is a combination of curves, tangents, and grades ; and it is the proper combination of these which makes a good location. If it be assumed that the line is to be ‘ included in the belt of country surveyed, then the map con- tains all the data necessary to enable the engineer to select the best arrangements of curves, tangents, and grades it is possible for him to obtain on this ground. This selection can be made 272 SUK VE YING. with much more certainty than is possible on the ground, where the view is generally obstructed, and where grades arc so deceptive. It is no part of this treatise to discuss the various problems that enter into the question of a location, but only to show how to proceed to make a location that may satisfy any given set of conditions, by means of the contour map. The contours themselves will enable the engineer to decide what the approxifnatc grades will have to be. Suppose a grade of 0.5 foot in 100 feet, or 26.4 feet to the mile, has been fixed upon. It is now known that the line should follow the gene- ral course of the contours, except that it should cross a lo-foot contour every 2000 feet. Spread the dividers to this distance, taken to scale, and mark off in a rough way these 2000-foot distances as far as this grade is to extend; and do the same for the successive grades along the line. Knowing the grade of the line at the beginning of the sheet, the problem is to ex- tend this line over the sheet so as to give the best location one can hope to get on this ground with the available means. * First, starting from the initial fixed point of line on the map, sketch in a line which will follow the contours exactly, crossing them, however, at such a rate as to give the necessary grade. This is the cheapest line, so far as cut and fill are con- cerned. Of course, where depressions or ridges are to be crossed, the line must cross over from a given contour on one side to the corresponding contour on the other, and then fol- low along the contour again. Second, mark out a series of tangents and curves which will follow this sketched line as nearly as it is possible for a rail- road to follow it. This will not be the final location, but it is valuable for study. This line will be faulty from having too many and too sharp curves, and too little tangent. Third, draw in a third line, as straight as possible, and with RAILROAD TOPOGRAPHY. 273 as low grade of curves as possible consistent with a reasonable amount of earthwork and a proper distribution of the same. For the purpose of deciding what degree of curve is best suited to the ground for a given deflection-angle, it is well to have a series of paper templets made, with the various curves for their outer and inner edges. Of course, these are cut with radii laid off to the scale of the drawing. It is still more con- venient to have these curves, laid off to scale, on a piece of isinglass, horn, or tracing-paper (not linen), so that this can be laid upon the map, and the curve at once selected which will follow the contours most economically. Fig. 66 shows such a series of curves drawn to a scale of 1600 feet to the inch. In this way the line is laid out over the map. The ques- tions of greater or less curvature have been balanced against a less or greater first cost, and greater or less operating expense. The question of shifting it laterally has also been examined, and finally a definite location fixed upon which seems to answer best to the case in hand. When this is done, it only remains to make up the location notes from which the line is to be staked out. 18 274 SURVEYING. The following is considered a good form for the location notes : Location Notes for ABC Railroad. From Map No Line. Azimuth and Deflection Angles. Length. Station. Remarks. T 260” 40' ft. 1020 10 20 P.C. 0 P + 18° 30' 617 lO -f 37 P.T. T 279° 10' 2670 43+7 P.C. 4° C.L. — 12° 20' 308 + 15 ] P.T.S. 46° 30' W. □ 12 320 ft. T 266° 50' 680 52 + 95 P.C. The first column designates the tangents and curves, and gives the degree of the curve, and the direction of its curva- ture, whether right or left. If it curve toward the right, the azimuth of the next tangent will be increased, and hence its sign is plus, and vice versa. The second column gives the azimuths of the tangents and the deflection-angles of the curves. Each azimuth is seen to be the algebraic sum of the two preceding angles. The third column gives the lengths of the tangents as meas- used from the map, and the lengths of the curves as determined by dividing the deflection-angle by the degree of the curve. Thus, 12° 20' — 12°. 33, and 12°. 33 4. — 308, which is the length of the curve in feet.* The fourth column gives the stations and pluses for the P.C.’s and the P.T.’s. These quantities are simply the con- tinued sum of those in the third column. The first, second, and fourth columns now give all the infor- * It is a great convenience to have at least one vernier, in railroad work, graduated to read to hundredths of a degree. The case here given is only one of many similar cases; but the principal advantage is in running the fractional parts of curves when the curve chosen is some even degree, as here taken. RAILROAD TOPOGRAPHY. 275 mation necessary to stake out the line. The stadia is no longer to be used, but a transit and chain, as is ordinarily done. The tangents need not be run out to their intersection ; but when the P.C. is reached, according to the location notes taken from the map, set up the instrument, and stake out the curve as far as possible, or around to the P.T. In either case, when the instrument is to be moved, make a note of the forward azimuth, and go forward and orient on the last station the same as when moving between two El’s. If the instrument be moved to the P.T. direct, then, after orienting back on the P.C., turn off to the azimuth given for the next tangent, and go ahead. The tangents could be run out to the intersection and the point occupied by the instrument, for a check, if thought desirable. The telescope is never reversed in laying out the line from the system of notes above given. With careful work, the line ought thus to be run out, and the curves put in at once. We have supposed there was no regular line cleared out on the preliminary, so the necessary clearing would all have to be done on the location. A levelling party follows the transit, and obtains the data for constructing a profile and for determining the exact grades. The stadia has served its purpose when it has enabled the engineer to select the most favorable position for the line. The transit, chain, and level must do the balance. It is not improbable that occasional modifications will be introduced in the field, even though the survey and the location have been made with the greatest possible care. 224. Another Method of making the preliminary survey from which to determine the final location is as follows : Run a transit and chain line, setting loo-foot stakes, as nearly on the line of the road as can be determined by eye. Follow this party by a level party which obtains the profile of the transit line. A third party of one or more topographers takes cross-sections at each loo-foot stake by means of a 276 SUR VE YING. pocket-compass, chronometer, and hand-level. These cross- sections show the ground on cither side of the line as far as desirable by slope and distance, these latter being either meas- ured by tape or paced. It is evident that contour lines could be worked out from these data, but these would not be needed if the distances and slopes were well determined, since these give a better cross-section than contours alone could do. The objections to this method are in the poor means it fur- nishes for accurate determination of either distances or slopes, and the haste with which it is usually done. There can be no question but that accurate distances and slopes on cross-sections 100 feet apart would give fuller data than even five-foot con- tours accurately drawn. But to be accurately determined the slope would have to change at all points — in other words, it would be a curve. As to whether the slopes and distances as they would probably be taken would give a better idea of the ground than five-foot contours determined by the stadia method, and the relative cost of the two systems, are matters of experience. Both systems are competent to give a good location when they are well executed. Note. — The further study of railroad surveying falls within the province of the various railroad field-books, which are printed in pocket form and contain the necessary tables for laying out a line of road. Having learned the con- struction and use of surveying-instruments, and the general methods of topo- graphical surveying and levelling, the special applications to railroad location given in the field-books are readily mastered. They will therefore not be further considered in this work. CHAPTER X. HYDROGRAPHIC SURVEYING. 225. Hydrographic Surveying includes all surveys, for whatever purpose, which are made on, or are concerned with, any body of still or running water. Some of the objects of such surveys are the determination of depths for mapping and navi- gation purposes ; the determination of areas of cross-sections, the mean velocities of the water across such sections, and the slope of the water surface ; the location of buoys, rocks, lights, signals, etc. ; the location of channels, the directions and ve- locities of currents, and the determination of the changes in the same ; the determination of the quantity of sediment car- ried in suspension, of the volume of the scour or fill on the bottom, or of the material removed by artificial means, as by dredging. A hydrographic survey is usually connected with an ex- tended body of water, as ocean coasts, harbors, lakes, or riv- ers. The fixed points of reference for the survey are usually on shore, but sometimes buoys are anchored off the shore and used as points of reference. All such points should be accu- rately located by triangulation from some measured base whose azimuth has been found. The buoys will swing at their moorings within small circles, these being larger at low tide than at high, but the errors in their positions should never be sufficient to cause appreciable error in the plotted positions of the soundings. Where soundings need to be located with great exactness, buoys could not be relied on. The triangula- tion work for the location of the fixed points of reference dif- fers in no sense from that for a topographical survey. In fact. 278 SURVEYING. a hydrographic survey is usually connected with a topographical survey of the adjacent shores or banks, the triangulation scheme serving both purposes. It is not uncommon, however, to make a hydrographic survey for navigation purposes sim- ply, wherein only the shore-line and certain very prominent features of the adjacent land are located and plotted. This is the practice of the U. S. Hydrographic Office in surveying for- eign coasts and harbors. In this case the work consists almost wholly in making and locating soundings for a certain limiting depth, as one hundred fathoms, or one hundred feet, inward to the shore, and along the coast as far as desired. The length and azimuth of a base-line are determined and the latitude ob- served by methods given in Chapter XIV. The longitude is found by observing for local time, and comparing it with the chronometer time which has been brought from some .station whose longitude was known. Whenever telegraphic com- munication can be obtained with a place of known longitude, the difference between the local times of the two places is found by exchanging chronographic signals. No special de- scription will be here given of the methods used in this part of the work, as they are all fully described in Chapter XIV. THE LOCATION OF SOUNDINGS. 226. Methods. — The location of a sounding can be found with reference to visible known points by (i) two angles read at fixed points on shore ; (2) by two angles read in the boat ; (3) by taking the sounding on a certain range, or known line, and reading one angle either on shore or in the boat ; (4) by sounding along a known range, or line, taking the soundings at known intervals of time, and rowing at a uniform rate ; (5) by taking the soundings at the intersections of fixed range lines ; (6) by means of cords or wires stretched between fixed HYDROGRAPHIC SURVEYING. 279 stations, these having tags, or marks, where the soundings are to be taken. These methods are severally adapted to differ- ent conditions and objects, and will be described in order. 227. Two Angles read on Shore. — If two instruments (transits or sextants) be placed at two known points on shore, and the angles subtended by some other fixed point, and the boat be read by both instruments, when a sounding is taken, the in- tersection of the two pointings to the boat, when plotted on the chart containing the points of observation duly plotted, will be the plotted position of the sounding. If three instruments are read from as many known stations, then the three point- ings to the boat should intersect in a point when plotted, thus furnishing a check on the observations. The objections to this method are that it requires at least two observers, and these must be transferred at intervals, as the work proceeds, in order to maintain good intersections, or in order to see the boat at all times. While an observer is shifting his posi- tion the work must be suspended. If there are long lines of off-shore soundings to be made and there are no fixed points or stations on shore of sufficient distinctness or prominence to be observed by the sextant from the boat, then this method must be used. When the angles are read on shore signals should be given preparatory to taking a sounding, and also when the sounding is made. If, however, the soundings are taken at regular intervals the preparatory signal may be omitted, and only the signal given when the sounding is taken. This usually consists in showing a flag. The instrument may be set to read zero when pointing to the fixed station. This reading need only be taken at intervals to test the stability of the instrument. 228. By Two Angles read in the Boat to three points on shore whose relative positions are known. This is called the “three-point” problem. Let A, C, and B be the three shore points, being defined by the two distances a and b and the angle 28 o SUf! VE Y/iVG. C. Let the two angles P and P' be measured at the point P, The problem is to find the distances AP BP, {a) Analytical Solution . — Let the un- known angle at A be ;if, and that at B be c B y. Then we may form two equations from which x and y may be found. For, _ a sin X b sin y ~ sin P ~ sin P' ' p Fig. 68. Also, ;r+J= 36 o°-(P+F+C) = ^. ... (2) From ( 2 ), y = R — X, and sin y = sin R cos x — cos R sin x. Substitute this value of sin_y in (i), reduce, and find a sin P b sin P cos R cot X — b sin P sin R ■ • (3) When X and y are found, the sides AP and BP are readily obtained. This is perhaps the simplest analytical solution of the problem. {U) Geometrical Solution . — The following geometrical solu- tion is of some interest, though it is seldom used : Let A, C, and B be the fixed points as before, and Pand P the observed angles. Having the points.^, .5, and C plotted H YDROGRAPHIC SUR VE YING. 281 in their true relative positions, draw from A the line AD, making with AB the angle P' (CPB), and from B the line BD, making with AB the angle P (APC), cutting the former line in D. Through A, D, and B pass a circle, and through C and D draw a line cutting the circum- ference again in P. The point P is the plotted position of the point of observa- tion from which the angles Pand P' were measured. For P must lie in the circumference through ADB by construction, otherwise would not be equal to as they are both measured by the same arc AD. The same holds for the angle P'. Also, the line PD must pass through C, other- wise the angle APC would be greater or less than P, which cannot be. The point Pis therefore on the line PP, and also on the circumference of the circle through ADB, whence it is at their intersection. This demonstration is valuable as showing when this method of location fails to locate, and when the location is poor. For the nearer the point D comes to C the more un- certain becomes the direction of the line CD, and when D falls at C — that is, when P is on the circumference of a circle through A, B, and C — the solution is impossible, inasmuch as P may then be anywhere on that circumference without 'changing the angles P and F . This is also shown by equation (3), above ; for if A, C, B, and Pall fall on one circumference, then -|-j^ — R — 180°; whence cot x— co X O, which is indeterminate. For cot P = — CO, and cos P = — i. Also a sin F — b sin P, both being equal to the perpendicular from C on AB. The equation then becomes cot X = 00(1 — i) = 00 X o. 282 SUR VE YING. (c) Mechanical Solution . — If the three known stations be plotted in position and the two observed angles be carefully set on a three-armed protractor,* then when the three radial edges coincide with the three stations, the centre of the pro- tractor circle corresponds to the position of the point of obser- vation. With a good protractor this method gives the posi- tion of the point as closely as the nature of the observations themselves would warrant. It is the common method of plot ting soundings when two sextant angles have been read from the sounding boat. The goniograph, described on p. 113, is designed to serve both for reading the angles and for plotting of point, replacing therefore both the sextant and protractor in this work. id) Graphical Solution . — The angles may be laid off on tracing-paper or linen by lines of indeSnite length, and this laid on the plot and shifted in position until the three radial lines coincide with the three stations, when their intersection marks the point of observation. This is the most ready method of plotting such observations when no three-armed protractor is available. The advantages of this method of locating soundings are that it requires but one observer, no time is lost in changing stations, and the party are all together, and hence there can be no misunderstandings in regard to the work. If the soundings are made in running water, so that the boat cannot be stopped long enough to read two sextant angles, two sextants are sometimes used with one observer, he setting both angles and reading them afterwards ; or two observers may be employed in the same boat and the angles taken simultaneously. 229. By one Range and one Angle. — The range may be two stations or poles set in line on shore, or it may consist of one point on shore and a buoy set at the desired position off- * For description, with cut, see p, 167. HYDROGRAPHIC SUR VE YING. 283 shore. If buoys are used they must be located by triangulation from the shore stations. A triangulation system along a rocky or wooded coast may consist of one line of sta- tions on shore and a corresponding line of buoys. The angles are read only from the shore stations, two angles in each triangle being observed. If the buoys are well set and the work done in calm weather, the results will be good enough for to- pographical or hydrographical purposes. The stations and buoys should be opposite each other, as in the figure, and readings taken to the two adjacent shore stations and to the three nearest buoys from each shore station. If the length of any line of this system be known, the rest can be found when the angles at A, B, C, and D are measured. In such a system the measured lines should recur as often as possible, ordinary chain- ing being sufficient. 230. Buoys, Buoy-flags, and Range-poles.— A conveni- ent buoy for this purpose may be made of any light wood, eighteen inches to three feet long in tideless waters, and long enough to maintain an erect position in tide-waters. It should be from six to ten inches in diameter at top, and taper towards the bottom. If the buoy is not too long, a hole may be bored through its axis for the flag-pole, which may then project two or three feet below the buoy and as high above it as desired. The buoy rope is then attached to the bottom end of the pole and made of such length as to maintain the pole in a vertical position in all stages of the tide. The anchor may be any suffi- ciently heavy body, as a rock or cast-iron disk. If the buoys are liable to become confused on the records, different designs may be used in the flags, as various combinations of red, white, and blue, all good colors for this purpose. The range-poles should be whitewashed so as to show up Fig. 70. 284 SUR VE YING. against the background of the shore. The ranges are desig- nated by attaching to the rear range-poles slats (barrel-staves would serve) arranged as Roman numerals when read up or down the pole. If range-poles are relied on, they must be very carefully located and plotted, in order to establish accurately a long line of soundings from a very short fixed base. The observed angle may be either from the boat or from a point on shore. In either case any other range-post of the series may be used either for the position of the observer, if on shore, or for the other target-point if the angle is read from the boat. 231. By one Range and Time-intervals. — This is a very common and efficient method, and quite satisfactory where soundings need not be located with the greatest accuracy and where there is no current. A boat can be pulled in still water with great uniformity of speed; and if the soundings be taken at known intervals with the ends of the line of soundings fixed, the time-intervals will correspond almost exactly with the space-intervals. If the ends of the line of soundings are not fixed by buoys or sounding-stations on shore, but the line sim- ply fixed by ranges back from the water’s edge, the positions of the end soundings may be fixed by angle-readings and the bal- ance interpolated from the time-intervals. 232. By means of Intersecting Ranges. — This method is only adapted to the case where soundings are to be repeated many times at the same places. When the object of the sur- vey is to study the changes occurring as to scour or fill on the bottom it is very essential that the successive soundings should coincide in position, otherwise discrepant results would prove nothing. Such surveys are common on navigable rivers and in harbors. Many systems of such ranges could be described, but the ingenious engineer will be able to devise a system adapted to the case in hand. 233. By means of Cords or Wires. — In the case of a fixed HYDROGRAPHIC SUR VE YING. 285 but narrow navigable channel, having an irregular bottom, or undergoing improvement by dredging, it may be found advis- able to set and locate stakes on opposite sides of the channel, to stretch a graduated cord or wire between them, and to locate the soundings by this. By such means the location would be the most accurate possible. MAKING THE SOUNDINGS. 234. The Lead is usually made of lead, and should be long and slender to diminish the resistance of the water. It should weigh from five pounds for shallow, still water, to twenty pounds for deep running water, as in large rivers. If depth only is required, the lead may be a simple cylinder something like a sash-weight for windows. If specimens of the bottom are to be brought to the surface at each sounding, the form shown in Fig. 71 may be used to advantage. An iron stem, /, is made with a cup, at its lower end. The stem has spurs cut upon it, or cross-bars attached to it, and on this is moulded the lead which gives the requisite weight. Between the cup and the lead is a leather cover sliding freely on the shank and fitting tightly to the upper edges of the cup. When the cup strikes the bottom, it sinks far enough to obtain a specimen of the same, which is then safely brought to the surface, the leather cover protecting the contents of the cup from be- ing washed out in raising the lead. A conical cav- ity in the lower end of the lead, lined with tallow, is often used, and it is found very efficient for in- dicating sand and mud. It is often very essential to know whether the bottom is composed of gravel, coarse or fine, sand, mud, clay, hard-pan, or rock, and this knowledge can be ob- tained with the cup device described above. 235. The Line should be of a size suited to the weight of Fig. 71. 286 StJRVEYING. the lead, and made of Italian hemp. It is prepared for use by first stretching it sufficiently to prevent further elongation in use after it is graduated. Probably the best way to stretch a line is to wind it tightly about a smooth-barked tree, securely fasten both ends, wet it thoroughly, and leave it to dry. Then rewind as before, taking up the slack from the first stretching, and repeat the operation until the slack becomes inappreciable. It may now be graduated and tagged. Sometimes it is fastened to two trees and stretched by means of a “ Spanish windlass,” and then wet. It is quite possible to stretch the line too much, for sometimes sounding-lines have shortened in use after being stretched by this method. Soundings at sea are taken in fath- oms. On the U. S. Lake Survey all depths over twenty- four feet (four fathoms) were given in fathoms, and all depths less than that limit were given in feet. On river and harbor surveys it is common to give depths in feet. Channel- soundings on the Western rivers made by boatmen are given in feet up to ten feet, then they are given in fathoms and quar- ters, the calls being “ quarter-less-twain,” “mark-twain,” “quar- ter-twain,” “half-twain,” “ quarter-less-three,” “mark-three,” etc., for depths of if, 2, 2f, 2f, 3, etc., fathoms respectively. If the line is graduated in feet leather tags are used every five feet, the intermediate foot-marks being cotton or woollen strips. The ten-foot tags are notched with one, two, three, etc., notches for the 10-, 20-, 30-, etc., foot points, up to fifty feet. The fifty-foot tag may have a hole in it, and the 60-, 70, 80-, etc., foot-marks have tags all with one hole and with one, two, three, etc., notches. The intermediate five-foot points have a simple leather tag unmarked. Sometimes the figures are branded on the leather tags, but notches are more easily read. The zero of the graduation is the bottom of the lead. The leather tags are fastened into the strands of the line ; the cloth strips may be tied on. The line should be frequently tested, and if it changes materially a table of corrections HYDROGRAPHIC SURVEYING. 287 should be made out and all soundings corrected for erroneous length of line. 236. Sounding-poles should be used when the depth is less than about fifteen feet. The pole may be graduated to feet simply, or to feet and tenths, according to the accuracy required. 237. Making Soundings in Running Water. — The sounding-boat should be of the “ cutter” pattern, with a sort of platform in the bow for the leadsman to stand on. If the current is swift, six oarsmen will be required and two ob- servers and one recorder. One of the observers may act as steersman. If the depth is not more than sixty or eighty feet, the soundings are made without checking the boat, the leads- man casting the lead far enough forward to enable it to reach bottom by the time the line comes vertical. When the depth and the current are such as to make this impossible, the boat is allowed to drift down with the current and soundings taken at intervals without drawing up the lead. The boat is then pulled back up stream and dropped down again on another line, and so on. In still water a smaller crew and outfit may be used, as the. boat may be stopped for each sounding if necessary. The record should give the date, names of observers, general locality, number or other designation of line sounded, the time, the two angles, the stations sighted, and the depth for each sounding, and the errors of the graduated lengths on the sounding-line. 238. The Water-surface Plane of Reference. — In order to refer the bottom elevations to the general datum-pHane of the survey, it is necessary to know the elevation of the water-, surface at all times when soundings are taken. In tidal waters the elevation of “ mean tide” is the plane of reference for both the topographical and hydrographical surveys, and then the- state of the tide must be known with reference to mean;tide.. 288 SUR VE YJNG. This is found from the hourly readings of a tide-gauge (pro- vided it is not automatic), the elevation of the zero of which, with reference to mean tide-water, has been determined. All soundings must then be reduced to what they would have been if made at mean tide before they are plotted. If the soundings are made in lakes, the datum is usually the lowest water-stage on record ; and here also gauge-readings are necessary, as the stage of the water in the lake varies from year to year. In this case the gauge need only be read twice a day. In rivers of variable stage the datum is either referred to mean or low-water stage, or else to the general datum of the map. If the stage is changing rapidly the gauge should be read hourly when soundings are taken, otherwise daily readings are sufficient. If the soundings are to be referred to the general datum of the map, then the slope of the stream must be taken into account. If they are referred to a particular stage of water in the river, then the slope does not enter as a correction, as the slope is assumed to be the same at all stages, although this is not strictly true. 239. Lines of Equal Depth correspond to contour lines in topographical surveys ; but to draw lines of equal depth with certainty the elevations of many more points are neces- sary than are needed for drawing contour lines, because the bottom cannot be viewed directly, while the ground can be. Where the ground is seen to be nearly level no elevations need be taken, while for a similar region of bottom a great many soundings would be required to prove that it was not irregular. 240. Soundings on Fixed Cross-sections in Rivers. — Where the same section is to be sounded a great many times, and especially when it is desirable to obtain the successive soundings at about the same points, it is best to fix range- posts on the line of the section (on both sides if it be a river) H YDROGRAPHIC SUR VE YING. 289 and then fix one or more series of intersecting ranges at points some distance above or below the section on one or both sides of the river. The soundings can then be made at the same points continuously without having to observe any angles at all. Such a system of ranges is shown in Fig. 72. AA' and BB' are range-poles on the section line. O and 0 ' are tall white posts set at convenient points on opposite sides of the river, either above or below the section. I., IL, III., etc., are shorter posts set near the bank in such positions that the in- tersection of the lines O-l., 6>-II., etc., with the section range BB' will locate the soundings at i, 2, etc., on this section line. The posts in the banks should be marked by strips nailed upon them so as to make the Roman numerals as given in the figure. Such a system of ranges as the above is useful also for fixing points on a section-line, for setting out floats, or for running current-meters for the determination of river discharge. 241. Soundings for the Study of Sand-waves. — In all cases where streams flow in sandy beds, the bottom consists of a series of wave-like elevations extending across the chan- nel. These are very gently sloping on the up-stream side 19 290 SURVEYING. and quite abrupt on the lower side. They are called sand- waves, or sand-reefs. They are constantly moving down- stream from the slow removals from the upper side and accre- tions on the abrupt lower face. They have been observed as high as ten feet on the Mississippi River, and with a rate of motion as great as thirty feet per day. In order to study the size and motion of these sand-waves, it is necessary to take soundings very near together, on longitudinal lines over the same paths at frequent intervals for a considerable period. The boat is allowed to drift with the current and the lead floats with the boat near the bottom. It is lowered to the bottom every few seconds and the depth and time recorded. About once a minute the boat is located by two instruments on shore or in the boat, and so the exact path of the boat located. A profile of the bottom can then be drawn for the path of the boat. A few days later the same line is sounded again in a similar manner and the two profiles compared. It will be found that the waves have all moved down-stream a short dis- tance, the principal waves still retaining their main charac- teristics, so that identification is certain."^ 242. Areas of Cross-section are obtained by plotting the soundings on cross-section paper, the horizontal scale be- ing about one tenth or one twentieth of the vertical. The horizontal line representing the water-surface is drawn, and the plotted soundings joined by a free-hand line. The enclosed area is then measured by the planimeter. If the horizontal scale is 50 feet to the inch and the vertical scale 5 feet to the inch, then each square inch of the figure represents 250 square feet of area. The planimeter should be set to read the area in square inches, and the result multiplied by * It is believed the author made the first successful study of the size and rate of motion of sand-waves, at Helena, Ark., on the Mississippi River, in 1879. Rep. Chief of Engineers, U. S. A., 1879, vol. iii., p. 1963. f See p. 143 for a description and theory of the planimeter. HYDROGRAPHIC SUR VE YING. 291 Areas of cross-section are usually taken in running water, and here great care must be taken to get vertical soundings, and to make the proper sounding-line corrections. They should be taken near enough together to enable the bottom line to be drawn with sufficient accuracy. BENCH-MARKS, GAUGES, WATER-LEVELS, AND RIVER-SLOPE. 243. Bench-marks should be set in the immediate vi- cinity of each water-gauge, and these connected by duplicate lines of levels with the reference-plane of the survey. If the gauge is not very firmly set, or if it is necessary to move it for a changing stage, its zero must be referred again to its bench- mark by duplicate levels, whenever there is reason to suspect it may have been disturbed. Such bench-marks as these are usually spikes in the roots of trees or stumps. 244. Water-gauges are of various designs, according to the situation and the purpose in view. For temporary use during the period of a survey, a staff gauge is best, consisting of a board painted white, of sufficient length, graduated to feet and tenths in black. Sometimes it is graduated to half-tenths, but this is useless unless in still water, and there is never any need of graduation finer than this. The gauge maybe read to hundredths of a foot if the water is calm enough. It should be nailed to a pile or to a stake driven firmly near the water’s edge. It is read twice a day, or oftener, if the needs of the service require. For the continuous record of tidal stages an automatic, or self-registering, gauge is employed. For rivers with widely varying stage an inclined scantling is fixed to stakes set from low to high water along up the sloping bank. It should be placed at a point where the bank is neither caving away nor growing by filling-in of new deposits. After the scantling is set (the slopes not necessarily the same throughout its length), the foot and tenth graduations are set by means of a level and 292 S UK VE Y INC. marked by driving copper tacks. The automatic gauge is described in Chap. XIV. ’ The staff gauge is the one generally used for engineering and surveying purposes. 245. Water-levels. — The surface of still water is by defi- nition a level surface. This fact is used to great advantage on the sea-coast, on lakes, ponds, and even on streams of little slope or on such as have a known slope. Thus, in finding the elevations of the Great Lakes above the sea-level, the elevation above mean tide-water of the zero of a certain water-gauge at Oswego, N. Y., on Lake Ontario, was determined. Then the rela- tive elevations of the zeros of certain gauges at Ports Dalhousie and Colborne, at the lower and upper ends of the Welland Canal respectively, were found by levelling between them, thus connecting Lake Ontario with Lake Erie. Lakes Erie and Hu- ron were joined in a similar manner by connecting a gauge at Rockwood, at the mouth of the Detroit River with one at Lake- port, at the lower end of Lake Huron. Lakes Michigan and Huron were assumed to be of the same level on account of the small flow between them and the very large sectional area of the Straits of Mackinac. F'inally, a gauge at Escanaba, on Lake Michigan, was joined by a line of levels with one at Mar- quette, on Lake Superior. This completed the line of levels from New York to Lake Superior, when sufficient gauge-read- ings had been obtained to enable water- levels \.o be carried from Oswego to Port Dalhousie, on Lake Ontario ; from Port Col- borne to Rockwood, on Lake Erie ; and from Lakeport, on Lake Huron, to Escanaba, on Lake Michigan. It was found that these water-levels were very accurate. Relative gauge- readings were compared for calm days, as well as for days when the wind was in various directions, and a final mean value found which in no case had a probable error as great as o.i foot.* * See Primary Triangulatior. of the U. S. Lake Survey. H YDROGRA PHIC S UR VE YING. 293 A line of levels run along a lake shore or canal in calm weather should be checked at intervals by reading to the water-surface, and in a topographical survey the stadia-rod should frequently be held at the water-surface, even when the body of water is a stream with considerable slope, as it gives a check against large errors even then, and at the same time gives the slope of the stream. Mean sea-level at all points on the sea-coast is universally assumed to define one and the same level surface. It is probable, however, that this is not strictly true. Wherever a constant ocean current sets stead- ily against a certain coast, it would seem that the water here must be raised by an amount equal to the head necessary to generate the given last motion. If the current flows into an enclosed space, as the equatorial current into the Gulf of Mexico, or the tides into the Bay of Fundy, the water-surface may rise much higher. There is some evidence that the ele- vation of mean tide in the Gulf of Mexico is two or three feet higher than that of the Atlantic at Sandy Hook.* The evi- dence on this point is as yet insuflicient to warrant any certain conclusion, however. 246. River Slope is a very important part of a river survey. Sometimes it is desirable to determine it for a given stretch of river with great care, in which case it is well to set gauges at the points between which the slope is to be found and con- nect them by duplicate lines of accurate levelling. The gauges are then read simultaneously every five minutes for several hours and the comparison made between their mean readings. This is always done in connection with the measurement of the discharge of streams when the object is to find what function the discharge is of the slope. It is now known, however, that in natural channels the discharge is no assignable function of * See paper by Prof. Hilgard, Supt. U. S. C. and G. Survey, in Trans. Am. Asso. Adv. Sciences, 1884, p. 446. 294 SUR VE YING. the slope, as is explained in section 259. For ordinary purposes the river slope may be determined with sufficient accuracy by simply reading the level or the stadia-rod at water-surface as the survey proceeds, daily readings of stage being made at permanent gauges at intervals of fifty miles or less along the river. In all natural channels the local slope is a very variable quantity. It is frequently negative for short distances in cer- tain stages, and over the same short stretch of river it may vary enormously at different stages, and even for the same stage at different times. It is determined by the local channel conditions, and these are constantly changing in streams flow- ing in friable beds and subject to material changes of stage. Great caution must therefore be exercised in introducing it into any hydraulic formulae for natural channels. It is usually expressed as a fraction, being really the natural sine of the angle of the surface to the horizon. That is, if the slope is one foot to the mile it is = 0.000189. THE DISCHARGE OF STREAMS. 247. Measuring Mean Velocities of Water Currents. — This is usually done only for the purpose of obtaining the discharge of the stream or channel, but sometimes it is done for other purposes, as for the location of bridge piers or harbor improvements. In the case of bridge piers the direction of the current at different stages must be known, so that the piers may be set parallel to the direction of the current. For find- ing the discharge of the stream or other channel the object may be : (1) To obtain an approximate value of the discharge at the given time and place. (2) To obtain an exact value of the discharge at the given time and place. (3) To obtain a general formula from which to obtain sub- H YDROGRAPHIC SUR VE YING. 295 sequent discharges at the given place, or to test the truth of existing formulae, or to determine the relative efficiency of certain appliances or methods. It will be assumed that the second object is the one sought, and modified forms of the methods used to accomplish this may be chosen for other cases. The mean velocity of a stream is by definition the total dis- charge in cubic feet per second divided by the area of the cross-section in square feet. This gives the mean velocity in feet per second. Evidently this is the mean of the veloci- ties of all the small filaments (as of one square inch in area) on the entire cross-section. If the velocities of these filaments could be simultaneously and separately observed and their mean taken, this would be the mean velocity of the stream. It is quite impossible to do this ; but the nearer this is approached, the more accurate is the final result. If, however, we could obtain by a single observation the mean velocity of all the fila- ments in a vertical plane, the number of necessary observations would be diminished without diminishing the accuracy of the result. There are two common methods of measuring the ve- locities of filaments at any part of the cross-section, and one for obtaining at once the mean velocity in a vertical plane. These are by sub-surface floats and current-meters, and by rod floats, respectively. 248. By Sub-surface Floats. — The ideal sub-surface float consists of a large intercepting area maintained at any depth in a vertical position by means of a fine cord joined to a sur- face float of minimum immersion and resistance, which bears a signal-flag. As good a form as any, perhaps, for the lower float, or intercepting plane, consists of two sheets of galvanized iron set at right angles, and intersecting in their centre lines, as shown in Fig. 73. There are cylindrical air-cavities along the upper edges and lead weights attached to the lower edges of the vanes. These serve to give the desired tension on the 296 S UK VE YING. connecting cord and to maintain the float in an upright posi- tion, even though the cord is drawn out of the vertical by faster upper currents. Tlie vanes should be from six to fifteen inches in breadth by from eight to twenty inches high, accord- ing to the size of the stream. The circular ribs serve simply to hold the vanes in place. The upper float is hollow, cylin- Fig. 73. drical in plan, and carries a small flag. The tension on the cord should be from one to five pounds, according to the size of the floats. The cord itself should be of woven silk and as small as possible, so as to exercise a minimum influence on the motion of the lower float. Wire is not suitable for this pur- pose, as it kinks badly in handling. The theory is that the lower float will move with the water which surrounds it, and that the upper float will be accelerated or retarded according H YDROGRAPHIC SUR VE YING. 297 as the surface current is slower or faster than that at the sub- merged float. The velocity of the current at any depth can thus be determined by running the lower float at this depth and observing the time required for the upper float to pass between two fixed range-lines at right angles to the direction of the current about two hundred feet apart. The floats are started about one hundred feet above the upper range-line, and picked up after having passed the lower range. Two transits are usually used for locating and timing the floats, one being set on each range. When the float approaches the upper range the observer on this line sets his telescope on range and calls “ ready” as the float enters his field of view. The other observer then clamps his instrument and follows the float with the aid of the slow-motion or tangent screw. When the float crosses the vertical wire of the upper instrument he calls “ tick,” and the lower observer reads his horizontal angle. He then sets his telescope on the lower range while the upper observer follows the float with his telescope, and the operation is re- peated to obtain an intersection on the lower range. One or two timekeepers are needed to note the time of the two '‘tick” calls, the difference being the time occupied by the float in passing from the upper to the lower range-line. Both these signals are sometimes transmitted telegraphically to a single timekeeper. When the angles are plotted the path of the float is also obtained. If the channel is not too wide, wires may be stretched across the stream and the float stations marked on these, or the float stations may be determined by means of fixed ranges on shore. The passage of the floats across the section lines may then be noted by a single individual without a transit, using a stop-watch and possibly a field-glass. He starts the watch when the float reaches the upper section, walks to the * lower section, and stops the watch when the float passes this range-line. The near range consists of a plumb-line, or wire 2g8 SUR VE YING. suspended vertically; and the observer stands several feet back of this, and brings it in line with the range-post on the opposite side of the stream. If several floats are started a few minutes apart at the same station and at the same depth, they will sometimes vary as Fig. 74. ’■nuch as twenty per cent in their times of passage, showing great irregularity in the velocity of different parts of the same filament. This is due to internal movements in the water, such as “ boils,” eddies, etc. It is for this reason that great refinement in such observations is useless. A float observation H YDROGRA PHIC SUR VE YING. 299 Fig. 75. gives only the velocity of a given small volume of water which surrounds the lower float, while a current-meter observation, as will be seen, gives the mean velocity of a given jilamejit of 300 SUJ^ VE YING. the stream of any required length. And as different portions of the same filament have very different longitudinal velocities, it requires a great many float observations to give as valuable information as may be obtained by running a current-meter in the same filament for one minute. If discharge observations are to be repeated many times at the same sections, then an auxiliary range should be established from which to start the floats; and if it is desirable to always run them over the same paths, these may be fixed by means of a system of intersecting ranges as described on p. 289. 249. By Current-meter. — This is the most accurate method of obtaining sub-surface velocities ever yet devised. Three patterns of current-meters are shown in Figs. 74 and 75. The first and third are shown in elevation, together with the electrical recording-apparatus. The second is shown in plan. The first has helicoidal and the other two conical cup-shaped * vanes. Neither has any gearing under water, the record being kept by means of an electrical circuit which is made and bro- ken one or more times each revolution. The cup vanes are better adapted to water carrying fibrous materials which tend to collect on the moving parts. The friction can also be made less on the cup meters, agate or iridium bearings being used. The recording-apparatus is kept on shore or in a boat, while the meter is suspended by proper appliances at any point of the section at which the velocity of the current is to be measured. In deep water a boat, or catamaran, is anchored at the desired * Invented by Gen. Theo. G. Ellis, and first used on the survey of the Connecticut River. The telegraphic attachment is due to D. Farrand Henry of Detroit, Mich. See Report of the Chief of Engineers, U. S. A., 1878, p. 308. The form shown in Fig. 75 is due to W. G. Price, and was specially de- signed to be used on the Mississippi River. It is very strong and well pro- tected against floating drift. The first two forms are manufactured by Buff & Berger, of Boston, while the Price meter is made by W. & L. E. Gurley, Troy. H YDROGRA PHIC S UR VE YING. 301 point, and a weight attached to the meter, which is then lowered to the requisite depth by means of a windlass. After it is in place the connection is made with the battery, and the record kept for a given period of time, as for two or three minutes. If the operation is to be repeated often at the same section a wire anchorage laid across the stream above the line would be found useful. This wire is anchored at intervals and is used both for holding the boat (or catamaran) in place and for pull- ing it back and forth across the stream. In large rivers a steam-launch may be required for handling the catamaran.* In this case the record begins and ends when the observer is brought on range, it being impossible to hold up steadily against the current. If only the discharge of the stream is sought, the meter is run at mid-depth at a sufficient number of points in the section. The mean velocity in a vertical section at a given point may be obtained by moving the meter at a uniform rate from sur- face to bottom and back again, noting the reading of the regis- ter for the two surface positions, and also for the bottom posi- tion. If the boat was stationary and the rates of lowering and raising strictly constant and equal, the number of revolutions in descending and in ascending should be equal. Either of these registrations, divided by the time, would give the mean regis- tration per second of all the filaments in that vertical plane. The mean of the downward and upward results may be used as giving the mean velocity in that vertical plane. This will not be quite accurate, since it is impossible to run the meter very close to the bottom, but the results will be found useful for comparison with the mid-depth results. Such observations are sometimes called integrations in a vertical plane. 250. Rating the Meter. — When any kind of current-meter * For a description of the latest methods used in gauging the Mississippi River see Report of the Miss. Riv. Com. for 1883, Appendix F. 302 sc//! VE YING. is used for determining the velocity of passing fluids, only the number of revolutions of the wheel carrying the vanes is ob- served for a given time. Before the velocity of the fluid in feet per second can be found, the relation between the rate of revo- lution of the wheel and the rate of motion of the fluid must be determined for all velocities that are to be observed. The de- termination of this relation is called rating the meter. It is usu- ally done by causing the meter to move through still water at a uniform speed, and noting the time occupied and the corre- sponding number of registrations made in passing over a given distance. It may be attached to the prow of a boat, as shown in Fig. 76, the electric register being in the boat. The dis- tance divided by the time gives the rate of motion or velocity of the meter through the water. The number of registrations (revolutions of the wheel) divided by the time gives the rate of motion of the wheel. The ratio of these two rates is the coefficient by which the registrations of the meter are trans- formed into the velocity of the current. This ratio is not a constant, but is usually a linear function of the velocity. Thus, if the observations be plotted, taking the number of registra- tions per second as abscissae and the velocities in feet per second as ordinates, they will be found to fall nearly in a right line, the equation of which is y — ax-\-b . . . . (i) H YDROGRA PHIC SUR VE YING. 303 Here x and j/ are the observed quantities, while a and b are constants for the given instrument. If these constants could be found, then the values of y (velocity) could be obtained for all observed values of x (registrations). There are two ways of solving this problem — one graphical and one analytical. Evi- dently any two observations at different speeds would give values of a and b\ but to find the best or most probable values of these constants a great many observations are taken, so that we have many more observations than we have unknown quan- tities. Each pair of observations would give a different set of values of a and b. The most convenient method of finding the most probable values of these functions, though somewhat approximate, is (i) The Graphical Method of Solution . — This consists simply in plotting the corresponding values of x and y on coordi- nate paper, and drawing the most probable straight line through the points. Then the tangent of the angle this line makes with the axis of x is and the intercept on the axis of y is b. One point on this most probable line is the point {x^y^, and y^ being the mean values of the coordinates of all the plotted points. This is shown by equation (3). Having determined this point, a thread may be stretched through it and swung until it seems to be in a position of equilibrium, when each point is conceived as an attractive force acting on the line, the measure of the force being the vertical intercept between the point and the line. The arms of these forces are evidently their several abscissae. Or the forces may be measured by their horizontal intercepts, and then their arms are their seve- ral ordinates. For the position of equilibrium the sum of the moments of these forces about the point {x^y^) would be zero.* Such a determination of a and b would be found sufficiently accurate for all practical purposes, but if desired the problem may be solved by * All this simply means to fix this most probable line by eye, through the point (jfo ya), giving greatest weight to the extreme points. 304 SURVEYING. (2) The Rigid or Analytical Method . — Equation (i) may be written b xa — y — o. Every observation may be written in this form, these being called the observation equations. It is probable that no given values of a and b would satisfy more than two of these obser- vations ; and if the most probable values be used, there would, in general, be no single equation exactly satisfied. If we let ;r„ etc., etc., and 2/,, 2/,, etc., be the several values of .r, j/, and the corresponding residuals for the several observa- tion equations, we would have h-\- x,a— b x,a - y,- v,\ (2) b-^x^a —y„ = Since b enters alike in all of them, it is evident that these equations are all of equal value for determining b. Also, since the properly weighted arithmetic mean is the most probable value of a numerously observed quantity, and since in this case the equations (or observations) have equal weight for deter- mining b, we may form from the given series of equations a single standard or “ normal " equation which will be the arith- metic mean of the observation equations ; put this equal to zero and say this shall give the value of b. If x^ and y^ be the mean values of the observed x*s and^s, we would then have, by add- ing the equations all together and dividing by their number b + x,a—y, = o,orb=y,—x,a. ... (3) HYDROGRAPHIC SURVEYING. 305 Substituting this value of b in equation (2), we have {x, — x,)a — (f, -jy,) = v,; (x, - x,)a - {y, - 7.) = V , : • • ( 4 ) {x„ — x,)a — (jj/„ —y„) = v„. We here have a series of equations involving one unknown quantity ; but they evidently are not of equal value in deter, mining the unknown quantity a, since its coefficients are very different. In fact, the relative value of these equations for de- termining a is in direct proportion to the size of this coefficient, so that if this coefficient is twice as large in one equation as in another, the former equation has twice the value of the latter for determining a. In other words, they should all be weighted in proportion to the values of these coefficients, and a conve- nient way of doing this is to multiply each equation through entire by this coefficient. The resulting multiplied equations then have equal weight, and may then be added together to produce another ‘‘ normal equation for finding a. This result- ing equation is lix-x.yy-lix-x, {y-y,)\=o, ... (5) where [ ] is a sign of summation. If we had divided this equation by the number of observation equations m, it would in no sense have changed it so far as the value of a is concerned. From equation (5) we can find the mean or most probable value of a, which when substituted in (3) gives the most prob- able value of b. These values should agree very closely with those found by the graphical method. The analytical method here given is precisely that by least squares, though arrived at through the conception of a properly weighted arithmetic mean, instead of by making the sum of the squares of the residuals a minimum. 20 3o6 SURVEYING. The following is an actual example from the records of the Mississippi l-^ver Survey: REX)UCTION OF OBSERVATIONS FOR RATING METER A, taken at Paducah, Ky., June 21, 1882. W. G. Price, Observer. L. L. Wheeler, Computer.* No. r t jr y X — JTo y-yo (x - JTo)’ (.r — j-q) ^ y - y < i ) Remarks.^ I 100 53 I 886 3-774 + O.II7 + 0.245 -f- 0.014 -j- 0.029 Observations 2 1 lOI 44 2.295 4-544 + 0.526 + I. 015 + 0.277 + 0.534 made with 3 ^OI 41 2.464 4.878 + 0.695 -h 1.349 + 0.483 4- 0.938 meter on vertical 4 96 124 0 774 1.613 - 0.995 — 1.916 + 0.990 4" I . 906 iron rod, five 5 94 152 0.618 1.316 - I.I5I — 2.213 + 1-325 4 - 2.548 feet in front of 6 90 193 0.466 1.036 - I.. 303 - 2.493 + 1-^97 + 3.249 bow of skiff, in 7 91 181 0.503 1. 105 - 1.266 - 2.424 + 1.603 4- 3-069 pond. 8 103 28 3.678 7.142 + 1-909 + 3.613 -h 3-644 4- 6.903 9 100 53 1.886 3-774 O.II7 + 0.245 -f- 0.014 4- 0.029 Length of base 10 98 73 1.342 2.740 - 0.427 — 0.789 -}- 0.182 4 - 0.337 = 200 feet. II 103 29 3-552 6.896 + 1.783 + 3-367 4 - 3-178 4- 6.002 w = O-o = 19.464 1.769 38.818 3-529 = /0 = 13-407 25.544= -[{x-x^){y-y^)'\ Normal Equations. b -f- 1.769a — 3529 = o ; Whence a — 1.905 ; 13.407a — 25.544 = o. ^=0.159. Equation for Rating, y = 1.905;!; -f 0.159. Even where the analytical method is to be used it is al- ways well to plot the observations for purposes of study. Then if any observations are especially discrepant, the fact will appear. By consulting column six of the computation it will * In the original computation the method by least squares was used and the probable errors of a and b found. H YDROGRAPHIC S UR VE YING. 307 be seen that observations of greatest weight were those taken at very high and at very low velocities. If the observations were taken in three groups about equally spaced, an equal number of observations in each group, the members of a group being near together, then the mean of each group could be used as a single observation. The middle group would serve to show whether or not the unknown quantities were linear functions of each other, since, if they were, the three mean observations should plot in a straight line. The value of a could be com- puted from the two extreme mean observations, and the value of b from the mean of all the observations as before. This would give a result quite as accurate as to treat them separately. If the observations do not plot in a straight line, draw the most probable line through them, and prepare a table of corresponding values of x and j/ from this curve. In any case, a reduction table should be used. The meter should be rated frequently if accurate results are required. In the rating the meter should be fastened several feet in front of the bow of the boat, and in its use it should be run at a sufficient distance from the boat or catamaran to be free from any disturbing influence on the current. 251. By Rod Floats. — These may be either wooden or tin rods, of uniform size, loaded at the bottom, and arranged for splicing if they are to be used in deep water. If the channel were of uniform depth, and the rod reached to the bottom with- out actually touching, then the velocity of the rod would be the mean velocity of all the filaments in that vertical plane,* * This is not strictly true, since the pressure of a fluid upon a body moving through it varies as the square of its relative velocity. The rod moves faster than the bottom filaments and slower than the upper filaments, but this differ- ence Is greatest at the bottom. Therefore, the retarding action of the bottom filaments will have undue weight, as it were, and so the velocity of the rod will really be about one per cent slower than the mean velocity of the current. See “Lowell Hydraulic Experiments,” by James B. Francis. 3o8 SUJ^ VE Y I NG. and this is the value sought. In practice the rod can never reach the bottom, even in smootli, artificial channels, while in natural channels the irregularities are usually such as prohibit its use within several feet of the bottom. The methods of observation are the same as with the double floats, and their velocity is the mean velocity of the water in that plane to the depth of immersion. I'or artificial channels, and for natural channels not more than twenty or thirty feet deep, rod floats may be advantageously used. Beyond that depth they cannot be made of sufficient length to give reliable results. The method is, therefore, best adapted to artificial channels of uni- form cross-section. The immersion of the rod should be at least nine tenths of the depth of the water, in which case, and for uniform channels, as wooden flumes, Francis found that the velocity of the rod required the following correction to give the mean velocity of the water in that vertical plane : Vm = Vr [I— 0.Il6(V7j-0.l)]. Where Vm = mean velocity in vertical plane ; Vr = observed velocity of rod ; _ depth of water below bottom of rod depth of water For natural channels, or for a less immersion than nine- tenths of the depth the formula cannot be used with certainty. The rods should be put into the water at least twenty feet above the upper section. 252. Comparison of Methods. — (i) The method by double floats is adapted to large and deep rivers, or rapid currents carrying much drift or impeded by traffic. It may be used in all cases, but it has the disadvantage of registering only the velocity of a small volume of water surrounding the lower float. H YDROGRAPHIC SUR VE YING. 309 (2) The method by meters is adapted to large or small streams. It records the mean velocity of a filament of indefinite length ; but it cannot be used where the water carries consider- able floating debris, or where the current is too swift to admit of a safe anchorage. (3) The method by rods is best adapted to small channels of uniform section ; it records the mean velocity in a vertical plane to a depth equal to its immersion, and it can be univer- sally used when the law of the velocities in a vertical plane is known, for then a proper coefficient could be derived for any depth of immersion. (4) One rod observation of sufficient immersion is prob- ably as good as several float observations, and a current-meter observation of two or three minutes is worth as much as twenty float observations for the same filament, provided the meter’s rate is constant and well determined. (5) The rods and floats are cheaper in first cost than the meter ; but if the work is to be prosecuted for a considerable period, the excess in the cost of the outfit will be more than balanced by the diminished cost of the work, by using the meter. On the whole, it may be said that the method by cur- rent-meter is the most accurate and satisfactory of any yet de- vised for measuring the velocity of running water. 253. The Relative Rates of Flow in Different Parts of the Cross-section. — (i) In a horizontal plane. If the cross- section of a stream were approximately the segment of a circle, then the relative rates of flow of the different filaments in any horizontal plane would be very nearly represented by the ordi- nates to a parabola, the axis of the parabola coinciding with the middle of the stream. If there .should be any shoaling in any part of this ideal section the corresponding ordinates would' be shortened, so that when the curve of the bottom is given the curve of velocities in a horizontal plane can be fairly pre- dicted. This applies only to straight reaches. If a portion of the section has a flat bottom line, the velocities over this por- 310 SURVEYING. tion will be about uniform. Where the depth is chanjrin^ rap- idly on the section, there the velocities will be found to change rapidly for given changes in positions across the section. It follows from this that observation stations should be placed near together where the section has a sloping bottom line, and they may be placed farther apart where the bottom line of the section is nearly flat. They are usually put closer together near the bank than near the middle ot the stream. (2) In a vertical plane. A great deal of time and talent has been spent in trying to find the law of the relative rates of flow in a vertical plane, but there is probably no law of universal application. The curve representing such rates of flow will always resemble a parabola more or less, the axis of which is always beneath the surface except when the wind is down stream at a rate equal to, or greater than, the rate of the cur- rent. That is to say, the maximum velocity is always below the surface except where the surface filaments are accelerated by a down-stream wind, and it is generally found at about one third the depth. The cause of this depression of the filament of maximum velocity is partly due to the friction of the air, HYDROGRAPHIC SURVEYING. 31I but mostly to an inward surface flow from the sides toward the centre, which brings particles having a slower motion towards the middle of the surface of the stream. This inward surface flow is probably due to an upward flow at the sides caused by the irregularities of the bank, which force the parti- cles of water impinging upon them in the direction of the least resistance which is vertical.* The curves in Fig. 77 represent the mean vertical curves of velocity observed at Columbus, Ky., on the Mississippi River and given in Humphreys and Scak ^ Ret n ( 2 _.i ^ etacM ■ ■ I ‘ Abbot’s Report. The left-hand vertical line is the axis of ref- erence, and the curves are found to fall between the seven- and eight-foot lines. That is, the velocity at all depths in this plane was between seven and eight feet per second. In this case double floats were used, and it is probable that the bottom velocities were not very accurately obtained. The effect of the wind is here shown in shifting the axis of the curve. It is to * See paper by F. P. Stearns before the Am. Soc. Civ. Engrs., vol. xii. p. 331. 312 SUI^VEY/NG. be observed that these curves all intersect at about mid-depth. That is to say, the velocity of the mid-depth filament is not affected by wind. This is why the mid-depth velocity should be chosen when the velocity of but a single filament is to be measured, and from this the mean velocity in the vertical sec- tion derived. It has also been found that the mid-depth veloc- ity is very near the mean velocity, being from one to six per cent greater, according to depth and smoothness of channel. In general, for channels whose widths are large as compared to their depths, a coefficient of from .96 to .98 will reduce mid-depth velocity to the mean velocity in that vertical plane. In Fig. 78* are shown the relative velocities in different parts of the Sudbury River Conduit of Boston. The velocity at each dot was actually measured by the current-meter. The lines drawn are lines of equal velocity, being analogous to con- tour lines on a surface, the vertical ordinates to which would represent velocities. The method of obtaining these velocities is shown in Fig. 79. is a pivoted sleeve through which the meter-rod slides freely. At A there is a roller fixed to the rod which runs on the curved tracks a a a. The graduations on these tracks fix the different positions of the meter, these be- ing so spaced that they control equal areas of the cross-section. Integrations were here taken in horizontal planes by moving the meter at a uniform rate horizontally. 254. To find the Mean Velocity on the Cross-section. — It is evident that this mean velocity cannot be directly ob- served. In fact, it can only be found by first finding the dis- charge per second and then dividing this by the total area of the section. That is to say, the mean velocity is, by definition. * This and ihe following figure are taken from the paper by F. P. Stearns, mentioned in foot-note on the previous page. H YDROGRAPHIC SUR VE YING. 313 MBTSi JLPPABJiTint nt rmarlof M»nhot« , Fig. 79. SURVEYING. 3H The area of the section is found by means of properly located soundings. The actual velocities of certain filaments crossing this section are then observed, and the section subdivided in such a way that the observed velocities will fairly represent the mean velocities of all the similar filaments (usually mid- depth) in that subsection. Each observed velocity is then reduced to the mean velocity in that vertical plane, and this is assumed to be the mean velocity in that subsection. These mean velocities, multiplied by the areas of their corresponding sections, give the discharges across these sections, and the sum of these partial discharges is the total discharge, Q, in the above equation. This may be shown algebraically as follows: Let Fa, Fg, etc., be the observed velocities ; C the coefficient to reduce these to the mean velocity in a vertical plane ; A^, etc., the partial areas of the cross-section corresponding to the observed velocities F„ F,, Fg, etc. ; A the total area of the cross-section = A^ A, A^^ etc. ; Gi, 2,, Qzr etc., the partial discharges; Q the total discharge ; zf the mean velocity for the entire section. Then 0, = CF,A , ; 0, = C^A,, etc. ; 0 = 01 + 03 + etc. = C{A,V, + .^,Fg + etc.); v = ^ = ^{A,V, + A,V, + ctc.). and HYDROGRAPHIC SUR VE YING. 315 Fig. 8q, 3i6 SURVEYING. It has been here assumed that observations are made at but one point in any vertical plane. The method is the same, how- ever, in any case, it only being necessary to apply such a co- efficient to the observed velocity as will reduce it to the mean velocity in its own sub-area. If these partial areas are made small, as in the case of the Boston Conduit, the observed ve- locities may be taken as the mean velocities in those areas ; and if these areas are all equal, which was also the case in this con- duit, then the mean velocity is the arithmetic mean of all the observed velocities. The partial and total areas are best found by means of the planimeter, the cross-section having been carefully plotted on coordinate paper. 255- Sub-currents. — It is often desirable to know the direction as well as the velocity of flow beneath the surface. The direction-meter,* Fig. 8o, is designed to give the direction of sub-currents, both horizontally and vertically. A magnetic needle swings freely until lifted and held by a magnet which is operated from above. The vertical direction is recorded in a similar manner on a small index-circle on the inside of the hollow sphere which always maintains an upright position. 256 . The Flow of Water over Weirs.f — The most ac- curate mode of measuring the flow through small open channels is by means of weirs. There are three kinds of weirs with which the engineer may have to deal in measuring the flow of water, — namely, sharp-crested weirs, wide-crested weirs, and submerged weirs. A sharp-crested weir is one which is entirely cleared by the water in passing over it, as in Fig. 81. A wide crest is shown * Invented by W. G. Price and manufactured by Gurley, Troy, N, Y. f The results given in this and the following article have been mostly taken from a paper by Fteley and Stearns before the Am. Soc. Civ. Engrs., vol. xii. (1883), describing experiments made in connection with the new Sudbury River Conduit, Boston, Mass. The paper was awarded the Norman medal of that society. HYDROGRAPHIC SURVEYING. 317 Fig. 84. Fig. 83. SURVEYING. 318 in Fig. 82, and its effect in increasing the depth on the weir for a given discharge. If the crest has a width equal to the line ab in Fig, 81, then the depth on the weir is unaffected, while if it has a less width, as in Fig. 83, and if the air has not free access to the interveiiing space beneath^ the water will soon fill this space, and the tendency to vacuum here will depress the overflowing sheet of water, thus diminishing the depth on the weir for a given flow. The dotted lines in Fig. 84 are those of normal flow, the full lines being the new positions assumed as a result of the partial vacuum below. A submerged weir is one at which the level of the water below the weir is above its crest, there being, however, a certain definite fall in passing the weir, as shown in Fig. 85. Here h — d — d' x’s, the fall in passing the weir. Velocity of Approach. — This is the velocity of the surface- water towards the weir at a distance above the weir equal to about two and one half times the height of the weir above the bottom of the channel. End Contractions. — These are the narrowing effects of the lateral flow at the ends of the weir. If this lateral component of the flow is shut off by a plank extending several feet up stream and from the water’s surface to several inches below the top of the weir, then there is no end contraction. This arrangement gives more accurate results, as the correction for end contraction involves some uncertainties. H YDROGRAPHIC SUR VE YING. 319 Depth of Water on the Weir . — This is the principal function of the discharge ; it is the difference of eleva- tion between the top of the weir and the surface of the water at a distance above the weir equal to about 2 \ times the height of the weir above the bottom of the channel. Evidently this is a quan- tity which cannot be directly measured. The best way of measuring this quantity is as follows: At a convenient point arrange a closed vertical box which connects by a free opening with the channel at about mid-depth at a point some six feet above the weir. The water will then stand in this box at its normal elevation, unaffected by the slope towards the weir. The elevation of this water-surface is determined by means of a hook-gauge^ Fig. 86, which consists of a metallic point turned upwards and made adjustable in height by means of a thumb-screw. When the point of the hook comes to the surface of the water it causes a distorted reflection. The eleva- tion of the water-surface can be found in this way with extreme accuracy. The difference of elevation between the point of the hook and the crest of the weir can then be determined with a level and rod. This difference is //"in the following formulse. 257. Formulae and Corrections. — For a simple sharp- crested weir, without end contractions and with no velocity of approach, the discharge in cubic feet per second is Q = Z.ZiLHi-{- 0.007 L, (i) where L is the length of the weir and H the depth of water upon it, both measured in feet. The weir must have a level crest and vertical ends ; it should be in a dam vertical on its Fig. 86. 320 SURVEYING. up-stream side ; the water on the down-stream side may stand even with the crest of the weir if it has considerable depth. The error is not more than one per cent when the water on the down-stream side covers fifteen per cent of the weir area, pro- vided H is then taken as the difference in elevation of the water-surface above and below the weir. In this case two- hook gauges would be needed. The crest of the weir should be at a height above the bottom of the channel on the up- stream side equal to at least twice the depth on the weir, to allow for complete vertical contraction. The following corrections apply to their respective condi- tions : For the velocity of approach, the depth on the weir, H in equation (i), is to be increased by 1.5//, where there is no end contraction, h being the head due to the velocity, or h = — . At sea-level this correction becomes c = ~ = 0.0234t/’ (2) This is to be added to H in equation (i), v being measured in feet per second. Where there is end contraction, the correction is _ 2.05^/* = 0.03 • • (3) For end contraction, the length of the weir, L in equation (l), is to be shortened by 0.1//" for each such contraction. This is a mean value, although it varies from o.oyH to 0.12H for different depths on the weir varying from i to 0.3 foot, the smaller correction applying to the greater depth on the weir. H YDROGRAPHIC SUR VE YING. 321 For wide crests the correction to the depth on the weir is sometimes positive and sometimes negative, as shown in fig- ures 82 and 84. The following correction is derived from care- ful experiments : C = 0.2016 -[-o.2I46z£;“ — 0.1876W, ... (4) where C is the correction to be added algebraically to the depth on the wide crest to obtain the depth on a sharp crest which will pass an equal volume of water ; w is the width of the crest ; y is the difference between o.Zo'jw and the depth on the crest. If the crest is narrower than the line ab. Fig. 81, then this correction is not to be applied unless the water adheres to the weir as in. Fig. 84. Up-stream edge of the weir rounded. If the up-stream edge of the weir is a small quarter-circle, add seven tenths of its ra- dius to the depth on the weir before applying the general weir formula. Submerged weir. When the water on the down-stream side rises above the level of the crest, use the formula for a submerged weir, which is Q — c/ (^d-j- - j Vh, (5) where Q is the discharge in cubic feet per second ; c is to be taken from the following table, its value varying d' with ^ ; I is the length of the weir in feet ; 21 322 SUA^ V a: yii\g. d is the depth on the weir in feet, measured from still water on the up-stream side ; d' is the depth to which the weir is submerged, measured from still water on the down-stream side ; h is the fall and equals d — d' . The value of d may be corrected for velocity of approach by formulas (2) and (3). There is no known correction for the velocity of discharge below the weir, and hence the formula can only be used for a channel of large capacity below, as com- pared with the discharge, so that the velocity here will be small. The following are the experimental values of c\ d ' d ‘ c. d ' d ‘ c. d ' d ' c. d ' d ■ c. O.OI 3-330 0.25 3-249 0.55 3.100 0.85 3.150 .05 3-360 -30 3-214 .60 3.092 .90 3.190 .oS 3-372 -35 3. 182 -65 3-089 -95 3-247 .10 3-365 .40 3-155 -70 3.092 1. 00 3-360 •15 3-327 -45 3-131 -75 3.102 .20 3.286 -50 3-I13 .80 1 3-122 This table is inapplicable to values of ^ less than 0.08, un- less the air has free access to the space underneath the sheet. The method of measuring discharge by means of sub- merged weirs is adapted to channels having very small slope. A fall as low as one half inch will give reliable results if it is accurately measured. 258. The Miner’s Inch. — This is an arbitrary standard both as to method and as to volume of water discharged. It rests on the false assumption that the volume discharged is proportional to the area of the orifice under a constant head above the top of the orifice. Its use grew out of the necessities H YDROGRAPHIC SUR VE YING. 323 of frontier life in the mining regions of the West, and should now be discarded in favor of absolute units. The miner’s inch is the quantity of water that will flow through an orifice one inch square, under a head of from four to twelve inches, according to geographical locality. Even if the head above the top of the orifice be fixed, and a flow of 144 miner’s inches be required, the volume obtained would be 3.3, 4.2, or 4.7 cubic feet per second, according as there were 144 holes each one inch square, one opening one inch deep and 144 inches long, or one opening twelve inches square, the tops of all the openings being five inches below the surface of the water. This simply illustrates the unreliable nature of such a unit. In some localities the following standard has been adopted : An aperture twelve inches high by twelve and three-quarter inches wide through one one- and one-half-inch plank, with top of opening six inches below the water-surface, is said to discharge two hundred miner’s inches. By this standard the miner’s inch is 1.5 cubic feet per minute, or 2160 cubic feet in twenty-four hours. Other standards vary from 1.39 to 1.78 cubic feet per minute.* When the miner’s inch can only be defined as a certain num- ber of cubic feet per minute, it is evidently no longer of ser- vice and should be abandoned. The method by weirs is more accurate, and could almost always be substituted for the method by orifices. 259. The Flow of Water in Open Channels. — For more than a century hydraulic engineers have labored to find a fixed relation between the slope and cross-section of a running stream and the resulting mean velocity. If such a relation could be found, then the discharge of any stream could be obtained at a minimum cost. It is now known that there is no such fixed relation. There certainly is a relation between the bed of a stream for a considerable distance above and below the section. * See Bowie’s “ Hydraulic Mining,” p. 126 (John Wiley & Sons, New York). 324 SUK VE YING. the surface slope, and the resulting velocity at the section ; but as no two streams have similar beds, nor the same stream in different portions of its length, and since the bed character- istics are difficult to determine, and, furthermore, are constantly changing in channels in earth, the function of bed cannot be incorporated into a formula to any advantage except for chan- nels of strictly uniform and constant bed, in which case the cross-section would sufficiently indicate the bed. Again, the slope cannot be profitably introduced into a velocity formula except where it is uniform for a considerable distance above and below the section, for the inertia of the water tends to produce uniform motion under varying slopes, and the effect is that the velocity at no point corresponds strictly to the slope across that section. For uniform bed and slope, how- ever, formulae may be often used to advantage. Let A — area of cross-section ; V = velocity in feet per second (= / for one second) ; p =z= wetted perimeter ; r — hydraulic mean radius = -- ; s — surface-slope =: sin / = ; Z = fall per length /; Q = quantity discharged in one second ; S = wetted surface in length /=://; /= coefficient of friction per unit area of S; p = weight of one cubic foot of water = density. Since the friction varies directly as the density and as the square of the velocity, we have for the frictional resistance on the mass covering the area 5, R = fftSv\ (I) HYDROGRAPHIC SURVEYING. 325 and the work spent in overcoming this resistance in one sec- ond of time is K — Rv — fpSv^ (2) If the velocity is constant, which it is assunjed to be, then this is also the measure of the work gravity does on this mass of water in pulling it through the height A' — h" — Z, which work is K = weight X fall — Zf>Q — ZpvA ; . . . (3) /. ZpvA = /pSv\ (4) or ( 5 ) A Z But S = pi; - = r; and - = sin z s ; P ^ T'U s or V ~ c . . c . (6) where c is an empirical coefficient to be determined. It is evi- dent that c is mostly a function of the character of the bed, and that it can, therefore, have no fixed value for all cases. Equation (6) is what is known as the Chezy formula. The most successful attempt yet made to give to the coefficient c a value suitable to all cases of constant flow is that of Kut- ter.'^ Kutter’s formula, when reduced to English foot-units, is * Kutter’s Hydraulic Tables, translated from the German by Jackson, »and published by Spon, London, 1876, 326 SURVEYING. S/rs = . , 1.811 , 0.00281 41.6 + ‘ ‘ s / . , 0.0028 1 \ 71 + ^41.6+— \/rs, . . (7) the total coefficient of the radical, in brackets, being the eval- uation of c in equation (6). Here v, r, and s are the same as before, and ii is a “ natural coefficient ” dependent on the nature of the soil, character of bed, banks, etc. Although it was the author’s intention to make a formula that would be applicable even to natural channels, it cannot safely be ap- plied to such unless they have great uniformity of bed and slope. The following values of n are given by Kutter: Planed plank. n — 0.008. Pure cement, n = .009. Sand and cement, n = .010 to .oil. Brickwork and ashlar. n — .012 “ .014. Canvas lining, n = .015. Average rubble, n = .017. Rammed gravel. n = .020. In earth — canals and ditches, n — .020 to .030, depending on the reg- ularity of the cross- section, freedom from weeds, etc. In earth of irregular cross-section, n = .030 to .040. For torrential streams, n = .050. In the last two cases the results are very uncertain. Kut- ter’s tables are evaluated for 7 i = 0.025, - 030 , and .035. HYDROGRAPHIC SUR VE YING. 327 The greatest objection to the use of this formula is the labor involved in evaluating the “r” coefficient. To facilitate the use of the formula this coefficient has been evaluated for a slope of o.ooi in Table* IX. This coefficient changes so slowly with a change in slope that the error does not exceed 3|- per cent if the table be used for all slopes from one in ten to one in 5280, which is a foot in a mile. These tabular co- efficients may therefore be used in all cases of ditches, pipe- lines, sewers, etc. The coefficients are seen to change rapidly for different values of n, so this value must be chosen with care. For brick conduits^ such as are used for water-supply and for sewers, the formula V — i27r°-6=.so-5 was found to represent the experiments on the Boston con- duit, shown in Figs. 78 and 79. This would correspond to a variable value of n in Kutter’s formula, being nearly 0.012 however, as given for brickwork. This conduit is brick-lined. Table X.* gives maximum discharges of such conduits as computed by Kutter’s formula, n being taken as 0,013. The results in heavy type include the working part of the table for sewers. All less than three feet per second when the depth of water is one eighth of the diameter, or when the flow is one fiftieth the maximum. This is as small a velocity as is consistent with a self-cleansing flow in sewers. All values below the heavy-faced type correspond to velocities more than fifteen feet per second when the conduit runs full, and this is as great a velocity as is consistent with safety to the structure. If the velocity is greater than this, the conduit should be lined with stone. * Taken from a paper by Robt. Moore and Julius Baier in Journal of the Association of Engineering Societies^ vol. v., p, 349. This table may also be used for tile drains. 328 SUR VE YING. The maximum flow does not occur when the conduit runs full, but when the depth is about 93 per cent of the diameter. A conduit or pipe will therefore not run full except under considerable pressure or head. The maximum velocity occurs when the depth is about 81 per cent of the diameter. The relative mean velocities and discharges of a circular conduit for varying depths is shown by the following table: Depth of Water. Relative Velocity. Relative Discharge. Depth of Water. Relative Velocity. Relative Discharge. .1 .28 .016 •7 .98 .776 .2 .48 .072 •75 •99 .850 .25 •57 .118 .8 •99 .912 •3 .64 .168 .81 1. 00 .924 •4 .76 .302 •9 .98 .992 •5 .86 •450 •93 .96 1. 000 .6 •93 .620 1. 00 .86 .916 260. Cross-sections of Least Resistance. — From equa- tion (6) of the preceding article it is apparent that for a given channel the velocity varies as the square-root of the hydraulic mean radius, r. But r = hence for a given area of cross- P section the velocity is greater as the wetted perimeter is less. The form of cross-section having a minimum perimeter for a given area is the circular, or for an open channel the semicircu- 7tl^ R lar. In both cases the hydraulic mean radius is r = — - = where R is the radius of the circle. Since it is not always con- venient to make the cross-section circular in the case of ditches and canals, it is evident that the more nearly a polygonal cross-section coincides with the circular form the less will be the resistance to flow. When a maximum flow is desired for HYDROGRAPHIC SUR VE YING. 329 Fig. 87. a given slope and cross-section, therefore, the .shape should conform as nearly as possible to that of a semicircle. To do this, construct a semicircle to scale of the required area of cross-section. Draw tangents for the sides of the section having the de- sired slope and join these by another tangent line at bottom, as in Fig. 87. This gives a little larger section- al area, but some allowance should be made for accumulations in the channel. If the slope is very great and it is desirable to re- duce the velocity of flow, it may be done by making the channel wide and shallow. 261. Sediment-observations. — It is often necessary in sur- veys of sediment-bearing streams to determine the amount of silt carried by the water in suspension. The work consists of three operations, namely: (i) obtaining the samples of water; (2) weighing or measuring out a specific portion of each, mix- ing these in sample jars according to some system, and setting away to settle ; (3) siphoning off the clear water, filtering, and weighing the sediment. Sometimes a fourth operation is re- quired, which is to examine the sediment by a microscope on a graduated glass plate, and estimate the percentages of differ- ent-sized grains. The sedimentary matter carried in suspen- sion may be divided into two general classes, — that in continu- ous suspension, and that in discontinuous suspension. The former is composed of very fine particles of clay and mud whose specific gravity is about unity, so that any slight dis- turbance of the water will prevent its deposition. This once taken up by a running stream is carried to its mouth or caught in stagnant places by the way. The matter in discontinuous suspension consists of sand, more or less fine according to the velocity and agitation of the current. This matter is con- stantly falling towards the bottom and is only prevented by the 330 SURVEYING. violent motions of the medium in which they are suspended. These particles are constantly being picked up where the ve- locity is greater, and dropped again where the velocity is less. A natural channel will therefore carry about the same per- centage of fine or continuous matter between two consecutive tributaries, but of the coarser material there will be no uniformity whatever in successive sections in this same stretch of river. In natural channels there are always alternate engorged and enlarged sections for any particu- lar stage of river, and the positions of these en- gorgements and enlargements are different for different stages. In fact, the engorged sections at high water are usually the enlarged sections at low water, and vice versa. If the bed is fria- ble the engorged section is always enlarging, and the enlarged section is constantly filling as a result of the discontinuous movement of sedi- mentary matter. The cause of these relative changes of position of engorged and enlarged sections is the great variation in width. It is the discontinuous sediment which is of principal significance to the engineer, for this is the material from which sand-bars are formed which obstruct navigation, and it is also the ma- terial from which he builds his great contraction works behind his permeable dikes. The water being partially checked behind these dikes at once drops the heavier sediment, and so artificial banks are rapidly formed. The continuous sediment is of little consequence to the engi- neer. * See paper by the author entitled “ Three Problems in River Physics,” be- fore the American Association for the Advancement of Science, Philadelphia meeting, 1884. H YDROGRAPHIC SUR VE YING. 331 262, Collecting the Specimens of Water. — It is neces- sary to take samples of water from various parts of the cross- section in order to obtain a fair average. Surface and bottom specimens should always be taken, and if great exactness is required specimens should also be taken at mid-depth. One of each of these should be taken at two or three points on the cross-section. A full set of specimens is collected once or twice a day. A special apparatus is required for obtaining samples from points beneath the surface. The requirements of such an apparatus are very well satisfied by the device shown in Fig. 88, which the author designed and used very successfully in a hydrographic survey of the Mississippi River at Helena, Ark., in 1879.^ C is a. galvanized iron or copper cup ; /an iron bar one inch square; L a mass of lead moulded on the bar at bottom ; B the bottom cup for bringing to the surface a specimen of the bottom, I being a leather cover; W the springing wire by which the lids a a are released and drawn together by the rubber bands b b when the apparatus strikes the bottom, or when this wire is pulled by an auxil- iary cord from above ; d d adjustable hinges allowing a tight joint on the rubber packing-disks c c when the lids are closed. In descending, the lids are open and the water in the can C is always a fair sample of the water surrounding the apparatus. When the lids are closed the sample is brought securely to the surface. The can when closed should be practically water- tight ; if it leaks at bottom some of the heavier sediment is likely to escape, for it settles very quickly. The bottom speci- men should be taken about a foot above the bottom to avoid getting an undue portion of sand which is at once stirred up by the apparatus striking the bottom. 263. Measuring out the Samples. — A given portion of each specimen by measure or by weight is selected for deposition. * See Report of Chief of Engrs., U. S. A., 1879, vol. iii., p, 1963. 332 SURVEYING. Great care must be exercised in obtaining the sample volume. It cannot be poured off, even after violent shaking, for the heavy sand falls rapidly to the bottom. A good way is to draw it from the vessel by an aperture in its side while the water is stirred within ; greater accuracy can be attained by weighing the sample of water than by measuring it. All the samples of a given kind are then put together in one jar, which is properly labelled, and set away to settle. Thus, all the sur- face samples are put into one jar, the mid-depth samples in another. The Mississippi and the Missouri River water re- quires about ten days’ settling to become clear. 264. Siphoning off, Filtering, and Weighing the Sedi- ment. — When the water has become quite clear it is carefully siphoned off, and the residue is filtered through fine filter paper (Munktell’s is best). Two papers are cut and made of exactly the same weight. One is used for filtering and the duplicate laid aside. The filter-paper containing the sediment and also its duplicate are then dried in an oven at a temperature not higher than 180*^. When quite dry the sediment paper is put in one pan of the balance, and the duplicate in the other and weights added to balance. The sum of the weights is the weight of the sediment. This divided by the weight of the sample of water, usually expressed by a vulgar fraction whose numerator is one, is the proportionate quantity sought. CHAPTER XI. MINING SURVEYING. 265. Definitions. — Mining Surveying, like all other classes of surveying, has for its object the determination pf the rela- tive positions of the different portions of the subject of the survey. The same principles which are employed in surveying on the surface govern the engineer in the prosecution of a mining survey. In fact, mining surveying may be considered as an extension of topographical surveying to the accessible portions beneath the surface of the earth, with certain modi- fications of the adjuncts of surface surveying, necessitated by the nature of the case. The parts of a mine included in a mining survey are the surface and surface-workings, shafts, tunnels, inclines or slopes, drifts, stopes, winzes, cross-cuts, levels, air-courses, entries, and chambers. Surface-workings include open cuts, pits, and other exca- vations of limited extent. A Shaft is a pit sunk from the surface more or less perpen- dicularly on the vein or to cut the vein. The inclination of the vein is called the Dip, or Pitch, and its direction across the country is called the Strike. A Tunnel is a horizontal excavation from the surface along the course of a vein, or across the course of known veins, for purposes of discovery. The approach to a tunnel is called an Adit. An Incline or Slope is a tunnel run at an angle to the hori- zontal. 334 SURVEYING. A Drift is a tunnel starting from an underground working such as a shaft. When there are a series of drifts at different depths, they are termed Levels; as first level, second level, or 50-foot level, lOO-foot level, etc. A Stope is the working above or below a level from which the ore is extracted. An overhand or back stope is the work- ing above a level ; an underhand stope is the working from the floor. A Winze is a shaft sunk from a level. A Cross-cut is a level driven across the course of a vein. An Air-course is a tunnel driven for the purpose of venti- lation. An Entry is a passage through a mine. A Chamber is a large room from which the ore is mined. The last two terms are used more especially in coal-mines, where the vein lies flat or nearly so. MINING SURVEYING. 335 The operations of a mining survey are conducted like those of a topographical survey. An initial point is selected usually from its importance to the object sought, and all the subse- quent stations are connected either directly or indirectly with it, and their positions with reference to it shown on the map of the survey. 266. Stations are occupied by candles or lamps constructed for the purpose, in place of poles, flags, etc., as on the surface. An illuminated plumb-line is a good substitute for a lamp, and gives the observer a greater vertical range, which is helpful in case the station is obscured by intervening objects. Owing to the peculiar nature of the survey, it is imprac- ticable and sometimes inexpedient to mark stations as on the surface ; recourse is therefore had to other devices, which must be employed to suit circumstances. It would not be advisable, even if it were practicable, to leave a station-mark on the bottom of any portion of a mine, as frequent passing would disturb or obliterate it. It is better therefore to leave the mark overhead, if acces- sible and not liable to be disturbed, either by driving a nail in a timber should one be convenient, or by drilling a hole in which may be inserted a wooden plug, properly marked, or simply by cutting a cross or other device on the exposed sur- face. Another method, where the above cannot be employed, is by marking points on the walls and measuring the respective distances from the station to them. 267. Instruments . — Steel tapes only should be used for measuring, being more convenient and less liable to inaccuracy than a chain. These may be of different lengths to suit the work on which they are to be employed ; sometimes, as in the case of coal-mine surveys, tapes of several hundred feet in length can be employed to advantage. The Compass., unless used as an angular instrument to de- flect from an established line, should not be employed in min- 33*5 SUK VE YING. ing surveys, as the variation between stations is so inconstant as to render it unreliable when used to deflect from the mag- netic meridian. The magnetic needle may be used, however, in connection with the transit as a check. The Transit alone should be used in important work, and certain additions to it for vertical pointings will be found indis- pensable. In sighting up or down a shaft the ordinary form becomes r.seless when the line of sight passes inside the upper plate of the instrument. A prismatic eye-piece. Fig. 90, will overcome this difficulty for upward sights, but the survey cannot be carried downward by its use> An extra telescope attached either to the top or side of the central telescope will overcome this difficulty. The attach- ment to the top is made as shown in Fig. 91 by coupling-nuts, Fig, 91. which fasten it firmly over the centre of the instrument. The attachment to the side, Fig. 92, is effected by means of a spindle from the attached telescope which fits into the hollow axis of the central telescope and is secured by means of a clip which MINING SURVEYING. 337 passes through both the axis and spindle. A counterpoise is similarly attached to the opposite side of the central telescope to preserve the equilibrium. This latter form of attachment is more com- pact than the former, the principal objection to it being that a correction must be applied to each reading of a horizontal angle equal to the tangent of an angle whose opposite side is the distance between the centres of the telescopes, and whose adjacent side is the horizontal dis- tance between stations. This objection, how- Fig. 92. Fig, 93. ever, is removed by a simple device. Two brass tubes, Fig. 93, about two inches long, are connected by an intermediate web, so that the distance between their centres shall exactly equal that between the centres of the telescopes. One of the tubes is of sufficient size to enclose a pike-staff graduated to fractions of a foot, upon which it can be easily moved to any desired 22 338 SUA' VE Y/NG. height, and the other large enougli to contain a candle, and has a light plumb-bob suspended below its centre the better to maintain the staff in a perpendicular position ; the staff now being placed over any station and the brass tube and candle set at the height of the instrument on the staff, which is held in a perpendicular position with the line between the tubes parallel to the horizontal axis of the telescope, a reading can be made to the flame of the candle which gives at once the true azimuth of the line and the dip of the shaft. In using this device the side telescope and tube carrying the candle should always be on the same side of the line. The transit must always be placed exactly over the point occupied by the foot of the staff ; and here it may be well to state that the greatest care and accuracy must be exercised in exactly centring the instrument over the station, as the courses are carried forward entirely by deflection angles, so that an error introduced at one station is carried through all and increased at each. Again, in sighting down a shaft, although the per- pendicular distance may be considerable, the horizontal dis- tance between stations must be small, so that even a slight error made in a shaft will be of considerable magnitude when carried out in the levels of a large mine. The transit should have an extension tripod. Fig. 94, so that one or more of the legs can be shortened, the better to place it over a station on a steep mountain-side or in a mine, or to lower the instrument to see under intervening objects, or to adapt it to different heights of the workings of the mine. The Mining Transit should be provided with the Solar At- tachment, that all lines of the survey may be referred to the meridian. In unimportant surveys the pitch of the shaft or the dip of the vein may be determined by the clinometer or by measuring the horizontal and perpendicular distances between any two conveniently located points of the foot or hanging wall and MINING SURVEYING. 339 calculating the pitch or dip from the measurements thus ob- tained. A plummet-lamp, Fig. 95, will also be found very convenient. Fig. 94. Fig. 95. 268. Mining Claims. — The first work of the surveyor upon a mining claim is its location. Mining claims are of different dimensions according to the local laws and customs of the 340 SUR VE YING. country ; varying from 50 feet to 600 feet in width and from 100 to 3000 feet in length. In the earliest days of Western mining, the dimensions of a claim were decided at a convention of all the miners in the district. Now the United States laws limit the length to 1500 feet, but the width still varies not only in different States, but in different counties in the same State. The form of a mining claim is essentially a parallelogram, being regulated by the U. S. mining laws, which prescribe that whatever the relative position of the side lines to each other, the end lines must be parallel. This is to prevent more than fifteen hundred feet of a vein or lode from being included in one claim. The side lines of a claim may be straight lines extending between the ends of op- posite end lines, or they may be broken lines to include the vein if it should be curved, so as to pass outside straight lines ; but in any case they can only include 1500 feet of the vein measured along the centre line of the claim. A mining claim is included between parallel vertical planes passed through the end lines ; but a miner has a right to follow his vein downward, although it so far passes from the perpendicular in its down- ward course as to extend beyond vertical planes passed through its side lines. The above are the essential features which govern the shape of a mining claim. The method of procedure in making the location is as fol- lows: When the discoverer of a mine has sunk his shaft ac- cording to law, so as to expose 10 feet of the vein, he is entitled to have his claim surveyed and recorded. He then decides how much of the 1 500 feet he desires to extend on either side of his discovery-shaft along the vein. He is governed in this by various considerations, such as his proximity to other claims, the promise of mineral in different portions of the lode, or the nature of the ground. The surveyor begins his survey for Fig. 96. 342 SUI^ VE YING. location at the point of discovery, and runs from it in opposite directions until he has measured off 15CX) feet. The survey has thus far been run on the centre line of the claim. On arriving at the ends of the line, the surveyor measures off half the width of the claim on each side of the centre line, generally at right angles to it if the claim is straight, and sets his corners at the ends of the end lines. He also places monuments on the side lines, midway between the corners, called the side-line centre monuments, the law re- quiring that the claim shall be distinctly marked upon the ground, so that its boundaries can be readily traced. This much of the survey being now completed, it remains to run a tie line from some corner of the claim to a well-known monu- ment. This must be a section corner of the Government surveys if the claim be on surveyed lands, otherwise to a prominent natural object, or to a locating monument estab- lished for the purpose. This is done to identify and locate the claim so that its locus may be a matter of record. An example of an actual plat of mining claims is shown in Fig. 96. The next survey of a mining claim is its survey for patent of the United States. The original location maybe made by any surveyor, and is sometimes made by the miner himself; but the survey on which the patent or title from the United States is issued must be made by a deputy of the U. S. Surveyor-general of the. Public Lands, who is thus known as a U. S. Deputy-mineral-surveyor. These officers give bonds to the Government in the sum of $10,000 for the faithful perform- ance of their work, and are required to pass an examination, that the Surveyor-general may be satisfied of their capability. The survey for patent must be made with the greatest care and accuracy. It must exactly locate the claim with reference to a corner of the public surveys, if such be within two miles, and must show the nature and extent of the conflict with other official surveys if it should conflict with any, or with MINING SURVEYING, 343 other mining claims not officially surveyed if it is desired to exclude from the claim the area in conflict. A specimen of the field-notes of a survey for patent issued for the instruction of the U. S. Deputy-surveyors of Colorado is given in Appendix B. The Surveyors-general of the different States and Territo- ries issue instructions to their deputies, and these, with a knowl- edge of the U. S. mining laws, must govern the surveyor in his work; but as they are more strictly legal than mathematical, it is not important to consider them in this chapter.* The foregoing surveys are strictly land surveys, and are only mentioned to illustrate the method of staking out a min- ing claim and to give some idea of the shape and size. UNDERGROUND SURVEYS. 269. Mining Surveying proper, or the underground work of the survey, will be considered in a few practical examples selected from actual cases. 270. To determine the Position of the End or Breast of a Tunnel and its Depth below the Surface at that Point. — Set up the instrument at a point outside the tunnel, so as to command as long a sight as possible into the tunnel and also the surface of the mountain above it. If the end of the tun- nel can be seen from the station a course and distance can be taken at once to the breast, and this course and distance dupli- cated on the surface. Vertical angles can then be measured to the points thus determined on the surface and in the tun- nel, and the calculation of the depth of the breast below the surface may be made from the data thus obtained. * Copies of the Instructions can be procured of any Surveyor-general on application. Those for Colorado are given in Appendix B. The U. S. mining laws, together with all State and Territorial laws and local mining rules and regulations, are compiled in vol. xiv. of the U. S. Census for 1880, 4to, 705 pp., 1885. This is a most valuable publication. Price $4 if not obtained through an M. C. 344 SURVEYING. In case the breast is not visible from the first station, take as long a sight as practicable to Station No. 2, and before re- moving the instrument reproduce Station No. 2 upon the sur- Fig. 97. face as in the preceding case, thus avoiding a resetting of the instrument at Station No. i when the underground work is completed. At the same time measure the vertical angle to Station No. 2 in the tunnel. Set the instrument at Station No. 2, and, after having obtained the back readings to Station No. I, measure the course, distance, and vertical angle to Sta- tion No. 3. Repeat the above operations at the different stations until MINING SUR VE YING. . 345 the breast is reached, taking any measurements of the dimen- sions of the work that may be necessary, and leaving station marks for future reference, as described in article 266. Set the instrument over Station No. 2 on the surface and very care- fully duplicate the courses and distances measured in the tun- nel, at the same time noting the vertical angles between the surface stations. The vertical angles can be measured most easily by sighting -to a point on a short staff at a height above the station equal to the height of the instrument. It is advisable to explore the tunnel before surveying it, as then any difficulties can be provided for and the stations selected more advantageously. Sometimes the course from Station No. I to Station No. 2 is assumed as a meridian of the survey and all courses deflected from it, but it is better to use the true solar course between these stations because the field notes can then be placed in the table for calculation without further reduction. Example. — Following is a specimen of field-notes of the survey of a tunnel both underground and surface: FIELD-NOTES. Station. Verticai in Tunnel. L Angles on Surface. Course. Distance. Remarks. S. 36° 50' W. S. 36° 50' W. 19. 1 ft. 99.1 to mouth of tunnel. 4- 1° 18' 4-10° 35' to Sta. No. 2. 2 4- 0° 31' +15“ 43' S. 49° 47' W. 104.2 “ “ 3. 3 4- 0° 45' 4-14° 27' S. 40° 0' W. 37.1 “ “ 4. 4 1 0 3 4-16° 17' S. 4°55'E. 56.5 “ “ 5. 5 4 - 3 ° 37' 4-12“ 21' S. 71“ 15' E. 46.0 “ “ 6. 6 4- 3° 30' 4-13° 56' S. 77° 30' E. 40.7 to breast of tunnel. Breast.. -17'’ 56' N. 16° 16' E. 266.57 from station on surface over breast of tunnel to Sta. No. I. 346 SURVEYING. The following table shows the method of reducing the survey. The first six columns represent the ordinary method of reducing a traverse to a straight line. The agreement between the resultant and the check course proves the accuracy of the field work. Columns 7 and 8 contain the vertical angles in the tunnel and the rise or fall in feet corresponding to them. Columns 9 and 10 similarly contain the vertical angles of the courses and distances on the surface and the difference of elevations between stations corre- sponding to them. The algebraic sum of the vertical heights in the tunnel gives the difference of level between the Station No. i and the breast ; and the sum of the differ- ences of elevation in column 10 gives the total difference of elevation between Station No. i and the point on the surface over the breast. The difference of columns 8 and 10 shows the depth of the breast of the tunnel below the surface. The tangent of the angle obtained by dividing the sum of the elevations in column 10 by the length of the resultant distance should agree with the vertical angle read to Station No. i from the point on the surface over the breast of the tunnel. The following is the form used in reducing the field-notes: OFFICE FORM. Course. Dist. Latitude. Departure. Vert. Angle. Rise or Fall. Vert. Angle. Rise or Fall. N. S. E. W. S. 36° 50' W. 99.1 79-32 59-41 + 1° 18' + 2-24 + 10® 35' 4 - 18.51 S. 49° 4/ w. 104.2 67.28 79-57 + 0° 31' + 0-94 + 15“ 43' + 29.32 S. 40° 00' w. 37-1 28.42 23.84 + 0° 45' + 0.48 + 14° 27' + 9-56 S. 4 ° 55 ' E. 56.5 56.29 4.84 - 0° 34' — 0.56 + 16° 17' + 16.50 S. 71° 15' E. 46 14.78 43-56 + 3 “ 37' + 2.91 4- 12® 21' + 10.07 S. 77° 30' E. 40.7 8.81 39-73 + 3 ° 30' + 2.49 + 13“ 56' 4- 10. 10 Resultant course, N. 16® 20' E. 265.62 254.90 74-69 Total -{- 8.50 Total -f- 94.06 254.90 254.90 162.82 162.82 8.50 Depth below surface = 85.56 Check : 85.56 ■+■ 265.62 = 0.3221 = tan 17° 51'. 271. Required, the Distance that a Tunnel will have to be driven to cut a Vein with a Certain Dip. — Case I. W/ien the direction of the tunnel is at right angles to the course, or par- allel to the pitch of the vem. The dip having been first ascer- MINING SURVEYING. 347 tained by sighting down a shaft sunk on the vein, or by any other practicable method, set up the instrument on the apex or outcrop of the vein directly over the line of the proposed tunnel and measure the vertical angle and horizontal distance to the mouth of the tunnel. From the results obtained calculate the depth at which the tunnel will intersect the vein, then from this depth and the angle of the dip, calculate the horizontal distance of the vein from a vertical line through the instrument station. This distance, added to or subtracted from the horizontal distance between the station and mouth of the tunnel, accord- ing as the dip is from or toward the mouth, will give the re- quired distance. Example. — A certain vein has a course of N. 45° E. and its pitch is N. 45° W. with a dip of is” from a vertical line. The horizontal distance to the mouth of a cross cut tunnel from the apex of the vein is 200 feet S. 45” E. and the vertical angle is — 25“. At what distance from the mouth will the tunnel intersect the vein? (Fig. 98.) 348 SURVEYING. Depth at which the tunnel will intersect the vein = 200 X tan. 25® = 93.26 ft. Distance of vein from vertical line at depth of 93,26 ft. = 93.26 X tan. 15“ = 24.99 Adding the last result to the horizontal measured distance, we have 24.99 -j- 200 = 224.99 ^he distance from the mouth of the tunnel to its intersection with the vein. Case II. IV/icn the direction of the tunnel is oblique to the course of the vein. Proceed as in Case I. and measure the hori- zontal distance from the instrument station to the mouth of the tunnel, the vertical angle and the dip, also the angle which the course of the vein and the line of the tunnel make with each other. Calculate the depth at which the tunnel will in- tersect the vein, and the distance of the vein at the tunnel level from a vertical line through the station, as in the previous case. Multiply this distance by the cosectant of the angle between the courses of the vein and tunnel and apply it to the meas- ured horizontal distance, as in Case I., and we have the re- quired result. Example, — A certain vein has a course of N. 45® E. and its pitch is N. 45® W. with a dip of 15° from a vertical line. The horizontal distance to the mouth of a cross-cut tunnel running due west is 200 ft. due east and the vertical angle is —25°. At what distance from the mouth will the tunnel intersect the vein ? (Fig. 99.) Depth at which the tunnel will intersect the vein = 200 X tan, 25° = 93.26 ft. Distance of vein from vertical line at depth of 93.26 ft, = 24.99 Angle between course of vein and line of tunnel = 45°. Multiplying, 24.99 X cosec. 45° = 35.34 ft. Add the result to the horizontal measured distance and we have 2004-35.34 = 235.34 ft., the required distance from the mouth of the tunnel to its intersection with the vein. 272. Required, the Direction and Distance from the Breast of a Tunnel to a Shaft, and the Depth at which it will cut the Shaft. — Make a survey of the tunnel and repro- duce it upon the surface, as in the first example. Calculate the depth of the breast below the surface. Set up the instru- MINING SURVEYING. 349 merit at the shaft and measure the vertical angle and hori- zontal distance to the point on the surface over the breast. Calculate their difference of level from the measurements ob- tained and add it to or subtract it from the depth of the breast below the surface. The result is the depth of tunnel below the mouth of the shaft. Survey the shaft to a point whose vertical depth is equal 350 SUK VE YING. to the depth of the tunnel level. Calculate the horizontal distance and direction of this point from the instrument station at the mouth of the shaft, and mark its position upon the surface. Connect it with the point marking the position of the breast of the tunnel, and we have the line required. From the information thus obtained range-lines can be sus- pended in the tunnel, to give the direction of the shaft from the breast. Example. — A shaft whose centre at the surface bears S. 65“ E. 73 ft., verti- cal angle 10° 20', from a point on the surface over the breast of the tunnel in the first example, has a pitch of 14° 30' N. 2° 15' W. from a vertical line. At what direction and distance from the breast of the tunnel will it cut the shaft, and at what depth ? (Fig. 97.) The following is the form of the field-notes: FIELD-NO'TES. Station, Vertical Angles. Course. Distance. •Remarks. In Tunnel. On Surface. I S. 36° 50' W. 19. 1 To mouth of tunnel. + 1° i8' + io» 35' S. 36° 50' W. 99.1 “ Station No. 2. 2 -f 0° 31' + 15° 43' S, 49° 47' W. 104.2 “ “ “ 3. 3 + 0° 45' + 14° 27' S. 40° 00' W. 37-1 4. 4 - 0° 34' 4- 16° 17' S. 4° 55' E. 56.5 5 + 3 ° 37' 4- 12° 21' S. 71“ 15' E. 46.0 “ “ “ 6. 6 + 3 ° 30' 4 - 13° 56' S. 77° 30' E. 40.7 “ breast of tunnel. Breast. Centre of In shaft. -|- 10® 20' S. 65° 00' E. 73 In shaft. “ centre of shaft. Shaft. - 75 ° 30' N. 2° 15' W. 102.12 “ point in shaft at tunnel level. The depth of the tunnel at the breast is determined as in the first example. The vertical distance between the point over the breast and the mouth of the shaft is determined by the equation; Difference of elevation = 73 X tan 10* 20' = 13.31 ft. Add this to the depth of the tunnel at the breast and we have 13.31 -|- 85.56 = 98.87 ft., the vertical depth at which the tunnel will cut the shaft. With this MINING SURVEYING. 351 depth and the pitch of the shaft, 14° 30', we obtain the depth (distance along the shaft) at which the tunnel will cut the shaft, measured along the dip, and also the horizontal distance from the instrument station, which is the intersection of the centre line of the shaft with the surface, to the point of intersection, by the following equations: Depth measured on the dip = 98.87 X sec 14° 30' = 102.12 ft. Horizontal distance = 98.87 X tan 14® 30' = 25.57 ft- Set a stake N. 2” 15' W. 25.57 ft. from the instrument, and connect it with the point over the breast of the tunnel, and we have the course and distance from the breast of the tunnel to the line of survey down the shaft, S. 85° 21' E. 65.22 ft. 273. To survey a Mine, with its Shafts and Drifts. — Set up the instrument at the top of the main shaft, and after having first obtained the meridian, take the bearing-distance and vertical angle to the point selected for the first station in the shaft. The distance is to be measured on a direct line be- tween the stations, and its horizontal and vertical components afterwards calculated from the data obtained. The stations in the shaft are to be selected with a view to the extension of the survey into the different levels and down the shafts, and, as in case of other underground surveys, it is well to explore the mine ahead of the work, that the stations may be selected advantageously. The field-notes of the survey of a mine are here given for illustration. The horizontal and vertical components of the distances measured down the shafts can be obtained by the use of a table of natural sines and cosines. (Figs. 100 and loi.) 352 SURVEYING. FIELD.NOTES. Sta. Course. Dist. Vertical Angles. N. 53° 57' W. 54-1 t 0 a N. 34« 52' E. 57-6 0® 00' S. 25» 13' W. 60.8 0° 00' N. 57» 46' W. 60.0 - 66® 24' 3 N. 30® 30' E. 73-0 0® 00' N. 69® 23' W. 47-5 - 77® 30' 4 N. 28® 24' E. 75 0® 00' S. 19° 59' W. 90 0® 00' N. 71® 15' W. 55 — 78® 00' 5 N. 37* 20' E. 75 0° 00' S. 28® 43' W. 64 0® 00' N. 72® oi' W. 55-6 1 0 0 to 0^ 6 S. 71® 50' E. 5-1 0° 00' S. 71® 50' E. 8.1 0 00' 7 N. 66° 15' W. 46.8 — 85® 00' 9 N. 32® 50' E. 60 0® 00' S. 24® 00' W. 51 0° 00' N. 55“ 03' W. 40 - 86® 46' 10 N. 34® 15' E. 39 0° 00 ! S. 37“ 45' W. 55 0® 00' N. 88® 30' W. 48.6 — 81® 00' 11 N. 34® 00' E. 54 0® 00' S. 24® 00' W. ^5 0® 00' Hori- zontal Com- ponent. ^ 3-37 10.76 ”•43 2.25 7.60 Vertical Com- ponent. 54-98 46-37 S 3 -So 54.67 46.62 39-93 48.0 Remarks. Begin at Sta. i at top of shaft. To Sta. 2 at 1st level in shaft. " air-shaft at end of ist level. “ centre of bottom of discov- ery-shaft, 50 ft. deep. “ Sta. 3 at 2d level in shaft. ‘‘ breast of 2d level, running N.E. Second level run- ning S.W. filled with de- bris, not accessible. “ Sta. 4 at 3d level in shaft. “ breast of 3d level, running N.E. “ breast of 3d level, running S.W. “ Sta. 5 at 4th level in shaft. “ 4*^^ level, running “ breast of 4th level, running S.W. “ Sta. 6 at 5th level in shaft. The vein here divides: the shaft follows the por- tion to the south. The shaft is chambered out at this point, being 10 ft. wide S.W. , and 20 ft. long S.E. from Sta. 6. “ Sta. 7 at top of shaft in chamber. “ Sta. 8 opposite drift, run- ning S. 24* 15' W., 102 ft. “ Sta. 9 at 6th level in shaft. “ breast of 6th level, running N.E. “ breast of 6th level, running S.W. “ Sta. 10 at 7th level in shaft. “ breast of 7th level, running N.E. “ breast of 7th level, running S.W. “ Sta. II at 8th level, at bot- tom of shaft. “ breast of 8th level, running N.E. “ breast of 8th level, running S.W. MINING SURVEYING. 353 Note. — The width of the levels of this mine are about four feet. The dimensions of the shaft are 4' X 10'. The line of survey down the shaft was Fig. ioo. run about 2 ft. from the north end of the shaft. In the levels the line of survey was to the centre of the breast. 23 354 SURVEYING. Fig. loi shows the plan, and Fig. too the longitudinal and transverse sec- tions of the mine as plotted from the field-notes. The plan is plotted from the courses and horizontal components of the measurements in the shaft and levels, as projected upon a horizontal plane. The longitudinal section is platted from the courses and vertical components of the measurements in the shaft, and the horizontal measurements in the levels as projected upon a vertical plane passing through Station i and at right PLAN * Fig. ioi. angles to a vertical plane passing through Stations i and ii, upon which the transverse section is plotted as projected from the vertical angles, courses, and measurements in the shaft. Thus it will be seen that for the full representation of an underground sur- vey, showing the relative position of the parts to each other, three planes are necessary, — two vertical planes at right angles to each other, and a horizontal plane. 274. Conclusion. — The above examples comprehend some of the more general cases arising in the practice of mining MINING SURVEYING. 355 surveying; any other cases which may arise will be found to be modifications or combinations of these. The problem to be considered can be solved by an application of the principles therein embraced, which the ^surveyor will find useful, also, in solving problems of mining engineering relating to the meas- urement of ore reserves, development, and systems of working. It has been shown that the following out of the underground workings of a mine corresponds to traversing when elevations are carried by means of vertical angles, as was fully described in the chapter on topographical surveying. The notes are also reduced in the same manner. It has been the object of this chapter to present the subject of mining surveying in as simple a form as possible, and divest it of all features which, although they may give it a distinctive aspect, serve only to render it more complex and give the reader an idea of difficulties which are only imaginary. It is useless, also, for the mining surveyor to encumber him- self with many paraphernalia. Good work can be done with a mining transit provided with an extra telescope for vertical pointings, one or two short rods, and a reliable steel tape, all of which can be carried by the surveyor on horseback over the rough mountainous roads. Any other adjuncts can be im- provised or be found at any well-conducted mine, and would prove more burdensome than useful. CHAPTER XII. CITY SURVEYING. 275. Land-surveying Methods inadequate in City Work. — The methods described in the chapter on Land-surveying are inadequate to the needs of the city surveyor. The value of the land involved in errors of work, with such a limit of er- ror as was there found practicable (see art. 175), is so great as to justify an effort to reduce this limit. Comparing the value of a given area of the most valuable land in large cities with the value of a like area of the least valuable land which a sur- veyor is ever called upon to measure, the ratio is more than a million to one. This view is emphasized by the manner of use. On farm lands the most valuable improvements are placed far within the boundary-lines, but the owner of the city lot is compelled by his straitened conditions to place the most costly part of his improvements on the limit-line. His neighbor’s wall abuts against his own. The surveyor, who should retrace this line and make but a small difference of location, would get his clients and himself into trouble. Both the value of the land and the manner of its use demand increased care. The modi- fications of the methods used in land-surveying to meet the requirements of work in the city will be treated in this chapter. Much of the work described furnishes data for the solution of engineering problems, but the obtaining of the facts falls en- tirely within the definition of surveyor’s work. CITY SURVEYING, 357 276. The Transit is used exclusively, but the common pat- tern may be very materially modifi^ed with obvious advantage. Seeing that the magnetic needle* is never precise and seldom correct, it should be wholly discarded in the construction of the city surveyor’s transit. The verniers can then be placed under the eye, the bubbles can be removed from the standards and placed upon the plate of the alidade, and the standards themselves can be more firmly braced. By these changes a steadier and more convenient instrument is secured, when the useless and somewhat costly appendage of a needle-box is out of the way. The adjustable tripod head and the levelling attachment are always convenient. For topographical work, the vertical circle, or a sector, and stadia wires are essential, otherwise the methods used must be primitive. The ther- mometer which is needed in order to make the proper correc- tions for temperature may be conveniently attached to one of the standards facing the eye-piece of the telescope. The danger of breaking the tube while handling the instrument may ‘be obviated by a guard sufficiently deep to protect the bulb, made open on the side toward the observer. 277. The Steel Tape is generally used for measuring. The legal maxim that “ distances govern courses,” when interpreted, means that, using customary methods, the intersection of two arcs of circles, centres and radii being known, is a more definite lo- cation of a point than the intersection of two straight lines whose origin and direction are likewise known. The fact is, the inter- sections are not more definite. The maxim grew into authority when the compass was pitted against the chain. With the transit to define directions of courses, and the chain still to measure the distances, such a maxim would not have voiced the results of experience, but would have been sheer nonsense. * The needle finds its proper place where checks are not so abundant, and in classes of work in which a close and rapid approximation is of more value than precision. 358 SUR VE YING. The ordinary chain has too many gaping links, and the brazed chain too many wearing surfaces, to be kept in very close ad- justment to standard length. Its weight is such as to make the “normal tension” (see p. 375) impracticable; hence the effect of slight variations of pull is much more marked than if the tape is used. Graduated wooden rods were used until i860 to 1870. They were unwieldy when twenty feet long, and were still so short that the uncompensated part of their compen- sating errors was a matter of considerable moment. Every time the pin is stuck or a mark made at the forward end of the tape or rod, the work is a matter of skill and involves an error dependent on the degree of skill attained. When the measure is brought forward, its proper adjustment in the new position is a matter requiring skill. These errors are compensating, but the resultant is not zero. The use of the plumb-line is another source of compensating errors which are reduced by an increase of length in the measure. First, the number of applications varies inversely as the length of the measure ; second, using the rod, it was necessary to work to the bottom of ravines and gul- lies and then work up again ; now the long tape spans them at a single application. The minus errors due to imperfect align- ment and inaccurate levelling of the two ends have a greater percentage of effect when the measure is short than when it is long. The longer tape brings with it some other sources of error. When used suspended at the ends there is a minus error on account of the sag of the intermediate parts, and a plus error from elongation due to tension ; there is also expan- sion by heat, which produces an error which may be plus or minus as the temperature at the time and place is above or below that for which the tape is tested. The effect of sag increases very nearly as the cube of the length when the ten- sion is constant. When, to counteract this increase, the pull is made greater than a man can apply uniformly under all conditions — at his feet or above his head — there come CITY SURVEYING. 359 irregularities from this cause. The limit of length of tape which it is practicable to use will be determined by the condi- tions of the work and should be such that the increase of length involves greater error than it eliminates. On account of convenience in keeping tally, 50-foot and lOO-foot lengths are generally used. In a level country the lOO-foot tape is pre- ferred. There are tapes made with the purpose to eliminate the errors which arise from the free-hand pull, the inclination of the tape, and the temperature. As seen by the writer, they carry a spring-balance marked for a pull of ten pounds, a bubble adjusted to the inclination of the end of the tape at that pull, and a thermometer graduated to such a scale that each division corresponds to one turn of a screw adjusting the whole length of the tape to the changes of temperature. The whole was connected by rings and swivels, eight or nine wearing surfaces, some of them conical, to a tape which carried no graduation. The effort is laudable ; but, probably on ac- count of the number and form of the wearing surfaces, they have not yet met with general favor. Further progress may be made in this direction. LAYING OUT A TOWN SITE. 278. Provision for Growth. — Cities grow. It is very rare that the considerations which should have governed have been given any place in determining upon the plan of the original town. The considerations first in importance are topographi- cal. What are the natural lines along which business will tend to distribute itself? To what form of subdivision can it adapt itself with the least resistance ? Where is the best harbor, the lake or river front, or the railway line ? Ordinarily the land immediately adjoining such natural features is not best used when used as a street, but when occupied by private 36 o SURVEYING. docks, or along a railway by warehouses and factories having switching facilities without crossing public streets. The streets parallel to such lines should be of ample width, easy grade, and continuous but not necessarily straight align- ment. Much of the heavy hauling will be along such streets. In the business part of the town the cross-streets should be so frequent as to make the blocks approximately square. In the residence portion alternate streets in one direction may with advantage be omitted: this saves the expense of unneces- sary streets, and permanently lightens the burden of taxation. Which fronts are on all accounts most desirable in the par- ticular locality will determine in which direction the blocks should be longest. 279. Contour Maps. — Another phase of topography de- mands attention. The sites of suburban towns may generally be best handled by laying out streets and lot lines in conformity to the undulations of the ground. Additions to the city may also have characteristic features that can be preserved with advantage. For all such cases a contour map is very useful to one who is able to interpret it. The making of all the ground available, and sightly points accessible, and at the same time so locating the streets as to secure economical grades, — in short, the judicious handling of the whole subject is facili- tated by the study of the contour map. 280. The Use of Angular Measurements in Subdivi- sions. — Shall subdivision lines be located by an angle with the street on which the lots front or by distances from the next cross-street ? Must distances govern courses, what- ever methods are used ? Let us sup- pose, for illustration, that it is re- ^ quired to locate lot o in the accom- FiG. 102 . . 1 T^. \ O panying sketch (Fig. 102). Suppose, farther, that it is possible to measure each of the lines ab eiA- -400- ' 50 2 3 4 5 6 7 8 9 S / -A -A s.SO'i ” ” .» II »» II CITY SURVEYING. 361 and dc with a maximum error of i in 5000 and that the conditions are such as to produce opposite errors in the two lines. Then, ist, the resulting error in locating the line be, i.e. {ab — dc') will be 3-0V0 X 400 X 2 = 0.16 feet. The sine of the angle by which the angle A' differs from A will be = .00107. Hence the change of direction on account of the errors in measurement is 3f minutes. 2d, the line ef must be distant from ab^\Y. 150 feet ^'550 feet, in order that, under like conditions, if it is measured instead of dc, the change in direction shall not exceed one minute. Or the loca- tion may be made by measuring the line ab, or a line near to it where favorable conditions exist, and then repeating ba the same man being fore-chainman ; the principle of reversal is thus applied to this measurement. Then measuring A' = A and repeating the angle, reading both verniers, the error is brought within the maximum error in the pointing power of the instrument. In order to locate be from ab parallel to ad, two monuments marking the line ab need to be known. The other method requires also a monument locating the line ae. It thus appears that when the side-lines of lots are located perpendicular, or at any other known angle with the street upon which the lot fronts, it is susceptible of more accurate location than by two (front and rear) measurements, unless the usual limit of error can be greatly reduced. While it is not likely that maximum errors of opposite character will fall to- gether affecting the work on the same lot, it is quite as im- probable that the maximum error in measuring an angle should vitiate the work of the transit. It is probably quite as easy to reduce the maximum error in measuring an angle to half a minute as it is to keep the maximum error in measur- ing distances down to i in 10,000. 281. Laying out the Ground. — The work of putting the plan upon the ground is a very important one. This is about the worst possible place to do hurried and inaccurate work. 362 SURVEYING. Fences or other styles of marking possession which limit the contour map cannot be relied upon as defining the property- lines. These lines must be accurately located in relation to the streets of the town or of the addition, in order to make practicable such exchanges or sales as may be necessary to ad- just property-lines to the new block-lines. This method is preferable to that which adjusts block-lines to the original property-lines.* As a framework for the whole survey an outline figure, generally a quadrilateral, of sufficient dimensions, and so placed that it can be permanently marked with monuments which will remain accessible when the town is built up, should be located with especial care. All lines should be measured, all angles observed, and all practicable checks introduced. This figure must close absolutely ; that is, the record of the work when completed should be mathematically consistent. Unreasonable errors are to be eliminated by retracing the work. In the adjustment which distributes the remaining errors each part of the work should be weighted (art. 174, Rule 2), for it is very rare that a land-survey is completed under such con- ditions that the man who does the work would be justified, while these conditions are fresh in his mind, in assuming that the probability of error is alike at all points. The angles ad- mit of adjustment independently of the length of the lines. That distribution of the angular errors which reduces the errors of measurement to a minimum has such weight that it can be overruled only by the most positive evidence that the cor- * In some places this idea of the private interest of the proprietor, some- times private spite, is carried to such an extent that it would seem that each man’s farm or garden patch was especially fitted to be a town by itself, laid out with utter disregard to the towns which others are in like manner laying out upon adjacent farms. In this practice the interests of the public for all time are neglected in order to secure a doubtful advantage for one. Where the custom prevails it is better honored in the breach than in the observance. CITY SURVEYING. 363 rections so indicated cannot be the true ones. The distances are then adjusted to the angles so determined. The re- mainder of the work of the subdivision is checked upon the adjusted outline, reasonable errors being distributed and un- reasonable ones retraced. 282. The Plat to be geometrically consistent.— The necessity that the recorded plat should be consistent lies in the use that is to be made of it. A parcel of ground de- scribed by reference to the plat of record should have but one location, not any one of two or more possible locations, as is the case when the plat contains errors on its face. In the course of years the lines of such parcels will be retraced proba- bly many times, at one time by one method, at another time by another equally in accord with the plat. If the plat is not > consistent with itself and with the monuments upon the ground, this error will be pretty sure to find its way into the lot location. When the fault is with the plat, it matters not how the monuments are placed upon the ground ; they cannot mark the chief points and all agree in such a way that if any two remain and the others are lost the relocation will in every case be the same. But this is just what the plat is for — to make a public record of the relation of each part of the sub- division to every other. 283. Monuments. — How many monuments shall be lo- cated, and where shall they be placed ? What material shall be used and how set? Answering the first question, it is plain that no more work should be attempted than can be done well. Better one point and an azimuth than points everywhere and no two agreeing either in distance or direction with the rela- tion described by the plat. But so much should be done well that the labor of locating any point in the subdivision from existing monuments shall not be excessive. The points chosen for placing monuments should be such as will continue to be accessible and will not be ambiguous. The centre lines 364 SURVEYING. of intersecting streets are sometimes marked, giving one monu^ ment to each intersection ; others choose the side-lines, giving four monuments to each intersection of streets. If the blocks are so long that intermediate points are desirable, points on the ridges should be selected. Stone is more often chosen than any other material ; iron bars, gun-barrels, gas-pipe, etc., are sometimes used, driven with a sledge ; cedar posts, say 4" X 4^, are quite durable, and hard-burned pottery is sometimes used. Whatever material is chosen, the foundation, which should be flat — not pointed — must reach below frost; and the centre of gravity is kept as low as possible, so that there shall be no tendency to settle out of place when the ground is soft in the spring. When the tops are much above the surface of the ground, there is a liability that they may be displaced by traffic. Probably the surveyor does not see any traffic, or the prospect of it, when he is doing his work, but the traffic must come before the work of the monument can be spared. It is better to bury the stone wholly and indicate where to dig for it by bearings than to run the risk of losing the whole work through indiscretion in placing the monument that marks it. In situations where every rain storm produces a slight fill it is safe to place the top consider- ably higher than would otherwise be reasonable. The stones to be set are so placed in the excavation, with the heavy end down, that when the top is in the proper position and before any earth is refilled there is no tendency to fall in any direction ; then while the earth is being refilled and thoroughly tamped about the stone, the top is kept in place. It is better that the mark denoting the point for which the stone stands should be cut upon before it is placed in the ground. When this is done, if the mark is worn off by traffic or knocked off by accident, the centre of that portion of the stone which remains is a very close approximation to the original point. A slovenly way of slighting this work is to tumble the stone into the excavation. CITY SURVEYING. 365 fill around it pretty much as it happens, push it to one side or another so that the point will come somewhere on the top, and then cut the mark wherever the point comes. Stones set in this way are liable to settle out of place after the first heavy rain, while frost and rain keep up their work till the stone lies flat upon its side. If by chance it should keep its place pretty well and the mark becomes defaced, it might as well be any loose bit of rock as a set stone, for its centre gives no idea of where the mark was placed. No one should be trusted to set corner-stones unwatched who is not familiar with the work and thoroughly reliable. Points are preserved temporarily by wooden stakes driven flush with the ground. The point, preserved by offsets while the stake is being driven, is marked by a nail. Witness-stakes driven alongside and standing above grass and weeds assist in finding the stakes when wanted. Made of half-decayed soft wood, e.g., old fence-boards, such stakes will hardly last a season ; while durable wood, well seasoned, will last much longer than any driven stake can be relied upon, since it does not go below frost, and is liable to be pushed by a passing wheel or be otherwise disturbed when the ground is soft. 284. Surveys for Subdivision. — The purpose of making a survey before recording a plat of a subdivision is twofold, — first, to get the information which it is desirable to record ; second, to leave such monuments as will make it easy to locate any portion when desired. The recorded plat should show sufficient facts to determine the relations of every part to the whole, and these relations should be shown by methods which involve the minimum of error, i.e., giving a location which may be retraced with least possible doubt. The current practice falls short of this standard at some points which are worthy of note. {a) Surveyors seem to have no doubt of the ability of their field-hands to measure a line, but very seriously doubt their 366 SURVEYING. own ability to measure an angle. Angles are measured dur- ing the progress of the work and are used for determining the lengths of lines ; these lengths are then made a part of the record, while the angles which determined them are omitted. Apparently some things which arc dependent have become more certain and fixed than that upon which they depend. A proper record of angles would show what lines are straight and where defictions are made. Defleections which are sufficient to very seriously affect the position of a brick wall do not show on the scale of the recorded plat. For example, an addition to a town extends from Fifth Street to Twelfth Street ; extreme points are well established, but intermediate monuments are missing; and it is required to establish at Eighth Street the line of a street which a ruler applied to the recorded plat sug- gests is a straight line. Custom approves that in such a case the surveyor should try a straight line, there being a mild pre- sumption in its favor; but if his straight line agrees with one wall and disagrees with two walls and a fence, he had better look further before he comes to a decision. No such doubt could have existed if the recorded plat had been properly made. (b) Very few recorded plats show what stones have been set by the surveyor, or indeed indicate that he has any knowl- edge that such monuments may ever be useful. If the custom were once established of noting upon the record the location and description of monuments, any monument found during a resurvey, but not shown on the record, would be discredited. As matters now stand it must be proved incorrect to be dis- credited — a thing not always easy, for a system of quadrilat- eral blocks whose angles are not recorded and whose street lines are not necessarily straight is not theoretically very rigid. (c) Many plats require measurements to be made along lines which are easily measured while the land is vacant, but which will become inaccessible as soon as the property is built up. The obstacles to be overcome before the result can be CITY SURVEYING. 367 reached by the method described on the record will each add to the doubt of the accuracy of that result. There are many ways in which plats are made, which are all justly subject to this criticism. Two examples will suffice. Irregularly shaped blocks are sometimes treated as in the annexed sketch, Fig. 103. The outline is subdivided mechanically, and proportional distances are given on interior lines which are not consistent with any trigonometrical relation of the exterior lines, much less with that which does exist but is not recorded. The point X has nine distinct locations directly from the plat. On the theory that ah and cd are straight lines, their intersection gives one; ab straight, the distances ax and bx give each one; cd straight, the distances cx and dx give two. Combine the dis- tances ax and cx, bx and cx, etc., and get four more. But this is not all, for the point x stands related to each of the ten other points along the line ab, and each of these has also nine loca^ tions which accord with the plat, and our point x may be lo^ cated from either of them or any combination of them when they have been located by any of the methods described. Besides the difficulty of determining how interior points should be located, this fan-like subdivision wastes ground in each lot which results in wedge-shaped remnants about the build- ings, which remnants would be valuable if thrown together into the corners, thus keeping the remaining lots rectangular at the 368 SUR VE YING. front. The attempt to reach a rectangular front sometimes fails through inattention to very simple matters, as in Fig. 104. Here no angles are recorded. The rear corners of the lots are located along the line ab by distances from aor b\ but the record-depths do not fall upon a straight line. The line ab should bisect the angle between the block-lines or be parallel to such bisection in order that with a constant distance along ab common to the series of lots on each side of that line their angles with their respective fronts may remain constant. In the case given every lot-line has an angle with the block-line upon which it fronts different from that of every other lot-line, and all dependent on some block-angle which is not recorded. If it is not desirable to bisect the block by the line ab, its di- rection may be chosen as desired, the distances along it are fixed by the fronts on one and the angular divergence from that side which is chosen, and the lot fronts on the other side of the block must be correspondingly increased or diminished. When alleys are laid out in a block so that the interior lines are accessible, it is very rare that after the block is improved these lines can be measured under the same conditions as the fronts. If alleys are not laid out, the difficulties are usually much greater. Location of lot-lines by angle from the front is CITY SURVEYING. 369 undoubtedly the most uniform and workmanlike method avail- able to the surveyor. Hence, distances on the rear lines of the corner lots should be omitted from the record, if their presence would leave any doubt as to which method of location is in- tended. It is not customary, nor is it desirable, that lot-lines or distances should be determined upon the ground before record- ing a subdivision, but they should be platted by a man who knows at least the first principles of trigonometry, and has an accurately measured basis for his work. ' 285. The Datum-plane. — Levels referred to a permanent datum are needed as soon as it is apparent that the town is to be a living reality and not simply a town on paper. The da- tum is a matter of some importance, and should have a simple relation to some natural feature of the locality which will re- tain a vital interest so long as the town exists. There is an individuality in town-sites which usually determines for each case very definitely what is best. High-water mark indicating the danger of overflow; the lowest available outlet for a drainage system in a flat country ; the average sea- or lake- level, as affecting commerce ; these are often chosen and may serve as examples. The datum selected has its value accu- rately determined and marked by a monument as enduring as the granite hills, or, if that is impossible, as near this stand-' ard as can be secured ; a block of masonry, with a single and durable cap-stone firmly bolted to its place, and bearing the datum, or a known relation to it, definitely marked and secured from abrasion is certainly possible for all. 286. The Location of Streets for which the most econom- ical and practical system of grades may be secured is to be considered when the plat is being prepared. Grades are usu- ally established from profiles taken along the centre lines of the street to be graded. This method is direct and protects the public fund, for the grade, which can be executed at minimum cost, the street being considered by itself, can be determined 24 370 SURVEYING. from such a profile. The method fails from the fact that it treats the fund raised by taxation as the sum total of the pub- lic interest. Parties representing abutting property appear before the legislative body which has final action and seek to amend the recommendation of the engineer, claiming that in- terests which should receive consideration are injured by the grades proposed. It seems plain that whatever is recommend- ed by the city’s officer should have the moral weight which attaches to an impartial consideration of all the interests which the city fathers are bound to recognize. But this involves a change of method. The contour map of the district involved seems to offer some help toward a solution. Methods by which a rapid approximation of the amount of cut and fill in- volved in any proposed grade may be arrived at are discussed in Chapter XIII., on the Measurement of Volumes. 287. Sewer Systems. — A well-devised sewer system touches very closely the public health. The information which is necessary in order to act intelligently involves, if storm-water is to be provided for, the area and slopes of the whole drainage-basin in which lies the area to be sewered. This will enable a close approximation to be made of the work required of the mains at the point of discharge. Each sub- district involves its own problem. The most economical method of reaching every point where drainage is necessary is learned by studying the details of topography. Borings along the lines of proposed work to determine the character of the soil and the depth of the bed-rock are necessary in order that contractors may bid intelligently. This species of under- ground topography sometimes modifies the location fixed by surface indications. 288. Water-supply. — The need of a water-supply fur- nishes new work to the surveyor. The distance and elevation of the source of supply, the topography of the country through which aqueducts or mains must be brought, eligible sites for CITY SURVEYING. 371 reservoirs, with their relation in distance and elevation to all points to be supplied, are to be furnished to the hydraulic engineer. The datum-plane for these maps and that of the town should correspond. 289. The Contour Map, which is so generally useful from the time the town is first planned until public improvements cease to be considered, if surveyed carefully at first, has no need to be retraced each time such a map is useful. It had best be drawn in sections of sufficient scale for a working-plan, and so arranged that when adjacent sections are placed side by side the contour lines will be continuous. If the contours of the natural surface are drawn in india-ink, and the contours showing the changes made by different kinds of public work be drawn in some color, the map may give a great amount of information without becoming confused. METHODS OF MEASUREMENT. 290. The Retracing of Lines comes with the private use of lots or blocks and with the execution of public improve- ments. The demand for this class of work comes not once only, but many times, and never ceases while there is life and growth. The changes to which these forces give rise furnish the main demand for knowing along what lines growth may proceed unchallenged. The man who first fences a lot in the middle of an unimproved block can ill afford to risk being com- pelled to move his fence for what a survey would cost. But the first attempt to go over any part of a subdivision and locate a lot-line raises the question, how nearly alike can a surveyor measure the same distance twice, or how nearly alike can two surveyors measure the same distance. If the distance noted on the recorded plat was not measured correctly, the resurvey must differ from it, or by chance make a mistake of the same amount. The difference which appears by compar- 372 SUJ^VEYING. ing results is not the error which exists in either the original or the resurvey; it may be more than either error, it may be less, being the algebraic difference of the two errors. If there is no difference it means that the work is uniform, and may be correct, but both may also be in error a like amount. It has happened in the days of twenty-foot rods and in a city of con- siderable size that every rod used by surveyors was too long. The change to steel tapes has not set matters wholly right. If a man compares steel tapes bearing the brand of the same manufacturer and offered for sale in the same shop, he soon ceases to be surprised at a very appreciable difference in length. 291. Erroneous Standards. — How long is a ten-foot pole or a hundred-foot tape is a pertinent and fundamental ques- tion. It cannot be ignored when deeds call for a distance from some other point, as fixing the beginning-point of the parcel conveyed. When the deed describes lot number — , as shown on the recorded plat, there is a theory in accordance with which uniformity is all that is required — a distribution of the distance between monuments in proportion to the figures of the record. Property is often laid out with a view to this theory of surveying. So long as block-boundaries are definitely marked, a degree of precision is very readily secured by this method which is rarely attained when surveyors attempt to measure standard distances. If the surveyor faithfully meas- ures the block through and every time distributes what he finds in proportion to the record, though his block distances may not agree with the record or with themselves, the lot-lines will be much more likely to be the same than if he measures his record distance and stops at the lot. This method assumes that the lots abut one upon another, and reach from one monu- ment to the other. But if this be true, the distances noted must often refer to some empirical standard peculiar to this block and not to the United States standard established by CITY SURVEYING. 373 law. But the courts recognize no standard, so far as the author knows, but that which is established by law. So that when a surveyor comes to mark lot one, finds the corner of the • block, and drives his stake by measuring from it the distance which the record assigns to lot one, it is hard to prove that he has not measured according to the subdivision, although he has given no thought to the distance which remains for the other lots. But trouble begins right here, for the theory which is correct for lot one cannot be very wrong for lot two ; con- tinue the process to lots six and eight, and give to another sur- veyor who has been doing the same kind of work at the other end of the block an order to survey lot seven. A conflict in this case is certain unless the surveyor who laid out the sub- division, and each of the others since, knew the length of his tape and knew how to measure. 292. True Standards. — The Coast Survey Department of the U. S. furnishes at small cost rods of standard length at a temperature which is stamped on the rods. Using a pair of these so as to measure by contact, a standard test-rod of any desired length can be laid off and only such marks retained as may be desired. This test-rod should be of the same ma- terial as the tape to be tested, in order that it may have the same coefficient of expansion by heat and may not be affected by humidity of the atmosphere. Care being taken that the tape is of the same temperature as the rod, be it 30°, 90°, or 60°, when the test is made, then the tape is correct at the tem- perature at which the rod is correct, and this is known by the U. S. stamp, and has no reference to the temperature at the time of the test. In some styles of tape the ring may be shaped to make the necessary adjustment to standard length. Where and how to construct a standard rod, and how to care for it so that it may be permanently reliable, each indi- vidual had best determine for himself. It should be fastened in its place in such a manner that it can expand and contract 374 SUR VE YING. freely, i.e., without any strain from its supports. If it is made of separate parts, these should be so joined together that there can be no lost motion between the pieces. The whole requires protection from the weather and to be so supported that it cannot be bent by a blow. The writer has solved this problem for himself in the following way : Bars of tool steel one inch wide and one fourth of an inch thick are joined, as shown in the sketch, to make the desired length 50 feet +; the whole is Fig. 105. fastened to the office floor by screws which hold the middle firmly, but each side of the middle the holes drilled for the screws are slotted sufficiently to allow for any possible change of temperature. The joints are so close that a light blow is necessary to bring the parts to place; the screws were set home and then withdrawn a little, so that they should not cause friction with the floor. After the fastening was com- pleted the standard marks were cut upon the rod. 293. The Use of the Tape. — It is one thing to have a tape of correct length ; it is another thing to be able to use it. In an improved town with curb-lines clear, perhaps the most obvious method is by a measurement along the grade with the same tension as that at which the tape is tested. It is then necessary to correct for temperature and to note all changes of grade, reducing the observed distance on each grade by the versed sine of the inclination or by the formula given in Chap. XIV. By this method the tape is supported for its entire length, and it is practicable to use a tape two or three hundred feet long to advantage provided there are enough assistants to keep it from being broken. A difficulty arises in the use of CITY SURVEYING. 375 this method from the fact that the town is not made for the convenience of surveyors, and curb-lines are not usually clear where measurements are needed, but are obstructed by piles of building matenal, bales of merchandise, etc., and in some towns the streets are so dirty that the graduation could not be seen long if a tape were used in this way ; it would also be so covered with drying mud that it could not be rolled in the box when out of use, hence would be frequently broken. Tapes that are wound on a reel, and have no graduations to speak of, could be used in the mud, but the other objections mentioned would still make the method of very limited appli- cation. It is further to be noted that the laying-out of the town, which is the basis of all later work, has all to be done before the streets are graded or the curbs set. This work must be done by some other method. The usual method is to keep the ends of the tape horizon- tal by using a plumb at that end of the tape where the surface is lowest, and often at both ends if the ground is so irregular or so covered with brush and weeds that the tape must be kept off the ground. The tape assumes a curved form, and the horizontal distance is something less than the length of the tape. There is also a tension in the tape which, on account of the elasticity of the metal, somewhat increases its length. As the tension increases the sag diminishes, hence there is a degree of tension such that its effect is equal and opposite to the effect of the sag. Call this the normal tension. If a line is measured with a pull less than the normal tension for the tape used, the tape will sag too much and there will be a minus error due to this excessive sag ; if the pull used exceeds the normal tension, there will be a plus error due to this excess. If the pull has been uniform the total error in either case is proportional to the length of the line ; but if the pull has not been uniform the error has varied irregularly with each length of tape and can most readily be calculated by retracing the line 376 SURVEYING. and using the proper tension. In practice the tape is tested with a known tension, and a tension so much above the “ nor- mal ” is adopted for field use that its plus error is equal to the plus error of the test. 294. To determine the Normal Tension in a tape sup- ported at given intervals. The tape forms a catenary curve, since it carries no load but its own weight and is of uniform section. Let P — horizontal tension (pull) ; w — weight of a unit’s length of tape ; e — base of Naperian logarithms ; s — length of curve from origin ; / = distance between supports ; W = wl = weight of tape ; X and^ = horizontal and vertical coordinates, origin at low- est point ; X = for cases considered. Then by mechanics,* y = D wx — {e~P-\-e 2W^ ‘ •wx ~p -2), and We observe (i), that if ^ is constant and s are constant for the same length of tape ; (2), if P be measured, say ten pounds. ■* The discussion here given is rigid, but both the development and the evalu- ation of the equations are laborious. If the curve be assumed to be a parabola, which It may when the sag is small, the development is much simpler. See the treatment of this subject in Chapter XIV. — J. B. J. CITY SURVEYING. 377 as a working condition, j/ and ^ will vary with the weight of PI P . every tape used, hence jp = —■ is the ratio which must be constant ; (3), if the surveyor can keep constant, the same conditions keep s constant, and if jy varies s must vary; (4), if P . 'WX x{= \l') varies, and -- varies in the same ratio, then is con- stant, hence the parts of the equations in parenthesis are con- P stant and y and s vary as I and — . ^ ^ w TABLES SHOWING NORMAL TENSION AND EFFECT OF VARIABLE TENSION. / = 100 feet. JT = 50 feet. Sag. Pull. P w' y- ( 2 ^ - /) — error. P JV' Elonga- tion -f- error. ft. ft. ft. 800 1.56 0.065 8 0.010 900 1-39 0.051 9 O.OII 1000 1.25 0.040 10 0.012 1 100 1. 14 0.033 II 0.014 1200 1.04 0.028 12 0.015 1300 0.96 0.023 13 0.016 1400 0.89 0.020 14 0.017 1500 0.83 0.017 15 0.019 1600 0.78 0.014 16 0.020 1800 0.70 O.OII ' 18 0.022 2000 0.62 0.009 20 0.025 2400 0.52 0.007 24 0.030 Resultants ± Error in 1 Error in 1000 ft. - + - + ft. 0.055 0.040 0.028 0.020 1 0.013 0.007 0.002 ft. ft. 0.55 0.40 0.28 0.20 0.13 0.07 0.02 ft. 0.002 0.006 O.OII 0.016 0.022 0.02 0.06 0. II 0. 16 0.22 378 SURVEYING. l — 50'. X = 25'. Sag. Puli.. Resultants ± P y. {is — 1 ) P IV ' Elonga- tion Error in / Error in looo ft. tv -f- error. - + - ft. ft. ft. 400 0.78 0.033 8 0.003 0.030 0.60 500 0.63 0.020 10 0.003 0.017 0. 34 600 0. 52 0.014 12 0.004 0.010 0.21 700 0.45 0.010 14 0.004 0.006 0. II 800 0-39 0.007 16 0.005 0.002 0.04 qoo 0.35 0.006 18 0.006 1000 0.31 0.004 20 0.006 0.002 0.03 1100 0.28 0.004 22 0.007 ' 0.003 0.06 1200 0. 26 0 . 004 . 24 0.008 1 0.004 0.08 I'^OO 0.24 0.003 26 0.008 0.005 0. 10 1400 0.22 0.003 28 0.009 0.006 0.12 1500 0.21 0.002 30 0.009 0.007 0.14 1600 0. IQ 0.002 32 0.010 0.008 0. 16 1700 0.18 0.002 34 O.OII 0.009 0.18 1800 0.17 0.001 36 O.OII 0.010 0.20 Assuminof values of — , the formulas are readily solved for any assumed distance between supports and the results tabu- lated ; seven-place logarithms are best for this work. The 100' tape is chosen because it furnishes a ready means of calculating a table for any other length of tape by a decimal reduction of the errors, per 1000', in proportion to the length P desired, and tabulated with values of — reduced in the same w proportion. There are those who use the 100' tape free-hand, with 16 to 20 pounds pull, and say they do the work uniformly. CITY SURVEYING. 379 In the ordinary formula for elongation, A PL * Ek^ we have the section k, a multiple of w. The foregoing tables are calcu- lated from the value w = 3.4^. The tension in the tape P *E is the modulus of elasticity in pounds to the square inch, and ^ is the area of the cross-section in square inches, Z being given in the same denomina- tion as A. 38 o SURVEYING. differs from the horizontal tension P, so thatP = P secant i {i — inclination to the horizontal), a second difference which is so small that it may be neglected. Let E = 27500000 (see Chapter XIV.), hence ^ — nearly. The same facts for 1000 feet distance are shown in Fig. 106. In the tables the plus and minus errors are shown separately for a single length of tape only, and combined for 1000' feet ; in the figure they are separated for the whole distance and the resultants of the table are the vertical intercepts between the curves (minus errors) and the straight line (plus errors). The sag for a single length of tape and cor- responding — IS w shown by dotted curved lines ; these are plotted to a reduced vertical scale which is shown at the right of the sketch. 295. The Working Tension. — In using these tables it is best to measure the sag until the necessary pull for the tape is learned. When the ends of the tape are at a known elevation above a level surface, a rule at the middle of the tape will show whether the pull is right. The fore chainman should learn to pull steadily, not with a jerk, as he sticks the pin. A more emphatic statement than the figure itself is of the worthlessness of an unsteady hand at the forward end of the tape it would be hard to make. A consciously constant pull, the same every time, is necessary for good work. To ob- serve the sag is the surveyor’s means, in the field, of knowing that the work is being done. He soon learns to judge with considerable accuracy whether the proper pull is constantly maintained. The proper pull is determined by the tension at which the tape is tested ; call this p. Then, having weighed the tape, . Seek the plus error from elongation for this w value of — ; then find the same plus error between the curve for w ^ CITY SURVEYING. 381 that length of tape and the straight line ; the corresponding — is right for field use. For example, a 50' tape weighs six ounces, and the pull, when tested, was five pounds; ^ ^ ^ = 666, and the tV elongation = o'.o83. The curve for a 50' tape marked — error from sag is distant from the line marked -j- error from P pull the same amount when— - = 1233. Whence P= 1233 X tV 50 — 9 i pounds, and the sag = o'.25. When a tape is to be suspended freely in use, the tension at the test,/, should not be such that the working tension Pwill be so great as to be impracticable ; but it is also to be noted that slight varia- tions of pull do not affect the result as much, when the tension is considerably above the normal, as the same variations would affect it if the tension were at or below the normal. 296. The Effect of Wind. — A very moderate wind has a marked effect on the sag of the tape ; the wind-pressure on the surface of tape exposed increases the sag and gives it a diago- nal instead of a vertical direction. The exposed surface of the tape constantly changes, and this results in vibrations which make it difficult to tell where either end of the tape is. The effect of its action, which is a minus error, varies approximately as the square of the length of tape exposed. The effect of winding up part of the tape so as to use a shorter length is to increase the use of the plumb, which is also affected by the wind, and the result is a loss of a part or all that is gained. A high working tension reduces the effect of the wind. But the only way to eliminate this source of error is to cease from any piece of work when the wind is so high that it cannot be done as it should be done. There are estimates, topography, etc., which do not require a high degree of precision and which can be done when other work cannot. 382 SUJ^ VE YING. 297. The Effect of Slope. — When the tape is used with its ends at different elevations, if it hangs freely its lowest point would not be in the middle, but nearer the lower end. The corrections for sag and pull still apply, however, with inappreciable error, for all practicable cases. The normal tension, therefore, remains the same as for a level tape. A correction must now be made, however, for the grade, the value of which is / vers, f, where / is the distance measured along the slope, and i is the angle with the horizontal. The measured distance is always too great by this amount.* The available means by which the tape may be kept level are: (i) The judgment of two field-hands. (2) On difficult lines, the presence of the surveyor standing at one side where his position has some advantages. A distant horizon often very sharply defines the horizontal. (3) Where streets are im- proved, although it may be impracticable to measure along the slope, the known fall per 100 feet will give the needed infor- mation. (4) Where none of these methods are sufficient, test the judgment by plumbing at different heights and correcting the pin if necessary. These methods will eliminate the worst errors; but where it is necessary to measure lengths of five or ten feet, and then plumb from above the head, the uncor- rected remnant will be considerable, probably that due an inclination of two per cent on the whole length of such lines, with very careful work to get so near. This difference in the character of lines is to be taken into the account in balancing the survey. Note that the resultant error is always minus. 298. The Temperature Correction. — The temperature of the tape at the time when the work is done affects the result. This is not the temperature in the shade that day, nor the * This question is fully discussed in Chapter XIV., where the correction is found in terms of the difference in elevation of the two ends. — J. B. J. CITY SURVEYING. 383 reading at the nearest signal station, but is the tempera- ture out on the line, under the conditions which exist there. A grass-covered slope, descending away from the sun, will often show at the same time as much as twenty or thirty degrees lower temperature than a bare hillside inclining toward the sun. The thermometer is needed with the work. If the co-efficient of expansion is not known, use 0.0000065 for 1° F. It is very desirable in a city-surveyor’s work that he be able to apply his corrections at once while in the field. If he goes out to measure any given distance, he must be able to fix his starting-point and drive his stake at the finish. If the weather is hot or cold, he knows what it differs from the temperature at which his tape is tested, and applies the correction at once to the whole distance. He watches that the pull is right, that the tape is kept horizontal, that the work stops when the wind is too severe, and that the checks show the desired accuracy. 299. Checks. — Every piece of work should be carried on till it checks upon other work, verifying its accuracy within desired limits. This method ties up every survey at both ends. In order to be prepared to do this expeditiously, the surveyor should lay out general lines which should be joined into a sys- tem embracing the town-site. The lines of leading streets and the boundary-lines of additions give most valuable information when made parts of such a system. This borders on the geo- detic idea, but it will generally be impracticable to determine the lengths of these lines by triangulation from a measured base, for the stations can very rarely be so chosen that the angles can be measured upon the whole length of the lines, or the diagonals be observed at all. Still, the angles should be measured upon the best base practicable. Permanent build- ings and existing monuments showing the lines of intersecting streets should be noted both for line and distance. 384 SUR VE YING. MISCELLANEOUS PROHLEMS. 300. The Improvement of Streets involves — (i) The estimation of the earthwork in the grading and shaping of the street. (2) The location of the improvements along the lines of the dedicated streets. City ordinances usually prescribe a cer- tain width of sidewalks and roadway for each width of street. (3) The location of improvements at the grade fixed by ordi- nance. (4) The estimation of materials furnished by contractors and used in the work. The position of monuments which will be disturbed during the progress of the work is preserved by witness-stakes driven beyond the limits of disturbance. When this precaution is neglected it results in all sorts of angles and offsets in the curb-lines, in cases where there is surplus or defi- ciency in the original survey. Take a case improved one block at a time, where the first block is established by record distance from the right, the second block by record distance from the left, and a third by running from this last point to a point established at the end of the third block by measuring again from the right, etc. The resulting lines of curb will not give a suggestion of where the street was laid out. Some sur- veyors are accustomed to replace from their witness-stakes the monuments on the new grade. Such a practice is certainly to be commended ; the small cost to the public treasury can well be borne for the public good. 301. Permanent Bench-marks. — In order to secure accu- racy and uniformity in elevations throughout a city, bench- marks are established by running lines of levels radiating from the directrix, and checking the work by cross-lines at conven- ient intervals, these cutting the whole territory into small par- cels, so that a standard bench-mark will never be far from any work which must be done.'^ This work is carried on as far as * These various lines of levels will form a network, such as that shown in Chapter XIV., which should be adjusted once for all as described in that chapter, CITY SURVEYING. 385 grades are established, and generally as far as the city officers are prepared to propose grades for adoption by ordinance. There is a view of what constitutes or is essential to accurate methods which would make every piece of work start from first principles, so that it may not depend in any way upon er- rors involved in work previously done. But work done on this plan does not have to be extended very far before the results will show plainly that there is a wide margin between the uni- formity attained and the accuracy attempted. 302. The Value of an Existing Monument is based (i) on the fact that it corresponds in character and position to a mon- ument described on the recorded plat ; (2) on the custom to place monuments upon the completion of a survey, and on the supposition that this monument in question was set in pursu- ance of such custom, although no monuments are noted on the plat ; (3) on recognition by surveyors and owners of land affected by it ; (4) on the knowledge that it was placed by a competent surveyor at a time when data were accessible which are not now in existence. The value of the evidence which establishes or tends to establish the reliability of the monument is primarily a question for the judgment of the surveyor. His decision must be reviewed and defended before courts and ju- ries when there is a difference of opinion. The monument is valueless, or less valuable in all degrees, when there is evidence that it has been disturbed. It some- times happens that there is no better way to establish a corner than to straighten up a stone which is leaning, but has not been thrown entirely out of the ground. Inquiry often brings out the fact that a stone, after being completely out of the ground, has been reset either by agreement of owners adja- :ind so one elevation obtained for each bench-mark. It is common for each bench-mark in a city to have numerous elevations differing by several tenths of a foot, and all of about equal credence. — J. B. J. 25 386 S UR VE YING. cent, or by the reckless individual who did the mischief, and is still pointed out as the stone the surveyor set. As a recog- nized corner such a stone has some value, i.e., it is to be sup- posed that it is somewhere in the right neighborhood; but if its position can be verified from other points which have not been disturbed the work should be retraced. If the original survey was made in a careless way or the corner-stones were badly set, they may help a careful man to come to an average line which shall correspond with the recorded plat. Monuments are sometimes moved or destroyed maliciously. It is wise for a surveyor to test discreetly everywhere, but to be especially careful where there has been quarrelling about lines. There is a principle, recognized to some extent by the courts, that the existing monument is the evidence of the orig- inal survey, whether or not it is called for by the recorded plat. The custom that the surveyor making the subdivision and the plat for record shall set corner-stones is so far fol- lowed that this is generally true, cases of accident, carelessness, and mischief, and such cases as that mentioned below, being somewhat exceptional, but many times very real. It is some- times attempted to go a step further and affirm that the re- corded plat is the record of the survey. This reverses the or- der of events in most cases, the survey being made in order to mark upon the ground the chief points of apian already fixed upon ; and as to all the main lines, the plat is not altered, how- ever carelessly the survey may be made. There are subdivisions where no monuments were set and where no certain evidence is in existence of how or where the original survey was made, or whether any survey was made at all, and yet there is a re- corded plat. A surveyor being called upon to make a survey of some parcel in such a subdivision, sets stones in order to se- cure recognition for his theory of the proper location. If he does his work carefully he undoubtedly does the public a ser- vice. Can any amount of ignorance of when or why these CITY SURVEYING. 387 stones were set ever make them evidence of the original sur- vey? In other cases some monuments maybe in existence, but more would be convenient, — points are determined from existing monuments in accordance with the recorded plat and stones are set. Another surveyor may feel a little nervous about manufacturing this sort of evidence of the original sur- vey, or more likely, may think it too much trouble and a dam- age to the business, for the more doubt the more work for the surveyor, so drives his stake. Then comes the owner who, desiring to secure a permanent corner, digs a hole about the stake without taking offsets, throws it out, and sets in a stone — an existing monument! This is no fancy sketch, nor are such facts so very rare. The young man who thinks he would like to be a surveyor, but has no eyes nor ears for facts like these, had better turn his attention to some other business. Surveying is an art — not an exact science.* 303. The Significance of Possession. — Possession has a value in reestablishing old lines where all monuments have disappeared. It is a species of perpetuating testimony of their positions. The average of a series of improvements will often give a very close determination of where the corner must have stood. The practised eye accustomed to sharply defined lines, every lot having very nearly its right quantity, which are cus- tomary where lines are well established, will notice at once the irregular possession, — gaps between houses, vacant spaces between fences and houses, too little for use, too much for ornament, which may be seen where lines are in doubt and every man expects the next surveyor to make a conflicting survey. Like the men of the present, most men in the past have preferred to be right — have made efforts to be right — have employed surveyors ; we can judge where these men in * Consult Judge Cooley’s paper on the Judicial Functions of the Surveyor, Appendix A. 388 SUR VE VING. the past worked from by seeing where their works arc. The legal principle has a bearing here, that “ he who would sue to dispossess another must first show a better title.” Tlic sur- veyor who attempts to dispute possession must show better evidence than possession of the right location of the lines he is employed to retrace. 304. Disturbed Corners and Inconsistent Plats. — The work of testing a corner that probably has been disturbed has many points of likeness to the work of reestablishing corners that have disappeared altogether. The recorded plat is in all cases the basis of the work. When it records the results of a survey it is to be presumed that the surveyor endeavored to do accurate work ; hence his work, if not absolutely correct, was probably uniform. Lines which are shown by the plat as straight lines are to be retraced as straight lines. Lines in- volve less liability to error than measurements, and are first to be considered. Determine as many points as possible by straight lines between existing monuments. Then test the measurements along the extreme lines and the streets which are the basis of the subdivision. If the measurements between undoubted corners agree with the plat so closely, or if they differ so uniformly that the presumption of accurate work is justified, corners that are out of line or out of proportionate distance have the burden of proof against them. He who would claim for them authority must show that they have not been disturbed, and that they are consistent with some ra- tional location. If there was no original survey, that fact is no excuse for careless Avork at a later time ; there is always some place to begin. The case when the recorded plat does not agree with itself presents more difficulties, such as the follow- ing: (i) The lines do not give the same points as the distance ; (2) The distances disagree among themselves; (3) The monu- ments disagree with both lines and distances impartially, or agree with one and disagree with the other, while the general CITY SURVEYING. 389 character of the work negatives the supposition that they were ever carefully set. The object to be sought is not to perpet- uate forever the blunders of the original survey, but to seek the most rational adjustment of all the evidence, so that all parts may be located with a minimum of conflict, and so that no one shall be able to prove your survey wrong, i.e., show a more reasonable location for any part. A consultation of surveyors before too many conflicting interests have developed is often advantageous. 305. Treatment of Surplus and Deficiency. — It is gen- erally a simpler problem to determine in which block differences of measurement, whether surplus or deficiency, belong than it is to know what to do with them in the matter of lot-location. There has never been any theory invented for the treatment of either surplus or deficiency which is able to stand the test of the courts against all combinations of circumstances. A few suggestions with the more probable limitations are all the help that can be offered: every case must be investigated for itself, (i) A distribution of the whole front in proportion to the record distances meets general approval, at least in cases of surplus, until it comes in conflict with possession. This is just the time when an owner of ground wants to know what his rights are, and it is also the time when no surveyor can tell him. A compromise, or the verdict of a petit jury, which passes foreknowledge, are the chief alternatives. The ^courts say that he who would sue for possession must show a better title. An examination shows that each has a better title than any other to so much ground as the plat assigns to his lot, but that no one has a better title than any other one to any part of the surplus. The surveyor does wisely to take note of possession and make, if he can, such a location as is in accordance with the record, and yet not in conflict with posses- sion. When this is not possible, let the map and certificate of survey be made in such a way that they are simply a state- SURVEYING, 390 merit of the facts. It is not a surveyor’s business to decide legal questions or give judgment in ejectment. (2) Because a suit for surplus will not lie, it has been thought that he who first took possession of the surplus would be secure if he were only careful to take it so that every other one might have his ground. Trouble with this view arises because it is not possi- ble to locate the surplus. When one man has appropriated all there is in the block, and the rest but one have appropriated each his proportionate share, then comes the last man. The more surplus in the block the more he is deficient ; he wants his ground, and he finds it easier to sue the one man than the twenty. Perhaps, in order to be sure of a case, he had better sue them all. The cases which arise in practice take on an infinite variety of complications and are not usually so simple as these described. (3) The fact is, that the idea that a subdivision ought to have a little surplus is irrational. The work should be so close to the standard that the surveyor who retraces the lines would testify : “ According to the best of my knowledge and belief, there is neither surplus nor deficiency there. In retrac- ing my own work, which is carefully executed, I observe as great discrepancies as any which I find in this subdivision, and I conclude that the small difference which I observed in this case was as likely to have been an error in my own work as to attach to the subdivision.” (4) Deficiency would seem to be easier to deal with than surplus; for when the last man has not his ground he has a valid claim against the original owner for a rebate on the purchase-price. But the burden of the difficulty in this case falls on the surveyor. When a man brings his deed and asks a survey of lot 9, while 8 and 10 are unsold and lots I to 7 are already in possession, he leaves lot 8 its ground and the deficiency in lot 10. Suppose it turns out that lot 10 is next sold, and that the surveyor reports it deficient, the seller, when waited on, may reply, “ I have not sold more ground in the block than I owned ; the surveyor has made a mistake in locat- CITY SURVEYING. 391 ing lot 9.” This liability attaches to every location which is made before every lot, between the one located and one corner of the block, is sold. (5) It is practicable for the original owner to so write his deeds as to locate surplus or deficiency. By beginning all deeds at the record distance from one street and continuing this uniformly through the block, the differ- ence goes in the lot farthest from the starting-point ; or he may continue the process up to any line which he may choose, and work from the other end of the block in deeding the re- maining lots ; then the difference falls upon the line chosen and falls to the share of the lot abutting upon that line which is last deeded. But to approve this method is to affirm the practicability of absolute accuracy in work. No one can tell how small a difference may cause trouble. 306. The Investigation and Interpretation of Deeds for the use of the land-surveyor, dealing with the harmony or conflict of the descriptions, is entirely a different work from that of the investigator of titles, which deals with the legal completeness of the conveyance. In the older parts of a town the deed of the present proprietor frequently does not give information sufficient to fix the correct location. The key may lie in some boundary in an early deed referring to a still earlier conveyance of adjacent property. Or the earlier deeds may give clearly defined locations, while the latter ones say “ more or less” at every point. In some cases the deeds are in such a condition that it is impossible to tell what they mean until it is known what the possession is. Skill in this work can only come after considerable experience ; local prac- tices must largely determine what is necessary. 307. Office Records. — The surveyor’s office when well planned is so arranged that no item of information which promises to be useful shall be lost. The customary methods of indexing, and of block-plats for keeping notes, do not take a very firm hold on general lines or the connections between 392 SUR VE YING. subdivisions; they fail, in fact, in that part of the work wliicli has the most vital relation to efforts at future improvement. It is advisable to add to the block-plats and indexes a general atlas of the whole town for office use, at a scale of say lOo' to the inch, so that an area nearly half a mile square may appear on the open pages. Such an atlas may show the notes of the general lines and their angles, the base-line measure- ments, the relation of subdivisions to one another, and a variety of other information which it is difficult to pick out in the widely scattered field-notes which first gathered the in- formation, and which, with their larger scale, the block-plats are not well adapted to show in a connected form. There are filed in connection with deeds many plats which do not appear on the record plat-books of the recorder’s office ; these need to be indexed, or, better, abstracted for office use. The field-notes, when prepared for the surveyor’s use in the field, should show in an accessible and portable form all the information which the office contains and which is rele- vant to the survey in hand. Labor spent beforehand in a thorough preparation of accessible information is labor saved. 308. The Preservation of Lines after the monuments have disappeared is accomplished by means of notes on build- ings, marks and notes on curbing, paving, fences, etc. Notes on buildings describe not only the character of the building, but the particular part noted, so that another man, years afterward, using the same note would have no doubt of the identity of the part. In a growing town the work of keeping up the notes goes on without ceasing, — buildings are remod- elled or rebuilt, streets reconstructed, destroying old marks. The old becomes the new so constantly that the surveyor who would preserve the information which he already has must be constantly employed at the work of renewal. There is no place either in the street or out of it where the surveyor CITY SURVEYING. 393 can place his mark and say to all comers, “ Touch not.” It follows that whenever it is necessary to use any mark, about the permanence of which there can be a shadow of a doubt, the permanence of the mark must be shown by some prac- ticable test ; it is careless to assume it. 309. The Want of Agreement between Surveyors arises from differences of information or of judgment, and in a less degree from differences of skill. These are all just as human elements as the lawyer deals with in his work. Testimony is affected by the interests of those who speak, and the judg- ment varies with the temperament of the individual. Per- haps one of the most difficult lines for a surveyor to draw is that which separates his confidence in his own skill in retrac- ing a survey which was confessedly inaccurate, from his re- liance on testimony which is evidently biassed as to the posi- tion or disturbance of monuments, and other facts which may help him to form a correct judgment. Errors in execu- tion may be kept within such limits that work which shows differences in closing of i in 5000 should be retraced, and the average observed differences in one surveying party’s work will not exceed i in 20000. Two sets of men working to reach the same standard may err in opposite directions, so that differences between two surveyors may reasonably be expected to be somewhat larger than either would tolerate in his own work. CHAPTER XIIL THE MEASUREMENT OF VOLUMES. 310. Proposition. — The volume of any doubly-truncated prism or cylinder^ bounded by pla 7 ie ends, is equal to the area of a right section into the length of the element through the centres of gravity of the bases, or it is equal to the area of either base into the altitude of the element joining the ceyitres of gravity of the bases, measured perpe^idicidar to that base. Let ABCD, Fig. 107, be a cylinder, cut by the planes OC and OB, the unsymmetrical right section EF being shown in plan in E' F . Whatever position the cutting planes may have, if they are not parallel they will intersect in a line. This line of intersection may be taken perpendicular to the paper, and the body would then appear as shown in the figure, the line of intersection of the cutting planes being projected at O. Let A — area of the right section ; A A — any very small portion of this area: X — distance of any element from O ; then ax = height of any element at a distance x from O. An elementary volume would then be axAA, and the total volume of the solid would be 'EaxAA. Again, the total volume is equal to the mean or average height of all the elementary volumes multiplied by the area of the right section. The mean height of the elementary volumes is, therefore, THE MEASUREMENT OF VOLUMES. 395 ^axAA . But is the distance from O to the centre of gravity, of the right section,* and a times this dis- tance is the height of the element LK through this point. Therefore, the mean height is the height through the centre of r., gravity of the base, and this into the area of the right section is the volume of the truncated prism or cylinder. The truth of the alternative proposition can now readily be shown. Corollary. When the cylinder or prism has a symmetrical cross-section, the centre of gravity of the base is at the centre of the figure, and the length of the line joining these centres is the mean of any number of symmetrically chosen exterior elements. For instance, if the right section of the prism be a regular polygon, the height of the centre element is the mean of the length of all the edges. This also holds true for paral- lelograms, and hence for rectangles. Here the centres of gravity * This is shown in mechanics, and the student may have to take it for granted temporarily. SUA' VE YANG. 396 of the bases lie at the intersections of the diagonals ; and since these bisect each other, the length of the line joining the in- tersections is the mean of the lengths of the four edges. The same is t^rue of triangular cross-sections. 31 1. Grading over Extended Surfaces. — Lay out the area in equal rectangles of such a size that the surfaces of the several rectangles may be considered planes. For common rolling ground these rectangles should not be over fifty feet on a side. Let Fig. 108 represent such an area. Drive pegs at 1222221 the corners, and find the elevation of the ground at each in- tersection by means of a level, reading to the nearest tenth of a foot, and referring the elevations to some datum-plane below the surface after it is graded. When the grading is completed, relocate the intersections from witness-points that were placed outside the limits of grading, and again find the elevations at these points. The several differences are the depths of excava- tion (or fill) at the corresponding corners. The contents of any partial volume is the mean of the four corner heights into the area of its cross-section. But since the rectangular areas were made equal, and since each corner height will be used as many times as there are rectangles joining at that corner, we have, in cubic yards. 4 X 27 [2/.. + 2^’4 + 32/^, + 4i'/«J. . . (I) THE MEASUREMENT OF VOLUMES. 397 The subscripts denote the number of adjoining rectangles the area of each of which is A. From this equation we may frame a Rule. — Take each corner height as many times as there are partial areas adjoining it, add them all together, and mul- tiply by one fourth of the area of a single rectangle. Tnis gives the volume in cubic feet. To obtain it in cubic yards, divide by twenty-seven. If the ground be laid out in rectangles, 30 feet by 36 feet, then — — — ■ = = 10; and if the elevations be taken to 4 X 27 108 the nearest tenth of a foot, then the sum of the multiplied corner heights, with the decimal point omitted, is at once the the amount of earthwork in cubic yards. This is a common way of doing this work. In borrow-pits, for which this method is peculiarly fitted, the elementary areas would usually be smaller. In general, on rolling ground, a plane cannot be passed through the four corner heights. We may, however, pass a plane through any three points, and so with four given points 1 3 3 3 3 4 1 on a surface either diagonal may be drawn, which with the bounding lines makes two surfaces. If the ground is quite irregular, or if the rectangles are taken pretty large, the sur- veyor may note on the ground which diagonal would most 398 SURVEYING. nearly fit the surface. Let these be sketched in as shown in Fig. 109. Each rectangular area then becomes two triangles, and when computed as triangular prisms, each corner height at the end of a diagonal is used twice, while the two other corner heights are used but once. That is, twice as much weight is given to the corner heights on the diagonals as to the others. In Fig. 109, the same area as that in Fig. 108 is A, shown with the diagonals drawn which best fit the surface of the ground. The numbers at the corners indicate how many times each height is to be used. It will be seen that each height is used as many times as there are triangles meeting at that corner. To derive the formula for this case, take a single rectangle, as in Fig. no, with the diagonal joining corners 2 and 4. Let A be the area of the rectangle. Then from the corollary, p. 395, we have for the volume of the rectangular prism, in cubic yards, _ ^ i 7^1 -|- I -h K + ^4 \ ~ 2 3 3 ] For an assemblage of such rectangular prisms as shown in Fig. 109, the diagonals being drawn, we have, in cubic yards, y — ^ + ; ... (3) where A is the area of one rectangle, and the subscripts denote the number of triangles meeting at a corner. THE MEASUREMENT OF VOLUMES. 399 As a check on the numbering of the corners, Fig. 109, add them all together and divide by six. The result should be the number of rectangles in the figure. In this case, if the rectangles be taken 36 feet by 45 feet, or, better, 40 feet by 40.5 feet, then the sum of ,the multiplied heights with the decimal point omitted is the number of cubic yards of earthwork, the corner heights having been taken out to tenths of a foot. The method by diagonals is more accurate than that by rectangles simply, the dimensions being the same ; or, for equal degrees of exactness larger rectangles may be used with diagonals than without them, and hence the work materially reduced. In any case some degree of approximation is neces- sary. 312. Approximate Estimates by means of Contours. — (A) Whenever an extended surface of irregular outline is to be graded down, or filled up to a given plane (not a warped or curved surface), a near approximation to the amount of cut or fill may be made from the contour lines. In Fig. in the full curved lines are contours, showing the original surface of the ground. Every fifth one is numbered, and these were the con- tours shown on the original plat. Intermediate contours one foot apart have been interpolated for the purpose of making this estimate. The figures around the outside of the bound- ing lines give the elevations of those points after it is graded down. The straight lines join points of equal elevation after grading; and since this surface is to be a plane these lines are surface or contour lines after grading. Wherever these two sets of contour lines intersect, the difference of their elevations is the depth of cut or fill at that point. If now we join the points of equal cut or fill (in this case it is all in cut), we ob- tain a new set of curves, shown in the figure by dotted lines, which may be used for estimating the amount of earthwork. The dotted boundaries are the traces on the natural surface of planes parallel to the final graded surface which are uniformly 400 SUf! VE YING. spaced one foot apart. These areas are measured by the planimeter and called^,, etc. Each area is bounded by the dotted line and the bounding lines of the figure, since on these bounding lines all the projections of all the traces unite, the slope here being vertical. For any two adjoining layers we have, by the prismoidal formula* as well as by Simpson’s one-third rule, 3 — 2 where h is the common perpendicular distance between the sections. * For the demonstration of the prismoidal formula see p. 403. THE MEASUREMENT OF VOLUMES. 401 For the next two layers we would have, similarly, (2) or for any even number of layers we would have, in cubic yards, where n is an odd number, h and A being in feet and square feet respectively. {E) Whenever the final surface is not to be a plane but a surface which may be defined by drawing the contour lines as they are to be when the grading is completed, the above method may still be used. Thus, suppose a given tract of ground, the contours of which have been carefully determined, is to be transformed into certain new outlines, as is often required in landscape-gardening and in the making of parks and cemeteries, the new contours may be traced on the plat containing the original contours by using a different-colored ink. The second set of contours are now curved instead of straight, as was the case in the preceding example. Otherwise there is no difference in the methods. The intersections of the two sets of contours are marked with the number of feet of cut or fill, the same as before, the cuts being designated by a plus and the fills by a minus sign. The curves of equal cut or fill are now drawn, preferably in an ink of a different color from the other two, and areas measured and the volume computed exactly as in the former case. It would also be well to desig- nate the cut and the fill curves by ink of different shades but of the same color. When a rectangular area, as a city block, is to be graded to 26 402 SU/^ VE y INC. a warped surface, which it generally is, the contours of this surface are readily obtained from the street-grades, and the above method used. For accurate measurements, such as should be made the basis of payment, the area should be di- vided into rectangles, as previously described. These approxi- mate methods serve well for preliminary estimates. They may be found useful in determining street-grades when it is desired to equalize the cuts and fills over the blocks rather than on the street-lines. 313. The Prismoid is a solid having parallel end areas, and may be composed of any combination of prisms, cylinders, wedges, pyramids, or cones or frustums of the same, whose bases and apices lie in the end areas. It may otherwise be defined as a volume generated by a right-line generatrix mov- ing on the bounding lines of two closed figures of any shapes which lie in parallel planes as directrices, the generatrix not necessarily moving parallel to a plane director. Such a solid would usually be bounded by a warped surface, but it can always be subdivided into one or more of the simple solids named above. Inasmuch as cylinders and cones are but special forms of prisms and pyramids, and warped surface solids may be divided into elementary forms of them, and since frustums may also be subdivided into the elementary forms, it is sufficient to say that all prismoids may be decomposed into prisms, wedges, and pyramids. If a formula can be found which is equally applicable to all of these forms, then it will apply to any com- bination of them. Such a formula is called 314. The Prismoidal Formula. Let A == area of the base of a prism, wedge, or pyramid ; Afnf Aj = the end and middle areas of a prismoid, or of any of its elementary solids ; /t = altitude of the prismoid or elementary solid. THE MEASUREMENT OF VOLUMES. 403 Then we have, For Prisms, + + (I) For Wedges, y — 5 (^1 + + ^2) (2) For Pyramids, F=y = §(A + 4^™ + ^,) (3) Whence for any combination of these, having all the common altitude h, we have F=g(^.+4^„+^,). ( 4 ) which is the prismoidal formula. It will be noted that this is a rigid formula for all prismoids. The only approximation involved in its use is in the assump- tion that the given solid may be generated by a right line moving over the boundaries of the end areas. This formula is used for computing earthwork in cuts and fills for railroads, streets, highways, canals, ditches, trenches, levees, etc. In all such cases, the shape of the figure above the natural surface in the case of a fill, or below the natural surface in the case of a cut, is previously fixed upon, and to complete the closed figure of the several cross-section areas only the outline of the natural surface of the ground at the section remains to be found. These sections should be located so near together that the intervening solid may fairly be as- 404 SU/^! VE Y I NG. sumed to be a prisinoid. They are usually spaced lOO feet apart, and then intermediate sections taken if the irregularities seem to require it. The area of the middle section is never the mean of the two end areas if the prismoid contains any pyramids or cones among its elementary forms. When the three sections are similar in form, the dimensions of the middle area are always the means of the corresponding end dimensions, d'his fact often enables the dimensions, and hence the area of the middle section, to be computed from the end areas. Where this can- not be done, the middle section must be measured on the ground, or else each alternate section, where they are equally spaced, is taken as a middle section, and the length of the prismoid taken as twice the distance between cross-sections. For a continuous line of earthwork, we would then have, in cubic yards, 1 + 4 -^ 2 + 2 ^ 3 + 4 ^ 4 + 2 ^ 6 + 4^ 6 • • +^n)> • (0 where / is the distance between sections in feet. This is the same as equation (3), p. 401. Here the assumption is. made that the volume lying between alternate sections conforms sufficiently near to the prismoidal forms. 315. Areas of Cross-sections. — In most cases, in practice at least, three sides of a cross-section are fixed by the conditions of the problem. These are the side slopes in both cuts and fills, the bottom in cuts and the top in embankments, or fills. It then remains simply to find where the side slopes will cut the natural surface, and also the form of the surface line on the given section. Inasmuch as stakes are usually set at the points where the side slopes cut the surface, whether in cut or fill, such stakes are called slope-stakes, and they are set at the time THE MEASUREMENT OF VOLUMES. 405 the cross-section is taken. The side slopes are defined as so much horizontal to one vertical. Thus a slope of to i means that the horizontal component of a given portion of a slope- line is times its vertical component, the horizontal com- ponent always being named first. The slope-ratio is the ratio of the horizontal to the vertical component, and is therefore always the same as the first number in the slope-definition. Thus for a slope of i| to i the slope-ratio is 316. The Centre and Side Heights. — The centre heights are found from the profile of the surface along the centre line, on which has been drawn the grade line of the proposed work. These are carefully drawn on cross-section paper, when the height of grade at each station above or below the surface line can be taken off. These centre heights, together with the width of base and side slopes in cuts and in fills, are the neces- sary data for fixing the position of the slope-stakes. When these are set for any section as many points on the surface line joining them maybe taken as desired. In ordinary rolling ground usually no intermediate points are taken, the centre point being already determined. In this case three points in the surface line are known, both as to their distance out from the centre line and as to their height above the grade line. Such sections are called “ three-level sections,” the surface lines being assumed straight from the slope-stakes to the centre stake. 317. The Area of a Three-level Section. Let d and d' be the distances out, and A and /i' the heights above grade of right and left slope- stakes, respectively; D the sum of d and d\ c the centre height, r the slope-ratio, w the width of bed. 4o6 SUR VE YING. Then the area ABCDE is equal to the sum of the lour trian- gles A£w, BCw, wCDy and wED. Or, (d d') c (Ji + h'')~ ^ = i (-) This area is also equal to the sum of the triangles FCD and FED^ minus the triangle AFB. Or, A = id 4 ^' ( 2 ) Equation (2) can also be obtained directly from equation (1) by substituting for h and h' in (i) their values in terms of d-^- ■ 2 d and w, h — , and then putting D = d-\- d'. Equation (2) has but two variables, c and D, and is the most convenient one to use. 318. Cross-sectioning. — It will be seen from Fig. 112 that in the case of a three-level section the only quantities to be determined in the field are the heights, h and h' , and the dis- tances out, d and d' , of the slope-stakes. These are found by trial. A levelling instrument is set up so as to read on the THE MEASUREMENT OF VOLUMES. 407 three points C, D, E, and the rod held first at D. The reading here gives the height of instrument above this point. Add this algebraically to the centre height (which may be negative, and which has been obtained from the profile for each station), and the sum is the height of instrument above (or below) the grade line. If the ground were level transversely, the distance out to the slope-stakes would be d= cr A — . ' 2 But this is not usually the case, and hence the distance out must be found by trial. If the ground slopes | | from the centre line in a -j i the distance out will evidently be more than that given by the above equation, and vice versa. The rodman estimates this distance, and holds his rod at a cer- tain measured distance out, d^. The observer reads the rod, and deducts the reading from the height of instrument above grade (or adds it to the depth of instrument below grade), and this gives the height of that point, above or below grade. Its w distance out, then, should he d = /qr + “ • If this be more than the actual distance out, d^, the rod is set farther in ; if less, it is moved out. The whole operation is a very simple one in prac- tice, and the rodman soon becomes very expert in estimating nearly the proper position the first time. In heavy work — that is, for large cuts or fills, and for irregu- lar ground — it may be necessary to take the elevation and dis- tance out of other points on the section in order to better determine its area. These are taken by simply reading on the rod at the critical points in the outline, and measuring the dis- tances out from the centre. The points can then be plotted 4o8 SUKVEVmG. on cross-section paper and joined by straight or by free-hand curved lines. In the latter case the area should be deter- mined by planimeter. 319. Three-level Sections, the Upper Surface con- sisting of two Warped Surfaces. — If the three longitudinal lines joining the centre and side heights on two adjacent three- level sections be used as directrices, and two generatrices, one on each side the centre, be moved parallel to the end areas as plane directers, two warped surfaces are generated, every cross- section of which parallel to the end areas is a three-level sec- tion. These same surfaces could be generated by two longi- tudinal generatrices, moving over the surface end-area lines as directrices. In this case the generating lines would not move parallel to a plane directer, but each would move so as to cut its directrices proportionally. The surface would therefore be a prismoid, and its exact volume would be given by the pris- moidal formula. The middle area in this case is readily found, since the centre and side heights are the means of the corre- sponding end dimensions. The prismoidal formula. could therefore be written This equation is derived directly from eq. (i) above, and eq. w (2), p. 406. The quantity — is the distance from the grade-plane THE MEASUREMENT OF VOLUMES. 409 to the intersection of the side slopes, and is a constant for any given piece of road. It would have different values, however, in cuts and fills on the same line. For brevity, let w Here K is the volume of the prism of earth, 100 feet long, in- cluded between the roadbed and side slopes. It is first in- cluded in the computation and then deducted. It is also a constant for a given piece of road. Equation (2) now becomes ' where and are the means of and respectively. This equation involves but two kinds of variables, c and D, and is well adapted to arithmetical, tabular, or graphical com- putation. Thus if / =: 100 ; ze/ = 18 ; and r = i-J ; then — and K — 200 ; and equation (3) becomes ^ [(^1 + A + A + ^)A + 4 (^m + 6)Z>J — 200 . (4) If the total centre heights (to intersection of side slopes) be represented by C^, C„ and C^, then eq. (3) becomes, in general. where K' = and is independent of width of bed and of slopes. For any given piece of road, the constants K, K\ and are known, and for each prismoid the Z”s and Z^’s are observed, hence for any prismoid all the quantities in eq. (5) are known. 410 SUR VE YING. 320. Construction of Tables for Prismoidal Computa- tion. — If a table were prepared giving the products K'CD for various values of C and it could be used for evaluating equation (3), which is the same as equation (5). The argu- ments would be the total widths (i?,), and the centre heights (6'i). Such a table would have to be entered three times for each prismoid, first with C, and ; second with and ; and finally with and D^. If four times the last tabular value be added to the sum of the other two, and K subtracted, the result is the true volume of the prismoid. VALUES OF (= AND K (= FOR VARIOUS WIDTHS \ 2;7 \ 4 X 27;/ AND SLOPES. Width Slopes. of Road- H to 1 . to 1 . to 1. 1 to 1 . tol. to 1 . 194 tol. 2 to 1. bed. Q AT Co AT Co AT Co AT Co AT Co AT Co A' Co AT 10 20 370 10 185 6.7 123 50 93 4.0 74 3-3 62 2.9 53 2-5 46 11 22 448 II 224 7-3 149 5-5 112 4.4 90 3.7 75 3-1 64 2.8 56 12 24 533 12 266 8.0 178 6.0 133 4.8 107 4.0 89 3-4 76 3.0 67 13 26 626 13 313 8.7 209 6.5 157 5-2 125 4-3 104 3-7 89 3-2 78 14 28 72s 14 363 9-3 242 7.0 181 5-6 145 4.7 121 4.0 104 3-5 91 15 30 833 15 417 10. 0 278 7-5 208 6.0 167 5.0 139 4-3 119 3-8 104 16 32 948 16 474 10.7 316 8.0 237 6.4 190 5.3 158 4.6 135 4.0 118 17 34 1070 17 535 II . 3 357 8.5 268 6.8 214 5-7 178 4.9 153 4.2 134 18 36 1200 18 600 12.0 400 9.0 300 7.2 240 6.0 200 5-1 171 4-5 150 19 38 1337 19 668 12.7 446 9-5 334 7.6 267 6.3 223 4.4 191 4.8 167 20 40 1481 20 740 13.3 494 10. 0 370 8.0 296 6.7 247 5-7 212 5-0 185 21 42 1633 21 816 14.0 544 10.5 408 00 327 7.0 272 6.0 233 5-2 204 22 44 1793 22 896 14.7 598 II .0 448 8.8 359 7-3 299 6.3 256 5-5 224 23 46 1959 23 980 15-3 653 II -5 490 9.2 392 7-7 326 6.6 280 5-8 245 24 48 2134 24 1067 16.0 711 12.0 534 9.6 427 8.0 356 6.9 305 6.0 267 25 50 2315 25 1158 16.7 772 12. 5 579 10. 0 463 8.3 386 7-1 331 6.2 264 26 52 2504 26 1252 17.3 835 13.0 626 10.4 501 8.7 417 7-4 358 6.5 313 27 54 2700 27 1350 18.0 900 13.5 675 10.8 540 9.0 450 7-7 386 6.8 338 28 56 2904 28 1452 18.7 968 14.0 726 II .2 581 9 3 484 8.0 415 7.0 363 29 58 3”5 29 1558 19.3 1038 14.5 779 II. 6 623 9.7 519 8-3 445 7.2 389 30 60 3333 30 1667 20.0 nil 15-0 833 12.0 667 10. 0 556 8.6 476 7-5 417 THE MEASUREMENT OF VOLUMES. All Table XL* is such a table, computed for total centre heights from I to 50 feet, and for total widths from i to 100 feet. In railroad work neither of these quantities can be as small as one foot, but the table is designed for use in all cases where the parallel end areas maybe subdivided into an equal number of triangles or quadrilaterals. Example i. Three-level Ground having two Warped Surfaces. — Find the volume of two prismoids of which the following are the field-notes, the width of bed being 20 feet, and the slopes to i. Station ii. Station 12. Station 12 -f- 56. 2S.9f 0 43-0 12.6 + 18.6 -{- 22.0 27.1 0 40.3 + 11.4 20.2 24-3 0 34-9 + 9-5 + 10.3 + 16.6 From the table, p. 410, giving values of Co and K, we find for w = 20, and r — i|, Co = 6-7, and K — 247. The computation may be tabulated as follows: Sta. Width, D^d^d'. Height, C = c + Cq. Partial Volume. Volume of Prismoid. II 71.9 25-3 562 M 69.6 23-4 503 X 4 = 2012 12 67.4 21.5 447 3021 — 247 2744 M 63-3 19.2 374 X 4 = 1496 12 + 56 59-2 17.0 311 .56 (2254 - 247) 1124 * Modeled somewhat after Crandall’s Tables, but adapted to give volumes by the Prismoidal Formula at once instead of by the method of mean end areas first and correcting by the aid of another table to give prismoidal volumes, as Prof. Crandall has done. f The numerators are the distances out, and the denominators are the heights above grade, + denoting cut and — fill. 412 SURVEYING. Entering^ the table (No. XI.) fora width of 71 and a height of 25, we find 548, to which add 7 for the 3 tenths of height, and 7 more for the 9 tenths in width, both mentally, thus giving 562 cu. yds. for this partial volume. Simi- larly for the width 67.4, and height 21.5, obtaining 447 cu. yds. The correspond- ing result for the middle area is 503, which is to be multiplied by 4, thus giving 2012 cu. yds. The sum of these is 3021 cu. yds., from which is to be subtracted the constant volume A”, which in this case is 247 cu. yds., leaving 2774 cu. yds. as the volume of the prismoid. The next prismoid is but 56 feet long, but it is taken out just the same as though it were full, and then 56 hundredths of the resulting volume taken. The data for the 12th station is used in getting this result without writing it again on the page. Example 2. Five-level Ground having four Warped Surfaces. — Find the volume of a prismoid of which the following are the field-notes, the width of bed being 20 feet, and the slopes to i : 28.9 150 0 20.0 43-0 + 12.6 + 12.0 + 18.6 + 21.0 + 22.0 27.1 12.5 0 18.5 40.3 + 11. 4 + 12.0 + 14.8 + 19.6 +20.2 This is the same problem as the preceding, with intermediate heights added. To compute this from the table, it is separated into three prismoids, as shown in Fig. 1 13. Fig. 113. Let ABDGCFE be the cross-section. This may be separated into the triangle ABC, and the two quadrilaterals BCGD and ACFE. The area of the triangle is icw. That of the right quadrilateral is, from Art. 179, p. 202, THE MEASUREMENT OF VOLUMES. \\2a c (^dic — — o) + — dky^ = — A)(^dk — + kdn ^ Similarly the area of the left quadrilateral is ^ The total area of the section then is + lid'n CW kdn + {f . . (I) If the interior side elevations be taken over the edges of the base, then *W 'W d'k and dk both become zero, and the first and last terms disappear. 2 2 Or if the centre and extreme side heights are the same, these terms go out. Experience shows that these terms can usually be neglected without material error. If they are retained, each partial volume will be composed of five terms, while if they are neglected there will be but three. The signs of these terms also must be carefully attended to. When the interior side readings are taken over the edges of the base, therefore, this equation becomes A = i {k'd'h + CW + kdh) ( 2 ) The tables are well adapted to compute the prismoidal volume for five-level sections by either of these formulae. Thus, if the adjacent section also has five points determined in its surface, its area may be represented by an equation similar to one of these, and from these end-area data mean values may be found for the corresponding middle-area points, and the volumes taken out as before. In this case the prism included between the road-bed and side- slopes, whose volume is K, is not included, and hence its volume is not to be deducted from the result. The computation by table XI. of equation (i) would be as follows : Sta. k '. d ' k . k '. d ' u - c . 4. k . dh - h . Partial Volumes. Total Volume. II 12.6 28.9 12.0 15.0 18.6 20.0 21.0 43-0 22,0 ■f9 + 108 -(- 114 -1-279— 10 = 500 M 12.0 28.0 12.0 13.8 16.7 19.2 20.3 41 .6 21 . I 4( -J- 6 -f- 104 + 102 -t- 260 - 1 2) = 1 840 12 II . 4 1 27.1 1 12.0 12.5 14.8 18. s 19.6 40-3 20.2 -1-3 + 100+ 90 + 242—13= 422 2762 412^ SUR VE YING. The use of the table is the same as before. First take out from the table the volume corresponding to (c ~ Ji) which when evaluated for section ii is (i8.6 — i2.6)(i5.o— io) = 6.0 x 5.0. This is positive, and the volume corre- sponding to a depth of 6.0 feet and a width of 5.0 feet is 9 cubic yards. Proceed to evaluate the remaining terms of eq. (i) in a similar manner, the last term coming out negative. The dimensions of the mid section are the means of the corresponding end dimensions, as before. If one end-area is a three-level section and the next a five-level section, the included prismoid is computed as a five-level prismoid, the vanishing points in the three-level section corresponding to the interior side elevations on the five-level section being indicated in the field. Par- tial stations, or prismoids, are first computed as though they were 100 feet long (for which the table is constructed), and then multiplied by their length and divided by 100 as before. If equation (2) may be used, the work is shortened very much. The columns in //', d\ , dk , and /^, may be omitted, and there will also be but three terms in each partial product. Thus, if sections ii and 12 had been taken with the interior elevations, each 10 feet from the centre line, we might have had something as follows : 28.9 lO.O 0 10. 0 43-0 -f- 12.6 + 15-4 -f i8.6 -b 19.8 -f- 22.0 27 »i 10. 0 0 10. 0 40.3 -f-11.4 + 12.5 + 14.8 + 17-4 -1-20.2 The computation then, by eq. (2), would have been : Sta. d\. k'. c. k. dh- Partial Volumes. Total Volume. II 28.9 15-4 18.6 19.8 43-0 137 + II4 + 263 = 514 M 28.0 14.0 16.7 18.6 41.6 4 (121 + 102 -f 239) = 1848 12 27.1 12.5 14.8 17.4 40.3 104 -f 90 -f 215 = 409 2771 By this method the computation of a five-level section is little more trouble THE MEASUREMENT OF VOLUMES. 413 than that of a three-level section, and yet the intermediate points taken at a dis- w tance of — from the centre, are apt to increase the accuracy considerably on ordinary rolling ground. 321. Three-level Sections, the Surface divided into four Planes by Diagonals. — If the surface included between two three-level sections be assumed to be made up of four planes formed by joining the centre height at one end with a side height at the other end sec- tion on each side the centre line (Fig. 1 14), these lines being called diagonals, an exact computation of the volume is readily made without computing the mid-area. Two diag- onals are possible on each side the centre line but the one is drawn which is observed to most nearly fit the surface. They are noted in the field when the cross-sections are taken. The total volume of such a prismoid in cubic * yards is V =■ ^ 2y ^1 Vl T (^2 d" ^2 )^2 d“ T D’ C* + + + V + + (I) where and h-l are the centre and side heights at one sec- tion and dx and d( the distances out, d^, and d<^ be- * For a demonstration of this formula see Henck’s Field-Book. 414 SUR VE YING. ing the corresponding values for the other end section. C and C are the centre heights, // and //' the side heights, and D and D' the distances out on the right and left diagonals. Although this formula seems long, the computations by it arc very simple. Thus let the volume be found from the following field-notes for a base of 20 feet and side slopes to i. 22 0 47*5 + 8 \ 4 - 8 \ + 25* 34 \ ° \ + 16 + 4 + 4* The upper figures indicate the distances out and those below the lines the heights, the plus sign being used for cuts. The computation in tabular form is as follows : Sta. d. h. c. h'. d'. d-\rd'. {d^d')c. DC. D’C. I 22 8 8 25 47-5 69.5 556 .... • • « • 2 34 16 4 4 16 50.0 200 88 128 h\ -|- h-i — 24 88 = 12 128 s = 65 X 10 = 650 6 ) 162200 27 ) 27033 1001 cu. yards. The great advantage of the method consists in the data> all being at hand in the field-notes. Hudson’s Tables* give volumes for this kind of prismoid. * Tables for Computing the Cubic Contents of Excavations and Embank- ments. By John R. Hudson, C.E. John Wiley & Sons, New York, 1884. THE MEASUREMENT OF VOLUMES. 415 They furnish a very ready method of computing volumes when this system is used, 322. Comparison of Methods by Diagonals and by Warped Surfaces. — Although the surveyor has a choice of two sets of diagonals when this method is used, the real surface would usually correspond much nearer the mean of the two pairs of plane surfaces than to either one of them. That is, the natural surface is curved and not angular, and therefore it is probable that two warped surfaces joining two three-level sec- tions would generally fit the ground better than four planes, notwithstanding the choice that is allowed in the fitting of the planes. More especially must this be granted when the truth of the following proposition is established. Proposition : The volume mdiidcd between two three-level sections having their corresponding surface lines joined by warped surfaces^ is exactly a mean betweeji the two volumes formed between the same end sections by the two sets of planes re- sulting from the two sets of diagonals which jnay be drawn. If the two sets of diagonals be drawn on each side the centre line and a cross-section be taken parallel to the end areas, the traces of the four surface planes on each side the centre line on the cutting plane will form a parallelogram, the diagonal of which is the trace of the warped surface on this cutting plane. Since this cutting plane is any plane par- allel to the end areas, and since the warped surface line bisects the figure formed by the two sets of planes formed by the diagonals, it follows that the warped surface bisects the volume formed by the two sets of planes. The proposition will there- fore be established if it be shown that the trace of the warped surface is the diagonal of the parallelogram formed by the traces of the four planes formed by the two sets of diagonals. Fig. 1 15 shows an extreme case where the centre height is higher than the side height at one end and lower at the other. Only the left half of the prismoid is shown in the figure. The 4i6 SUJ? VE Y TNG. cutting plane cuts the centre and side lines and the two diago- nals in cfgh on the plane, and in c'fg'h' on the vertical projection. For the diagonal the surface lines cut out are e'f' and f'h'. For the diagonal they are e'g' and g'h\ For the warped surface the line cut out is e'h\ this being an Fig. 115. element of that surface. It remains to show that e'fdi'g' is a parallelogram. Since the cutting plane is parallel to the end planes all the lines cut are divided proportionally. That is, if the cutting plane is one of I from then it cuts off one of all the lines cut, measured from that end plane. But if the lines are divided proportionally, the projections of those lines are divided proportionally, and hence the points e' ,f divide THE MEASUREMENT OF VOLUMES. 417 the sides of the quadrilateral proportionally. But it is a proposition in geometry that if the four sides of a quad- rilateral, or two opposite sides and the diagonals, be divided proportionally and the corresponding points of subdivision joined, the resulting figure is a parallelogram. Therefore ef'H g' is a parallelogram, and e'li is one of its diagonals and hence bisects it. Whence the surface generated by this line moving along and parallel to the end areas bisects the volume formed by the four planes resulting from the use of both di- agonals on one side the centre line. Q. E. D. It is probable, therefore, that the warped surface would usually fit the ground better than either of the sets of planes formed by the diagonals. Furthermore, the errors caused by the use of the warped surface (Table XL) are compensating errors, thus preventing any marked accumulation of errors in a series of prismoids.* There are extreme cases, however, such as that given in the example, Fig. 114, which are best computed by the method by diagonals. 323. Preliminary Estimate from the Profile. — If the cross-sections be assumed level transversely then for given width of bed and side slopes, a table of end areas may be pre- pared in terms of the centre heights. From such a table the * The two methods here discussed are the only ones that have any claims to accuracy. The method by “ mean end areas,” wherein the volume is assumed to be the mean of the end areas into the length, always gives too great a volume (except when a greater centre height is found in connection with a less total width, which seldom occurs), the excess being one sixth of the volume of the pyramids involved in the elementary forms of the prismoid. This is a large error even in level sections, and very much greater on sloping ground, and yet it is the basis of most of the tables used in computing earthwork, and in some States it is legalized by statute. Thus in the example computed by Henck’s method on p. 414 the volume by mean end areas is 1193 cu. yards; by the prismoidal formula u is 1168 cu. yards, while by the method by diagonals it was only 1001 cu. yards. This was an extreme case, however, and was selected to show the adaptation of the method by diagonals to such a form. 27 4i8 SURVEYING. end areas may be rapidly taken out and plotted as ordinates from the grade line. The ends of these ordinates may then be joined by a free-hand curve, and the area of this curve found by the planimeter. The ordinates may be plotted to such a scale that each unit of the area, as one square inch, shall represent a convenient number of cubic yards, as looo. The record of the planimeter then in square inches and thou- sandths gives at once the cubic yards on the entire length of line worked over by simply omitting the decimal point. Evi- dently the scale to which the ordinates are to be drawn to give such a result is not only a function of the width of bed and side slopes, but also of the longitudinal scale to which the pro- file line is plotted. The area of a level section is A = we rc^, . . • • (I) where w., and r are the width of base, centre height, and slope-ratio respectively. Now if li — the horizontal scale of the profile, that is the number of feet to the inch, and if one square inch of area is to represent lOOO cu. yards, the length of the ordinate must be hA h {zve -|- rc"^) ( 2 ) lOOO X 27 27,CXX) If values be given to //, and r, which are constants for any given case, then the value of y becomes a function of c only, and a table can be easily prepared for the case in hand. Since y is a. function of the second power of e, the second dif- ference will be a constant, and the table can be prepared by means of first and second differences. Thus if c takes a small increment, as i foot, then the first difference is (3) THE MEASUREMENT OF VOLUMES. 419 But this first difference is also a function of c, and hence when c takes an increment this first difference changes by an amount equal to A'y = h 27000 2r, (4) which is constant. An initial first difference being given for a certain value of a column of first differences can be obtained by simply adding the A"y continuously to the preceding sum. With this column of first differences the corresponding column of values of y may be found by adding the first differences con- tinuously to the initial value of y for that column.* TABULAR VALUES OF jy IN EQUATION (2) FOR w=2o, r=ii, AND h — 400. c o.'o O.'l 0.'2 o .'3 o .'4 o .'5 o .'6 o.'l o .'8 o .'9 in. in. in. in. in. in. in. in. in. in. 0 0.00 0.03 0.06 0.09 0.12 0.15 0. 19 0.22 0.25 0.28 I •32 •35 -39 .42 .46 -49 -.53 -57 .61 .64 2 .68 .72 -76 .80 .84 .88 .92 .96 1. 00 1.05 3 1.09 1-13 1-17 1.22 1.26 1-31 1-35 1.40 1-45 1-49 4 1-54 1.59 1.63 1 .69 1-73 1.78 1.83 1.88 1.93 1-99 5 2.04 2.09 2.14 2.19 2.24 2.30 2.36 2.41 2.47 3-52 6 2.58 2.63 2.69 2.75 2.80 2.87 2.92 2.98 3-04 3.10 7 316 3.22 3.28 3-35 3-41 3-47 3-54 3.60 3.66 3-73 8 3-79 3.86 3-92 3-99 405 4-13 4-19 4.26 4.33 4-40 9 4-47 4-54 4.60 4.68 4-75 4.82 4-89 4-97 5-04 5.11 10 5.18 5.26 5-33 5-40 5 48 5-56 5-64 5-72 5-79 5-87 II 5-95 6.03 6.10 6.18 6.26 6.35 6-43 6.51 6.59 6.67 12 6.76 6.84 6.92 7.00 7-09 7.18 7.26 7.35 7-43 7-52 13 7.61 7.70 7.78 7.86 7.96 8.05 8.14 8.23 8.32 8.41 14 8.50 8.60 8.68 8.77 8.87 8.97 9.06 9.16 9-25 9-35 15 9.44 9-54 9-63 9-73 9-83 9-94 10.03 10.13 10.23 10.33 16 10.43 10-53 10.62 10.73 10.83 10.94 11.04 11.15 11.25 11-35 17 11.46 11.56 11.66 11-77 11.88 12.00 12.10 12.21 12.31 12.42 18 12.53 12.64 12.75 12.86 12.97 13-09 13.20 13.32 13.42 13-54 19 13-65 13-77 13.87 13-99 14.10 14 23 14.34 14-47 14.58 14-70 20 14.81 14.93 15-04 15.16 15.29 15-42 15.53 15.66 15.78 15.90 * For a further exposition of this subject, see Appendix C. 420 SURVEYING. The preceding table was constructed in this manner, for w — 20 feet, r = \\\ and h — 400 feet to the inch. 324. Borrow-pits are excavations from which earth has been “ borrowed ” to make an embankment. It is generally preferable to measure the earth in cut rather than in fill, hence when the earth is taken from borrow-pits and its volume is to be computed in cut, the pits must be carefully staked out and elevations taken both before and after excavating. The meth- ods given in art. 31 1 are well suited to this purpose, or they may be computed as prismoids by the aid of Table XL, if pre- ferred. To use the table it is only necessary to enter it with such heights and widths as give twice the elementary areas (triangles or quadrilaterals) into which the end sections are divided, and then multiply the final result by the length and divide by 100. The table is entered for both end-area dimen- sions and also the mid-area dimensions, four times this latter result being taken the same as before. 325. Shrinkage of Earthvvork. — Excavated earth first increases in volume, when removed from a cut and dumped on a fill, but it gradually settles, or shrinks, until it finally comes to occupy a less volume than it formerly did in the cut. Both the amounts, initial increase, and final shrinkage depend on the nature of the soil, its condition when removed, and the man- ner of depositing it in place. There can therefore be no gen- eral rules given which will always apply. For ordinary clay and sandy loam^ dumped loosely, the first increase is about one twelfth, and then the settlement about one sixth of this increased volume, leaving a final volume of about nme tenths of the original volume in cut.^ Thus for 100 cubic yards of settled embankment in cubic yards in cut would be required. But a contractor should have * See paper by P. J. Flynn in Trans. Tech. Soc. of the Pacific Coast, vol. ii. p. 179, where all the available experimental data are given. THE MEASUREMENT OF VOLUMES. 421 his stakes or poles set one fifth higher than the corresponding fill, so that when filled to the tops of these, a settlement of one sixth will bring the surface to the required grade. These changes of volume are less for sand and more for stiff, wet clay. For rock the permanent increase in volume is from 60 to 80 per cent, the greater increase corresponding to a smaller average size of fragment. 326. Excavations under Water. — It is often necessary to determine the volume of earth, sand, mud, or rock removed from the beds of rivers, harbors, canals, etc. If this be done by soundings alone, it is likely to work injustice to the con- tractor, as he would receive no pay for depths excavated below the required limit ; and besides, foreign material is apt to flow in and partially replace what is removed, so that the material actually excavated is not adequately shown by soundings within the required limits. It is common, therefore, to pay for the material actually removed, an inspector being usually furnished by the employer to see that no useless work is done beyond the proper bounds. The material is then measured in the dumping scows or barges. The unit of measure is the cubic yard, the same as in earthwork. There are two general methods of gauging scows, or boats. One is to actually meas- ure the inside dimensions of each load, which is often done in the case of rock, and the other is to measure the displacement of the boat, which is the more common method with dredged material. When the barge is gauged by measuring its dis- placement, the water in the hold must always be pumped down to a given level, or else it must be gauged both before and after loading and the depth of water in the hold observed at each gauging. A displacement diagram (or table) is prepared for each barge, from its actual external dimensions, in terms of its mean draught. There should always be four gaugings taken to determine the draught, at four symmetrically located points 422 SURVEYING. on the sides, these being one fourth the length of the barge from the ends. Fixed gauge-scales, reading to feet and tenths may be painted on the side of the barge, or if it is flat-bot- tomed, a gauging-rod, with a hook on its lower end at the zero of the scale, may be used and readings taken at these four points. Any distortion of the barge under its load, or any unsymmetrical loading, will then be allowed for, the mean of the four gauge-readings being the true mean draught of the boat. To prepare a displacement diagram, the areas of the sur- faces of displacement must be found for a series of depths uni- formly spaced. This series may begin with the depth for no load, the hold being dry. They should then be found for each five tenths of a foot up to the maximum draught. If the boat has plane vertical sides and sloped ends these areas are rec- tangles, and are readily computed. If the boat is modelled to curved lines, the water-lines can be obtained from the original drawings of the boat, or else they must be obtained by actual measurement. In either case they can be plotted on paper, and their areas determined by a planimeter. These areas are analogous to the cross-sections in the case of railroad earth- work, and the prismoidal formula may be applied for comput- ing the displacement. Thus, Let A^, A^, etc., be the areas of the displaced water surfaces, taken at uniform vertical distances h apart. Then for an even number of intervals we have in cubic yards F= + . (I) If the total range in draught be divided into six equal por- tions, each equal to h, then Weddel’s Rule * would give a For the derivation of this rule see Appendix C. THE MEASUREMENT OF VOLUMES 423 nearer approximation. With the same notation as the above we would then have, in cubic yards, = ~ [^0 + + ^4 + ^6 + 5 (^i + ^3 + ^5) + ^3]- • (2) These rules are also applicable to the gauging of reservoirs, mill-ponds, or of any irregular volume or cavity. After the displaced volume of water is found, the corre- sponding volume of earth or rock is found by applying a proper constant coefficient. This coefficient is always less than unity, and is the reciprocal of the specific gravity of the material. This must be found by experiment. In the case of soft mud it is nearly unity, while with sand and rock it is much more. When rock is purchased by the cubic yard, solid rock is not implied, but the given quality of cut or roughly-quarried rock, piled as closely as possible. When rock is excavated, solid rock is meant. A measured volume of any material put into a gauged scow will give the proper coefficient for that material. Thus if the measured volume V give a displacement of F, V then — C \s the coefficient to apply to the displacement to give the volume of that material. CHAPTER XIV. GEODETIC SURVEYING. 327. The Objects of a Geodetic Survey are to accurately determine the relative positions of widely separated points on the earth’s surface and the directions and lengths of the lines joining them ; or to accurately determine the absolute (in latitude, in longitude from a fixed meridian, and in eleva- tion above the sea-level) of widely separated points on the earth’s surface and the directions and lengths of the lines join- ing them. In the first case the work serves simply t© supply a skeleton of exact distances and directions on which to base a more de- tailed survey of the intervening country ; in the second, the re- sults furnish the data for computing the shape and size of the earth, in addition to their use in more detailed surveys. It is usually desirable also to have some knowledge of the latitude and longitude of the points determined in the first case, but a very accurate knowledge of these would not be es- sential to the immediate objects of the work. In both cases the points determined form the vertices of a series of triangles joining all the points in the system. One or more lines in this system of triangles and all of the angles are very carefully measured, and the lengths of all other lines in the system computed. The azimuths of certain lines are also determined, and, if desired, the latitudes and longitudes of some of the points. I"rom this data it is then possible to compute the latitudes and longitudes of all the points in the system and GEODETIC SURVEYING. 425 the lengths and azimuths of all the connecting lines. The work as a whole is denominated triangulation. The measured lines are called base-lines, the points deter- mined are triangulation-stations, and those points (usually tri- angulation-stations) at which latitude, longitude, or azimuth is directly determined are called respectively latitude, longi- tude, or azimuth stations. The latitude of a station and the azimuth of a line are determined at once by stellar observations at the point. The longitude is found by observing the differ- ence of time elapsing between the transit of a star across the meridian of the longitude-station and the meridian of some fixed observatory whose longitude is well determined. An ob- server at each station notes the time of transit across his merid- ian, and each transit is recorded upon a chronograph-sheet at each station. This requires a continuous electrical connection between the two stations. This difference of time, changed into longitude, gives the longitude of the field-station with ref- erence to the observatory. 328. Triangulation Systems are of all degrees of magni- tude and accuracy, from the single triangle introduced into a course to pass an obstruction, up to the large primary systems covering entire continents, the single lines in which are some- times over one hundred miles in length. The methods herein described will apply especially to what might be called secondary and tertiary systems, the lines of which are from one to twenty miles in length, and the accu- racy of the work anywhere from i in 5000 to i in 50,000. Al- though the methods used are more or less common to all sys- tems, yet for the primary systems, where great areas are to be covered and the highest attainable accuracy secured, many refinements, both in field methods and in the reductions, are introduced which would be found useless or needlessly expen- sive in smaller systems. If it is desired to connect two distant points by a system 426 SURVEYING. of triangulation at the least expense, then use system I.,sliown in Fig. 1 16. This S3^stcm is also adapted to the fixing of a double row of stations with the least labor. If such distant points are to be joined, or such double system of stations established, with the greatest attainable accuracy, then system III. should be used. This system is also best adapted to secondary work, where it is desired to simplify the work of reduction. Each quadrilateral is independently re- duced. If the greatest area is to be covered for a given degree of accuracy or cost, then system II. is the one to use. System I. consists of a single row of simple triangles, sys- I. tern II. of a double row of simple triangles or of simple tri- angles arranged as hexagons, and system III. of a single row of quadrilaterals. A quadrilateral in triangulation is an arrange- ment of four stations with all the connecting lines observed. This gives six lines connecting as many pairs of stations, over which pointings have been taken from both ends of the line. GEODETIC SURVEYING. 42^ For the same maximum length of lines we have the follow- ing comparison of the three systems : * System. Composition. Distance Covered. No. of Sta- tions. Total Length of Sides. Area Covered. No. of Conditions. I. Equilateral triangles. 5 II 19 4.5 O' II 1 II. Hexagons. 5.2 17 34 9 5 III. Quadrilaterals (squares). 4.95 16 29 3-5 2 n — 4 = 28 Thus, for the same distance covered, the number of sta- tions to be occupied and the total length of lines to be cleared out are about one half more for systems II. and III. than for system I. The area covered by system II. is twice that by system I., but the number of conditions is much greater in system III. than in either of the others. Since almost all the error in triangulation comes from erroneous angle-measure- ments, the results will be more accurate according to our ability to reduce the observed values of the angles to their true values. The “ conditions” mentioned in the above table are rigid geometrical conditions, which must be fulfilled (as that the sum of the angles of a triangle shall equal 180°), and the more of these geometrical conditions we have, the more neaily are we able to determine what the true values of the angles are. The work will increase in accuracy, therefore, as the number of these conditions increases, and this is why sys- tem III. gives more accurate results than systems I. and II. This will be made clear when the subject of the adjustment of the observations is considered. 329. The Base-line and its Connections. — The line whose length is actually measured is called the base-line. The * Taken from the U. S. C. and G, Survey Report for 1876. SURVEYING. lengths and distance apart of sucli lines depend on the charac- ter of the work and the nature of the ground. Primary base- lines are from three to ten miles in length, and from 200 to Slint Tnine>r‘ Morgan. Park Fig. 117.* 600 miles apart. In general, in primary work, the distance apart has been about one hundred times the length of the base. Secondary bases are from two to three miles in length, Otis Fig. 118.* and from fifty to one hundred and fifty miles apart, the dis- tance apart being about fifty times the length of base. Ter- tiary bases are from one half to one and a half miles in length * Taken from professional papers, Corps of Engineers U. S. Army, No. 24, being the final report on the Triangulation of the United States Lake Survey. GEODETIC SURVEYING. 429 and from twenty-five to forty miles apart, the distance apart being about twenty-five times the length of base.* The location of the base should be such as to enable one side of the main system to be computed with the greatest accuracy and with the least number of auxiliary stations for a given length of base. In flat open country the base may be chosen to suit the location of the triangulation-stations in the main system ; but in rough country some of the main stations must often be chosen to suit the location of the base-line. In Fig. 1 17 the location of the base-line is almost an ideal one, being taken directly across one of the main lines of the sys- tem. By referring to Fig. 118 it will be seen that the line Willow Springs — Shot Tower is one of the fundamental lines of the main system, and the base is located directly across it. Here the ground is a flat prairie, and the base was chosen to suit the stations of the main system. The station at the middle of the base is inserted in order to furnish a check on the measurements of the two portions as well as to increase the strength of the system by increasing the number of equations of conditions. Sometimes it is neces- sary to use one or more auxiliary stations outside the base before the requisite expansion is obtained. Thus suppose the stations Morgan Park and Lombard were the extremities of the line of the main system whose length was to be computed from this base, then the stations Willow Springs and Shot Tower might have been occupied as auxiliary stations from which the line Morgan Park — Lombard could be computed. 330. The Reconnaissance. — A system of triangulation having been fixed upon, of a given grade and for a given pur- * These intervals between bases are in accordance with the practice that has hitherto been followed. The new method of measuring base-lines with a steel tape, described on p. 450, will probably change this practice by causing more bases to be measured, leaving much shorter intervals to be covered by angular measurement. 430 SUR VE Y I NG. pose, the first thing to be done is to select the location of the base-line and the position of the base-stations. The base should be located on nearly level ground, and should be favorably sit- uated with reference to the best location of the triangulation- stations. These stations are then located, first for expanding from the base to the main system, and then with regard to the general direction in which the work is to be carried, and to the form of the triangles themselves. No triangle of the main system should have any angle less than 30° nor more than 120°. Although small angles can be measured just as accurately as large ones, a given error in a small angle, as of one second, has a much greater effect on the resulting distances than the same error in an angle near 90°. In fact, the errors in distance are as the tabular differences in a table of natural sines, for given errors in the angles. These tabular differences are very large for angles near 0° or 180°, but reduce to zero for angles at 90°. The best-proportioned tri- angle is evidently the equilateral triangle, and the best-propor- tioned quadrilateral is the square. In making the reconnais- sance the object should be to fulfil these conditions as nearly as possible. The most favorable ground for a line of triangles is a valley of proper width, with bald knobs or peaks on either side. Sta- tions can then be selected giving well-conditioned triangles, with little or no clearing out of lines, and with low stations. In a wooded country the lines must be cleared out or else very tall stations must be used. In general, both expedients are re- sorted to. Stations are built so as to avoid the greater portion of the obstructions, and then the balance is cleared out. So much depends on the proper selection of the stations in a system of triangulation, as to time, cost, and final accuracy, that the largest experience and the maturest judgment should be made available for this part of the,work. The form of the triangles ; the amount of cutting necessary to clear out the lines and the probable resulting damage to private interests ; GEODETIC SURVEYING. 431 the height and cost of stations, and the accessibility of the same ; the avoidance of all sources of atmospheric disturbance on the connecting lines, as of factories, lime- or brick-kilns, and the like, which might either obstruct the line by smoke or in- troduce unusual refraction from heat ; the freedom from dis- turbance of the stations themselves during the progress of the work, and the subsequent preservation of the marking-stones — these are some of the many subjects to be considered in de- termining the location of stations. It is the business of the reconnaissance party not only to locate the stations, but to determine the heights of the same. A station that has been located is temporarily marked by a flag fastened upon a pole, and this made to project from the top of a tall tree in the neighborhood. In selecting a new station it is customary to first select from the map the general locality where a station is needed, and then examine the region for the highest ground available. When this is found, the tallest trees are climbed and the horizon scanned by the aid of a pair of field-glasses to see if the other stations are visible. If no tree or building is available for this purpose ladders may be spliced together and raised by ropes until the desired height is obtained. 331. Instrumental Outfit. — The reconnaissance party re- quires a convenient means of measuring angles and of determ- ining directions and elevations. For measuring angles a pocket sextant would serve very well, provided the stations are distinct or provided distinct range-points in line with the stations may be selected by the aid of field-glasses. A prismatic pocket- compass will often be found very convenient in finding back stations which have been located and whose bearings are known. An aneroid barometer is desirable for determining approxi- mate relative elevations. For methods of using it in such work, see Chapter VI., p. 136. If to the above-named instru- ments we add field-glasses, and creepers for climbing trees, the instrumental outfit is fairly complete. 432 SURVEYING. 332. The Direction of Invisible Stations. — It often hap- pens that one station cannot be seen from another on account of forest growth, which may be cleared out. In such a case the station may be located and the line cleared from one station or from both, the direction of the line having been determined. This direction may always be computed if two other points can be found from each of wliich both stations and the other auxiliary point are visible. Thus in Fig. 119 let AB be the line to be cleared out, and let C and D be two points from which all the stations may be sighted. Measure the two angles at each station and call the distance CD unity. Solve the triangle BCD for the side BC, and the triangle ADC for the side AC. We now have in the triangle ABC two sides and the included angle to find the other angles. When these are found the course may be aligned from either A or B. It will often happen that either C or D or both can be taken at regu- lar stations. Of course a target must be left at either C or D to be used in laying out the line from A or B. The above is a modification of the problem given in art. 1 10, p. 107. A use of this expedient will often greatly facilitate the work. 333. The Heights of Stations depend on the relative heights of the ground at the stations and of the intervening region. If the surface is level, then the heights of stations depend only on their distance apart. In any case the dis- tance apart is so important a function of the necessary height that it is well to know what the heights would have to be for level, open country. The following table* gives the height of one station when the other is at the ground level, for open, level country: * Taken from Report of U. S. Coast and Geodetic Survey for 1882. GEODETIC SURVEYING. 433 DIFFERENCE IN FEET BETWEEN THE APPARENT AND TRUE LEVEL AT DISTANCES VARYING FROM i TO 66 MILES. Dis- tance, miles. Difference in feet for — Dis- tance, miles. Difference in feet for — Curvature. Refraction. Curvature and Refraction. Curvature. Refraction. Curvature and Refraction. I 0.7 0. 1 0.6 34 771-3 108.0 663.3 2 2.7 0,4 2.3 35 817.4 II4.4 703.0 3 6.0 0.8 5-2 36 864.8 I 2 I. I 743-7 4 10.7 15 9.2 37 913-5 127.9 785.6 5 16.7 2-3 14.4 38 963.5 134.9 828.6 6 24.0 3-4 20.6 39 1014. 9 142. 1 872.8 7 32.7 4.6 28.1 40 1067.6 149-5 918.1 8 42.7 6.0 36-7 41 1121.7 157.0 964.7 9 54-0 7-6 46.4 42 1177. 0 164.8 1012. 2 10 66.7 9-3 57-4 43 1233.7 172.7 1061. 0 II 80.7 ir -3 69-4 44 1291.8 180.8 IIII.O 12 96.1 13-4 82.7 45 1351. 2 189*. 2 1162.0 13 112.8 15.8 97-0 46 1411.9 197.7 1214. 2 14 130.8 18.3 112.5 47 1474.0 206.3 1267.7 15 150. 1 21.0 129. 1 48 1537.3 215.2 1322. I 16 170.8 23-9 146.9 49 1602.0 224.3 ^ 377-7 17 192.8 27.0 165.8 50 1668 . I 233.5 1434.6 18 216.2 30.3 185.9 51 1735.5 243-0 1492.5 19 240.9 33-7 207.2 52 1804,2 252.6 1551.6 20 266.9 37-4 229.5 53 1874.3 262.4 1611.9 21 94-3 41.2 253-1 54 1945.7 272.4 1673-3 22 322.9 45-2 277-7 55 2018.4 282.6 1735.8 23 353-0 49-4 303.6 56 2092.5 292.9 1799.6 24 384-3 53-8 330.5 57 2167.9 303.5 1864.4 25 417.0 58.4 358.6 58 2244.6 314.2 1930.4 26 451. 1 63.1 388.0 59 2322.7 325.2 1997-5 27 486.4 68.1 418.3 60 2402 . I 336.3 2065.8 28 523-1 73-2 449.9 61 2482,8 347.6 2135.2 29 561.2 78.6 482.6 62 2564.9 359-1 2205.8 30 600.5 84.1 516.4 63 2648.3 370.8 2277-5 31 641.2 89.8 551-4 64 2733.0 382.6 2350.4 32 683.3 95-7 587.6 65 2819.1 394.7 2424.4 33 726.6 loi .7 624.9 66 2906 . 5 406.9 2499.6 28 434 SURVEYING. square of distance urvature _ diameter of earth ’ Log curvature = log square of distance in feet — 7.6209807 ; Refraction = where K represents the distance in feet, R the mean radius of the earth (log R = 7*3199507), and m the coefficient of refraction,* assumed at .070, its mean value, sea- coast and interior. Curvature and refraction = (i — 2w) Or, calling h the height in feet, and K the distance in statute miles, at which a line from the height h touches the horizon, taking into account terrestrial refraction, assumed to be of the same value as in the above table (.070), we have \ni •7575’ 1.7426* The following examples will serve to illustrate the use of the preceding table : I. Elevation of Instrument required to overcome Curvature and Refraction. — Let us suppose that a line, A to B, was 18 miles in length over a plain, and that the instrument could be elevated at either station, by means of a portable tripod, to a height of 20 or 30 or 50 feet. If we determine upon 36.7 feet at A, the tangent would strike the curve at the distance rep- resented by that height in the table, viz., 8 miles, leaving the curvature (decreased by the ordinary refraction) of 10 miles to be overcome. Opposite to 10 miles we find 57.4 feet, and a * See discussion on refraction, under Geodetic Levelling, this chapter. GEODETIC SURVEYING. 435 signal at that height erected at B would, under favorable refraction, be just visible from the top of the tripod at or be on the same apparent level. If we now add 8 feet to tripod and 8 feet to signal-pole, the visual ray would certainly pass 6 feet above the tangent point, and 20 feet of the pole would be visible from A. 11 . Elevations required at given Distances . — If it is desired to ascertain whether two points in the reconnaissance, esti- mated to be 44 miles apart, would be visible one from the other, both elevations must be at least 278 feet above mean tide, or one 230 feet and the other 331 feet, etc. This sup- poses that the intervening country is low, and that the ground at the tangent point is not above the mean surface of the sphere. If the height of the ground at this point should be 200 feet above mean tide, then the natural elevations should be 478 or 430 and 531 feet, etc., in height, and the line is barely possible. To insure success, the theodolite must be elevated at both stations to avoid high signals. Since the height of station increases as the square of the distance, it is evident that the minimum aggregate station height is obtained by making them of equal height. Or, if the natural ground is higher at one station than the other, then the higher station should be put on the lower ground — that is, when the intervening country is level. If, however, the obstruction is due to an intervening elevation, the higher station should be the one nearer the obstruction. Sometimes a very high degree of refraction is utilized to make a connection on long lines. Thus on the primary trian- gulation of the Great Lakes three lines respectively 100, 93, and 92 miles in length were observed across Lake Superior, which could not have been done except that the refraction was found sometimes to exceed twice its average amount. The line from station Vulcan, on Keweenaw Point, to station Tip-Top in Canada, was 100 miles in length. The ground at statior, 436 SUItVEVINC. Vulcan was 726 feet above the lake, and the observing station was elevated 75 feet higher, making 801 feet above the surface of the lake. The station at Tip-Top was 1523 feet above the lake, the observing tripod being only 3 feet high. From the above table we find that the line of sight from Vulcan would become tangent to the surface of the lake at a distance of 37.4 miles, and that from Tip-Top at a distance of 51.5 miles, thus leaving a gap of about eleven miles between the points of tangency, for ordinary values of the refraction. If this inter- val were equally divided between the two stations and these raised to the requisite height, we would find from the table that Tip-Top would have to be elevated some 340 feet and Vulcan some 260 feet. Since this was not done, we must con- clude that an occasional excessive value of the refraction was sufficient to bend these rays of light by about these amounts in addition to the ordinary curvature from this source. In other words, the actual refraction when one of these stations was visible from the other must have been more than double its mean amount. The following is a synopsis of the heights of the stations built for the observation of horizontal angles in the primary triangulation of the Great Lakes: Total number of stations * 243 Combined height of stations 14,100 feet Average height of stations 58" “ Average height of stations from Chicago to Buffalo 81.3 “ Number of stations less than 10 feet high 22 “ “ from 10 feet to 24 feet in height 18 “ “ “ 25 “ 49 “ “ 50 “ “ “ 50 “ 74 “ “ 71 “ “ “75 “ 99 “ “ 47 “ “ “ 100 “ 109 “ “ 18 “ “ “ no “ 119 “ “ 15 “ “ “ 120 “ 124 “ “ 2 *Only stations built expressly for the work are here included. Sometimes buildings or towers were utilized in addition to these. GEODETIC SURVEYING. 437 The heights above given are the heights at which the in- strument was located above the ground. The targets were usually elevated from 5 to 30 feet higher. The excessive heights of the stations from Chicago to Buffalo are due to the country being very heavily timbered, and the surface only gently rolling. In the vicinity of Lake Superior they averaged only about 35 feet high, while from Buffalo to the eastern end of Lake Ontario they averaged 51 feet in height. 334. Construction of Stations. — If it is found necessaiy to build tall stations, two entirely separate structures must be SCAFFOLD OBSERVING TRIPOD Fig. 120. erected, one for carrying the instrument and one for sustain- ing the platform on which the observer stands. These should have no rigid connection with each other. These structures are shown in plan and elevation in Figs. 120 and 121. The inner station is a tripod on which the instrument rests; this is sur- rounded by a quadrangular structure, shown separately in ele- vation to prevent confusion. Both structures are built entirely of wood, the outer one being usually carried up higher than 438 SURVEYING. the tripod (not shown in the drawinp^), and the target fixed to its apex. This upper framework serves also to support an awning to shade the instrument from the sun. For lower sta- tions a simpler construction will serve, but the observer’s plat- form must in all cases be separate from the instrument tripod. The wire guys and wooden braces shown in Fig. 120 were not used on the U. S. Lake Survey stations. For stations less than about 15 feet in height the design ■-©- /i /' |\'\ ">< \\ / '''-1 \ '"''4 \ \ I ! AWt i 1'4 GROUND PLAN Scale ^ Fig. 121. ' shown in Figs. 122 and 123 may be used. Here the outer platform on which the observer stands is entirely separate from the tripod which supports the instrument. For ground stations a post firmly planted serves very well, or a tree cut off to the proper height. The common instrument tripod will seldom be found satisfactory for good work. Sometimes extra heavy and stable tripods of the ordinary pattern have given excellent re- sults. 335. Targets. — The requisites of a good target are that it shall be clearly visible against all backgrounds, readily bisected, GEODETIC SURVEYING. 439 rigid, capable of being accurately centred over the station, and so constructed that the centre of the visible portion, whether in sun or in shade, shall coincide with its vertical axis. It is not easy always to fulfil these conditions satisfactorily. To make it visible against light or dark backgrounds, it is well to paint it in alternating black and white belts. For ready bi- section it should be as narrow as possible for distinctness. This 440 SURVEYING. is accomplished by making the width subtend an angle of from two to four seconds of arc. Since the arc of one second is three tenths of an inch for one-mile radius, an angle of four seconds would give a target one tenth of a foot in diameter for one-mile distances, or one foot in diameter for ten-mile dis- tances. Something depends on the magnifying power of the telescope used. The design shown in Fig. 124 will satis- Fig. 124. factorily satisfy the conditions as to rigidity and convenience of centering. Of course it should stand vertically over the station so that a reading could be taken on any part of its height. The last condition is not so easily satisfied. If a cylinder or cone be used the illuminated portion only will appear when the sun is shining, and a bisection on this portion may be several inches to one side of the true axis. GEODETIC SURVEYING. 441 The target is then said to present a phase, and corrections for this are sometimes introduced. It is much better, however, to use a target which has no phase. If the target is to be read mostly from one general direction, a surface, as a board, may be used ; but if the target is to be viewed from various points of the compass, then from those stations which lie nearly in the plane of the target it would not be visible, from its width being so greatly foreshortened. In this case two planes could be set at right angles, one above the other. One or both would then be visible from all points, and since their axes are coincident, either one could be used. The objection to this would be that the upper disk would cast its shadow at times on the lower one, leaving one side in sun and the other in shade, thus giving rise to the very evil it is sought to eliminate. A very satisfactory solution of this problem was made on the Mississippi River Survey by means of the following device (Fig. 125): Four galvanized-iron wires, about three-sixteenths inch in diameter, are bent into a circle of, say, four inches in diameter, and soldered. To these four circles are attached four vertical wires about one fourth inch in diam- eter and four feet long, as shown in the accompanying figure. All joints to be securely soldered, the size of the wire increas* ing with the size of the target. The target is now divided into a number of zones by stretching black and white canvas alter- nately and in opposite ways between the opposing uprights, making diametral sections. If there are more than two zones, those marked by the same color should have the canvas cross- ing in different ways, so that if one plane is nearly parallel to any line of sight the other plane of this color will be nearly at right angles to it. This target has no phase, is visible against any background, and readily mounted. A wooden block may be inserted at bottom, with a hole in the axis of the target. This may then be set over a nail marking the station. The target is held at top by wire guys leading off to stakes in the 442 SUR VE YING. ground. Such a target could be mounted on top of the pole shown in Fig. 124, if it should be found necessary to elevate it. 336. Heliotropes. — When the distance between stations is such that, owing to the distance, the state of the atmosphere, or the small size of the objective used, a target would appear indistinct, or perhaps not be visible at all, the reflected rays of the sun may be made to serve in place of a target. This limit- ing distance is usually about twenty miles. Any device for accomplishing this purpose may be called a heliotrope. In Figs. 126 and 127 are two forms of such an instrument. That shown in Fig. 126 is a telescope mounted with a vertical and horizontal motion. This is turned upon the station occupied by the observer, and is then left undisturbed. On the tele- scope are mounted a mirror and two disks* with circular open- ings. The mirror has two motions so that it can be put into any position. Its centre is coincident with the axis of the disks, in all positions. The mirror may be turned so as to * The disk next to the mirror is unnecessary. GEODETIC SURVEYING. 443 throw a beam of light symmetrically through the forward disk, in which position the reflected rays are parallel to the axis of the telescope, and hence fall upon the distant point. The heliotrope shown in Fig. 127 is to be used in conjunc- tion with a single disk, which may be a plain board mounted on a plank with the mirror. The silvering is removed from a small circle at the centre of the mirror. The disk has a small hole through it as high above its base as the clear space on the mirror is above the plank. The operator points the apparatus by sighting, through the clear spot on the mirror and the open- ing in the disk, to the distant station. If the plank be fas- tened in this position the attendant now has only to move the mirror so as to keep the cone of reflected rays symmetrically covering the opening in the disk, and the light will be thrown to the distant station. Since the cone of incident rays subtends an angle of about thirty-two minutes, the cone of reflected rays subtends the same angle. The base of this cone has a breadth of about fifty feet to the mile distance, or at a distance of twenty miles the station sending the reflection is visible over an area in a vertical plane 1000 feet in diameter. The alignment of the heliotrope need not, therefore, be very accurate. This align- ment may vary as much as fifteen minutes of arc on either side of the true line. This is nearly o.oi of a foot in a distance of two feet. If the bearing, or direction, of the distant station is once determined, it may be marked on the station by some means within this limit, and a very rude contrivance used for sending the reflected ray, or flash, as it is called. Thus, a mir- ror and a disk with the requisite movements may be mounted on the ends of a board or pole from five to twenty feet long, and when this is properly aligned it serves as well as any other more expensive apparatus. The hole in the disk should usually subtend an angle at the observer’s station of something less than one second of arc, which is a width of three-tenths of an 444 SURVEYING. inch to the mile distance. On the best work with large instru- ments it should subtend an angle of less than one half a second, the minimum effective opening depending almost wholly on the condition of the atmosphere.^' Whatever form of heliotrope is used, an attendant is re- quired to operate the apparatus. Evidently it can be used only on clear days, whereas cloudy weather is much better adapted to this kind of work, since the atmosphere then trans- mits so much clearer and steadier an image. The heliotrope can be used as a means of communication between distant stations by some fixed code of flashing sig- nals, and it has been so used very often with great advantage to the work. The attendant on the heliotrope, usually called a flasher, can thus know when the observer is reading his sig- nals, when he is through at that station, and, in general, can re- ceive his instructions from his chief direct from the distant station. 337. Station Marks. — If the triangulation is to serve for the fixing of points for future reference, then these points must be marked in some more or less permanent manner. In this case the station has been chosen with this in view, so that if possible it has been provided that even the surface for a few feet around the station shall remain undisturbed. To insure against disturbance from frost or otherwise, the real mark is usually set several feet underground. Many different means are employed to mark these points. The underground mark is to serve only when the superficial marks have been dis- turbed, there being always left a mark of some kind projecting above ground. On the U. S. Lake Survey, “ the geodetic point is the centre of a J-inch hole drilled in the top of a stone * Reflected sunlight has been seen a distance of sixty miles, through an opening one inch in diameter, which then subtended an angle of but one eigh- teenth of one socond of arc at the instrument. This would require a very clear atmospliere. GEODETIC SURVEYING. 445 two feet by six inches by six inches, sunk two and one-half feet below the surface of the ground. When the occupation of the station is finished, a second stone post, rising eight inches above the ground, is placed over the first stone. Three stone reference-posts, three feet long, rising about a foot above the ground, are set within a few hundred feet of the station, where they are the least likely to be disturbed. A sketch of the topography within a radius of 400 metres about the sta- tion is made, and the distances and azimuths of the reference- marks are accurately determined.” When the station is located in natural rock a copper bolt may be set to mark the geodetic point. On the Mississippi River survey, stations had to be set on ground subject to overflow. These were to serve both for geodetic points and for bench-marks, both their geographical position and their elevation being accurately determined. Both the rank growth and the sedimentary deposits from the annual overflows would soon obliterate any mark which was but slightly raised above the surface. After much study given to the subject, the following method of marking such points was adopted : A flat stone eighteen inches square and four inches thick, dressed on the upper side, has a hole drilled in the centre, into which a copper bolt is leaded, the end project- ing a quarter of an inch above the face of the stone. The U S stone is marked thus, ^ and is placed three feet under ground. On this stone, and centred over the copper bolt, a cast-iron pipe four inches in diameter and five feet long is placed, and the dirt tamped in around it. The pipe is large enough to admit a levelling-rod. The top is closed with a cap, which is fastened to the pipe by means of a bolt. The eleva- tions of both the top of the pipe and of the stone are de- termined. Fig. 128 .— Four-Metre Contact-slide Base Apparatus, after the Design of Prof. J. E. Hilgard, Superintendent U. S. C. and G. Survey. GEODETIC SURVEYING. 447 MEASUREMENT OF THE BASE-LINE. 338. Methods. — The methods heretofore employed in meas- uring a base-line have depended on the degree of accuracy requisite. If an accuracy of one in one million was desired, then the most elaborate primary apparatus has been used, such as may be found described in the U. S. Coast and Geodetic Survey Reports for 1873 and 1882, or in the Primary Triangulation of the U. S. Lake Survey.* For an accuracy of one in fifty thousand or one in one hundred thousand, more simple appli- ances have been used, such as that shown in Fig. 128. This apparatus is fully described and illustrated in the U. S. Coast and Geodetic Survey Report for 1880, Appendix No. 17. It consists essentially of a four-metre steel bar, with zinc tubes on either side of it. One of these zinc tubes is attached to the steel bar at one end and the other at the other end. Since the expansion of zinc is about two and a half times that of steel, it is evident that the corresponding ends of the zinc bars will have a relative motion with reference to each other as the temperature changes. This relative mo- tion is observed by means of the vernier scales attached to the ends of the zinc tubes. When the absolute length of the steel bar and the coefficients of expansion of both the steel and zinc bars are determined, and the readings of the vernier scales for a given temperature, then any other temperature will be in- dicated by the scale-readings. This combination thus becomes a metallic thermometer, from which the temperature of the steel rod may be accurately known while in use in the field. This assumes that the steel and zinc rods are at the same temperature at all times, and that the changes in length due to changes in temperature occur simultaneously with the tempera- *This is a large quarto volume of 920 pp. and 30 plates, describing the methods and results of the geodetic work of the U. S. Lake Survey. It is a most valuable contribution to the science of geodesy, and is No. 24 of the Professional Papers of the Corps of Engineers of the U. S. Army, 1882. 448 S UR VE YING. turc changes. Unfortunately, this latter condition is not ful- filled in the case of zinc. From elaborate observations on the relative expansions of steel and zinc bars on the United States Lake Survey, it was found that zinc is like glass in that its volume-change is not wholly coincident with its corresponding temperature-change, a residual portion of its change of volume requiring a consid- erable time for its completion. In other words, the volume- change lags behind the temperature-change, so that its volume is not truly indicated by its temperature, it being rather a function of the changes in temperature for an indefinite pre- vious period. Zinc is, therefore, not a fit metal to use in the most accurate measurements, although it is sufficiently reliable for a secondary apparatus. When two combinations of bars described above are prop- erly protected from sun, wind, and from too sudden and varia- ble temperature-changes, and when they are mounted in such a way as to enable them to be aligned both horizontally and vertically, with suitable provision for making exact contacts between the ends of the steel bars, they then form a base apparatus. Sometimes simple wooden or iron rods have been used in this way, but then the great source of error is in not knowing the mean temperature (and hence length) of the rods at any time. If mercurial thermometers are used, these may be many degrees warmer or cooler than the bar, since the mercury bulb is so much smaller in cross-section than the bar, and therefore responds more quickly to changes in temperature. The steel- zinc combination is an ideal one, and would be practically per- fect if zinc were as reliable a metal as steel. The best metals for metallic thermometers are probably steel and brass, the coefficient of expansion of the latter being about 1.5 times the former.* * Mr. E. S. Wheeler, U. S. Asst. Engr., who has had a very large experi- perience in the measurement of primary base-lines on the U. S. Lake Survey, GEODETIC SURVEYING. 449 The Steel Tape furnishes the most convenient, rapid, and economical means for measuring any distance for any desired degree of accuracy up to about one in three hundred thousand, and if the most favorable times are chosen, an accuracy of i in 1,000,000 may be attained. It is probable, therefore, that all engineering measurements, even including primary base- lines, will yet be made by the steel tape or by steel and brass wires. The conditions of use depend on the accuracy re- quired. Let us suppose the absolute length, coefficient of expansion, and modulus of elasticity have been accurately determined. Any distance can then be measured in absolute units within an accuracy of one in one million, by taking due precautions as to temperature and mechanical conditions. The length of the tape for city work is usually fifty feet, and its cross-section about \ inch by A inch. That used in New York City is inch wide by-^^^ inch thick. For mining, topo- graphical, and railroad surveying a length of one hundred feet, with a cross-section of about \ by inch, is most convenient. For base-line measurement the length should be from three hundred to five hundred feet, and its cross-section from two to three one-thousandths of a square inch. For an accuracy of one in five thousand the tape may be used in all kinds of weather, held and stretched by hand, the horizontal position and amount of pull estimated by the chainmen. The tempera- ture may be estimated, or read from a thermometer carried along for the purpose. On uneven ground, the end marks are given by plumb-line. For an accuracy of one in fifty thousand the mean tem- perature of the tape should be known to the nearest degree Fahrenheit, the slope should be determined by stretching over stakes, or on ground whose slope is determined, and the pull recommends the use of a single bar packed in ice, with micrometer microscopes mounted on iron stands to mark the end positions of the bar. By this means a constant length of standard can be obtained. This has never yet been done, however. 29 450 SURVEYING. should be measured by spring balances. The work could then be done in almost any kind of cloudy weather. For an accu- racy of one in five hundred thousand, extreme precautions must be taken. The mean temperature must be determined to about one fifth of a degree F., the slope must be accurately determined by passing the tape over points whose elevations above a given datum are known, the pull must be known to within a few ounces, and all friction must be eliminated. The largest source of error is apt to be the temperature. On clear days, the temperature of the air varies rapidly for varying heights above the ground, and, besides, the temperature of the tape would neither be that of the air surrounding it, nor of the bulb of a mercurial thermometer. In fact, there is no way of determining by mercurial thermometer, even within a few degrees, the mean temperature of a steel tape lying in the sun, either on or at varying heights above the ground. The work must then be done in cloudy weather, and when air and ground are at about the same temperature. There should also be no appreciable wind, both on account of its mechanical action on the tape, and from the temperature- variations resulting therefrom. 339. Method of Mounting and Stretching the Tape. — To eliminate all friction, the tape is suspended in hooks about two inches long, these being hung from nails in the sides of “ line-stakes" driven with their front edges on line. These stakes may be from twenty to one hundred feet apart. The nails may be set on grade or not, as desired ; but if not on grade, then each point of support must have its elevation deter- mined. A low point should not intervene between two higher ones, or the pull on the tape may lift it from this support. “ Marking-stakes" are set on line with their tops about two feet above ground, at distances apart equal to a tape-length, say 300 feet. Zinc strips about one and one half inches wide are tacked to the tops of these stakes, and on these the tape-lengths are GEODETIC SURVEYING. 451 marked with a steel point. These strips remain undisturbed until all the measurements are completed, when they n preserved for future reference. In front of the marking-stake three “ table-stakes” are driven, on which to rest the stretching apparatus, and in the rear a “ straining-stake” to which to at- tach the rear end of the tape. These auxiliary stakes are set two or three feet away from the marking-stake, and enough lower to bring the tape, when stretched, to rest on the top of the marking-stake. The stretching apparatus is shown in Fig. 129.* A chain is attached to the end of the tape, and this is hooked over the * This figure, and the method here described, are taken from the advance- sheets of the Report of the Missouri River Commission for 1886. The work was in charge of Mr. O. B. Wheeler, U. S. Asst. Engr., who first used this method on the Missouri River Survey in 1885. The author had previously developed and used the general method, except that he stretched his tape by a weight hung by a line passing through a loop which was kept at an angle of 45“ with the vertical, and his end marks were made on copper tacks driven into the tops of the stakes. He had also used spring balances for stretching the tape. 452 SURVEYING. staple K which is attaclied to the block KHK' . This block is hinged on a knife-edge at //, and is weighed at K' by the load P, The hinge bearing at H is attached to a slide which is moved by the screw S working in the nut N. The whole ap- paratus is set on the three table-stakes in front of the marking- stake, the proper link hooked over the staple, and the block brought to its true position by the screw. This position is shown by the bubble L attached to the top of the block. If the lever-arms HK and HK' are properly proportioned, the pull on the tape is now equal to the weight P. To find this length of the arm HK, let HK — k ; HK' = k ' ; the horizontal distance from the knife-edge H to the centre of gravity of the block —g\ and the weight of block “ B. Then, taking moments about H, we have Pk — Pk' Bg or k = k' g. . . . . (i) When equation (i) is fulfilled then the pull on the tape is just equal to the weight P, when the bubble reads horizontally. The centre of gravity of the block is found by suspending it from two different axes and noting the intersection of plumb-lines dropped from these axes. At the rear end the tape is held by a slide operated by an adjusting screw similar to that shown in Fig. 129. This slide rests on the straining-stake, and the rear-end graduation is made to coincide exactly with the graduation on the zinc strip which marked the forward end of the previous tape- length. The rear observer gives the word, and the forward end is marked on the next zinc strip. The thermometers are then read, and the tape carried forward. The measurement is duplicated by measuring again in the same direction, the zinc strips being left undisturbed. In obtaining a profile of the line the level rod is held on the suspension nails and on a block, equal in height to the length of the hooks, set on top of the marking-stakes. GEODETIC SURVEYING, 453 For transferring the work to the ground, or to a stone set beneath the surface, a transit is mounted at one side of the line and the point transferred by means of the vertical motion of the telescope, the line of sight being at right angles to the base-line. 340. M. Jaderin’s Method. — Prof. Edward Jaderin, of Stockholm, has brought the measurement of distances by wires and steel tapes to great perfection. He uses a tape 25 metres in length, and stretches it over tripods set in line, as shown in Fig. 130. On the top of the tripod head is a fixed graduation. At the rear end of the tape there is a single grad- uation, but at the forward end a scale ten centimetres in length is attached to the tape, this being graduated to millimetres on a bevelled edge. The middle of this scale is 25 metres from the graduation at the other end of the tape. The tripods are set as near as may be to an interval of 25 metres, but it is evident that the reading may be taken on them if this interval is not more than 5 centimetres more or less than 25 metres. The reading is taken to tenths of millimetres, the tenths being estimated. The tape is stretched by two spring balances, a very stiff spring being used at the rear end and a very sensi- tive one at the forward end. The rear balance simply tells the operator here when the tension is approximately right, the measure of this tension being taken on the forward balance, which is shown in the figure. 454 SURVEYING. If a single steel wire or tape be used, Mr. Jadcrin also finds that the work must be done in cloudy and calm weather, or at night, if the best results arc to be obtained. But he finds that if two wires be used, one of steel and the other of brass, he can continue the work during the entire day, even in sunshine and wind, and obtain an accuracy of about one in one million in his results.* The wires are stretched in succes- sion over the same tripods, by the same apparatus, one wire resting on the ground while the other is stretched. More ac- curate results could doubtless be obtained if both wires are kept off the ground constantly, the wire not in use being held by two assistants, or if stakes and wire hooks are used, both wires might be stretched at once in the same hooks. The two wires form a metallic thermometer, the difference between the readings of the same distance by the two wires determining the temperature of both wires, when their relative lengths at a certain temperature and their coefficients of expansion are known. This method is similar in principle to that of the Coast Survey apparatus, where steel and zinc bars are used, shown in Fig. 128. In such cases the true length of line is found by equation (5), p. 461. At least three thermometers should be used on a 300-foot tape, and they should be lashed to the tape or suspended by it at such points as to have equal weight on determining its tempera- ture. Thus if the tape is 300 feet long the thermometers should be fastened at the 50, 150, and 250 foot marks. They should of course have their corrections determined by comparison with some absolute standard or with other standardized thermom- eters. * See “ Geodatische Langenmessung mit Stahlbanden and metalldrahien,” von Edv. Jaderin, Stockholm. 1885. 57 pp. Also, “ Expose el^mentaire de la nouvelle Methode de M. Edouard Jaderin pour la mesure des droites ge'ode- siques au moyen de Bandes d’Acier et de Fils metalliques,” par P. E. Bergstrand, Ingenieur au Bureau central d’Arpentage, i Stockholm. 1885. 48 pp. GEODETIC SURVEYING. 455 If the appliances above outlined be used with a single tape or wire, and the work be done on calm and densely cloudy days, or at night, or with two wires used even in clear weather, it is not difficult to make the successive measurements agree to an accuracy of one in five hundred thousand. There still re- mains, however, the errors in the absolute length, in the coeffi- cient of expansion, in the modulus of elasticity, in the measure of the pull, and in the alignment, none of which would appear in the discrepancies between the successive measurements. 341. The Absolute Length is the most difficult to deter- mine. The best way of finding it would be to compare it with another tape of known length. This may be impracticable, since all the so-called “ standards” are more or less discrepant on coming from the makers.* If an absolute standard is not available, then the length may be found by measuring a known distance, as a previously measured base-line, and computing the temperature at which the tape is standard. Or the tape may be compared with a shorter standard, as a yard or metre bar, by means of a com- parator furnished with micrometer microscopes.f * The absolute length of the 300-foot steel tape belonging to the Mississippi River Commission, the coefficient of expansion and the modulus of elasticity of which the author himself determined in 1880, has now been obtained. This was done by measuring a part of the Onley Base Line with this tape, using the method herein outlined. This base is situated in Southern Illinois, and forms the southern extremity of U. S. Lake Survey primary triangulation-system. The probable error in the length of the base, from the original measurements, was about one one-millionth. The recent tape-measurements are remarkably accor- dant, so the length of this tape is now very accurately known. A similar tape belonging to the engineering outfit of Washington University has been com- pared with this one at different temperatures, and its absolute length and coeffi- cient of expansion found. The 50-foot subdivisions have also been carefully determined. f Such an apparatus is used in the physical laboratory of Washington Uni- versity, which, in conjunction with a standard metre bar which has been com- pared with the European standards, enables absolute lengths to be determined to the nearest one-thousandth of a millimetre. 456 SUR VE YING. 342. The Coefficient of Expansion may be taken any- where from 0.0000055 to 0.0000070 for 1° F.* If the tape is used at nearly its standard temperature, then the coefficient of expansion plays so small a part that its exact value is unim- portant. If it is used at a temperature of 70° F. from its standard temperature, and if the error in the coefficient used be twenty per cent, the resulting error in the work would be one in ten thousand. This is probably the extreme error that would ever be made from not knowing the coefficient of ex- pansion, some tabular value being used. If nothing is known of the coefficient of expansion, probably o.cmX)Oo 65 would be the best value to use. It is evident, however, that for the most accurate work the coefficient of expansion of the tape used must be carefully determined. * The author made a series of observations on a steel tape 300 feet long, the readings being taken at short intervals for four days and three nights. The tape was enclosed in a wooden box, and supported by hooks every sixteen feet. The observations were taken on fine graduations made by a diamond point, there being a single graduation at one end, but some fifty graduations a millimetre apart at the other end. The readings were made by means of micrometer microscopes mounted on solid posts at the two ends. The range of temperature was about 50° F., and the resulting coefficient of expansion for 1° F. was 0.00000699 ± 3 in the last place. The coefficient for the Washington University tape is 0.00000685. Prof. T. C. Mendenhall found from six or eight experiments on steel bands used for tapes, a mean coefficient of 0.0000059. Steel standards of length have coefficients ranging from 0.0000048 to 0.0000066. Mr. Edward Jaderin, Stockholm, has obtained a mean value of 0.0000055, from a number of very careful determinations, both from remeasuring a primary base-line, and from readings in a water-bath. Several steel wires were tested, and their coefficients all came very near the mean as given above. For brass zvires he found a mean coefficient of 0.0000096 F. The 15-foot standard brass bar of the U. S. Lake Survey has a coefficient of 0.0000100, while tabular values are found as high as 0.0000107 F. There is some evidence that cold-drawn wires have a less coefficient of expan- sion than rolled bars and tapes. Coefficients of expansion have seldom been found with great accuracy, the coefficients of the “ M6tre des Archives,” the French standard, having had an erroneous value assigned to it for ninety years GEODETIC SURVEYING. 457 343. The Modulus of Elasticity is readily found by ap- plying to the tape varying weights, or pulls, and observing the stretch. The correction for sag will have to be applied for each weight used, in case the tape is suspended from hooks, which should be done to eliminate all friction. Let be the maximum load in pounds ; Pq “ “ minimum load in pounds ; a “ “ increased length of tape in inches due to the increased pull ; L “ “ length in inches for pull P^, or the graduated length of tape ; S “ “ cross-section in square inches ; E ‘‘ “ modulus of elasticity ; d “ “ distance between supports ; w “ “ weight of one inch of tape in pounds ; s “ “ shortening effect of the sag for the length L ; V “ “ sag in inches midway between supports. Then we have (^. - p :)L Sa (I) But for the pull Pj, the shortening from sag is much less than for the pull P^. We must therefore find the effect of the sag in terms of the pull. 344. Effect of the Sag. — Where the sag is small, as it always is in this work, the curve, although a catenary, may be considered a parabola without an appreciable error. If we pass a section through the tape midway between sup- ports, and equate the moments of the external forces on one side of this section, we obtain, taking centre of moments at the support, wd d wd'' or Pv = — .-- ^ 24 8 zvd'" ZP' (0 458 SURVEYING. $ If the length of a parabolic curve be given by an infinite series, and if all terms after the second be omitted, which they may when ^ is small, then we may write — Length of curve := d If we now substitute for v its value as given in equation (i), we have Length of curve = ^ | ^ If we call the excess in length of curve over the linear dis- ^ tance between supports the effect of the sag., we have d I wdV 24 \PJ (3) for one interval between supports. If there are n such inter- vals in one tape-length, then 7id = Z, and the effect of the sag in the entire tape-length is (4) If 5, and be the effects of the sag for the pulls Z, and (5i<5o forP,>Po), then the total movement at the free end due to the pull being increased from P^ to P, would he a -\- (S, — 5,). If this total movement be called M, then we would have £ = S(M-\ + S.) P. m {wdf \L~ 24 V p^p: j) ( 5 ) GEODETIC SURVEYING. 459 Example. Let P\ = 6o pounds; /’o = 10 pounds; ^i} = 0.00055 pound per inch of tape; d — 300 inches = 25 feet; S — 0.002 square inch; M — 3.2 inches; Z = 3600 inches = 300 feet. To find E. From equation (5) we have E = — — 28,500,000. 3.2 0.027/ 3500 3600 24 UfioocK 0.002 From the same data, we find from eq. (4) the effect of the sag to be 0.040 inch for the ten-pound pull, and 0.001 inch for the sixty-pound pull. Evidently, if the tape is stretched by the same weight when its absolute length is found, and when used in measuring, the stretch, or elongation from pull, would not enter in the computation, and so the modulus of elasticity would be no function of the problem. Again, the stretch per pound of pull may be observed for the given tape, and then neither E nor S, the cross-section, would enter in the computation. 345. Temperature Correction. — If mercurial thermome- ters are used, their field-readings must first be corrected for the errors of their scale-reading, each thermometer having, of course, a separate set of corrections. Then the mean of the corrected readings may be taken for all the whole tape-lengths in the line measured, and the correction for the entire line obtained at once. Thus, let L = length of line ; 7*0 = temperature at which the length of the tape is given for the standard pull this usually being the tem- perature at which its true length is its graduated length for that standard pull ; Tm = the mean corrected temperature of the entire line ; a = coefficient of expansion for 1° ; Q = correction for temperature. 460 SURVEYING. Then C,^^a[T^-T:)L (i) The temperature correction for a part of a tape-length is com- puted separately. If the value of 01 for the tape used is not known, it may be taken at 0.0000065. If a metallic thermometer is used, as a brass and a steel wire, or a brass and a steel bar as in the U. S. C. and G. S. apparatus shown on p. 446, then we have the following: 346. Temperature Correction when a Metallic Ther- mometer is used. Let / = length of wire or tape used, as 300 feet ; 4 = absolute length of the steel wire at the standard temperature of, say, 32° F. ; 4 = same for brass wire ; L — total length of line for whole tape-lengths ( = nl approximately) ; n — number of lengths of the standard measured ; Tg — mean value of all the scale-readings on steel wire for the entire line = same for scale-readings on brass wire ; as = coefficient of expansion for the steel wire a^ — “ “ “ “ brass “ 4 = mean temperature for the entire line. Then we have Z = K4 + r,)(i+(/,- 32°K)) = »(4 + ^,)(i+(;.-32>,)) • • • • W Since the temperature correction is relatively a very small quantity, we may put 4 -|- = 4 + ^6 = 4 the length of the tape to which the temperature correction is applied. GEODETIC SURVEYING. 461 We then have from (2) { t . - 32“) = (4 "h ^s) ~ (4 ~f~ . . ( 3 ) Substituting this value of the temperature in (2), we obtain L — n[ls + ^ ((4 + ^s) — (4 + ^b))]- • ( 4 ) If we put 4 -f = 5s and 4 + ^& = ‘S'5, we have £ = .[i. + (5,-5.)5^J' • • • ( 5 ) From either of the equations (5) we may compute the length of the line as corrected for temperature. If, however, it is desired to find the temperature correction separately, in order to combine it with the other corrections, we have — ^(‘5's 55 ) , ..... (6) — ^3 for the temperature correction to be applied to the measured length by the steel wire, or ( 7 ) as the temperature correction to be applied to the measured length by the brass wire. 462 SUI^ VE YING. These formulae all apply only to the entire tape-lengths. Any fractional length would have to be computed separately, or else a diminished weight given to their scale-readings in obtain- ing the mean values, r, and 347. Correction for Alignment, both horizontal and ver- tical. — The relative elevations of the points of support are found by a levelling instrument, and the horizontal alignment done by a transit or by eye. An alignment by eye will be found sufficiently exact if points be established on line by transit every 500 or locx) feet. The suspending nails and hooks afford considerable latitude for lateral adjustment when the tape is stretched taut ; hence the horizontal deviation will be practically zero unless the stakes are very badly set, and the relative elevations of any two successive supports should be determined to less than 0.05 foot. If no care is taken to have more than two suspension points on grade, then each section of the tape will have a separate correction. Usually a single grade may as well extend over several sections, in which case the portion on a uniform grade may be reduced as a single section. Let /,, 4 , 4 , etc., be the. successive lengths of uniform grades, and h^y h^y etc., the differences of elevation between the extremities of these uniform grades ; then for a single grade we would have the correction C = / _ IP p _ 2CI+C -h\ or But since is a very small quantity as compared with 4 jp we may drop the O, whence we have C = for a single grade. The exact value of Cy in ascending powers of hy is (0 GEODETIC SURVEYING. 463 For the entire line, if all but the first term be neglected, the correction is If the /’s are all equal, as when no two successive suspen- sion points fall in the same grade, then we have G — “ + ^^3" + • • • • (3) Since the relative elevations are determined, and not the angles of the grades, these formulae are more readily applied than one involving the grade angles. The error made in rejecting the second power of C in the above equations is given in the table on the following page, where / and h are taken in the same unit of length.* If the grades are given in vertical angles, as they always are with the ordinary base apparatus, then we have for the correction to each section whose length is /, and whose grade is 6 above or below the horizon. Cg = —l{i — COS 6) 2/ sin" -. 2 If 6 be expressed in minutes of arc, and if the grade angle is less than about six degrees, or if the slope is less than one in ten, we may write • sin I 2 = —0.00000004231 OH; * From Jaderin’s Geodiitische Liingenmessung. 464 SUR VE YING. or by logarithms, log Cg — const, log 2.626422 + 2 log 0 log /. TABLE OF RELATIVE ERRORS IN THE FORMULA Cg=z 2/ Length of Relative Error = Grade. 1 1 1 0.00005 0.00015 0.00025 0.00035 1 0.00045 /. h = Rise or Fall in Length /. I 0.14 0. 19 1 2 .24 .31 1 “2 .32 .42 i . .40 • 53 •47 .62 D 6 . 54 .71 0.81 7 .61 .80 .91 i 8 .67 •88 1 .00 9 •73 .97 1 . 10 1. 19 10 •79 1.05 1. 19 1.29 II • 85 1 . 12 1.28 1-39 12 .91 1.20 1.36 1 .48 13 •97 1.27 1-45 1-57 j 1.67 14 1.02 1-35 1-53 1.66 1-77 15 1.08 1.42 1 .61 1-75 1.86 16 1. 13 1.49 1.69 1.84 1.96 17 1. 18 1.56 1-77 1.92 i 2.05 18 1.24 1.62 1.85 2.01 2.14 19 1.29 1.69 1.92 1 2.09 2.23 20 1-34 1.76 2.00 i 2.17 2-31 21 1-39 1.82 2.07 2.25 2.40 22 1.44 1.89 2.15 2-33 2.48 23 1.48 1-95 2.22 2.41 2-57 24 1-53 2.02 2.29 2.49 2.65 25 1.58 2.08 2.36 2.57 2-73 26 1.63 2.14 2.43 2.65 2.82 27 1.67 2.20 2.50 2.72 2.90 28 1.72 2.26 2-57 2.80 2.98 29 1.77 2.32 2.64 2.87 3 -06 30 1. 81 2.38 2.71 2.95 3-14 GEODETIC SURVEYING. 465 ( 348. Correction for Sag.— From equation (4), p. 458, we have L fwd'^ (4) If the standard length be given with the pull and the distance between supports while in the field the pull P and distance d between supports be used, then the correction for sag is LzP fd," 24 \ p : L_ 24 . ( 5 ) where Z, d, and are taken in the same unit of length, and w is the weight of a unit’s length of tape in the same units used forP. 349. Correction for Pull.— From equation (i), p. 457, we may write at once , {p-p:)L SE ■ Here P is taken in pounds, Z and Cp in inches, and S in square inches, since E is usually given in inch-pound units. If E has not been determined by experiment, it may be taken at 28000000. The cross-section .S is best found by weighing the tape and computing its volume, counting 3.6 cubic inches to the pound. Knowing the length, the cross-section can then be found. If the stretch has been observed for different weights, and the value of E computed, the value of .S is of no conse- quence, provided the same value be used for both observations. 350. Elimination of Corrections for Sag and Pull. — Since the correction for sag is negative and that for pull is positive, we may make them numerically equal, and so elimi- 30 466 SUR VE YING. nate them both from the work. If this be done, the normal or standard length of the tape should be obtained for no sag and no pull, and its normal or standard temperature found such that at this temperature, and for no sag and no pull, its gradu- ated length is its true length. If Z’o is the temperature at which the tape is of standard length for the pull and the distance between supports, and if I is the length of the tape, then we have. Shortening from sag = - l-r,-) , 24 ' / Lengthening from x degrees F. == xaL If, therefore, the effects of sag and pull were eliminated, the tape would be of standard length at a temperature degrees above where • • (I) where all dimensions are in inches and weights in pounds. The standard temperature for no sag and no pull would be, therefore. 7; = 7; + (2) We will call this the normal temperature. GEODETIC SURVEYING. 467 In order that the corrections for sag and pull shall balance each other, we must have or EL which we will call the normal tension. If the stretch in inches is known for one pound of pull for the given tape, we may call this and we will have e — or SE = SE e Also, Iw — W — weight of entire tape between end graduations, W or w = I And -^=71 = number of sags in the tape. Substituting these values in (3), we obtain where W = weight of entire tape in pounds ; I = length of tape in inches ; e = elongation of tape for a one-pound pull ; I n = number of sags in tape = If the tape has no intermediate supports, then n—i, and we have for the normal tension (5) 468 scr/c VE YING. Example. — For the 300-foot steel tape, whose constants the author deter- mined, we have IV = 2 lbs., / = 3600 inches, e = 0.066 inch. If the supports are 30 feet apart, n = 10, whence, from eq. (4), Pn = 4.48 pounds. If n = 6, or if the supports were placed 50 feet apart, we would find I'n = 6.32 pounds. If n = 3, or if the supports are too feet apart, Pn = 10.03 pounds. In the last case, the sag would be ten inches midway between supports. 351. To reduce a Broken Base to a Straight Line.— It is sometimes nece.ssary or convenient to introduce one or more angles into a base-line. These would never deviate much from 180°. Let the difference between the angle and 180° be 6 , and let the two measured sides be a and b, to find the side c. If 6 be expressed in minutes of arc and if it is not more than about 3°, the following approximate formula will prove suf- ficiently exact : side c = a-\- b — ab&^ a-\-b a b — 0.00000004231 abe^ a-\-b' If S is greater than from 3° to 5°, the triangle would have to be computed by the ordinary sine formula. 352. To reduce the Length of the Base to Sea-level. — In geodetic work, all distances are reduced to what they would be if the same lines were projected upon a sea-level surface by radii passing through the extremities of the lines. It is not necessary, however, to reduce all the lines of a trian- gulation system in this manner, since if the length of the base- line is so reduced the computed lengths of all the other lines of the system will be their lengths at sea-level. The angles that are measured are the horizontal angles, and are not affected by the differences of elevation of the various stations. It is GEODETIC SURVEYING. 469 necessary, therefore, to know the approximate elevation of the base above sea-level. Let r = mean radius of earth ; a =: elevation above sea-level ; B — length of measured base ; b — length of base at sea-level. Then r a \ r :: B : b, or b = B r r-\- a The correction to the measured length is always negative, and is C=b-B= - B B ^ \ Since a is very small as compared to r, we may write The mean radius* in feet is mean r = 20926062 + 20855121 2 = 20890592 feet, log r (in feet) = 7*3199507. 353. Summary of Corrections. — For the significance of the notation used in the following equations, see the preceding articles where they are derived. The corrections are all for * Rigidly, we should use ihe length of the normal for the given latitude, but the mean radius as above found is sufficient for most cases. 4/0 SUK VE Y I NG. the entire line measured, or rather for that portion of it com- posed of entire tape-lengths, and are to be applied with the signs given to the measured length. I. Correction for Temperature. For a single standard with mercurial temperatures, C, = + «(7'„- 7;)Z (I) For metallic thermometer-readings, as found from steel and brass standards, for instance, the correction to be applied to the length as found by the steel wire, or standard, is = ( 2 ) 2. Correction for Grade. In terms of the difference of elevation of grade, points at a common distance, /, apart, = (3) In terms of the grade angles, expressed in minutes of arc, Cg = — o.oooocxx)423i.2’6'V. (4) 3. Correction for Sag. For the standard length given for a pull and a distance between supports while P and d are used in the field-work, ( 5 ) GEODETIC SURVEYING. 471 For the standard length given for no pull and no sag, L (wdV 24 \P 1 ’ ( 6 ) 4. Correction for Pull. P-P. SE ( 7 ) or C^—{P— Po)en. ( 8 ) 5. To reduce Standard Temperature to Normal Temperature. When the temperature of the tape {T^ is known at which the graduated is the true length for the pull P^ and distance between supports to find the corresponding temperature for no pull and no sag, this being called the nor^nal temperature {Tn), we have, in degrees. _! Izudiy- 2i\PjJ • • (9) 6. To eliminate Correctioyis for Sag and Pull. or Make the pull P^ = Pn=^ (10) 00 For no intermediate supports to tape. 472 SUA^ VE YING. is called the normal tc 7 isio 7 i. 7. Correction for Broke 7 i Base. If a and b are the two measured sides which make an angle of 180° — the correction to be added to a b to get the distance between their extremities, 6 being less than 5'’, and expressed in minutes of arc, is abO^ a 8. Correctio 7 i to Sea-level. where L is the length of the measured base at an altitude a above sea-level. log r (in feet) = 7*3199507. 354. To compute any Portion of a Straight Base which cannot be directly measured. — It sometimes is convenient A x> o Fig. 131. to take a base-line across a stream or other obstruction to di- rect measurement. In such a case a station may be chosen GEODETIC SURVEYING. 473 as O in Fig. 131, and the horizontal angles A OB = P, BOC = Q, and COD = R measured. If the parts AB and CD lie in the same straight line, and AB = a and CD — b are known, then BC = x may be found by measuring only the angles at O. Thus in the triangles ABO and ACO we have CO _ X -^a sin P m ~ sin (P+"0’ also from the triangles BDO and CDO we have CO _ b sin {Q + R) BO ~ X b sin P Let K = P-\- Q and L— Q + P, then by equating the above values of -SS we have whence {x+a){x + b) = ab (sin K sin L) sin P sin P ^ a-\-b jabism K sin L) , fa — bV ^ = — r- ±V + {-^1 • Evidently only the positive result is to be taken. The points A, O, and D should be chosen so as to give good intersections at A and D. 355. Accuracy attainable by Steel-tape and Metallic- wire Measurements. — The following results have been at- tained by using the methods herein described : I. In Sweden, Mr. Edw. Jaderin measured a primary base-line two kilometres in length three times, by means of steel and brass wires 25 metres long, in ordinary summer 474 SUJ^VEV/NG. weather, mostly clear, with a probable error of a single deter- mination of I in 600,000, and a probable error of the mean re- sult of I in 1,000,000, as compared with the true length of the line as obtained by a regular primary base apparatus.* 2. On the trigonometrical survey of the Missouri River, in 1885, O. B. Wheeler, U. S. Asst. Engineer, obtained the following results, using one steel-tape 300 feet long: First measurement. Second “ Glasgow Base. 7923-237 feet. 7923-403 “ Mean In this case the sun was shining more or less on both measurements. The probable error of a single result is i in 100,000, and of the mean of two measurements i in 140,000. First measurement. Second “ Benton Base. 9870-443 feet. Mean The probable error of a single measurement is i in 380,000, and of the mean, i in 533,000. First measurement. Second “ Trovers Point Base. 9711.892 “ Mean 971 1.904 ± 0.0078 feet. * For title of Mr. Jaderin’s pamphlet describing his methods and results, see foot-note, p. 454. GEODETIC SURVEYING. 475 The probable error of a single measurement is i in 900,000, and of the mean it is i in 1,250,000. Olney Base. First measurement 10821.9658 feet. Second “ 10821.9665 “ Mean 10821.9662 ± 0.0002 feet. This base had been measured by the U. S. Lake Survey Repsold base apparatus, with a probable error of about i in 1,000,000. This portion of it, about half the entire base, was remeasured with the tape in order to determine the absolute length of the tape. The work was done on both the tape- measurements in a drizzling rain, so that the temperatures were obtained with great accuracy. The mean tempera- tures of the two measurements differed, however, by several degrees, so that the two sets of graduations on the zinc strips were quite divergent, and it was only after the final reduc- tion that the two results were known to be so nearly identical.* 3. The author has measured a number of bases about one half mile in length, in connection with students’ practice sur- veys, by the methods given above, and in each case obtained a probable error of the mean of three or four measurements of less than one-millionth part of the length of the line. The work was always done on densely cloudy days, all the con- stants of tape and thermometers being well determined. * From advance-sheets of the Report of the Missouri River Commission, 1886. 476 SUR VE YING. Fig. 132. GEODETIC SURVEYING. A77 MEASUREMENT OF THE ANGLES. 356. The Instruments used in triangulation are designed especially for the accurate measurement of horizontal angles. This demands very accurate centring and fitting at the axis, and strict uniformity of graduation. It was formerly supposed that the larger the circle the more accurate the work which could be done. It is now known that there is no advantage in having the horizontal limb more than ten or twelve inches in diameter. There are two general methods of reading fractional parts of the angle, smaller than the smallest graduated space on the limb. One is by verniers, the other by micrometer micro- scopes. Verniers may be successfully used to read angles to the nearest ten or twenty seconds of arc, but if a nearer ap- proximation is desired microscopes should be employed. Fig. 132 shows a high grade of vernier transit, capable also of reading vertical angles to 70°. Its horizontal limb is 8 inches in diameter and reads by verniers to ten seconds. It may be used as a repeating instrument, and used either with or without a tripod. To mount such an instrument upon a station or post, a trivet, made of brass and shown in Fig. 135, is used. The pointed steel legs are driven into the station, the centre of the opening being over the station point. The arms have angular grooves cut in their upper surface. On this trivet may be set any three-legged instrument, so long as the radius of its base is not greater than the length of the trivet arms. In Fig. 133 is shown a theodolite (not a transit since the telescope does not revolve on its horizontal axis) designed for the measurement of horizontal angles exclusively. Here mi- crometer microscopes are used. The horizontal limb is from eight to twelve inches in diameter. There is no vertical circle or arc, so that no vertical angles can be read. Since the rela- tive heights of triangulation-stations are usually determined * See p, 484 for explanation of this term. 478 SUR VE YING. GEODETIC SURVEYING. 479 I'iG. 134 480 SUR VE YING. from their relative angular elevations or depressions, it is usual- ly necessary to have a vertical limb. This instrument could not be used as a repeater. In Fig. 134 is shown an altazimuth instrument, or an in- strument designed for accurately measuring altitudes as well as the azimuths of points or lines. Both horizontal and vertical limbs are read by means of micrometer microscopes. Such an instrument is designed especially for astronomical observations for latitude and azimuth, but may also be used as a meridian or transit instrument for observ- ing time as well as for measuring horizontal and vertical angles in triangulation. It is in fact the universal geodetic instrument, just as the complete engineer’s transit is the universal instrument in ordinary surveying. In almost all cases where micrometers are used in reading the angles the limbs are graduated to five or ten minutes and the readings made to single seconds. 357. The Filar Micrometer* is used for the accurate meas- urement of small distances or angles, when the required exact- ness is greater than can be obtained by means of a vernier scale. It is usually combined with a microscope, the microme- ter threads and scale lying in the plane of the image produced by the objective. This image is always larger than the object itself in microscopes, and therefore a given movement of the wires in the micrometer corresponds to a ver}^ much less dis- tance on the object sighted at, according to the magnifying power of the objective. * From fdum, thread; micros, small, and metros, measure. The thread is in this case a spider’s web, or scratches on glass. GEODETIC SURVEYING. 481 The frame holding the movable wires has a screw with a very fine thread working in it, called the micrometer screw. This screw has a graduated cylindrical head, or disk, attached to it, there usually being sixty divisions in the circumference when used in angular measurements. The number of whole revolutions are recorded by noting how many teeth of a comb- scale are passed over, this scale being nearly in the plane of the wires and therefore in the focus of the eye-piece. The frac- tional parts of a revolution are read on the graduated screw- head outside. These micrometer attachments are shown on the two microscopes in Fig. 133 and on the five in Fig. 134. h Fig. 136. Fig. 136 is a sectional view of a filar micrometer. The graduat- ed head Ji is attached to the milled head m, forming a nut into which the micrometer-screw a works. This screw is rigidly at- tached to the frame b, to which are fastened the movable wires f . The comb-scale s and fixed wire f are attached to the frame c, which is adjusted to a zero-reading of the graduated head by the capstan-screw d. The lost motion on both of these frames is taken up by springs. The complete revolutions of the screw are counted on the comb-scale, and the fractional part of a revolution on the graduated head. The reading is made by bringing the double wires symmetrically over a grad- uation, the space between the wires being a little more than the width of the graduation, when the exact number of revolu- tions and sixtieths are read on the comb-scale and on the head. 31 482 SURVEYING. If the limb is graduated to ten minutes and cacli revolution corresponds to one minute, then if the reading is taken on the nearest graduation, the number of revolutions need never ex- ceed five. If, however, the reading be always taken to the last ten-minute mark counted on the limb, then ten revolutions may have to be read on the screw. The movement of the threads is as they appear to be, tliere being no inversion of image be- tween wires and eye. The movement on the limb is, however, opposite from the apparent motion. If the limb is graduated to ten minutes, and a single revo- lution of the screw corresponds to the space of one minute, then just ten revolutions of the screw should move the wires from one graduation to the next. If this is not exactly true, then the value of a ten-minute space should be measured a number of times, by running the wires back and forth, the mean result taken, and from this the value of one revolution of the screw determined. This value is called the “ run of the screw,” and a correction is applied to the readings, which are always made in degrees, minutes, and seconds, counting one revolution a minute and one division on the head a second of arc. This correction is called “correction for run,” and should be determined for all parts of the screw used. If the value of one revolution is not exactly what it is designed to be, it can be adjusted by moving the objective of the microscope in or out a little, or the whole microscope up or down with refer- ence to the limb, thereby changing the size of the image. Even when this adjustment is accurately made, there may be still a correction for run on account of the screw-threads not being of uniform value. In this case the value of each revolu- tion of the screw is determined independently, these values tabulated, and the correction for run from this source deter- mined for any given reading. Again, as the microscope re- volves around the limb with the alidade, the plane of the graduations may not remain at a constant distance from the GEODETIC SURVEYING. 483 objective, in which case the size of the image would vary to a corresponding degree. To determine this, the values of ten- minute spaces are determined on various parts of the limb, and if these are not constant, then a table of corrections for run may be made out for different parts of the circle. For reading on graduated straight lines the double threads give better results than either the single thread or the inter- secting threads. The space between the threads should be a little greater than the width of the image of the graduation- line, so that a narrow strip of the limb’s illuminated upper surface may appear on either side of the graduation and inside the wires. The setting is then made so as to make these illu- minated lines of equal width. It is conceded that such an ar- rangement will give more exact readings than any other that has been used. The magnifying power of the microscope is from thirty to fifty. 358. Programme of Observations. — There are two gen- eral methods of reading angles in triangulation work. One method consists in measuring each angle inde- pendently, usually by repeating it a number of times by successive additions on the limb, and then reading this multiplied angle, which is di- vided by the number of repetitions to give the o true value of the angle. In the other method the readings are made on the several stations in order, as A, B, C, D, and Ey in the figure, and the angles found by taking the difference between the successive readings. Each method has its advantages and disadvantages. If the instrument has an ac- curate fitting in the axis, clamps which can be set and loosened without disturbing the positions of the plates, is provided with verniers which have a coarse reading, as twenty or thirty sec- onds, and accurate work is desired, and if such an instrument 484 SUR VE YING. is mounted on a low, firm station, then the method by repeti- tion would give superior results. If any of these conditions are not fulfilled, and especially if the instrument is provided with micrometer microscopes, whereby readings may be taken to the nearest second of arc, it is much more convenient, cheaper, and generally more accurate to read the stations continuously around the horizon, back and forth, until a sufficient number of readings have been obtained. 359. The Repeating Method. — This method was for- merly used almost exclusively, but the other is the only one now used with the most accurate instruments. It was found that systematic errors were introduced in the method by repetition of a single angle, due largely to the clamping appa- ratus. If this method is used the repetitions should be made first towards the right and then towards the left ; the number of repetitions making a set should be such as to make the mul- tiplied angle a multiple of 360°, as nearly as possible, so as to eliminate errors of graduation on the limb. Thus, for an angle of 60° repeat it six times and then read. For the second set repeat six times in the opposite direction, and with telescope inverted. If triangulation work is to be done with the ordi- dary engineer’s transit, which reads only to 30 seconds or one minute, this method may give very fair provided there is no movement of circles from the use of the clamping apparatus and no lost motion in the axes. The programme would be as follows : PROGRAMME. Telescope Normal. Set on left station, and read both verniers. Unclamp above and set on right station. below above below above etc., left right left right etc., GEODETIC SURVEYING. 485 until the entire circle has been traversed, then read both ver- niers while pointing to right station. The total angle divided by the number of repetitions is the measure of the angle sought. I. 2. 3 * 5 - 6 . Telescope Reversed. Set on right station, and read both verniers. Unclamp above and set on left station. “ below “ ‘ above below above etc.. right left right left etc.. until the entire circle has been traversed by each vernier, when both verniers are read on the left station. The repetition in opposite directions is designed to elimi- nate errors from clamp and axis movements, and the revers- ing of the telescope is designed to eliminate errors arising from the line of sight not being perpendicular to the horizon- tal axis, and from the horizontal axis not being perpendicular to the vertical axis of the instrument.* As many such sets of readings may be made as desired, but there should always be an even number, or as many of one kind as of the other. It will be observed that two pointings are taken for each measurement of the angle, but compara- tively few readings are made. 360. Method by Continuous Reading around the Hori- zon. — By this method the limb is clamped in any position, and * In case the instrument used is a theodolite, and its telescope cannot be revolved on its horizontal axis, it should be lifted from the pivot bearings and turned over end for end, leaving the pivots in their former bearings. If this cannot be done conveniently, then the limb should be shifted by (see next ft page) each time, and this will result in mostly eliminating these same errors of collimation and inclination of horizontal axis 486 SUR VE YING. left undisturbed except between tlie different sets of readings. The pointings are made to the stations in succession around the horizon, and both verniers, or microscopes, read for each pointing. Thus, if the instrument were at o, h'ig. 137, the pointings would be made to ‘'in d E. If the telescope is now carried around to the right until the line of sight again falls on A, and a reading taken, the observer is said to close the horizon ; that is, he has moved tlie telescope continuously around in one direction to the point of beginning. If the two readings here do not agree, the error is distributed among the angles in proportion to their number, irrespective of their size. It is questionable whether such an adjustment adds much to the accuracy of the angle values, and therefore it is common to read to the several stations back and forth without closing the horizon. Sum-angles can afterwards be read if desired. Thus, after the regular readings have been taken on the sta- tions, the angle AOE, or AOC, and COE, may be read, and so one or more equations of condition obtained. If the station is tall, there is always a twisting of its top in clear weather in the direction of the sun’s movement. This twisting effect has been observed to be as much as 1" in a, minute of time on a seventy-five-foot station. To eliminate this action the readings are taken both to the right and to the left. The reading of opposite verniers, or microscopes, elimi- nates errors of eccentricity, the inverting of the telescope elimi- nates errors of adjustment in the line of collimation and hori- zontal axis, and to eliminate periodic errors of graduation each angle is read on symmetrically distributed portions of the limb. To accomplish this the limb is shifted after each set of read- I So^ ings an amount equal to ,* where n is the number of sets of readings to be taken. The following is the For exception, see foot-note on previous page. GEODETIC SURVEYING. 487 PROGRAMME. 1ST Set. Telescope normal. Read to right. Read to left. Telescope inverted. Read to right. Read to left. Shift the Limb. 2D Set. Telescope inverted. Read to right. Read to left. Telescope normal. Read to right. Read to left. Shift the Limb. Evidently each set is complete in itself, and as many com- plete sets may be taken as desired, but no partial sets should be used. If the work is interrupted in the midst of one set of readings, the partial set of readings should be rejected, and when the work is resumed another set begun. In reducing the work, if one reading of any angle is so erroneous as to have to be rejected this should vitiate that entire set of readings of that angle. If preferred, the telescope may be inverted between the right and left readings, and then two readings on each mark would constitute a complete set, when the limb could be shifted again. If this were done, the readings at o. Fig. 137, would be : 1ST Set \ Telescope Normal — Read ABODE. ( “ Inverted “ EDCBA. % Shift the Limb. 2D Set i Telescope Inverted — Read ABODE, \ “ Normal “ EDOBA. Shift the Limb. 361. Atmospheric Conditions. — In clear weather not even fair results can be obtained during the greater part of the day. From sunrise till about four o’clock in the afternoon in sum- mer the air is so unsteady from the heated air-currents th^»- 488 SURVEYING. any distant target is either invisible or else its image is so un- steady as to make a pointing to it very uncertain. From about four o’clock till dark in clear weather, and all day in densely cloudy weather with clear air, good work can be done. If heliotropes are used, the work is limited to clear weather. It has often been proposed to do such work at night, but the lack of a simple and efficient light of sufficient strength has usually prevented. The higher the line of sight above the ground the less it is affected by atmospheric disturbances. 362. Geodetic Night Signals. — Mr. C. O. Boutelle, of the U. S. Coast and Geodetic Survey, made a series of experiments in 1879 Sugar Loaf Mountain, Maryland, for the purpose of testing the efficiency of certain night signals and the compara- tive values of day and night work. His report is given in Ap- pendix No. 8 of the Report of the U. S. C. and G. Survey for 1880. It seems that either the common Argand or the “ Elec- tric” coal-oil lamp, assisted by a- parabolic reflector or by a large lens, gives a light visible for over forty miles. His con- clusions are : 1. That night observations are a little more accurate than those by day, but the difference is slight. 2. That the cost of the apparatus is less than that of good heliotropes. 3. That the apparatus can be manipulated by the same class of men as those ordinarily employed as heliotropers. , 4. That the average time of observing in clear weather may be more than doubled by observing at night, and thus the time of occupation of a station proportionately shortened ; “clear- cloudy” weather, when heliotropes cannot show, can be utilized at night. 363. Reduction to the Centre. — It sometimes happens that the instrument cannot be set directly over the geodetic point, as when a tower or steeple is used for such point. In this case two angles of each of the triangles meeting here may GEODETIC SURVEYING. 489 be measured and the third taken to be 180° minus their sum, or the instrument maybe mounted near to the geodetic point and all the angles at this station measured from this position. These angles will then be very nearly the same as though measured from the true position, and may readily be reduced to what they would have been if the true station point had been occupied. Thus in Fig. 138 let C be the true station to which pointings were taken from other stations, and C the posi- tion of the instrument for measuring the angles at this station. The line AB is a side of the system whose length has been found. From the measured angles at A and B the approximate value of the angle C is found and the lengths of the sides a and b computed. At C the angle AC’ B is measured with the same exactness as though it were the angle C itself and the angle CC ' B — a is measured by a single ob- servation. The distance CC' = r is also a found. Since the exterior angle at the inter- Fig. 138. section d.sADB, is equal to the sum of the opposite interior angles, we have C-{-j^=C'-\-x, or C=C' + {x~j/). . . (i) In the triangle ACC we have the sides b and r and the angle ^ 6"^ ^7 known, whence rsin{C'-\-a) sm ;i: = ^ ; b k . , . . . (2) . r sm a similarly sm/ = — ^ — . Since x and / are very small angles, their sines are propor- tional to their arcs, and we may write sin x — x sin where 490 SUR VE Y I NG. X is expressed in seconds ; similarly sin ^ sin i", and equa- tions (2) become r sin {C' -j- b sin i" ’ X — y = r sin a a sin i"‘ Substituting these values in (i) we have r /sin C-\- a sin ^ ^ ^ sini" \ ~ b a (3) • (4) where the correction to C is But since the angles x and equal to their arcs, and we ha given in seconds of arc. The signs of the trigonometrical functions of the angle a must be carefully attended to, as it is measured continuously from B around to the left to 360°. The following is another so- lution of the same problem : Measure the perpendiculars from C upon AC and BC, Fig. 139, calling them m and n re- spectively. Then from equation (i) above we have C=C+{x- y). y are very small, their sines are ^e, in seconds of arc, m X = whence b sin i" C=C and y = -7; a sin I I (m ii\ ihrF' \Jb ~ hr (5) GEODETIC SURVEYING. 491 There are four cases corresponding to the four positions of C\ as shown in Fig. 140. For these several cases we have C= c: I hn n\ sin i" \b a) ’ 1 11 I (m n\ sin i" \J -a!' C = C3' + I (m sin i" 11 1 I (m sin i" \d J ADJUSTMENT OF THE MEASURED ANGLES. 364. Equations of Conditions. — When any continuous quantity, as an angle or a line, is measured, the observed value is always affected by certain small errors. Indeed, it would not be possible even to express exactly the value of a contin- uous quantity in terms of any unit, as degrees or feet and fractional parts of the same, even though this value could be exactly determined. If, therefore, the measured values of the three angles of a triangle be added together, the sum will not be exactly 180°. But we know that a rigid condition of all tri- angles is that the sum of the three angles is 180°. An equation which expresses a relation between any number of observed quantities which of geometrical necessity must exist is called an equation of condition, or a condition equation. Thus, in the above case, if A' , B\ and C be the mean observed values of the angles, and A, B, and C their true values, we would have for our condition equation A + B+C= 180* (>) 492 S UR VE YING. We would also have A'+a' = = C'-^c=C, where a\ b' , and c' are small corrections to the measured B values A', B\ and C' which are to be determined. Let us suppose that the length of the side b has been exactly meas- ured,* then when the true values of the angles are found we may com- pute the other two sides. If the sides b and c have both been measured, the length of the side c as computed from b must agree with its measured length, and so we might write the condition equation b sin {C'+ c') ~~ sin {B' + b') ' ( 2 ) Again, if the side a had been measured and its exact length found, we would obtain the third condition equation, _ b sin {A' + a') ^ ~~ sin {B' -f- b') We now have three independent equations involving three unknown quantities, and can, therefore, find the quantities a\ b\ and c' . But if only one side had been measured, we should have had but one equation from which to determine three un- known quantities. Evidently there is an infinite number of * This assumption is made in regard to the measured base-lines in a trian- gulation-system, since its exactness is so much greater than can be obtained in the angle-measurements. GEODETIC SURVEYING. 493 sets of values of a’ , b\ and c\ which would satisfy this equation. If we now impose the condition that the corrections shall be the most probable ones, then there is but one set of values that can be taken. Equation (i) is called an angle equation^ since only angles are involved ; while equations (2) and (3) are called side equa- tions^ since the lengths of the sides are also involved. 365. Adjustment of a Triangle. — The finding and ap- plying of the most probable corrections to the measured values of the angles of a system of triangulation is called adjusting the system. In the case of a single triangle with one known side and three measured angles, we have seen that there is but one equation of condition. If the three angles have been; equally well observed, then it is most probable* that they are all equally in error, and hence this condition of highest proba.-. bility gives us the probability equation a' = b' = c' (4) which enables the corrections to be determined. Thus, let A' + C - iSo^ = a' + b' + c' = ' then from (4) we have where V is the error of closure of the triangle. • • (5) * That is, this relation is more probable than are any other smg/^ relation that can be assigned, but of course it is not more probable than all other eases combined. 494 SUR VE YING. ADJUSTMENT OF A QUADRILATERAL. 366. The Geometrical Conditions. — In the quadrilateral in Fig. 142 there are eight observed angles,^,, /i,, etc. The geometrical conditions which must here be fulfilled are : {a) The sum of all the angles of any triangle must be 180° plus the spherical excess* and the opposite angles at the intersection of the diagonals must be equal. (b) The computed length of any side, as DC^ when obtained from any other side, as AB, through two independent sets of triangles, as ABC, BDC, and ABD, ADC, shall be the same in both cases. The probability condition is that the set of corrections ap- plied to the several angles shall be more probable than any other one of the infinite number of sets of corrections which would satisfy the other condition. The condition given in {a) gives rise to the angle equations, and that given in {p) gives one side equation. There are evidently eight unknown corrections to be de- termined. 367. The Angle-equation Adjustment. — In the quadri- lateral ABCD we have four triangles in which all the angles have been observed, two sets of opposite angles where the other two angles of the corresponding triangles have been ob- served, and the quadrilateral itself in which all the angles have *It is not necessary to take account of the spherical excess in computing a single triangle or quadrilateral ; but if azimuth is to be carried over a series of triangles it is necessary that all the angles be spherical angles. In this place spherical excess will be omitted ; but if it is desirable to introduce it, it is in- serted in equations (i), (2), and (3), the right members then becoming A ) + ^5 — (s' + ^Z i+a-(|+4) + A - 1 ^8 1 + ^8 ~ ('f+f: 0 11 0 CO 1 ... (6) * The errors in the mean observed values of the angles are supposed to re- sult from the small incidental errors and approximations made in pointing, GEODETIC SURVEYING. 497 Or if Vs, etc., be the total corrections to the several mean observed angles for angle-equations, we have 4 - 2 /, v, = Vs= g , 4 - 24 8 ’ V, = V,r= V, — Vs - 4 + 2/ ^ 8 ’ 4 24 8~’ ( 7 ) 368. The Side-equation Adjustment. — In the quadri- lateral shown in the figure, let AB be the known side, and CB the required side, which is to be computed through two inde- pendent sets of triangles. Let A/, BJ, etc., be the several angles corrected for angle conditions by the corrections found in eq. (7). As computed through the first set of triangles, we have DC = BC sin Bs sin Ds AB sin A^ sin B^ sin sin ( 8 ) • Similarly DC ^ AD sin As sin Cs AB sin Bs sin A^ sin Cs sin D^ (9) Whence sin sin B^ _ sin sin A^ sin Cl sin Dl ~ sin Cl sin Dl ’ reading, etc. ; in other words, they are supposed to be errors of observation and not instrumental errors, these latter having been eliminated by the method of making the observations. Since the sources of the errors of observation are the same for small as for large angles, it follows that they should be credited with equal portions of the aggregate error of any combination of angles, re- gardless of the size of the angles themselves. 32 498 SC/A' VE YING. or sin sin sin sin Z?/ _ sin sin sin sin ~~ ’ which is called the side-equation. It is evident that in any case where the angles have all been observed, even after they have been adjusted for the angle-conditions, this equation will not hold true, the value of the left member being a little more or less than one. When put into the logarithmic form for computation, therefore, we will have log sin A^ log sin + log sin + ^^g sin D^' —'log sin B^ — log sin — log sin Z?/ — log sin A^ = /^, (i i) where ^ is the logarithmic residual due to erroneous observa- tions. We must now distribute this residual among the log sines according to the most probable manner of the occurrence of the errors which caused it. For a given small error, as i", in any angle, the effect on the log sine is measured by the loga- rithmic tabular difference for for that angle. This tabular difference varies for different angles, being large for angles near zero or i8o°, and small for angles near 90°. Let 7^/, 7^/, , etc., be the corrections to be made to the angles A^, B^, B^', etc., for the side-equation (ii), and let <^3, etc., be the corresponding logarithmic tabular differences for i". Now, the influences on 4 of the small angular errors were in direct proportion to the tabular differences of the correspond- ing log sines ; therefore the corrections should be in proportion to the corresponding tabular differences. In other words, the GEODETIC SURVEYING. 499 corrections are weighted in proportion to their tabular differ- ences.* We therefore have the numerical relation: \ d^w :: etc., or, paying attention to signs, d’^ d^ d^ ( 12 ) But since the log-sine correction is the angular correction multiplied by the tabular difference, and since the sum of these would equal 4, we would have v^d,—v^d^-\-v^d,-v^d.-{-v,'d,-v'd.-\-v,'d,-v^d,= — l.. . (13) From equations (12) and (13) we are to find the side-equation corrections 7/3', etc. Dividing eq. (13) by eq. (12), term by term, we have + + + + + d: I// + V,' -^- = +etc. * To illustrate this principle more fully, let us suppose that for a ^change of i" in the angles Ax and the corresponding changes in the log sines are i for Ax and 2 for A-i\ then for a given error of i in log sin Ax -j- log sin A-i — I there are two chances that it came from to one chance that it came from Ax, when these angles were equally well observed. If the error is to be divided between the angles Ax and y^a, therefore, we should make the correction to y^a i • y , , . Vx V ON ♦H VO VO On On On 0 » m 0 fO S. 2"8 M o VO VO CO « 10 ro fO 0 0 0 0 + + + + 0000 till 0000 + +-P 0000 1 ! I I f) N 00 >0 d »A M ro o >0 + + + + + + + + o^ o> o> 0 t-N ro 00 VO \o 00 o 00 fO tx O' ov O' ov O 3 .2 J!- ^ H' . " 52 «= S ! c cro rx 00 0 H S' ^ tx ro CO M 0 0 0 I I I 0 0 O H 1 I I I . ^ .r II ^ "O Iw S 00000 + + + + + !l II II II II H H H II II II ^ ^ ;j II II II II II c ^ c c c ro ro W Ov VO 00 10 M 0 CO w in 00 ov M m ^ yf 00 Ov 00 CO ro O O VO M CO O M o VO CO N 10 tv fO tv M fO + + + II II II II C CQ U Q CQ U Q < 5o6 SURVEYING. ADJUSTMENT OF LARGER SYSTEMS. 370. Used only in Primary Triangulation. — The simul- taneous adjustment of all the angles in an extended system of triangles with one measured base which is taken as exact, is a very complicated problem. The methods of least squares must here be applied, so that a discussion of this problem belongs rather to a treatise on geodesy than to one on surveying. The adjustments of a triangle and of a quadrilateral will be found sufficient for all secondary work, or such as is intended to serve only for topographical or geographical purposes. Especially is this true if the stations be so selected that the observed lines will form a series of quadrilaterals. The adjustment of these quadrilaterals by the rigid method given above gives nearly as good results as could be obtained by reducing the work as a single system. For a discussion of the least-square methods of adjustment of an extended system of triangles the student is referred to “Primary Triangulation of the U. S. Lake Sur- vey,” being Professional Papers, Corps of Engineers U. S. A., No. 24; Report of the U. S. Coast and Geodetic Survey for 1875; Clarke’s Geodesy; and especially to Wright’s “Adjust- ment of Observations.” The facility and accuracy with which base-lines may now be measured by means of long steel tapes will result in actually measuring many more lines than has heretofore been done, and so errors from angular measurements will not be allowed to accumulate to any great extent. It is not improbable that geodetic methods will be materially influenced by this new method of accurate measurement. 371. Computing the Sides of the Triangles.— After the angles of the system are adjusted, the sides of the triangles are computed by the ordinary sine ratio for plane triangles. If the system consist of simple triangles, then one side is known and the other two sides computed from it. In this case there is no check on the computation except what the computer GEODETIC SURVEYING. 507 carries along with him, or what may be obtained from a dupli- cate computation. If the system be made up of a series of quadrilaterals, then the line which is common to two successive quadrilaterals is computed through two sets of triangles from the previous known side. Thus if the quadrilateral of Fig. 142 be one of a series, the lines in common being AB and CD, then AB is computed in duplicate from the previous quadrilateral, and the mean of the two results taken. In the triangle ABD compute AD, and then in the triangle compute in the triangle compute BC, and then in the triangle BCD compute again. There are thus obtained two independent values of DC, as computed from AB. If the adjustment had been exact these values would have agreed exactly, but the adjusted angles were computed only to the nearest second, or tenth of a second ; hence the two values of DC will agree only to a corresponding exactness. If the system be composed of quadrilaterals and the adjustment be made to the nearest second, then the two values of DC would probably differ in the fifth or sixth signifi- cant figure. If the adjustment be made to the nearest tenth of a second, and a seven-place logarithmic table be used, then the two values of DC should begin to differ in the sixth or seventh place. Of course the adjusted values are not the true values of the angles, but simply the most probable values. If the angles were not accurately measured the adjusted values may still be considerably in error, but any such errors would not prevent the two values of CD from agreeing, since this agreement is one of the conditions which the adjustment is made to satisfy. The disagreement between the two computed values of CD comes only from the inexactness of the computed corrections to the angles, an angle, like a length, being an in- commensurable quantity, and hence some degree of approxi- mation is necessary in its expression. If the true computed values of CD differ by more than the amounts above signified, 508 SUR VE YING. then it is probable that an error has been made in the com- putation. LATITUDE AND AZIMUTH. 372. Conditions. — In the methods here given for obtaining latitude, azimuth, and time, the instrument used may either be an ordinary field transit mounted on its tripod, or a more elabo- rate altazimuth instrument, such as shown in Figs. 132 and 134. The accuracy sought is only such as is sufficient for topographi- cal or geographical purposes. Both the field methods and the office reductions are of the simplest character; but all large errors are eliminated, so that the results will be found as accu- rate as it is possible to obtain with anything less than the regu- lar field astronomical instruments. This higher grade of work falls within the sphere of the astronomer rather than of the surveyor. 373. Latitude and Azimuth by Observations on Cir- cumpolar Stars at Culmination and Elongation. — When latitude and azimuth are to be found to a small fraction of a minute, or as accurately as can be read on the instrument used, if this be an ordinary field transit, the most convenient method is by means of observations on circumpolar stars. The observa- tion for latitude is made on such a star when it is at its upper or lower culmination, since it is then not changing its altitude, and the observation for azimuth is made at elongation, since then the star is not changing its azimuth. At these times a number of readings may be taken on the star, thus eliminating instrumental constants by reversals, since a half hour may be utilized for this work without the star sensibly changing its position so far as the use it is serving is concerned. Two close circumpolar stars have been chosen whose right ascensions differ by about five hours and thirty minutes. They therefore always give a culmination and an elongation about thirty min- utes apart. This is very convenient, since it allows observations GEODE TIC S UR VE YING. 509 to be made for latitude and azimuth at one setting with a suf- ficient intervening interval to complete one set of observations before commencing the next. The two stars selected are Polaris (a Ursse Minoris), which is of the second magnitude, and 51 Cephei, which is of the fifth magnitude. Their relative positions are shown in Fig. 143. The position of 51 Cephei may be described with reference to the line joining “ the pointers,” in the constellation of the Great Bear, with Polaris. Thus, 51 Cephei is to the right of this line, when looking towards the pole-star along the line, at a distance of about three times the sun’s disk from the line, and of about five times the sun’s disk from Polaris, in the direction of the pointers. In case 51 Cephei is not visible to the naked eye, as it may not be on moonlight nights, or with a slightly hazy atmos- phere, it may be found, when near elongation, by the tele- scope, as follows : Having carefully levelled the instrument, turn upon Polaris. When 5 1 Cephei is near its eastern elongation Polaris is near its upper culminatiorn, and when near its western elongation Polaris is near its lower culmination. To find 51 Cephei at eastern elongation, therefore, after taking a pointing on Pola- 510 SUR VE YING. ris, lower the telescope (diminish the vertical angle) by about one degree (if the time is about twenty minutes before elonga- tion), and then turn off towards the east about two and a half degrees. This will bring the cross wires approximately upon the star. To find it at western elongation, simply reverse these angles ; that is, increase the vertical angle one degree, and turn off to the west two and one half degrees. The following table gives the times of the elongations and culminations of these two stars for 1885 latitude 40°, which may be used for observing azimuth and latitude. The times given are for the nights following the dates named in the first column. TIMES OF ELONGATION AND CULMLNATION, 1885. LATITUDE, 40°. ! Polaris (a Urs. Min.). 51 Cephei. Date. Elon- ga- tion. Time. Cul- mina- tion. Time. Elon- ga- tion. Time. Cul- mina- tion. Time. Jan. I W I2’>24"*.6 A.M. U 6 ** 29'®.9 P.M. W 5 '' 48“.3 A.M. U ii>>58'”.6 P.M. Feb. I 44 10 22 .2 P.M. L 4 25 .6 A.M. “ 3 46 .4 “ “ 9 56 .7 “ Mar. I 44 8 31 .8 “ 44 2 35 -I “ I 56 .1 “ “ 8 6 .4 “ April I 44 *6 29 .7 “ 44 12 33 .1 “ “ 11 54 .0 P.M. “ *6 4 -3 “ May 1 E *4 36 .6 A.M. 44 10 35 .2 P.M. “ 9 55 .9 “ L 4 4 .2 A.M. June I ii 2 37 .0 “ 44 8 33 -7 “ (( *7 53 -9 “ “ 2 2 .2 “ July I “ 12 39 .0 “ 44 *6 36 .2 “ E *6 12 .6 A.M. 44 12 4 .2 “ Aug. I “ 10 38 .1 P.M. U 4 32 .8 A.M. “ 4 10 .8 “ 10 2 .4 P.M. Sept. I “ 8 36 .6 “ 44 2 31 -3 “ “ 2 9 .1 “ 8 0 .8 “ Oct. I <( *6 38 .9 “ 44 12 33 .6 “ (( 12 n .4 “ 44 *6 3 .1 “ Nov. I W 4 26 .4 A.M. 44 10 31 .7 P.M. “ 10 9 .8 P.M. U 3 59 .5 A.M. Dec. I “ 2 28 .2 “ 44 8 33 -5 “ 8 12 .0 “ 2 I .8 “ * Probably not visible to the naked eye. From the above table it is evident that both an elongation and a culmination of one of these stars can always be obtained. For other days than those given in the table, either inter- GEODETIC SURVEYING. 51I polate, or find by allowing 3”".94 for one day, remembering that each succeeding day the elongation occurs earlier by this amount. For other years than 1885, take from the table the time cor- responding to the given month and day, and add 0^.35 for each year after 1885 ; also. Add if the year is the second after leap-year. Add 2^ if the year is the third after leap-year. Add 3™ if the year is leap-year before March i. Subtract if the year is leap-year after March i. For the first year after leap-year there is no correction ex- cept the periodic one of 0'".35 per annum. For other latitudes than 40°, add 0^.14 for each degree south of 40° north latitude, or subtract o"^. 18 for each degree north of 40° north latitude for Polaris, and 0^.29 and o°^39 for the corresponding correction for 51 Cephei. The following table gives the pole distances of Polaris and 51 Cephei for Jan. i of each third year from 1885 to 1930: POLE DISTANCE (90° — Declination). Star. 1885. 1888. 1891. 1894. 1897. 1900. 1903. 1906. Polaris. . . . 51 Cephei.. i°i8'i6" 2 46 35 2 4647 i ® i 6'23" 2 47 00 i ° i 5'26" 2 47 13 i°i4'3o" 2 47 26 1OJ3/33// 2 47 40 i°i 2 ' 37 " 2 47 54 i°ii'4i" 2 48 09 Star. 1909. 1912. 1915. 1918. 1921. 1924. 1927. 1980. Polaris 51 Cephei.. i°io'45'' 2 48 24 1° 9 ' 49 " 2 48 39 1° 8 ' 53 " 2 48 55 1“ 7 ' 58 '' 2 49 12 1° 7' 2" 2 49 28 1° 6 ' 7" 2 49 45 1° 5'i2" 2 50 01 1° 4'i6" 2 50 18 To find the pole distance for any intermediate time, make a linear interpolation between the two adjacent tabular values. 512 SUR VE YING. To observe for latitude no knowledge of the geographical position is needed. 374. The Observation for Latitude consists simply in observing the altitude of a circumpolar star at upper or lower culmination and correcting this altitude for the pole distance of the star and for refraction. Let 0 = latitude ; d = polar distance; r — refraction ; h — altitude ; then (p = h^^d—r\ (i) the minus sign being used for upper, and the plus sign for lower, culmination observations. The value of r is taken from the following table of mean refractions computed for barometer 30 inches, and temperature ^0° F. TABLE OF MEAN REFRACTIONS. Altitude. Refraction. Altitude. Refraction. 10° 5' 19” 20 ° 2' 39" II 4 51 25 2 04 12 4 28 30 I 41 13 4 07 35 I 23 14 3 50 40 I 09 15 3 34 45 0 58 16 3 20 50 0 4<9 17 3 08 60 0 34 18 2 58 70 0 21 19 2 48 80 0 10 GEODETIC SURVEYING. 513 The index error of the vertical circle is eliminated by read- ing with the telescope direct and reversed, providing the verti- cal circle is complete. If the vertical limb is but an arc of 180° or less, the index error cannot be eliminated in this way. In this case the second method is recommended. 375. First Method. — Mount the instrument firmly, pre- ferably on a post, and adjust carefully the plate-bubbles, especially the one parallel to the plane of the vertical circle. About five or ten minutes before the star comes to its culmi- nation read the altitude of the star with telescope direct. Revolve the telescope on its horizontal axis and also on its vertical axis, relevel the mstrument if the bubbles are not in the middle, but do not readjust the bubbles, and bring the tele- scope upon the star. Make two readings in this position. Revolve the telescope and instrument again about their axes, relevel, and read again in first position. This gives two direct and two reversed readings taken in such a way as to eliminate the error from collimation, the index error of vertical circle, and also the error of adjustment of the plate-bubbles. The result, when corrected for refraction and the pole distance of the star, should be the latitude of the place within the limits of accuracy and exactness of the vertical circle-readings. 376. Second Method. — An “artificial horizon,” formed by the free surface of mercury in an open vessel, may be used in conjunction either with the transit or a sextant. If the former is used two pointings are made — one to the star and the other to its image in the mercury surface. The angle measured is then twice the apparent altitude of the star. The position of the vessel of mercury will be on a line as much below the horizontal as the star is above it. The instrument is first set up and then the artificial horizon put in place. The surface of the mercury must be free from dust. If the mercury is not clean it may be strained through a chamois-skin or skimmed by a piece of cardboard. Any open vessel three or more 33 514 SUJ^VEYING. inches in diameter may be used for holding the mercury. It should be placed on a solid support and protected from the wind. The observations with a transit would then consist in taking a reading on the star just before culmination, two readings on the image, and then one on the star. The index error of the vernier on the vertical circle will then be eliminated, since both plus and minus angles have been read, and their sum taken for twice the altitude of the star. This method is adapted to transits with incomplete vertical limbs. The Sextant may also be used with the artificial horizon and will give more accurate results than can be obtained with the ordinary field transits. The double altitude angle is then measured at once by bringing the direct and reflected images of the star into coincidence. In both cases the observed angle is 2^, and the latitude is found from equation (i), as before. If there is much wind the mercury basin may be partially covered, leaving only a narrow slit in the vertical plane through instrument and star, or the regular covered mercurial horizon may be used. This is covered by two pieces of plate-glass set at right angles to each other like the roof of a house. If the opposite faces of these glasses are not parallel planes, an error is introduced. This is eliminated by reversing the horizon apparatus on half the observations. It is best, however, to avoid the use of glass covers, if possible. If tin-foil be added to the mercury an amalgam is formed, whose surface remains a perfect mirror, which is not readily disturbed by wind. As much tin-foil should be used as the mercury will unite with. Observations may then be made in windy weather without the aid of a glass cover. 377. Correction for Observations not on the Meridian. If the star is more than five or ten minutes of time from the meridian, it is necessary to apply a correction to the observed altitude to give the altitude at culmination. The following GEODETIC SURVEYING. 515 approximate rule gives these corrections for the two circum- polar stars here used, with an error of less than i" of arc when the observation is taken not more than 18 minutes of time from the star’s meridian passage, and the error is less than 10" of arc when the observation is made 32 minutes of time from the meridian. Rule for reducing circumrneridian altitudes to the altitude at culmination. For Polaris : Multiply the square of the time from meridian passage, in minutes, by 0.0444, and the product is the correc- tion in seconds of arc. For 51 Cephei : Multiply the square of time from meridian passage, in minutes, by 0.1143, and the product is the correc- tion in seconds of arc. The correction is to be added to the observed altitude for upper culmination, and subtracted for lower culmination. By using these corrections an observation for latitude may be made at any time for a period of about one hour, near the time of culmination. 378. The Observation for Azimuth is made on one of the two stars here chosen when it is at or near its eastern or western elongation, for the same reason that latitude observa- tions are taken at culmination. The azimuth of a star at elongation is found from the formula, . . , sine of polar distance sine of azimuth = ^ ^ ^ . cosine 01 latitude • (I) This formula is so simple that it is hardly necessary to give a table of values of azimuths for various latitudes. Such a table is given for Polaris, however, on p. 33. The pole distances are given on p. 511, and the latitude is found by observation. It is not necessary to know the azimuth of the star at elonga- 5i6 SURVEYING. tion before making the observations. This can be computed afterwards from the observed latitude. The observation for azimuth consists simply in measuring the horizontal angle between the star and some conveniently located station, marked by an artificial light. The operation is in no sense different from the measurement of the horizontal angle between two stations at different elevations. The great source of error is in the horizontal axis of the telescope. If this is not truly horizontal then the line of sight does not de- scribe a vertical plane, and since the two objects observed have very different elevations, the angle measured will not be that subtended by vertical planes passing through the objects and the axis of the instrument. To eliminate this error the tele- scope is reversed, and readings taken in both positions. The following programme is recommended: PROGRAMME FOR OBSERVING FOR AZIMUTH ON A CIRCUM- POLAR-STAR AT ELONGATION. Instrument. Time of Observation. Reading on Direct lo min. before elongation Mark. Reversed / ••••••••• 4 “ “ Star. ii 2 “ “ << Direct 2 min. after “ << 4 “ “ ti 7 “ “ Mark. Reversed lO “ “ < i The instrument should always be relevelled after reversing, but the bubbles should not be readjusted after the observa- tions have begun. If this be done and the above programme followed, all instrumental errors will be eliminated except GEODETIC SURVEYING. 517 those of graduation. Of course both verniers are to be read each time. Having found the latitude, the azimuth of the star at elon- gation is found from equation (i) above. This is then added to or subtracted from the horizontal angle between mark and star, as the case may be, to give the azimuth of the mark from the north point. If the azimuth is to be referred to the south point, which it generally is, we must add or subtract 180°. 379. Corrections for Observations near Elongation. — As in the case of observations for latitude, we may have an approximate rule for reducing an observed azimuth when near elongation to what it would have been if taken at elongation. The limits of accuracy are also about the same, but the factors are slightly different. Ride for reducing azimuth observations on Polaris and 51 Cephei near elongation to their true values at elongation, for latitude 40°. For Polaris, multiply the square of the time from elonga- tion in minutes by 0.058, and the product will be the correction in seconds of arc. For 51 Cephei, multiply the square of the time from elonga- tion in minutes by 0.124, and the product will be the correction in seconds of arc. The formula for reduction, when near elongation, is c — 1 12.5 f sin i" tan A, where c = correction to observed azimuth in seconds of arc ; t — time from elongation in seconds of time; A = azimuth of star at elongation. log 1 12.5 sin i" = 6.7367274. From this formula and that of equation (i) we may compute the coefficients for the above approximate rules for any latitude. 518 SURVEYING. Thus, for latitude 30° we have azimuth of Polaris, 1885, 3o'.4, whence the coefficient of reduction for elongation of Polaris in latitude 30° is found to beo.052, and for latitude 50° it is 0.069. For 51 Cephei, this coefficient for latitude 30° is o.iio, and for latitude 50°, 0.148. Ph'om the above data the corrections for an observation of a circumpolar star near elongation may be computed. If azimuth be reckoned from the south point, as is common in topographical and other geodetic work, and if it increase in the direction S.W.N.E., then a star at western elongation has an azimuth of less than 180°, and at eastern elongation its azimuth is more than 180°. The corrections to reduce to elongation, as above com- puted, should be added to the computed azimuth of the star at western elongation, and subtracted when at eastern elongation. 380. The Target. — This may be a sort of box, in which a light may be placed. A narrow vertical slit should be cut, sub- tending an angle, at the instrument, from one to two seconds of arc. This should be set as far from the instrument as conven- ient, as from a quarter of a mile to one mile. The width of slit desired may be computed for any given angular width and distance by remembering that the arc of one second is three-tenths of an inch for a mile radius. The target should be sufficiently distant to enable it to be seen with the stellar focus without appreciable parallax, as the instrument should not be refocused on the target. This target may be set on any convenient azimuth from the observation-station, as upon one triangulation station when the observations are taken at another, thus obtaining directly the azimuth of this line. 381. Illumination of Cross-wires. — Various methods are used to illuminate the wires, the crudest of which is, perhaps, to hold a bull’s-eye lantern so as to throw light down the tele- scope-tube through the objective, taking care not to obstruct the line of sight. GEODETIC SURVEYING. 519 A very good reflector may be made from a piece of new tin, cut and bent as in Fig. 144. The straight strip is bent about the object end of the tele- scope tube, leaving the annular elliptic piece projecting over in front. This is then bent to any desired angle, preferably about forty-five degrees, and turned so that an attendant can '44- reflect light down the tube by illuminating the disk from a convenient position. This position should be so chosen that the lantern may throw the light from the observer, rather than towards him. If the reflecting side of the disk be whitened, the effect is very good. The opening should be about three-fourths or seven-eighths inch in its shorter diameter, the longer diameter being such as to make its vertical projection equal to the shorter one. There is, of course, no necessity of limiting or of making true the outer edges of the disk. TIME AND LONGITUDE. 382. Fundamental Relations. — In all astronomical compu- tations the observer is supposed to be situated at the centre of the celestial sphere and the stars appear projected upon its surface. Their positions with respect to the observer may be fixed by two angular coordinates. The most common plane of reference for these coordinates is that of the celestial equator, and the coordinates referring to it are known as Right Ascen- sion and Declination — corresponding to Longitude and Lati- tude on the earth’s surface. Right ascension is counted on the equator from west tow- ards east. As a zero of right ascension the vernal equinox is taken. Declination is counted on a great circle perpendicular to the equator, and is called positive when the star is north and negative when south. 520 SURVEYING. In Fig. 145 P is the pole ; Z is the zenith of the observer ; 5 is the star ; Then R. A. star = VPS — arc VE ; Dec. star = SE. These coordinates are fixed, varying only by slow changes due to the shifting of the reference-plane. Another system of coordinates is often used in fixing the place of a star, namely: Hour-angle and Declination. Hour- angle is the angle at the pole between the meridian and the great circle passing through the star and the pole perpendicu- z lar to the equator. Hour-angle will of course be constantly changing each instant. In Fig. 145 hour-angle = ZPS. 383. Time. — The motion of the earth on its axis is perfect- ly uniform. We obtain, therefore, a uniform measure of time by employing the successive transits of a point in the equator across the meridian of any place. The point naturally chosen is the vernal equinox. A Sidereal Day is the interval of time between two succes- GEODETIC SURVEYING. 521 sive upper transits of the vernal equinox over the same merid- ian. The Sidereal Time at any instant is the hour-angle of the vernal equinox at that instant reckoned from the meridian westward from o’" to 24*". Thus, when the vernal equinox is on the meridian, the hour-angle is o'" o® and the sidereal time is o™ o®. When the vernal equinox is west of the merid- ian the sidereal time is o'"o®. We have in Fig. 145 Hour-angle of ver. eq. = ZPV = 6 = sidereal time; Right asc. of star = VPS = a ; Hour-angle of star = ZPS = H; Whence S — a — H, (i) From this equation, knowing the sidereal time and the R. A. of the star, the hour-angle may always be computed. When H —o^ i.e., when the star is on the meridian, 0 = a, or, in other words, the R. A. of any star is equal to the true local sidereal time when the star is on the meridian. By noting the exact time of transit of any star whose R. A. is known, the local sidereal time will be at once known. An Apparent Solar Day is the interval of time between two successive upper transits of the true sun across the same meridian. Apparent or True Solar Time is the hour-angle of the true sun. Owing to the annual revolution of the earth, the sun’s right ascension is constantly increasing. It follows, therefore, that a solar day is longer than a sidereal day. In one year the sun moves through 24^'^® of right ascension. There will be, therefore, in one tropical year (which is the interval be- 522 SURVEYING. tween two successive passages of the sun through the vernal equinox) exactly one more sidereal day than solar days ; or, in other words, in a tropical year the vernal equinox will cross the meridian of any given place once more than the sun will.^ The solar days will, however, be unequal for two reasons : 1st. The sun in its apparent motion round the earth does not move in the equator, but in the ecliptic. 2d. Its motion in the ecliptic is not uniform. On account of these inequalities the true solar day cannot be used as a convenient measure of time. But a mean solar day has been introduced, which is the mean of all the true solar days of the year and which is a uniform measure of time. Suppose a fictitious sun to start out from perigee with the true sun, to move uniformly in the ecliptic, returning to peri- gee at the same moment as the true sun. Now, suppose a second fictitious sun moving in the equator in such a way as to make the circuit of the equator in the same time that the first fictitious sun makes the circuit of the ecliptic, the two fic- titious suns starting together from the vernal equinox and re- turning to it at the same moment. The second fictitious sun will move uniformly in the equator and will be therefore a uniform measure of time. This second fictitious sun is known as the Mean Sun. A Mean Solar Day is therefore the interval between the upper transits of the mean sun over the meridian of any place. Mean Solar Time at any meridian is the hour-angle of the mean sun at that meridian counted from the meridian west from to 24''^®. The Equation of Time is the quantity to be added to or subtracted from apparent solar time to obtain mean time. The equation of time is given in the American Ephemeris for Washington mean and apparent noon of each day. If the value is required for any other time it can be interpolated be- tween tlie values there given. GEODETIC SURVEYING. 523 384. To convert a Sidereal into a Mean-time Interval, and vice versa. — According to Bessel, the tropical year con- tains 365.24222 mean solar days, and since the number of side- real days will be greater by one than the number of mean solar days, we have 365.24222 mean sol. days = 366.24222 sid. days ; I mean sol. day = 1.00273791 sid. days. Let /„ = mean solar interval ; Is = sidereal interval ; k = 1.00273791. Thus 4 = IJ: = /„+ IJk - i) = 4 + 0.00273794 ; Im-J = 4 - 4 (i - = 4 - 0.00273044. By the use of these formulae the process of converting a sidereal interval into a mean-time interval, and vice versa, is made very easy. It is rendered more easy by the use of Tables II. and III. of the Appendix to the American Ephem- eris and Nautical Almanac, where the quantity IJ^k — i) is given with the argument I^, and 4 with the argument 4. Example . — Given the sidereal interval 4 = 15^ 40"" 50^.50, find the corresponding mean-time interval. Is — 15^ 40™ 50^.50 Table II. gives for 15*" 40"^ 2 33.996 “ 50^50 0.138 /«.= i 5 38 16.37 524 SURVEYING. 385. To change Mean Time into Sidereal. — Referring to Fig. 145, suppose 5 to represent the mean sun. Then ZPS = hour-angle of mean sun = mean-time = T ; VPE — R. A. of mean sun = ; 6 = sidereal time. From equation (i), p. 521, 6 = a^-\- T. The right ascension of the mean sun is given in the Ameri- can Ephemeris both for Greenwich and Washington mean noon of each date. It is called ordinarily the sidereal time of mean noon, which is of course the right ascension of the mean sun at noon, since at mean noon the mean sun is on the meridian and its right ascension is equal to the sidereal time. Since the sun’s right ascension increases 360° or 24^” in one year, it will change at the rate of 3"^ 56®*555 in one day, or 9^.8565 in one hour. Suppose = sid. time of mean noon at Greenwich; = “ “ “ “ “ “ the place for which T is known ; L — longitude west of Greenwich. Then — OJ -4- 9^.8565^, where L is expressed in hours and decimals of an hour. In this way the sidereal time of mean noon may be obtained for the meridian of observation. Substituting for its equivalent, and reducing the mean- time interval to sidereal, i = d, + T+ T{k - I). Example. — Longitude of St. Louis, 6^ o™ 49^.16 = 6^.0136. Mean time, 1886, June 10, 25™ 25^.5. Required correspond- ing sidereal time. GEODETIC SURVEYING. 52s From Amer. Ephem., p. 93 : (for Greenwich) 6.0136 X 9-8565 T e T{k- I), Table 111.,= = s'" 15" 3 ’- 30 = o 59-27 = 5 16 2.57 = 10 25 25.50 = I 42.74 = 15 43 10.81 It should be remarked that the quantity 59^.27 will be a constant correction, to be added to the sid. time of mean noon at Greenwich to obtain the sid. time of mean noon at St. Louis. 386. To change from Sidereal to Mean Time. — This process is simply the reverse of that for changing from mean to sidereal time. Using the same notation as before, we shall have Subtracting from the given sidereal time {Q) the sidereal time of mean noon (^0), we have the sidereal interval elapsed since mean noon, and this needs simply to be changed into a mean-time interval. Example. — Given 1886, June 10, 15^ 43"" io’.8i sidereal time, to find the corresponding mean time. ^ = 15 43 10.81 (as before) = 5 16 2.57 — 6 '^ = 10 27 8.24 I — -i) (Table II.) = i 42.74 7 ^= 10 25 25.50 526 SURVEYING. 387. The Observation for Time, as here described,* consists in observing the passage, or transit, of a star across the meridian. The direction of the meridian is supposed to have been determined by an azimuth observation. If the in- strument be mounted over a station the azimuth from which to some other visible point is known, the telescope can be put in the plane of the meridian. An observation of the passage of a star across the meridian will then give the local time, when the mean local time of trajisit of that star has been computed. In order to eliminate the instrumental errors at least tv/o stars should be observed, at about the same altitude. If the instru- ment has no prismatic eye-piece, then only south stars can be observed with the ordinary field-transits; that is, only stars having a south declination, if the observer is in about 40° north latitude. Stars near the pole should not be chosen, since they move so slowly that a small error in the instrument would make a very large error in the time of passage. 388. Selection of Stars. — The stars should be chosen in pairs, each pair being at about the same altitude, or declination. It is supposed that the American Ephemeris is to be used. The “ sidereal time of transit, or right ascension of the mean sun,” is its angle reckoned easterly on the equatorial from the vernal equinox. This is given in the Ephemeris for every day of the year. Similarly, the right ascension of many fixed stars is given for every ten days of the year, under the head of Fixed Stars, Apparent Places for the Upper Transit at Wash- ington.” These latter change by a few seconds a year, from the fact that the origin of coordinates, the vernal equinox itself, changes by a small amount annually. If, therefore, the hour- angle, or right ascension, of both the mean sun and a fixed * It is assumed that the engineer or surveyor has only the ordinary field- transit, without prismatic eye-piece, so that he can only read altitudes less than 60°. The accuracy to be attained is about to the nearest second of time. GEODETIC SURVEYING. 527 star be found for any day of the year, the difference will be the sidereal interval intervening between their meridian passages, the one having the greater hour-angle crossing the meridian much later than the other. When this interval is changed to mean time the result is the mean or clock time intervening between their meridian passages. If a fixed star is chosen whose right ascension is eight hours greater than that of the mean sun for any day in the year, then this star will come to the meridian eight hours (sidereal time) after noon, or at 7^ 58™ 411364 after noon of the civil day indicated in the Nau- tical Almanac. If, therefore, one wishes to make his observa- tions for time from 8 to 10 o’clock P.M. he should select stars whose hour-angles, or right ascensions, are from 8 to 10 hours greater than that of the mean sun for the given date. In the following table such lists are made out for the first day of each month for the year 1888. The mean time of transit is given for the meridian of Washington to the nearest minute, as well as its mean place for the year. None of these values will vary more than three or four minutes from year to year, and therefore the table may be used for any place and for any time. The table merely enables the observer to select the stars to be observed. After these are chosen their local mea 7 i time of transit must be worked out with accuracy from the Nautical Almanac.* For any other day of the month we have only to remember that the star comes to the meridian 3“ 56® earlier (mean time) each succeeding day, so that for n days after the first of the month we subtract 3.93 n minutes from the mean time of transit given in the table, and this will give the approximate mean time of transit for that date. If we take n days before * Even this trouble may be avoided by using Clarke's Transit Tables (Spon, London). Price to American purchasers less than one dollar. They are pub- lished annually in advance, and give the Greenwich mean time of transit of the sun and many fixed stars for every day in the year. They are computed for pop- ular use from the Nautical Almanac. LIST OF SOUTHERN TIME-STARS FOR EACH 528 SU/^ VE YING. LIST OF SOUTHERN TIME-STARS FOR EACH n—Co7ttinued. GEODETIC SURVEYING. 529 August i. Approx. Mean Time of Transit. e N >0 ■>r M 00 »© CO 'ro M N CO CO ■ October i. S VO ro VO N w to 0 fO ^ N CO 00 00 00 On Ov December i. B O' ^ M On fO Cl to fO 0 ''t- to 00 00 On 0 0 0 c . ^0 0 10 H o> M N iO 0 H M N 00 00 00 00 ^ tx 0 VO w VO CO 'tf w N M H ■Q 0 M H N N W C4 Cl N W W B 00 00 VO tv HI ro M CO fO ja 0 0 W M W « .Sc y.S -«(*- 10 tn VO On 0 VO 0 N 0 ‘O W 0 W 10 00 N CH fO N W 1 1 1 1 1 1 1 "0 "T 3- 0 ■'T PC 10 0 M m ‘U O' N VO 00 M Pt PI 1 1 1 1 1 1 'tv VO VO fO On VO Cl ro to 0 0 O' 00 00 0 0 Cl 1 1 1 T I + bD rt S . m to ro fO CO -•i- to ^ CO CO ro M ro ro to K H C/3 9 Ophiuchi b Ophiuchi 72 Sagitt Sagitt Tj Serpentis A Sagitt I Aquilae 0 12; B *-i to Cl Cl tv 0 0 Cl to 10 M CXD 00 00 00 00 O' w 0 0 ^ On 00 fO M M to to 0 fo 0 10 to VO VO VO TO M M VO VO W M CO to 0 M 00 On O' O' 0 0 M M M M d W ^ M On 00 <00 to 0 fO 0 N S e? c? rt •S c U.2 CJ w Q 00 00 0 H 0 to to IH CO N H N fO 0 00 N On <0 NO 0 to W M N M M 1 1 1 1 1 1 1 O' VO M O' fo Cl 0 H 0 0 to 0 VO ^ C» w CJ 1 1 1 1 1 1 '%-! ro O' ^ to 0 M M ro to Cl 0 00 0 VO oo 00 VO ro H w 1 1 1 1 1 1 tij n S N N W fO M fO fO Cl 10 to to fO CO '.t- M Tf to to « H in ^ Librae 5 Scorpii / 3 * Scorpii 5 Ophiuchi a Scorpii ^ Ophiuchi 7} Ophiuchi cr Sagitt d Sagitt K Aquilae c Sagitt & Aquilae a2 Capri A Aquarii a Pis. Aus (f) Aquarii z'l Aquarii 5 Sculp 33 Piscium 34 530 SUK VE Y INC. a date in the table, add 2.93 n minutes to the corresponding time of transit to find the approximate time of transit for the given date. This table is therefore a mere matter of conve- nience to assist in selecting the stars to be used. They arc nearly all southern stars, since these only can be observed with the ordinary field-transit. 389. Finding the Mean Time of Transit. — As explained above, the mean or clock time of transit is simply the sidereal interval between the mean sun and star for the given place and date, reduced to mean time. To find this interval we find the right ascension of both mean sun and star, and fake their dif- ference. But the right ascension or sidereal time of the mean sun or mean noon is given for the meridian of Greenwich, whereas by the time the sun has reached the given American meridian its right ascension or sidereal time has increased somewhat, the hourly increase being 9^8565. To find the '•sidereal time of mean noon” for the given place, therefore, we take the value for the given date for Greenwich and add to it 9^8565 for every hour of longitude the place is west of Greenwich. This then gives the “ local sidereal time of mean noon.” The right ascension of the star, or the sidereal time of its meridian passage, is then found. This changes only by a few seconds in a year, and is given for every ten days in the Washington Ephemeris. This, therefore, needs no correction to reduce it to its local value for any place. The difference between the “ local sidereal time of mean noon” and the sidereal time of the star is the sidaral interval of time elapsing between local mean noon and the transit of the star. When this sidereal interval is changed to a mean-time interval, which is effected by means of a table at the back of the Nautical Almanac, the result is the local mean time of transit of the star. GEODETIC SURVEYING. 531 Example. — Compute the local mean time of transit of e Eridani at St. Louis on Jan. 16, 1888. Sidereal time of mean noon at Greenwich = 19'' 41“ 0 00 Correction for longitude 6.05^ west = + 59 -63 Local sidereal time of mean noon = 19 42 27 .43 Right ascension e Eridani Jan. 16 = 3 27 39 -21 Sidereal interval after mean noon = 7 45 II .78 Correction to reduce to mean time = — I 16 .21 Local mean time of transit = 7^ 43 ™ ‘ 55®.57 390. Finding the Altitude. — The relation between lati- tude, declination, and altitude is shown by Fig. 146, which rep- resents a meridian section of the celestial sphere. Let PF be the line through the earth’s axis ; QQ the plane of the equa- tor; Z the zenith, and HH' the horizon. Then H'P—ZQ=.

, and the azimuth of the line DC from D, COMPUTATION OF L M Z. z C to A 80° 00' 00". 0 z. ACD = Cs (see p. 495) 39 48 06 .1 z> C toD ... II9 48 06 . 1 dZ 0 AQ . ^ *T7 • D 180° z 180 Z? to C 299 38 16 .6 V dL L 40® + 00' 6 00'^. 000 41 .8j7 C 25000 metres. D M’ dM 90® + 00' 15 00' \ 000 i6 .019 40 06 41 .847 M 90 15 16 .019 ist term — 402". 8 53 2d and 3d terms 4“ i — dL — 401 .847 ■^m = 40° °3^ 22" B K cos Z' h 8.5108517 4.3979400 9.6963560 C sin* Z 1.32833 8.79588 9.87679 D 2.3863 5.2103 2.6051477 0.00100 I .0023 7.5966 0.0039s A K sin Z cos L (a.c.) dM 8.5091156 4.3979400 9.9383948 0.1164540 dM sin Zn, - dZ 2.96190 9.80857 2.9619044 -f- 916". 019 2.77047 -f 589^.48 540 SURVEYING. GEODETIC LEVELLING. 395. Geodetic Levelling is of two kinds : (A) Trigonomet- rical Levelling 7 i\\d {B) Precise Spirit-levelling. In trigonomet- rical levelling the relative elevations of the triangulation-sta* tions are determined by reading the vertical angles between the stations. When these are corrected for curvature of the earth’s surface and for refraction it enables the actual difference of elevation to be found. In precise spirit-levelling a special type of the ordinary spirit or engineer’s level is used, and great care taken in the running of a line of levels from the sea-coast inland, connecting directly or indirectly with the triangulation stations and base-lines. Both these methods will be described. {A) TRIGONOMETRICAL LEVELLING. 396. Refraction. — If rays of light passed through the atmos- phere in straight lines, then in trigonometrical levelling we should have to correct only for the curvature of a level surface at the locality. It is found, however, that rays of light near the sur- face of the earth usually are curved downwards — that is, their paths are convex upwards. This curve is quite variable, some- times being actually convex downwards in some localities. It GEODETIC SURVEYING. 541 has its greatest curvature about daybreak, diminishes rapidly till 8 A.M., and is nearly constant from 10 A.M. till 4 P.M., when it begins to increase again. The curve may be considered a circle having a variable radius, the mean value of which is about seven times the radius of the earth. 397. Formulae for Reciprocal Observations. — In Fig. 147 the dotted curve represents a sea-level surface. Let H — height of station ^ above sea-level ; H' = height of station A above sea-level ; C = angle subtended by the radii through A and B ; Z = true zenith distance of A as seen from B\ Z' — true zenith distance of B as seen from A ; d — true altitude of A as seen from B — 90° — Z ; = true altitude of B as seen from A — 90° — Z' \ h = apparent altitude of A as seen from ^ = (J -|- re- fraction ; h' — apparent altitude of B as seen from A = d' re- fraction. d = distance at sea-level between A and B ; r = radius of the earth ; m = coefficient of refraction. In the figure join the points A and ^ by a straight line. This would be the line of sight from A to ^ if there were no refraction. Through A and B draw the radii meeting at C, ex- tending them beyond the surface.* Take the middle point of the line AB, as H, and draw HC. Take .^^'perpendicular to HC, and EE through H and perpendicular to HC. Extend A A' to meet a perpendicular to it from B. Then do we have A'C — AC\ E'E=.AD\ and * In reality these are the normals at A and B, but will here be taken as the radii. 542 SUR VE YING. Neither of these three relations is quite exact, because HC does not quite bisect the angle C. The figure is greatly exaggerated as compared to any possible case in practice, for the angle C would never be more than i° in such work. The error in practice is inappreciable. From the geometrical relations shown in the figure we have H - IT ^ A'n=z DB scc^ (i) But since ^7 is never more than i°, and usually much less, we may say H-H' =A'B = DB = AD\:7inBAD.. . . (2) But AD = E'E = distance between the stations reduced to their mean elevation above sea-level = d' ; also BAD = - Z ') ; ... tan i{Z - Z') (3) But since d = distance between stations at sea-leveh. we have or ,, . , H+H' d : d :: r-\ — : r. d' =4 1+ 2 H+H ' 2r . . (4j whence we have, for reciprocal observations at A and B, = . . (5) GEODETIC SURVEYING. 543 or, in terms of d and 6' , = . . ( 6 ) where attention must be paid to the signs of <5^ and 8'. The effect of refraction is to increase <5^ and by equal amounts (presumably), whence their difference remains unaf- fected. Equations ( 5 ) and ( 6 ) are therefore the true equations to use for reciprocal observations at two stations. Since the refraction is so largely dependent on the state of the atmos- phere, the observations should be made simultaneously for the best results. This is seldom practicable, however, and therefore it is highly probable that a material error is made in assuming that the refraction is the same at the two stations when the observations are made at different times. 398. Formulae for Observations at One Station only. — If the vertical angle be read at only one of the two stations, then the refraction becomes a function in the problem. Since the curve of the refracted ray is assumed to be circular (it probably is not when stations have widely different elevations), the amount of angular curvature on a given line is directly pro- portional to the length of the line or to the angle C. The dif- ference at ^ ox B between the directions of the right line AB and the ray of light passing between them is one half the total angular curvature of the ray ; that is, it is the angle between the tangent to the curved ray at A and the cord AB. The ratio between this refraction angle at ^ or B and the angle {7 is a constant for any given refraction curve ; that is, this ratio does not change for different distances between sta- tions. This ratio is called the coefficient of refraction, and is Q here denoted by m. The true angle BAD Is equal to d' but since the observed altitude is increased by the amount of 544 SUK VE YIXG. the refraction, we have for the apparent altitude of B, as seen from Ji' — d' viC\ r whence BAD — h'-\--^ — mC. (7) Using this value of the angle BAD in equation (2), we obtain H-H’ = d tan (// + ^ - mc) = d tan [ti + f - w/C) (i + • (8) where It! is positive above and negative below the horizon. Equation (8) is used where the vertical angle is read from one station only. Since the total angular curvature of the ray of light between A and B is 2mC^ and the curvature of the earth is we may write C : 27nC \\ r' \ r, or r' = — , . . . (o) 2m where r' is the radius of the curve of the refracted ray. Since the curvature of the ray is of the same kind as that of the earth, but less in amount, the total correction for curva- C C ture and refraction is for an angle equal to mC= —(i-- 2 m)* 2 2 Also, since C is always a small angle, we may put C (in seconds of arc) = — f - — j,. ^ r sm I If the mean radius is used, we have, in feet, log r = 7.32020, and log sin i" = 4.6855749, GEODETIC SURVEYING. 545 whence in seconds of arc and distance in feet we have or log C — log d — 2.00577 d 101.34 • • ( 10 ) or the curvature is approximately equal to for 100 feet in distance. The following table gives computed values of the combined mean corrections for curvature and refraction for short dis- tances, either for horizontal or inclined sights. Both the dis- tance d and the correction are in feet, except for the last column, where the distance is given in miles. For a more ex- tended table for long distances, see page 433. CORRECTION FOR EARTH’S CURVATURE AND REFRACTION. d Cn d Cn d Cn d Cn d Cn miles. 1 Cn 300 .002 1300 •035 2300 . 108 3300 223 4300 •379 I •571 4 (X) .003 1400 .040 2400 .118 3400 •237 4400 •397 2.285 500 .005 1500 .046 2500 . 128 3500 .251 4500 •415 3 5-142 600 .007 1600 .052 2600 •139 3600 .266 4600 •434 4 9.141 700 .010 1700 •059 2700 .149 3700 .281 4700 •453 5 14.282 800 .013 1800 .066 2800 .161 3800 .296 4800 .472 6 20.567 900 .017 1900 .074 2900 . 17 ? 3900 .312 4900 •492 7 27.994 1000 .020 2000 .082 3000 . 184 4000 .328 5000 .512 8 36-563 1100 .025 2100 .090 3100 .197 4100 •345 5100 •533 9 46.275 1200 .030 2200 .099 3200 .210 4200 .362 5200 •554 10 57.130 J 399. Formulae for an observed Angle of Depression to a Sea Horizon. — In Fig. 148 let A be the point of observa- tion and 5 the point on the sea-level surface where the tangent from A falls. Then we have H=AD=^ASt^nASD C = r tan C tan - (ii) 35 * Let the student prove this relation. 546 S UK VE Y INC. Since tlie angle C is always very small, wc may let the arc equal its tangent, whence //=-tan’(r. . (12) If the observed angle of de- pression ho. h = C — mCy then and or — m H — - tan 2 H <' 3 > = (-) where /i is expressed in seconds of arc. Log - tan’* i" = 6.39032 for distances in feet. 400. To find the Value of m we have whence Z = 90° — h-\- mC, Z' — 90° — h' + 'tnC ; Z + Z' = 180° + 6'= 180° — k — k' 2 mC, GEODETIC SURVEYING. 547 where It and k are the observed altitudes above the horizon. It is evident that every pair of reciprocal observations at two stations will give a value for m. The mean values of as found from observations on the United States Coast Survey in New England, were: Between primary stations, . . , . m — 0.071 For small elevations, m 0-075 For a sea horizon, m — 0.078 On the New York State Survey the value from 137 ob.ser- vations was m — 0.073.* H ~\- In this work also the term — — — in equations (4) to (8) never affected the result by more than of its value. PRECISE SPIRIT-LEVELLING. 401. Precise Levelling differs from ordinary spirit-level- ling both in the character of the instruments used and in the methods of observation and reduction. It is differential levelling over long lines, the elevations usually being referred to mean sea-level. In order that the elevations of inland points, a thousand miles or more from the coast, may be de- termined with accuracy, the greatest care is required to pre- vent the accumulation of errors. In order that triangulation distances may be reduced to sea-level, the elevations of the bases at least must be found. It is impossible to carry eleva- tions accurately from one triangulation-station to another by means of the vertical angles, on account of the great variations in the refraction. Barometric determinations of heights are also subject to great uncertainties unless observations be * See pages 435 and 436 for a case of excessive refraction profitably utilized. 548 SUR VE YING. made for long periods. The only accurate method of finding the elevations of points in the interior above sea-level is by first finding what mean sea-level is at a given point by means of automatic tide-gauge records for several years, and then run- ning a line of precise spirit-levels from this gauge inland and connecting with the points whose elevations are required. Most European countries have inaugurated such .systems of geodetic levelling, this work being considered an integral part of the trigonometrical survey of those countries. In the United States this grade of work was begun on the U. S. Lake Survey in 1875, by carrying a duplicate line of levels from Albany, N. Y., and connecting with each of the Great Lakes. The Mississippi River Commission has carried such a line from Biloxi, on the Gulf of Mexico, to Savannah, Ilk, along the Mississippi River, and thence across to Chicago, connect- ing there with the Lake Survey Elevations.* The U. S. Coast and Geodetic Survey is carrying a line of precise f levels from Sandy Hook, N. J., across the continent, passing through St. Louis, their line here crossing that of the Mississippi River Commission. On all these lines permanent bench-marks are left at intervals of from one to five miles, whose elevations above mean sea-level are determined and published. 402. The Instruments used in precise levelling differ in many respects from the ordinary wye levels used in America. The levelling instrument prescribed by the International Geo- detic Commission held in Berlin in 1864 is shown in Fig. 149. These instruments are made by Kern & Co., of Aarau, Switzerland, and this illustration is almost an exact representa- tion of the instruments used on the U. S. Lake and Mississippi River Surveys.^ The bubble is enclosed in a wooden case (metal case in the cut), and rests on top of the pivots or rings; * The author had charge of about 600 miles of this work, f Called on that service geodesic levels. X This cut is from Fauth’s Catalogue, Washington, D. C. GEODETIC SURVEYING, 549 it is carried in the hand when the instrument is transported. A mirror is provided which enables the observer to read the bubble without moving his eye from the eye-piece. There is a thumb-screw with a very fine thread under one wye which is used for the final levelling of the telescope when pointed on the rod. There are three levelling-screws, and a circular or box level for convenience in setting. The telescope bubble is very Fig. 149. delicate, one division on the scale corresponding to about three seconds of arc. The bubble-tube is chambered also, thus al- lowing the length of the bubble to be adjusted to different temperatures. The magnifying power is about 45 diameters. There are three horizontal wires provided, set at such a distance apart that the wire interval is about one hundredth of the dis- tance to the rod. The tripod legs are covered with white 550 SURVEYING. cloth to diminish the disturbing effects of the sun upon them. The level itself is always kept in the shade while at work. The levelling-rod is made in one piece, three metres long, of dry pine, about four inches wide on the face, and strengthened by a piece at the back, making a T -shaped cross-section. The rods are self-reading, that is, they are without targets, and are graduated to centimetres. An iron spur is provided at bottom which fits into a socket in an iron foot-plate. The end of the spur should be flat and the bottom of the socket turned out to a spherical form, convex upwards. A box-level is attached to the rod to enable the rodman to hold it vertically, and this in turn is adjusted by means of a plumb-line. Two handles are provided for holding the rod, and a wooden tripod to be used in adjusting the rod-bubble. The decimetres are marked on one side of the graduations and the centimetres on the other, all figures inverted since the telescope is inverting. 403. The Instrumental Constants which must be accu- rately determined once for all, but re-examined each season, are — 1. The angular value of one division on the bubble-tube. 2. The inequality in the size of the pivot-rings. 3. The angular value of the wire-interval, or the ratio of the intercepted portion on the rod to the distance of the rod from the instrument. 4. The absolute lengths of the levelling-rods. These constants may be determined as follows: The value of one division of the bubble may be readily found by sighting the telescope on the rod, which is set at a known distance from the instrument, and running the bubble from end to end of its tube, taking rod-readings for each posi- tion of the bubble. The bubble-graduations are supposed to be numbered from the centre towards the ends. GEODETIC SURVEYING. 551 Let = mean of all the eye-end readings of the bubble when it was run to the eye-end of its tube ; ^2 = same for bubble at object-end of tube ; = mean of all the object-end readings when bubble was at eye-end of tube ; (^2 = same for bubble at object-end of tube; = mean reading of rod for bubble at eye-end , i^2 == same for bubble at object-end ; D — distance from instrument to rod ; V — value of one division of the bubble (sine of the angle) at a unit’s distance. Then V = D R.-R. 'E, - O, E, (0 In seconds of arc we would have V (in seconds) = sin I' E, -O, E,~ O, V ■ ( 2 ) If a table is to be prepared for corrections to the rod-read- ings for various distances and deviations of the bubble from the centre of its tube, then the value as given by equation (i) is most convenient to use. The value of one division of a level bubble should be constant, but it is often affected by its rigid fastenings, which change their form from changes in tempera- ture. The inequality in the size of the rings is found by revers- ing the bubble on the rings, and also reversing the telescope in the wyes. The bubble is reversed only in order to eliminate its error of adjustment. The following will illustrate the method of making and reducing the observations: 552 SURVEYING. BunnLR-RRADINCS. Tel. eye-end north. Lev. direct. North. 4.3 South. 5-5 it ti it “ reversed. 4-7 5-2 Tel. eye-end south Lev. direct. 9.0 0.2 (-1.7) — 0.42 10.7 3.7 it it it “ reversed. 6.6 3-3 Tel. eye-end north Lev. direc^. 12.8 4.4 (+5.8) + I -45 7-0 5-5 “ “ south “ reversed. 5-2 Mean reading north “ “ south North minus south = — 0.40 = + 1-45 = - 1.85 9.2 (-1.5) — 0.38 10 7 That is to say, the bubble moves 1.85 divisions towards the object-cud when the telescope is reversed in the wyes. This is evidently twice the inequality of the pivot-rings ; and since the axis of a cone is inclined to one of its elements by one half the angle at the apex, so the line of sight is inclined to the tops of the rings by one fourth of 1.85 divisions, or 0.46 divi- sions of the bubble. It is also evident that the eye-end ring is the smaller, and that therefore when the top surfaces of the rings are horizontal the line of sight inclines downward from the instrument. The correction is therefore positive. This is called the pivot-correction^ and changes only with an unequal wear in the pivot-rings. The angidar value of the wire-interval is found by measur- ing a base on level ground of about 300 feet from an initial point -f-/* in front of the objective. Focus the telescope on a very distant object, and measure the distance from the inside of the objective to the cross-wires, this being the value * See art. 205 for the significance of these terms, as well as for the theory of the problem. GEODETIC SURVEYING. 553 of y for that instrument. Measure the space intercepted on the rod between the extreme cross-wires. \{ d— length of base, counting from the initial point ; s = length of the intercepted portion of the rod ; r ='C = constant ratio of distance to intercept; then r = - ; and for any other intercept s' on the rod we have d' = rs' +/+C (3) When r, /, and c are found, a table can be prepared giving distances in terms of the wire-intervals. T/ie errors m the absolute lengths of the rods affect only the final differences of elevation between bench-marks. This correction is usually inappreciable for moderate heights. 404. The Daily Adjustments. — The adjustments which are examined at the beginning and close of each day’s work are as follows : 1. The collimation, that is, the amount by which the line of sight, as determined by the mean reading of the three wires, deviates from the line joining the centres of the rings. 2. The bubble-adjustment — that is, the inclination of the axis of the bubble to the top surface of the rings. 3. The rod-level. This is examined only at the beginning of each day’s work, and made sufficiently perfect. The first two adjustments are very important, since it is by means of these (in conjunction with the pivot-correction, determined once for the season) that the relation of the bubble 554 SUA' VE Y INC. to the line of siglit is found. It is not customary in this work to try to reduce these errors to zero, but to make them reason- ably small, and then determine iJieir values and correct for them. It is evident that if the back and fore sights be kept exactly equal between bench-marks, then the errors in the instrumental adjustments are fully eliminated ; and in any case these errors can only affect the excess in length of the sum of the one over that of the other. It is to this excess in length of back-sights over fore-sights, or vice versa, that the instru- mental constants are applied ; but in order to apply them their values must be accurately determined. The curvature of a level surface would also enter into this excess, but it is usually so sm.all a residual distance, that the correction for curvature is quite insignificant. There are, however, three instrumental corrections to be applied for the amount of the excess, namely, the corrections for collimation, inclination of bubble, and in- equality of pivots, designated respectively by e, i, and p. Since three horizontal wires are read on the rod, the wire-intervals can be used in place of the distances, for they are linear func- tions practically, and so a record is kept of the continued sum of the lengths of the back and fore sights, and from these the final difference is found. The colliniation-correction is taken out for a distance of one unit (the metre has been universally used in this kind of levelling), and then the correction for any given case found by multiplying by the residual distance. Let = rod-reading for telescope normal ; “ inverted ; d — distance of rod from instrument. 2d Then (I) GEODETIC SURVEYING. 555 « The correction for the inclination of the bubble to the tops of the rings is found by reversing the bubble on the telescope and reading it in both positions. In such observations the initial and final readings are taken with the bubble in the same position, thus giving an odd number of observations. Usually two direct and one reversed reading are taken. The correction is found in terms of divisions on the bubble, the correction in elevation being taken from the table prepared for that purpose. Let — mean of the eye-end* readings for level direct ; E, = “ “ a a “ “ reversed ; 0, = “ “ object (( “ “ direct ; 0 , = “ “ u (( “ “ reversed ; then ( 2 ) The pivot correction has already been found, and is sup* posed to remain constant for the season. If E be the excess of the sum of the back-sights over that of the fore-sights, then the final correction for this excess is ( 3 ) where v is taken from eq. (i), p. 551. Evidently, if the fore- sights are in excess, the correction is of the opposite sign. 405. Field Methods. — The great accuracy attained in pre- cise levelling is due quite as much to the methods used and precautions taken in making the observations as to the instru- mental means employed. Aside from errors of observation and instrumental errors, we have two other general classes of * By eye-end is always meant the end towards the eye-end of the telescope, whether in a direct or a reversed position. 556 SURVEYING. errors, which can be avoided only by proper care being used in doing tlie work. Tliese two classes arc errors from unstable supports and atmospheric errors. Any settling ot the rod between the fore and back readings upon it will result in the final elevation being too high, while any settling of the instrument between the back and fore readings from it will also result in too high a final elevation. Such errors are therefore cumulative, and the only way in which they can be eliminated is to duplicate the work over the same ground in the opposite dircctio 7 i. As a general pre- caution, the duplicate line should always be run in the opposite direction. This will result in larger discrepaficies than if both are run in the same direction, but the mean is nearer the truth. Atmospheric errors may come from wind, heated air-cur- rents causing the object sighted to tremble or “dance,” or from variable refraction.' For moderate winds the instrument may be shielded by a screen or tent, but if its velocity is more than eight or ten miles an hour, work must be abandoned. To avoid the evil effects of an unsteady atmosphere the length of the sights is shortened ; but when a reading cannot be well taken at a distance of about 150 feet, or 50 metres, it would be better to stop, since the errors arising from the number of stations occupied would make the work poor. At about 8 o’clock A.M. and 4 P.M. very large changes in the refraction have been observed on lines over ground which is passing from sun to shade, or vice versa, when the image was apparently very steady. In clear weather not more than three or four hours a day can be utilized for the best work, and sometimes, with hot days and cool nights, it is impossible to get an hour when good work can be done. In making the observations the bubble is brought exactly to the centre of its tube, the observer being able to do this by means of the thumb-screw under one wye, and the mirror which reflects the image of the bubble to the observer at the GEODETIC SURVEYING. 557 eye-piece. If there is no mirror to the bubble, then it is brought approximately to the centre, and the recorder reads it while the observer is reading the three horizontal wires. In any case the bubble-reading is recorded in the note-book, and if it was not in the middle a correction is made for the eccen- tric position by means of a table prepared for the purpose. The mean of the three wire-readings is taken as the reading of that rod, the observer estimating the tenths of the centi- metre spaces, thus reading each wire to the nearest millimetre. The wires should be about equally spaced so that the mean of the three wires coincides very nearly with the middle wire. The differences between the middle and extreme wire-readings are also taken out to give the distance, as well as to check the readings themselves by noting the relation of the two intervals. If they are not about equal, then one or more of the three readings is erroneous. This is a most important check, and constitutes an essential feature of the method. It has been found economical to have two rodmen to each instrument, so that no time shall be lost between the back and fore sight readings from an instrument-station. Since but a small portion of the day can generally be utilized, it is desira- ble to make very rapid progress when the weather is favora- ble. When two rodmen are used, and the air is so steady that lOO-metre sights can be taken,* it is not difficult for an expe- rienced observer to move at the rate of a mile an hour. On the U. S. Coast and Geodetic Survey a much more laborious method of observing than the one above outlined has been followed. There a special kind of target-rod has been employed, the target being set approximately and clamped. The thumb-screw under the wye is used as a mi- crometer-screw, and two readings are taken on it one when * This is about the limiting length of sight for first-class work, even under the most favorable conditions. 558 SURVEYING. the bubble is in the middle and the other when the centre wire bisects the target, the bubble now not being in the middle, since the target’s position was only approximate. The bubble is then reversed, and two more readings of the screw taken. The telescope is now revolved in the wyes, and read- ings taken again with bubble direct and reversed. Thus there are four independent readuigs taken on the rod, each necessi- tating two micrometer-readings. The reduction is also very complicated, each sight being corrected for curvature and re- fraction as well as for instrumental constants. The duplicate line is carried along with the first one by having two sets of turning-points for each instrument-station. The instrument, however, is set but once, so that the lines are not wholly inde- pendent. The alternate sections are run in opposite directions, thus partly obviating the objection to running both lines in the same direction. The method first described was used on the U. S. Lake and Mississippi River surveys, and is also the method used on most of the European surveys of this char- acter. The instrument is always shaded from the sun, both while standing and while being carried between stations. It is abso- lutely necessary to do this in order to keep the adjustments approximately constant, and the bubble from continually moving. 406. Limits of Error. — On the U. S. Coast and Geodetic Survey the limit of discrepancy between duplicate lines is 5mm f 2K where K is the distance in kilometres. On the U. S. Lake Survey the limit was 10"^™ and on the Mississippi River Survey it was f K. These limits are respectively 0.029 0.041, and 0.021 feet into the square root of the distance in miles. If any discrepancies occurred greater than these the stretch had to be run again. The probable error” of the mean of several observations on the same quantity is a function of the discrepancies of the GEODETIC SURVEYING. 559 several results from the mean. If etc., be the several residuals obtained by subtracting the several results from the mean, and if ^[vv] be the sum of the squares of these residu- als, and if m be the number of observations, then the probable error of the mean is 7 ? = ± -6745 '^\yv\ m{m — i) This is the function which is universally adopted for meas- uring the relative accuracy of different sets of observations. If there be but two observations this formula reduces to R^±W. where V is the discrepancy between two results. The European International Geodetic Association have fixed on the following limits of probable error per kilometre in the mean or adopted result: 4 ; 3““ per km. is tolerable; ± per km. is too large ; ± 2"^“ per km. is fair ; and' d: i™"' per km. is a very high degree of pre'cision. On the U. S. coast and geodetic line from Sandy Hook to St. Louis, a dis- tance of 1009 rniles, the probable error per kilometre was dh 1.2™"".* For the 670 miles of this work on the Mississippi River Survey, of which the author had charge, the probable error of the mean for the entire distance was 23.5™”" (less than one inch), and the probable error per kilometre was ± 0.7“™.t Of course very little can be predicated on these results as to the actual errors of the work, since the number of observations on each value was usually but two ; but they may fairly be used for the purpose of comparing the relative accuracy of different lines where this function has been computed from similar data. 407. Adjustment of Polygonal Systems in Levelling. — If * Report U. S, Coast and Geodetic Survey, 1882, p. 522. f Reports of the Miss. Riv. Commission for the years 1882, 1883, and 1884. 560 SU/^ VE Y I NG. a line of levels closes upon itself the summation of all the differ- ences of elevation between successive benches should be zero. If it is not, the residual error must be distributed among the several sides, or stretches, composing the polygon, according to some law, so that the final corrections which arc applied to the several sides shall be independent of all personal considera- tions. These corrections should also be the most probable corrections. There are two general criterions on which to found a theory of probabilities. One may be called a prioriy and the other a posteriori. By the former we would say that the errors made are some function of the distance run, as that they are directly proportional to this distance, or to the square root of this distance, etc.; while by the latter, or a posteriori method, we would say the errors made on the several lines are a function of the discrepancies found between the duplicate measurements on those lines, or to the computed “ probable error per kilometre,” as found from these discrepancies. Both methods are largely used in the adjustment of observations. These laws of distribution are equivalent to establishing a method of weighting the several sides of the system, a larger weight implying that a larger share of the total error is to be given to that side. When any system of weights is fixed upon, then the corrections may be computed by the methods of least squares so as to comply with the condition that the corrections shall be the most probable ones for that system of weighting. The most probable set of corrections is that set the sum of whose squares is a minimum. If the system includes more than a few polygons, this method of reduction is exceedingly laborious, while the increased accuracy is very small over that from a much simpler method. Fig. 150 represents the Bavarian network of geodetic levels, there being four polygons. Every side has been levelled, and the difference of elevation of its extremities found. These ele- vations must now be adjusted so that the differences of eleva- GEODETIC SURVEYING. 561 tion on each polygon shall sum up zero. When these sums are taken the following residuals are found : L, -|- 20.2““ ; II., + 39 - 3 “"'; III., — 25.2"'"' ; and IV., -f- 108. It was sup- ^ posed that an error of one deci- metre had been made in the fourth polygon, but in the ab- sence of any knowledge in the case this error must be distrib- uted with the rest. The method which the au- q( thor would recommend is a modification of Bauernfeind’s, ^ ,, ' riG. 150. in that the errors are to be made proportional to the square roots of the lengths of the sides in- stead of the lengths of the sides directly. Since the errors in levelling are compensating in their nature they would be ex- pected to increase with the square root of the length of the line, and it is the author’s experience that the error is much nearer proportional to the square root of the distance than to the distance itself. Instead of treating the four polygons as one system and solving by least squares, the polygon having the largest error of closure is first adjusted by distributing the error among its sides in proportion to the square roots of the lengths of those sides. Then the polygon having the next largest error is ad- justed, using the new value for the adjusted side, if it is con- tiguous to the former one, and distributing the remaining error among the remaining sides of the figure, leaving the previously adjusted side undisturbed. The adjustment pro- ceeds in this manner until all the polygons are adjusted. The Bavarian system is worked out on this plan in the following tabulated form : 562 SUR VE Y INC. ADJUSTMENT OF THE BAVARIAN SYSTEM OF LEVEL POLYGONS. No. Side. Length. Sq. Root of Length = A. No. Polygon. 2 A. Difference of Elevation. Error of Closure Cor- rected Error of Closure Cor- rection. Corrected Difference of Elevation. I km. 125.8 II .2 I. 24.6 m. + 35-8723 mm. 20.2 + 3»-3 - 14.3 + 35-8580 2 179.0 13-4 I. — 217.5062 - 17.0 - 217.5232 3 147-3 12. 1 II. ± 181.6541 + 39.3 + 39-3 — II. I ± 181.6652 4 60.6 7.8 II. 43-1 + 32.0958 - 7.1 -f- 32.0887 5 174.0 13-2 II. -f- 179-5981 — 12.0 + 179.5861 6 lOI . I 10. 0 II. 20.9 T 30.0005 — 25.2 + 19-9 - 9-1 T 30.0096 7 134-9 II. 6 III. — 38.6644 — 11.0 - 38.6754 8 80. 1 9.0 IV. T 48.8053 — 36.0 ± 48.7693 9 87.0 9-3 III. + 57-4440 - 8.9 + 57-4351 10 96.8 9.8 IV. 27.0 — 100.1619 108.0 + 108.0 - 39.2 — 100.2011 II 67.9 8.2 IV. + 51-4646 — 32.8 + 51-4318 Beginning with polygon IV., we find its error of closure to be -j- loS.o'"™, this being distributed among the three sides so that goes to side 8,/^ to side 10, and to side ii. The corrected values for these sides are now found. Next take the polygon having the next largest error of closure, which is number II., and distribute its error in like manner. This leaves polygons I. and III. to be adjusted, one side of the former and two of the latter being already adjusted. The corrected errors of closure for these polygons are 31.3'"™ and respectively, the former to be di.stributed between the sides I and 2 and the latter between the sides 7 and 9. The resulting corrected values cause all the polygons to sum up zero. The sum of the squares of the corrections here found is 50.02 square centimetres, whereas if the differences of eleva- tion had been weighted in proportion to the lengths of the sides and the system adjusted rigidly by least squares the sum of the squares of the corrections would have been 49.65 square centimetres, showing that the method here used is practically GEODETIC SURVEYING. 563 as good as the rigid method which is commonly used. It has been found in practice to give, in general, about the same sized corrections as the rigid system. 408. Determination of the Elevation of Mean Tide. — To determine accurately the elevation of mean tide at any point on the coast requires continuous observations by means of an automatic self-registering gauge for a period of several years. The methods of making these observations with cuts of the instruments employed are given in Appendix No. 8 of the U. S. Coast Survey Report for 1876. A float, inclosed in a perforated box, rises and falls with the tide, and this motion, properly reduced in scale by appropriate mechanism, is re- corded by a pencil on a continuous roll of paper which is moved over a drum at a uniform rate by means of clockwork. An outer staff-gauge is read one or more times a day by the at- tendant, who records the height of the water and the time of the observation on the continuous roll. This outer staff is connected with fixed bench-marks in the locality by very careful levelling, and this connection is repeated at intervals to test the stability of the gauge. To find from this automatic record the height of mean tide, ordinates are measured from the datum-line of the sheet to the tide-curve for each hour of the day throughout the entire period. This period should be a certain number of entire lunar months. The mean of all the hourly readings for the period maybe taken as mean tide. It maybe found advisable to reject all readings in stormy weather, in which case the entire lunation should be rejected. CHAPTER XV. PROJECTION OF MAPS, MAP-LETTERING, AND TOPO- GRAPHICAL SYMBOLS. I. PROJECTION OF MAPS. 409. The particular method that should be employed in representing portions of the earth’s surface on a plane sheet or map depends, yfrj/, on the extent of the region to be repre- sented ; second, on the use to be made of the map or chart ; and third, on the degree of accuracy desired. Thus, a given kind of projection may suffice for a small region, but the approximation may become too inaccurate when extended over a large area. It is quite impossible to represent a spherical surface on a plane without sacrificing something in the accuracy of the relative distances, courses, or areas ; and the use to which the chart is to be put must de- termine which of these conditions should be fulfilled at the expense of the others. A great many methods have been proposed and used for accomplishing various ends, some of which will be described. 410. Rectangular Projection. — In this method the merid- ians are all drawn as straight parallel lines ; and the parallels are also straight, and at right angles with the meridians. A central meridian is drawn, and divided into minutes of latitude according to the value of these at that latitude as given in Table XI. Through these points of division draw the paral- lels of latitude as right lines perpendicular to the central meridian. On the central parallel lay off the minutes of PROJECTION OF MAPS. 565 longitude, according to their value for the given latitude, by Table XL; and through these points of division draw the other meridians parallel with the first. The largest error here is in assuming the meridians to be parallel. For the latitude of 40°, two meridians a mile apart will converge at the rate of about a foot per mile. A knowl- edge of this fact will enable the draughtsman to decide when this method is sufficiently accurate for his purpose. Thus, for an area of ten miles square, the distortion at the extreme cor- ners in longitude, with reference to the centre of the map as an origin of coordinates, will be about twenty-five feet. At the equator this method is strictly correct. In this kind of projection, whether plotted from polar or rectangular coordinates, or from latitudes and longitudes, all straight lines of the survey, whether determined by triangula- tion or run out by a transit on the ground, will be straight on the map ; that is, the fore and back azimuth of a line is the same, or, in other words, a straight line on the drawing gives a constant angle with all the meridians. This is the method to use on field-sheets, where the survey has all been referred to a single meridian. 411. Trapezoidal Projection. — Here the meridians are made to converge properly, but both they and the parallels are straight lines. A central meridian is first drawn, and grad- uated to degrees or minutes ; and through these points paral- lels are drawn, as before. Two of these parallels are selected ; one about one fourth the height of the map from the bottom, and the other the same distance from the top. These paral- lels are then subdivided, according to their respective lati- tudes, from Table XI. ; and through the corresponding points of division the remaining meridians are drawn as straight lines. The map is thus divided into a series of trapezoids. The parallels are perpendicular to but one of the meridians. The principal distortion comes from the parallels being drawn as 566 SUR VE YING, straight lines, and amounts to about thirty-two feet in ten miles in latitude 40°, and is nearly proportional to the square of the distance east or west from the central meridian. The work should be plotted from computed latitudes and longitudes. The method is adapted to a scheme which has a system of triangulation for its basis, the geodetic position of the stations having been determined. These conditions would be fulfilled in a State topographical or geological survey for the separate sheets, each sheet covering an area of not more than twenty-five miles square. 412. The Simple Conic .P/oiection.— In this projection, points on a spherical surface are first projected upon the sur- face of a tangent cone, and then this conical surface is devel- oped into the plane of the map. The apex of the cone is taken in the extended axis of the earth, at such an altitude that the cone becomes tangent to the earth’s surface at the middle parallel of the map. When this conical surface is de- veloped into a plane, the meridians are straight lines converg- ing to the apex of the cone, and the parallels are arcs of con- centric circles about the apex as the common centre. The sheet is laid out as follows: Draw a central meridian, and graduate it to degrees or minutes, according to their true values as given in Table XI. Through these points of divi- sion draw parallel circular arcs, using the apex of the cone as the common centre. For values of the length of the side of the tangent cone, which is the length of the central parallel above, see Table XI. The rectangular coordinates of points in these curves are also given in the same table. On the middle parallel of the map the degrees or minutes of longitude are laid off, and through these are drawn the re- maining meridians as straight lines radiating from the apex of the tangent cone. It will be seen that the latitudes are correctly laid off, and the degrees of longitude will be sufficiently accurate for a map PROJECTION OF MAPS. 567 covering an area of several hundred miles square. The merid- ians and parallels are at right angles. In this projection the degrees of longitude on all parallels, except the middle one, are too great ; and therefore the area represented on the map is greater than the corresponding area on the sphere. The chart should be plotted from computed latitudes and longitudes. 413. De ITsle’s Conic Projection. — This is very similar to the above, except that two parallels, one at one fourth, and one at three fourths the height of the map, are properly grad- uated, and the meridians drawn as straight lines through these points of division. The parallels are drawn as concentric cir- cles, as in the simple conic projection. This is therefore but a combination of the second and third methods, and is more accurate than either of them. The cone here is no longer tan- gent, but intersects the sphere in the two graduated parallels. In this case the region between the parallels of intersection is shown too small, and that outside these lines is shown too large ; so that the area of the whole map will correspond very closely to the corresponding area on the sphere. When these parallels are so selected that these areas will be to each other exactly as the scale of the drawing, then it is called “ Mur- doch’s projection.” 414. Bonne’s Projection. — This differs from the simple conic in this — that all the parallels are properly graduated, and the meridians drawn to connect the corresponding points of division in the parallels. These latter are, however, still concentric circles. The meridians are at right angles to the parallels only in the middle portion of the map. The same scale applies to all parts of the chart. There is a slight dis- tortion at the extreme corners, from the parallels being arcs of concentric circles. The proportionate equality of areas is 568 SU/^VEVING. preserved. A rhumb-line appears as a curve ; but when once drawn, its length may be properly scaled. It will be noted that the last three methods involve the use of but one tangent or intersecting cone. 415. The Polyconic Projection. — For very large areas it is preferable to have each parallel the development of the base of a cone tangent in the plane of the given parallel. This projection differs from Bonne’s only in the fact that the parallels are no longer concentric arcs, but each is drawn with a radius equal to the side of the cone which is tangent at that latitude. These, of course, decrease towards the pole ; and therefore the parallels diverge from each other towards the edge of the chart. The result of this is, that a degree of latitude at the side of the map is not equal to a degree on the central meridian ; or, in other words, the same scale cannot be applied to all parts of the map. These defects ap- pear, however, only on maps representing very large areas. The whole of North America could be represented by this method without any material distortion. This method of projection is exclusively used on the Unit- ed States Coast and Geodetic Survey, and for all other maps and charts of large areas in this country. Extensive tables are published by the War and Navy Departments, and also by the Coast Survey, to facilitate the projection of maps by the polyconic system. Table VIII. gives in a condensed form the rectangular coordinates of the points of intersection of the parallels and meridians referred to the intersection of the sev- eral parallels with the central meridian as the several origins. 416. Formulae used in the Projection of Maps.* — The fundamental relations on which the method of polyconic pro- jection rests are given in the following formulae : * See Appendix D for the derivation of equations (i) and (2). PROJECTION OF MAPS. 569 Normal, being the radius of curvature of a section perpendicular to the ^ meridian at a given point N = 7 (i) ^ ^ (l — rsm*Z)^’ where Re is the equational radius, e is the eccentricity, and L is the latitude. Radius of the meridian /p _ pj-k} f) (2) Radius of the parallel (3) Degree of the meridian • • • (4) =: 36ooZ;« sin i'\ Degree of the parallel = IIS'"- • • • (5) = 3600^^ sin Radius of the developed parallel, or side of tangent cone r = iVcot Z. . . . (6) If n be any arc of a parallel, in degrees, or any difference of longitude from the central meridian of* the drawing, and if 6 be the corresponding angle, in degrees, at the vertex of the tangent cone, subtended by the developed parallel, then since the angular value of arcs of given lengths are inversely as their radii, we have 6 n Rp sin L, or 6 = n sin L, ( 8 ) 570 SURVEYING. Since the developed parallels are circular arcs, the rectangu- lar coordinates of any point an angular distance of d from the central meridian is, Meridian distance, d^n — x = r sin < 9 . "j Divergence of parallels, dp = y — r vers 6 . V. . (9) = X tan ) For arcs of small extent, the parallel may be considered coincident with its chord ; but the angle between the axis of x and the chord is If, then, the length of the arc, which is 7iDpj be represented by the chord, we may write d^ — meridian distance x = iiDp cos ^ and dp = divergence of parallels = y = nDp sin ^ 6 . j If, now, dtn-, = meridian distance for i degree of longitude, and d„ift = meridian distance for 71 degrees of longitude. we have d,j^ _ 7 iDp cos \en dm\ Dp cos But 6 n sm Z, so that = 1° X sin L = 38' for latitude 40^ Therefore cos = cos 19' = I, nearly; PROJECTION OF MAPS. 571 For L = 30°, we have sin L = Therefore, for latitude 30°, = n cos \n= n cos (0.25;^), nearly. If we have obtained the meridian distance, for i degree of longitude, and wish to obtain it for n degrees in latitude 30°, we have but to multiply the distance for i degree by n cos (0.2 5 ?2). 417. In Table VIII. the meridian distances are given, at vari- ous latitudes, for a difference of longitude of one degree. To find the meridian distance for an}^ number of degrees or parts of degrees, multiply the distance for one degree by the factor there given for the given latitude. The factor given in the table for latitude 30° is n cos (0.288/2), in place of 71 cos (0.25/2), as obtained above. The difference is a correction which has been introduced to compensate the error ‘made in assuming that the chord was equal in length to its arc. The corrected factors enable the table to be used without material error up to 25 degrees longitude either side of the central meridian. To obtain the divergence of the parallels for differences of longitude more or less than one degree, multiply the diver- gence for one degree by the square of the number of degrees. It is evident that this rule is based on the assumption that the arc of the developed parallel is a parabola, and so it may be considered for a distance of 25 degrees either side of the cen- tral meridian between the latitudes 30° and 50° without mate- rial error. If the whole of the United States were projected by this table, using the factors given, to a scale of i to 1,500,000, thus giving a map some 8 by 10 feet, the maximum deviation of the meridians and parallels from their true positions (which would be at the upper corners) would be but about 0.02 inch. 572 S UK VE Y/iVG. Thus, for a map of this size, covering 20 degrees of lati- tude and 50 degrees of longitude, the geodetic lines would have their true position within the width of a fine pencil line, by the use of Table VIII. Fig. 151 will illus- trate the use of the table in project- ing a map by the polyconic method. The map covers 30 degrees in lati- tude (30° to 60°) and 60 degrees in ^ longitude. The straight line 0 ^ 0 ^ is first drawn through the centre of the map, and graduated ac- cording to the lengths of one degree of latitude, as given in the second column of Table VIII. Through these points of di- vision the lines in\ are drawn in pencil at right angles to the central meridian. On these lines the points etc., are laid off by the aid of the first part of Table VIII. This ta- ble gives the meridian distances when n is less than one degree, as well as when 71 is greater. From the points 7 nx^ 777^, etc.^ the divergence of the parallels is laid off above the lines by the aid of the second portion of Table VIII., thus obtaining the positions of the points etc. The points p mark the intersection of the meridians and parallels ; and these may be drawn as straight lines between these points, provided a sufficient number of such points have been located. The map is then to be plotted upon the chart from computed latitudes and longitudes. 418. Summary. — We have seen that there are, in general, two ways of plotting a map or chart, and two corresponding uses to which it may be put: First. We may plot by a system of plane coordinates, either polar (azimuth and distance) or rectangular (latitudes and departures). This gives a map from which distance, azimuth (referred to the meridian of the map), and areas are correctly determined. PROJECTION OF MAPS. 573 Second. We may plot the map by computed latitudes and longitudes, and determine from it the relative position of points in terms of their latitude and longitude. The first system is adapted to small field sheets and detail charts for which the notes were taken by referring all points to a single point and meridian. For this purpose the system of rectangular projection should be selected, as long as the area of the chart is not more than about one hundred square miles. If it be larger than this, the trapezoidal system should be used. In case this is done, the work is still plotted as before, provided it has all been referred to a given meridian in the field work, and then converging meridians are drawn as described above. From such a chart, not only the azimuth (referred to the central meridian) and distance may be deter- mined, but the correct longitude and nearly correct latitude are given. In the case of topographical charts, based on a system of triangulation, each sheet is referred to a meridian passing through a triangulation-station on that sheet, or near to it, and the rectangular system used. In the case of a survey of a long and narrow belt, as for a river, railroad, or canal, if the survey was based on a system of triangulation, the convergence of meridians has been looked after in the computation of the geodetic positions of these stations, and each sheet is plotted by the rectangular system, being referred to the meridian through the adjacent triangulation-station. When many of these are combined into a single map on a small scale, then they must be plotted on the condensed map by latitudes and longitudes, these being taken from the small rectangular projections, and plotted on the reduced chart in polyconic projection ; the meridians and parallels having been laid out as shown above. In case the belt extends mostly east and west, and is not based on a triangulation scheme, then observations for azimuth 574 SU/i! VE Y I NG. should be made as often as every fifty miles. It will not do to run on a given azimuth for this distance, however; for there has been a change in the direction of the parallel (or meridian) in this distance, in latitude 40°, of about 40 minutes. Accord- ing to the accuracy with which the Avork is done, therefore, when running wholly by back azimuths, the setting of the in- strument must be changed. Thus, if in going i degree (53 miles), cast or west, in latitude 40°, the meridian has shifted 40', then in going 13 miles cast or west the meridian has changed 10'; and this is surely a sufficiently large correction to make it worth while to apply it. When running west, this correction is applied in the direc- tion of the hands of a watch, and when running east, in the opposite direction; that is, having run west 13 miles by back azimuth, then the pointing which appears north is really 10' west of north, and the telescope must be shifted 10' around to the right. If the azimuth be corrected in this way, a survey can be carried by back azimuth an indefinite distance, and still have the entire survey referred to the true meridian. 419. The Angle of Convergence of Meridians is the angle 6 in the equations given in the above formula. Then 6 = 11 sin Z,* where n is the angular change in degrees of longitude, and L is the latitude of the place. For Z = 30°, sin or, in latitude 30° a change of longitude of one degree changes the direction of the meridian by 30 minutes. For Z = 40°, sin Z = 0.643 ; or, a change of longitude of one degree changes the direction of the meridian by 0.643 of 60 minutes, or 38.6 minutes, being approximately 40 minutes. For Z = 50°, sin L— o.y 66 \ or, in going east or west one * From Eq. (G), p. 621, when cos \ A L\s taken as unity. MAP.LETTERING AND TOPOGRAPHICAL SYMBOLS. 575 degree, the meridian changes 0.766 X 60 minutes == 46 min- utes, or approximately 50 minutes. Therefore we may have the approximate rule, that a change of longitude of one degree changes the azimuth by as many minutes as equals the degrees of latitude of the place. This rule gives results very near the truth between plus and minus 40° latitude, that is, over an equatorial belt 80 degrees in width. II. MAP-LETTERING AND TOPOGRAPHICAL SYMBOLS. 420. Map-Lettering. — The best-drawn map may have its appearance ruined by the poor skill or bad taste displayed in the lettering. The letters should be simple, neat, and dignified in appearance, and have their size properly proportioned to the subject. The map is lettered before the topographical symbols are drawn. When a map is drawn for popular display, some ornamentation may be allowed in the title ; but even then, the lettering on the map itself should be plain and simple. When the map is for official or professional use, even the title should be made plain. On Plate IV. are given several sets of alphabets which are well adapted to map work. Of course the size should vary according to the scale of the map and the subject, as shown on Plate V. It is a good rule to make all words connected with water in italics. The small letters in stump writing will be found very useful, and these should be practised thorougjily. The italic capitals go with these small letters also. In place of the system of letters above described, and which has heretofore been almost exclusively used for map- ping purposes, a new system, called “ round writing,” may be used. A text-book on this method, by F. Soennecken, is pub- lished by Messrs. Kueffel & Esser, New York. This work is done with blunt pens, all lines being made with a single stroke. 576 SU/^VEV/NG. It looks well when well done, and requires but a small fraction of the time required to make the ordinary letters, h'or work- ing drawings and field maps it is especially adapted. 421. Topographical Symbols, — In topographical repre- sentation, where elevations have been taken sufficiently num- erous and accurate, the outline of the ground should be rep- resented by contours rather than by hachurcs, or hill shading, which simply gives an approximate notion of the slope of the ground, but no indication of its actual elevation. Where the ground has so steep a slope that the contour lines would fall one upon another, it is well here to put in shading-lines, as shown on Plate III. The water surfaces and streams may be water-lined in blue, or left white. The contour lines over al- luvial ground should be in brown (crimson and burnt sienna), while those over rocky and barren ground should be in black. This is a very simple and effective method of showing the character of the soil. The practices of the government surveys should be fol- lowed in the matter of conventional surface representation, such as meadow, swamp, woodland, prairie, cane-brake, etc., with all their varieties. Some of these are given in the United States Coast Survey Report for 1879 and 1883, while Plate III. shows most of those used on the Mississippi River Survey. Those shown in Plate II. are adapted to higher latitudes, and are those used in the field-practice surveys at Washington University. This plate is an exact copy of one of the annual maps made from actual surveys by the Sophomore class. On these the contours are all in black, for the purpose of photo- lithographing. PLATE 1. Ining space each year, except on the Pacific coast, where ISOCONIC CHART FOR 1885 . Reduced from U. S. Coast ard Geodetic PLATE 1 litu(le_ Vj&st- from j Greenu/ich. fOVAl. / '?VENn\ LincoiIn ICOCOY ENVER LOUIS ^es^oni SCALE OF STATUTE mI RAn\mC/^ALLY a CO., iA/GR’S, CHICAGO. NOTE.-AII isogonic lines are moving towards the left westerly) , at an average rate of one-twentieth ^1-20) the intervening space each year, except on the Pacific coast, where there is a very slow movement in. the opposite direction. 1 ' — 1 1 TOPOGRAPHICAF. PRACTICE SURVEY 1886 SW EET SPRINGS MO. by the CLASS POL Y 1’E( H N 1 C SCIK ) OL of WASl 1 lNGr( )N I'N f VERSI T Y »giS^j a^&g * fOe© 'i viT-r® .'A ^.'« It b;«5ii»!» ‘“Ml '"Gt,^S^". (. (earxn^ £iir-»;fcfeg«Kyk 1“.' t 'V “v V " ' •■'■'v '“'/(f^ V'V'C j|''Va ■■^■. ' ^ ,. ^ ”» "i “fli'®' ^ a"," .. ,,* ® 4 :^^^:>>>>>v •/.■4 ' \ ^,*'* ".'* ^ "'* '^T 3 ■ '-j.-^ima' ^ \ ' > '%''i :_z 5 ’■^''i' "% '% '^’-i'l f • ■?)'‘\ \ v‘j i. u'^T^ % '■» "» "% "'i "^1 [£_ I " ^ ^ ^ ]^.''l p \\ \ Yotloh ^ "'^‘■v'l It:.. ar'5V ■■3’'} ! "a ^Vv%^4AjcXi-^ -» '3.'l 1^ % \‘'%-‘V»"3."V', »'j ;J;U • ":j 5,"^ '» '»• 'v '^'l ^ f "^'ij. W \'n ' V ''i '+'1 ^t; I ^'^''‘i'A j^j "4 A "1 “n '» ( py. MTid, /‘as In f-e; ConveiitioTial Si^iis Tor T 01^0 GRAPHICAL MAPS, SCAI>E 1: 10000 . T) e signed for Photo -Ptho^aphin^. PLATE m. :«iva»utKi;iiDa(foit Pvavna and engrave ^ bj EDWARD MOLITOR, T.E. APPENDICES. APPENDIX A. THE JUDICIAL FUNCTIONS OF SURVEYORS. BY JUSTICE COOLEY OF THE MICHIGAN SUPREME COURT. When a man has had a training in one of the exact sciences, where every problem within its purview is supposed to be susceptible of accu- rate solution, he is likely to be not a little impatient when he is told that, under some circumstances, he must recognize inaccuracies, and govern his action by facts which lead him away from the results which theoreti- cally he ought to reach. Observation warrants us in saying that this re- mark may frequently be made of surveyors. In the State of Michigan all our lands are supposed to have been surveyed once or more, and permanent monum.ents fixed to determine the boundaries of those who should become proprietors. The United States, as original owner, caused them all to be surveyed once by sworn officers, and as the plan of subdivision was simple, and was uniform over a large extent of territory, there should have been, with due care, few or no mistakes; and long rows of monuments should have been perfect guides to the place of any one that chanced to be missing. The truth unfortunately is that the lines were very carelessly run, the monuments inaccurately placed ; and, as the recorded witnesses to these were many times wanting in permanency, it is often the case that when the monument was not correctly placed it is impossible to determine by the record, with the aid of anything on the ground, where it was located. The incorrect record of course becomes worse than useless when the witnesses it refers to have disappeared. It is, perhaps, generally supposed that our town plats were more ac- curately surveyed, as indeed they should have been, for in general there can have been no difficulty in making them sufficiently perfect for all practical purposes. Many of them, however, were laid out in the woods; some of them by proprietors themselves, without either chain or com- pass, and some by imperfectly trained surveyors, who, when land was cheap, did not appreciate the importance of having correct lines to deter- mine boundaries when land should become dear. The fact probably is that town surveys are quite as inaccurate as those made under authority of the general government. It is now upwards of fifty years since a major part of the public sur- veys in what is now the State of Michigan were made under authority of 58 o SUN VE YING. the United States. Of tlie lands south of Lansing, it is now forty years since the major part were sold and the work of improvement begun. A generation has passed away since tliey were converted into cultivated farms, and few if any of the original corner and quarter stakes now re- main. The corner and quarter stakes were often nothing but green sticks driven into the ground. Stones might be put around or over these if they were handy, but often they were not, and the witness trees must be relied upon after the stake was gone. Too often tlic first settlers were careless in fi.xing their lines with accuracy while monunuMits remained, and an irregular brush fence, or something equally unirustwortliy, may have been relied upon to keep in mind where the blazed line once was. A fire running through this might sweep it away, and if nothing were sub- stituted in its j)lace, the adjoining proprietors might in a few years be found disputing over their lines, and perhaps rushing into litigation, as soon as they had occasion to cultivate the land along the boundary. If now the disputing parties call in a surveyor, it is not likely that any one summoned would doubt or question that his duty was to find, if possible, the place of the original stakes which determined the boundary line between the proprietors. However erroneous may have been the original survey, the monuments that were set must nevertheless govern, even though the effect be to make one half-quarter* section ninety acres and the one adjoining but seventy; for parties buy or are supposed to buy in reference to those monuments, and are entitled to what is within their lines, and no more, be it more or less. Mclver ?/. lVa//cer,4 Whea- ton’s Reports, 444; Land Co. v. Satinders, 103 U. S. Reports, 316; Cot- tingha 7 n v. Parr, 93 111 . Reports. 233; Bimtoji v. Cardwell, 53 Texas Re- ports, 408: lVaiso 7 i V. Jones, 85 Penn. Reports, 117. While the witness trees remain there can generally be no difficulty in determining the locality of the stakes. When the witness trees are gone, so that there is no longer record evidence of the monuments, it is remarkable how many there are who m.istake altogether the duty that now devolves upon the surveyor. It is by no means uncommon that we find men whose theoretical education is supposed to make them experts who think that when the monuments are gone, the only thing to be done is to place new monuments where the old ones should have been, and where they would have been if placed correctly. This is a serious mis- take. The problem is now the same that it was before : to ascertain, by the best lights of which the case admits, where the original lines were. The mistake above alluded to is supposed to have found expression in our legislation ; though it is possible that the real intent of the act to which we shall refer is not what is commonly supposed. An act passed in 1869, Compiled Laws, § 593, amending the laws re- specting the duties and powers of county surveyors, after providing for the case of corners which can be identified by the original field-notes or other unquestionable testimony, directs as follows ; “ Second. Extinct interior section-corners must be re-established at the intersection of two right lines joining the nearest known points on the original section lines east and west and north and south of it. APPENDIX A. 581 ''Third. Any extinct quarter-section corner, except on fractional lines, must be re-established equidistant and in a right line between the section corners; in all other cases at its proportionate distance between the nearest original corners on the same line.” The corners thus determined, the surveyors are required to perpetu- ate by noting bearing trees when timber is near. To estimate properly this legislation, we must start with the admit- ted and unquestionable fact that each purchaser from government bought such land as was within the original boundaries, and unquestionably owned it up to the time when the monuments became extinct. If the monument was set for an interior-section corner, but did not happen to be “ at the intersection of two right lines joining the nearest known points on the original section lines east and west and north and south of it.” it nevertheless determined the extent of his possessions, and he gained or lost according as the mistake did or did not favor him. It will probably be admitted that no man loses title to his land or any part thereof merely because the evidences become lost or uncertain. It may become more difficult for him to establish it as against an adverse claimant, but theoretically the right remains; and it remains as a poten- tial fact so long as he can present better evidence than any other person. And it may often happen that, notwithstanding the loss of all trace of a section corner or quarter stake, there will still be evidence from which any surveyor will be able to determine with almost absolute certainty where the original boundary was between the government subdivisions. There are two senses in which the word extinct may be used in this connection : one the sense of physical disappearance ; the other the sense of loss of all reliable evidence. If the .statute speaks of extinct corners in the former sense, it is plain that a serious mistake was made in supposing that surveyors could be clothed with authority to establish new corners by an arbitrary rule in such cases. As well might the stat- ute declare that if a man lose his deed he shall lose his land altogether. But if by extinct corner is meant one in respect to the actual location of which all reliable evidence is lost, then the following remarks are per- tinent; 1. There would undoubtedly be a presumption in such a case that the corner was correctly fixed by the government surveyor where the field -notes indicated it to be. 2. But this is only a presumption, and may be overcome by any satis- factory evidence showing that in fact it was placed elsewhere. 3. No statute can confer upon a county surveyor the power to “estab- lish ” corners, and thereby bind the parties concerned. Nor is this a question merely of conflict between State and Federal law ; it is a ques- tion of property right. The original surv^eys must govern, and the laws under which they were made must govern, because the land was bought in reference to them ; and any legislation, whether State or Federal, that should have the effect to change these, would be inoperative, because disturbing vested rights, 4. In any case of disputed lines, unless the parties concerned settle the controversy by agreement, the determination of it is necessarily a 582 S UR VE YING. judicial act, and it must proceed upon evidence, and ^ivc full oppor- tunity for a hcarinjT. No arl)itrary rules of survey or of evidence can be laid down whereby it can be adjudged. The general duty of a surveyor in such a case is plain enough. lie is not to assume that a monument is lost until after he has thoroughly sifted the evidence and found himself unable to trace it. Kven then he should hesitate long before doing anything to the di.sturbance of settlerl possessions. Occupation, especially if long continued, often alTords very satisfactory evidence of the original boundary when no other is attain- able : and the surveyor should inquire when it originated, how, and why the lines were then located as they were, and whether a claim of title has always accompanied the po.ssession, and give all the facts due force as evidence. Unfortunately, it is known that surveyors sometimes, in supposed obedience to the State statute, disregard all evidences of occu- pation and claim of title, and plunge whole neighborhoods into quarrels and litigation by assuming to “establish ” corners at points with which the previous occupation cannot harmonize. It is often the case that where one or more corners are found to be extinct, all parties concerned have acquiesced in lines which were traced by the guidance of some other corner or landmark, which may or may not have been trustworthy; but to bring these lines into discredit when the people concerned do not question them not only breeds trouble in the neighborhood, but it must often subject the surveyor himself to annoyance and perhaps discredit, since in a legal controversy the law as well as common-sense must declare that a supposed boundary line long acquiesced in is better evidence of where the real line should be than any survey made after the original monuments have disappeared. Stewart vs. Carleto?i, Mich. Reports, 270; Diehl vs. Zaiiger, 39 Mich. Reports, 601 ; Dupont vs. Starrhig, 42 Mich. Reports, 492. And county surveyors, no more than any others, can conclude parties by their surveys. The mischiefs of overlooking the facts of possession must often appear in cities and villages. In towns the block and lot stakes soon disappear; there are no witness trees and no monuments to govern except such as have been put in their places, or where their places were supposed to be. The streets are likely to be soon marked off by fences, and the lots in a block will be measured off from these, without looking farther. Now it may perhaps be known in a particular case that a certain monument still remaining was the starting-point in the original survey of the town plat; or a surveyor settling in tlie town may take some central point as the point of departure in his surveys, and assuming the original plat to be accurate, he will then undertake to find all streets and all lots by course and distance according to the plat, measuring and estimating from his point of departure. This procedure might unsettle every line and every monument existing by acquiescence in the town ; it would be very likely to change the lines of streets, and raise controversies everywhere. Yet this is what is sometimes done ; the surveyor himself being the first person to raise the disturbing questions. Suppose, for example, a particular village street has been located by acquiescence and use for many years, and the proprietors in a certain APPENDIX A. 583 block have laid off their lots in reference to this practical location. Two lot-owners quarrel, and one of them calls in a surveyor that he may be sure that his neighbor shall not get an inch of land from him. This surveyor undertakes to make his survey accurate, whether the original was, or not. and the first result is, he notifies the lot-owners that there is error in the street line, and that all fences should be moved, say, one foot to the east. Perhaps he goes on to drive stakes through the block ac- cording to this conclusion. Of course, if he is right in doing this, all lines in the village will be unsettled ; but we will limit our attention to the single block. It is not likely that the lot-owners generally will allow the new survey to unsettle their possessions, but there is always a prob- ability of finding some one disposed to do so. We shall then have a lawsuit; and with what result.? It is a common error that lines do not become fixed by acquiescence in a less time than twenty years. In fact, by statute, road lines maybe- come conclusively fixed in ten years; and there is no particular time that shall be required to conclude private owners, where it appears that they have accepted a particular line as their boundary, and all concerned have cultivated and claimed up to it. McNamara vs. Seato?i, 82 111 . Re- ports, 498; Biince vs. Bidivell, 43 Mich. Reports, 542. Public policy re- quires that such lines be not lightly disturbed, or disturbed at all after the lapse of any considerable time. The litigant, therefore, who in such a case pins his faith on the surveyor, is likely to suffer for his reliance, and the surveyor himself to be mortified by a result that seems to im- peach his judgment. Of course nothing in what has been said can require a surveyor to conceal his own judgment, or to report the facts one way w^hen he be- lieves them to be another. He has no right to mislead, and he may rightfully express his opinion that an original monument was at one place, when at the same time he is satisfied that acquiescence has fixed the rights of parties as if it were at another. But he would do mischief if he were to attempt to “ establish” monuments which he knew would tend to disturb settled rights; the farthest he has a right to go, as an officer of the law, is to express his opinion where the monument should be, at the same time that he imparts the information to those who em- ploy him, and who might otherwise be misled, that the same authority that makes him an officer and entrusts him to make surveys, also allows parties to settle their own boundary lines, and considers acquiescence in a particular line or monument, for any considerable period, as strong, if not conclusive, evidence of such settlement. The peace of the com- munity absolutely requires this rule. Joyce vs. Williams, 26 Mich. Re- ports, 332. It is not long since that, in one of the leading cities of the State, an attempt was made to move houses two or three rods into a street, on the ground that a survey under which the street had been located for many years had been found on more recent survey to be erroneous. From the foregoing it will appear that the duty of the surveyor where boundaries are in dispute must be varied by the circumstances. i. He is to search for original monuments, or for the places where they were 584 SURVEYING, originally located, and allow these to control if he finds thetn, unless he has reason to believe tliat agreements of the parties, express or implied, have rendered them unimportant. By monuments in the case of gov- ernment surveys we mean of course the corner and quarter stakes: blazed lines or marked trees on the lines are not monuments; tliey are merely guides or finger-posts, if we may use tlie expression, to inform ns with more or less accuracy where the monuments may be found. 2. If the original monuments are no longer discoverable, the question of loca- tion becomes one of evidence merely. It is merely idle for any State statute to direct a surveyor to locate or “establisli ’ a corner, as the place of the original monument, according to some inflexible rule. The sur- veyor on the other hand must inquire into all the facts ; giving due prom- inence to the acts of parties concerned, and always keeping in mind, first, that neither his opinion nor his survey can be conclusive upon parties concerned ; second, tliat courts and juries may be required to fol- low after the surveyor over the same ground, and that it is exceedingly desirable that he govern his action by the same lights and rules that will govern theirs. On town plats if a surplus or deficiency appears in a block, when the actual boundaries are compared with the original figures, and there is no evidence to fix the exact location of the stakes which marked the division into lots, the rule of common-sense and of law is that the surplus or deficiency is to be apportioned between the lots, on an assumption that the error extended alike to all parts of the block. O' Brien vs. McGra 7 ie, 29 Wis. Reports, 446 ; Qumnm vs. Reixers, 46 Mich. Reports, 605. It is always possible when corners are extinct that the surveyor may usefully act as a mediator between parties, and assist in preventing legal controversies by settling doubtful lines. Unless he is made for this pur- pose an arbitrator by legal submission, the parties, of course, even if they consent to follow his judgment, cannot, on the basis of mere consent, be compelled to do so: but if he brings about an agreement, and they carry it into effect by actually conforming their occupation to his lines, the action will conclude them. Of course it is desirable that all such agree- ments be reduced to writing; but this is not absolutely indispensable if they are carried into effect without. Meander Lines. — The subject to which allusion will now be made is taken up with some reluctance, because it is believed the general rules are familiar. Nevertheless it is often found that surveyors misapprehend them, or err in their application; and as other interesting topics are somewhat connected with this, a little time devoted to it will probably not be altogether lost. The subject is that of meander lines. These are lines traced along the shores of lakes, ponds, and considerable rivers as the measures of quantity when sections are made fractional by such waters. These have determined the price to be paid when government lands were bought, and perhaps the impression still lingers in some minds that the meander lines are boundary lines, and all in front of them remains unsold. Of course this is erroneous. There was never any doubt that, except on the large navigable rivers, the boundary of the owners of the banks is the middle line of the river; and while some APPENDIX A. 585 courts have held that this was the rule on all fresh-water streams, large and small, others have held to the doctrine that the title to the bed of the stream below low-water mark is in the State, while conceding to the owners of the banks all riparian rights. The practical difference is not very important. In this State the rule that the centre line is the bound- ary line is applied to all our great rivers, including the Detroit, varied somewhat by the circumstance of there being a distinct channel for navigation in some cases with the stream in the main shallow, and also sometimes by the existence of islands. The troublesome questions for surveyors present themselves when the boundary line between two contiguous estates is to be continued from the meander line to the centre line of the river. Of course the original sur- vey supposes that each purchaser of land on the stream has a water-front of the length shown by the field-notes ; and it is presumable that he bought this particular land because of that fact. In many cases it now happens that the meander line is left some distance from the shore by the gradual change of course of the stream or diminution of the flow of water. Now the dividing line between two government subdivisions might strike the meander line at right angles, or obliquely ; and in some cases, if it were continued in the same direction to the centre line of the river, might cut off from the water one of the subdivisions entirely, or at least cut it off from any privilege of navigation, or other valuable use of the water, while the other might have a water-front much greater than the length of a line crossing it at right angles to its side lines. The effect might be that, of two government subdivisions of equal size and cost, one would be of very great value as water-front property, and the other comparatively valueless. A rule which would produce this re- sult would not be just, and it has not been recognized in the law. Nevertheless it is not easy to determine what ought to be the correct rule for every case. If the river has a straight course, or one nearly so, every man’s equities will bp preserved by this rule ; Extend the line of division between the two parcels from the meander line to the centre line of the river, as nearly as possible at right angles to the general course of the river at that point. This will preserve to each man the water front which the field-notes Indicated, except as changes in the water may have affected it, and the only inconvenience will be that the division line be- tween different subdivisions is likely to be more or less deflected where it strikes the meander line. This is the legal rule, and it is not limited to government surveys, but applies as well to water lots which appear as such on town plats. Bay City Gas Light Co. v. The I Jidnstrial Works, 28 Mich. Reports, 182. It often happens, therefore, that the lines of city lots bounded on navigable streams are deflected as they strike the bank, or the line where the bank was when the town was first laid out. When the stream is very crooked, and especially if there are short bends, so that the foregoing rule is incapable of strict application, it is sometimes very difficult to determine what shall be done; and in many cases the surveyor may be under the necessity of working out a rule for himself. Of course his action cannot be conclusive; but if he adopts one 586 SU/^ VE YING. that follows, as nearly as the circumstances will admit, the j:^cncral rule above indicated, so as to divide as near as may be the bed of the stream amonj^ the atljoininj^ owners in proportion to their lines u[)on the shore, his division, beinp^ that of an expert, macle upon the ground aiul with all available lights, is likely to be adopted as law for the case. Judicial de- cisions, into which the surveyor would find it ])rudent to look under such circuitistances, will throw lit(ht upon his duties anrl may constitute a suf- ficient ^uide when peculiar cases arise. Each riparian lot-owner ought to have a line on the legal boundary, namely, the centre line of the stream, proportioned to the length of his line on the shore; and the problem in each case is, how this is to be given him. Alluvion, when a river imper- ceptibly changes its course, will be apportioned by the same rules. The existence of islands in a stream, when the middle line constitutes a boundary, will not affect the apportionment unless the islands were surveyed out as government subdivisions in the original admeasurement. Wherever that was the case, the purchaser of the island divides the bed of the stream on each side with the owner of the bank, and his rights also extend above and below the solid ground, and are limited by the peculiarities of the bed and the channel. If an islanrl was not surveyed as a government subdivision previous to the sale of the bank, it is of course impossible to do this for the purposes of government sale afterwards, for the reason that the rights of the bank owners are fixed by their purchase: when making that, they have a right to understand that all land between the meander lines, not separately surveyed and sold, will pass with the shore in the government sale ; and having this right, anything which their purchase would include under it cannot afterward be taken from them. It is believed, however, that the federal courts would not recog- nize the applicability of this rule to large navigable rivers, such as those uniting the great lakes. On all the little lakes of the State which are mere expansions near their mouths of the rivers passing through them — such as the Muskegon, Pere Marquette and Manistee — the same rule of bed ownership has been judicially applied that is applied to the rivers themselves ; and the divi- sion lines are extended under the water in the same way. Rice v. Ruddi- man, lo Mich., 125 . If such a lake were circular, the lines would con- verge to the centre: if oblong or irregular, there might be a line in the middle on which they would terminate, whose course would bear some relation to that of the shore. But it can seldom be important to follow the division line very far under the water, since all private rights are sub- ject to the public rights of navigation and other use, and any private use of the lands inconsistent with these would be a nuisance, and punishable as such. It is sometimes important, however, to run the lines out for some considerable distance, in order to determine where one may law- fully moor vessels or rafts, for the winter, or cut ice. The ice crop that forms over a man’s land of course belongs to him. Lo7'ina7i v. Be7iso7i, 8 Mich., 18 ; People s Ice Co. v. Stea777er Excelsior, recently decided. What is said above will show how unfounded is the notion, which is sometimes advanced, that a riparian proprietor on a meandered river may lawfully raise the water in the stream without liability to the proprietors APPENDIX A. 587 above, provided he does not raise it so that it overflows the meander line. The real fact is that the meander line has nothing to do with such a case, and an action will lie whenever he sets back the water upon the proprie- tor above, whether the overflow be below the meander lines or above them. As regards the lakes and ponds of the State, one may easily raise questions that it would be impossible for him to settle. Let us suggest a few questions, some of which are easily answered, and some not : 1. To whom belongs the land under these bodies of water, where they are not mere expansions of a stream flowing through them ? 2. What public rights exist in them ? 3. If there are islands in them which were not surveyed out and sold by the United States, can this be done now.? Others will be suggested by the answers given to these. It seems obvious that the rules of private ownership which are applied to rivers cannot be applied to the great lakes. Perhaps it should be held that the boundary is at low-water mark, but improvements beyond this would only become unlawful when they became nuisances. Islands in the great lakes would belong to the United States until sold, and might be surveyed and measured for sale at any time. The right to take fish in the lakes, or to cut ice, is public like the right of navigation, but is to be exercised in such manner as not to interfere with the rights of shore owners. But so far as these public rights can be the subject of ownership, they belong to the State, not to the United States ; and, so it is believed, does the bed of a lake also. Pollard v. Hagan, 3 Howard’s U. S. Reports. But such rights are not generally considered proper subjects of sale, but, like the right to make use of the public highways, they are held by the State in trust for all the people. What is said of the large lakes may perhaps be said also of many of the interior lakes of the State ; such, for example, as Houghton, Higgin.s, Cheboygan, Burt’s, Mullet, Whitmore, and many others. But there are many little lakes or ponds which are gradually disappearing, and the shore pr opi ietor ship advances pari j^>assu as the waters recede. If these are of any considerable size — say, even a mile across — there may be ques- tions of conflicting rights which no adjudication hitherto made could settle. Let any surveyor, for example, take the case of a pond of irregu- lar form, occupying a mile square or more of territory, and undertake to determine the rights of the shore proprietors to its bed when it shall totally disappear, and he will find he is in the midst of problems such as probably he has never grappled with, or reflected upon before. But the general rules for the extension of shore lines, which have already been laid down, should govern such cases, or at least should serve as guides in their settlmeent. Hole. — Since this address was delivered some of these questions have received the attention of the Supreme Court of Michigan m the cases of Richardson.v, Prentiss, 48 Mich. Reports, 88, and Backus V. Detroit, Albany Law Journal, vol, 26, p. 428, Where a pond is so small as to be included within the lines of a pri- vate purchase from the government, it is not believed the public have any rights in it whatever. Where it is not so included, it is believed they have 588 SUR VE YING. rights of fishery, rights to take ice and water, and rights of navigation for business or pleasure. Tliis is tlie common belief, atifl probably the just one. Shore rights must not be so exercised as to disturb these, and tlie States may pass all proper laws for their protection. It would be easy with suitable legislation to preserve these litlle bodies of water as perma- nent places of resort for the pleasure and recreation of the people, and there ought to be such legislation. If the State should be recognized as owner of the beds of these small lakes and ponds, it would not be owner for the purpose of selling. It would be owner only as a trustee for the public use; and a sale would be inconsistent with the right of the bank owners to make use of the water in its natural condition in connection with their estates. Some of them might be made salable lands by draining ; but the State could not drain, even for this purpose, against the will of the shore owners, unless their rights were appropriated and paid for. Upon many questions that might arise between the State as owner of the bed of a little lake and the shore owners, it would be presumptuous to express an opinion now, and fortunately the occasion does not require it. I have thus indicated a few of the questions with which surveyors may nowand then have occasion to deal, and to which they should bring good sense and sound judgment. Surveyors are not and cannot be judicial officers, but in a great many cases they act in a gtiasi judicial capacity with the acquiescence of parties concernerJ : and it is important for them to know by what rules they are to be guided in the discharge of their judicial functions. What I have said cannot contribute much to their enlightenment, but I trust will not be wholly without value. APPENDIX B. INSTRUCTIONS TO U. S. DEPUTY MINERAL SURVEYORS. FOR THE DISTRICT OF COLORADO. (1886.) GENERAL RULES. 1. All official communications must be addressed to the Surveyor-Gen- eral. You will always refer to the date and subject-matter of the letter to which you reply, and when a mineral claim is the subject of corre- spondence, you will give the name, ownership and survey number. 2. You should keep a complete record of each survey made by you, and the facts coming to your knowledge at the time, as well as copies of all your field-notes, reports and official correspondence, in order that such evidence may be readily produced when called for at any future time. 3. Field-notes and other reports must be written in a clear and legible hand, and upon the proper blanks furnished by this office. No cut sheets, interlineations or erasures will be allowed ; and no abbreviations or sym- bols must be used, except such as are indicated in the specimen field- notes. 4. No return by you will be recognized as official unless made in pur- suance of a special order from this office. 5. The claimant is required, in all cases, to make satisfactory arrange- ments with you for the payment for your services and those of your assistants in making the survey, as the United States will not be held responsible for the payment of the same. You will call the attention of applicants for mineral-survey orders to the requirements of the circular of this date in the appendix. 6. You will promptly notify this office of any change in your post-office address. Upon permanent removal from the State, you are expected to resign your appointment. NOT TO ACT AS ATTORNEY. 7. You are precluded from acting, either directly or indirectly, as at- torney in mineral claims. Your duty in any particular case ceases when you have executed the survey and returned the field-notes and prelimi- nary plat, with your report to the Surveyor-General. You will not be al- lowed to prepare for the mining claimant the papers in support ol his 590 VE YING. application for patent, or otherwise perform the duties of nn attorney before the land office in connection with a mininjr claim. You arc not permitted to combine the duties of surveyor and notary-public in tlic same case by administering oaths to the parties in interest. In short, you must have absolutely nothing to do with the case except in your official capacity as surveyor. You will make no survey of a mineral claim in which you hold an interest. THE FIELD-WORK. 8. The survey made and reported must, in every case, be an actual sur- , vey on the ground in full detail, made by you in person after the receipt of the order, and without reference to any knowledge you may have pre- viously acquired by reason of having made the location-survey or other- wise, and must show the actual facts existing at the time. If the season of the year, or any other cause, renders such personal examination im- possible, you will postpone the survey, and under no circumstances rely upon the statements or surveys of other parties, or upon a former exami- nation by yourself. The term survey in these instructions applies not only to the usual field-work, but also to the examinations required for the preparation of your affidavits of five hundred dollars expenditure, descriptive reports on placer-claims and all other reports. SURVEY AND LOCATION. 9. The survey must be made in strict conformity with, or be embraced within, the lines of the record of location upon which the order is based. If the survey and location are identical, that fact must be clearlv and distinctly stated in your field-notes. If not identical, a bearing and dis- tance must be given from each established corner of the survey to the corresponding corner of the location. The lines of the location, as found upon the ground, must be laid down upon the preliminary plat in such manner as to contrast and show their relation to the lines of the survey. 10. If the record of location has been made prior to the passage of the mining act of May 10, 1872, and is not sufficiently definite and certain to enable you to make a correct survey therefrom, you are required, after reasonable notice in writing, to be served personally or through the United States mail on the applicant for survey and adjoining claimants, whose residence or post-office address you may know, or can ascertain by the exercise of reasonable diligence, to take testimony of neighboring claimants and other persons who are familiar with the boundaries there- of as originally located and asserted by the locators of the claim, and after having ascertained by such testimony the boundaries as originally established, you will make a survey in accordance therewith, and trans- mit full and correct returns of the survey, accompanied by the copy of the record of location, the testimony, and a copy of the notice served on the claimant and adjoining proprietors, certifying thereon when, in what manner, and on whom service was made. 11. If the location has been made subsequent to the passage of the APPENDIX B. 591 mining act of May 10, 1872, and the law has been complied with in the mat- ter of marking the location on the ground and recording the same, and any question should arise in the execution of the survey as to the iden- tity of monuments, marks, or boundaries which cannot be determined by a reference to the record, you are required to take testimony in the man- ner hereinbefore prescribed for surveys of claims located prior to May 10, 1872, and having thus ascertained the true and correct boundaries origi- nally established, marked and recorded, you will make the survey accord- ingly. 12. In accordance with the principle that courses and distances must give way when in conflict with fixed objects and monuments, you will not, under any circumstances, change the corners of the location for the purpose of making them conform to the description in the record. If the difference from the location be slight, it may be explained in the field- notes, but if there should be a wide discrepancy, you will report the facts to this office and await further instructions. INSTRUMENT. 13. All mineral surveys must be made with a SOLAR transit, or other instrument operating independently of the magnetic needle, and all courses must be referred to the true meridian. It is deemed best that a solar transit should be used under all circumstances. The varia- tion should be noted at each corner of the survey. CONNECTIONS. 14. Connect corner No. i of your survey by course and distance with some corner of the public survey or with a United States location-mon- ument, if the claim lies within two miles of such corner or monument. If both are within the required distance, you will connect with the near- est corner of the public survey. LOCATION-MONUMENTS. 15. In case your survey is situated in a district where there are no corners of the public survey and no monuments within the prescribed limits, you will proceed to establish a mineral monument, in the location of which you will exercise the greatest care to insure permanency as to site and construction. 16. The site, when practicable, should be some prominent point visi- ble for a long distance from every direction, and should be so chosen that the permanency of the monument will not be endangered by snow, rock or land slides, or other natural causes. 17. The location-monument should consist of a post eight feet long and six inches square set three feet in the ground, and protected by a well-built conical mound of stone three feet high and six feet base. The letters U. S. L. M. followed by a na 7 ne must be scribed on the post and also chiselled on a large stone in the mound, or on the rock in place that may form the base of the monument. There is no objection to the establishment of a location-monument of larger size, or of other material of equally durable character. 18. From the monument, connections by course and distance must 592 SUJ^ VE YING. be taken to two or three bearing trees or rocks, and to any well-known natural and permanent objects in the vicinity, such as the confluence of streams, prominent rocks, buildings, shafts or mouths of adits. Bearings should also be taken to prominent mountain-peaks, and the approximate distance and direction ascertained from the nearest town or mining camp. A detailed description of the location-monument must be in- cluded in the field- notes of the survey for which it is established. CORNERS. 19. Corners may consist of First — A stone at least twenty-four inches long by six inches square set eighteen inches in the ground. Second— K post at least four and a half feet long by four inches square set twelve inches in the ground and surrounded by a mound of stone or earth two and a half feet high and five feet base. Third — A rock in place. 20. All corners must be established in a permanent and workmanlike manner, and the corner and survey number must be neatly chiselled or scribed on the sides facing the claim. When a rock in place is used its dimensions above ground must be stated, and a cross chiselled at the ex- act corner-point. 21. In case the point for the corner be inaccessible or unsuitable, you will establish a witness-corner, which must be marked with the letters W. C. in addition to the corner and survey number. The witness-corner should be located upon a line of the survey and as near as practicable to the true corner, with which it must be connected by course and distance. The reason for the establish m.ent of a witness-corner must always be stated in the field-notes. 22. The identity of all corners should be perpetuated by taking courses and distances to bearing trees, rocks, and other objects, as pre- scribed in the establishment of location-monuments. If an official sur- vey has been made within a reasonable distance in the vicinity, you will run a connecting line to some corner of the same, and connect in like manner with all conflicting surveys and claims. TOPOGRAPHY. 23. Note carefully all topographical features of the claim, taking dis- tances on your lines to intersections with all streams, gulches, ditches, ravines, mountain ridges, roads, trails, etc., with their widths, courses and other data that may be required to map them correctly. If the claim lies within a town-site, locate all municipal improvements, such as blocks, streets and buildings. 24. You are required also to locate all mining and other improve- ments upon the claim by courses and distances from corners of the sur- vey, or by rectangular offsets from tlie centre line, specifying the dimen- sions and character of each in full detail. CONFLICT.S. 25. If in running the exterior boundaries of a claim, you find that two surveys conflict, you will determine the courses and distances from the APPENDIX B. 593 established corners at which the exterior boundaries of the respective surveys intersect each other, and run all lines necessary for the determi- nation of the areas in conflict, both with surveyed and unsurveyed claims. You are not required, however, to show conflicts with unsur- veyed claims unless the same are to be excluded. 26. When a placer-claim includes lodes, or when several lode-loca- tions are included as one claim in one survey, you will preserve a con- secutive series of numbers for the corners of the whole survey in each case. In the former case you will first describe the placer-claim in your field-notes. PLACER-CLAIMS AND MILL-SITES. 27. The exterior lines of placer-claims cannot be extended over other claims, and the conflicting areas excluded as with lode-claims, it being the surface ground only, with side lines taken perpendicularly downward for which application is made. The survey must accurately define the boundaries of the claim. The same rule will apply to the survey of mill- sites. 28. If by reason of intervening surveys or claims a placer or mill-site survey should be divided into separate tracts, you will also preserve a consecutive series of numbers for the corners of the whole survey, and distinguish the detached portions as Lot No. i. Lot No. 2, etc., connect- ing by course and distance a corner of each lot with some corner of the one previously described. LODE AND MILL-SITE. 29. A lode and mill-site claim in one survey will be distinguished by the letters A and B following the number of the survey. The corners of the mill-site will be numbered independently of those of the lode. Cor- ner No. I of the mill-site must be connected with a corner of the lode claim as well as with a corner of the public survey or U. S. location- monument. FIELD-NOTES. 30. In order that the results of your survey may be reported to this office in a uniform manner, you will prepare your field-notes and pre- liminary plat in strict conformity with the specimen field-notes and plat, which are made part of these instructions. They are designed to furnish you with all needed information concerning the manner of describing the boundaries, corners, connections, intersections, conflicts and improve- ments, and stating the variation, area, location and other data con- nected with the survey of mineral claims, and contain forms of affidavits for the deputy-surveyor and his assistants. In your first reference to any other mineral claim you will give the name, ownership, and if surveyed, the survey-number. 31. The total area of a lode-claim embraced by the exterior bounda- ries, and also the area in conflict with each intersecting survey or claim should be so stated, that the conflicts with any one or all of them may be included or excluded from your survey. 'I'his will enable the claim- ant to state in his application for patent the portions to be excluded in express terms, and to readily determine the net area of his claim. 594 SURVEYING. 32. You will state particularly whether the claim is upon surveyed or unsurveyed public lands. J?iving in the former case the quarter-section, township and range in which it is located, and in the latter the township, as near as can be determined. 33. The field-notes must contain the post-office address of the claim- ant or his authorized agent. EXPENDITURE OF FIVE HUNDRED DODLAR.S. 34. The claimant is required by law, either at the tim^ of filing his application, or at any time thereafter, within the sixty days of publica- tion, to file with the Register the certificate of the Surveyor-General that five hundred dollars’ worth of labor has been expended or improvements made upon the claim by himself or grantors. The information upon which to base this certificate must be derived from the deputy who makes the actual survey and examination upon the premises, and such deputy is required to specify with particularity and full detail the char- acter and extent of such improvements. See also Sec. 8. 35. When a survey embraces several locations or claims held in com- mon, constituting one entire claim, whether lode or placer, an expendi- ture of five hundred dollars upon such entire claim embraced in the sur- vey will be sufficient and need not be shown upon each of the locations included therein. 36. In case of a lode and mill-site claim in the same survey, an ex- penditure of five hundred dollars must be shown upon the lode-claim only. 37. Only actual expenditures and mming improvements, made by the claimant or his grantors, having a direct relation to the development of the claim, can be included in your estimate. 38. The expenditures required may be made from the surface, or in running a tunnel for the development of the claim. Improvements of any other character, such as buildings, machinery or roadway.s, must be excluded from your estimate unless you show clearly that they are asso- ciated with actual excavations, such as cuts, tunnels, shafts, etc., and are essential to the practical development of the survey-claim. 39. You will give in detail the value of each mining improvement in- cluded in your estimate of expenditure, and when a tunnel or other improvement has been made for the development of other claims in con- nection with the one for which survey is made, your report must give the name, ownership and survey-number, if any, of each claim to which a proportion or interest is credited, and the value of the proportion or interest credited to each. The value of improvements made upon other locations or by a former locator who has abandoned his claim cannot be included in your estimate. 40. In making out your certificate of the value of the improvements, you will follow the form prescribed in the specimen field-notes. 41. Following your certificate you will locate and describe all other improvements made by the claimant or other parties within the bounda- ries of the survey. 42. If the value of the labor and improvements upon a mineral claim APPENDIX B. 595 is less than five hundred dollars at the time of survey, you are authorized to file your affidavit of five hundred dollars expenditure at any time before the expiration of the sixty days of publication, but not afterwards unless by special instructions. DESCRIPTIVE REPORTS ON PLACER-CLAIMS. 43. By General Land Office circular, approved September 23, 1882, you are required to make a full examination of all placer-claims at the time of survey, and file with your field-notes a descriptive report in which you will describe — {a) The quality and composition of the soil, and the kind and amount of timber, and other vegetation. {b) The locus and size of streams, and such other matters as may ap- pear upon the surface of the claims. (0 The character and extent of all surface and underground workings, whether placer or lode, for mining purposes. {d) The proximity of centres of trade or residence. {e) The proximity of well-known systems of lode deposits or of indi- vidual lodes. (/’) The use or adaptability of the claim for placer-mining, and whether water has been brought upon it in sufficient quantity to mine the same, or whether it can be procured for that purpose. (g) What works or expenditures have been made by the claimant or his grantors for the development of the claim, and their situation and location with respect to the same as applied for. {k) The true situation of all mines, salt-licks, salt-springs, and mill- seats, which come to your knowledge, or report that none exist on the claim, as the facts may warrant. (/) Said report must be made under oath, and duly corroborated by one or more disinterested persons. 44. Descriptive reports upon placer-claims taken by legal subdivisions are authorized only by special order, and must contain a description of the claim in addition to the foregoing requirements. PRELIMINARY PLAT. 45. You will file with your field-notes a preliminary plat on drawing- paper or tracing-muslin, protracted on a scale of two hundred feet to an inch, on which you will note accurately all the topographical features and details of the survey in conformity with the specimen plat herewith. Pencil sketches will not be accepted. REPORT. 46. You will also submit with your return of survey a report upon the following matters incident to the survey, but not required to be embraced in the field-notes. 47. If the meridian from which your courses were deflected was estab- lished by other means than by the solar apparatus attached to your transit, you will state in detail your observations and calculations for the establishment of such meridian. 596 SURVEYING. 48. If any of the lines of the survey were determined by triangulation or traverse, you will give in full detail the calculations whereby you ar- rived at the results reported in your field-notes. You will also submit your calculations of areas of placer and mill-site claims or other irregular tracts. 49. You will mention in your report the discovery of any material errors in prior official surveys, giving the extent of the same. ERRORS. 50. Whenever a survey has been reported in error, the deputy-sur- veyor who made it will be required to promptly make a thorough exami- nation, upon the premises, and report the result under oath to this office. In case he finds his survey in error, he will report in detail all discrep- ancies with the original survey, and submit any explanation he may have to offer as to the cause. If, on the contrary, he should report his survey correct, a joint survey will be ordered to settle the differences with the surveyor who reported the error. JOINT SURVEY. 51. A joint survey must be made within ten days after the date of order, unless satisfactory reasons are submitted, under oath, for a post- ponement. 52. The field-work must in every sense of the term be a joint and not a separate survey, and the observations and measurements taken with the same instrument and chain, previously tested and agreed upon. 53. The deputy-surveyor found in error, or if both are in erhor, the one who reported the same will make out the field-notes of the joint sur- vey, which, after being duly signed and sworn to by both parties, must be transmitted to this office. 54. The surveyor found in error will be required to pay all expenses of the joint survey and preliminary examinations incident thereto, includ- ing ten dollars per day to the surveyor whose work is proved to be sub- stantially correct. 55. Your field-work must be accurately and properly performed, and your returns made in conformity with the foregoing instructions. Errors in the survey must be corrected at your own expense, and if the time re- quired in the examination of your returns is increased by reason of your neglect or carelessness you will be required to make an additional deposit for office work. You will be held to a strict accountability for the faith- ful discharge of your duties, and will be required to observe fully the re- quirements and regulations in force as to making mineral surveys. If found incompetent as a surveyor, careless in thedischarge of your duties, or guilty of a violation of said regulations, your appointment will be promptly revoked. 56. All former instructions inconsistent with the foregoing are hereby recalled. APPENDIX. 597 SPECIMEN PRELIMINARY PLAT. SPECIMEN FIELD-NOTES. Survey No. 4225 A and B. District No. 3. FIELD-NOTES. Of the survey of the claim of The Argentum Mining Company, upon the Silver King and Gold Queen lodes, and Silver King Mill site, in Alpine Mining District, Lake county, Colorado. Surveyed by George Lighifoot, U. S. Deputy Mineral Surveyor. .Survey begun April 22d, 1886, and completed April 24111, 1886. Address of claimant, Wabasso, Colorado. FEET. 1242. 1365.28 152. 300 - SURVEY NO. 4225 A. SILVER KING LODE. Beginning at Cor. No. i. Identical with Cor. No. i of the location. A spruce post, 5 ft. long, 4 ins. square, set 2 ft. in the ground, with mound of stone, marked whence The W. 1 cor. Sec. 22, T. 1 1 S., R. 81 W. of the 6th Prin- , cipal Meridian, bears S. 79° 34' W. 1378.2 ft. Cor. No. i,Gottenburg lode funsurveyed), Neals Mattson, claimant, bears S. 40° 29' W. 187.67 ft. A pine 12 ins. dia., blazed and marked B. T. A, bears S. 7° 25' E. 22 ft. Mount Ouray bears N. 11° E. Hiawatha Peak bears N. 47° 45' W. Thence S. 24^" 45' W. Va. 15° 12' E. To trail, course N. W. and S. E. To Cor. No. 2. A granite stone 25x9x6 ins., set 18 ins. in the ground, chiselled 4/^ A, whence Cor. No. 2 of the location bears S. 24° 45' W. 134.72 ft. Cor. No. I, Sur. No. 2560, Carnarvon lode, David Davies et al., claimants bears S. 3° 28' E. 116. 6 ft. North end of bridge over Columbine creek bears S. 65® 1 5' E. 650 ft. Thence N. 65® 15' W. Va. 15® 20' E. Intersect line 4-1, Sur. No. 2560, at N. 38® 52' W., 231.2 ft. from Cor. No. i. To Cor. No. 3. APPENDIX B. 599 FEET. A cross at corner-point, and A chiselled on a granite rock in place, 20 x 14 x 6 ft. above the general level, whence Cor. No. 3 of the location bears S. 24° 45' W. 134.72 ft. 734 A spruce 16 ins. dia., blazed and marked B. T. A, bears S, 58° W. 18 ft. Thence N. 24° 45' E. Va. 15° 20' E. Intersect line 4-1 Sur. No. 2560 at N. 38° 52' W. 396.4 ft. from Cor. No. i. 150. 237 . 1000.9 Intersect line 6-7 of this survey. To trail, course N. W. and S. E. Intersect line 2-3, Gottenburg lode, at N. 25° 56' W. 76.26 ft. from Cor. No. 2. 1365.28 To Cor. No. 4. Identical with Cor. No. 4 of the location. A pine post 4.5 ft. long 5 ins. square, set one foot in the ground, with mound of earth and stone, marked whence 28.5 A cross chiselled on rock in place, marked B. R. A, bears N. 28° 10' E. 58.9 ft. Thence S. 65° 15' E. Va. 15° 12' E. Intersect line 4-1, Gottenburg lode, at N. 25° 56' W 285.15 ft. from Cor. No. i. 65. 300. Intersect line 5-6 of this survey. To Cor. No. I, the place of beginning. 285. 315 - GOLD QUEEN LODE. Beginning at Cor. No. 5, A pine post 5 ft. long, 5 ins. square, set 2 ft. in the ground, with mound of earth and stone, marked A, whence Cor. No. I of this survey bears S. 14° 54' E. 370.16 ft. A pine 18 in. dia. bears S. 33° 15' W. 51 ft., and a silver spruce 13 ins. dia. bears N. 60° W. 23 ft., both blazed and marked B. T. A. Thence S. 24° 30' W. Va. 15° 14' E. Intersect line 4-1 of this survey. Intersect line 4-1. Gottenburg lode, at N. 25° 56' W. 237.78 ft. from Cor. No. i. 688.3 Intersect line 1-2, Gottenburg lode, at N. 64^04' E. 12.23 from Cor. No. 2. 1438. 1500. To trail, course N. W. and S. E. To cor. No. 6, s ! A granite stone 34 x 14x6 ins., set one foot in the ground to bedrock, with mound of stone, chiselled j/g-j A, whence A cross chiselled on ledge of rock marked B. R. bears due north 12 ft. 6oo SURVEYING. FEET. 70.3 223.37 300. 38.43 165. 1043.73 1432.90 1500. 300. Thence N. 65° 30' W. Va. 15° 20' E. Intersect line 3-4 of this survey. Intersect line 4-1, Sur. No. 2560 at N. 38° 52' W. 567.28 ft. from Cor. No. i. To Cor. No. 7. A cross at corner-point and^’g-y A chislled on a granite boulder 12 x6 x 3 ft. above ground, whence A cross chiselled on vertical face of cliff, marked H. R. ^^27 hears N. 72° W. 56.2 ft. A pine 14 ins. dia., blazed and marked B. T. hears N. 10° E. 39 ft. Thence N 24® 30' E. Va. not determinerl on account of local attraction. Intersect line 4-1, Sur. No. 2560, at N. 38® 52' W. 653 ft. from Cor. No. I. To trail, course N. W. and S. E. Intersect line 2-3, Gottenburg lode, at N. 25° 56' W. 379.06 ft. from Cor. No. 2. Intersect line 4-1, Gottenburg lode, at N. 25® 56' W. 626.94 ft. from Cor. No. i. To Cor. No. 8. A spruce post 6 ft. long, 5 ins. square, set 2.5 ft. in the ground with mound of stone, marked A, whence A cross chislled on rock in place, marked B. R. bears S. 9° 12' E. 1 5.8 ft. A pine, 20 ins. dia., blazed and marked B. T. hears N. 83® E.28.5 ft. Thence S. 65° 30' E. Va. 15® 16' E. To Cor. No. 5, the place of beginning. Area. Total area of Silver King lode 9-403 acres Less area in conflict with Sur. No. 2560 124 acre Gottenburg lode 1*363 “ 1.487 acres Net area of Silver King lode 7*9i6 acres Total area of Gold Queen lode 10.331 acres Area in conflict with Sur. No. 2560 034 “ Gottenburg lode 2.679 “ Silver King lode 1.887 “ Silver King lode (exclusive of conflict of said Silver King lode with the Gottenburg lode) 1.309 APPENDIX B. 6oi FEET. 90. 208. 504.8 351 - 3944 15. 40. 370. 647.2 Total area of Gold Queen lode 10.331 acres Less area in conflict with Sur. No. 2560 034 acre Gottenburg lode 2.679 “ Silver King lode i -309 “ 4- 022 acres Net area of Gold Queen lode 6. 309 acres “ “ Silver King lode 7-9i6 Net area of lode claim 14.225 acres SURVEY NO. 4225 B. SILVER KING MILL-SITE. Beginning at Cor. No. i, A gneiss stone 32x8x6 ins., set 2 ft. in the ground, chis- elled B, whence W. 4 cor. Sec. 22, T. ii S, R. 81 W. of the 6th Principal Meridian, bears N. 80° W. 1880 ft. Cor. No. I, Sur. No. 4225 A, bears N. 40° 44' W. 760.2 ft. A cottonwood 18 ins. dia., blazed and marked 4 ^Vt bears S. 5° 30' E. 17 ft. Thence S. 34° E. Road to Wabasso, course N. E. and S. W. Right bank of Columbine creek, 75 ft. wide, flows S. W. To Cor. No. 2, An iron bolt 18 ins. long, i in. dia., set one foot in rock in place, chiselled 4^3- B, whence A cottonwood, blazed and marked B. T. bears E. 182 ft. Thence S. 56° W. Left bank of Columbine creek. To Cor. No. 3, A point in bed of creek, unsuitable for the establishment of a permanent corner. Thence N. 34° W. Right bank of Columbine creek. To witness-corner to Cor. No. 3, A pine post 4.5 ft. long, 5 ins. square, set one foot in ground, with mound of stone, marked W. C. 4^25 whence A cottonwood 15 ins. dia. bears N. ii" E. 16.5 ft. and a cottonwood 19 ins. dia. bears N. 83° W. 23 ft., both blazed and marked B. T. W. C. B. Road to Wabasso, course N. E. and S. W. To Cor. No. 4, A gneiss stone 24 x 10x4 ins., set 18 ins. in the ground, chiselled 72W whence A cross chiselled on ledge of rock, marked B. R. B, bears N. 85° 10' E. 26.4 ft. Thence N. 48^43' E. 6o2 SURVEYING. FEET. I 125.5 I To Cor. No. 5, I A gneiss stone 30x8x5 ins., set 2 ft. in the grounrl, I cliisellecl 13 . Thence S. 34° E. 1 58.3 To Cor. No. 6, A pine post 5 ft. long, 5 ins. square, set 2 ft. in the ground with mound of earth and stone, marked whence A pine 12 ins. dia., blazed and marked I>. 13 , bears S. 33“ E. 63.5 ft. Thence N. 56° E. 270. To Cor. No. I, the place of beginning. Containing 5 acres. Variation at all the corners, 15° 20' E. The surveys of the Gold Queen lode and Silver King mill site are identical with the respective locations. LOCATION. This claim is located in the \V. ^ Sec. 22, T. 1 1 S., R. 81 W. EXPENDITURE OF FIVE HUNDRED DOLLAR.S. I certify that the value of the labor and improvements upon this claim, placed thereon by the claimant and its grantors, is not less than five hundred dollars, and that said improvements consist of The discovery shaft of the Silver King lode, 6x3 ft., 10 ft. deep in earth and rock, which bears from Cor. No. 2 N. 6° 42' W. 287.5 ft. Value S80. An incline 7x5 ft., 45 ft. deep in coarse gravel and rock, timbered, course N. 58° 15' W., dip 62°, the mouth of which bears from Cor. No. 2 N. i5°37'E. 908 ft. Value S550. The discovery shaft of the Gold Queen lode. 5x5 ft., 18 ft. deep in rock, which bears from Cor. No. 7 N. 67° 39' E. 219.3 ft., at the bottom of which is a cross-cut 6.5x4 ft. running N. 59^26' W. 75 ft. Value of shaft and cross-cut, $1,000. A log shaft-house 14 ft. square, over the last-mentioned shaft. Value $100. Two-thirds interest in an adit 6.5 x 5 ft., running due west 835 ft., timbered, the mouth of which bears from Cor. No. 2 N. 61° 15' E. 920 ft. This adit is in course of construction for the development of the Silver King and Gold Queen lodes of this claim, and Sur. No. 2560, Carnarvon lode, David Davies et al., claimants, the remaining one-third interest therein having already been included in the estimate of five hundred dollars expenditure upon the latter claim, Total value of adit, $13,000. APPENDIX B. 603 A drift 6.5 x4 ft. on the Silver King lode, beginning at a point in adit 800 ft. from the mouth, and running N. 20° 20' E. 195 ft., thence N, 54° 15' E. 40 ft. to breast. Value $2,800. I further certify tliat no portion of the improvements claimed have been included in the estimate of five hundred dollars expenditure upon any other claim. OTHER IMPROVEMENTS. A log cabin 35x28 ft., the S. W. corner of which bears from Cor. No. 7 N. 30° 44' E. 496 ft. A dam 4 ft. high, 50 ft. long, across Columbine creek, the south end of which bears from Cor. No. 2 of the mill-site N. 58° 20' W. 240 ft. Said cabin and dam belong to The Argentum Mining Company. An adit 6x4 ft., running N. 70° 50' W. 100 ft., the mouth of which bears from Cor. No. 5 S. 58° 12' W. 323 ft. belong- ing to Neals Mattson, claimant of the Gottenburg lode. INSTRUMENT. The survey was made with a Young & Sons mountain transit No. 5322, with Smith’s solar attachment. The courses were deflected from the true meridian as determined by solar observations. The distances were measured with a 50 ft. steel tape. employe’s certificate. List of the names of individuals employed to assist in running, meas- uring and marking the lines and corners described in the foregoing field- notes of the survey of the claim of The Argentum Mining Company upon the Silver King and Gold Queen lodes and Silver King mill-site, in Alpine Mining District, Lake County, Colorado. William Sharp, Robert Talc. We hereby certify that we assisted George Lightfoot U. S. Deputy Mineral Surveyor, in surveying the exterior boundaries and marking the corners of the claim of The Argentum Mining Company upon the Silver King and Gold Queen lodes and Silver King mill-site in Alpine Mining District, Lake County, Colorado, and that said survey has been in all re- spects, to the best of our knowledge and belief, well and faithfully sur- veyed and the boundary monuments planted according to the instruc- tions furnished by the Surveyor-General. William Sharp, Robert Talc. Subscribed and sworn to by the above-named persons before m-e, this 26th day of April, 1886. [Seal] John Doolittle, Notary Public. 6o4 VE Y INC. surveyor’s oath. I, George Lightfoot, U. S. Deputy Mineral Surveyor, do solemnly swear that in pursuance of an order from Jas. A. Dawson, Surveyor-Gen- eral of the public lands in the State of Colorado, bearing date the 30th day of March 1886, and in strict conformity with the laws of the United States, and instructions furnished by said Suiweyor-General, I have faithfully surveyed the claim of The Argentum Mining Company upon the Silver King and Gold Queen lodes and Silver King mill-site in Alpine Mining District, Lake County, Colorado, and do further solemnly swear that the foregoing are the true and original field-notes of such survey, and that the improvements are as therein stated. George Lightfoot, U. S. Deputy M incral Surveyor. Subscribed by said George Lightfoot, U. S. Deputy Mineral Surveyor, and sworn to before me this 26th day of April, 1886. [Seal] John Doolittle, Notary Public. APPENDIX C FINITE DIFFERENCES. THE CONSTRUCTION OF TABLES. In the accompanying figure the ordinates are spaced at the uniform distance / apart. Let the successive values of these ordinates, and their several orders of differences, be represented by the following notation: Values of the function, hi, hi, hz, kz, kz. First order of differences, A' A' A' A' h^, A'h^, A'^^. Second “ “ A"Ao, A" k^, A"h^, A" h^, A"h^. Third “ “ ' A’''hz. A'"a„ A'"/^,. Fourth “ “ etc., etc. 6o6 SCrji^ VE YING. We may now write hi ■=. ^'Aq’, 1 hi = hi -j- z/'/;, = ho + z/'Ao + ^'Ao + ^"Ao = ^^0 + 2^'//o + ^"Ao'. h% — hi -f- A ' = ho + 3'^Vi’o 4" 3-^’V/o + Ao\ h\ = ho -j- 4‘^'Ao 4“ ^^ "Ao 4“ 4-^ "ho 4“ ^^''ho\ * hn = /^o 4- ho 4 T72~^ VT^~3 ^ ^0 4- etc. (0 It is to be observed that the coefTicients follow the law of the bino- mial development. It is also seen that the first of the successive orders of differences are alone sufficient to enable any term of the function to be computed. We will now proceed to find these first terms of the several orders of differences for any given equation. Almost all functions of a single variable can be developed by the aid of Maclaurin’s Formula, in the form y'o — Co CiXo -(- CiXo"^ 4“ ^3-^0^ 4“ CiiXo^ -f- etc (2) If X take an increment A^, thus becoming Xi, the cha^ige in yo will be represented by A' and its value will be the new value of the function minus its initial value, or A'y^ =yi —yo. By putting x + Aj^ for x in the above equation, developing, subtracting the original equation, and re- ducing, we would obtain — yo = ^Vo = (^1 4- 2 CiXo 4" sCsXo^ 4" 4 4“ (C2 4" 3^3^0 4“ ^C 4 Xo^)A'^^ -j- (C3 -f- 4C4Xo)A^x CiA^x , • (3) assuming that the function stops with C4X0*. If jfi should now take another increment equal to the previous one, we would have Xi — Xi + z/^ and yi = yi 4- A'y^_ Now A' is the value A'y when Xo has become Xu and the difference between A' and A'y^ is the change in the value of A' y^ due to this change in .r. Hence z/'ji — = 4 /'Vo- To find the value of substitute x Ax for Jir in equation (3), develop, subtract equation (3), reduce, and obtain A" y^ — (2C2 -f- 6C3X0 -f- I2C4.Yo*)zf'^^ T" (6C3 4“ 24^^4 >^o)zf®^ (4) Similarly we find z/''Vo = (6a 4- 24a^o)^3^4- 3 (>C 4 ^^x (5) A^'^y^ = 24az/'‘^(a constant) ( 6 ) APPENDIX C. 607 From the above development we see — 1. Til at ihe iiianber of orders of differences is equal to the highest ex- ponent of the variable involved, the last difference being a constant. 2. That if nny initial value of the variable be taken, the first of the several orders of differences can be obtained in terms of this initial value, its constant increment, and the constant coefficients. This fur- nishes a ready means of computing a table of values of the function, if it can be represented in the form of equation (i). Evidently if the ini- tial value of the variable (xo) be taken as zero, the evaluation for the several initial differences is much simplified, for then all the terms in x disappear. If the constant increment be also taken as unity, the labor is still further reduced. Example. — Construct a table of values of the function / = 50 — 40X -|- 20x^ -|- 4x'^ — (7) Let the initial value of the variable be zero and the increments unity. Evaluating the initial differences by equations (3) to (6), we find, for a-q = o, and = i, yo = -j- 50; Vo2 — Cl -j- Ca -j- C3 -{- C4 = - 17; A' j'q = iCi -j- 6(73 ~1“ 14C4 = + 50; A"'yO — 6C3 — j— = — 12; A'^'^yo = 24 C4 = - 24. From these initial values we may readily construct the following table : Values of X. Values of ist Differences. AV 2d Differences. A-,. 3d Differences. A-V. 4th Differences. Aivy. 0 1 2 3 4 5 6 7 8 etc. 50 33 66 137 210 225 98 — 279 — 1038 etc. - 17 + 33 + 71 + 73 + 15 - 127 - 377 - 759 etc. + 50 + 38 + 2 — 58 — 142 — 250 — 382 etc. — 12 — 36 — 60 — 84 — 108 — 132 etc. .- 24 - 24 - 24 - 24 - 24 etc. * Fig. 152 is the locus of this curve, the ordinates being taken from th^.s column. 6o8 SUR VE YING. The initial values in all the columns beiii," ^iven, the table is made by continual additions, one column after anollier, workinj:!^ from right to left. Thus, the 4th difTerence being constant, the initial value, — 24, is simply repeated indefinitely. The column of 3d clifTcrcnccs is now com-, puted by adding continuously —24 to the preceding value. The column of 2d differences is next made out, the quantity to be added each time being the intervening 3d difference, which is not constant. In a similar manner proceed with the column of ist differences, and finally with the values of the function itself. The above formulae apply to all functions of a single variable not higher than the fourth degree. Evidently any of the C’ coefficients may be zero, and so cause one or more of the powers of .r to entirely disappear. If the variable is involved to a higher degree than the fourth, a new de- velopment may be made, or the initial values of the successive orders of differences may be determined by simply evaluating the function for a series of successive values of the variable, one more in number than the degree of the equation, and then working out the successive columns of differences from these until the last, or constant, difference is found. The table may then be continued by combining these differences, as be- fore. Thus in the above example the first five values of j/ might have been found by direct evaluation of the function for the corresponding values of x, and then the successive differences taken out until the con- stant fourth difference, — 24, was found. This can always be done with- out resorting to any algebraic discussion as given above. THE EVALUATION OF IRREGULAR AREAS. The ordinates to any curve, as that in Fig. 152 for instance, may be represented by such an equation as the last of equations, (i ), where the length of any ordinate is given in terms of its number from the initial or- dinate, the value of this first ordinate, and the first of the successive orders of differences. This equation is , , , ~ i) hn = /lo /iq -j- ^ /i’o n {n — i) {it — 2) 1.2.3 ^'"y^o-f-etc., where h„ is the ;zth, and therefore any ordinate to the curve. The con- stant distance between the ordinates apparently does not enter the equa- tion, but it is really represented in the several /I’s. By the calculus the area of any figure included between any curve, the axis of abscissas, and two extreme ordinates is A -j: hdx, where h is the general value of an ordinate, = hn in the above equation, where it is shown to be a function of n. Also x = nl where / is the constant distance between ordinates, whence dx = ld 7 i. ues of /2 and dx, we have Substituting these val- APPENDIX C. 609 0 Integrating this equation, we obtain From the schedule of differences on p. 605 we may at once find the initial values of the several orders of differences in terms of the succes- sive values of the function. Thus A'Ao = hx — h^\ A " — A ' — A' — hi — 2h\ A'"/to = + A'/,^ - hi — 2,hi + 2>hi - ho; AKho = A"'h^ — A'" ho = A”h., — 2A"h^ -f A" ho = A’ ho — sA'h^ + 3^'Aj —A'ho — hi — 4'^3 6^2 — 4^1 Again, the coefficients follow the law of the binomial development, and we may write n{n — i){n — 2) 1.2.3 /^«-3 -(- etc. . (10) By the aid of this equation we may now substitute for the several initial differences in equation (9) their values in terms of the successive values of the function. Also for any area divided into n sections by or- dinates, uniformly spaced a distance / apart, equation (9) will give the area in terms of /, 71, and the several ordinates, when these latter are sub- stituted for the z/’s by means of eq. (10). Thus, for n = \, equation (9) becomes (II) 39 6io SURVEYING. which is the Trapezoidal Rule. Fer n = 2, ^ =/(2/i. + 24'^. + (5- = /(i/, 0+ 3/,,+ i/;,) = ^(//.+ 4/S, + //,), (12) which is called Simpson s \ Rule. If r — 'll = total length of figure, this formula becomes /' ^ (■^^0 -j- 4^0 - j - hi), ( l 2 fl ) which is the well-known form of the Prismoidal T'ormula, and it would be that formula if areas were substituted for ordinates. If n = 3, ^ ~ H" 3-^^! 3 hi + hi), (13) which is called Simpson s f Rule. If n = 4, A — - [7 (^0 -f- hi) 32 {Ji\ -f- hi) -j- 12,^2]. 45 • . (14) If n = 6, A = l[ 6 ho iSA'/i^ -j- 2 yA"/tQ is,A"' J iq + If now the coefficient of be changed from to which would not affect curves of a degree less than the sixth, the resulting equation, when the //s are substituted for the A’s, takes the following very simple form ; A = ~ [//o + h-2 hi -{- liQ S {hi -{- -^3 + hi) -|- hi\, . . (15) which is called IVeddel’s Ride. For a greater number of ordinates than seven, it is best to use either equation (12), (13), or (15) several times, as the formulae become very complicated for 7 t > 6. APPENDIX D. DERIVATION OF FORMULiE FOR COMPUTING GEOGRAPH- ICAL COORDINATES AND FOR THE PRO- JECTION OF MAPS* Let Fig. 153 represent a distorted meridian section of the earth. Let a = the major and b the minor semi-axes. Then = the ellipticity. The eccentricity is given by — IP' o = 5 — , whence \ The line nm = A" is the normal to the curve at n ; the angle ncd = A is the geocentric latitude ; while nld = Z is the geodetic latitude. The geodetic latitude is always understood, ^s it is the latitude ob- tained from astronomical observations. It is desirable to find the length of the line nl, of the normal nm, and of the radius of curvature /V', all in terms of e, L, and a. Also to find the geocentric latitude in terms of a, b, and L. To find nl, we have nl—^/ nd' -f dl^= |/ ^ For the ellipse, whence dy _ b’^x ^ dx al^y' -j- (i — e^yx^ (I) ( 2 ) * See Chapters XIV. and XV. for the use of the formulae. 6i2 SUR VE YING. But the equation of the ellipse in terms of its eccentricity is ,4 whence nl — V y’^e^ -f- ~ Fig. 153. Squaring, remembering thatj^ = sin L, we have, after reducing, • (i — sin^ L)i To find the length of the normal nm = N, we have nm \ nl w x\ dl. But whence dl = nd tan dnl ■= y~—^— x — — e^)x\ dx nm = N = 1 — (i — sin'-' L)i APPENDIX D. 613 To find the geocentric latitude in terms of a, b, and Z, we have A = ncd ; L — nld. Since both have the common ordinate nd, we may write tan A : tan L :: dl : dc. b^ But dl = —X from (4), and dc = x, b^ whence tan A = ^ tan Z. . . (C) To find the radius of curvature, Z, we have, in general, (5) For the ellipse, dy dx whence h^x_ dy _ b* d^ y dx^ ~ a-y ’ ( 6 ) To get this in terms of a, e, and Z, we have, from Fig. 153, d^ (i — sin*'^ Z nl sin® Z = sin® Z Also from the equation of the ellipse in terms of its eccentricity we have 'I — 2 _ _Z!__ _ — sin® Z) I — ^® I — ^® sin® Z * We may now find d^lA I — ^® sin® Z’ a^y^ 4 " — 6i4 SUA' VE YAXC. or (ay + ^^^5)5 = (i — sin‘^ Z)3' ( 7 ) Substituting this in (6), wc obtain ^ ^ _ a — e'‘') a ’ (i — e'^ sin'' L)l (i — e'^ sin'' L)^ . . (D) The radius of curv^aturc of the meridian, R, and the radius of curva- ture of the great circle perpendicular to a given meridian at the point where they intersect, wliicl) is the normal, N, are the most important functions in geodetic formulae. We will now derive the equations used on the U. S. Coast and Geodetic Survey for computing geodetic positions from the results of a primary triangulation. In Fig. 154, let A and B be two points on the surface of the earth, which were used as adjacent triangulation-stations. The distance between them, the azimuth of the line AB at one of the stations, and the latitude APPENDIX D. 615 and longitude of one station are supposed to be known ; the latitude and longitude of the other station, and tlie back azimuth of the line joining them, are to be found. Let L' = known latitude of B ; Z = unknown latitude of A ; K — known length of line AB reduced to sea-level; s — length of arc AB — Z — known azimuth of BA at Z; Z — unknown azimuth oi A B z.\. A\ M' — known longitude of B\ M = unknown longitude of A. The angle APB formed by the two meridional planes through A and B is the difference of longitude M— M' — AM. The difference of latitude is, L — L' — AL — Bl in the figure. Al is the trace of a parallel of latitude through A and I is its intersection with the meridian through B. AP' is the trace of a great circle through A perpendicular to the meridian through B, and P' is the point of its inter- section with tliat meridian. The normals are B71' = N' and Ati = N. The radii of curvatvre are Br — R' and Ar = R. The latitude and longitude of A, and the azimuth of the line AB from A towards B, can now be found by solving the spherical triangle APB. Thus Z = 90° - ; M ^ M' - Al\ and Z = 180° - PAB. Although the line AB lies on the surface of a spheroid, if a sphere be conceived such that its surface is tangent internally to the surface of the spheroid on the parallel of latitude passing through the middle point of the line AB, then this line will lie so nearly in the surface of the sphere, that no appreciable error is made by assuming it to be in its surface. The triangle ABP then becomes a triangle on the surface of the tangent sphere, and hence is a true spherical triangle. The sphere is defined by taking its radius equal to the normal to the meridian at the mean lati- tude of the points A and B. Since this mean latitude is unknown, the formulse are first derived for the latitude of B, Z', and then a correction applied to reduce it to the mean latitude. THE DIFFERENCE OF LATITUDE. Let it first be required to find Z from Z', or find AL = Z — Z'. If we write /, for the co-latitudes of Z, Z', and z' for 180° — Z', we liave, from the spherical triangle ABP, cos I = cos I' cos s -}- sin I' sin s cos z' , ( 8 ) 6i6 SUR VE YING. By means of Taylor’s Formula we may find the value of / in ascending powers of s, and since s is always a very small arc in terms of the radius, usually from 20 to 60 minutes, the series will be rapidly converging By means of Taylor’s Formula, we may at once write /=/' + I d^-r I dH etc. (9) We will use but the first three terms of this development, the fourth term being used only in the largest primary triangles. The derivation of the successive dilferential coefficients of / with respect to s is the most difficult portion of this general development. If s be supposed to var3% then / and z both must vary, and they are all im- plicit functions of each other. These coefficients are therefore best found geometrically, as follows : in Fig. 155, Let AB — BC = ds = differential portions of the line AB = s in Fig. 154 ; AD — — dt — change in AB {= /') due to the change + ds in s. Let the angle BAB = z' and BBC = z", z" being greater than z' by the convergence of the meridians shown by the angle AB' B. APPENDIX D. 617 The lines BD and CE are parallels of latitude through the points B and C They cut all meridians at right angles. Since the triangle ABD is a differential one on the surface of the sphere, it may be treated as a plane triangle, and we may at once write dt ds AD AB — cos 0', (10) the minus sign indicating that I and s are inverse functions of each other. Differentiating this equation and dividing both sides by ds we obtain dN . ,dz' . . (II) Now the angle dz’ is the angle AP'B, subtended by the arc BD with radius BP'. But this arc, with radius gives the angle ds sin z'. Therefore dz' — sin z ds BN BP' — sin dds tan D = sin z cot /Vj, dz . , or — = sin z cot / . ds . Substituting this in (ii) we obtain d'^l' — sin®^' cot /' (12) Substituting these values in (9), we have I — I = — s cos 2 -f- sin^ 2 cot / etc. Now, replacing /, and z, by L, L\ and Z, we have D — L — s cos Z -|- z tan L (13) Here s is expressed in arc to a radius of unity. 6i8 SI//? VE YING. Referring it now to the radius N, we have j where A' is the length of the arc s in any unit, iV being the length of the normal 7 ivi in Fig. 153, given in the same unit. Substituting these in (13), w'e have j, j. _ A' cos Z , I A' 2 sin2 Z tan L ” “ ~N ^2 ' (14) This gives the difference of latitude in units of arc in terms of radius N. But differences of latitude are measured on a sphere wliose radius is the radius of curvature of the meridian at the middle latitude. Since we do not yet know the middle latitude, we can use the known latitude L' and afterwards correct to — . 2 Changing to a sphere whose radius is R, and dividing by the arc of i" in order to get the result in seconds, we have L' - L = ^ SL = J? arc i' cos Z If we let B = R arc 1"’ and C = 2 I?i\' arc I tan A - sin® Z tan L. (15) 2 AW arc i"’ we may write — oL = A" cos Z-B AT® sin® Z-C. (16) To reduce this to what it would be if the mean latitude had been used we have to correct it for the difference in the radii of curvature. Rl and Rm, at the latitude L and the middle latitude respectively. If ZL be the true difference of latitude when Rm is used, and <5Z be the difference when Rl is used, we would have ZL : 8L :: Rl : Rm, AL = = 5Z (i ^ -A 'itt ' Rr )-“(■+ a- To reduce 8L to ZL, therefore, we must add the quantity . ciRr Rm Now R (I a{\ — D)-Y sm^ Z-C. . . . (E') THE DIFFERENCE OF LONGITUDE. In the triangle APB, Fig. 154, the three sides and the angle at the known station B are known. To find AM = angle APB, we have, therefore, sin PA : sin :: sin PBA : sin APB, or sin / ; sin s sin z : sin AM. * In the U. S. Coast and Geodetic Survey Report for 1884, Appendix 7, p. 326, this term is given with its denominator raised to the f power, and the tabular values of D are computed accordingly. The development there given is laborious and approximate, but the error is not more than 0.001 of the value of this term, which is itself very small. 620 SURVEYING. But j = ^ where N is the normal An, Fig. 154; and if we assume that the arc s is proportional to its sine, we have AM = K sin Z N cos L arc i"’ where AM is expressed in seconds of arc. T If we put A = . (18) this equation becomes Am = N arc i'" A' sin Z . A cos L fF) In order to correct for the assumption that the arc is proportional to its sine, a table of the differences of the logarithms of arcs and sines is given in the U. S. C. and G. Report for 1884, p, 373, with instructions for its use on p. 327. THE DIFFERENCE OF AZIMUTH. In the spherical triangle APB, Fig. 154, we have, from spherical trigonometry, or cot \{PBA + PAB)^ = tan \BPA cos \{BP AP) cos \{BP — AP)' cot \{z' -\~ z) = tan ^ (— AM\) cosi(/'-f-/) cosi(/'— /) , ^ , sin i(Z' -f Z) • = - tan iAM 7777^— cos i(Z — Z) But z = 180° — Z, therefore cot i(i8o° — Z z') = tan 4 (Z — z') = tan \{AZ), .sin W ■\-L) whence tan ^AZ = tan i AM- cos i(Z' - Z)* . (19) *Chauvenet’s Spherical Trigonometry, eq. (127). f Increments of M are measured positively towards the west. APPENDIX D, 621 It will be seen that since the azimuth Z of a line is measured from the south point in the direction S.W.N.E., the azimuth of the line BA from B towards A (forward azimuth) is the angle PBA + 180° = Z', while the azimuth of the same line from A is 1^0° — PAB = Z. Also, that ZZ = Z+ 180° - Z'. Assuming that the tangents ^AZ and ^AM are proportional to their arcs, and putting Lm for the middle latitude, we have - AZ- Am sin Lm cos ^AL (G) The U. S. Coast Survey Tables are based on the following semi- diameters: a = 6 378 206 metres, = 6 356 584 “ or a\b w 294.98 : 293.98. See Appendix No. 7, U. S. Coast and Geodetic Survey, for tabulaf values of constants and forms for reduction. TABLES. TABLE I. Trigonometric Formulae. Trigonometric Functions. Let A (Fig. 107) = angle BAG = arc BF, and let the radius AF = AB — AH=\. We then have sin ^ = BG • cos^ ' = AG tan A — DF cot A = HG sec A = AD cosec A = AO versin A - CF = BE covers A = BK ~ HL exsec A = BD coexsec A = BG chord A — BF chord 2 A = BI = 2BC Fig. 107. In the right-angled triangle ABC (Fig. 107) Let AB = c, AC = b, and BG = a. We then have : 1. sin A = a c = cosB 11. a = c sin A = b tan A 2. cos A — b c = sin B 12. b = c cos^ = a cot A 3. tan A = a F = cotB 13. c _ a _ & ~ sin A “ cos J. 4. cot A = b a = tan B 14. a = c cos B = b cot B 5. sec A = c F = cosec B 15. b = c sin B = a tan B 6. cosec A = c a = sec B IG. c a b cos B ~ sin B 7. vers A = c — b c — covers B 17. a = V(c+6)(c-0) 8. exsec A = c - b b = coexsec B 18. b = 4^ (c -f a) (c - a) 9. covers A = c — a c = versin B 19. c = Vaa-iT^i 10. coexsec A = c — a a = exsec B 20. C = 00» = ^ + 2? 21. area ah 626 SURVE YING. TA'BLE I. — Continued. Tfuoonometric Formul.^. Solution oir Oblk^uk Trianoles. GIVEN. SOUGHT. FORMUL.1!:. 1 1 22 A. D, a C, b, c c = 180° - (y 1 -1- 77), b = - . sin 77. sin A j 23 A, a, 6 1 r:;i n = J,, C= ISO" - (A + 1!). j c = sin C. sin A 24 C, a, h 1 4- B) y. (^ + 77) = C0° - 14 C 25 tan ^ M - 77) = tan 14 M + 77) 26 A, B A = 14 {A + 77) -f 14 M - B), B^y {A + B)-y (A - B) 27 c ‘ Qosy{A—B) ^ sin y(A — B) 28 area 77 = 34 rt & sin C. 29 a, by c A Let s = 34 (a 6 -f c) ; sin 34 % c" ”” 30 1 / .1 /s(s — a) . . /(s-b)(s-c) cos ^ ; tan .1 ^ 81 sin A ~ C-5— — c)_ ~ be ' 2 (.9 -b) (s- c) vers ^ = , be 32 ® area K = Vs (s — a) (s — 6) (.; — c) 33 Ay B, C, a area sin 77 . sin C ~ 2 sin TABLES. 627 TABLE I. — Contmued. TrigonOxMetric Formula. general formula. 34 sin = VI — cos2 A = tan A cos A 36 37 38 39 40 41 42 43 41 45 40 47 48 49 50 61 sin A sin A cos A cos^ cos A tan A — tan A = tan A = cot A ~ cot A = cot A =s cosec A 2 sin 14 A cos A = vers A cot 14 Y 54 vers 2 A = (1 — cos 2 A) 1 = V 1 — sin2 A = cot A sin A sec A 1 — vers A = 2 cos^ = 1 — 2 sin^ 14 ^ cos^ y^A — sin2 yA = V cos 2 A 1 sin A cot A cos A = V seo^ A — I — 1 cos^ A 1 — cos 2 A vers 2 A V ^ — cos‘^ ^4 _ sin 2 A cos A i cos 2 A sin 2 A sin 2 A = exsec A cot J4 A. cos A — — j = — — / = i/cosec^J. — 1 tan A sin A sin 2 A 1 — cos 2 A _tan J4 A exsec A sin 2 A vers 2 A 1 + cos 2 A sin 2 A vers A = 1 — cos A = sin A tan yA = 2 sin^ y A vers A = exsec A cos A exsec A = sec A — 1 = tan A tan yA = cos A sin yA sin 2 A = 2 sin A cos A /I — cos A / versyl ~ y 2 ~ y 2 cos Yi A s /'- + cos^ 2 cos 2 A = 2 cos^ A — 1 = cos^ A — sin^ yl = 1 — 2 sin^' A 628 SURVEYING. 53. 54 . 65 . 56 . 57 . 68 . 69 . 60 . 61 . 62 . 63 . 64 . 65 . 66 . 67 . 68 . 69 . 70 . TABLE I. — Continued. Trigonometric Formula. General FoRMCLiE. . . , . tan A tan 7—r ^ = cosec 1 4- sec A A ^ A 1 ^ A / ^ A — cot A = . . — - = - 4 / j- sm ^ r I — cos 4- cos tan 2 A = cot. = cot 2 ^ = vers ^ A ■ 2 tan A 1 — tan^ A _Bln A__ _ 1 4 - cos A vers k “ sin A ^ot2 A — 1 2 cot .4 ^ vers A cosec A — cot A 1 — cos A 1 4- — 34 vers A 24 - ^2 (1 4- cos A) vers 2^ = 2 sin^ A exsec }^A=- — cos A exsec 2 A = (1 4- cos A) + 4/2 (1 4- cos A) tan^ A 1 — tana ^ sin ± S) = sin A . cos 5 ± sin .B , cos A cos {A ± B) = cos A . cos B T sin ^ . sin 5 sin ^ 4- sin j 5 = 2 sin (4. 4- B) cos 34 (4 — B) sin 4 — sin B = 2 cos 34 (4 4- B) sin 34 (4 — B) cos 4 4- cos B = 2 cos y^{A-\- B) cos 34 (4 — B) cos B — cos 4 = 2 sin 34 (-4 4- -S) sin ^(A — B) sina 4 — sina B = cosa B — cosa 4 = sin (4 4- -B) sin (4 — cosa 4 — sina ^ = ^os (4 4* i?) cos (4 — B) sin (A 4- B) tan 4 4- tan B = tan 4 — tan B cos 4 . cos B sin (4 B) cos 4 . cos B TABLES, 629 TABLE 11. For Converting Metres, Feet, and Chains. Metres to Feet. Feet TO Metres and Chains. Chains TO Feet. Metres. Feet. Feet. Metres. Chains. Chains. Feet. I 3.28087 I 0.304797 O.OI51 0.01 0.66 2 6.56174 2 0.609595 -0303 .02 1.32 3 9.84261 3 0.914392 -0455 -03 1.98 4 13.12348 4 1.219189 .0606 .04 2.64 5 16.40435 5 1.523986 -0758 -05 3-30 6 19.6S522 6 I .828784 .0909 .06 3.96 7 22.96609 7 2.133581 . 1061 -07 4.62 8 26.24695 8 2.438378 .1212 .oS 5.28 9 29-52732 9 2.743175 .1364 .09 5.94 10 32 . 80869 10 3.047973 -1515 . 10 6.60 20 65.61739 20 6.095946 .3030 .20 13.20 30 98.42609 30 9.143918 -4545 -30 19.80 40 131.2348 40 12. 19189 .6061 .40 26.40 50 164.0435 50 15.23986 .7576 -50 33.00 60 196.8522 60 18.28784 .9091 .60 39.60 70 229.6609 70 21.33581 1.0606 -70 46.20 80 262.4695 80 24-38378 1 .2121 .80 52.80 90 295.2782 90 27-43175 1.3636 .90 59-40 100 328.0869 100 30.47973 I. 5151 I 66.00 200 656.1739 100 60.95946 3-0303 2 132 300 984.2609 300 91.43918 4-5455 3 198 400 1312.348 400 121.9189 6 . 0606 4 264 500 1640.435 500 152.3986 7.5756 5 330 600 1968.522 600 182.8784 9 . 0909 6 396 700 2296.609 700 213-3581 10.606 7 462 8 00 2624.695 800 243.8378 12. 121 8 528 900 2952.782 900 274-3175 13.636 9 594 1000 3280.869 1000 304 • 7973 15-151 10 660 2000 6561.739 2000 609.5946 30.303 20 1320 3000 9842.609 3000 914.3918 45-455 30 1980 4000 13123.48 4000 1219. 189 60 . 606 40 2640 5000 16404.35 5000 1523.986 75.756 50 3300 6000 19685.22 6000 1828.784 90.909 60 3960 7000 22966.09 7000 2133-581 106.06 70 4620 8000 26246.95 8000 2438.378 121 .21 80 5280 9000 29527.82 , 9000 2743-175 136.36 90 5940 630 S UR VE YJNG. TABLE III. Logarithms of Numbers. § 173. c/i 0 rt 0 1 a 3 4 6 G 7 8 9 Proportional Parts. >' 2 1 .1 \ i 1 G 1 1 7 H 0 10 .0000 .0043 .0086 .0128 .0170 .0212 •0253 .0294 •0334 i •0374 4 Is t2 . : i 25 29 33 37 II .0414 .0453 .0492 •0531 .056a .0607 .0645 .0682 .0719 •0755 4 8 I I i 5>9 23 26 30 34 12 .0792 .0828 .0864 .0899 •0934 .0969 . 1 CX 34 . 1038 . 1072 . 1 106 3 1 7 10 14 >7 21 24 28 3 > .”39 .1173 .1206 .1239 .1271 •1303 • 1335 .1367 .1399 .1430 3 6 10 • 3 , iq 23 26 M . 1461 .1492 • 1523 • 1553 .1584 . 1614 .1644 .1673 .1703 •»732 3 6 9 12 15 18 21 24 '27 1 15 .1761 .1790 .1818 .1847 • 1875 .1903 • 1931 .1959 .1987 .2014 3 6 1 8 1 1 14 17 20 22 125 16 .2041 .2068 • 2095 .2122 .2148 •2175 .2201 .2227 .2253 .2279 3 5 8 1 1 »3 16 18 21 '24 >7 .2304 .2330 •2355 • 2380 .2405 .243d •2455 .2480 .2504 .2529 2 5 1 7 10 15 »7 20 22 18 .2553 •2577 .2601 .2625 .2648 .267s •2695 .2718 •2742 .2765 2 5 1 7 1" '4 16 ,»9 21 19 .2788 .2810 .2833 .2856 .2878 .2900 .2923 •2945 .2967 .2989 2 4 1 ^ 9 1 1 '3 16 18 20 i 20 .3010 .3032 • 3054 • 3075 .3096 .311S •3139 .3160 • 318. • 3201 2 4 l6 8 1 1 '3 '.5 17 19 21 .3222 •3243 •3263 • 3284 •3304 •3324 •3345 •3365 •3385 • 3404 2 4 ‘ 6 8 10 12 M '16 j8 22 •3424 •3444 .3464 • 3483 .3502 •3522 • 354 > •3560 •3579 •3598 1 2 4 1 6 8 10 12 14 15 '7 23 •3617 .3636 .3655 •3674 .3692 • 37 ” •3729 •3747 .3766 •3784 i 2 4 6 7 9 1 1 l '3 !‘5 '7 24 .3802 .3820 •3838 .3856 •3874 .3892 •3909 •3927 •3945 .3962 1 ^ 4 5 7 9 [ 1 16 i 25 •3979 •3997 .4014 .4037 .4048 • 4065 .4082 .4099 .4116 •4133 i ^ 3 5 7 9 10 1 12 1 '5 26 .4150 .4166 .4183 .4200 .4216 .423. •4249 .4265 .4281 .4298 2 3 5 7 8 10 II 13 15 27' •4314 •4330 .4346 .4362 •4378, •4393 .4409 •4425 .4440 .4456 1 3 5 6 8 9 II 13 M 28 .4472 •4487 .4502 .4518 •4533 •4548 .4564 •4579 •4594 .46^ 3 5 6 8 9 11 12 l «4 29 .4624 •4639 .4654 .4669 .46^3 .4698 •4713 .4728 .4742 •4757 j ' 3 4 6 7 9 10 12 1 ,13 30 •4771 .4786 .4800 .4814 .4829 •4843 •4857 .4871 .4886 .4900 3 4 6 7 9 10 1 II 13 3 ^ .4914 .4928 .4942 • 4955 .4969 •4983 •4997 .5011 .5024 .5038 3 4 6 7 8 10 11 ,12 32, .5051 .5065 •5079 .5092 •5105 • 5”9 •5132 •5145 •5159 •5172 ; I 1 3 4 5 7 8 9 II 12 33 .5185 .5198 .5211 • 5224 •5237 .5250 .5263 .5276 .5289 .5302 I 3 i ^ 5 6 8 9 '10 12 34 •5315 •5328 •5340 •5353 .5366 •5378 •5391 •5403 .5416 •5428 I 1 1 ^ 4 5 6 8 9 II 35 •5441 •5453 •5465 • 5478 •5490 •5502 •5514 •5527 •5539 •5551 I 2 4 5 6 7 9 10 36, •5563 •5575 •5587 •5599 .5611 .5623 •5635 •5647 .5658 .5670 I 2 ! 4 5 6 7 8 10 II 37 .5682 •5694 • 5705 • 5717 •5729 •5740 •5752 •5763 •5775 •5786 I 2 3 5 6 7 8 9 10 38, •5798 .5809 .5821 • 5832 •5843 •5855 .5866 •5877 .5888 •5899 I 2 3 5 6 7 i 8 9 10 39 • 59 ” .5922 •5933 • 5944 •5955 .5966 •5977 .5988 •5999 .6010 I 2 3 4 5 7 8 9 10 40 .6021 .6031 .6042 • 6053 .6064 .6075 . 6085 .6096 .6107 .6117 I 2 I 3 4 5 6 8 9 10 41 .6128 .6138 .6149 .6160 .6170 .6180 .6191 .6201 .6212 .6222 1 2 3 4 5 6 7 8 9 42 .6232 .6243 .6253 .6263 .6274 .6284 .6294 .6304 •6314 .6325 I 2 3 4 5 6 7 8 9 43 •6335 •6345 .6355 •6365: •6375 .6385 •6395 .6405 .6415 .6425 I 2 3 4 5 6 7 8 9 44 •6435 .6444 • 6454 .6464 .6474 .6484 •6493 .6503 •6513 • 6522 2 3 4 5 6 7 8 9 45 •6532 .6542 • 6551 • 6561 •657’ .6580 .6590 •6599 .6609 .6618 I 2 3 4 5 6 7 8 9 46 .6628 .6637 .6646 .6656 .6665 .6675 .6684 .6693 .6702 .6712 I 2 3 4 5 6 7 7 8 47 .6721 .6730 • 6739 .6749 .6758 .6767 .6776 .6785 .6794 .6803 I 2 3 4 5 5 6 7 8 48, .6812 .6821 .6830 .6839 .6848 .6857 .6866 .6875 .6884 .6893 I 2 3 4 4 5 6 7 8 49 .6902 .6911 .6920 .6928 •6937 .6946 •6955 .6964 .6972 .6981 I 2 3 4 4 5 6 7 8 50 .6990 .6998 .7007 .7016 .7024 •7033 .7042 .7050 •7059 .7067 I 2 3 3 4 5 6 7 8 51 .7076 .7084 • 7093 .7101 .7110 .7118 .7126 •7135 •7143 •7152 I 2 3 3 4 5 6 7 8 52 .7160 .7168 .7177 • 7183 •7193 .7202 .7210 .7218 .7226 •7235 I 2 2 3 4 5 6 7 7 53 •7243 .7251 • 7259 .7267 •7275 •7284 .7292 .7300 .7308 •7316 I 2 2 3 4 5 6 6 7 54' •7324 •7332 • 7340 .7348 . 735 f •7364 •7372 .7380 .7388 •7396 I 2 2 3 4 5 6 6 7 TABLES. 631 TABLE III. — Continued. Logarithms of Numbers. Nat. Nos, 1 0 1 3 3 4 5 6 7 8 9 Proportional Parts. 1 2 3 4 5 6 7 8 9 ,S 5 .7404 • 74*2 • 74*9 • 7427 •7435 •7443 • 745 * •7459 .7466 •7474 1 2 2 3 4 5 5 6 7 5 t> .7482 .7490 •7497 •7505 •75*3 •7520 • 7528 •7536 •7543 • 755 * I 2 2 3 4 5 5 6 7 57 •7559 .7566 •7574 • 7582 •7589 •7597 .7604 .7612 .7619 •7627 I 2 2 3 , 4 5 5 6 7 58 •7634 .7642 .7649 •7657 .7664 .7672 .7679 .7686 .7694 .770*1 I I 2 3 4 4 5 6 7 59 .7709 • 77*6 •7723 • 773 * •7738 •7745 •7752 .7760 •7767 • 7774 : 1 2 3 4 4 5 6 7 60 .7782 •7789 .7796 •7803 .7810 .7818 •7825 •7832 •7839 .7846 I I 2 3 4 4 5 6 6 61 •7853 .786c .7868 •7875 .7882 .7889 .7896 •7903 • 79*0 • 79*7 I I 2 3 4 4 5 6 6 62 .7924 • 793 * •7938 •7945 •7952 •7959 .7966 ■7973 .7980 .7987 I I 2 3 j 3 4 5 6 6 63 •7993 .8000 .8007 .8014 .8021 .8028 •8oj5 .8041 .8048 •8055 I I 2 3 3 4 5 5 6 64 .8062 .8069 •8075 .8082 .8089 .8096 .8102 .8109 .8116 .8122 I 2 3 3 4 5 5 6 65 .8129 .8136 .8142 .8149 .8156 .8162 .8169 .8176 .8182 .8189 I I 2 3 3 4 5 5 6 66 •8195 .8202 .8209 .8215 .8222' .8228 •8235 . .8241 .8248 •8254 I I 2 3 3 4 5 5 6 67 .8261 .8267 .8274 .8280 .8287' .8293 .8299 • 8306 • 8312 •8319 I I 2 3 3 4 5 5 6 68 •8325 ■8331 •8338 1 -8344 •835* •8357 •8363 •8370 .8376 ■83821 I I 2 3 3 4 4 5 6 69 .8388 •8395 .8401 .8407 .8414 0 00 .8426 •8432 •8439 •8445 I 2 2 3 4 4 5 6 70 .8451 •8457 .8463 .8470 .8476 . 8482 .8488 •8494 . 8500 ■ 8506 I I 2 2 3 4 4 5 6 71 • 85'3 •8519 •8525 •853' •8537, • 8543 •8549 •8555 • 8561 •85671 I I 2 2 3 4 4 5 5 72 •8573 ■•8579 •8585 •859* •8597, .8603 .8609 .8615 .8621 .8627' I I 2 2 3 4 4 5 5 73 .8633 .8639 .8645 .8651 .8657 .8663 .8669 .8675 .8681 .8686, I I 2 2 3 4 4 5 5 74 .8692 .8698 .8704 .8710 .8716 .8722 .8727 •8733 •8739 •8745'; I 2 2 3 4 4 5 , 5 75 •8751 .8736 .8762 .8768 •8774 • 8779 •8785 ■879* •8797 .8802* T r 2 2 3 3 4 5 5 76 .8808 .8814 .8820 1 .8825 •8831 •8837 .8842 .8848 • 8854 •8859^ I I 2 2 3 3 4 5 5 77 .8865 .8871 .8876 .8882 .8S87 •8893 .8899 .8904 .8910 •8915' I I 2 2 3 3 4 4 5 78 .8921 •8927 .8932 • 8938 •8943 • 8949 •8954 .8960 .8965 .89711 I I 2 2 3 3 4 4 5 79 .8976 .8982 .8987 •8993 .8998 .9004 .9009 .9015 .5020 •9025 I 2 2 3 3 4 4 5 80 .9031 .9036 .9042 .9047 •9053 .9058 .9063 .9069 .9074 • 9079 ] I I 2 2 3 3 4 4 5 81 .9085 .9090 .9096 .9101 .9106 .9112 • 9**7 .9122 .9128 ■91.33! I I 2 2 3 3 4 4 5 82 .9138 • 9*43 • 9*49 • 9*54 •9159' .9165 .9170 • 9*75 .9180 .9186^ I 1 2 3 3 4 4 5 83 .9191 .9196 .9201 .9206 .9212 • 9217 .9222 •9227 •9232 •9238I I ■ I 2 2 3 3 4 4 5 84 •9243 .9248 •9253 I -9258 .9263 .9269 .9274 •9279 .9284 •9289 ^1 I 2 2 3 3 4 4 5 85 .9294 •9299 •9304 .9309 •93*5 .9320 •9325 ■9330 •9335 1 •9340^ 1 1 1 I 2 2 3 3 4 4 5 86 •9345 •9350 •9355 .9360 •9365, •9370 ■9375 •9380 •9385 ■9390 ^1 I 2 2 3 3 4 4 5 87 •9395 .9400 .9405 • 94*0 • 94 * 5 ‘ .9420 •9425 •9430 •9435 ■9440' 0; I I 2 2 3 3 4 4 83 •9445 •9450 •9455 1 . 9460 •9465 .9469 •9474 ■9479 .9484 •9489' I I 2 2 3 3 4 4 89 •9494 •9499 .9504 ! -9509 • 95*3 • 95*8 •9523 •9528 •9533 •9538, 0! 1 2 2 3 3 4 4 90 •9542 •9547 •9552 •9557 .9562 .9566 • 957 * .9576 •9581 •9586 1 0 I I 2 2 3 3 4 91 •9590 •9595 .9600 .9605 . 9609 .9614 .9619 .9624 .9628 ■96.33; 0 I I 2 2 3 3 4 4 92 .9638 •9643 .9647 9652 •9657 .9661 .9666 .9671 •9675 .9680 0 I I 2 2 3 3 4 4 93 .9685 .9689 .9694 .9699 •9703 .9708 • 97*3 • 97*7 •9722 •9727 0 I 1 2 2 3 3 4 4 94 • 973 * •9736 .9741 •9745 •9750 •9754 •9759 •9763 .9768 •9773 0 I 2 2 3 3 4 4 95 •9777 .9782 .9786 .9791 •9795 .9800 .9805 .9809 .9814 .9818 0 I I 2 2 3 3 4 4 96 .9823 .9827 •9832 .9836 .9841 9845 .9850 ■9854 •9859 .9863 0 I I 2 2 3 3 4 4 97 .9868 .9872 •9877 .9881 .9886 .9890 •9894 •9899 •9903 .9908 0 I I 2I 2 3 3 4 4 98 • 99*2 • 99*7 .9921 .9926 •9930 •9934 •9939 •9943 .9948 •9952 0 I I 2 2 3 3 4 4 99 • 995 ^ .9961 •9965 .9969 •9974 •9978 •9983 •9987 • 999 * •9996 I 2 2 3 3 3 4 Logarithmic Traverse Table. § 173. Zero angle at South Point, and increasing to W. (90“), N. (180®), E. (270®). 632 SU/>! VE YING. TABLES. 633 A 000 m m n M A oco r^vo W MMMW I 0^00 10 • o ^0 VC VO VO 0 0 0-0^ O O' C> 3 ' O On OV O' Q c 5 vovovovovo'o \f\\r\\r\ O' O' O 0"0 O' O' Os o> ^ ^ w- w w ^ O'O O'O'O'O'OO'O' v 3 O'O'O'O'O'O'O'OsO' v 3 O'OsOsO'O'O'OsO'O' ^ O'O'OsO'OsO'OsO'O' ^ O 0 ' 0 s 0 'c> 0 ' 0 s 0 s 0 ' 0 s O O'. OsO'O'OsOsO'OsO' Q O' O' O'CO rts r-- O roso CN fO ro ro Cv VC VO 'O VO O ro O' .1** roso 00 •'4- I- c ‘ O ro-.w . ^ 10 10 •O'O 'O w t^oo 00 00 00 ^ O VO'OvOvO'O'OvO'OVO ^ 'OvOVOvO'O'OsO'OvO ^ 00 lOMVOMVOOVOO^aQ VO O' - '»^sO O' >-» 1-vO ^ O'O'OOOO-^-* H vovo VC r^co O' o »-• €v w C 4 (N - VX^ ^ 0 CO CO fo fo CO cn fo rc CO lovo t>.oo O' 0 O'CO t^vo 10 • O'CO t^vo 10 fO N H 0 O'©© tN.VO 10 CO N : ^ . i o O' O' O' O' ^ O' O' O' O' i: O' O' O' O' w 5 O' O' O' OS Q CO 00 00 00 00 00 00 00 O© Ci O'OO'O'OS'O'O'O'O' ^ O'O'O'OsO'O'O'OsO' ^ OsO'O'O'O'OsOsO'Os ^ 00 00 ?0 00 00 00 00 00 CO ffi O'O'Osa'OsOsO'O'Os j: O'O'O'O'O'O'O'O'O' vS OO'O'OvO'O'O'OvO. gi O'O'OsD'O'^'O'O'O' iT O'O'O'O'C'O'OsO'O' w 5 <>c>(>c^ 0 ' 0 ><> 0 > 0 > 01 C^ t>.00 00 00 O' O' o O O O' t^vo « CO O' M VC O '^00 •- loioiotnioioioioj VO t'^oo O' 0 « N C M fo CO CO fo fo CO \o c^oo o^ o w .2 © 000 10 Tj- CO N M Q 0 P~ t". tv. VO .o O' »ooo O O' 0000 q6 ?> tN. 0 M M Q\VO H SO Cl CO f:. O'CO VO H O' C^ t>»00 Os 0 M *-i (s cc H M ww>-«WNCr, <- 'Q 0 f'l'O 00 O' O 00 f- ^ O lO - t-.roo'-ro moo moo <^oo w.oo moo S r- m >3 ~ m P. - m t' 3 \0 >0 ^mm m •'*■-r■'1•'^••'^'^■'^■■^••»-'r■'r-^•*^■'^■r^■mmmmmmmmmmmmmfom^'■. mmw « « ri O' CT> O' C7> o d. d> Cos. Dif. for MNmN-^-'^•'^>r' uo'O ,0 t^oo O'O'00«ciMm'<--»- lO'O rx r^oo O 0 « « m io>o vO '0^0 'O 'O Log. sin. (Dep.) 'O 0 »A>0'fi'0 O' rovo O' O' O' ^ 1^00 *- -f* r>. 0 mvo o*- 0 M M Cl M in'O 'C'O rN.r>.f>.oocooo O'OOO'O O 0 0 — ^ ^ 00 CO 00 00 C^OO O05^0^000''> O'O'O'O'O'O'O'O'O. O'O'O'O'O'O'O'O'O'O'O O'O'O^O'O' O' O 0‘0'000'0'0‘0'0'0» O' O' O' O O' O' o* Arc ist and 3 d Quad- nints. x 1 A* 1 “-I 1 X 1 1 So 1 CJ •“i ^ ' n o o ' i.o ' o ' 1.0 N 0 0 O 0 0 ;j 0 C 0 0 0 ct OOOOqwOOOOOejOOOOOOjOOOOOO? k \ i 1 ^ if: 1 ^ \ h Arc 2d and 4th Quadrants. O O 0 O o o o o^lo 1? 1? Ici lo lo 0000^00000’^ 00000 *^ 00000^00000^^ 00000"^ ^ io*^(r)c. l>. tvvo vO'O'OvOvO'OvC'O'O'O'OvO'O'OvO'OvO'O'O^O'O'C'O'O'CvO'C'O'O O' O' O' O' O' O' o Cos. Dif. for i'. M M w csi M w c« Ci mci fOCi mvo m'O 'O vO 'O 'O ^s CiWCiWWWCiWC'iCiNNNWWdMCiNWMWeiCiWNWWCICiCiClClWCIN bi cL c - ^ J Q moovO m«oo mroO t^-i-'^oo mN 0"C ro 0''0 m O'VO c* O' m cs co -t 0 rs. rr O- m, - fO ro ro mvo vO r^oo O'O'O ^ - Ci rom-tmirvo t^r^co O'O'O ^ ^ Ci rr m-t'-t if.\o vo mroromrororomrom'^ -r •^-i-'^-i-«^-i-mmmiommmmmmmm 0'0'0'0'0'0'0'0'0'0'0'0'0'0'0'c^0'0'0'0'0'0'0'c^0'0*0'0'0'0'0'0'0'0 0'0*0' O' O' O' O' O' o* o» 1 Arc ist and 3d Quad- rants. 0 0 O O O 0 0 Ci 1 O 1 iH 1 1 ^ 1 •it 1 O ^ '•it •f ^ *0 0000^00000^00000^00000 A? OOOOOO^OOOOOJ^ o e 0 o o o o Ci 1 o 1 - 1 1 CO 1 •n 1 LO O O O C C O Arc 2d and 4th Quadrants. 0 o o o o o o tH 1h |h |iH |h|h Io '^''0 0000^00000^00000^00000 ‘^ooooo'^ooooo*^ ^ m'^rociH- m^rocitn lO'^fCCin-^ lO’^rrociM^ rnTj-fOWM^ 0 O O O 0 o o O 1 1 1 1 1 ® 1 Ci Mleoicoifc'MiMle? iH H tH fH H iH H bi (/) -t; 00 h4 <-> d- mci to^-oo m*-co -t - t^mO'mM moo ^O'mo iomvo ^vo 0 mo ^ Os O'OO vo m m - O ov t^vo m m- ci m O'oo vo m ci m O'oo vo m ci m 0*00 vo m ro ci 0 co ! COOOOOCOOOOOOOOOOOOOOOOOOOCOOOCCOOOOOOCOOOOOOOOOOOOOCOOOCOOOOOOOOOOOOOOO O' O' 0 ©v O' O' O' Cos. Dif. for 1'. mmN mmmm-i-mm-^m-^m'>»-'<#-->i-T)-'i-'j-'!fioTFu-i'<-ir)io-^ioio mvo m lovo lo Log. sin. (Dep.) m 0 Ci m r'^ O' Ci '^tvo 00 O' - m mvo 00 O' Ci m mvo t^oo OvO 0 « Ci roro-^’»i'-tmm O' 0 Ci m mvo 00 O' 0 Ci mo r^co O' h ci rn mvo r^oo 0 — m m mvo »'voo O' 0 10 m to m m m m mvo vovcvovovovovovo t^co cooooooooooooooocjo O' COCOOOOOOOODOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOCOOOOOCOOOCOOOOOOOOOOOC5000 O' O' O' O' O' O' O' Arc ist and 3d Quad- rants. 0 1 © 1 1°- 1 So 1 © 1 0 1 H M 0 0 0 0 OQt 0 0 0 0 oi^ 0 0 0 0 oei 0 0 0 0 on 0 0 0 0 oet 0 0 0 0 OQi W M m -< 1 - W M m -,»■ 10^ « N m m o m - 4 - ia'*^ m m m ^ ia'*' ►- m m -a- ia^^ o o o o o o o »!5|0|*’IX|©|0|H tJI tJI Fjl Fjl 'U5 'IQ iti TABLES, ^37 ^ 1 ^ ^ ® , So oo|ool*-l»- ISjl^loISjl^l; O Oi O O O ( I I inBiiiti i iiHi mimmmiimmmmimm VOVOOOCO 0 w foro liOVO ( 'Cvovo'O t^oo 0000(300000 c^osc>a.aN0 0 0 O h m m n w w ^ ^ ^ iTi ir)\o t^oo 2^8 S ?? Illllirilllillllillr |l|lSl|l|l|l|l|l£ « S S S^S-S,” 2 § S,^S,« 2 S S-S-a" 2 S s.as," 2 S 8>42.« 2 § S.iS.'' 2 g g.3.S.»i 2 g 2,JS,« 2>2-2,82„ 2>2-2,82„ 2,^2,82„ a2-2,82„ 2.2-2>82„ S,^2,82„ S.2-?o82„ 2,^2>8 I \ ^ I ® I ^ fS I H S.S>foS-'i^°^2'cg'2;=S'?.^?oS-f2^g'2.?2 2.5^8^^"^ 8^ i? f: 85 8 vS J::cS ^8 ^ vSvSvo^vSvSvB^I lOOOOOOOO C^a^C^O Os o O 0 m w m 01 IN cn rr) m c^t ^ ^ ^ ^ iTiVO VO 'O -o VOVO CO t^CO 00 OV Ov o 0 O I d d d d d d d rod rO(^(^roc^rorofOfOrocofOrorofOfOcofOfOcorr>corofOrorororororoco'T^'^'^ 1 ” ‘ 1(5 |l|l£ISl|l|lglg w 2 8 8>2-2>« 2 8 8,^S.« 2 8 2 8 2 8 2 8 8,2-8,« 2 8 2 8 |l|l|l|l|l|lil| ii|i®i|i|i®i|i|i| 8 , 8 - 8,8 2 8 >^ 8,8 2 ; 8 ,^ 8 , 82 , 8 , 8 - 8 , 82 ^ 8 , 8 - 8 , 82 ; 8 , 8 - 8 , 82 ^ 8 , 8 - 8,8 2 „ 8 ,^ 8,8 2 g I I I ^ I cl I ^ I I I ^ \0 vO VO tovo t*^vo VO t^vo t^vo l oco osoo ovo'CvCvOovavooooooMO ^'2 ?'8'8,?ro,8§'Rc?s,8'§ 0'0,0'0>0'0,0,5,0'0,0'0'0'^0,0,0,0'0'0,0,0'0'^wSSS I g I g I 3 S I I I I « 2 8 8,8-8« 2 8 8,8-8« 2 8 8,8-8« 2 8 8,8-8,« 2 8 8,8-8,« 2 8 8,8-8« 2 8 8,8-8« 2 8 8,8-8,« I •'5 ' 1(5 I I 638 SUKVEYINC. I-) pa < H w (/3 ps! W < P2! H u <5 K H 5 < o o H-I Arc 2d and 4th Quadrants. 0 r- ^ 1 1 1 C? 0*00 KV© \n ^ rri r* Q c>00 lA ^ « M ^ OvfJO »A ^1 1 1 0 « 0 X . a‘000»A»>P. 0 ►- fve 00 O' ^ m mv© VO h' f^'C 0 fO n M - 0 OuO r^f^O iO’^r*-)fi-00'OOl^;0 ‘A^fO^O C 4 NCIMMH-m«^ ^mmmmmOOOq OOCOO OC CO X tPd hJ Q oooo* OOOO- OOOO- OOOO’ 1 OOOO- 0000*01 OOOO’ CXXX3’ OOOO- OOOO’ OOOO OOOO’ OOOO’ OOOO’ OOOO’ i 0000*01 OOOO- 1 oooooi 6666-6 6666- 6666- 6666- 6666- 6666- 6666- OGGG’G Arc ist and 3d Quad- rants. 1 . 1 1 1 Ci M M m vovo 1*^00 o o ft fA lA^o »-^oo o o •- fi ro -t- in o 1 it g ' 1 1 Arc 2d and 4th Quadrants. h ^ \ 1 1 ^>00 \n ^ N M © C'co r^i© m ft m o 0*00 t-^'O m ?t 1 1 1 « ^ (n CO *-) u d CO © C4 moo •- mvo f'^O'O in «-OOoo»^mmO ^ O'oo v< O'm^co-fO'ONC'Ij - r^mOfOvOc^oo^ c-fOm- ^ mr<*;rrMMc<-«0© OC' COO ocoor^f^vo O mmm*^*^ jai^ mmmmmmmmmin XXX fcr cL Q QO CC jm. r-x t^oo 00 00 00 00 00 rs cooooooooooooooooo ^ croooooooo w oo^O'O'O'a'Ovoa'i* ca'Oo-.'^O'a'O'C']^ oaoc>a» C* O'O'C'OOO'C'OvC'O O'OO'O'C'C'OO'O'CC O'O'O'^^Oi ^ ovC'Ova'O'a'C'O'O'^ O'oa'O'oova'OvO'c^ O'O'O'O'Oi C 5 d o' Arc isi and 3d Quad- rants. 1. ' 1 ' ? 1 S g ' 1 1 Arc 2d and 4th } Quadrants. | k ^ \ 1 1 ^ *<>00 t^vo m ro N M ^ ooo f^v© m m w m a ooo t^vo m lommmmmmmin^ mromenm 1 1 1 o ••'j r Co ^ H-I u d QC O fX •^t-omO'O'-vo^m^ ThOfOf'-O'ifooM'^j-^ owmt^ov ‘*v \o M O'vo - a"0 ^ *-cov©ro'-oommo' mci C"© m ^ MMMOOOOOO'O O'oo cooooof^r^i^r’^C ©©mmm ^ t*^ r-. f^© © ©©)©vo©vo©©© ^ ©vo©©© o6 » X be A d. ►J ” Q to ic ^ iAinmmmm''^mmj-s mm m© © 0'a'0v0N0'0va''^0‘X: oo'O'Cnoo'O'ovo^ oocvoo O O'OvOO'OO'OC'O' Cv O0v0vO'C'0'<^0*0v © O'O'O'O'OV 0>0>OvOnOvOvOvOvOv^ CvO'C'O'O'O'O'QvO'^ C'OvC'OvOi odd Arc ist and 3d Quad- rants. 1 £ . 1 1 1 H w cn ^ m© t-%00 o> o m w m T^ ir© t^oo o o « w m m O 1 ^««h,m»m»m-^nc O ro O ■'J-'O VO -^vo VO ro N M O O' rj- t^vo VO 0 00 00 00 00 0 O' > 2 O' cv O' O' Di O' O' O' o> ^ 0 Q'O'O'O'O'O'O'O'O' O'O'O'Q'rTNa'C'O'O' ^ O'O'O'CT'O'a'O'O'a' w O'O'O'O'O'O'O'O'O' ^ O'O'O'O'O'O'O'O'O' ft O'OvO'O'O'O'O'O'O' ^ O'O'O'O'O'O'O'C'O' WJ O) <> O) O' O) O) O) O) ,25 0 O'O'O'O'O'O'O'O'O' ft O'O'O'O'O'O'O'O'OV ^ O'O'O'O'O' '•Q'O'O' 0 OVO'O'O'O'O'O'O'O' ft © © © ©’ 1 1 1 ® VO hvoo O' © W W W N ^ M N ro » 0 V 0 t'^oo O' ft ro ro ro fO ro fO ro ro ro ^ 1 H PI ro ^ vnvo t^oo O) 0 1 M PI (V) Tp v/-))© t^oo "o' rv> lovovoioiovovovovo 'V 1 1 Tf ro N M ^ cororofo ^ 1 000 hvvo m ro N *-• ft WNMWMNCiNW it 1 1 O'CO r^vo lo ^ ro Ci M 0 MWMMMMriMM ^ 0 1 « 0)00 C^VO VO -vp ro PI "1 fi 1 1 © C 5 VO - 1 - -i- -I- C 9 VO )© VO 'O Q CO O' O' O' 0 0 O' O'CO /-S VO ro 0 VO N 00 10 Ci i: ro ro fO CJ Ci N - M M 0 vO'OvOvO'OvO'O'Ovo © VO»C)rr)NCJ,r^ir,IMft, V0root^rr)0t'--1-0 ' 0 0 0 O' O) 0)00 00 00 VOVOVO vovovoiOvovo 00 CiOO’^OvowinO'^^ '^0r^*f0r^ro0'0 ^ t^NVO VO VO 10 m lO T^ ^ lovoiou^ioiomirjio jjii^ CO 00 06 00 VO VO VO VO ft O' O' O) O) g O' O' O' © <> (^ <> <> 0 VO'O'O'CVOVOVOVOVO ft O'O'O'O'O'O'O'O'O' ^ O'O'O'O'O'OnQnO''^ Cs O'O'O'O'O'O'O'O'O' ^ VOVO ft O'O'O'O'O'O'O'O'O) " O'O'O'O'O'C'O'r^O' © O'O'O'O'O'O'O'O'O) ft t^c*shs.r>.hshs.r*«ir>.r*«k ft O'O'O'no'O'O'rT'O ^ O'O'O'O'O'O'O'O'O' wi O-O'O'O O'O'O'O'O' ft ci ©* ©' © 1 1 1 ^ . © w PI ( 1 ) -p VOVO 1^00 O' M loiovovovoioioiolo 0 1 ^ VO r>.co O' 0 CV C M cv ® M Cl ro to'O O' ft ro ro ro ro ro ro ro ro ro 1 M Ci ro lovo r^oo O' ft 1 640 SURVEY INC. TABI.E V. Horizontal Distances and Elevations from Stadia Readings. § 204, I Minutes. 00 1 0 2° 30 Ilor. Dinr. llor. Dinr. Hor. Dinr. Hor. Dinr. Dist. Elcv. Dist. Elcv. Dist. Idcv. Dist. Elcv 0 . . 100.00 0.00 99-97 1.74 99.88 3-49 99-73 5-23 2 . . “ 0.06 “ 1.80 99.87 3-55 99.72 5.28 4 • • a 0.12 ii 1.86 “ 3.60 99.71 5-34 6 . . u 0.17 99.96 1.92 <( 3-66 “ S-40 8 . . 4 ( 0.23 “ 1.98 99.S6 3-72 99.70 5-46 10 . . ° Hor. Dinr. Hor. Diir. Hor. Iiiff. Hor. DifT. Dist. Elev. Dist. Elcv. Dist. Elcv. Dist. Elcv. 0 . . 92.40 26.50 91.45 27.96 90-15 29.39 ' 89.40 30.78 2 . . 92-37 26.55 91.42 28.01 90.42 29.44 ' 89.36 1 30-83 4 • • 92-34 26.59 9 '-39 28.06 90.38 29.48 89-33 30.87 C . . 92.31 26.64 91-35 28.10 90-35 29-53 89.29 30.92 8 . . 92.28 26.69 91.32 28.15 90.31 29-58 89.26 30-97 10 . . 92.25 26.74 91.29 28.20 90.28 29.62 89.22 31.01 12 . . 92.22 26.79 91.26 28.25 90.24 29.67 89.18 31.06 14 . . 92.19 26.84 91.22 28.30 90.21 29-72 89.15 31.10 i6 . . 92.15 26.89 91.19 28.34 90.18 29.76 89.1 1 3 i-'S i8 . . 92.12 26.94 91.16 28.39 90.14 29.81 89.08 31-19 20 . . 92.09 26.99 91.12 28.44 90.1 1 29.86 89.04 3'-24 22 . . 92.06 1 27.04 91.09 28.49 90.07 29.90 89.00 31.28 24 . . 92.03 27.09 91.06 28.54 90.04 29.95 88.96 3''33 26 . . 92.00 27-13 91.02 28.58 90.00 30.00 88.93 31-38 28 . . 91.97 27.18 90.99 28.63 89.97 30.04 88.89 31.42 30 . . 91-93 27.23 90.96 28.68 89.93 30.09 88.86 3'-47 32 . . 91.90 27.28 90.92 28.73 89.90 30.14 88.82 31-5' 34 . • 91.87 27-33 90.89 28.77 89.86 30.19 88.78 3'-56 36 . . 91.84 27.38^ 90.86 28.82 89.83 30-23 88.75 31.60 38 . . 91.81 27-43' 90.82 28.87 89.79 30.28 88.71 31-65 40 . . 91.77 27.48 90.79 28.92 89.76 30-32 88.67 31.69 42 : . 91.74 27-52 90.76 28.96 89.72 30-37 88.64 3'-74 44 • • 91.71 27.57 90.72 29.01 89.69 30.41 88.60 31-78 46 . . 91.68 27.62 90.69 29.06 89.65 30.46 88.56 31-83 48 . . 91.65 27.67 90.66 29.1 1 89.61 30-51 88.53 3 '-87 50 . . 91.61 27.72 90.62 29.15 89.58 30-55 88.49 31.92 52 . . 91.58 27-77 90-59 29.20 89-54 30.60 88.45 31.96 54 . • 9 t -55 27.81 90-55 29-25 89.51 30-65 88.41 32-01 56 . . 91.52 27.S6 90.52 29-30 89-47 30.69 88.38 32-05 58 . . 91.48 27.91 90.48 29-34 89.44 30-74 88.34 32.09 60 . . 91-45 27.96 90-45 29-39 89.40 30.78 88.30 32-14 ^ = 075 0.72 0.21 0.72 0.23 0.71 0.24 0.71 0.25 c = 1 .00 0.86 0.28 0.95 0.30 0-95 0.32 0.94 0-33 «rz= 1.25 1.20 0-35 1. 19 0.38 1. 19 0.40 1. 18 0.42 TABLES. 645 TABLE V. — Contintud. Horizontal Distances and Elevations from Stadia Readings. Minutes. 0 0 21° 22° 23 ° Hor. Dist. Diff. Elev. Hor. Dist. Diff. Elev. Hor. Dist. Diff. Elev. Hor. Dist. Diff Elev. 0 . . 88.30 32.14 87.16 33-46 85-97 34-73 84-73 35-97 2 . . 88.26 32.18 87.12 33-50 85-93 34-77 84.69 36.01 4 • • 88.23 32.23 87.08 33-54 85.89 34.82 84.65 36.05 6 . . 88.19 32.27 87.04 33-59 85.85 34.86 84.61 36.09 8 . . 88.15 32.32 87.00 33-63 85.80 34-90 84-57 36-13 10 . . 88.11 32.36 86.96 ' 33-67 85-76 34.94 84.52 36.17 12 . . 88.08 32.41 86.92 33-72 85.72 34.98 84.48 36.21 14 . . 88.04 32.45 86.88 33-76 85.68 35-02 84-44 36.25 16 . . 88.00 32-49 86.84 33-80 85.64 35-07 84.40 36.29 18 . . 87.96 32.54 86.80 33-84 85.60 35-11 84.35 36-33 20 . . 87-93 32.58 86.77 33-89 85.56 35-15 84.31 36.37 22 . . 87.89 32-63 86.73 33-93 85.52 35-19 84.27 36.41 24 . . 87.85 32.67 86.69 33-97 85.48 35-23 84.23 36.45 26 . . 87.81 32-72 86.65 34.01 85-44 35-27 84.18 36.49 28 . . 87.77 32.76 86.61 34-06 8540 35-31 84.14 36.53 30 . . 87.74 32.80 86.57 34.10 85.36 35-36 84.10 36.57 32 . . 87.70 32.85 86.53 34-14 85.31 35-40 84.06 36.61 34 • . 87.66 32.89 86.49 34.18 85.27 35-44 84.01 36-65 36 . . 87.62 32-93 86.45 34-23 85-23 35-48 83-97 36.69 38 . . 87.58 32.98 86.41 34-27 85.19 35-52 83-93 36.73 40 . . 87.54 33-02 86.37 34-31 85-15 35-56 83.89 36.77 42 . . 87-51 33-07 86.33 34-35 85.11 35-60 83.84 36.80 44 • • 87.47 33-11 86.29 34-40 85-07 35-64 83.80 36.84 46 . . 87-43 33-15 86.25 34-44 85.02 35.68 83.76 36.88 48 . . 87-39 33-20 86.21 34-48 84.98 3572 83-72 36.92 50 . . 87-35 33.24 86.17 34-52 84-94 35-76 83.67 36.96 52 . . 87-31 C..J to 00 86.13 34-57 84.90 35-80 83-63 37-00 54 • • 87.27 33-33 86.09 34.61 84.86 35-85 83-59 37-04 56 . . 87.24 33-37 86.05 34-65 84.82 35-89 83-54 37-08 58 . . 87.20 33.41 86.01 34-69 84-77 35-93 83-50 37.12 Co . . 87.16 33.46 85-97 34-73 84-73 35-97 83-46 37-16 5 .997.54 59 2 .00058 One. .01803 .9998.4 . 0 : 3 . 518 , . 999:57 .05292 .99800 . 070:41 .997.52 58 3 .00087 One. .018.32 .99983 . 0 : 3 . 577 ! .999:56 .05.321 .998.58 .(OOO-l .9!)7.5<) 57 4 .00116 One. .01862 .9998:3 .0:3606: . 999:35 . 0 . 53 . 50 ! .998.57 .07(K)2 .99748 56 6 .00145 One. .01891 .99982 .0.36:35 . 999 : 5-4 . 05 : 379 , .998.55 .07121 .99746 55 C .00175 One. .01920 .99982 .0.3664! . 999 : 3.3 .054081 .998.51 .071.51 .99744 .54 7 .00204 One. .01949 .99981 .0:3693 .99932 .0.31.37' .998.52 .07179 .99742 53 8 .00233 One. .01978 .99980 . 0 : 372:3 .99931 .0.5466! .99851 : .07208 .99740 52 y .00262 One. .02007 .99980 .0.37.52 . 999:30 .0.319.5I .99.849 .072371 [.997.38 51 10 .00291 One. .02036 .99979 .03781 .99929 .05524 1 .9984i .07266 .99736 50 11 .00320 .99909 ' .02065 .99970 .03810 .90927 .0.5.5.53 .99846' .07295 .997.34 49 12 .00.349 . 99999 .02094 .99978 .0:3839 .99926 .0.5.582 .99844 . 07:121 .997.31 48 13 .00378 .99999 .02123 .999771 .0.3868 .99925 .0.5611 .99842 . 07 . 3 . 5.3 .99729 47 14 .00407 .99999 .021.52 .999171 .0.3897 .99924 .0.5640 .99841' .07:182 .99727 46 15 .00436 .99999 .02181 .99976' .0.3926 .99923 .05669 .99839 .07411 .9972.5 45 IG .00465 .02211 .99976 ' . 0 : 39 . 5.5 .99922 .05698 .99838 .07440 . 9972:1 44 17 .00495 .99999 .02;240 .99975 .0.3984 .99921 .05727 .99836 .07469 .99721 4.3 18 .00.524 .99999 .02269 .99974 1.04013 .99919 .0.57.56 .99834 .074;i8 .99719 42 19 .00.553 .99998 .02298 .99974 1.04042 .99918' .0.5785 .9983.3 .07.527 .99716 41 20 .00582 . 02:327 .99973 1.04071 .99917, .05814 .99831 .07556 .99714 140 21 .00611 .99998 ,02.356 .99972 1.04100 .99916 .0.5844 .99829 .07.5a5 .99712 I 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .05873 .99827 .07614 .99710 ! 38 23 .00669 .99998 .02414 .90971 .041.59 .99913 .0.5902 .99826 .(■7643 .99708 37 24 .00698 OQdOU .02443 .999701 .04188 .99912 .0.5931 .99824 .07672 .99705 36 25 .00727 .99997 .02472 .999691 .04217 .99911 .0.5960 .99822 .07701 .99703 35 2G .00756 .99997, .02.501 .999691 .04246 .99910 .05989 .99821 .077.30 .99701 34 27 .00785 .99997 .02530 .99968 .04275 .99909 .06018 .99819 .07759 .99699 33 28 00814 .99997 .02560 .99967 .01.304 .99907 ! .06047 .99817 .07788 .99696 1 32 29 .00844 .99996 i .02.589 1.99966 .04.3.33 .99906 ' .06076 .99815 .07817 .99694 1 31 30 .00873 ,.99996 .02618 .99966 1.04362 .99905 1 .06105 .99813 .07846 .99692 30 31 .00902 L 99996 .02647 .99965 .04391 .99904 ! .06134 .99812 .07875 .99689 29 32 .00931 1.99996 ! .02676 .90964 .04420 .99902 ' .06163 .99810 .07904 .99687 28 33 .00960 .99995 .02705 .99963 .04449 .99901 1 06192 .99808 .079.33 .99685 27 34 .00989 .99995 .02734 .99963 .04478 .99900 ! .06221 .99806 ' .07962 .996a3 26 35 .01018 .99995 .02763 .99902 .04507 .99898 1 .06250 .99804 : .07991 .99680 25 36 .01047 .99995 .02792 .99961 .04536 .99897 1 .06279 .99803 .08020 .99678 24 37 .01076 .99994 .02821 .99960 .04565 .99896 .06.308 .99801 .08049 .99676 23 38 .01105 .99994 .02850 .99959 .04594 .99894 ! .06337 .99799 .08078 1.99673 22 39 .01134 .99994 .02879 .99959 .04623 .99893 1 .06366 .99797 .08107 1.99671 21 40 .01164 .99993 .02908 .99958 .04653 .99892 1 .06395 .99795^ .08136 .99668 20 41 .01193 .99993 .02938 .99957 .04682 .99890 [.06424 . 99793 ' .08165 .99666 19 42 .01222 .99993 .02967 .99956 .04711 .99889 ' .06453 .99792 .08194 .99664 18 43 .01251 .99992 .02996 .99955 .04740 .99888 [ .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 1.99886 .06511 .99788 .08252 .99659 16 45 .01309 .99991 .0:3054 .99953 .04798 .99885 .06540 .99786 .08281 .99657 15 46 .01338 .99991 .03083 .99952 .04827 .99883 .06569 .99784 .08310 .99654 14 47 .01367 .99991 .03112 .99952 .04856 .99882 .06598 .99782 .O& 3.39 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .99990 .03170 .99950 [.04914 .99879 .06656 .99778 .0^397 .99647 11 50 .01454 .99989 .03199 .99949 .04943 .99878 .06685 .99776 .0&42G .99644 10 51 .01483 .99989 .03228 .99948 .04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989 .032.57 .99947 .05001 .99875 .06743 .99772 .08484 .99639 8 53 .01542 .99988 .03286 .99946 .0.5030 .99873 .06773 .99770' .08513 1.99637 7 54 ,01571 .99988 .03316 .99945 .0.50.59 .99872 .06802 .99768- .08542 .99635 6 55 .01600 .99987 .03.345 .99944 .0.5088 .99870 .06831 .997661 .08571 .99632 5 56 .01629 .99987 . 0 : 3:374 .99943 .05117 .99869 .06860 .997641 .08600 .99630 4 57 .01658 .99986 .0:3403 .99942 .05146 .99867 .06889 .997621 .08629 .99627 3 58 .01687 .99986 . 0 : 34:32 .99941 .05175 .99866 .06918 .99760! .08658 .99625 2 59 .01716 .99985 .0.3461 .99910 .0.5205 .99864 .06947 .99758! .08687 .99622 1 60 .01745 .99985 .0.3490 .99939 .0.5234 .99863 .06976 .997.56 .08716 .99619 / Cosin 1 Sine Cosin Sine Cosin Sine Cosin Sine 0 0 M B 1 Sine / 89“ 0 00 00 87“ 86“ 0 m GO TABLES. 649 TABLE VI. — Continued . ^ Natural Sines and Cosines. 5 0 6 7 » 8 9 9 0 Sine Cosin Sine Cosin Sine Cosln Sine Cosin Sine Cosin / 0 .08716 .99619 . 10453 .99452 .12187 .992M .13917 T99CW .15643 . 98709 1 .08745 .99617 .10482 .99449 .12216 .99251 .13946 .99023 .15672 .987641 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .13975 .99019 ’ .1.5701 .98760 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004 .99015 .15730 .98755! 57 4 .08831 .99609 .10,569 .99440 . 12:302 .99240 .140.33 .99011 .157'58 .98751 56 5 .08860 .99607 .10597 . 994:37 .12331 .992.37 .14061 .99000 .15787 .98746: 55 6 .08889 .99604 .10626 . 994:341 .12360 .99233; .14090 .99002 .15816 .98741! .54 7 .08918 99602 .10655 . 994:31 .12389 .99230 .14119 .98998 .15845 .987.371 53 8 .08947 .99599 .10684 .99428' .12418 .99226' .14148 .98994 .1.5873 GO 52 9 .08976 .99596 .10713 .994241 .124471.992221 .14177 .98990 .15902 .98728 51 10 .09005 .99594 .10742 .99421 1 .12476 .99219 .14205 .98986 j .15931 .98723 50 11 .09034 .99591 .10771 . 99413 ! .12504 .99215 .14234 .98982 ! .15959 .98718 49 12 .09063 .99588 .10800 .99415 .12533 .99211 .14263 .98978 i .15988 .98714 48 13 .09092 .99586 .10829 .99412' .12562 .99208 .14292 .98973 ! .16017 .98709 47 14 .09121 .99583 . 10858 .994091 .12591 .99204 .14320 .98969 .16046 .98704 46 15 .09150 .99580 .10887 .99406 .12620 .99200 .14349 .98965 .16074 .98700' 45 16 .09179 .99578 .10916 .99402 .12049 .99197 .14378 .98961 .16103 .98095' 44 17 .09208 .99575 .10945 .99399 .12678 .9919.3 .14407 .989.57 .16132 .98690 43 18 .09237 .99572 .10973 .99.396 .12706 .99189 .14436 .98953 .16160 .98680 42 19 .09266 .99570 .11002 . 9939:3 .12735 .99186 .14464 .98948 .16189 .98681 41 20 .09295 .99567 .11031 .99390 .12764 .99182 .14493 .98944 .16218 . 98676 1 40 21 .09324 .99564 .11060 .99386 .12793 .99178 .14522 .98940 .16246 .98671 39 22 .09353 .99562 .11089 .99383 .12822 .99175 .14551 .98936 .16275 .98667' 38 23 .09382 .99559 .11118 .99380 .12851 .99171 .14580 .98931 .16304 .98662; 37 24 .09411 . 99556 .11147 .99377 .12880 .99107 .14608 .98927 .16333 .98657: 36 25 .09440 .99553 .11176 . 99.374 .12908 .99163 .14637 .98923 .16361 .98652 35 26 .09469 .99551 .11205 .99370 . 129:37 .99160 .14666 .98919 .16390 .986481 34 27 .09498 .99548 .11234 .99,367 .12966 .99156 .14695 .98914 .16419 .980431 33 28 .09527 .99545 .11263 .99364 .12995 .99152 .1472.3 .98910 .10447 .986381 32 29 .09556 .99542 .11291 .99,300 . 1:3024 .99148 .14752 .98906 .16476 .98633 31 30 .09585 .99540 .11320 .99357 .13053 .99144 .14781 .98902 .16505 .98629 30 31 .09614 .99,537 .11349 .99354 .13081 .99141 .14810 .98897 .16533 .98624 29 32 .09642 . 99 . 5:44 .11.378 . 99:351 .13110 .99137 .14838 .98893 .16562 .98619 28 33 .09671 .99531 .11407 .99347 .1.3139 .99133 .14867 .98889 .10591 .98614 27 34 .09700 .99528 .11436 . 99:344 .13168 .99129 .14896 .98884 : .16620 .98609 26 35 .09729 .99526 .11465 .99341' .13197 .90125 .14925 1 .98880 1 .16648 .98604 25 36 .09758 .99523 .11494 . 99 . 3.37 .1,3226 .99122 .14954 1.98876 ^ .16677 .98600 24 37 .09787 .99520 .11523 .99.3.341 .13254 .99118 .14982 .98871 , .16706 .98595 23 38 .09816 .99517 .11552 .99.331 .1.3283 .99114 .15011 .98867 i .16734 .98590 22 39 .09845 .99.514 .11580 .99.327 .13312 .99110 .15040 .98863 .16763 .88585 21 40 .09874 .99511 .11609 .99324 .13341 .99106 .15069 .98858 .16792 .98580 20 .09903 .99.508 .11638 .99320 .1,3370 .99102 .15097 .98854 .16820 .98575 19 42 .09932 .99.506 .11667 .993171 .13.399 .99098 .15126 .98849 .16849 .98570 18 43 .09961 .99503 .11696 .993141 .13427 .99094 .15155 .98845 .16878 .98565 17 44 .09990 .99500 .11725 . 99.310 .13456 .99091 .15184 .98841 .16906 .98561 16 45 .10019 .99497 .117.54 •99307 .13485 .99087 .15212 .98836 .16935 .98556 15 46 .10048 .99494 .1178:3 .99.3031 .13514 .99083 .15241 .98832 .16964 .98.551 14 47 .10077 .99491 .11812 .99.300, .13543 .99079 .15270 .98827 .16992 .98546 13 48 .10106 .99488 .11840 .99297 .1.3.572 .99075 .15299 .98823 I .17021 .98.541 12 49 .10135 .99185 .11869 .99293' .13600 .99071 .15327 .98818 .17050 .98536 11 50 .10164 .99482 .11898 .99290 .13029 .99067 .15356 .98814 .17078 .98531 10 51 .10192 .99479 .11927 .99286' .136.58 .9906.3 .1.5385 .98809 .17107 .98.526 9 52 .10221 .99476 1 .119.56 .99283 .1.3687 .99059 .15414 .98805 .17136 .98521 8 53 1 . 10250 .99473 1 .11985 .99279 .13716 .990.55 .1.5442 1.98800 .17164 .98516 7 54 .10279 .99470 .12014 .99276 .1.3744 .99051 .1.5471 .98796 .17193 .98511 6 55 i .10308 .99467 .12043 .99272 .1.3773 .99047 .1.5.500 i. 98791 .17222 .98.506 5 56 1 .10337 .99464 .12071 .99269 .1.3802 .99043 .1.5.529 i. 98787 .17250 .98.501 4 57 10366 ,.99461 .12100 .99265 .i:i83l .99039 .1.5.5.57 .98782 .17279 .98496 3 58 .10395 .994.58 s .12129 .99262 .13860 .990.35 .1.5.586 .98778 .17.308 .98491 2 59 ' .10424 .994.55 .121.58 .992.58 .1.3889 .99031 .1.5615 .98773 .17:336 .98486 1 60 .10453 ,.994.52 .12187 . 992.55 j . 1 : 191 7 .99027 .15643 i. 98769 .17.365 .98481 0 / Cosin 1 Sine Cosin Sine ! Cosin Sine Cosin 1 Sine Cosin Sine t 84» 1 83“ 1 82“ 81“ 80“ 650 SUR VE YING. TABLE VI. — Continued. Natural Sines and Cosines. 1 10° 11 ° 12 ° 13“ II H’ 1 Rino Cosin Sine Cosin ' Sine Cosin Sine 1 Cosin 1 Sine (’osIn 0 I.173G5 .9.8181 .19)81 .98] 0:3 1 .2079i '.97)^) >2495 ■97437 721192 !):o:3»! (T) 1 1.17393 .98470 .19109 .981.57 1 .2))M'20 .97809 .22.52.3 .974:30 11 .242-2) .!)7^)^•5 .50 2 ! .17422 .9.8471 .I9i:}8 .981.52 1 .20848 1.97803 .22.5.52 .97421 .2124 - .07() .97772 .22093 .97:i.)l [ .21:500 .00080' .53 8 1.17591 .98110 .19:309 .93118 .21019 .9770() 22722 .97:381 .21118 .90!)73: .52 9 .17021 .9.8135 .193.3,8 .93112 .21017 .95700 .227^ .97:578 .21110 , 00000 1 .51 10 .17051 .98430 .19360 .9.3107 .2107(5 .97754 .22778 .97:371 .21174 .0,60.501 .50 11 .17GS0 1.98125 .19395 .08101 .2iiai .97748 . 2-2807 .97.305 .24.503 .009.52! 40 12 .17708 .93120 .19423 .930.)3 .21132 .97742 .22835 . 97 : 35 s! .21.531 .!Mi01.5| IS 13 .17737 .98111 .194.52 .98.:90 .21101 . 977 : 3.5 . 22 . 30:3 .97:351 1 .24.5.5!) .0(l!):i7 47 14 . 1 7703 .98100 .19481 .93034 .21189 .97729 .22892 . 97:3451 .24587 .000.30 1 40 15 .17791 .98401 i .19.509 .98070 .21218 .97723 .22920 .97:338 .21015 .0!i0».3| 45 IG .17811 .9830.) ! .19.5.38 .98073 .21240 .97717 .22948 .973:31 ' .24C44 .000101 44 17 .17852 .98391 .19500 .9.8017 .21275 .97711 .22977 .9732.5 .21072 .OOOiX)! 4.3 18 .17880 .98380! .19595 .98031 .21.303 .97705 .23005 .97318 .21700 .000v)2j 42 10 .17900 .98381! .19023 .9'8).53 .21331 .97098 .2:30:3:3 .97.311 .24728 .008041 41 20 .1793 7 .98378 .19052 .9.3050 .21300 .97092 .23062 .97304 .247.50 .00887! 40 21 .17900 .93373 .19080 .93044 .21.388 .97aso* .2.3090 .97208 .21784 . 90880 ' 39 22 .179.15 .93308 .19700 .98030 .21417 .97080 .2.3118 .97-291 .24813 .9087,5: 88 23 .18021 1 [.98302 i .19737 .9 5033 .21415 .97073 .2,3140 .97284 .24841 .90,3001 ‘37 24 .18052' .93357 1 .19700 .93027 .21474 .97007 .2.3175 .97278 .24800 .90.8.58: 30 25 .18081 .93.352 I .19791 .93021 .21502 .97061 .2.3-20.3 .97271 .24897 .90851 ‘ ,35 23 .18100 .93347 ! .19823 .93013 .21.530 .97655 .2.3231 .97'204 .24025 .90844 31 27 .18138 .98.341 .19351 .93010 .21559 .97613 .2.3200 .97257 .24954 .90.337 3:3 23 .18100 .9.8.330 .19830 .93001 ' .21.537 .97312 .23288 .97251 .24982 .90829 32 29 .18195 .93331 .19908 .97003 1 .21010 .97633 .28316 .97244 .25010 .90822, 31 30 .18224 .98335 .19937 .97002 .21641 .97630 .23345 .97237 .25038 .90815; 30 31 .18252 .98320 .19905 .970378 .21672 .97023 .2a37.3 .97230 .25066 .90807 j 29 32 .18281 .93315 .19994 .97031 .21701 .97617 .23401 .97223 1 .2.5094 .9(3800; 23 33 .18300 .93.310 .20022 .9707.5 .21720 .97611 .2.3420 .97217; 1 .25122 .907931 27 34 .18338 .90.304 .20051 .97030 .21753 .97301 .23458 .97210! 1 .25151 .90780, 26 35 .18307 .93290 .20079 .97033 .21733 .97.503 .23486 .97203! 1 .25179 . 90778 i 25 36 .18395 .93204! ! .21103 .97053 .21314 .97502 .23.514 .97190 ! .25207 .90771 24 37 .18121 .93238: i .20130 .97052 .21843 .97535 .23542 .97189 i .25235 .98764 : 2:3 38 .184521 .9.L’33' ! .20105 .97013 .21871 .97570 .23571 .97182; j .2.5263 .9C7-.50l22 39 .18431 ' .93277! ! .20103 .97040 .21300 .97573 .23599 .97170, .25291 .96749! 21 40 .18509 .98272! ! .20220 .97034^1 .21923 .9 7563 j .23627 .97169 .25320, .96742 20 41 .18538 .932071, ; .20250 .97023' .21953 .97560 [ .23656 .97162' 1 .2.5348 .907341 19 42 .18507 .932011 1 .20270 .97022 ' .21035 .97553 .23684 .97155; .25:370' .907271 18 43 .18595 .93250! .2J307 .97013 ; .2201-3 .97547 .23712 .97148 .25404 1 .96719 17 44 .18624 .93250 .20330 .97010 i .22041 .975411 .23740 .97141! ' .254:32! .90712! 10 45 .18052 ' .93215 .20331 .97.005 .22070 .97534 1 .2.3769 .97134 ; .25460 .90705 15 4G .18081 1 .93240. .20303 .97300 .22003 .97.523' .23797 .97127 1 .2.5488! .90097 14 47 .18710 .9323L .20121 .973.03 .22126 .97521; .2:3825 .97120 ! .25516' .90090 13 48 .18738 .93220, .20450 .97337 .22155 .97.515: .2:3853 .97113 .255451 .90082! 12 49 .18707 .932231 .20173 .97331 .221831 .97503: .2.3882 .97100 .2.5.573! .9(3075 11 50 .187’95 .98213, .20507 .97375 ! .22212 .97502 j .2:3910 .97100, 1 .2-5001 . 90007 1 10 51 .18824 .98212 1 .20535 .97869'! .22240 .974961 .23938 .97093 .25029 . 96060 1 9 52 .18852 .98207 ; .20503 .97333,1 .22268 .97489 .2:3966 .970801 1 .2.5657 .900.>ii 8 53 .18881 .98201 1 ! .20.502 .978.5711 .22207' .97483 .2:3995 .97079 1 .25085 .90015' 7 54 .18910 .98190| ! .203201.978514 .223251 .97476 .24023 .97072 .2.5713 .900.38: 0 55 .18938 .981901 .200401 .97345 j .22.353 .97470 .24051 .97065 .2.5741 .90630' 5 5G .18907 .98185! ! .200771 .978:30! .22.382, .97403! .24079 . 97058 .25769 .90023! 4 57 .18995 .981791 I .20706' .97833' 1 .22410; .97457; .24108 .97051 .2.5798 .9(3015 3 58 .19024 .981741 1 .207.34 .97827 1 .224.38! .97450; .24130 .97044 .2.5820 .90008 2 59 .19052 .981081 i .207'03 .97821! .22467 .97444; .24104 .970:37 .2,58.54 .90000 1 GO .190811.98103! 1 .20701 .2‘2495 .97437 .24192 .970:30 .2.5882 .90.593 0 Cosin 1 Sine Cosin ' Sine 1 Cosin Sine Cosin Sine Cosin Sine ; 1 / 79° 1 i 78° '! 1 770 76° 75° ! TABLES. 651 TABLE VI. — Continued. Natural Sines and Cosines. 15“ 16° 17° 18° 1 19° Sine Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine Cosin T .25882 .96593 .27564 .96126 .29237 .95030 730902 79510(5 .32557 114^52 60 1 .25910 .96585 .27592 .96118 .29265 .95622 .30929 .9.5097 .32584 .94542 59 2 .25938 .96578 .27620 .96110 .29293 .95613 .30957 .9.5088 .32612 .94533 58 3 .2.5966 .96570 .27648 .96102 .29321 .95605 .30985 .9.5079 ! .32639 .94.523 57 4 .25994 .96562 .27670 .96094 ,29:348 .95596 .31012 .95070 .32067 .94514 56 5 .26022 . 90555 .27704 .96086 .29:376 .95588 .31040 .95061 .32694 .94504 55 G .2G050 .90547 .27731 .96078 .29404 .95579 .31068 .950.52 .32722 .94495 54 .2G079 .90540 .27759 •96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .20107 .90532 .27787 .96062 .29460 .95562 .31123 .9503.3 .32777 .94476 52 9 .26135 .90.524 .27815 .96054 .29487 .95554 .31151 .9.5024 .32:04 .94406 51 10 .2G1G3 .90517 .27843 .96040 .29515 .95545 .31178 .95015 ^ .32832 .94457 50 11 .2G191 .90509 .27871 .90037 .29.543 .95536 .31200 .95000 ' .32859 .94447 4S 12 .2621 9 .90502' .27899 .96029 .29571 .95528 .312.33 .94997 : .32887 .94438 48 13 .2G247 .90194' .27927 .98021 .29599 .95519 .31201 [.94988 .32914 .94428 47 14 .20275 .96480! .27955 .90013 .29626 .95511 .31280 '.94979 .32942 .94418 46 15 .2G303 .90479 .27983 .96005 .29654 .95502 .31:310 .94970 .32909 .94409 45 IG .20331 .96471' .28011 .95997 .29682 .95493 .31344! .94901 .32997 .94399 44 17 .26359 .90403 .28039 .95989 .29710 .95485 .31372 .94952 .33021 .94390 43 18 .26387 . 90450 .28067 .95981 .29737 .95476 .31399 .94943 .33051 .94380 42 19 .2G415 .90448 .28095 .95972 .29705 .95407 .31427 .94933 : .33079 .94370 41 20 .26443 .90440 .23123 .95964 .29793 .95459 .31454 .94924 : .33100 .94301 40 21 .2G471 .90-4.33' .28150 . 95950 .29821 .95450 .31482 .94915 ' .33134 .94351 39 22 .2G500 .90425 .28178 .95948 .29849 .95441 .31510 .94900 .£3101 .91342 38 23 .20528 .90417 .28200 .95940 .29876 .95433 .31537 .94897 i .33189 .94332 37 24 .20556 .90410 .23234 .95931 .29904 .95424 .31505 .94888 .33216 .94322 36 25 .20584 .90402 .28202 .95923 .29932 .95415 .31593 .94878 [ .33244 .94313 35 2G .20612 .90304 .28290 .95915 .29900 .95407 .31020 .94809 .33271 .94303 34 27 .20040 .90380 .23318 .95907 .29987 .95398 .31648 .94800 1 .33298 .94293 33 28 .26003 .90379 .23340 .95898 .30015 .95389 .31675 .94851 j .33320 .942^ 32 29 .20096 .90371 .23374 .95890 .30043 .95:380 .31703 .94842 ! .33353 .94274 31 30 .26724 .90303 .23402 .95882 .30071 .95372 .31730 .94832 j .33331 .94204 30 31 .20752 .90355 .23429 .95874 .30098 .95363 .31758 1.94823 ’ .33408 .94254 29 32 .26780 .90347 .23457 .95805 .30120 .95354 .31780 ,.94814 .3:3430 .94245 28 33 .26808 .90340 .28485 .95857 .30154 .95345 .31813 .94805 i .33403 .94235 27 34 .26836 .90332 .23513 .95849 .30182 .95337 .31841 .94795 1 .33490 .94225 26 35 .26864 .96324 .23541 .95841 .30209 .95328 .31808 .94780 -33518 .94215 25 36 .26892 .90310 .28569 .95832 .30237 .95319 .31896 .94777 .3tiL)4o .94206 24 37 ,20920 .90303 .28597 .95824 .30205 .95310 .31923 .94708 ' .33573 .94196 23 38 .26948 .96301 .28625 .95810 .30292 .95301 .31951 .917'58 1 .33000 .94186 22 39 .20976 .90203 .28652 .95807 .30320 .95293 .31979 .94749 .33027 .94176 21 40 .27004 .90285 .28080 .95799 .30348 .95284 .32006 .94740 .33055 .94167 20 41 .27032 .9027?' .28708 .95791 .30376 .95275 .32034 .94730 .33082 .94157 19 42 .27000 .90200 .28730 .95782 .30403 .95266 .32001 .94721 .33710 .94147 18 43 .27088 .90201 .28704 .95774 .30431 .95257 .32089 .94712 .33737 .941:37 17 44 .27116 .90253, .28792 .9.5706 .30459 .95248 .32116 .94702 .33764 .94127 16 45 .27114 .90246; .28820 .95757 .30486 .95240 .32144 .94093 .33792 .94118 15 4G .27172 .902.38' .28847 .9.5749 .30514 .95231 .32171 .94084 .33819 .94108 14 47 .27200 .962.30 .28875 .95740 .305421 .95222 .32199 .94074 .33846 .94098 13 48 .27228 90222 .28903 .9.57.32 .30570! .95213! .32227 .94665 .33874 .94088 12 49 .27250 ; 90214 .28931 [.9.5724 .30.597 .95204! .32254 .94050 33901 .94078 11 50 .27284 90200 .28959 .95715 .30025 .95195 .32282 .94046 1 .33929 .94008 10 51 .27312 '.90198 .2898?! .9.5707 .30653 .95186 .32309 .94637 1 .a3956 .94058 9 52 .27340 .90190 .29015: .95698 .30080 .95177 .32337 .94027 i .33983 .91019 8 53 .27368 .90182 .29042 .9.5090 .30708 .95108 .32304 .94018 .34011 .91039 7 54 .27396 .96174 .29070 .9.5681 .30730 .951.59 .32392 .94009 1 .34038 .94029 6 55 .27424 .90I0ii .29098 .9.5073 .30703 .951.50 .32410 .94599 1 .34005 .94019 5 5G .27452 .901.58 .29126 .95004 i .30791 .95142 .32447 .94.590 1 .34093 .94009 4 57 .27480 .90150 .291.54 .9.50.50 .30819 .951:3:3 .32474 .93580 i .34120 .93999 3 58 .27508 .90142 .20182 .9.5047 .30846 .95124 .32.502 .94571 1 .34147 .93989 2 59 .27.536 .061.34 .20209 .9.50.39 .30874 .95115 .32529 .94501 ! .341751 .93979 1 GO 27.504 .9(5126 .292.37: .9.5030 .30902 .95106 .32.5.57 .94552 1 .34202| .93909 0 / L’osin Sine 1 Cosin 1 Sine Cosin Sine 1 Cosin Sine j Cosin 1 Sine / 740 1 73° 1 72° 1 71° 0 0 452 TABLE VI. — Contiuued. Natural Sines and Cosines. 20“ 21» 1 22" 23" 11 e Sine Cosin Sino Cosin Sino Cosin Sine Cosin Sin*» C’osin 1 0 .34202 .93909 .:4.5a37 .9a‘4.58 .:4740i .9*2718 .39073 .9*20.5) I .40074 .9i:i55' 00 1 .34229 .939.59 ..‘4.")8(;4 .9:4:448 .374H8 . 9*2707 1 ..39I(X) .920:49 1 .40700 .91:313 59 2 ..342.57 .9:4949 ..3.5891 .9:4:i‘i7 .37515 .9*2097 ..‘491*27 .920*28 1 .40727 .91:331 58 3 ..312,S4 .9:49.39 .:3.5918 .9:4:427! ..‘47512 .9*20801 ..‘491.54 .9*2010 , .407.53 .01319 57 4 ..34311 .93929 .:3594.5 .9:4:410' ..37.509 .9*2075 ..‘49180 .92005 ' .40780 .91.307 .V, 5 .343.39 .9:4919 .:3.5973 .9.a‘406 .37.595 .9*2004 ..39*207 .91991 1 .40800 .91295 55 G .34.300 .9:4909 ..3(5000 .9:4295 .37022 .920.53 ' ..392‘M .91982 ' .40833 .91253 .51 7 ..‘34.393 .9.3899 .:3(‘>()27 .9:42R5 .370-19 .9*20-12', .;49*200 .91971 ! .40800 .91 ‘272 i .53 8 ..‘34121 .9:4889; ..‘50().->4 . 9:4274 > .37070 .9*20:41 ' j ..39*287 .919.59 1 .40880 .91200; 52 9 ..‘344 48 .9:4879 .:50081 .9:4204 .3770:4 .9*2020 ! ..39:111, .91918 ; .40913 .91 218 1 51 10 .34475 .93809 .:30108 .9.4253 .377:40 .92009 ■ ..39311 .919.30 1 .409.39 .912361 50 11 ..34.503 .93a59 ..30ia5 .9.3213 .377.57 .92.598 .39307 .91925 ! .409001 .91221! 49 12 ..‘345.‘30 .9:4,849 ..‘40102 .9:4*2:42 ..37781 .9*2587 , .39:494' .91911 1 .409!)2| .91212 48 13 .34,5.57 .9:4839 1 .30190 .93222 .37811 .9*2,570 .:494 21, .91902 .41019| .91200 47 14 .34.584 .9:4829 1 .:30217 .9:4211 .37848 .92505' ..39418; .91891 , .410451 .91188 40 15 .34012 .9.3819 I ..3024 4 .9:4201 ..3780.5 .92554; .39474 .91879, .41072! .91170 10 ..340.‘59 .93809 ! .30271 .93190 .37892 .92543' ' .;49.501 .91808 , .41098; .911(34 ! 41 17 .34000 .9:4799 1 .30298 .93180, ..37919 .92532 ..39528 .91K50 i .41125, .91152 43 18 .31094 .93789 ; ..30.325 .93109 .37940 .92521 ' ..395.55 .91M.5 j .411.511 .91140 42 19 .34721 .93779 i .30.3.52 .93159 .37973 . 9*2510 i ..‘49.581 '.918.53 .41178; ,.91128 41 20 .34748 .93709 1 .36379 .93148 .37999 .92499, .39008 .9182*2 .41204 .91110 40 21 .34775 .937.59 ! .36406 .931.37 ..38020 .92188 .390.a5 .91810 .41231 .91101 39 22 .31803 .93748 1 ..304:34 .93127 .380.5.3 .924771 .:49001 .91799 ; .41257 1.91092 38 23 .34830 .9:47:48 .30401 .9:4116 .38080 .92400 .39688 .91787 .41284 '.91080 37 24 .34857 .9.3728 ! .30488 ! .9.3100 .38107 .92455 .39715 .91775 .41.310 .91008 30 25 .34834 .93718 1 .30515 .9.3095 .381.34 .9244-41 .39741 .91704 ! .41.537 .91056 35 26 .34912 .93708 ..30,542 ' .9.3084 1 .38101 .924321 ..39768 .91752 ; .41.303 .91044 34 27 .34939 .93098 .30509 .93074 .38188 .92421 .39795 .91741 .41.390 .910.32 33 28 .34966 .9.3688 .30590 .9.3003 .38215 ,.92410 .39822 .91729 1 .41416 .91020 32 29 .34993 .93077 .30023 .9.3052 .38241 ,.92399 .39848 .91718 .41443 .91008 31 30 .35021 .93007 ,30650 .93042 .38268 ,.92388 .39875 .91700 .41409 .90996 130 31 .35048 .9.36.57 .36677 .9.3031 .38295 !. 92377 .39902 .91694 .41496 .90984 ' 29 32 .35075 .9:4047 .367(44 .93020 .38322 ; .92300 .39928 .91083 .41522 .9097-2 28 33 .35102 .9.3037 .36731 .9:4010 i .38819 i .92355 ' .39955 .91671 .41549 .90960 27 34 .351.30 .9:3026 .36758 .92999 .38376 ;. 92343 .39982 .91600 ,41575 .90948 20 35 .35157 .9:3616 .36785 .92988 .38403 .92232 .40008 .91648 .41602 .90936 25 30 .35184 .9.3606 .36812 .92978 .38430 .92321 .400.35 .91636 .41628 .90924 24 37 .35211 .93596 .36839 .92907 .38456 .92310 .40002 .91625 .416.55 .90911 23 38 .35239 . 93585 .36807 .92950 .38483 .92299 .40088 .91613 .41681 .90899 22 39 .35266 . 9:3575 .36894 .92945 .38510 .92287 .40115 .91601 ; .41707 .90887 1 21 40 .35293 . 93505 .36921 .92935 .38537 .92276 .40141 .91590 1 .41734 .90875 20 41 .35320 .93555' .36948 .92924' .38564 .92265' .40168 .91578 .41760 .90863 ! 19 42 .35347 .93544 .36975 .92913 .38591 .92254; .40195 .91566 1 .41787 .90851 18 43 .35375 .93534 .37002 .92902 .38017 .92243 .40221 .91555 1 .41813 .90839 17 44 .35402 .93524 .37029 .92892 .38644 .922311 .40248 .91543 ! .41840 .90826 16 45 .35429 .93514 .37056 .92881 .38671 .922*20 .40275 .91531 ! .41866 .90814 15 46 .35456 .93503 .37083 .92870 .38698 . 92209 1 .40301 .91519 1 .41892 .90802 14 47 .35484 .93493 .37110 .92859 .38725 .92198 .40328 .91.508 j .41919 .90790 13 48 .355111 .93483 .371.37 .92849 .38752 .92186 .40355 .91496 .41945 .90778 12 49 .35538 .9:3472 . .37104 .92838 .38778 .92175 .40381 .91484 .41972 1.90766 11 50 .35505 .93462 .37191 .92827 .38805 .92164 .40408 .91472 1 .41998 .90753 10 51 .35.592 .9.3452 .37218 .92816 .38832 .92152 .40434 .91461 .42024 .90741 9 52 .a5619 .9:3441 .37245 .92805 .38859 .92141 .40401 .91449 I .42051 .90729 8 53 .35647 .9:3431 .37272 .92794 .38886 .92130 .40488 .91437 j .42077 .90717 7 54 .35674 .9:3420 .37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 6 55 .3.5701 .9:4410 .:37.326 .92773 .38939 .92107 .40541 .91414 .42130 .90692 5 56 ..35728 .93400 .37353 .92762 .38966 .92096 .40567 .91402 .42156 .90680 4 57 ..3.57.55 .93:389 ..37.380 .92751 .38993 .92085 .40594 .91390 .42183 .90668 3 68 ..35782 .9.3:379 .37407 .92740 ,39020 .92073 .40621 .91378 .42209 .90655 2 59 ..3.5810 .93:308 .37434 .92729 .39046 .92062 .40647 .91366 .42235 .90643 1 60 ..35837 .93:358 .:47401 .92718 .39073 .920.50 .40074 .913.55 .42262 .90631 0 / Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine / 69 » 68 " 67 " 66 ° 65 " TABLES. 653 TABLE VI. — Coniinued. Natural Sines and Cosines. 25 ° 26 “ 27 “ I 00 29 ° Sine Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine Cosin 0 .42262 .90631 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 60 1 .42288 .90618 .43863 .89867 .45425 .89087 .46973 .88281 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 .87434 58 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 .90.569 .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 .905.32 .44046 .89777 .45606 .88995 .47153 .88185 .48684 .87349 52 9 .42499 .90520 .44072 .89764 .45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578 .90483 .44151 .89728 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278: 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87.207 42 19 .42762 .90.390 .44333 .896.36 .45891 .88848 .47434 .8803-1 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45908 .88803 .47511 .87993 .49040 .87150 38 23 .42867 .90346 .44437 .89584 .45994 .88795 .47537 .87979 .49065 .87136 37 24 .42894 . 903:14 .44464 .89571 i .40020 .88782 .47502 .87905 .49090 .87121 36 25 .42920 .90321 .44490 .89558! .40046 .88708 .47588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .40072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .40097 .88741 .47039 .87923 .49166 .87079 33 28 .42999 .90234 .44568 .89519 .46123 .88723 .47665 .87909 .49192 .87064 32 29 .43025 .99271 .44594 .89506 .40149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .44672 .89467 .40220 .88074 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .46252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90208 .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196 .44750 .89428 .46304 .8863-1 .47844 .87812 .49369 .86964 25 36 .43209 .90183 .44776 .89415 .40330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88607 .47895 .87784 .49419 .86935 23 38 .43261 .901.58 .44828 .89389 .46381 .88593 .47920 .87770 .49445 .86921 22 39 .43287 .90146 .44854 .80376 .40407 .88580 .47946 .87756 .49470 .86906 21 40 .43313 .90133 .44880 .89363 .40433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 .88553 .47997 .87729 .49521 .86878 19 42 .43366 .90103 .44932 .89337 .46484 .88539 .48022 .87715 .49546 .86863 18 43 .43392 .90095 .44958 .89324 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90082 .44984 .89311 .46536 .88512 .48073 .87687 .49596 .86834 16 45 .43445 .90070 .45010 .89298 .46561 .88499 .48099 .87673 .49622 .86820 15 46 .43471 .90057 .45036 .89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 .90045 .45062 .89272 .46613 .88472 .48150 .87045 .49672 .86791 13 48 .43523 .90032 .45088 .89259 .46639 .88458 .48175 .87031 .49697 .86777 12 49 .43549 .90019 .45114 .89245 .40064 .88445 .48201 .87617 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .40690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89931 .45192 .89206 .46742 .88404 .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390 .48:303 .87501 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86690 6 55 .43706 .89943 .45269 .89167 .46819 .88363 .48354 .87532 .49874 .86675 5 66 .43733 .89930 .45295 .89153 .46844 .88349 .48379 .87518 .49899 .86661 4 67 .43759 .89918 .45321 .89140 .46870 .88336 .48405 .87504 .49924 .86646 3 58 .43785 .89905 .45347 .89127 .46896 .88322 .484:30 .87490 .49950 .86632 2 59 . 4:1811 .89892 .4.5373 .89114 .46921 .88:308 .48456 .87476 .49975 .86617 1 60 .43837' .89879 . 45:599 .89101 .46947 .88295 .48481 .87462 .50000 .86603 _0 $ Cosin 1 Sine Cosin Sine 1 Cosin Sine Cosin Sine Cosin Sine ! / 63 “ 1 62 “ 61 “ 60 ° 654 SURVEYING. TABLE V\.—CvuiiutuE Natural Sines and Cosines. CO o 31“ i' |co 0 83° II 34^ 1 Sine Cosin Sine Cosi n Sine Cosin ^ Sine ' CosfnII Bine ICosIn 0 ..50000 .86603 .51.504 .857171 752992 ; 84805 j '.M424 .83867,' .55919! .82904 60 1 ..50025 .86.588 .51.529 .8.5702, .5;K)17 .84789 : ..5-1 IKH. .83851 , .6r)913' .82887 .59 2 .50050 .86573 .51.5.54 .8,5687 .5.3041 .84774 ..5-1.513 .85835" ..5.5!HWl ,82.871 ' .58 3 ..50076 .86.5.59 .51.579 .8.5672:1 ..5.3066 ,847.59 .. 54 . 5:37 .83819 ..259921 .82855 .’•,7 4 ..50101 .86514 .51604 .8.56.57; .5:5091 .81743 .5-1.561 .83804, .56016 .828:39 6 ..50126 .86.5.30 .51628 .a5642 .r).3115! .84728 ..54.')K6 .83788 ..56(M0 .82822 . 5.5 6 ..50151 .86.515 .516.53 .8.5627 .53140 .84712 ..5-1610 .83772 ..'•>6061 .82800 54 7 ..50176 .86.501 .51678 .a5612 ,.531M .81697 .,546.35 .817.56 ..560H8 .82790 .53 8 .50201 .86486 .51703 .85597 .!>31H9 .84681 ..516.59 . 83710 ; ..56112 .827'7:3i f -J 9 ..50227 .86171 .51728 .R5.5S2 ..53214 .84066 .51(583 .83724 , ..561.36 .82757 5 . 10 .50252 .86457 .51753 .85567 i .53238 .84650 .54708 .837081 , .56160 .827411 50 11 .50277 .80412 .51778 .8.5.5.51 ' ..5.3263 .846.35' .. 54732 ' .83692' ..56184 . 82724 ' 49 12 .50302 .86427 .51803 .a5.5.36| .. 5:4288 .81619 ..517.50 .85676 ..56208 .82708 48 13 ..50327 .86113 .5182.8 .a5.521 1 ..5.'5312! .81001 ..51781 ' .83(360 .5(7'32 .82692 47 14 ..503.52 .86.391 .518.52 .a5.506 .5.33.37 .81.588 ..54^05' .83615 ..':-32.56 .82675 46 15 .50377 .86.384 .51877 .85491 .,5:4.361 .84;>73 .548291 .83029 .56280 .826.59 45 10 .50403 .86.369 .51902 .a5470 .5.2386 ,81.5.57 .51854' .83(313 .,56.305 .82013 44 17 ..50428 .86.354 .51927 .85101 1 ..53411 .81.542 ..51878 . 83.597 1 ; .56.329 .82026 43 18 ..504.53 .86.340 .519.52 .8.54-16 . 5 : 44.35 .84.526 .54902 .82581 1 ! ..56:3.53 .82610 42 19 .50478 .86325 .51977 .a5431 ..53-160 .84511 ..'.4927 .8:3.5651 .56.377 .82.593 41 20 .50503 .80310 .52002 .85416 .58484 .84495 .54951 .83249 .56401 .82577, 40 21 ..50528 .86295 ..52020 .a5401 .53.509 .84480 ..54975 .8.3523 .56425 .8256r 39 22 .50.553 .80281 ..5,20.51 .85335 . 5.3534 .81404 i ..54999 .83517 .56449 .82544 .38 23 .50578 .802.50 . .520 ( 6 ' .^370 .53558 .81418 j .5.5024 .82501 ..50473 .82528 37 24 ..50603 .862.51 .52101 . 85.3.55 ! .5:4583 . 814:23 1 .5.5048 .83485 .56497 .82.511 36 25 .50628 .802.37 1 .52126 .a5340. .5.3007 .81417 1 .55072 .82169 .56.521 .82-495 35 26 .500.54 .86222 ..52151 .8.5325. .5.3032 .81402 ..5.5097 .84453 .56.545 .82478 34 27 ..50679 .86207 ..52175 .a53l0 ..5.3056 .84280 .55121 .834.37 .50.569 .82462 33 28 .50704 .80192 ..52200 .a5294 .53681 .81:570 .55145 .83421 .56.593 .8:^6 32 29 .50729 .861731 ..52225 .a5279 .53705 .813.55 .5.5109 .82405 .56617 .82-129 31 30 .50754 .80103 .52250 .85204. .53730 .84339 .55194 . 83389 1 .56641 .8;«i:3 30 31 .50779 .80148' .52275 .85249 ..53754 .84.324 .55218 .82373 .56665 .82.396 ' 29 32 .50804 .801331 .52209 .a5234 .53779 .84303 .55242 .83356 .56089 .82:380 28 33 .50829 .80119 .52324 .85218 .53804 .84292 .t52CG .83340 .56713 .82363 27 U .50854 .8010L .52349 .85203 .53828 .84277 .55291 .83.3241 .56730 .82347 26 35 .50879 .80089 .52374 .85188 .53853 .84261} .55315 .82308 .56760 .82.3.30 25 36 .50904 .80074, .52399 .85173 .53877 .84245 .55239 .8.3292 .56784 .82314 24 37 .50929 .80059' .52423 .85157 .53902 .84230 .55.363 1.83276 .56808 .82297 23 38 .50954 .86045 j .52-443 .85142 .53926 .84214 .55388 1.8.3260 .56832 .82281 22 39 .50979 .86030; .52173 .85127 .53951 .84198 .55412 i .8:3244, .56856 .82264 21 40 .51004 .86015} .52493 .85112 .53975 .84182 .55436 .83228 .50880 .82248 20 41 .51029 .86000 .52522 .85096 .54000 .84167 ,55460 .832121 .56904 .82231 ' 19 42 .51054 .85985 .52547 .85031 .54024 .84151 .55484 .83195! .56928 .82214 18 43 .51079 .85970 .52572 .85066 .54049 .84135 .55509 .831791 .56952 82198 . 17 44 .51104 .85956 .52597 .85051 .54073 .84120 .55533 .831631 .56970 .82181 1 16 45 ..51129 .85941 .52021 .85035 .54097 .84104 .55557 .831471 .57000 .82165 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .831.31 1 .57024 .82148 14 47 .51179 .85911 .52071 .85005 1 .54146 .84072 .55605 .83115: .57047 .82132 13 48 .51204 .8.5896 .52096 .84989 .54171 .84057 .55630 .83098; .57071 .82115 12 49 .51229 .85881 .52720 .84974 .54195 .84041 .55654 .83082, .57095 .82098 ; 11 50 .51254 .85860 .52745 .84959 .54220 .84025 . 55678 .83066, .57119 .82082 10 51 ..51279 .8.5851 .52770 .84943 .54244 .84009 .55702 .830501 .57143 .82065 9 52 .51304 .85330 .52794 .84928 ,54269 .83994 .55726 .83034: .57167 .82048 8 53 .51329 .85821 .52819 .84913 .54293 .83978 .55750 .83017; .57191 .82032 7 54 .51.354 .85800 .52844 .84897 .54317 .83962 .55775 .83001 1 .57215 .82015 6 55 .51379 .85792 .52869 .84882 .54.342 .8.3946 . 55799 .82985, ..57238 .81999 5 56 .51404 .8.5777 .52893 .84866 .54.366 .83930 .55823 .829091 .57262 .81982 4 57 .51429 .8.5762 .52918 .84851 .54.391 .83915 ..55847 .82953} .57286 .81965 3 58 .514.54 .8.5747 .52943 .84836: .54415 .83899 .5.5871 .829.36 .57310 .81949 8 59 .51479 .85732 .52907 .848201 .54440 .8:3883 ..55895 .82920 .573.34 .819:32 1 .51.504 .85717 ..52992 .84805 .54404 .83867 .5.5919 .82904 .573.58 .81915 1 J / Cosin Sine Cosin 1 Sine Cosin Sine Cosin Sine Cosin Sine 1 f 69° 68° 67° 66“ 65“ 1 1 TABLES. 655 TABLE VI. — Continued. Natural Sines and Cosines. 35 ° 36 ° 37 ° 0 CO CO 39 ° Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin / 0 .57358 .81915 .58779 .80902 .60182 .7‘9864 761506 778801 762932 .77715 1 .57381 .81899 ,58802 .80885 .60205 .79846 .61589 .78783 .62955 .77696 59 2 .57405 .81882 .58826 .80867 .60228 .79829 .01012 .78705 .62977 1.77678 58 3 .57429 .81865 .58849 .80850 .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729 .63022 .77'641 56 5 .57477 .81832 .58896 .80816 .60298 .79776 .61081 .78711 .63045 1.77623 55 6 .57501 .81815 .58920 .80799 .60321 .79758 .01704 .78694 .63068 1.77605 54 7 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 1.77586 53 8 .57548 .81782 .58967 .80705 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .77531 50 11 .57619 .81731 .59037 .80713 .60437 .79671 .61818 .78604 .63180 .77513 40 12 .57643 .81714 .59061 .80096 .60460 .79653 .61841 .78586 .03203 .77494 48 13 .57067 .81698 .59084 .80679! .60483 .79035 .61864 .78508 .03225 .77476 47 14 .57091 .81081 .59108 .80002 .60506 .79018 .61887 .78550 .03248 .77458 46 15 .57715 .81004 .59131 .80644 .60529 .79000 .61909 .78532 .63271 i. 77439 45 16 .57738 .81047 .59154 .80627 ! .60553 .79.583 .61932 .785141 .63293 .77421 44 17 .57762 .81631 .59178 .80610, .60576 .79505 .61955 .78496; .03310 .77402 43 18 .57786 .81014 .59201 .80593 .60599 .79547 .61978 .78478 .03338 .77384 42 19 .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .77366 41 20 .57833 .81580 .59248 .80558 1 .60645 .79512 .62024 .78442 .03383 .77347 40 21 .57857 .81563 .59272 .805411 .60668 .79494 .62046 .78424 .03406 .77329 39 22 .57881 .81546 .59295 .80524 .60091 .79477 .62069 .78405 .63428 .77310 38 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387 .63451 .77292 37 24 .57928 .81513 .59342 .80489 .60738 .79441 .02115 .78309 .63473 .77273 36 25 .57952 .81496 .59305 .80472 .60761 .79424 .62138 .78351 .63496 .77255 35 23 .57976 .81479 .59389 .80455' .60784 .79406 .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438 .60807 .79388 .62183 .78315 .03540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 .62206 .78297 .03563 .77199 32 29 .58047 .81428 .59459 .80403 60853 .79353 .62229 .78279 .03585 .77181 31 SO .58070 .81412 .59482 .80380, .60876 .79335 .62251 .78201 .03608 .77162 30 31 .58094 .81395 .59506 .80368' .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .81378' .59529 .80351 .60922 .79300 .62297 .78225 .63653 .77125 28 33 .58141 .813611 .59552 .80334 .60945 .79282 1 .62320 .78206 .63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 26 35 .58189 .81327 .59599 .80299: .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .590221 .80282; .61015 .79229 .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59046 .80264! .61038 .79211 .62411 .78134 .63765 .77033 23 38 .58200 .81276 .59009 .80247 .61001 .79193 .62433 .78116 .63787 .77014 22 39 .58283 .81259 .59093 .80230 .61084 .79176 .02450 .78098 .63810 .76996 21 40 .58307 .81242 .59710 .80212 .61107 .79158 .62479 .78079 .63832 .76977 20 41 .5a330 .81225 1 .59739 .80195' .61130 .79140' .62502 .78001! .03854 .76959 19 42 .58354 .81208 .59763 .80178 .61153 . 79122 : .62524 .78043 .63877 .76940 18 43 .58378, .811911 .59780 .80160 .61176 .79105 .62547 .78025 .03899 .76921 17 44 .584011 .811741 .59809 .80143 .61199 .79087 .62570 .78007 .63922 .76903 16 45 .584251 .811571 .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 15 46 .58449, .81140' .59856 .80108 .61245 .79051 .02615 .77970 .63966 .76866 14 47 .58472, .811231 .59879 .80091 1 .61268 .79033 .62638 .77952 .63989 .76847 13 48 .58496 .81106; .59902 .80073' .61291 .79016 .62060 .77934 .04011 .76828 12 49 ..58519 .81089 .59926 .81)056 .61314 .78998 .02083 .77916 .04033 .76810 11 50 .58543 .81072 .59949 . 80038 j .61337 .78980 .62706 .77897 .64056 .76791 10 61 ..5a507 .81055 .59972 .80021' .61360 .78962 .62728 .77879 .64078 .76772 9 62 ..58.590 .81038 .59995 80003 .61383 .78944 .62751 .77801 .64100 .76754 8 53 .58014 .81021 .00019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58037 .81004 .60042 .79908 .61429 .78908 .62796 .77824 .64145 .76717 6 55 , .58061 .80987 .60005 .79951 .61451 .78891 .62819 .77800 .64167 .76698 5 66 .58684 .80970 .60089 .79934 .61474 .78873 .62842 .77788 .04190 .76679 4 57 .58708 .809.53 .60112 .79916 .61497 .788.55 .62864 .77709 .64212 .76661 3 58 .58731 .80936 .601% .79899 .61.520 .78a37 .62887 .77751 .64234' .76642 2 59 ..587.55 .80919 .601.58 .79881 .61.543 .78819 .62909 .77733 . 642.56 .76623 1 60 .58779 .801X12 .60182 .79864 .61.566 .78801 .62932 .77715 .64279 .76604 _0 / Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine / 64 ° 63 ° 62 ° 61 ° 60 ° 656 SUR VE YING. TAliLE W\.— Continued. Natural Sines and Cosines. 40° 41° 0 1 43* I 44° Sine Cosin Sine Cosin Sine Cosin 1 Sine ( !osin ' Sine Cosln 0 .61279 .76604 . 6,5606 .T.Wl 766913 .74314 . 6.8-21 K) 7731 : 3.5 .6916<5 .710.31 60 1 .61301 ' . 76586 .6.5628 .7.’>4.52 .66935 .74295 .68221 .7-3116 .(i91Hr .71911 .59 2 .61323 . 76.567 .6.56.50 . 7 . 5 - 1:13 .669.56 .741276 .68212 . 7:3096 .71891 TA 3 .61316 .70.548 .6.5672 .7.5414 .66978 .742.56 .68261 .73ir,V, .71873 57 4 .64368 .76,5.30 .6.5691 . 7.5395 .66999 .742.37 .68285 .7.3056 .69.519 .71K\3 56 5 .61390 .76.511 .6.5716 .75.375 .67021 .74217 .6h:«x> . 7 : 10:56 .69.570 .718.33 55 G .61112 .76192 .657:18 .7.5856 .67013 .74198 .68.327 .7.3016 .69.591 .71813 .5-4 7 .61135 .76473 .6.5759 . 75 : 1.37 .670(11 .74178 .6H;>19 .72996 .(39612 .71792 . 5.3 8 .61157 .761.55 .6.5781 .75318 .67036 .741.59 .68.370 .72970 .(;'.)(;:3:i .71772 52 9 .61179 .764:16 .6.5803 .7.5299 .67107 .741.39 .68391 .729:57 .(:;)(3.5i .717.52 51 10 .61501 .76417 .65825 .75280 .67129 1.74120 .68412 .72937 .69675 ^ .71732 50 11 .64.524 .76398 .6.51347 .7.5261 .671.51 .74100 .68434 .72917 .69606 .71711 40 12 .61.546 .76.380 .6.5369 .7.5241 .67172 .74080 .681.55 .72897 .69717 .71691 48 13 .61.568 .76.361 1 .65391 .75222 .67194 .74061 ,68176 .72877 .69737 .71671 47 11 .64.590 70312 1 .6.5913 .7.5203 .67215 .74011 .68497 .72857 .69758 .71650 46 15 .61612 .76323 .659.3,5 .75184 .672.37 .74022 .68518 .728.37 .69779 .716.30 45 16 .616.35 .76301 .6.5956 .75165 .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64657 .76286 .65978 .75146 .67280 .7.398.3 .68561 .72797 .69821 .71590 4.3 18 .61679 .76267 .66000 .75128' i .67301 .73963 .6.8582 .72777 .69842 .71.569 42 19 .61701 .76218 .66022 .75107 i .67323 .7.3911 .68003 .72757 .69862 .71.549 41 20 .64723 .76229 .66041 .75088 .67344 .73924 .08024 ,72737 .69883 .71529 40 21 .61746 .76210 .66066 .75069 I .67366 .73904 .0-8045 .72717 .69004 .71508 30 22 .61768 .76192 .66038 .7.5050 .67087 .7.38a5 .680(30 .72097 .69925 .71488 38 23 .61790 .76173 .63109 .750.30 .67409 .73865 .08688 .72077 .69946 .71468 87 24 .64812 .76154 .66131 .75011 .67430 .73846 .68709 .72057 .69900 .71447 36 25 .64834 .761:35 .681.33 .74992 1 .67452 .73826 .68730 .72037 .69987 .71427 a5 26 .61856 .76116 .66175 . 74973 ' .67473 .73806 .68751 .72017 .70008 .71407 S4 27 .64878 .76097, .66197 .74953 .67493 .73787 .68772 .72597 .70029 .71386 a3 28 1 .64901 .76078 .68218 .74934 .67516 .73767 .68793 .72.577 .70049 .71366 '32 29 .64923 . 76059 .66210 .74915 .67.5.38 . 7.3747 .68814 .7'2.557 .70070 .71345 31 30 .64945 .76041 .66262 .74896 .67559 .73728 .68835 .72537, .70091 .71325 30 31 .64967 .76022 .66284 .74876 .67580 .73708 .68857 .72517' .70112 .71305 29 32 .64989 .76003 . 63 108 .74857 .67692 .73683 .08878 .72497 70132 .71284 28 33 I .65011 .75934 .66327 .74833 .67623 .73669 .63899 .72477! .70153 .71264 27 34 1 .65033 . 75965 .66349 .74818 .67645 .7.3649 .63920 .724571 .70174 .7124.3 26 35 . 65055 .75946, .66371 .74799 .67666 .73329 .68941 .72437 .70195 .71223 35 36 I .65077 .75927! .68393 .74780 .67688 .73610 .63962 .7'2417! ,70215 .7im 24 37 1 .65100 .75903 .66414 .74760 .67709 .7.3590 .68983 .72397 .70236 .71182 23 38 1 .65122 .75889 .66433 .74741 .67730 73570 .69004 .72377 .70257 .71162 22 39 i .65144 .75870 .634.58 .74722 .67752 .7-3551 .69025 .72357 .70277 .71141 21 40 .65166 .75851: .66480 .74703^ ,67773 .73531 .69046 .72337 .70298 .71121 20 41 .65188 .75832 .66501 .74683 .67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 .75813' .66523 .74634 .67816 .73491 .69088 .72297 .703.39 .71080 18 43 .65232 .75794' .66545 .74644 .67837 .73472 .69109 .72277 .70360 .71059 17 44 . 65254 . 75775 .68566 .74625 .67859 .73452 .69130 .72257 .70381 .71039 16 45 .65276 .75756 .66588 .74606 .67880 .73432 .69151 .72236 .70401 .71019 15 46 .65298! . 75738 i .66610 .74586 .67901 .73413 .69172 .72216 .70422 .70998 14 47 .65320 .75719' .66632 .74567 .67923 .73393 .69193 .72196 .70443 .70978 13 48 .65342 .75700 .66653 .74548 .67944 .73373 .69214 .72176 .70463 .70957 12 49 .65.364 .7.5630 . 66675 .74528 .67965 .73353 .69235 .72156 .70484 . 709:17 11 50 .65386 .75661 .66697 . 74509 .67987 .73333 .69256 .72136 .70505 .70916 10 51 .65408 .75642 .66718 .74489! .68008 .73314 .69277 .72116 .70525 .70896 9 52 .65430 .75623 .66740 .74470 .68029 .7.3294 .69298 .72095 .70546 .70875 8. 53 .65452 .75604 .66762! .74451 i .680.51 .7:3274 .69319 .72075 .70567 .70a55 7 54 .65474 ' . 75585 .66783: .744:31: .68072 . 7:3254 .69340 .720.55 .70.587 .70834 6 55 .65496 1.75.566! .66805! .74412 .68093 .73234 .69:361 .72035 .70608! .70813 5 56 .65518 . 75.547 i .66827 .74392 .68115 .7:3215 .69382 .72015 .70628; .70793 4 57 .65540 . 75528 1 .66848 1.74373: .681:36 .73195 .69403 .71995 .70649! .70772 3 58 .65562 . 7.5.509 i .66870 ! .743.53 .68157 .73175 .69424 .71974 .70670, .70752 2 59 .6.5.584 .75490 .66891 .743:34: .68179 .73155 .69445 .719.54 .70690! .70731 1 60 .65606 .7.5471 .66913 !. 74314 1 .68200 .731.35 .69460 .71934! .70711' .70711 0 / Cosin Sine Cosin 1 Sine 1 Cosin Sine Cosin Sine 1 Cosin 1 Sine / i 49° i 1 48° 1 47° 46° 1 45° TABLES. 657 TABLE VII. Natural Tangents and Cotangents. 0“ ] “ 1 i ^ 1“ ! 3“ Tang Cotang Tang Cotang Tang Cotang i Tang Cotang $ 0 .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .0.3550 28.1664 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 7 .00204 491.106 .01949 61.3032 .0.3696 27.0566 .05445 18.3655 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2677 52 9 .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 61 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47.7395 .03842 26.0307 1.05591 17.8863 48 13 .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 16 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 42 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 1.56.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 38 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 20 .00756 132.219 .02502 39.9655 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 33 28 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 32 29 .00844 11 8.. 540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 SO .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 |28 a3 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 !27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 26 35 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 36 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 38 .01105 90.4633 .02851 35.0095 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.71.51 .04628 21.60.56 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 42 .01222 81.8470 .02908 33.6935 .0-4716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .03026 as. 0452 .04774 20.9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 15 46 .Class 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .00613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 49 .01425 70.1533 .03172 31.5284 .04920 20.32.53 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .00700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.10.55 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 61.8580 .03288 30.4116 .05037 19.a546 .06788 14.7317 7 54 .01.571 63.6.567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .OlfXK) 62.4992 .aa346 29.8823 .05095 19.6273 .06847 14.60.59 5 56 .01629 61.. 3829 .03376 29.0245 .05124 19.51.56 .06876 14.54;i8 4 57 .016.58 60.30.58 .03405 29.. 3711 .05153 19.40.51 .06905 14.4823 3 58 .01687 .59. 26.59 .03434 29.1220 .05182 19.29.59 .069.34 14.4212 2 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 60 .01746 57.2900 .03492 28,6.363 .05241 19.0811 .00993 14.3007 0 Cotang Tang Cotang Tang Cctnng Tang Cotang 1 Tang / 89“ 88“ 1 87“ 86“ 658 SUR VE YING. TABLE VW. — Coutinued. Natural Tangents and Cotangents. 4° 5“ e 7 Tang Cotang Tang Cot an g Tang Cotang Tang 1 Cotang 0 .06993 14.:10()7 .08749 11 ,4:«)1 .10510 1 9.514:36 .12278 8.144.-15 1 .07022 14.2411 .08778 11 ..3919 .10.540 9.48781 . 12:308 8.12181 59 2 .07051 14.1821 1 .0Kh07 ll.:3;>40 .10.569 9.46141 .123-38 8.10.5.36 58 8 .07080 ]4.12;i5 1 .088:17 11.3163 .10.599 9 . 4:3515 .12367 8.08600 .57 4 .07110 14.0655 1 .08866 11.2789 .10628 9.40'K)1 . 12:397 8.06674 .56 5 .07139 14.0079 .08895 11.2117 1 .10557 9.38307 , .12426 8.047.56 .\5 0 .07168 13.9507 .08925 11.2048 ' .10687 9.35724 ! .12456 8.02818 .54 7 .07197 13.8940 ! .089.54 11.16.81 1 .10716 9.. 33 1.55 .12485 8.00948 .53 8 .07227 13.8378 .08983 11.1316 .10746 9.30.599 .12515 7.990.58 52 9 .07256 13.7821 .OIK) 13 11.09.54 .1077'5 9.28058 .12.514 7.97176 51 10 .07285 13.7267 ,09042 11.0594 .10805 9.25530 .12574 7.9.536.2 50 11 .07314 13.6719 .09071 11.0237 .10R-V4 9.23016 .12608 7. 9.34.38 49 12 .07:144 13.6174 , .09101 10.9882 .10853 9.20516 .126.^3 7.91.5H2 48 13 .07373 13.56:i4 ! .091 :10 10,9.529 .10893 9.18028 .12662 7.897.34 47 14 .07402 13.5098 1 .091.59 10.9178 .10922 9.15.5.54 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12722 7.86004 45 IG .07461 13.4039 .09218 10. m3 .10981 9.106-16 .12751 7.84242 44 17 .07490 13.3515 .09217 10.8139 .11011 9.08211 .12781 7.82128 43 18 .07519 13.2996 .09277 10.7797 .11040 9.0.5789 .12810 7.80622 42 19 .07548 13.2480 .09.306 10.74.57 .11070 9.03.379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 1 .12869 7.770:i5 40 21 .07607 13.1461 .09.365 10.6783 .11128 8.98598 ! .12899 7.7.5254 39 22 .07636 13.0958 .09394 10.6450 .111.58 8.96227 .12929 7.7.-1480 38 23 .07665 13.0458 .09423 10.6118 .11187 8.9.3867 1 .12958 7.71715 .37 24 .07695 12.9902 .09453 10.5789 .11217 8.91520 .12988 7.69957 .36 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 7.68208 .35 26 .07753 12.8981 .09511 10.51.36 .11276 8.86862 ! .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 1 .13076 7.647.32 .33 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.&3005 32 29 .07841 12.7536 .09600 10.4172 .11364 8.79964 ' .13136 7.61287 31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .|13165 7.59575 30 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 1 .13195 7.57872 29 82 .07929 12.6124 .09688 10.3224 .11452 8.73172 ' . 1:3224 7.56176 28 33 .07958 12.5660 .09717 10.2913 .11482 8.70931 . 1:3254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .1.3284 7.52806 26 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .1.3313 7.511.32 25 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 . 1.3343 7.49465 24 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .ia372 7.47806 23 38 .08104 12.3390 .09864 10.1.381 .11629 8.59893 .1.3402 7.46154 22 39 .08134 12.2946 .09893 10.1080 .11659 8.57718 .13432 7.44509 I 2 I 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 .11718 1 53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 7.36.389 16 45 .08309 12.0346 ,10069 9.93101 .11836 8.44896 .13609 7.34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 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 7.28442 11 60 .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 9.7.3217 .12042 8.. 30406 .13817 7.23754 8 63 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 64 .08573 11.6645 .10:334 9.67680 .12101 8.26355 .1.3876 7.20661 6 55 .08602 11.6248 .10363 9.649:35 .12131 8.24:345 .13906 7.19125 5 56 .086:12 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20a52 .13965 7.16071 3 58 .08690 11 5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 69 .08720 11.4685 .10481 9.54106 .12249 8.16:398 .14024 7.1.3042 1 60 .08749 11.4:101 .10510 9. 51 436 .12278 8.1 44^5 .14054 7.115.37 0 Cotang Tang Cotang Tang Cotang Tang 1 Cotang Tang 85» 0 CO 83° ! 82° TABLES. 659 TABLE VII. — Continued. Natural Tangents and Cotangents. 8° 1 9“ >-* I 11 “ Tang Cotang Tang Cotang Tang Cotang Tang Cotang / 0 .14054 7.11537 .15838 6.31375 .17633 5.67128 .19'438 5.14455 m 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19.529 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .195.59 5.11279 56 5 .14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24321 .17813 5.61397 .19019 5.09704 54 7 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 0.22003 .17873 5.59511 .19680 5.08139 52 9 .14321 6.98268 .16107 0.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.95385 .16167 0.18.559 1 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 0.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.042G7 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 . 14559 6.86874 .10346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.9S188 39 22 .14707 6.79936 .16495 0.06240 .18293 5.40048 .20103 4.97438 38 23 .14737 6.78564 .16525 C. 05143 .18323 5 . 4.JI 51 .20133 4.96690 37 24 .14767 6.77199 .16555 6.04051 .18353 5.44857 .20104 4.95945 36 25 .14796 6.75838 .16585 6.02962 .18384 5.43966 .20194 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 .34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 28 .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.93046 .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14973 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18054 5.36070 .20466 4.88605 20 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5.89151 .18775 5.32631 .20588 4.85727 22 39 .15213 6.57339 .17004 5.88114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.. 30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.275.53 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15.511 6.44720 .17303 5.77936 .19106 5.2.3391 .20921 4.77978 11 50 .15640 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 10 51 .1.5.570 6.422.53 .17303 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41020 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .156.30 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 7 54 .15660 6.3a587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.. 37.374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .1.5719 6.. 361 65 .17513 5.71013 .19317 5.17071 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72490 3 58 .15779 6.. 3.3761 .17573 5.69064 .19378 5.160.58 .21195 4.71813 2 59 .15809 6.. 32.566 .17603 5.68094 .19408 5.1.52.50 .21225 4.71137 1 60 .1.58.38 6.31375 .17033 5.67128 .19438 5.144.55 .21256 4.70463 0 / Co tang Tang Cotang Tang Cotang Tang 1 Cotang Tang / 81 “ 1 80 “ 1 79 “ ' 78 “ 66o SUK VE YING. TABLE VII, — Con ti)i Kcd. Natural Tangents and Cotan(;ents. 12° 13° 1 14° 15° Tanpr Cotang Tang Cotang 1 Tang ' Cotang Tang 1 Cotang r 0 .2125(5 4.7046:1 .23087 4.:i3r48 , . 249 : 1.3 r-i.uiW .26795 60 1 .21286 4.69791 .23117 4.32573 .24964 4.00.5H2 .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.000H(; .2(5857 ; 3.72338 68 3 .21347 4.684.52 .2.3179 4 . 311:10 .25026 3.99.592 .26888 3.71907 67 4 .21377 4.67786 .2:}209 4.30860 .250.56 3.99099 .26920 , 3.71476 66 6 .21408 4.67121 .2:1240 4.30291 .25087 3.9(3(507 .2(/951 3.71016 55 6 .21438 4.66458 .2:1271 4.29724 .25118 3.98117 .26982 3.70616 64 7 .21469 4.65797 . 2 : 1.301 4.29159 .25149 3.97627 .27013 3. 701 88 63 8 .21499 4.651.38 .23332 4.28595 .25180 3.97139 .27(M4 3.69761 62 « .215^9 4.61480 .23363 4.280;i2 .25211 3.96651 .27(^6 3.69.335 51 10 .21560 4.63825 .23393 4.27471 .25212 3.96165 .27107 3.08909 50 11 .21590 4.6.3171 .23124 4.26911 ^ .2.5273 3.9.56.S0 .27138 3.CR4R.5 49 12 .21621 4.62518 .23455 4.263.52 1 .25.304 3.95196 .271(59 3.(5i;0(51 48 13 .21651 4.61868 .2:1485 4.2.5795 1 .2.5.235 3.91713 .27201 3.(57(; .8 47 14 .21682 4.61219 .2.3516 4.25239 1 .25.366 3.942.32 .272.52 t 3.(57217 i46 15 .21712 4.60.572 .21517 4.24685 ' .25.397 3.9.3751 .27263 3.(56796 45 16 .21743 4.59927 .21578 4.241.32 .25428 3.9.3271 .27294 3.66376 44 17 .21773 4.. 59283 .2.3608 4.23580 .254.59 3.1r2793 !2.'32G 3.6.5957 43 18 .21804 4.58641 . 2:1039 4.23630 .25190 3.92316 27.357 3.65538 42 ly .21834 4.58001 .23070 4.22481 .25521 3.91839 ! 27388 3.6.5121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91304 .27419 3.04705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 22 .21925 4.56091 .23762 4.20f«2 .2.5614 3.90417 .27482 3.03874 38 23 .21956 4.55458 .23793 4.20298 .2r>645 3.89945 .27513 3.6.3461 37 24 .21986 4.64826 .23823 4.19756 .25676 3.89474 .27515 3.63048 36 25 .22017 4.54196 .23354 4.19215 .25707 3.89004 .27576 3.62636 35 26 .22047 4.53568 .23885 4.18675 .257.38 3.88536 .27607 3.622^ 34 27 .22078 4.02941 .23916 4.181.37 .2.5769 3.880C8 .276:18 [ 3.61814 a3 28 .22108 4.52316 .23946 4.17600 2.5800 3.87601 .27670 1 3.61405 32 29 .22139 4.51093 .23977 4.17064 ; 25831 3.87136 .27701 1 3.60996 31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 1 3 C0588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.C0181 29 32 .22231 4.49832 .24069 4.15465 .2.5924 3.85745 .27795 3.59770 23 33 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 34 .22292 4.48600 .24131 4.14405 .2.5986 3.84824 .278.58 3.589G6 26 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 36 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.. 581 60 24 37 .22383 4.46734 .24223 4.12825 .26079 3.83449 .27952 3.57758 23 38 .22414 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57357 22 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 19 42 .22536 4.43735 .24377 4.10216 .262.35 3.81177 .28109 3.55761 18 43 .22567 4.43134 .24408 4.09699 .26286 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 IG 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 46 .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3..5417'9 14 47 .22689 4.40745 .24532 4.07639 .28390 3.78931 .23266 3.53765 13 48 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 .22750 4.39560 .24593 4.06616 .26452 3.78040 .28329 3.53001 11 50 .22781 4.38969 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 52 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 8 53 .22872 4.37207 .24717 4,04586 .26577 3.76268 .28454 3.51441 7 54 .22903 4.36623 .24747 4.04081 .26608 3.75828 .28486 3.51053 6 55 .22934 4.36040 .24778 4.03578 .26639 S. 75388 .28517 3.50G66 5 56 .22964 4.35459 .24809 4.03076 .26670 3.74950 .28549 3.. 50279 4 57 .22995 4.31879 .24840 4.02574 .26701 3.74512 .28580 3.49894 3 58 .23026 4.34300 .24871 4.02074 .26733 3.74075 .28612 3.49509 2 59 .23056 4.3:1723 .24902 4.01576 3.73640 .28643 3.49125 1 ra .23087 4.33148 .24933 4.01078 .26795 3.73205 _^28675 3.48741 0 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 77° 1 76° 75° 740 TABLES. 66l TABLE VTI. — Continued. Natural Tangents and Cotangents. 16“ 17° oo 0 19° Tang Cotang Tang Cotang Tang Cotang Tang Cotang 9 0 .28675 3.48741 .30573 3.27085 ■ .32492 3.07768 .34433 2.90421 1 .28706 3.48359 .30605 3.26745 .32524 3.07464 .34465 2.90147 59 2 .28738 3.47977 .30637 3.26406 .32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.26067 .32588 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 .32621 3.06554 .34563 2.89327 56 5 .28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 55 6 .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 54 7 .28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.88511 53 8 .28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 9 .28958 3.45327 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2 87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 12 .29053 3.44202 .30955 3.23048 .32878 3.04152 .34824 2.87161 48 13 .29084 3.43829 .30987 3.22715 .32911 3.03854 .34856 2.86892 47 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 15 .29147 3.43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 IS .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .33233 8.00903 .35183 2.84229 37 24 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.83965 36 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 26 .29495 3.39042 .31402 3.18451 .33330 3.00023 .35281 2.83439 34 27 .29526 3.38679 .31434 3.18127 .33363 2.99738 .35314 2.83176 33 28 .29558 3.38317 .31466 3.17804 .33395 2.99447 .35346 2.82914 32 29 .29590 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29685 3.36875 .31594 3.16517 .33524 2.98292 .35477 2.81870 28 33 .29716 3.36516 .31626 3.16197 .33557 2.98004 .35510 2.81610 27 34 .29748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 26 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .20843 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 23 38 .29875 3.34732 .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 , 3.33670 .31882 3.13656 .33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .35805 2.79289 18 43 .30033 3.329G5 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .a3945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 4C .30192 3.31216 .32106 3.11464 .34043 2.93748 ' .36002 2.77761 12 49 .30224 3.30868 .32139 3.11153 .34075 2.93468 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 .52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 8 53 .30:i51 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 56 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 4 57 .30478 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 3 58 .30509 3.27767 .32428 3.0a379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 1 60 .30573 3.27085 _^32492 3.07768 .34433 2.90421 .36397 2.71718 0 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang f CO o to 71° i o 662 SURVEYING. TABLE VII. — Con tin ned. Natural Tangents and Cotangfcnts. 1 20 » 21 » 22 “ CO o 1 Tanp: Cotang Tang Cotang Tang Cotang 1 Tang Cotang 0 .36397 2.74748 .38;i86 .40403 2.47.509 1 .V24i7 2.3558T 60 1 .36430 2.74499 .31W20 2.60283 .404.36 2.47302 .42182 2. 3.6395 59 2 .36163 2.742.51 .384.53 2.600.57 .40470 2.47095 1 .42516 2.85205 .58 3 .36196 2.74004 .38487 2.. 59831 .40504 2.46888 .42,651 2.. 650 15 57 4 .36529 2.737.56 .a8520 2.. 59606 .40.5.'i8 2.46682 1 .42.585 2. 61825 56 6 .36562 2.7.3509 .38.5.53 2.. 59381 .40572 2.46116 .42619 2,. 616.36 .55 6 .36595 2.73263 .38.587 2.591.56 .40606 2.46270 .426.54 2. 344 17 54 7 .36628 2.73017 .38620 2.. 5.89.32 .40640 2.46065 .42688 2.. 612.58 53 8 .36661 2.72771 ..386.54 2.. 587 08 .40674 2.4.5.860 .42722 2. 64069 52 9 ! .36691 2.72.526 ..186.87 2.. 584 84 .40707 2.4.5655 ! 42757 2..6‘1881 51 10 ; .36727 2.72281 .38721 2.58261 .40741 2.45151 .42791 2.33693 50 11 .36760 2.720.36 ..38754 2.580.38 .40775 2.4.5246 .42826 2.66505 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45013 .42860 2.86317 48 13 .36826 2.71548 ..33821 2.. 57.593 .40843 2.41839 .42891 2. 631 .30 47 14 .368.59 2.71305 ..18854 2.57371 .4087^ 2.41636 .42929 2.. 32943 46 15 .36892 2.71062 .3.8888 2.. 571.50 .40911 2.414.33 .42963 2. 327.56 45 16 .36925 2.70819 ..38921 2.. 56928 .40915 2.412.30 .42998 2. 32570 44 T7 .36953 2.70.577 ..189.55 2.56707 .40979 2.41027 .4.30.32 2. 32.383 43 18 .36991 2.70335 .38988 2.. 56487 .41013 2.4.3825 .4.3067 2.. 321 97 42 19 .37024 2.70091 .39022 2.. 56266 .41017 2.4.3623 .43101 2. .32012 41 20 .37057 2.69853 .39055 2.56046 .41081 2.43*122 .43136 2.31826 40 21 .37090 2.69612 .39089 2.. 55827 .41115 2.4.3220 .4.3170 2.316*11 39 22 .37123 2.69371 .39122 2.53608 .41149 2.4.3019 .43205 2.31456 38 23 .37157 2.69131 .39156 2.55389 .41183 2. 42.81 9 .4.3239 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 36 25 .37223 2.68653 .39223 2.. 54952 .412.51 2.42418 .4.3.308 2.. 30902 35 26 .37256 2.68414 ..39257 2.54734 .41285 2.42218 .4.3.343 2.. 3071 8 .34 27 .37289 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.. 30534 83 28 .37322 2 679,37 .39324 2.54299 .41:3.53 2.41819 .4.3412 2.. 38351 32 29 .37355 2.67700 ..39357 2.. 540.82 .41387 2.41620 .4.3447 2.. 301 67 31 30 .37383 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2.669S9 .39458 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 .39492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40629 .43620 2.29254 126 35 .37554 2.66281 .39559 2.52786 .41592 2.404.32 .43654 2.29073 j 25 36 .37588 2.66046 ' .39593 2.52571 .41626 2.402.35 .43689 2.28891 ^ 24 37 .37621 2.65811 , .39626 2.52357 .41660 2.40038 .43724 2.28710 1 23 38 .37654 2.65576 1 .39660 2.52142 .41694 2.39341 .43758 2.28528 1 22 39 .37687 2.65342 ' .39694 2.51979 .41728 2.39645 .43793 2.28348 1 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 120 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2.64642 .39795 2.51289 .41831 2.39058 .43897 2.27806 18 43 .37820 2.64410 ..39329 2.51076 .41865 2.38863 .43932 2.27626 17 44 .37853 2.64177 .39862 2.50364 41899 2.38663 .43966 2.27447 16 45 .37887 2.63945 .39896 2.50652 .419^3 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .41968 2.38279 .44036 2.27088 14 47 .379.53 2.63483 i .39963 2.50229 .42002 2.38084 .44071 2.26909 13 48 .37986 2.63252 I .39907 2.50018 .42036 2.37891 .44105 2.26730 12 49 .38020 2.63021 .40031 2.49807 .42070 2.-37697 .44140 2.26552 11 50 .38053 2.62791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .38086 2.62.561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 .38120 2.623;32 .40132 2.49177 .42173 2.37118 .44214 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 ..38186 2.61874 .40200 2.48758 .42242 2.. 36733 .44314 2.25663 6 55 .38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 2 48340 .42310 2.36349 .44384 2.25309 d 57 ..38286 2.61190 .40.301 2.48132 .42345 2.36158 .44418 2.25132 3 58 .38.320 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24956 2 59 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44488 2 24780 1 60 .38386 2.60.509 .40403 2.47509 .42447 2.3.5.585 .44523 2.24604 0 Cotang Tang Cotang Tang Cotang Tang Cotang 1 Tang 69 “ C5 CO o 67 " 1 60 “ TABLES. 663 TABLE VII. — Continued. Natural Tangents and Cotangents. 24 » 25 “ 1 26 “ 27 “ Tang Cotang Tang Cotang Tang Cotang Tang Cotang / 0 .44523 2.24604 .46631 2.14451 .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 1.96120 59 2 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 1.95698 56 5 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.95557 55 6 .44732 2.23553 .46843 2.13477 .48989 2.04125 .51173 1.95417 54 7 .44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 1.95277 53 8 .44802 2.23204 .46914 2.13154 .49062 2.03825 .51246 1.95137 52 9 .44837 2.23030 .46950 2.12993 .49098 2-. 0.3675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12832 .49134 2.03526 .51319 1.94858 50 11 .44907 2.22683 .47021 2.12671 .49170 2.0a376 .51356 1.94718 49 12 .44942 2.22510 , .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 ! .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 j .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .45047 2.21992 1 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 1.94023 44 17 .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 1.93885 43 18 .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 1.9,3746 42 19 .45187 2.21304 .47305 2.11392 .49459 2.02187 .51651 1.93608 41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 1.93195 38 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.93057 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 1.92920 36 25 .45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 1.92782 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 1.92645 34 27 .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47626 2.09969 .49786 2.00862 .51983 1.92371 32 29 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 32 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 1.91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47912 2.08716 .50076 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.08405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.9047'2 18 43 .46030 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 .50368 1.98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.167.51 1 .48270 2.07167 .50441 1.98253 .52650 1.899,35 14 47 .46171 2.16585 1 .48306 2.07014 .50477 1.98110 .52687 1.89801 13 48 .46206 2.16420 ! .48342 2.06860 .50514 1.97966 .52724 1.89667 12 49 .46242 2.16255 i .48378 2.06706 .50550 1.97823 .52761 1.89.533 11 50 .46277 2.16090 1 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 61 .46312 2.1.5925 .48450 2.06400 .50623 1.97538 ..52836 1.8926b 9 62 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 63 .46383 2.1.5.596 .48521 2.06094 .50696 1.97253 .52910 1.89000 7 64 .46418 2.154.32 .48557 2.05942 .50733 1.97111 .,52947 1.88867 6 55 .464.54 2.15268 .48.593 2.05790 .50769 1.96969 .52985 1.88734 5 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 .5,3022 1.88602 4 67 .46525 2.14940 .48665 2.0.5485 .50843 1.96685 .5,30.59 1.88469 3 68 .46560 2.14777 .48701 2.0.53:« .50879 1.96.544 .53096 1.88337 2 69 .46595 2.14614 .48737 2.05182 ..50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2,0:50.30 .50953 1.96261 .53171 1.88073 _0 / Cotang Tang Cotang j Tang 1 Cotang Tang Co tang Tang / 65 “ 64 “ 1 63 “ 62 “ 464 664 SUR VE YJNG. TABLE VII, — Contiiincd. Natural Tangents and Cotangents, 28« ! 29° 0 0 CO 31° Tariff Cotanff Tang Cotang Tang ! Cotang _Tang 1 Cofang / 0 ,53171 1.88073 .5.5431 1.80405 .677.35 1.7:320.'r .GOOHC, 1 .6(i42H' 1 .53208 1.87941 ..5.5469 1.80281 ..67774 1.7:3089 .60126 ; 1.6(i;31H 59 2 .63246 1.87-809 ..5.5.507 1.801.58 .67813 1.72973 .6016.-) 1 1 rT)2(H) r)8 3 .6328:4 1.87677 ..5.5.545 1.80034 .67851 1.728.67 .60205 1 1 WKIOO 57 4 .63320 1.87546 ..5.5.5K3 1.791)11 ,57890 1.72741 .00245 1 1,6.5990 .56 5 .53358 1.87415 ..5.5621 1.79788 ..67929 1.72625 .G02K1 1 1.6.5K81 55 C .63:495 1.87283 .55659 1 . 7 966.5 ..67968 1.72509 .60:321 1.6.5772 54 7 .5:4432 1.87152 ..5.5697 1.79.542 ..68(X)7 1.72393 .60.361 1.6.566.3 .53 8 .5:4470 1.87021 ..557:10 1.79419 .58046 1.72278 .60103 j 1 . 6.5.5.54 52 9 .53507 1.86891 ..5.5774 1.79296 . .6808.5 1.72163 .60113 ! 1.6.5-145 51 10 .53545 1.86760 .55812 1.79174 1 .58124 1.72047 1 .60483 1.65337 50 11 ..53.582 1.86630 ..5.5a50 1.79a51 1 .58162 1.710,32 .60522 1 1.6.5228 49 12 ..5:4620 1.86499 ..5.588.S 1.78929 1 .58201 1.71817 .60.562 1 1.65120 48 13 ..5.3657 1.86369 , .5.5926 1.78807 ' .58240 1.71702 . , .60602 1 1.6.5011 47 14 ..5.3694 1.862.39 ..5.5964 1.78685 .58279 1.71588 , 1 .60642 1 1.6.19tt3 46 15 ..5:4732 1.86109 ..50003 1.78503 1 .68318 1.71473 1 .00681 1 1.61795 45 16 .53769 1.85979 ..56041 1.78441 i .58357 1.7ia68 1 .60721 1 1.64687 44 17 ..5.3807 1 . 85850 ..50079 1.7a319 .58396 1.71244 ; 1 .6076] 1.64.579 43 18 .53844 1.85720 ..56117 1.78198 , .58435 1.71129 i .60801 1.64171 42 19 .5.3882 1.8.5591 .501.56 1.78077 ..68474 1.71015 .60841 1.64.363 41 20 .53920 1.85462 .50194 1.77955 j .58513 1.70901 .66881 1.04256 40 21 .53957 1.853.33 ..50232 1.77a34 : ..6,8552 1.70787 .60921 1.64148 39 22 .53995 1.85204 1 .50270 1.77713 ..68591 1.7007'3 .609fX) 1.64041 38 23 .540:42 1.85075 .56309 1.77592 ..686:31 1 .70560 .61000 1.0.39.34 37 .54070 1.84946 .56347 1.77471 .58670 1.70446 .61040 1.6.3826 36 25 .54107 1.84818 .56.385 1.77.3,51 .58709 1.70.3.32 .61080 1.6.3719 35 26 .54145 1.84689 .56424 1.77230 .58748 1.70219 .61120 1.63012 34 27 .54183 1.84561 ..50462 1.77110 .58787 1.70106 1 .61160 1.6.3.505 33 28 .54220 1.84433 .50501 1.70990 .58826 1.C9992 .61200 1.0.3.398 32 29 ..54258 1.84305 .56639 1.70869 .58805 1.69879 .61240 1.6.3292 31 30 .54296 1.84177 .50577 1.76749 .58905 1.69766 .61280 1.63185 30 31 .54333 1.84049 .56616 1.76629 .58944 1.69653 .61.320 1.63079 29 32 .54371 1.83922 .56654 1.76510 .58983 1.69541 .61360 1.62912 28 33 .54409 1.83794 .50693 1.76390 .59022 1.69428 .61400 1.62860 27 34 .54446 1.83667 .56731 1.76271 .59061 1.69316 ; .61440 1.62760 26 35 .54484 1.83540 .56769 1.76151 .59101 1.69203 .61480 1 .62654 25 36 .54522 1.83413 .56808 1.76032 .59140 1.69091 .61520 1.62.548 24 37 ..54560 1.83286 .56846 1.75913 1 .591';9 1.68979 .61561 1.62442 23 38 .54597 1.83159 .56885 1 75794 .59218 1.68866 .61601 1.62a36 22 39 .54635 1.83033 .56923 1.75675 i .59258 1.68754 .61641 1.622.30 21 40 .54673 1.82906 ^6962 1.75556 .59297 1.68643 .61681 1.02125 20 41 54711 1.82780 .57000 1.75437 .59336 1.68531 .61721 1.62019 19 42 .54748 1.82654 .57039 1.75319 .59376 1.68419 .61761 1.01914 18 43 .54786 1.82528 .57078 1.75200 .59415 1.68308 .61801 1.61808 17 44 .54824 1.82402 .57116 1.75082 .59454 1.68196 .61842 1.61703 16 45 .54862 1.82276 .57155 1.74964 .59494 1.68085 .61882 1.61598 15 46 .54900 1.82150 .57193 1.74846 .59533 1.67974 .61922 1.61493 14 47 .54938 1.82025 .57232 1.74728 .59573 1.67863 .61962 1.61388 13 48 .54975 1.81899 .57271 1.74610 .59612 1.67752 .62003 1.61283 12 49 .55013 1.81774 .57309 1.74492 .59651 1.67641 .62043 1.61179 11 50 .55051 1.81649 .57348 1.74375 1 .59691 1.67530 .62083 1.61074 10 51 .5.5089 1.81.524 .57386 1.74257 .59730 1.67419 .62124 1.60970 9 52 ,.5.5127 1.81399 .,57425 1.74140 1 .59770 1.67309 .62164 1.60865 8 53 .55165 1.81274 ..57404 1.74022 1 .59809 1.67198 .62204 1.60761 7 54 .55203 1.81150 .57503 1.7.3905 i .59849 1.67088 .62245 1.60657 6 55 .5.5241 1.81025 .57.541 1.73788 1 .59888 1.66978 .62285 1.60553 5 56 .5.5279 1.80901 .57,580 1.7:3671 .59928 1.66867 .62325 1.60449 4 57 .55317 1.80777 .57619 1.7:3555 .59967 1.66757 .62366 1.60345 3 58 .5.5.355 1.806.53 .576.57 1.7:3438 ! .60007 1.66647 .62406 1.60241 2 59 .5.5393 1.80.529 .57696 1.73321 .60046 1.66538 .62446 1.60137 1 60 .5.5431 1 .H0405 ..57735 1.73205 .60086 1.66428 .62487 1.60033 0 Colang Tang Cotang 1 Tang Cotang 1 Tang Cotang 1 Tang / 61» 0 0 0 1 69° 1 68° TABLES. 665 TABLE VII. — Continued. Natural Tangents and Cotangents. 32 ° 33 ° CO 35 ° Tang Cotang Tang Cotang Tang Cotang 1 Tang Cotang 0 .62487 1.60033 .64941 1.53986 .67451 1.48256 .70021 1.42815 1 .62527 1.59930 .64982 1.53888 .67493 1.48163 1 .70064 1.42726 .62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 3 .62608 1.59723 . 65065 1.53693 .67578 1.47977 .70151 1 .42550 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 5 .62689 1.59517 .65148 1.53497 .07603 1.47792 .70238 1.42374 6 .62730 1.59414 .65189 1.53400 .67705 1 .47699 .70281 1.42286 7 .62770 1.59311 .65231 1.53302 .67748 1.47607 .70.325 1.42198 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 I .70412 1.42022 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 11 .62933 1.58900 .65397 1.52913 1 .67917 1.47238 .70499 1.41847 12 .62973 1.58797 .65438 1.52816 ; .67960 1.47146 .70542 1.41759 13 .63014 1.58695 .65480 1.52719 .68002 1.47053 .70586 1.41672 14 .63055 1.58593 .65521 1.52622 ' .68045 1.46962 .70629 1.41584 15 .63095 1.58490 .05503 1.52525 .68088 1.40870 .70673 1.41497 16 .63136 1.58388 .05604 1.52429 .68130 1.46778 .70717 1.41409 17 .63177 1.58286 .05646 1.52332 .08173 1.40686 .70760 1.41322 18 .63217 1.58184 .05688 1.. 52235 .68215 1.40595 .70804 1.41235 19 .63258 1.58083 .65729 1.52139 .08258 1.46503 .70848 1.41148 20 .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 21 .63340 1.57879 .65813 1.51946 .68343 1.40320 .70935 1.40974 22 .63380 1.57778 .65854 1.51850 .68386 1.40229 .70979 1.40887 23 .63421 1.57676 .65896 1.51754 .08429 1.40137 .71023 1.40800 24 .63462 1 .57575 .65938 1.51658 .08471 1.46046 .71066 1.40714 25 .63503 1.57474 .65980 1.51502 .08514 jL . 4o9od .71110 1.40627 26 .63544 1.57372 .06021 1.51460 .08557 1.45864 .71154 1.40540 27 .63584 1.57271 .66063 1.51370 .08600 1.45773 .71198 1.40454 28 .6362.5 1.57170 .60105 1.51275 .08642 1.4.5682 .71242 1.40367 29 .63666 1.57069 .66147 1.51179 .68685 1.45592 .71285 1.40281 30 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 31 .63748 1 .56868 .66230 1.50988 .68771 1 .45410 .71373 1.40109 32 .63789 1.. 56767 .66272 1.50393 .08814 1.45320 .71417 1.40022 33 .63830 1.56067 .66314 ,1.50797 .68857 1.45229 .71461 1.39936 34 .63871 1.56566 .66350 1.50702 .63900 1.45139 .71505 1.39850 35 .63912 1.50466 .66398 1.50607 .68942 1.45049 .71549 1.39764 36 .63953 1.. 50306 .00440 1.50512 .08985 1 .44958 .71593 1.39679 37 .63994 1.. 56205 .66482 1.50417 .69028 1.44808 .71637 1.39593 38 .64035 1.. 56105 .66524 1.50322 .69071 1.44778 .71681 1.39507 39 .64076 1.56065 .66566 1.50228 .69114 1.4-1088 .71725 1.39421 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 41 .641.58 1.. 55806 .66650 1.50038 .69200 1.44.508 .71813 1.39250 42 .64199 1 . 55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 43 .64240 1.55666 .66734 1.49849 .69:230 1.44329 .71901 1.39079 44 .64281 1.55567 .66776 1 .49755 .69329 1.44239 .71946 1.38994 45 .64322 1.55467 .00818 1.49661 .09372 1.44149 .71990 1.38909 46 .64363 1.55.368 .60860 1.49506 .09416 1.44060 .72034 1.38324 47 .64404 1.5.5269 .66902 1.49472 .09459 1.43970 .72078 1.38738 48 .64446 1.5.5170 .06944 1.49378 .09502 1.43881 .72122 1.38653 49 .64487 1.5.5071 .66986 1.49284 .69545 1.43792 .72167 1.38568 50 .64528 1.54972 .07028 1.49190 .69588 1.43703 .72211 1.38484 51 .64.569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 52 .64610 1.54774 .67113 1.49003 .69675 1 . 4:3525 .72299 1.38314 53 ; .64652 1., 54675 ; .07155 1.48909 .69718 1 . 4 : 34:36 .72344 1.38229 54 : .64693 1.54.576 ! .67197 1.48816 .69761 1.4.3.347 .72388 1.38145 55 p .647:44 1.. 54478 . 672:49 1.48722 ! .69804 1.432.58 .72432 1.38060 5e i . 647’i 5 1.. 54:479 i .67282 1.48629 i .69847 1.4.3169 .72477 1.37976 51 ' .64817 1.54281 1 .67:424 1.48536 ! .69891 1.43080 .72.521 1.37891 5f. ! .648.58 1.54183 1 .67:466 1.48442 1 .69934 1.42992 .72565 1.37807 5i > .64899 1.540a5 ‘ .67409 1.48:449 .69977 1.42903 .72610 1.37722 C( ) .64941 1.. 5:4986 .67451 1.48256 ' .70021 1.42815 .72654 1.37638 / Cotang 1 Tang Cotang 1 Tang i Cotang Tang , Cotang Tang 57 ° 1 56 ° i 65 ° il 51 ° / GO 59 58 57 56 55 54 53 52 51 50 49 43 47 40 45 44 43 42 41 40 39 33 37 36 35 34 33 32 31 30 29 28 27 28 25 24 23 22 21 20 19 10 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 _0 _ / 666 SUR VE Y TNG. TABLE VW. — Continued. Natural Tangents and Cotangents. 36“ 87° 38° 39° Tanpr J Cotiingf Taiih' C'otang Tang Cotang j Tang < 'ot arig 0 .72651 1 .376;48 .7.53.55 1.. 32701 ; .78129 Tr2799 r .H0!)78 1 T.vsiTio 60 1 .7269'.) 1.37.554 .7r)401 1.32624 .78175 1.27917 1 .81027 1 21116 .59 2 .72743 1.37470 .7.5147 1.32514 .78222 1.27811 .81075 1 . 2 : 1 : 11.3 .58 3 .72788 1.37380 .75192 1.. 32164 .78269 1.27764 1 .81123 1.21270 57 4 .72a32 1 ..37.302 .7.5.538 1.323H4 .78316 1.27GS8 j .81171 1.23196 G6 5 .72877 1.37218 .7.5.58.4 1.32304 .78363 1.27611 .81220 1.21123 .55 f) .72921 1.. 371. 34 .7.5629 1.32224 .78110 1.27.535 .81268 1.2)0.50 .54 7 .72966 1.370.50 .75675 1.32144 .784.57 1.274.58 .81316 1.22977 •53 8 .73010 1 ..36967 .7.5721 1.32064 .78504 1.27;W2 1 .813W 1.22901 52 9 .730.55 1.368S3 .75707 1.319S4 .78551 1.27306 .81113 1.22831 .51 10 .73100 1.. 36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 .50 11 .73144 1.. 3671 6 .75R58 1.31825 .78045 1.271.53 .81.510 1.22685 49 12 .73189 1.36633 .7.5904 1.31745 .78692 1.27077 .81.5.58 1.22612 48 13 .73231 1.. 36549 .75950 1.31606 .7-8739 1.27001 .81606 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .816.55 1.22*167 40 15 .73323 1.. 36383 .76012 1.31507 .78834 1.26849 .81703 1.22394 4.5 IG .73368 1.36.300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 17 .73413 1.36217 .761.44 1.31343 .78923 1.26698 .81800 1.22219 4.3 18 .734.57 1.361.34 .76180 1.31269 1 .78075 1.26622 .81849 1.22176 42 19 .73502 1.330.51 .76226 1.31190 .79022 1.26.546 .81898 1.22104 41 20 .73547 1.35'J63 .76272 1.31110 .79070 1.26471 81946 1.22031 40 21 .73592 1.3.5885 .76318 1.31031 .79117 1.26395 .81995 1.21959 39 22 .73637 1.S.5C02 .76364 1.30952 .79164 1.26319 .82044 1.21886 ;38 23 .73681 1.35719 .76410 1.30073 .79212 1.26244 .82092 1.2I8I4 37 'iA .73726 1.35637 .764.56 1.30795 .79259 1.26169 .82141 1.217-42 36 25 .73771 1.355.54 .76502 1.30716 .79306 1.26093 .82190 1.21670 85 2G .73816 1.35472 .76548 1.30637 .79354 1. 2601 8 .82238 1.21598 34 27 .73861 1.35389 .7-6594 1 .30558 .79401 1.25943 .82287 1.21526 .83 £8 .73906 1.35307 .70640 i.saiBO .79449 1.25867 .82336 1.214.54 32 29 .73951 1.35224 .76686 1.39401 .79496 1.2.5792 .82385 1.21382 31 30 .73993 1.35142 .76733 1.30323 .79544 1.25717 .824.*^ 1.21310 30 31 .74041 1.3.5060 .76779 1.30244 .79591 1.2.5642 .82483 1.21238 20 32 .74033 1.34978 .76825 1.30166 .79639 1.25567 .82531 1.21166 23 33 .74131 1.31896 .76871 1.30087 .79036 1.25492 .82580 1.21004 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25-117 .82629 1.21023 26 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .74267 1.34650 .77010 1.29653 .79829 1.25268 .82727 1.20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.207.36 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.1^969 .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74533 1.34160 .77289 1.29385 .80115 1 .24820 .83022 1.204.51 18 43 .74583 1.34079 .77-335 1.29307 .80103 I.a4r46 .83071 1.20379 17 44 .74628 1.3.3998 .77382 1.29229 .80211 1.24672 .83120 1 .20308 16 45 .74674 1.33916 .77423 1.29152 .80258 1.24597 .83169 1 .20237 15 46 .74719 1.33835 .77475 1.29074 .80306 1.24523 .83218 1.20166 14 47 .74764 1.. 33754 .77521 1.28997 .80354 1.^49 .63268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .63317 1.20024 12 49 .74855 1.33.592 .77615 1.28842 .80450 1.24301 .83366 1.19953 11 50 .74900 l.:43511 .77661 1.287G4 .80498 1.24227 .83415 1.19882 10 51 .74946 1.334.30 .77708 1.28687 .80546 1.24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28G10 .80594 1.24079 .83514 1.19740 8 53 .75037 1.33263 .77801 1.28533 .80642 1.24005 .83564 1.19669 7 54 .75082 1.33187 .77848 1.28456 .80090 1.23931 .83613 1.19599 6 55 .75128 1.. 33107 .77895 1.28379 .80738 1.23858 .83662 1.19528 5 56 .75173 1.3.3026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.. 32946 .77988 1.28225 .80834 1.23710 .83761 1.19387 3 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .83811 1.19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.2.3563 .83860 1.19246 1 GO .75355 1.. 32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 0 / Cotaiif' 1 Tang [Cotang Tang Cotang Tang Cotang Tang / 63° 62° 61° 1 60 ° TABLES. 667 TABLE Y\\.— Continued. Natural Tangents and Cotangents. 0 41° 42° 43° Tang 1 Cotang Tang Cotang Tang Cotang Tang 1 Cotang 0 .83910 1.19175 .86929 1.15037 .90040“ 1.11061 .932.52 1.07237 60 1 .83960 1.19105 .86980 1.14969 .90093 1.10996 .93306 1.07174 59 2 .84009 1.19035 .87031 1.14902 .90146 1.10931 .93360 1.07112 58 3 .84059 1.18964 .87082 1.14834 .90199 1.10867 .93415 1.07049 57 4 .84108 1.18894 .87133 1.14767 .90251 1.10802 .93469 1.06987 56 5 .84153 1.18824 .87184 1.14699 .90304 1.10737 .93524 1.06925 55 6 .84208 1.187.54 .87236 1.14632 .903.57 1.10672 .9.3578 1.06862 54 7 .84258 1.18684 .87287 1.14565 .90410 1.10607 .93633 1.06800 53 8 .81307 1.18614 .87338 1.14498 .90463 1.10543 .93688 1.067.38 52 9 .84357 1.18544 .87389 1.14430 .90516 1.10478 .93742 1.06676 51 10 .84407 1.18474 .87441 1.14363 .90569 1.10414 .93797 1.06613 50 11 .84457 1.18404 .87492 1.14296 .90621 1.10349 .93852 1.06551 49 12 .84507 1.18334 .87543 1.14229 .90674 1.10285 ! .93906 1.06489 43 13 .84556 1.18264 .87595 1.14162 .90727 1.10220 1 .93961 1.06427 47 14 .84606 1.18194 .87646 1.14095 .90781 1.101.56 1 .94016 1.06365 46 15 .84656 1.18125 .67698 1.14028 .90834 1.10C91 i .94071 1.06303 45 IG .84706 1.18055 .87749 1.13961 .90887 1.10027 .94125 1.06241 44 17 .84756 1.17986 .87801 1.13894 .90940 1.00903 .94180 1.06179 43 18 .&4806 1.17916 .87852 1.13828 .90993 1. 09899 .94235 1.06117 42 19 .84856 1.17846 .87904 1.137G1 .91046 1 .09834 .94290 1.06056 41 20 .84906 1.17717 .87955 1.13694 .91099 1.09770 .94345 1.05994 40 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 1.0,59.32 39 22 .85006 1.17638 .88059 1.13561 .01206 1.09642 .944.55 1.05870 33 23 .85057 1.17569 .88110 1.13494 .91259 1.09578 .94510 1.0,5809 37 24 .85107 1.17500 .88162 1.13423 .91313 1.09514 .94565 1.05747 36 25 .85157 1.17430 .88214 1.13361 .91-366 1.09450 .94620 1.05685 35 20 .85207 1.17361 .83265 1.13295 .91419 1.09386 .94676 1.05624 34 27 .85257 1.17292 .88317 1.13223 .91473 1.09322 .94731 1.0.5.562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 1.05501 32 29 .85358 1.17154 .88421 1.1 3096 .91580 1.09195 .94841 1.054.39 31 30 .85408 1.17085 .8847'3 1.13029 .91633 1.09131 .94896 1.05378 30 31 .85458 1.17016 .88524 1.12963 .91687 1.C90C7 .94952 1.05317 29 32 .85509 1.16947 .88576 1.12897 .917-40 1.C9003 .95007 1.05255 28 33 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 1.05194 27 31 .85609 1.16209 .88680 1.12765 .91847 1.0887'6 .9.5113 1.0.5133 26 35 .85660 1.16741 .88732 1.12699 .91901 1.08813 .95173 1.05072 25 3G .85710 1.16672 .88784 1.12633 .91955 1.08749 .9.5229 1.0.5010 24 37 .85761 1.16603 .88836 1.12567 .92008 1.08686 .9.5284 1.04949 23 38 .85811 1.16535 .88888 1.12501 .92162 1.08622 .9.5340 1.04888 22 39 .a58C2 1.1G4C6 .88940 1.12435 .92116 1.03559 .9.5.395 1.04827 21 40 .85912 1.16398 .68992 1.12369 .92170 1.08496 .95451 1.04766 20 41 .85963 1.16329 .89045 1.12303 .92224 1.08432 .95.506 1.04705 19 42 .86014 1.1 6261 .89097 1.12238 .92277 1,08369 .95.562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92331 1.08306 .95618 1.04583 17 44 .86115 1.16124 .89201 1.12106 .92385 1.08243 .9.5673 1.04.522 16 45 .86166 1.16056 .89253 /I. 12041 .92439 1.0817'9 .95729 1.04461 15 4G .8621« 1.15987 .89306 1.1 1975 .92493 1.08116 .9.57-85 1.04401 14 47 .86267 1.15919 .89358 1.11909 .92:547 1.08053 .9.5841 1.04.340 13 48 .86318 1.15851 .89410 1.11844 .92601 1 .07990 .95897 1.04279 12 49 .86368 1.15783 .89463 1 11778 .92655 1.07927 .9.5952 1.04218 11 50 .86419 1.15715 .89515 1.11713 .92709 1.07864 .96008 1.04158 10 51 .86470 1.15647 .89567 1.11648 .92763 1.07801 .96064 1.04097 9 52 .86521 1 . 15579 .89620 1.11.582 .9»j817 1.07738 .96120 1.04036 8 53 .86.572 1.15511 .89672 1.11.517 .92872 1.07676 .96176 1.03976 7 r>4 .86623 1.15443 .89725 1.11452 .92926 1.07613 .962.32 1.03915 6 55 .86674 1.15375 .89777 1.11387 .92980 1.07,5.50 .96288 1.03855 5 5G .86725 1.1.5308 .89830 1.11321 .9:4034 1. 07487 .96.344 1.03794 4 57 .86776 1.1.5240 .89883 1.112.56 .93088 1.07425 .96400 1.0.3734 8 58 .86827 1.15172 .899^5 1.11191 .93143 1.07362 .964.57 1.03674 2 59 .86878 1.15104 .89988 1.11126 .93197 1.07299 .96.513 1.0.3613 1 GO .86929 1.1.5037 .‘K)040 1.11061 .932,52 1.07237 .96.569 1 03.5.53 0 / Cotang i Tang j Cotang I Tang Cotang Tang Cotang 'I'ang 49“ 0 CO i 47° ! i 46° 668 SURVEYING. TABLE VII. — Continued. Natural Tangents and Cotangents. 9 440 . |, 440 / 1 1! . 440 / Tang Cotang 1 Tang Cotang Tang Cotang 0 .96569 1.03.5.53 60 20 .97700 1.02855 40 ' !40 .98843 1.01170 20 1 .96625 1.0:3493 59 21 .977.56 1.02295 1 '^>9 1 1 41 .98901 1 01112 19 2 .96681 1.034.33 58 22 .97813 1.022:56 1 .381 42 .089.58 1 010.53 18 3 .96738 1.0;3172 57 23 ,97870 1.02176 1 37 i 4.3 .99016 1.00994 17 4 .96791 1.0:3312 56 24 .97927 1.02117 ; 36 : 44 .9(K)7.3 1.009:35 16 5 .90a50 l.a32.52 55 25 .9798'4 1.020.57 1 .85! 45 .99131 1.00876 15 6 .96907 1.03192 54 26 .98041 1.011H)8 .34 ! 46 .99180 1.00818 14 7 .96963 1.031.32 53 27 .98098 1.019.30 :33 47 .99217 1.00750 13 8 .97020 1 .0:1072 52 28 .981.55 1.01879 32 48 .99.304 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 49 .99.362 1.00642 11 10 .97133 1.02952 50 :30 .98270 1.01761 30 50 .99120 1.00583 10 11 .97189 1.02802 40 31 .08327 1.01702 20 51 .90178 1.00.525 9 12 .97246 1.028.32 48 32 .93384 1.01612 , 281 52 .99.536 1.00467 8 13 .97302 1 .02772 47 .33 .98441 1.01.583 I 27 1 .5:3 .99.594 1.00408 7 14 .97359 1.02713 46 :34 .08190 1.01.521 ' 26 1 51 .996.52 1.00.350 6 15 .97416 1.026.53 45 .35 .9,85.56 1.01465 25 > 55 .99710 1.00291 5 16 .97472 1.02.503 44 .36 .98613 1.01406 24 .56 .997(W 1.00*233 4 17 .97529 1.02533 43 37 .08671 1.01.347 1 23! ' 57 .998*26 1.00175 3 18 .97586 1.02474 42 .38 .98728 1.01288 22 ' 58 .09884 1.00116 2 19 .97643 1.02414 41 .30 .98786 1.01*220 21 50 .9994*2 1.00058 1 ! 20 .97700 1.02:355 40 40 .98843 1.01170 20' 60 1 .00000 1 l.OOOOO 0 1 Cotang 1 Tang / / Cotang Tang 1 / Cotang 1 Tang 45° 45° 1 45° = Dumber degrees of longitude between the given meridian and the prime meridian of the map. TABLES. 669 Co-ordinates of Points of Intersection of Parallels and Meridians in Polyconic Projection. § 417. Giving Values of C in Kutter’s Formula when j‘ = p.ooi. § 259. 670 SURVEYING. c ^ « CO ^ >5 C X © © e> -t « X © e» -? © X © -f X ?> c © ^ ^ ^ ^ et it et n X r. t -I- o' ro in 00 0 0 lO « m fx in N VO Os 000 0"0 0 m tx tx tx CO © SO M 10 0 M Ct N ^ m n m m'O m m m m CO ^ M txoo O' 0 0 ^ sr ^ m c« m t m'O m m m m m 0 M ro r*-i M m m 0 -t Os M W O'© 0 w m c* 00 CO m rx tx CO 0 0 VO 0 N Cl fO m ro 00 0 (s m -t m ^ tx 0 •- c< t ^ ^ m m m m'O cxco O' m m m m m n r« rs* in sc VO so VO vo iio C "0 ^ 0 N c* 00 M M m 0 m O' « fi m m 0 moo 0 M © in moo n m w CO CO ^ 00 0 c« m 1/. ^ m m m m 00 0 w -t m mvo VO VO VO soooao- so VO so tx Cx m t m txoo tx tx tx tx tx IS e» e? c 0 'O so -^00 m O' 0 00 m m cv 0"0 -t |X 0 •-' w « 0 m Os « m 0 00 -too CO ro '•t Tf so f SO 0 0 m m mso vo m O' cn so VO VO «tvO txOO Os rx Cx fx rx Cx •- ri m mvo 00 CO 00 00 GO © in N 00 N 0 0 -t m m 0 m ft Os m.so 0 « m w m 't-'O tt 0 »o in SO Q CO m mo \0 ^ tx. Ov « n tx tx. txOO 00 -t m rxoo O' CO CO CO CO CO O' O' O' Os'os so W '«t00 00 ^ m O' N 0 Cl m'O ''t 0 'too 0 M CO mvo Os 0 •^vo moo mvo VO VO O' '^'O 1^00 00 00 <» O' O'^'os Q >-• Cl -t m 00000 ^<3 3:2 b, 0 in Id D .J < > I .015 N CO 0 mio m 0 m m'O c^oo 00 00 0 VO 0 M 00 N -t c^ Os 00 Os O' Os Os tx cx w m. m f» moo 0 N 0 0 0 « -i 0 -too 0 Cl t m-o CO O' m 0 VO O' ci M m m tx W d M Cl W CO 0 N (N W N m t^vo w m tx c< tx m m O' CO IX O' 0 CO -stco 0 c m 0 0 VO *-• VO CO Os O' 0 moo ^ ’^t'O 0 0 « « M 0 mvo 00 0 c< cs c< c* m Cl m m'O tx m rr. m m, m 0 HI ro -^vo -t -St -t -^r tv « 00 0 w m m M m cx. ^ ^xoo tx m.oo M m, -t m 0 tx H 0 ; W C'OO VO « r^oo O' 0 « m O' c< m M »-i CJ N M xj- fx O' >-• m m m m m \0 00 O' M m. mvo tx m m m, u-i m N 0 0 0 00 m O' M 00 m VO OsCO W m. 0 moo N CO Cl 00 mso r=l © c< 0 •- 00 m 00 0 M M o» 00 ►- m hx 0 ci m m m ''I- 'St cx 0 m m xt m m m tx O' 0 •- m m m\c VO VO m txoo 0 VC VO VO IX tx 0 CO M 00 mvo m ''too w •♦00 m'O m Cl CO w ^ txvo moo M 0 m. m m N 00 Ov ^ C 4 m m m C'x 0 co*o •'T ''i* m m m 0 't Cx O' M SO VO VO VO Cx m m.vo CO O' tx tx tx tx rx M m mvo CO cc 00 CO 00 00 0 m moo moo O' w C^OO "t 0 VO tx N 'St mo t- 0 rn tx tx '.t 0 m © © 00 O' 0 VO 0 cs m m M VO O' 01 m so VO VO 0 mvo O' ^ CO 00 00 00 Os m. mvo 00 O' O' O' O' O' O' M m m txoo 00000 Cl Cl Cl Cl Cl r in feet. H N CS Tjl LS ©t'X©© N^OX© HHHHM N'i'XX© «W««N '!j‘ X e? © © . p: ec ^ ^ ‘-'5 TABLE X. § 25 q. Giving Diameters in Feet of Circular Brick Conduits for Various Inclinations and Rates of Discharge. Conduit full to point of maximum discharge. (By Kutter’s formula.) § 239. TABLES. 671 0) •a - a H«CO'i0 » h °. H ” ” ” ” ” N N « N N N N N CO CO CO CO CO rji U t^oo 0 •- M moo 0 « 'CiO CD t'* ® H CO rj< ® ® H O ® Tji 0 mo 'd-oo m m < „ „ M M M M Cl N ej;' {*}■ ((j' e,’ e^' {,' ^ ^ 1010 '°'® C' C' 00 0 O' M N mvo Ov M m mvo r^ovo Pc^lC?'*Xeii.O®®^OOHl.O®»-0® c^ HMMHiHcicicicicicim roj^ CO M M "ji ^ LO LO UO 0 CD r- r* X °° °° ^ c 00 0 Cl m ^co M mmc^OvO w mmi^o w dcecjNvjjcs^Qo^acMXCOr^H CO „ „ „ „ „ d Cl Cl Cl Cl H © 00 H m■'^mo^mmt^ovM ci r^m^^o « r^vo o mvo 05Tf(®«r’^®»0®U5Ci N H M M M M d d d d 1H HH 10 O' M m mvo 0 -vJ-vo O' M m -vj-vo c^ o ci -.i-vo oo d vo O' moo Xf^rHX»0®®iHia H mmhhwWNN ^ rH H H H 0 Os c* Th\o t^NVO Ow fOiot^OO ro irioo O Cl m ^co ro o* moo IC O'? X ^ Ci ^ iM H M M N w 0 lovo 'O \o t^oo oo ^ q ^ ^ ^ rH H iH H flj *0 u a HNMTj(>a®»D®».'5®>'i®»D®®®®®®W®»«®®®®®®0®©®© 0 V 0 r-IH0JNXM^^»0©i''XO®«l0?'®»0O»D®®®®®®® !5 CL 0 r1i-ltHHW0lMMT)(ID®r-X®® 3 u t/) H 672 S UR VE YING. TAIU.E XI. Volumes by the Prism'jid/l Formula. § 320. Widths. Heights. Corrections for ti nths in height. 1 2 3 5 6 7 8 9 10 1 0 1 1 1 2 0 2 2 3 3 . I 0 2 1 1 0 2 3 3 4 5 6 6 .2 0 » 1 2 3 4 5 6 6 7 8 9 .3 0 4 1 2 4 5 6 7 9 10 11 12 .4 1 6 2 —3 — 5 -6 —8 —9 —11 —12 —14 —15 .5 1 6 2 4 6 7 9 11 13 15 17 19 .6 1 7 2 4 G 9 11 13 15 17 19 22 .7 1 8 2 5 7 10 12 15 17 20 22 25 .8 1 {) 3 0 8 11 14 17 19 22 25 28 •9 1 10 3 0 9 12 15 19 22 25 28 31 11 3 7 10 14 17 20 24 27 31 34 .1 0 12 4 7 11 15 19 22 26 30 :« 37 .2 1 13 4 8 12 16 20 24 28 32 36 40 .3 1 14 4 9 13 17 22 26 30 35 39 43 • 4 2 16 —5 —9 —14 —19 —23 —28 —32 — 37 —42 —46 • 5 2 10 6 10 15 20 25 30 35 40 44 49 .6 3 17 5 10 16 21 26 31 37 42 47 52 .7 3 18 6 11 17 22 28 33 39 44 50 56 .8 4 19 6 12 18 23 29 ;i5 41 47 53 59 •9 4 20 6 12 19 25 31 37 43 49 56 62 21 6 13 19 26 32 39 45 52 58 65 , t 1 22 7 14 20 27 34 41 48 54 61 68 .2 2 23 7 14 21 28 35 43 50 57 64 71 .3 2 24 7 15 22 30 37 44 52 59 67 74 .4 3 25 —8 — 15 —23 -31 —39 —46 —54 —62 —69 —77 .5 4 26 8 16 24 32 40 48 56 64 72 80 .6 5 27 8 17 25 33 42 50 58 67 75 83 .7 5 28 9 17 26 35 43 52 60 69 78 86 .8 6 29 9 18 27 36 45 54 63 72 81 90 •9 7 80 9 19 28 37 46 56 65 74 83 93 81 10 19 29 38 48 57 67 77 86 96 .1 1 82 10 20 30 40 49 59 69 79 89 99 .2 2 83 10 20 31 41 51 61 71 81 92 102 .3 3 84 10 21 31 42 52 63 73 84 94 105 •4 4 35 —11 -22 —32 —43 —54 —65 —76 —86 -97 —108 • 5 5 36 11 22 33 44 56 67 78 89 100 111 .6 6 37 11 23 34 46 57 69 80 91 103 114 .7 8 38 12 23 35 47 59 70 82 94 106 117 .8 9 39 12 24 36 48 60 72 84 96 108 120 •9 10 40 12 25 37 49 62 74 86 99 111 123 41 13 25 38 51 63 76 89 101 114 127 .1 1 42 13 26 39 52 65 78 91 104 117 130 .2 3 43 13 27 40 53 66 80 93 106 119 133 .3 4 44 14 27 41 54 68 81 95 109 122 136 •4 6 45 —14 -28 —42 —56 —69 -83 —97 -111 —125 —139 .5 7 46 14 28 43 57 71 85 99 114 128 142 .6 8 47 15 29 44 58 73 87 102 116 131 145 .7 10 48 15 30 44 59 74 89 104 119 133 148 .8 11 49 15 30 45 60 76 91 106 121 136 151 .9 13 60 15 31 46 62 77 93 108 123 139 154 1 2 3 4 5 6 7 8 9 10 . I .2 •3 •4 ■ 5 .6 •7 .8 •9 Corrections for tenths in width. 0 0 0 1 1 1 1 1 TABLES. 673 TABLE XI. — Contimied. Volumes by the Prismoidal Formula. Heights. Corrections 1 1 2 3 4 5 6 7 8 9 10 in height. 61 16 31 47 63 79 94 110 126 142 1.57 .1 2 62 16 32 48 64 80 96 112 128 144 160 .2 3 63 16 33 49 65 82 98 115 131 147 163 • 3 5 64 17 33 50 67 83 100 117 133 1.50 167 •4 7 65 —17 -31 —51 —68 —85 —102 —119 —1.36 —153 —170 .5 8 66 17 35 52 69 86 104 121 138 156 173 .6 10 67 18 35 53 70 88 106 123 141 158 176 .7 12 68 18 36 54 72 90 107 125 143 161 179 .8 14 69 18 36 55 73 91 109 127 146 164 182 •9 15 60 19 37 56 74 93 111 130 148 167 185 61 19 38 56 75 94 113 1.32 151 169 188 .1 2 62 19 38 57 77 96 115 134 153 172 191 .2 4 63 19 39 58 78 97 117 136 156 175 194 .3 6 64 20 40 59 79 99 119 138 158 178 197 ■4 8 65 —20 -40 —60 —80 —100 —120 —140 —160 —181 -201 .5 10 66 20 41 61 81 102 122 143 163 183 204 .6 12 67 21 41 62 83 103 124 145 165 186 207 .7 14 68 21 42 63 84 105 126 147 168 189 210 .8 16 69 21 43 64 85 106 128 149 170 192 213 •9 18 70 22 43 65 86 108 130 1.51 173 194 216 71 23 44 66 88 100 131 153 175 197 219 .1 2 72 22 44 67 89 111 133 156 178 200 222 .2 5 73 23 45 68 90 113 135 1.58 180 203 225 .3 7 74 23 46 69 91 114 137 160 183 206 228 •4 9 75 —23 —46 —69 —93 -116 —139 —162 — 185 —208 —231 .5 12 76 23 47 70 94 117 141 164 188 211 235 .6 14 77 24 48 71 95 119 143 166 190 214 238 .7 16 78 24 48 72 96 120 144 169 193 217 241 .8 19 79 24 49 73 98 122 146 171 195 219 244 •9 21 SO 25 49 74 99 123 148 173 198 222 247 81 25 50 75 100 125 1.50 175 200 225 250 3 82 25 51 76 101 127 152 177 202 228 253 .2 5 83 26 51 77 102 128 154 179 205 231 256 .3 8 84 26 52 78 104 130 156 181 207 233 259 •4 10 85 —26 —52 —79 —105 —131 —1.57 -184 —210 —236 —262 .5 13 86 27 53 80 106 133 159 186 212 239 265 .6 16 87 27 54 81 107 134 161 188 215 242 269 .7 18 88 27 54 81 109 136 163 190 217 244 272 .8 21 89 27 55 82 110 137 165 192 220 247 275 •9 24 90 28 56 83 111 139 167 194 222 250 278 91 28 56 84 112 140 169 197 225 253 281 .1 3 92 28 57 85 114 142 170 199 227 256 284 .2 6 93 29 57 86 115 144 172 201 230 258 287 .3 9 94 29 58 87 116 145 174 203 232 261 290 •4 12 95 -29 —59 -88 —117 —147 -176 —205 —235 —264 —293 .5 15 90 30 59 89 119 148 178 207 2:37 267 296 .6 18 97 30 60 90 120 150 180 210 240 269 299 .7 21 98 30 60 91 121 151 181 212 242 272 302 .8 23 99 31 61 92 122 153 183 214 244 275 306 •9 26 100 31 62 93 123 151 185 216 247 278 309 1 2 i 3 4 5 6 7 8 1 9 10 . I .2 1 “ •4 •5 .6 •7 .8 •9 / 0 0 1 “ 1 1 1 1 1 1 tenths in width. 674 SURVEYING. TAHLE XI. — ContiftiicJ. Volumes ijy the Prismoidal Formula. (/) x: Heights. Correction* for tenths 11 12 1,3 11 15 10 ' 1 ' i 18 19 20 in height. 1 3 4 4 4 5 5 5 0 0 0 .T 0 a 7 7 8 9 9 10 10 11 12 12 .3 0 3 10 11 12 13 14 15 10 17 18 19 .3 0 4 1 1 15 10 17 19 20 21 22 23 25 .4 1 r> —17 —10 —20 00 —23 -25 —20 —28 —29 -31 .5 1 (> 20 oo 21 20 28 30 31 33 35 37 .6 1 1 24 20 28 30 32 .3.5 37 39 41 43 .7 1 8 27 30 32 35 37 40 42 44 47 49 .8 1 1) 31 33 36 39 42 44 47 .50 .53 56 •9 1 10 34 37 40 43 40 49 52 50 59 02 11 37 41 44 48 51 54 58 01 05 08 0 12 41 44 48 52 50 59 63 07 70 74 .2 1 13 44 48 52 50 on 04 08 72 70 80 .3 1 14 48 52 50 00 05 09 73 78 82 80 •4 2 15 —51 —50 —00 —05 —09 —74 —79 —83 —88 —93 .5 2 10 54 59 04 09 74 79 84 89 94 99 .6 3 17 58 03 08 73 79 84 89 94 100 105 .7 3 18 01 07 70 78 8.3 89 94 100 106 in .8 4 10 05 70 70 82 88 94 100 106 111 117 •9 4 20 08 74 80 80 93 99 105 111 117 12.3 21 71 78 84 91 97 104 110 117 123 130 .1 1 22 75 81 88 95 102 109 115 122 129 1.36 .2 0 23 78 85 92 99 106 114 121 128 13.5 142 .3 2 24 81 89 90 104 111 119 120 133 141 148 •4 3 25 -85 -93 —100 —108 —116 —123 131 -139 -147 —1.54 .5 4 20 88 96 104 112 120 128 130 144 152 100 .6 5 27 92 100 108 117 125 133 142 1.50 1.58 107 .7 5 28 95 104 112 121 130 1.38 147 1.50 164 173 .8 0 20 98 107 116 125 134 143 : 152 101 170 179 •9 7 80 102 111 120 130 139 148 i 157 107 176 185 81 105 115 124 134 144 153 103 172 182 191 .1 1 32 109 119 128 138 148 1.58 168 178 188 198 .2 2 33 112 122 132 143 1.53 163 173 183 194 204 .3 3 84 115 126 136 147 1.57 168 178 189 199 210 •4 4 35 —119 —130 —140 —151 —102 —173 —184 —194 -205 -216 .5 5 30 122 133 144 156 167 178 189 200 211 222 .6 6 37 126 137 148 100 171 183 194 200 217 228 .7 8 38 129 141 152 164 176 188 199 211 223 235 .8 9 30 132 144 1.50 109 181 193 205 217 229 241 •9 10 40 130 148 160 173 185 198 210 222 235 247 41 139 152 165 177 190 202 215 228 240 258 . I 1 42 143 156 169 181 194 207 220 233 246 259 .2 3 43 146 159 173 186 199 212 226 239 252 205 .3 4 44 149 163 177 190 204 217 231 244 258 272 •4 6 45 —153 -167 -181 —194 -208 -222 -236 —250 —264 —278 .5 7 40 156 170 185 199 213 227 241 256 270 284 .6 8 47 160 174 189 203 218 232 247 261 276 290 .7 10 48 163 17-8 193 207 222 237 252 267 281 290 .8 11 40 160 181 197 212 227 242 257 272 287 302 •9 13 60 170 185 201 216 231 247 262 273 293 309 11 12 13 14 15 16 17 18 19 20 .1 .2 ■3 •4 •5 .6 •7 .8 •9 Corrections for 0 1 1 0 2 3 3 4 4 tenths in width. TABLES. 675 TABLE XI. — Con tin ued. Volumes by the Prismoidal Formula. ■5 Heights. Corrections for tenths in height. 11 12 13 14 15 16 17 18 19 20 51 173 ISO 205 220 236 252 268 283 299 315 . I 2 52 177 193 209 225 241 257 273 289 305 321 .2 3 53 180 196 213 229 245 262 278 294 311 327 .3 5 54 183 200 217 233 250 267 283 300 317 333 •4 7 55 —187 -204 —221 —238 —255 -272 —289 —306 —323 -340 .5 8 50 190 207 225 242 259 277 294 311 328 346 .6 10 57 104 211 229 246 264 281 299 317 334 352 .7 12 58 197 215 233 251 269 286 304 322 340 358 .8 14 59 200 219 237 255 273 291 310 328 346 364 •9 15 60 204 222 241 259 278 296 315 333 352 370 61 207 226 245 264 282 301 320 339 358 377 . I 2 62 210 230 249 268 287 306 325 344 364 383 .2 4 63 214 233 253 272 292 311 331 350 369 389 .3 6 64 217 237 257 277 296 316 336 356 375 395 •4 8 65 —221 -241 —261 -281 -301 —321 -341 -361 —381 -401 .5 10 66 224 244 265 285 306 326 346 367 387 407 .6 12 67 227 248 269 290 310 331 352 37'2 393 414 .7 14 68 231 252 273 294 315 336 357 378 399 420 .8 16 69 234 256 277 298 319 311 362 383 405 426 •9 18 70 238 259 281 302 324 346 367 389 410 432 71 241 263 285 307 329 351 373 394 416 438 . I 2 72 244 267 289 311 333 356 378 400 422 444 .2 5 73 248 270 293 315 338 360 383 406 428 451 .3 7 74 251 274 297 320 313 365 388 411 434 457 ■4 9 o_-55 — 278 —301 —324 —347 —370 —394 417 —440 4(33 12 i 0 76 258 281 305 328 352 375 399 422 446 469 • 5 .6 14 77 2(il 285 309 333 356 380 404 428 452 475 .7 16 78 2(55 289 • 313 337 361 385 409 433 457 481 .8 19 79 2(58 293 317 341 366 390 415 439 463 488 •9 21 80 272 296 321 346 370 395 420 444 469 494 81 275 300 325 350 375 400 425 450 475 500 . I 3 82 278 304 329 354 380 405 430 456 481 506 .2 5 83 282 307 333 359 384 410 435 461 487 512 .3 8 84 285 311 337 363 389 415 441 467 493 519 •4 10 85 —289 —315 -341 -367 -394 —420 —446 -472 —498 -525 .5 13 86 292 319 345 372 398 425 451 478 504 531 .6 16 87 295 322 349 376 403 430 456 483 510 537 • 7 18 88 299 326 353 380 407 435 462 489 516 543 .8 21 89 303 330 357 385 412 440 467 494 522 549 •9 24 90 306 333 361 389 417 444 472 500 528 556 91 309 337 365 393 421 449 477 506 534 562 .1 3 92 312 311 369 398 426 454 483 511 540 568 .2 0 93 316 314 373 402 431 459 488 517 545 574 .3 9 94 319 348 377 406 435 464 493 522 551 580 •4 12 95 -323 —352 -381 —410 —440 -469 —498 -528 -557 —586 .5 15 . 96 326 356 385 415 444 474 504 533 563 593 .6 18 97 329 359 389 419 449 479 509 539 569 599 .7 21 98 333 3(>3 393 423 454 484 514 514 575 605 .8 23 99 336 367 397 428 458 489 519 550 581 Oil •9 26, 100 340 370 401 432 463 494 525 556 586 617 11 12 13 i 14 ' 15 16 17 18 19 . I .2 l_^ 1 -4 •5 iIL!| •7 .8 •9 Corrections for 0 1 1 ‘ 2 2 3 3 ^ 1 4 tenths in width. 676 SUR VE YJNG. TABLE XI. — Continued. Volumes by the Prismoidal Formula. Widths. 1 1 RIGHTS. (Correct ions for tcnilis III lici(;lit. 21 22 23 a, 25 j 26 27 28 20 30 1 6 7 7 7 8 8 9 1 9 9 . I 0 2 13 14 14 15 15 10 17 17 j 18 19 . a 0 8 19 20 21 22 23 21 2.5 20 1 27 28 .3 0 4 20 27 28 30 31 32 33 35 30 37 ■4 1 5 —32 —34 -ao -37 —39 —40 —42 —43 —45 —46 .5 1 0 39 41 43 44 40 48 50 52 54 50 .6 1 7 45 48 50 52 54 50 58 60 63 05 .7 8 52 54 57 59 02 04 07 09 72 74 .8 1 9 58 01 04 07 09 72 75 78 81 83 .9 10 05 08 71 74 77 80 83 80 90 93 11 71 75 78 81 a5 88 92 95 98 102 . I 0 12 78 81 85 89 93 90 100 104 107 in .3 1 18 84 88 92 90 100 114 108 112 no 120 .3 1 14 91 95 99 104 108 112 117 121 125 130 ■4 2 15 -9? —102 -100 —111 —no —120 -125 -130 - 1:44 —139 .5 2 10 104 109 114 119 123 128 133 1.38 143 148 .6 3 17 110 115 121 120 131 130 142 147 1.52 1.57 .7 3 18 117 122 128 133 139 144 150 1.50 161 107 .8 4 19 123 129 135 111 147 1 52 1.58 104 170 170 .9 4 20 130 130 142 148 154 100 107 173 179 185 21 130 14? 149 150 102 109 175 181 188 194 .1 1 22 143 149 150 103 17’0 177 183 190 197 204 .2 2 28 149 150 103 170 177 185 192 199 200 213 .3 2 24 150 103 170 178 185 193 200 207 215 222 •4 3 25 —102 —170 —177 -185 -193 -201 -208 —210 —224 -231 .5 4 20 109 177 185 193 201 209 217 225 233 241 .6 5 27 175 183 192 200 208 217 225 233 242 2.50 .7 5 28 181 190 199 207 210 225 2733 242 251 259 .8 6 29 188 197 200 215 224 233 242 251 260 269 •9 7 30 194 204 213 222 231 241 250 259 209 278 31 201 210 220 230 239 249 258 208 277 287 .1 1 32 207 217 227 237 247 257 267 277 286 290 .2 2 38 214 224 234 214 255 205 275 285 295 31 lO .3 3 34 220 £31 241 252 202 273 283 294 304 315 .4 4 35 —227 —238 —248 —259 —270 —281 —292 —302 —313 -324 .5 5 30 233 244 250 207 278 289 300 311 322 333 .6 6 37 210 251 203 274 285 297 308 320 331 .343 .7 8 38 240 258 270 281 293 305 317 328 340 352 .8 9 39 253 205 277 289 3'1 313 325 337 349 361 •9 10 40 259 272 284 290 309 321 333 340 358 370 41 200 278 291 304 310 329 342 354 367 380 I 1 42 272 285 298 311 324 337 350 363 376 389 .2 3 43 279 292 305 319 332 345 358 372 385 398 .3 4 44 285 299 312 320 340 353 307 380 . 394 41 '7 .4 6 45 —292 -300 —319 —333 —347 —301 —375 -389 —403 —417 .5 7 40 298 312 327 341 355 309 383 398 412 420 .6 8 47 305 319 334 348 303 377 392 406 421 435 .7 10 48 311 320 341 350 370 385 400 415 430 444 .8 11 49 318 333 348 303 37'8 393 408 423 4.39 4.54 •9 13 50 324 340 355 370 380 401 417 432 418 463 21 22 23 24 25 26 27 28 29 30 . I .2 •3 •4 •5 .6 •8 •9 Corrections for 1 2 2 3 4 5 5 6 7 tenths in width. TABLES. 677 TABLE XL — Continued, Volumes by the Prismoidal Formula. CO Heights. Corrections 21 22 23 24 25 26 27 28 29 30 111 height. 61 331 346 362 378 394 409 425 441 456 472 .1 2 52 337 353 369 385 401 417 433 449 465 481 .2 3 53 344 360 376 393 409 425 442 458 474 491 • 3 5 54 350 367 383 400 417 433 450 467 483 500 •4 7 55 —356 —373 -390 —407 —424 —441 -458 —475 —492 —509 • 5 8 56 363 380 398 415 432 449 467 484 501 519 .6 10 57 369 387 405 422 440 457 475 493 510 528 .7 12 68 376 394 412 430 448 465 483 501 519 537 .8 14 59 382 401 419 437 455 413 492 510 528 546 •9 15 60 389 407 426 444 463 481 500 519 537 556 61 395 414 433 452 471 490 508 527 546 565 .1 2 62 402 421 440 459 478 498 517 536 555 574 .2 4 63 403 428 447 467 486 506 525 544 564 583 .3 6 64 415 435 454 474 494 514 533 553 573 593 •4 8 65 —421 —441 —461 —481 —502 -522 —542 -562 —582 —602 .5 10 68 428 418 469 489 509 530 550 570 591 611 .6 12 67 431 455 476 496 517 538 558 579 600 620 • 7 14 68 441 462 483 504 525 546 567 588 609 630 .8 16 69 447 469 490 511 532 554 575 596 618 639 •9 18 70 454 475 497 519 540 562 583 605 627 648 71 460 482 504 526 548 570 592 614 635 657 .1 2 72 467 459 511 533 556 578 600 622 644 667 .2 5 ^ 0 i 0 473 496 518 541 563 586 608 631 653 676 .3 7 74 430 502 525 548 571 594 617 640 662 685 .4 9 75 —486 —509 —532 -556 —579 -601 —625 -648 —671 -694 • 5 12 76 493 516 540 563 586 610 633 657 680 704 .6 14 77 499 523 547 570 594 618 642 665 689 713 • 7 16 78 506 530 554 578 602 626 650 674 698 722 .8 19 79 512 536 561 585 610 634 658 683 707 731 •9 21 80 519 543 568 593 617 642 667 691 716 741 81 525 550 575 600 625 650 675 700 725 750 .1 3 82 531 557 582 607 633 658 683 709 734 759 .2 5 83 538 564 589 615 640 666 692 717 743 769 • 3 8 84 544 570 596 622 648 674 700 726 752 778 •4 10 85 — 551 — 577 -603 -630 -656 —682 —708 —735 —761 -787 • 5 13 86 557 584 610 637 664 690 717 743 770 796 .6 16 87 564 591 618 644 671 698 725 752 779 806 .7 18 88 570 598 625 652 679 706 733 760 788 815 .8 21 89 577 004 632 659 687 714 742 769 797 824 •9 24 90 583 611 639 667 694 722 750 777 806 833 91 590 618 646 674 702 730 758 786 815 843 . I 3 92 596 625 653 681 710 738 767 795 823 852 .2 6 93 603 631 060 689 718 746 775 804 832 861 • 3 9 94 609 638 667 696 725 754 783 812 841 870 • 4 12 95 —616 -045 -674 —704 -733 — 762 —792 -821 —850 —880 .5 15 96 022 652 681 711 741 770 800 830 859 889 .6 18 97 G29 659 689 719 748 778 808 838 868 898 • 7 21 00 C35 665 696 726 756 786 817 847 877 907 .8 23 09 C42 672 703 733 764 794 825 856 886 917 •9 26 ICO 048 679 710 741 772 802 833 864 895 926 21 22 23 24 25 26 27 28 29 30 , I .2 ■3 •4 •5 .6 •7 .8 •9 Corrections for 1 2 2 3 4 5 5 6 7 tenths in width. 678 SUR VE VI NG. TABLE yA. — Contifiucd. Volumes by the Prismoidal Formula. ) Widths. 1 Heights. Corrections for tciulis in heiKhl. 31 32 33 34 35 36 37 38 39 1 1 10 10 10 10 11 11 11 12 12 12 , 0 10 20 20 21 22 22 23 23 21 25 .2 0 ll 2 i ) 30 31 31 32 .33 34 35 36 ; 37 .3 0 4 38 40 41 42 43 44 46 47 48 49 4 1 5 —48 —49 — 51 —52 —54 —56 —57 -.59 —60 1 —62 ,5 1 0 57 59 61 63 65 67 68 70 72 i "4 .6 1 7 67 69 71 73 76 78 80 82 81 86 .7 1 8 ' t 7 79 81 84 86 89 91 94 ! 96 , 97 .8 1 9 86 89 92 94 97 100 103 106 108 1 111 .9 1 10 96 99 102 105 108 111 114 117 120 : 123 11 10.5 109 112 115 119 122 126 129 1.32 I 136 . I 0 12 115 119 122 126 1.30 133 137 141 1 14 148 .2 1 IS 124 128 132 136 140 144 148 1.52 1.56 ■ 160 .3 1 14 134 138 143 147 151 1.56 160 164 169 ' 173 .4 2 15 —144 —148 —1.53 —1.57 —162 —167 —171 —116 — IHl 1—185 .5 2 16 1.53 1.58 163 168 173 178 183 188 193 198 .6 3 17 163 168 173 178 183 189 194 199 205 210 .7 3 18 172 178 183 189 194 200 206 211 217 222 .8 4 19 182 188 194 199 205 211 217 223 229 235 •9 4 20 191 198 204 210 216 222 228 235 241 247 21 201 207 214 220 227 233 240 246 253 259 . I 1 22 210 217 224 231 238 244 251 2.58 265 272 .2 2 23 220 227 234 241 248 256 263 270 277 284 .3 2 24 230 237 244 2.52 259 267 274 281 289 296 •4 .3 25 -239 —247 —25.5 —262 —270 —278 — 285 —293 —301 —.3119 .5 4 26 249 257 265 273 281 289 297 .305 313 321 .6 5 27 2.58 267 275 283 292 300 .308 317 325 333 .7 5 28 268 277 285 294 302 311 320 328 .337 346 .8 6 29 277 286 29.5 304 313 322 .331 340 349 .358 •9 7 30 287 296 306 315 324 333 343 352 361 370 81 297 306 316 325 335 344 354 .364 373 383 .1 1 32 306 316 326 336 346 356 365 375 385 395 .2 0 33 316 326 336 346 356 367 377 387 397 407 •3 3 34 325 336 346 357 367 378 388 .399 409 420 •4 4 35 -.335 —346 —356 —367 —378 —389 —400 ^10 —421 —432 .5 5 36 344 356 367 378 389 400 411 422 433 444 .6 6 37 354 365 377 388 400 411 423 434 445 4.57 .7 8 38 364 375 387 399 410 422 434 446 457 469 .8 9 39 373 .385 397 409 421 433 445 457 469 481 •9 10 40 383 395 407 420 4.32 444 457 469 481 494 41 392 405 418 4,30 443 456 468 481 494 .506 j 1 42 402 415 428 441 454 467 480 493 506 519 .2 3 43 411 425 438 451 465 478 491 504 .518 .531 .3 4 44 421 4.35 448 462 475 489 502 516 5.30 .543 ■4 6 45 —431 —444 —4.58 -472 —486 —500 —514 —528 — 542 —556 .5 7 46 440 4.54 469 483 497 511 .525 540 .554 568 .6 8 47 450 464 479 493 508 522 537 .551 566 580 .7 10 48 4.59 474 489 504 519 533 548 56.3 578 593 .8 11 49 469 484 499 514 .529 .544 560 575 590 605 .9 13 60 478 494 .509 525 540 556 571 586 602 617 31 32 33 34 35 36 37 38 39 40 . I .2 •3 •4 •5 .6 •7 .8 •9 Corrections for 1 2 3 4 5 6 8 9 10 tenths in width. TABLES. 679 TABLE XI. — Continued. Volumes by the Prismoidal Formula. CA -C Heights. Corrections for tenths i 31 32 33 34 35 36 37 38 39 40 in height. 61 488 504 519 535 551 567 582 598 614 630 . I 2 52 498 514 530 546 562 578 594 610 626 642 .2 3 53 507 523 540 556 573 589 005 622 638 654 .3 5 64 517 533 550 567 583 600 617 633 650 667 •4 7 65 — 526 —543 —560 — 577 —594 —611 —628 —645 -662 —679 .5 8 oC 536 553 570 588 605 622 640 657 674 691 .6 10 67 545 563 581 598 616 633 651 669 686 704 .7 12 58 555 573 591 609 627 644 662 680 698 716 .8 14 69 565 583 601 619 637 656 674 692 710 728 •9 15 60 574 593 611 630 648 667 685 704 722 741 6] 584 602 621 640 659 678 697 715 734 753 .1 2 62 593 612 631 651 670 089 708 727 746 765 .2 4 63 603 622 642 661 681 700 719 739 758 778 .3 6 64 613 632 652 672 691 711 731 751 770 790 •4 8 65 —623 —642 —662 —682 —702 'J'OO —742 —762 —782 —802 .5 10 66 631 652 672 693 713 733 754 77’4 794 815 .6 12 67 641 662 682 703 724 744 765 786 806 827 .7 14 68 651 672 693 714 735 756 777’ 798 819 840 .8 16 69 660 681 703 724 745 767 788 809 831 852 •9 18 70 670 691 713 735 756 778 799 821 843 864 71 679 701 723 745 767 789 811 833 855 877 .1 2 72 689 711 733 756 778 800 822 844 867 889 .2 5 73 698 721 744 766 789 811 834 856 879 901 .3 7 74 708 731 754 777 799 822 845 868 891 914 •4 9 76 —718 —741 —764 —787 —810 —833 —856 —880 —903 —926 .5 12 76 727 751 774 798 821 844 868 891 915 938 .6 14 77 737 760 784 808 832 856 879 903 927 951 .7 16 78 746 710 794 819 843 867 891 915 939 963 .8 19 79 756 780 805 829 853 878 902 927 951 975 ! -9 21 80 765 790 815 840 864 889 914 938 963 988 81 775 800 825 850 875 900 925 950 975 1000 .1 3 82 785 810 835 860 886 911 936 902 987 1012 .2 5 83 794 820 845 871 897 922 948 973 999 1025 .3 8 84 804 830 856 881 907 933 959 985 1011 1037 •4 10 85 —813 —840 —866 —892 —918 —944 —971 —997 —1023 —1049 .5 13 86 823 849 876 902 929 956 982 1009 1035 1062 .6 16 87 832 859 886 913 940 967 994 1020 1047 1074 .7 18 88' 842 809 896 923 951 978 1005 1032 1059 1086 .8 21 89 852 87'9 906 . 934 961 989 1016 1044 1071 1098 •9 24 90 861 889 917 944 972 1000 1028 1056 1083 nil 91 871 899 927 955 983 1011 1039 1067 1095 1123 .1 3 92 880 909 937 965 994 1022 1051 1079 1107 1136 .2 6 93 890 919 917 976 1005 1033 1062 1091 1119 1148 .3 9 94 899 928 957 986 1015 1044 1073 1102 1131 1160 •4 12 95 —909 —938 —968 —997 —1026 -1056 —1085 -1114 —1144 —1173 .5 15 96 919 948 978 1007 1037 1067 1096 1126 1156 1185 .6 18 97 928 958 988 1018 1048 1078 1108 1138 1168 1198 .7 21 98 938 968 998 1028 1059 1089 1119 1119 1180 1210 .8 23 99 947 978 1008 1039 1069 1100 1131 1161 1192 1 222 •9 26 100 957 988 1019 1049 1080 1111 1142 1173 1204 1235 31 32 33 34 35 36 33 39 40 . I .2 •3 •4 •5 .6 •7 .8 ■9 Corrections for 1 2 3 4 5 6 8 9 10 tenths in width. 68o sun VE YtE G. TABLE XI. — Continued. Volumes by the Prismoidal Formula. Widths. I 1 Heights, j Corrections j for tenths ' in height. 41 42 43 44 45 46 47 48 49 50 1 13 13 13 14 14 14 15 15 15 15 ' J 0 2 25 20 27 27 28 28 29 80 30 31 • 2 0 8 38 39 40 41 42 43 44 44 45 40 ; .3 0 4 51 52 53 54 50 57 58 59 (K) 02 .4 1 6 —03 —05 —00 —08 —09 —71 —73 —74 —76 —77 • 5 1 G 70 78 80 81 83 85 87 89 91 93 .6 1 7 89 91 93 95 97 99 102 104 106 108 .7 1 8 101 104 106 109 111 114 110 119 121 123 ; .8 9 111 117 119 122 125 128 131 133 136 139 i 1 10 127 130 133 136 139 142 145 148 151 154 ! 11 139 113 1 10 149 153 156 160 103 106 170 . I 0 12 152 156 159 103 107 170 174 178 181 185 .2 I 18 105 109 173 177 181 185 189 193 197 201 1 -3 1 14 117 181 186 190 194 199 203 207 212 216 1 .4 2 l.> —190 —194 —199 —204 —208 —213 —218 —222 —227 —231 .5 2 IG 203 207 212 217 222 227 232 237 242 247 .6 3 17 215 220 226 231 236 211 247 252 257 262 .7 3 18 228 233 239 244 250 250 201 207 272 218 .8 4 19 210 216 252 258 204 270 270 281 2.S7 293 •9 4 20 253 259 205 272 278 284 290 290 302 309 21 266 272 279 285 292 298 305 311 318 324 . I 1 22 278 285 292 299 306 312 319 320 333 340 .2 2 23 291 298 305 312 319 327 334 341 318 355 .3 2 24 304 311 319 326 333 341 348 356 363 370 •4 3 25 —316 -324 —332 — :340 -347 —355 —363 —370 -378 —386 .5 4 2G 3J9 337 315 353 361 369 377 385 393 401 .6 5 27 342 310 358 367 375 383 392 400 408 417 .7 5 28 354 303 372 380 389 398 406 415 423 432 .8 6 29 367 376 385 394 403 412 421 430 439 448 •9 7 80 380 389 393 407 417 426 435 444 454 463 31 392 402 411 421 431 410 450 459 409 478 . I t 82 405 415 425 435 444 454 464 474 484 494 .2 2 33 418 428 438 448 458 469 479 489 499 509 .3 3 84 430 441 451 402 472 483 493 504 514 525 •4 4 35 —443 —454 — 405 — 175 -486 —497 —508 — 519 —529 —540 .5 5 36 456 407 478 489 500 511 522 533 544 556 .6 6 37 408 480 491 502 514 525 537 548 560 671 .7 . 8 38 481 493 504 516 528 540 551 563 575 586 .8 9 39 494 506 518 530 542 554 506 578 590 602 • -9 10 40 506 519 531 543 556 568 580 593 605 617 41 519 531 544 557 569 582 595 607 620 633 . I 1 42 531 544 557 570 583 596 609 622 635 648 .2 3 43 544 557 571 584 597 610 624 637 650 664 .3 4 44 557 570 584 598 Oil 625 638 652 665 679 •4 6 45 —509 —583 —597 —Oil -625 —639 — 653 —667 —681 —094 .5 7 4G 582 596 010 625 639 053 607 681 696 710 .6 8 47 595 009 624 038 653 007 082 096 711 725 .7 10 48 007 622 037 052 667 081 096 711 726 741 .8 11 49 020 035 050 005 681 090 710 726 741 756 •9 13 60 633 018 004 019 694 710 725 741 756 772 41 42 44 45 46 47 48 49 50 .1 . 2 ■3 •4 ■5 .6 ■7 .8 •9 Corrections for 1 3 4 0 7 8 10 11 13 tenths in width. TABLES. 68 1 TABLE XI. — Continued. Volumes by the Prismoidal Formula. cn Heights. Corrections 41 42 43 44 45 46 47 48 49 60 in height. 61 645 661 677 693 708 724 740 756 771 787 2 52 658 674 690 706 722 738 754 770 786 802 .2 3 53 671 687 703 720 736 752 768 785 802 818 • 3 5 54 683 700 717 733 750 767 783 800 817 833 *4 7 65 —696 —713 —730 —747 —764 —781 —798 —815 -832 —849 • 5 8 66 709 726 743 760 778 795 812 830 847 864 .6 10 57 721 739 756 774 792 809 827 844 862 880 • 7 12 58 734 752 770 788 806 823 841 859 877 895 .8 14 59 747 765 783 801 819 833 856 874 892 910 •9 15 60 759 778 796 815 833 852 870 889 907 926 61 772 791 810 828 847 866 885 991 923 941 . I 2 62 785 804 823 842 861 880 899 919 938 957 .2 4 63 797 817 836 856 875 894 914 933 953 972 • 3 6 64 810 830 849 869 889 909 928 948 968 988 •4 8 65 —823 -843 —863 —883 —903 —923 -943 —963 —983 —1003 • 5 10 66 835 856 876 896 917 937 957 978 998 1019 .6 12 67 848 869 889 910 931 951 972 993 1013 1034 . 7 14 68 860 881 902 923 944 965 986 1007 1028 1049 .8 16 69 873 894 916 937 958 980 1001 1022 1044 1065 •9 18 70 886 907 929 951 972 994 1015 1037 1059 1080 71 898 920 942 964 986 1008 1030 1052 1074 1096 . I 2 72 911 933 956 978 1000 1022 1044 1067 1089 nil .2 3 73 924 946 969 991 1014 1036 1059 1081 1104 1127 • 3 7 74 936 959 982 1005 1028 1051 1073 1096 1119 1142 •4 9 75 —949 —972 —995 —1019 —1042 —1065 -1088 —nil -1134 —1157 • 5 12 76 962 985 1009 1032 1056 1079 1102 1126 1149 1113 .6 14 77 974 998 1022 1046 1069 1093 1117 1141 1165 1188 • 7 16 78 987 1011 1035 1059 1083 1107 1131 1156 1180 1204 .8 19 79 1000 1024 1048 1073 1097 1122 1146 1170 1195 1219 •9 21 80 1012 1037 1062 1086 nil 1136 1160 1185 1210 1235 81 1025 1050 1075 1100 1125 1150 1175 1200 1225 1250 . I 3 82 1038 1063 1088 1114 1139 1164 1190 1215 1240 1265 .2 5 83 1050 1076 1102 1127 1153 1178 1204 1230 1255 1281 • 3 8 84 1063 1089 1115 1141 1167 1193 1219 1244 1270 1296 •4 10 85 —1076 —1102 —1128 —1154 —1181 —1207 —1233 —1259 —1285 —1312 • 5 13 86 ' 1088 1115 1141 1168 1194 1221 1248 1274 1301 1327 .6 16 87 1 1101 1128 1155 1181 1208 1235 1262 1289 1316 1343 • 7 18 88 1114 1141 1168 1195 1222 1249 1277 1304 1331 1358 .8 21 89 1126 1154 1181 1209 1236 1264 1291 1319 1346 1373 •9 24 90 1139 1167 1194 1222 1250 1278 1306 1333 1361 1389 91 1152 1180 1208 1236 1264 1292 1320 1348 1376 1404 .1 3 92 1164 1193 1221 1249 1278 1306 1335 1363 1391 1420 .2 6 93 1177 1206 1234 1263 1292 1320 1349 1378 1406 1435 • 3 9 94 1190 1219 1248 1277 1306 1335 1364 1393 1422 1451 • 4 12 95 —1202 —1231 —1261 -1290 —1319 —1349 —1378 -1407 —1437 —1466 5 15 96 1215 1244 1274 1304 1333 1363 1393 1422 1452 1481 .6 18 97 1227 1257 1287 1317 1347 1377 1407 1437 1467 1497 • 7 21 98 1240 1270 1301 1331 1361 1391 1422 1452 1482 1512 .8 23 99 1253 1283 1314 1344 1375 1406 1436 1467 1497 1528 •9 26 100 1265 1296 1327 1358 1389 1420 1451 1481 1512 1543 41 42 43 44 45 46 47 48 49 60 .1 .2 •3 ■4 •5 .6 •7 .8 •9 1 Corrections for 3 4 6 7 8 10 11 13 tenths in width. INDEX. Abney Level and Clinometer Accuracy of the Stadia Method Attainable by Steel Tapes, and Metallic Wires in Measurements Adjustments, Method of Studying General Principle of Reversion of Compass of Level Precise of Plane Table of Sextant of Solar Compass Attachment of Transit of Angles in Triangulation Systems Triangle Quadrilateral.. . Larger Systems of Polygonal Systems in Leveling Agreement, Want of, between .Surveyors Alignment, Corrections for, in Base-line Measurements to invisible Stations Altitude of a Heavenly Body Aneroid Barometer Angle Measurement in Triangulation Angles Measured by Chain Angular Measurements in Subdivision Areas of Cross Sections in Rivers PAGE 141 263 473 4 15 15 63 553 119 Ill 41 102 86 491 493 494 506 559 393 462 432 531 127 477 to 488 12 360 290 684 INDEX. PAGE Areas of Land 179 by Triangular Subdivision 180 from boundary Lines . . 181 from Rectangular Coordinates of the Corners 200 of Irregular Figures 208 Formulx for Derived 605 Azimuth Defined ii and Latitude by Observations on Circumpolar Stars 508 to 518 of Polaris at Elongation, Table of 33 Balancing a Survey 190 Barometer, Aneroid 127 use of the Aneroid 136 Barometric Formula Derived 128 Tables I 33 Base-line and its Connections 427 Measurement 447 to 465 Broken, Reduction to a Straight Line .... 468 Reduction to Sea Level 468 Computation of Unmeasured Portion 472 Summary of Corrections to 469 Bed Ownership in Water Fronts 586 Bench-marks 74, 291 in Cities 384 in Triangulation 445 Borrow Pits 420 Bubble, Value of one Division of 58, 550 Bubbles, Level 55 Construction of Tube 5 ^ Propositions Concerning 57 Use of, in Measuring Small Vertical Angles 5^ Angular Value of one Division found in three ways 5 ^ General Considerations 59 Buoys and Buoy Flags 283 Catenary Effect with Steel Tapes 457 to 465 Chain, Engineer’s 5 Gunter’s 5 Erroneous Lengths of - 6 Testing of 6 Permanent Provision for 7 INDEX. 685 PAGE Chain, Standard Temperature 7 Use of 8 On Level Ground 8 On Uneven Ground 8 Number and use of Pins g Exercises with ii Chaining over a Hill il Across a Valley ii Random Lines ii Check Readings in Topographical Surveying 253 Circumpolar Stars, Times of Elongation and Culmination 510 Pole distances of 511 Azimuth of Polaris at Elongation 33 City Surveying 356 Land Surveying Methods Inadequate 356 The Transit 357 The Steel Tape 357 Laying out a Town Site 359 Provision for Growth 359 Contour Maps 36a,, 371 Angular Measurements in Subdivision 360 I^aying out the Ground 36 r Plat to be Geometrically Consistent 363 Monuments 363 Surveys for Subdivision 365 Datum Plane 369 Location of Streets 369 Sewer Systems 370 Water Supply 370 Methods of Measurement 371 Retracing Lines 371 Erroneous Standards 372 True Standards 373 Use of Tape 374 Normal Tension 376 Working Tension 380 Effect of Wind 381 Effect of Slope 382 Temperature Correction 382 Checks 383 686 INDEX. PACK City Surveying — Miscellaneous Problems 384 Improvement of Streets 384 Permanent Bench-marks 384 Value of an Existing Monument 385 Significance of Possession 387, 582 Disturbed Corners and Inconsistent Plats 388 Surplus and Deficiency 389, 584 Investigation and Interpretation of Deeds 391 Office Records 391 Preservation of Lines 392 Want of Agreement between Surveyors 393 Clinometer 141 Coefficient of Expansion of Steel Tapes 456 of Brass Wires 456 Compass, Needle, Description of 13 Adjustments 15 Use of 34 Setting of the Declination 36 Local Attractions, Sources of 36 Tests of 37 To Establish a Line of a Given Bearing 37 To Find a True Bearing of a Line 37 To Retrace an Old Line 37 Exercises with Compass and Chain 38 Compass, Prismatic Pocket 38 Compass, Solar 39 Adjustments of 41 Use of 44 Finding the Declination of the Sun 44 Errors in Azimuth due to Errors in Declination and Latitude 49 Table of such Errors 51 Time of Day Suitable for Observations 52 Exercises with the Solar Compass 53 Convergence of Meridians 176, 574 Contour Lines, Propositions Concerning 260 Found by Transit and Stadia 259 Clinometer 275 Used in Computing Earth Work 379 Contour Maps in City Work. ... 360, 371 Coordinate Protractor 168 Corner Monuments in Land Surveying 178, 580 INDEX. 687 PAGE Comer Monuments in Land Surveying — Cannot be Established by Surveyors. 581 Cross-sectioning in Earth-work 406 Cross-sections, Areas of, in Rivers 2go of Least Resistance 328 in Earth-work 404 Cross-wires, Illumination of 518 Setting of 236 Current-meters 300 Rating of 301 Use of, in Streams 300 Conduits 313 Curvature and Refraction, Tables of 433, 545 Datum Planes in Cities 369 Declination of Magnetic Needle 20 Variations in 20 The Daily Variation 20 The Secular Variation 21 Other Variations : 29 To Find the Declination with the Compass and an Observation on Po- laris 29 Lines of equal Declination in United States, or Isogonic Lines 23 Formulae for finding the Declination at 82 Points in the United States and Canada 25 Declination of the Sun 44 Method of Finding 44 Correction for Refraction 45 Table of Corrections 48 Deeds, Investigation and Interpretation of 391 Deficiency, Treatment of, in City Work 389, 584 Differences, Finite, Method of 605 Construction of Tables 605 Derivation of Formulae for Evaluating Irregular Areas 608 Direction Meter 316 Discharge of Streams 294 Measuring Mean Velocities of Water Currents 294 Submerged Floats 295 Current Meter 300 Rating the Meter 301 Rod Floats 307 Comparison of Methods 308 688 INDEX. PAGE Discharge of Streams — Relative Rates of Flow in Different Parts of the Cross- section 309 Computation of the Mean Velocity over the Cross-section 312 Sub-currents 316 Flow over Weirs 316 F'ormulae and Corrections 319 Miner’s Inch 322 Flow of Water in Open Channels, Formula? for 323 Rutter’s Formula?... 326 Formula? for Brick Conduits 327 Cross-sections of Least Resistance 328 Sediment Observations 329 Collecting the Specimens 331 Measuring out the Samples 331 Siphoning off, Filtering, Weighing, etc 332 Disturbed Corners 388 Dredging 421 Earth-work, See Volumes. Earth-work Tables 410 Elevation of Stations in Triangulation 432 Elongation of Polaris, Times of ' 32 EiTor, Proportionate 2 Errors, Compensating and Cumulative 2 in Precise Leveling 558 Estimates, Preliminary in Earth-work 399, 417 Excavations under Water. 421 Excess, Spherical 494 Expansion, Coefficient of 456 Field Notes, Changes in 3 in Land Surveying 182 in Differential Leveling 75 in Profile Leveling 7 ^ in Topographical Surveying 249 Filar Micrometer 480 Floats, Submerged 295 Flow of Water in Open Channels 323 in Brick Conduits 327 Cross-sections of Least Resistance 328 See also Discharge of Streams. INDEX. 689 PAGE Gauge Hook 319 Water 2gi, 563 Geodetic Leveling, Trigonometrical and Spirit 540 to 563 Geodetic Positions, Computation of 535 Derivation of Formulae for 61 1 Geodetic Surveying 424 Triangulation Systems 425 Base-line and its Connections 427 Reconnaissance 429 Instrumental Outfit for 431 Direction of Invisible Stations 432 Heights of Stations 432 Construction of Stations 437 Targets .... 438 Heliotropes 442 Station Marks 444 Measurement of Base-lines 447 Use of Steel Tape in 449 Method of Mounting and Stretching 450 M. Jaderin’s Method 453 Absolute Length 455 Coefficient of Expansion 456 Modulus of Elasticity 457 Effect of Sag 457 Temperature Correction 459 with Metallic Thermometer 460 Correction for Alignment 462 Sag 465 Pull 465 Reduction of Broken Base to a Straight Line 468 Reduction to Sea Level 468 Summary of Corrections 469 Computation of an Unmeasured Portion 472 Accuracy attainable with Steel Tapes and Metallic Wires 473 Measurement of the Angles 477 Instruments 477 Filar Micrometer 480 Programme of Observations 483 Repealing Method 484 Continuous Reading around the Horizon 485 Atmospheric Conditions 487 44 690 INDEX, PACK Geodetic wSurveying — Geodetic Night Signals 488 Reduction to the Center 488 Adjustment of the Measured Angles 491 Equations of Condition 491 Adjustment of a Triangle 493 Spherical Excess 494 Adjustment of a Quadrilateral 494 Geometrical Conditions 494 Angle Equation Adjustment 494 Side Equation Adjustment 497 Rigorous Adjustment for Angle and Side Equation 501 Example 504 Adjustment of Larger Systems 506 Computing the Sides of the Triangles 506 Latitude and Azimuth 508 Conditions of the Discussion 508 Found by Observations on Circumpolar Stars at Elongation and Cul- mination 508 Observation for Latitude, Two Methods 512 Correction to the Meridian 514 Observation for Azimuth 515 Correction to Elongation 517 The Target » 518 Illumination of Cross-wires 518 Time and Longitude 519 Fundamental Relations 519 Sidereal to Mean Time 523 Mean to Sidereal Time 524 Change from Sidereal to Mean Time 525 Observation for Time 526 Selection of Stars 526 List of Southern Time Stars 528 Mean Time of Transit 530 Altitude of Star 531 Making the Observations 532 Programme of Observations 534 Computing the Geodetic Positions 535 Table of L. M. Z. Coefficients 537 Example 539 Geodetic Leveling 540 Trigonometrical Leveling 54® INDEX. 691 PAGE Geodetic Leveling — Formulae for Reciprocal Observations 541 Observations at one Station only 543 an observed Angle of Depression 545 Value of the Coefficient of Refraction 546 Precise Spirit Leveling 547 Instruments 548 Instrumental Constants 550 Daily Adjustments 553 Field Methods 555 Limits of Error 558 Adjustment of Polygonal Systems 559 Determination of the Elevation of Mean Tide.. 563 Grade, Leveling for 81 Grading over Extended Surfaces 396 Hand Level, Locke’s 81 Heights of Stations in Triangulation 432 Heliotropes 442 Hook Gauge 319 Horizontal Angle Measurement 93 Hydrographic Surveying .... 277 Location of Soundings 278 Two Angles read on Shore 279 in the Boat 279 One Range and one Angle 282 Buoys, Buoy Flags, and Range Poles 283 One Range and Time Intervals 284 Intersecting Ranges 284 Cords or Wires 284 Making the Soundings 285 Lead. 285 Line 285 Sounding Poles 287 Soundings in Running Water 287 Water-surface Plane of Reference 287 Lines of Equal Depth 288 Soundings on Fixed Cross-sections in Rivers 288 Soundings for the Study of Sand-waves 289 Areas of Cross-section 290 Bench-marks 291 Water Gauges 291 692 INDEX. PACK Hydrographic Surveying — Water Levels 292 River Slope 293 Finding the Discharge of Streams (See Discharge of Streams) 294 Illumination OF Cross-wires 518 Inaccessible Object. Distance to and Elevation oT 105 Length and Bearing of a Line joining two such 107 Integrations with Current Meter 301 Isogonic Lines in the United States 23 and PI. 11 . Judicial Functions of Surveyors 579 Kutter’s Formul.,® 326 Lakes, Riparian Rights in 587 Land Surveying 172 Laying out Land 172 United States Method 173 Origin of 173 Reference Lines 173 Division into Townships 174 Division into Sections 175 Convergence of Meridians 176 Corner Monuments 178 Areas of Land 179 by Triangular Subdivision 180 by use of Chain alone 180 by use of Compass or Transit and Chain 180 by use of Transit and Stadia 181 from Bearing and Length of Boundary Lines i8i Field Notes 182 Computing the Area 185 The Method stated 185 Latitudes, Departures and Meridian Distances 185 Computing Latitudes and Departures 187 Balancing 190 Rules for Balancing 192 Error of Closure 193 Form of Reduction. 194 Area Correction Due to Erroneous Length of Chain 197 INDEX. 693 PAGE Land Surveying — From Rectangular Coordinates of Corners 200 Conditions of Application 200 M'ethod Stated 201 Form of Reduction 203 Supplying Missing or Erroneous Data 203 Bearing and Length of one Course unknown 205 Bearing of one Course and Length of Another unknown 205 Two Bearings unknown 206 Lengths of two Courses unknown 206 Plotting the Field Notes 208 Areas of Irregular Figures 208 Offsets at Irregular Intervals 208 Regular Intervals 2io Subdivision of Land 213 To cut off by a Line through a given Point 213 in a given Direction 215 Exercises 220 Latitude, Geocentric and Geodetic 61 1 Latitude and Azimuth 508 Leads used in Soundings 285 Length, Standards of 372 Absolute, of Steel Tapes 455 Lettering on Maps 575 Level Bubbles 55 Level, Hand 81 Leveling, Ordinary 7 1 Precise Spirit 547 Trigonometric 540 Leveling Rods 70 Levels, Water 292 Level Surface ... 55 Level, The Engineer’s 60 Adjustments 63 Relative Importance of 68 Focussing and Parallax 68 Use of the Level 71 Back and Fore-sights. 71 Differential Leveling 72 Length of Sights 73 Bench-marks 74 The Record 75 694 INDEX. PACE Level, DifTerential Leveling — The P'ield Work 76 Trofilc Leveling 77 The Record 78 Leveling for Grade 81 Exercises 82 Level Trier 59 Line, Sounding 285 Lines, Clearing out 432 Preservation of, in Cities 392 L, M. Z. Coefficients, Table of 537 Formuloe Derived 61 1 Location of Railroad on Map 271 Longitude, Determination of 519 Map Lettering 575 Maps in Topographical Survey 262 in Railroad Surveying 267 Projection of 564 Meander Lines, Extension of, in Boundaries 584 Mean Tide, Water, Determination of Elevation of,, 563 Mean Velocities of Water Currents 294 Measurement of Volumes 394 Meridians, Convergence of 176, 574 Metallic Thermometer Temperature Corrections 460 Micrometer, Filar 480 Mineral Surveyors, Instructions to ■ 5^9 Mining Surveying 333 Definitions 333 Stations 335 Instruments 335 Mining Claims 339 Under-ground Surveys 343 To Fix the Position of the End or Breast of a Tunnel 343 To Find the Length of Tunnel to cut a given Vein 346 To Find Direction and Distance from a Tunnel to a Shaft 348 To Survey a Mine 35 1 Missing Data, Supplying of 203 Bearing and Length of One Course unknown 205 of One Course and Length of Another unknown 205 Two Bearings unknown 206 Two Lengths unknown 206 INDEX. 695 PAGE Modulus of Elasticity of Steel Tape 457 Monuments at Section Corners 178 in City Work 363, 385, 388 in Triangulation 444 Night Signals IN Triangulatio.n 488 Normal Tension of Tape in City Work 376 Notes, Field, Changes in 3 Obstacles, Passing with Chain alone ii Transit and Chain 105 Odometer 139 Office Records 391 Offsets in Land Surveying 208 at Regular Intervals 208 at Irregular Intervals 210 Optical Square 142 Parallax, How Removed 68 Parallel Ruler 169 Pantograph, Theory of 161 Varieties 164 Use of 165 Pedometer 137 Pivot Correction in Leveling 551 Plane Table 117 Adjustments 119 Use of 120 Location by Resection 123 Resection on Three Points 123 Resection on Two Points 124 Use of Stadia 125 Exercises 126 Planimeters 143 Theory of the Polar Planimeter 144 To Find Length of Arm 150 Suspended Planimeter 152 Rolling Planimeter 152 Theory of 154 Test of Accuracy of Planimeter Measurements 157 Use of the Planimeter 158 696 INDEX. PACB rianimeters — Accuracy of Planimcter Measurements 160 Used in Computing Earth-work 417 Plats, to be Geometrically Consistent 363 Inconsistent 388 Platting (See Plotting). Plotting in Land Surveying 208 Topographical Surveying 252, 254 Railroad Surveying 2O9 Plumb-line, its great Utility 55 Use of, in chaining 9 Deviations of 56 Polaris, Times of Elongation of 32 Azimuth of, at Elongation 33 Possession, Significance of 3S7, 582 Preservation of Lines 392 Prismoid, The Warped Surface 408 The Henck’s. . 413 Prismoidal Forms 402 Formulae 402 Tables 410 Precise Spirit Level 5<^7 Projection of Maps 564 Rectangular Projection 564 Trapezoidal Projection 565 Simple Conic Projection 566 De ITsle’s Conic Projection 567 Bonne’s Conic Projection 567 Poly conic Projection 568 Formulae used in 568 Derivation of Formulae 61 1 Table of Constants. 571, 681 Summary 572 Convergence of Meridians 574 Protractors 166 Three-armed 167 Paper Protractor 167 Coordinate .... 168 Topographical 255, 257 Public Lands in the United States 173 (See also Land Surveying). INDEX. 697 PAGE Railroad Topographical Surveying 265 Objects of the Survey 265 P'ield Work 265 Another Method 275 Maps 267 Plotting the Survey 269 Making the Location on the Map 271 Ranges and Range Poles in Sounding 284 Reconnaissance in Triangulation 429 to 444 Records, Office, in City Work 391 P eduction to the Center in Triangulation 488 Refraction and Curvature, Table of Values of 433, 545 Refraction, Table of Mean Values 512 Tabular Corrections to Declination for, with Solar Compass 47 in Trigonometrical Leveling 540 Coefficient of 546 Repeating, Method in Triangulation 484 Results, Number of Significant Figures in 3 Retracing Lines in City Work 371 Riparian Rights in Water Fronts 584 to 587 River Slope 293 Rod Floats 307 Ruler Parallel 169 Sag Effect with Steel Tapes 457, 465 Sand Waves, Study of 289 Scales 169 Sections in Land Surveying 175 Sediment Observations 329 Sewer Systems * 370 Sextant 108 Theory no Adjustments in Use 112 Exercises 112 Shrinkage of Earth-work 420 Sides, Computation of, in Triangulation 506 Simpson’s Rules Derived 610 Slope of River Surface 293 Solar Attachments 99 The Saegmuller Attachment 102 698 INDEX. PAf.E Solar Attachments — Adjustments 102 Solar Compass (See Compass, Solar). Soundings, Location of 278 Making 285 Spherical Excess 494 Stadia Metliods, Accuracy of 263 Stadia Rod, Craduation of 237 Stadia Surveying (See Topographical Surveying). Standards of Length in City Work 372, 373 Stars for Time Determinations, List of 528 Circumpolar, Times of Elongation and Culmination of 510 Pole Distances of 511 Stations, Direction of Invisible 432 Heights of in Triangulation 432 Construction in Triangulation 437 Marks at in Triangulation 444 Steps, Length of Men’s 138 Steel Tapes 9 In City Work 357, 372 to 383 In Base-line Measurement 449 to 465 Straight Lines Run by Transit 95 Streams, Discharge of 294 Streets, Location 369 Improvement of 384 Stretch of Steel Tapes 465 Subdivision of Land 213 Cutting off by a Line from a given Point 213 in a given Direction 215 Subdivision of Town Plats 365 Submerged Floats 295 Surplus, Treatment of, in City Work 389, 584 Surveying Land (See Land Surveying). Surveyors, Want of Agreement - 393 Judicial Functions of 579 Cannot Change Original Monuments 580 The Location of Lost Monuments 580 . Re-location of Extinct Interior Comers 580 Cannot “ Establish ” Corners. 581 Significance of Possession 582 Surplus and Deficiency 584 Meander Lines, Extension 584 INDEX. 699 PAGE Surveyors, Meander Lines, not Boundary Lines 584 Extension of Water Fronts 585 Bed Ownership in Water Fronts 586 Riparian Rights in Small Lakes 587 Tables, Construction of 605 List of I. Trigonometric Formulae 625 II. For Converting Meters, Feet and Chains 629 III. Logarithms of Numbers to Four Places 630 IV. Logarithmic Traverse Tables, Four Places 632 V. Stadia Tables 640 VI. Natural Sines and Cosines 648 VII. Tangents and Cotangents 656 VIIL Coordinates in Polyconic Projections. 669 IX. Value of Coefficient C in Kutter’s Formulae 670 X. Diameters of Brick Conduits 671 XL Volumes by the Prismoidal Formulae 672 Tape, Steel (See Steel Tape). Targets in Triangulation 438 Temperature Correction in Tapes 459 Tension of Tape in City Work 376, 380 Tide Water, Determination of Elevation of Mean 563 Time and Longitude, Determination of 519 to 534 Time, Sidereal and Mean 523 Time Stars, List of 528 Three-point Problem, Four Solutions 280 Topographical .Surveying 223 Transit and Stadia Method 224 Fundamental Relations 224 Adaptation to Inclined Sights 231 Reduction Tables and Diagrams 235 Instrumental Fixtures 236 Setting the Cross Wires 236 Graduating the Stadia Rod 237 The Topography 241 Field Work 241 Reduction of Notes .. 249 Plotting the Stadia Line 252 .Side Readings 254 Check Readings. 253 700 INDEX. PAGB Topographical Surveying — Contour Lines 259 The I'inal Map 262 Topographical Symbols 1C3 Accuracy of the Stadia Method 263 Topographical vSymbols 263, 576 Topography, Railroad (Sec R. R, Topographical Surveying). Townships in Land Surveying 174 Town Site, Laying out 359 Transit, The Engineer’s 83 General Description 83 Adjustments 86 Relative Importance of Adjustments 89 Eccentricity in Horizontal Circle 90 Inclination of Vertical Axis 91 Horizontal Axis 92 Collimation Error 93 Use of the Transit 93 Measurement of Horizontal Angles 93 Vertical Angles 94 Running out Straight Lines 95 Traversing 97 The Sola*- Attachment 99 Adjustments of Saegmuller Attachment 102 The Gradienter Attachment 104 Care of the Transit 104 Exercises 105 Transit in City Work 357 in Mining Work 336 in Topographical Work 236 Triangulation, Instruments used in 477 Programme of Observations 483 Adjustment of Angles 491 Computing Sides 506 Latitude and Azimuth 508 Time and Longitude 5^9 Computation of Geodetic Positions 535 Triangulation Systems 425 Traversing 97 Trigonometer (See Coordinate Protractor). Trigonometrical Leveling 540 Formulae 625 INDEX. 701 PAGE Variation of Magnetic Needle (See Declination). Velocities of Water Flow 294 in Vertical Planes 301, 310 in Horizontal Planes 309 Verniers 18 The Smallest Reading of 20 Rule for Reading 20 Vertical Angle Measurement 94 Volumes, Measurement of 394 The Elementary Form 394 Grading over Extended Surfaces 396 Approximate Estimates by Means of Contours 399 Prismoid 402 Prismoidal Formula 402 Areas of Cross Section 404 Center and Side Heights 405 Area of Three-level Sections 105 Cross Sectioning 406 The Warped Surface Prismoid 408 Construction of Tables for 410 The Henck Prismoid 413 Comparison of the Henck and Warped Surface Prismoid 415 Preliminary Estimates from the Profile 417 Borrow Pits 420 Shrinkage of Earth-work 420 Excavations under Water 421 Water Currents, Mean Velocity of 294 Sub-surface 316 Water Fronts, Riparian Rights in 584 Water Gauges 291 Water Levels 292 Water Supply, Surveys for 370 Weddel’s Rule Derived 610 Weirs Flow Over 316 Formulae and Corrections 319 Wire Interval, Value of 552 ALPHABETS AND NUMERALS. w H I— i Ph o <1 ViH o IS] P H zn G? Pw O hP M Q Q W ai . jy s an o o C <5^ r>«o r^ •"D HH M oSj O O O pO Ph rO e •isi ^T\ O r<:) e GO ao iG) Ci ro — H CO <> r^ ,— t CO o •iO CO % oH JS ^»>>o •S o CT 005 f ^ o ^■s 10 sp) 7o2 I II III lY V VI VII VIII IX X MAP LETTERING. 703 UNIVERSITY Of H.LINOW URBAN* 3 0112 067496940 \'>\\V'\'^VV\V'\'\>'