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^, 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
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