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REPAIRED BY 

WISCONSIN 

W P A 

BOOK REPAIR 

PROJECT NO. 

.11.^1: 

Date..|^^^.ii 



%■ A* 



IN PREPARATION 



ADVANCED SURVEYING 



C. B. BREED AND G. L. HOSMER 



THE 

PRINCIPLES AND PRACTICE 

OF 

SURVEYING 



BY 
CHARLES B. BREED and GEORGE L. HOSMER 

N 
INSTRUCTORS IN CIVIL ENGINEERING, MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY 



First Edition 



NEW YORK 
JOHN WILEY & SONS 



London : CHAPMAN & HALL, Limited 
1906 



Copyright, 1906 

BY 

CHARLES B. BREED and GEORGE L. HOSMER 



Stanbope prcM 

r. H. OILSON COMPANT 
BOSTON. U. •. A. 



105155 

APR 2 31907 

SUB 



PREFACE 



In the preparation of this volume, it has been the authors' 
chief purpose to produce a text-book which shall include the 
essentials of a comprehensive knowledge of practical surveying 
and at the same time be adapted to the use of teachers and 
students in technical schools. In this book, which is essentially 
an elementary treatise, such subjects as stadia, plane table, 
hydrographic and geodetic surveying, are entirely omitted, these 
sabjects being left for a later volume. 

Considerable stress is laid upon the practical side of sur- 
veying. The attempt is made not only to give the student a 
thorough training in the fundamental principles and in approved 
methods of surveying, computing, and plotting, but also to 
impress upon him the importance of accuracy and precision in 
all of his work in the field and the drafting-room. In carrying 
out this purpose it has seemed necessary to lay particular stress 
upon some points which to the experienced engineer or the 
advanced student may appear too obvious to require explanation, 
but which teaching experience has shown to be most helpful to 
the beginner. The most common errors and mistakes have 
therefore been pointed out and numerous methods of checking 
have been explained. Every effort has been made to inculcate 
right methods even in minor details, and for this purpose a large 
number of examples from actual practice have been introduced. 

In arranging the subject matter of the work, the four parts 
are presented in what appears to be a logical sequence. First, 
the use, adjustment, and care of instruments are taken up ; then 
the next three parts, surveying methods, computations, and 
plotting, are taken in the order in which they are met in the 
daily practice of the surveyor. To show more clearly the steps 
in the process, the notes which are used as illustrations in sur- 
veying methods are calculated in the computation section, and 



iv PREFACE 

are treated again under the methods of plotting, finally appearing 
as a completed plan. 

While the authors recognize fully their indebtedness to those 
who have preceded them in this field, they hope that they have 
made some useful contributions of their own to the treatment 
of the subject. Thus in the section on Surveying Methods, 
many practical suggestions have been inserted which they have 
found of value in their own work and which, so far as they are 
aware, now appear in a text-book for the first time. On the sub- 
ject of Computations, much emphasis is laid upon the proper use 
of significant figures and the arrangement of the work, matters 
which heretofore have not been adequately treated in books on 
surveying. The section on Plotting contains many hints refer- 
ring particularly to surveying drafting, which are not given in the 
published books on drawing and lettering. It is hoped also that 
the complete set of original illustrations which have been in- 
troduced throughout the book will aid materially in making the 
text clear. ^ 

A comprehensive cross-reference system giving the page as 
well as the article number has been adopted : this, together with 
the complete index at the end of the book and the many practi- 
cal hints throughout the volume will, it is hoped, render it tise- 
ful to the practical surveyor as a reference book. 

The authors desire to acknowledge their indebtedness to their 
various associates in the teaching and engineering professions 
who have kindly responded to requests for information and as- 
sisted in the preparation of this work, particularly to Blamey 
Stevens, M. Sc, of Ellamar, Alaska, who supplied the entire 
chapter on Mining Surveying, They are also under obligations 
for the use of electrotype plates of tables : to W. H. Searles for 
Tables IV, V, and VI ; to Professor J. C. Nagle for Tables II 
and III ; and to Professor Daniel Carhart for Table I ; all of these 
plates were furnished by John Wiley & Sons. The authors are 
under special obligation to Professors C. F. Allen, A. G. Rob- 
bins, and C. W. Doten of the Massachusetts Institute of Tech- 
nology, and to H. K. Barrows, Engineer U. S. Geological Sur- 
vey, who have read the entire manuscript and who have offered 
many valuable suggestions in preparing the work for the press. 



PREFACE V 

The authors also desire to express their appreciation of the ex- 
cellent work of W. L. Vennard, who made the drawings for illus- 
trations. 

No pains has been spared to eliminate all errors, but the 
authors cannot hope that their efforts in this line have been com- 
pletely successful, and they wiirconsider it a favor if their atten- 
tion is called to any which may be found. 

Boston, Mass., September, 1906. C. B. B. 

G. L. H. 



CONTENTS. 

PART I. 

USE, ADJUSTMENT, AND CARE OF INSTRUMENTS. 

Chapter I. — General Definitions. — Measurement of Lines. 

Art. Paob 

1. Definition 3 

2. Purposes op Surveys 3 

3. Horizontal Lines 3 

instruments for measuring lines. 

4. The Chain 3 

5 Metric Chain 5 

6. The Tape 5 

7. Steel Tapes 5 

8. The Stadia 6 

9. Other Instruments; — Wooden Rods, Two-Foot Rule, Odome- 

ter 6 

MEASUREMENT OF LINES. 

^ 10. Measurement of a Horizontal Line with a Chain 7 

> II. Measurement of a Horizontal Line wtth a Cblain on Slop- 
ing Ground 9 

^ 12. Measurebcent with a Steel Tape 10 

13. Common Sources of Error in Measurement of Lines .... 11 

14. Common Mistakes in Reading and Recording Measurements ii 

15. Avoiding Mistakes 11 

16. Accuracy Required 12 

17. Amount of Different Errors 12 

18. Pull 12 

19. Temperature 13 

20. AuGNMENT 13 

21. Sag 13 

22. Effect of Wearing on Length of the Chain 14 

23. Accuracy of Measurements Z4 

Problems 15 

Chapter II. — Measurement of Direchon. 

24. The Surveyor's Compass 16 

25. The Pocket Compass 18 

96. Method of Taking a Magnetic Bearing x8 

vii 



vm CONTENTS 

Akt. Paob 

27. The Earth's Magnetism, — Dip op the Needle 19 

28. Decunation of the Needle 19 

29. Variations in Declination ao 

3a IsoGONic Chart 25 

31. Observations for Declination 25 

ADJUSTMENTS OF THS COMPASS. 

33. Adjustment of the Bubbles 25 

34. Detecting Errors in Adjustment of the Needle 25 

35. To Straighten the Compass-Needle 26 

36. To Center the Pivot-point 27 

37. To Remagnetize the Needle 28 

38. Common Sources of Error in Compass Work 28 

39. Common Mistakes 28 

40. Detecting Local Attractions op the Needle 29 

41. Calculating Angles from Bearings 30 

Problems 30 

Chapter III. — Measurement of Angles. 

THE transit. 

42. General Description of the Transit 31 

45. The Telescope 34 

46. The Objective 35 

47. Cross-Hairs 36 

48. Eyepiece 37 

49. Magnifying Power 37 

51. Field of View 38 

52. The Vernier 38 

53. Verniers Used on Transits • 39 

54. Eccentricity 44 

use of the transit. 

56. Setting up the Transit 45 

58. To Measure a Horizontal Angle 47 

59. To Measure an Angle by Repetition 48 

61. To Lay Off an Angle by Repetition 50 

62. Running a Straight Line — One Point Visible from the Other 51 

63. Neither Point Visible from the Other 51 

64. Prolonging a Straight Line 52 

65. Methods of Showing Sights 52 

66. Signals 53 

67. To Measure a Vertical Angle 54 

68. Precautions in the Use op the Transit 55 

adjustments of the transit. 

70. Adjustment of the Plate Bubbles 56 

71. Adjustment of the Cross-Hairs 57 



CONTENTS ix 

AsT. Pacb 

73. Adjustment op the Standards 59 

74. Adjustment of the Telescope Bubble 59 

75. Adjustment op the Auxiliary Level on the Vernier op the 

Vertical Arc 59 

76. Adjustment of the Vernier of the Vertical Circle .... 60 

77. Adjustment of the Objective Slide 60 

78. Shop Adjustments 61 

79. How TO Eliminate the Effect of Errors of Adjustment in 

THE Transit 61 

80. Care op Instruments 61 

81. Common Sources of Error in Transit Wore 63 

82. Common Mistakes in Transit Work 62 

THE SOLAR ATTACHMENT. 

83. Description of Solar Attachment 65 

84. The Celestial Sphere 65 

85. Observation on the Sun for Meridian with Solar Attachment 66 

86. Computation op Declination Settings 68 

87. Comstock's Method op Finding the Refraction 69 

89. Mistakes in Using the Solar Attachment 70 

ADJUSTMENTS OF THE SOLAR ATTACHMENT. 

90. Adjustment of the Polar Axis 70 

91. Adjustment of the Cross-Hairs 70 

92. Adjustment of Telescope Bubble 71 

Problems 71 

Chapter IV. — Measurement of Differences of Elevation. 

93. Level Surface 72 

94. The Spirit Level 72 

95. Angular Value of One Division of the Level Tube .... 73 

THE LEVEL. 

97. The Wye Level 73 

98. The Dumpy Level 74 

99. Comparison of Wye and Dumpy Levels 77 

100. The Locke Hand Level 77 

LEVELING rods. 

102. Boston Rod 78 

103. New York Rod 80 

104. Philadelphia Rod 81 

105. Special Self-Reading Rods 81 

106. Tape Rod 81 

107. Precise Level Rod 82 

108. Advantages op the Self-Reading Rod 82 

109. Attachments to the Rod for Plumbing 83 

no. Effect of Heat and Moisture 83 



X CONTENTS 

USE OF THE LEVEL AND ROD. 

AsT. Faob 

112. To Level the Instrument ... 83 

113. To Take a Rod-Reading 84 

114. Waving the Rod 84 

115. Signals 85 

116. Differential Leveling 85 

117. The Proper Length of Sight 87 

118. Effect of the Earth's Curvature and of Refraction on 

Leveling 87 

119. Precautions in Level Work 88 

adjustments of the level. 

I. adjustments of the wye level. 

120. Adjustment of the Cross-Hair 88 

122. Adjustment of the Level Tube 89 

123. Adjustment of the Level Tube by Indirect Method ... 89 

125. Adjustment of the Wyes 90 

II. adjustments of the dumpy level. 

126. Adjustment of the Cross-Hairs 91 

127. Adjustment of the Bubble Tube 91 

128. The Direct or "Peg" Adjustment 91 

129. Adjustment of the Locke Hand Level 93 

130. Common Sources of Error in Leveling 94 

131. Common Mistakes in Leveling 94 

Problems 94 



PART II. 
SURVEYING METHODS. 

Chapter V. — Land Surveying. 

132. Surveying for Area 99 

133. Surveying for Area with Compass and Chain 99 

SURVEY OF FIELD WITH TRANSIT AND TAPE. 

134. Survey of a Field by a Traverse loi 

137. Irregular Curved Boundaries 105 

138. Survey of a Field with a Single Set-up of the Transit ... 105 

139. Survey of a Field with a Tape only 106 

140. Selecting the Corners 106 

141. Method of Procedure 107 

142. Ties 107 

143. Measurement of the Angles of the Traverse 108 

144. Measurement of Azimxtth Angles 108 

145. Checking the Fieldwork 109 



CONTENTS xi 

AiT. Page 

146. Accuracy Required m 

147. Organization op Transit Party m 

148. Note-keeping 112 

149. Survey of a Field for a Deed 114 

150. Deed Description ^ 115 

151. JuDiCTAL Functions of the Surveyor 116 

152. Rerunning Old Surveys from a Deed 118 

153. How TO Look up a Recorded Deed 120 

THE UNITED STAIBS SYSTEM OF SURVEYING THE PUBLIC LANDS. 

154. The System 121 

155. Initial Points .• 124 

156. Base Line 124 

157. Principal Meridian 126 

158. Standard Parallels 126 

159. Guide Meridians 126 

160. Township Exteriors 127 

161. Method of Subdividing 128 

162. Meandering 134 

163. Summary of Objects and Data Intersected by the Line or 

IN Its Vicinity, to be Noted 139 

164. Prescribed Limits for Closings and Lengths of Lines ... 141 

165. Field Notes 142 

166. Marking the Corners 147 

167. To Establish a Parallel of Latitude 148 

168. The Secant Method 148 

169. Tangent Method 150 

170. Convergence of the Meridians 154 

Chapter VI. — Traverse Lines — Location of Buildings — 
Miscellaneous Surveying Problems, 
traverse lines. 

171. Traverses which do not Form Closed Figures 156 

172. Method of Checking Traverses which do Not form 

Closed Figures, — Checking by Astronomical Method . 156 

173. Checking by Cut-Off Lines 157 

174. Checking by Angles to a Distant Object 158 

175. Checking by Connecting with Tri angulation Points .... 158 

location of buildings from transit line. 

176. Methods of Locating Buildings 159 

177. Geometric Principles 159 

178. Ties, Offsets, Swing Offsets, and Range Lines 159 

179. General Suggestions 160 

180. Typical Cases 160 

181. Building near Transit Line and Nearly Parallel to it. . . 160 

Plotting 161 

182. Building near Transit Line, Making a Slight Angle with 

IT 161 

Plotting 162 



xii CONTENTS 

An. Page 

183. Building Located Entirely by Direct Ties 162 

Plotting 163 

184. Building Located at a Considerable Skew to the Transit Line 163 

Plotting 164 

185. Buildings at a Long Distance from the Transit Line ... 164 

Plotting 164 

186. Buildings Located from Other Buildings 165 

187. Buildings of Irregular Shape 166 

188. Large City Buildings 166 

189. Location of Buildings by Angles and Distances 167 

190. Location of Buildings and Fences from Transit Line ... 167 

miscellaneous surveying problems. 

191. Random Line 169 

192. Obstacles on Line 170 

193. Offsetting Transit Line 170 

194. Short Transit Sights 172 

195. Bisection Method 172 

196. Measuring Around a Small Obstacle 173 

197. Equilateral Triangle Method 174 

198. Inaccessible Distances , . . , 174 

199. By Right Triangle Method 174 

200. Intersecting Transit Lines 175 

201. By Swing Offset Method 175 

202. By Tangent Offset Method 176 

203. By Oblique Triangle Method 177 

204. To Obtain the Distance Between Two Inaccessible Points 

BY Observation from Two Accessible Points 178 

205. To Obtain the Inaccessible Distance Between Two Accessible 

Points by Observation on Two Inaccessible Points of 

Known Distance Apart 179 

Chapter VII. — Observation for Meridian and Latitude. 

observations for meridian. 

206. To Establish a True Meridian Line by Observation on Polaris 

WITH THE Transit 180 

207. Observation for Meridian on Polaris at Elongation . . . 182 

208. Observation for Meridian on Polaris at Culmination ... 185 

209. To Find the Standard Time of Culmination and Elongation. 186 

210. Meridian Observations on Polaris with the Compass ... 187 

211. Meridian Observation on Polaris AT ANY Time WITH THE Transit 188 

212. Solar Observations 190 

213. Observation for Meridian by Equal Altitudes of the Sun in 

THE Forenoon and Afternoon 190 

214. Observation for Meridian by a Single Altitude of the Sun 193 

215. Observation for Meridian by Means of the Solar Attachment 195 



CONTENTS xiii 

OBSERVATIONS FOR LATITUDE. 

Akt. Page 

216. By the Altitude of Polaris at Upper or Lower Culmination 196 

217. By the Altitude of the Sun at Noon 196 

Problems 197 

Chapter VIII. — Leveling. 

218. Definitions 198 

219. Leveling to Establish Bench Marks 198 

222. Double Rodded Lines 201 

224. Bench Marks and Turning Points 202 

225. Leveling for Profile 203 

227. Cross-Sectioning 206 

228. Use of the Tape Rod in Cross-Section Work 206 

229. Cross-Sectioning for Earthwork 207 

230. Road Cross-Sections 207 

231. Setting Slope Stakes 208 

232. Earthwork Notes for Road Cross-Sections 209 

233. Cross-Sections for Borrow-Pits 210 

234. Cross-Sections for Trench Excavation 210 

235. Leveling to Establish a Grade Line 210 

236. "Shooting in" a Grade Line 211 

237. To Establish a Datum Plane by Means of Tidal Observations 211 

238. The Staff Gauge 212 

239. Leveling Across a River 213 

Problems 215 

Chapter IX. — City Surveying 

240. Instruments Used 216 

241. Tapes and Tape Measurements 216 

242. Transits and Levels 217 

243. City Standard 218 

CITY LAYOUTS. 

245. Streets 219 

246. Location op Streets 220 

247. Size of Blocks and Lots '. 222 

248. Width of Streets 222 

249. Street Grades 224 

250. The Datum Plane 226 

251. Establishing Bench Marks 226 

252. Water and Sewer Systems 227 

staking out aTY WORK. 

253. Staking out a New District 227 

254. Monuments 228 

255. Setting Stone Bounds 230 



xiv CONTENTS 

Art. Pack 

256. Curved Layouts 232 

257. Elements of a Circular Curve 233 

258. Staking Out Circular Curves 234 

259. Deflection Angles 234 

260. Keeping the Notes 237 

261. When the Entire Curve Cannot be Laid Out from One End 238 

262. First Method 238 

263. Second Method 239 

264. Curved Street Corners 239 

265. One Street Line Straight, the Other Curved 239 

266. Both Street Lines Curved 240 

367. Staking Out Street Grades 241 

268. Vertical Curves 242 

269. Cross-Section of Street 243 

270. Gutters at Same Elevation 244 

271. One Gutter Higher than the Other 244 

273. Irregular Shaped Blocks 247 

274. Staking out City Lots 248 

275. Staking Out Curb Lines and Grades 249 

276. Staking Out Sewers 251 

277. Staking Out Street Railway Tracks 251 

278. Rerunning Street Lines and Grades 251 

279. Revising Street Lines 252 

280. Revising Street Grades 252 

281. Setting Batter-Boards for a Building 253 

282. City Plans and Records 255 

RECTANGULAR COORDINATE SYSTEM OF SURVEYING CITIES. 

283. General Description 255 

284. Triangulation Scheme 256 

285. Measurement of Base-Line 261 

286. Measurement of Angles 261 

287. Adjustment of the Angles 262 

288. Azimuth 262 

289. Secondary and Tertiary Triangulation 263 

290. Traverses 263 

291. Method of Locating Property Lines and Buildings .... 263 

Chapter X. — Topographical Surveying. 

293. Triangulation for Control 264 

294. Location of Points from the Transit Line 265 

295. Contour Lines 265 

296. Characteristics of Contours 268 

297. Rel.\tion Between Contour Map and Profile 270 

298. Relation Between Contour Map and Side Elevation or Pro- 

jection 271 

299. Drainage Areas 273 

300. Sketching Contours from Streams and Summits 273 

301. Sketching Contours from Known Elevations 276 



CONTENTS XV 

Abt. Page 

302. Mistakes in Sketching Contours 276 

303. Locating Contours 277 

304. Locating Contours by Cross-Sections 278 

305. Locating Contours by Profiles 278 

306. Locating Points on Contours 278 

307. Locating Contours by Means of the Hand Level .... 278 

308. Location of Streams and Shore Lines 280 

309. Contour Problems 280 

310. Intersection of Plane with Surface of Ground 280 

311. Intersection of Curved Surface with Surface of Ground . 281 

31 2. Intersection of Side Slopes of Road with Surface of Ground 283 

Chapter XI. — Mining Surveying. 

313. General Remarks 285 

314. Definitions of Mining Terms 285 

MINING INSTRUMENTS. 

315. Mining Transits 287 

316. Side Telescope 287 

317. Top Telescope 287 

318. Adjustments of Side Tei^^scope 287 

319. Adjustment of Top Telescope 290 

320. Interchangeable Side and Top Telescope 290 

321. Eccentric Bearing Telescope 290 

322. Combined Solar Attachment and Top Telescope 292 

324. Use of the Ordinary Transit in Mining Surveying .... 292 

325. Compasses used in Mines 293 

UNDERGROUND SURVEYING. 

326. Transferring a Meridian into a Mine by use of the Transit 294 

327. Plumbing the Meridian down a Shaft 296 

328. Transferring a Meridian into a Mine when there are Two 

Shafts 298 

329. Underground Traverses 299 

330. Establishing Station Points 300 

331. Notes of a Mine Traverse 305 

332. Plotting a Mine Traverse 305 

333- Underground Leveling 306 

334. Mine Maps and Charts 306 

335. Laying out Mining Work 307 

336. Underground Surveying Problems 307 

337. Vertical Angle Correction for Eccentricity of the Top 

Telescope 308 

338. Vertical Angle Correction for Eccentric Bearing Tele- 

scope 308 

339. To Establish a Boundary Ln^E of the Claim Underground 309 

340. To Lay Out a Connection in a Mine 309 

341. Hydraulic Surveying for Mines 310 

342. Testing for Ore by Electric Currents 311 



xvi CONTENTS 

SURFACE SURVEYING. 

Art, Paoe 

343. SUIUFACE StJRVEYINO IN RuGGED MOUNTAIN REGIONS 311 

344. Mine Boundaries. Appropriations Under United States Laws 312 

345. Surveying for Patent 314 

346. The Surveying of Boreholes 315 

347. Staking out the Probable Apex of a Vein 316 

348. Economic Principles 316 

Problems 318 



PART III. 

COMPUTATIONS. 

Chapter XII. — General Principles — Miscellaneous Prob- 
lems — Earthwork Computations. 

349. General Remarks 323 

354. Logarithmic or Natural Functions 327 

355. Short Cuts 328 

357. Arrangement of Computations 330 

358. Checks 330 

359. Slide Rule 330 

360. Thacher Slide Rule 333 

361. Reducing the Field Notes for Computations 333 

362. Curved Boundary by Offsets 333 

363. Trapezoidal Rule 334 

364. Simpson's One-Third Rule 334 

365. Straightening Crooked Boundary Lines 336 

366. Area by Triangles 336 

367. Area of a Quadrilateral by Triangles 337 

368. Area of a Curved Corner Lot 338 

369. Rough Checks on Areas 339 

370. Planimeter 339 

371. Deflection Angles and Chords for a Circular Curve ... 341 

372. Computations of Observations 342 

computation of volume. 

373. BoRROw-Prrs 342 

374. Volume of Prismoid 344 

375. End Area Formula , 344 

376. Prismoidal Formula 344 

378. Estimates for Grading 345 

382. Rough Estimates 349 

Problems 350 



CONTENTS xvii 

Chapter XIII. — Area by Double Meridian Distances — 

Coordinates. 

Art. Pace 

383. Computation of Ajkea 352 

384. AiiEA BY Double Meridian Distance Method 352 

386. Computation or Area of Compass Survey . . •. 356 

387. Balancing a Chain and Compass Traverse 358 

390. Double Parallel Distance 359 

391. Error op Closure 360 

392. Computation of Area of a Transit and Tape Survey. . . . 360 

395. Balancing a Transit and Tape Survey 364 

396. Fractional Areas 365 

397. Supplying Missing Data 366 

401. Detecting Mistakes 368 

402. The Subdivision of Land 368 

40 jl To Cut Off from a Traverse a Given Area by a Straight 

Line starting from a Known Point on the Traverse . . . 369 

404. To Cut Off from a Traverse a Given Area by a Line Running 

in a Given Direction 369 

405. To Find the Area Cut Off from a Traverse by a Line run- 

ning in a Given Direction from a Given Point en the Trav- 
erse 370 

CALCULATIONS RELATING TO TRAVERSES WHICH DO NOT CLOSE. 

406. To Calculate the Total Distance between End Points . . 371 

407. Cut-Off Lines 371 

408. Computation of Azimuths when Checking Angles to a Dis- 

tant Object 372 

409. Calculation of Triangulation 372 

4ia Coordinates 373 

411. To Determine the Area of a Field by Rectangular Co- 
ordinates 374 

Problems 376 



PART IV. 

PLOTTING. 

Chapter XIV. — Drafting Instruments and Material. 

engineering drafting instruments. 

415. Straight Edge . 381 

416. Engineer's Scale 382 

417. Protractor 382 

418. Semictrcular Protractor 382 

419. Full-Circle Protractor 384 

421. Three-Armed Protractor 384 



xviii CONTENTS 

Aet. Pjm» 

423. Pantcx>raph . 385 

424. Parallel Ruler 386 

425. Beam Compass 386 

426. Contour Pen 386 

427. Proportional Dividers 387 

429-31. Railroad Curves, French Curves, Flexible Curves, akd 

Spline 387 

DRAWING PAPERS. 

433. Drawing Paper por Plans 388 

434. Tracing Paper and Tracing Cloth 389 

436. Cross-Section and Profile Papers 390 

438. Process Papers, — Blue-Prints 391 

441. Vandyke Solar Paper 393 

442. Electrical Printing Frames 395 

443. Inks and Water-Colors 395 

Chapter XV. — Methods of Plotting. 

444. Laying Out a Plan 397 

445. Scale 397 

methods of plotting traverses. 

446. Plotting by Protractor and Scale 398 

447. Checks 398 

448. Protractor and T-Square 400 

449. Plotting by Rectangular Coordinates 401 

454. Checks 405 

455. Plotting by Tangents 405 

456. Checks 407 

457. Plotting by Chords 408 

458. Use of the Sine 409 

459. Checks 409 

method of plotting details. 

460. Buildings, Fences, Streams, Etc. 410 

461. Contours 410 

462. Cross-Sections 411 

463. Profiles 4" 

467. Checks 413 

Problems 413 

Chapter XVI. — Finishing and Filing Drawings. 

468. What Should Appear on a Drawing 415 

469. Traverse Lines 415 

470. Physical Features 416 

471. Topographic Conventional Signs 416 



• CONTENTS xix 

Akt. Page 

473. Lettering 422 

474. Titles 423 

475. Notes 426 

476. Border Lines 427 

477. Meridians 427 

478. Scales 428 

479. Shrinkage of Drawing Papers 428 

480. Maps op Large Extent 429 

481. Inking in a Profile 430 

482. Cleaning Drawings 430 

483. Filing Drawings 431 

484. Indexing Drawings 432 

485. Filing Note-books 432 

486. Indexing Notes 433 

487. Other Records 433 



TABLES. 

I. Logarithms of Numbers 437 

11. Logarithmic Sines and Cosines 455 

III. Logarithmic Tangents and Cotangents 470 

IV. Natural Sines and Cosines 485 

V. Natural Tangents and Cotangents 494 

VI. Lengths of Circular Arcs: Radius — i 506 

VII. Mean Refractions in Declination 507 

Vni. Trigonometric and Miscellaneous Formxtlas 511 

IX. Circular Curve Formulas 513 

X. Geometric Formulas 513 

XI. Linear Measure 514 

XII. Square Measure 514 

Xin. Linear Measure — Metric System 514 

XIV. Square Measure — Metric System 514 

XV. Constants 515 

Greek Alphabet 516 



THE PRINCIPLES AND PRACTICE OF 
SURVEYING. 



PART I. 
USE, ADJUSTMENT, AITD CASE OF INSTRUHENTS. 



CHAPTER I. 

OBNBRAL DEFINITIONS. — MEASXTRISltfENT OF LINES. 

1. Definition Surveying is the art of measuring and 

locating lines and angles on the surface of the earth. When 
the survey is of such limited extent that the effect of the earth's 
ciu-vature may be safely neglected it is called Plane Survey- 
ing. When the survey is so large that the effect of curvature 
of the earth must be taken into account as, for instance, in the 
survey of a state or a country, it is called Geodetic Surveying, 

2. Purposes of Surveys, — Surveys are made for a variety 
of purposes such as the determination of areas, the fixing of 
boundary lines, and the plotting of maps. Furthermore, engi- 
neering constructions, such as waterworks, railroads, mines, 
bridges, and buildings, all require surveys. 

$• Horizontal Lines. — In siuveying, all measurements of 
lengths are horizontal or else are subsequently reduced to hori- 
zontal distances. As a matter of convenience, measurements 
are sometimes taken on slopes, but the horizontal projection is 
afterward computed. The distance between two points as 
shown on a map' then is always this horizontal projection. 

INSTRUMENTS FOR MEASURING LINES. 

4. The Chain. — There are two kinds of chain in common use, 
the Surveyor^ s {or Guntet's) Chain, and the Engineer's Chain 
(Fig. i). Gunter's chain is 66 feet long, and its use is confined 
chiefly to land surve)dng on account of its simple relation to the 
acre and to the mile. 

I Gunter's Chain = 4 Rods= 100 Links. 

I Mile =80 Chains. 

I Acre =10 Square Chains. 

Evidently each link is ^^ of a foot (or 7.92 inches) long. 
The inch, however, is never used in surveying fieldwork. 

3 



MEASUREMENT OF LINES 



[Chap. I. 



The engineer's chain is icx) feet long and is divided into one 
hundred hnks of one foot each. Each end hnk is provided with a 
handle, the outside of which is the zero point, or end, of the chain. 
In these chains, every tenth link counting from either end is 
marked by a brass tag having one, two, three, or four points 
corresponding to the number of tens which it marks. The 
middle of the chain is marked by a round tag. In the en- 
gineer's chain then the lo-ft. and 90-ft. points, the 20-ft. and 
80-ft. points, etc., are marked alike ; hence it is necessary to ob- 




Engineer's Chain. Gunter's Half-Chain. 

Fig. 1. 

serve on which side of the 50ft. point a' measurement falls in 
order to read the distance correctly. Distances measured with 
the surveyor's chain are recorded as chains andlinks^ (or in chains 
and decimals) ; while those measured with the engineer's chain 
are recorded zsfeet and decimals. 

On account of the large number of wearing surfaces and the 
consequent lengthening with use, the chain should be frequently 
compared with a standard of length (Art. 243, p. 218). It may 
be adjusted to agree with the standard, by means of a nut at the 



THE CHAIN AND TAPE 5 

handle, which allows the length of the chain to be altered by 
lengthening or shortening the end link. 

5. Metric Chain. — The Metric Chain is usually 20 meters 
long and is divided into one hundred links, each 2 decimeters 
long. 

6. The Tape. — There are three kinds of tape in common 
use, — cloth, metallic J and steel. Cloth tapes stretch so easily 
that they are of little use in surveying. The so-called metallic 
tapes are cloth tapes havmg very fine brass wires woven into 
them to prevent stretching. They are usually graduated into 
feet, tenths, and half -tenths and are made in lengths of 25 ft., 
50 ft., and 100 ft. When precise results are required a steel 
tape should be used. While a steel tape varies a slight 
amount in length with the temperature and with the pull, it is 
possible to determine the amount of these variations and hence 
to arrive at accurate results. 

7. Steel Tapes. — Steel tapes may be obtained in lengths 
up to 500 ft., but the most common in use are the 50-ft. and 
loo-ft. lengths. While the shorter tapes are usually made of 
thin steel ribbon the longer ones are of sufficiently large cross- 
section to withstand hard usage. These heavy tapes are gen- 
erally marked every 10 ft. by a brass tag, the lo-ft. length at 
one end of the tape being marked at every foot, and the last 
foot divided into tenths. Some of these tapes are marked every 
foot throughout their entire length. The light tapes are divided 
throughout their entire length into feet, tenths, and hundredths, 
each line being etched on the steel. The numbering is contin- 
uous from o ft. to icx) ft. These tapes are more convenient* to 
handle than the heavy ones, but are not suited to very rough 
work as they are easily kinked and broken. They can be readily 
mended, however, by riveting to the back of the tape a piece of 
tape of the same width. 

Since the surveyor's measurements are usually in feet and 
decimals, they are not in convenient form for use by mechanics 
in construction work. It is therefore often necessary to con- 
vert decimals of a foot into inches and vice versa. The follow- 
ing table shows the general relation between these two and is 
sufficiently close for most work 



6 MEASUREMENT OF LINES [Chap. I. 

TABLE 1. 
Decimals of Foot in Inches. 

Dbcimal op Foot. Inchbs. 

.01 - i- 

.08 := I- 

.17 = 2 -f 

.25 « 3 (exact) 

.50 — 6 (exact) 

.75 - 9 (exact) 

Decimals of a foot can easily be converted mentally into 
inches, by use of the equivalents in the above table, for example, 
0.22 ft. = .25 — .03 = 3'' — |"= 2f". 

In surveying farms, timber lands, or other property of low 
value, chain measurements are usually of sufficient accuracy and 
the chain is well adapted to work in rough country. In city 
surveys, and in fact in all surveys where great accuracy is de- 
manded, the steel tape is indispensable. In preliminary railroad 
surveys the engineer's chain, which formerly was *used exclusively, 
is gradually being replaced by the long heavy tape which, while 
adapted to rough work, will at the same time give accurate 
results. 

8. The stadia. — Where it is desired to measure distances 
with great rapidity but not with very great accuracy the stadia 
method is coming to be very generally used. The distance is 
obtained by simply sighting with a transit instrument at a grad- 
uated rod held at the other end of the line and noting the space 
on the rod included between two special cross-hairs set in the 
instrument at a known distance apart. From this observed in- 
terval on the rod the distance from the transit to the rod can be 
easily calculated. 

9. OTHER Instruments. — Wooden Rods^xQ used in certain 
kinds of work for making short measurements, usually less than 

15 ft. 

The Two-Foot Rule divided into tenths and hundredths of a 
foot is very convenient for short measurements. 

The Odometer is an instrument which may be attached to a 
carriage in such a manner as to register the number of revolutions 
of one of the wheels. The circumference of the wheel being 
known the approximate distance traversed is easily determined. 



CHAINING A HORIZONTAL LINE ^ 

MEASUREMENT OF LINES. 

10. MEASUREMENT OF A HORIZONTAL LINE WITH A CHAIN. 

— This work is done by two chainmen using a chain and a set of 
eleven steel marking pins. One man, called the head-chainman, 
carries ten of the marking pins and the front end of the chain. 
The rear-chainman takes the eleventh pin and the other end of 
the chain. The head-chainman then goes forward keeping as 
nearly on the line as he can. The rear-chainman holds his end 
of the chain just to one side of the initial point, as in Fig. 2, so 
that any jerking of the chain will not disturb the pin at which he 
is holding. The rear-chainman, with his eye over the point, 
places the head-chainman in line with some object, such as a 




Head-chainman. Rear-Chainman. 

Fig. 2. Measuring a Horizontal Line with a Chain. 

sighting-rod, which marks the other end or some point on the 
line. When the head-chainman is nearly in line he takes a pin 
and, standing to one side of the line, holds it upright on the 
ground a foot or so short of the end of the chain and the rear- 
chainman motions him to the right or left until his pin is on the 
line. When the head-chainman has the pin in line he stretches 
the chain taut, seeing that there are no " kinks ** and that no ob- 
structions cause bends in the chain. The rear-chainman at the 
same time holds his end of the chain at his pin and when he 
calls out, "All right here," the head-chainman stretching the 



8 MEASUREMENT OF LINES [Chap. I. 

chain past his line pin, removes this line pin, places it at the end 
of the chain, as in Fig. 2, and presses it vertically into the 
ground. When the chainmen are experienced the pin may be 
set for both line and distance at the same time. When the pin 
is in place the head-chainman calls, "All right," the rear-chain- 
man takes the pin left at his end of the line and they proceed to 
the next chain-length. The pin that the rear-chainman has is a 
record of the first chain-length. Just before reaching the second 
pin the rear-chainman calls out,- "Chain," to give the head-chain- 
man warning that he has nearly reached a chain-length. The 
process of lining in the head-chainman and measuring a 
chain-length is then repeated. When the third pin is stuck in 
the ground the rear-chainman pulls the second pin ; in this way 
the number of pins the rear-chainman holds is a record of the 
number of chain-lengths measured. There is always one pin 
in the ground which simply marks the distance and is not 
counted. 

When lo chains have been measured the head-chainman will 
be out of pins and calls to the rear-chainman, who brings for- 
ward lo pins. The pins are then counted by both chainmen. 
Every time lo chains are measured a record of it is made in 
note-books kept by both men and the process is repeated until the 
end of the line is reached. 

In measuring the fraction of a chain the head-chainman holds 
his end of the chain at the required point and the fractional 
distance is read by the rear-chainman at the last pin. In some 
kinds of work, however, it is more convenient to draw the chain 
ahead past the end point and, while the rear-chainman holds his 
end of the chain at the last pin, the head-chainman reads the 
fractional measurement. The links are read by counting from 
the proper tag and the tenths of a link are estimated. Great 
care should be taken to count the tags from the proper end of 
the chain since the loft. points each side of the center, as has 
been explained, are marked alike. 

It can be easily shown that if a pin is placed a few tenths of 
a foot to the right or left of the line the resulting error in the 
distance is very small and consequently " lining in " by eye is 
accurate enough, so far as the distance is concerned. But when 



MEASURING ON SLOPING GROUND 9 

any side measurements or angles are to be taken the points 
should be set accurately on line by means of a transit instrument. 

The chain should always be kept stretched out full length ; 
it should never be doubled back on itself as it may become 
tangled and the links bent. 

Much time can be saved if the head-chainman will pace the 
chain-length and then place himself very nearly in the line by 
means of objects which he knows to be on line as, for example, 
the instrument, a pole, or the last pin. The beginner should 
pace, several times, some line of known length so as to deter- 
mine approximately how many steps he takes in lOO ft. In 
doing this he should take his natural step and avoid any attempt 
to take steps just 3 ft. long. 

II. Measurement of a Horizontal Line with a Chain on Slop- 
ing Ground. — If the measurement is not on level ground the 
chain must be held horizontal and the distance transferred to 
the ground by means of a plumb-line. This is difficult to do 
accurately and is a fruitful source of error. Beginners usually 
hold the downhill end of the chain too low. Horizontal lines 
on buildings are very useful in judging when the chain is level. 
Since it is supported only at the ends its weight will cause 
it to sag so that the distance between the ends is less than 
a chain-length. The pull exerted on the chain should be such 
that it will stretch enough to balance as nearly as possible the 
shortening due to sag. 

Whenever a slope is so steep that the chainman on the lower 
end cannot plumb high enough to keep the chain horizontal the 
measurement must be made in sections, 50-ft., 20-ft., or even 10- 
ft lengths being used. Mistakes will be avoided if the rear-chain- 
man comes forward at each measurement and holds the same 
fractional point on the chain that the head-chainman held, and 
so on until a whole chain-length has been measured. In this 
way it will be unnecessary to count the fractional distances, but 
care should be taken that these pins which marked the inter- 
mediate points are returned to the head-chainman so that the 
count of the chain-lengths will not be lost. Chaining downhill 
will, in general, give more accurate results than chaining uphill, 
because in the former case the rear end is held firmly at a point 



lO MEASUREMENT OF LINES [Chap. I. 

on the ground so that the head-chainroan can pull steadily on 
the chain and transfer the distance to the ground by means of 
the plumb-line ; in the latter case the rear-chainman is plumbing 
his end of the chain over the point and it is difficult to hold it 
steady. The result is that the head-chainman cannot easily judge 
where the pin should be placed. 

12. MEASUREMENT WITH A STEEL TAPE. — In measuring 
with the steel tape the process is similar to that described for 
the chain. As the tape is used for more precise work than the 
chain it is necessary to employ more exact methods of marking 
the intermediate points. In some cases stakes are driven into 
the ground and tacks or pencil marks used to mark the points. 
A small nail pressed into the ground so that the center of the 
head is in the proper position makes a good temporary mark, but 
of course is easily lost. In measuring on the surfaces of hard 
roads, spikes are used for permanent marks. 

Measurements of important lines which are not checked by 
some geometric test should be checked by repeating the meas- 
urement, and in such a way as not to use the same intermedi- 
ate points taken in the first measurement.* 

Where distances are to be measured continuously from the 
initial point of a line without regard to angles in the line, as in 
railroad surveys, it is. customary to establish the loo-ft. points. 
Mistakes will often be avoided by setting the lOO-ft. points as fol- 
lows: — suppose an angle to occur at 870.1 ft. from the point of 
beginnmg; this would be called "Station 8 + 70.1." To set 
" Station 9 " the 70. i-ft. point of the tape should be held on stake 
8 -h 70.1 and the stake at station 9 placed at the lOO-ft. point 
of the tape. This is preferable to making a measurement of 
29.9 ft. from the zero end of the tape. 

* In measuring with the tape some prefer to make a series of measure- 
ments between points set in the ground a little less than lOo ft. apart, summing 
up the partial measurements when the end of the line is reached. This guards 
against the mistake of omitting a whole tape-length. Another advantage is that 
it is easier to read the distance to a fixed point than to set a point accurately at 
the end of the tape ; this is especially true in measurements where plumbing is 
necessary. This method takes less time than the usual method, but it is not 
applicable when it is necessary to mark the loo-ft. points on the line. 



ERRORS AND MISTAKES IN MEASURING II 

13. COHMOir SOURCES OF ERROR IN MEASUREMEUT OF 
LDIES. — 

1. Not pulling chain or tape taut. 

2. Careless plumbing. 

3. Incorrect alignment. 

4. Effect of wind. 

5. Variation in temperature. 

6. Erroneous length of chain or tape. 

14. COHMOir MISTAKES IN READING AND RECORDING 
MEASUREMENTS. — 

1. Failure to observe the position of the zero point of the tape. 

(In some tapes it is not at the end of the ring.) 

2. Omitting a whole chain- or tape-length. 

3. Reading from wrong end of chain, as 40 ft. for 60 ft., or in 

the wrong direction from a tag, as 47 ft. for 53 ft. 

4. Transposing figures, e.g., 46.24 for 46.42 (mental) ; or read- 

ing tape upside down, e.g., 6 for 9, or 86 for 98. 

5. Reading wrong foot-mark, as 48.92 for 47.92. 

15. Avoiding mistakes. — Mistakes in counting the tape- 
lengths may be avoided if more than one person keeps the tally. 
Mistakes of reading the wrong foot-mark may be avoided by 
noting not only the foot-mark preceding, but also the next fol- 
lowing foot-mark, as, "46.84 ... 47 feet," and also by holding 
the tape so that the numbers are right sidC' up when being read. 

In calling off distances to the note keeper, the chainman 
should be systematic and always call them distinctly and in such 
terms that they cannot be mistaken. As an instance of how 
mistakes of this kind occur, suppose a chainman calls, " Forty- 
nine, three ;" it can easily be mistaken for "Forty-nine feet." 
The note keeper should repeat the distances aloud so that the 
chainman may know that they were correctly understood. It 
is frequently useful in doubtful cases for the note keeper to use 
different words in answering, which will remove possible ambi- 
guity. For example, if the chainman calls, "Thirty-six, five," 
fhe note keeper might answer, "Thirty-six and a half." If the 



12 MEASUREMENT OF LINES [Chap. L 

chainman had meant 36.05 the mistake would be noticed 
The chainman should have called in such a case, "Thirty-six 
naught five." The following is a set of readings which will 
be easily misinterpreted unless extreme care is taken in calling 
them off. 

40.7 — " Forty and seven." 
47.0 — " Forty seven naught." 
40.07 — " Forty, — naught seven." 

All of these might be carelessly called off, " Forty-seven.** 

In all cases the chainmen should make mental estimates of 
the distances when measuring, in order to avoid large and ab- 
surd mistakes. 

16. ACCURACY REQUIRED. — If, in a survey, it is allowable 
to make an error of one foot in every five hundred feet the chain 
is sufficiently accurate for the work. To reach an accuracy of i 
in 1000 or greater with a chain it is necessary to give careful 
attention to the pull, the plumbing, and the deviation from the 
standard length. With the steel tape an accuracy of i in 5000 
can be obtained without difficulty if ordinary care is used in 
plumbing and aligning, and if an allowance is made for any con- 
siderable error in the length of the tape. For accuracy greater 
than about i in 10,000 it is necessary to know definitely the tem- 
perature and the tension at which the tape is of standard length 
and to make allowance for any considerable variation from these 
values. While the actual deviation from the U. S. Standard 
under ordinary conditions may be i in 10,000, still a series of 
measurements of a line taken under similar conditions may 
check themselves with far greater precision. 

17. Amount of different errors. — The surveyor 
should have a clear idea of the effects of the different errors on 
his results. For very precise work they should be accurately 
determined, but for ordinary work it is sufficient to know 
approximately the amount of each of them. A general idea of 
the effect of these errors will be shown by the following. 

* i8. Pull. — At the tension ordinarily used, the light steel 
tape will stretch between 0.0 1 and 0.02 ft. in 100 ft. if the pull 
is increased 10 pounds. 



AMOUNT OF DIFFERENT ERRORS 1 3 

19. Temperature. — The average coefficient of expansion for 
a sted tape is nearly o.ocx>oo63 for i° F. Hence a change of 
temperature of 15® produces nearly 0.01 ft. change in the length 
of the t2q>e. Tapes are usually manu&ictured to be of standard 
length at 62^ F., with a pull of 12 lbs. on them while supported 
throughout their entire length. 

20. Alignment. — The error in length due to poor alignment 
can be calculated from the approximate formula 

f^ • 
^— tf = — 

2C 

where h is the distance of the end of the tape from the line, c is 
the length of the tape, and a is the distance along the straight 
line. For example, if one end of a loo-ft. tape is held i ft. to 
one side of the line the error produced in the length of the line 

wiD be — — = 0.005 ft-> (about -^ inch). The correction 

to be applied to the distance when the two ends of the tape are 
not at the same level is computed in the same way. 

21. Sag. — If a tape is suspended only at the ends it will hang 
in a curve which is known as the " catenary." On account of this 
curvature the distance between the end points is evidently less 
than the length of the tape. The amount of this shortening, 
called the effect of sag, depends upon the weight of the tape, the 
distance between the points of suspension, and the pull exerted 

* In the right triangle, 

{c + fl) (r - a) - k\ 
assoming c ^ a and applying it to the first parenthesis only, 

ze {c — a) ^ k^ (approximately) ^--^^ 

i. tf — tf— —(approximately) ^^^00^"'^'^ A-j^ 

Similariy ^ ~" ^ "" 2tf (approximately) 

It is evident that the smaller ^ is in comparison with the other two sides the 
more exact will be the results obtained by this formula. This formula is even cor- 
rect to the nearest ^ ft. when A -^ 14 ft and tf — 100 ft, or when A — 30 ft. 
and a — 300 ft. 



14 MEASUREMENT OF LINES [Chap. I. 

at the ends of the tape. With a I24b. pull on an ordinary loo- 
ft. steel tape supported at the ends the effect of sag is about 
O.OI ft. 

22. Effect of Wearing on Length of the Chain. — When a 
chain is new it is very nearly the standard length. During 
its first use the links become bent and the chain thus shortened 
But there are nearly six hundred wearing surfaces and before 
long the small amount of wear on each surface lengthens the 
chain an appreciable amount. It is very common to find chains 
which, after considerable use, have lengthened 0.3 ft. or more. 

23. ACCURACY OF MEASUREMENTS. — In surveying we are 
dealing entirely with measurements. Since absolute accuracy 
can never be attained, we are forced to make a careful study of 
the errors of measurement. Extremely accurate measurements 
are expensive, and the cost of making the survey usually limits 
its accuracy. On the other hand, if a given degree of accuracy 
is required, the surveyor must endeavor to do the work at a 
minimum cost. In most surveys certain measurements are far 
more important than others and should therefore be taken With 
more care than the relatively unimportant measurements. 

The surveyor should distinguish carefully between errors 
which are of such a nature that they tend to balance each other 
and those which continually accumulate. The latter are by far 
the more serious. Suppose that a line 5000 ft. long is measured 
with a steel tape which is 0.0 1 ft. too long and that the error in 
measuring a tape-length is, say, 0.02 ft., which may of course be a 
+ or a — error. There will then be 50 tape-lengths in the scxx> 
ft. line. A study of the laws governing the distribution of ac- 
cidental errors (Method of Least Squares) shows that in such a 
case as this the number of errors that will probably remain uncom- 
pensated is the square root of the total number of opportunities 
for error, Le., in the long run this would be true. Hence 
the total number of such uncompensated errors in the line is 7 ; 
and 7 X 0.02 = 0.14 ft., which is the total error due to inaccuracy 
in marking the tape-lengths on the ground. Since the error due 
to erroneous length of tape increases directly as the number of 
measurements, and since these errors are not compensating, the 
total error in the line due to the fact that the tape is 0.0 1 ft. too 



ACCURACY OF MEASUREMENTS 1 5 

long is 50 X 0.01 = 0.50 ft. The small (0.0 1) accumulative 
error is therefore seen to have far greater eiSect than the larger 
(0.02) compeiisating error. 

PROBLEMa 

1. A distance is measiired with an engineer's chain and found to be 7964 ft. 
The chain when compared with a standard is found to be 0.27 ft too long. What 
is the actual length of the line ? 

2. A metallic tape which was originally 50 ft. is found to be 50.14 ft. long. 
A house 26 ft. X 30 ft. is to be laid out. What measurements must be made, 
using this tape, in order that the house shall have the desired dimensions ? 

5. A steel tape is known to be 100.000 ft. long at 62° F. with a pull of 12 lbs. 
and supported its entire length. Its coefficient of expansion is 0.0000063 for i^ F. 
A line was measured and found to be 142.67 ft. when the temperature was Bf* 
below zero. What is the true length of the line ? 

4. In chaining down a hill with a surveyor's chain the head-chainman held 
his end of the chain 1.5 ft. too low. What error per chain-length would this 
produce ? 

5. In measuring a line with a loo-ft. tape the forward end is held 3 ft. to the 
side of the line. What is the error in one tape-length ? 



CHAPTER IL 

BCBASURBMENT OF DIRBCTZON. 

24. The surveyor's compass. — -The surveyor's compass 
(Fig. 3) is an instrument for determining the direction of a line 
with reference to the direction of a magnetic needle. The needle 
is balanced at its center on a pivot so that it swings freely in 
a horizontal plane. The pivot is at the center of a horizontal 
circle which is graduated to degrees and half -degrees, and num- 
bered from two opposite zero points each way to 90*^. The zero 
points are marked with the letters N and S, and the 90® points 
are marked E and W. The circle is covered with a glass plate 
to protect the needle and the graduations, the part enclosed 
being known as the compass-box, A screw is provided for 
raising the needle from the pivot by means of a lever. The 
needle should always be raised when the compass is lifted or 
carried, to prevent dulling the pivot-point ; a dull pivot-point 
is a fruitful source of error. Both the circle and the pivot 
are secured to a brass frame, on which are two vertical sights 
so placed that the plane through them also passes through 
the two zero points of the circle. This frame rests on a tripod 
and is fastened to it by means of a ball-and-socket joint. On 
the frame are two spirit levels at right angles to each other, 
which afford a means of leveling the instrument. This ball-and- 
socket joint is connected with the frame by means of a spindle 
which allows the compass-head to be revolved in a horizontal 
plane, and to be clamped in any position. 

The magpietic needle possesses the property of pointing in a 
fixed direction, namely, the Magnetic Meridian, The horizontal 
angle between the direction of this meridian and of any other 
line may be determined by means of the graduated circle, and 
this angle is called the Magnetic Bearing of the line, or simply 
its Bearing, By means of two such bearings the angle between 
two lines may be obtained. Bearings are reckoned from o® to 90% 

16 



THE SURVEYOR'S COMPASS 



17 



the o® being either at the N or the S point and the 90® either 
at the E or the W point. The quadrant in which a bearing 
falls is designated by the letters N.E., S.E., S.W., or N.W. 
For example, if a line makes an angle of 20^ with the meridian 
and is in the southeast quadrant its bearing is written S 20® E. 
Sometimes the bearing is reckoned in a similar manner from 




Fig. 8. Surveyor's Compass. 
the geographical meridian, when it is called the true bearing. 
In general this will not be the same as the magnetic bearing. 
True bearings are often called azimittfts^ and are commonly 
reckoned from the south point right-handed (clockwise) to 360° ; 
i.e., a line running due West has an azimuth of 90°, a hne due 
North an azimuth of 180° Sometimes, however, the azimuth 



i8 



MEASUREMENT OF DIRECTION 



[Chap. n. 



is reckoned from the north as in the case of the azimuth of the 
Pole-Star (Art. 206, p. 180). • 

25. The Pocket Compass. — The pocket compass is a small 
hand instrument for obtaining roughly the bearing of a line. 
There are two kinds, the plain and the prismatic. The former 
is much like the surveyor's compass, except that it has no sights. 
In the prismatic compass the graduations, instead of being on 
the compass-box, are on a card which is fastened to the needle 
(like a mariner's compass) and which moves with it. This com- 
pass is provided with two short sights and the bearing can be 
read, by means of a prism, at the same instant that the compass 
is sighted along the line. 

26. METHOD OF TAKING A MAGNETIC BEARING. — The 
surveyor's compass is set up (and leveled) at some point on the 
line whose bearing is desired. The needle is let down on 
the pivot ; and the compass is turned so that the sights point 
along the line. While looking through the two sights the sur- 





Bcorlng of AB Bearing cf AB Bearing of AB 

seo'E seo'w n45'w 

Diagram Illustrating Reading of Bearings. 
veyor turns the compass-box so that they point exactly at a 
lining pole or other object marking a point on the line. The 
glass should be tapped lightly over the end of the needle to be 
sure that the latter is free to move. If it appears to cling to 
the glass this may be due to the glass being electrified, which 
condition can be removed at once by placing the moistened fin- 
ger on the glass. The position of the end of the needle is then 
read on the circle and recorded. Bearings are usually read to 
the nearest quarter of a degree. 

Since the needle stands still and the box turns under it, the 
letters E and W on the box are reversed from their natural 
position so that the reading of the needle will not only give the 



TAKING A MAGNETIC BEARING 1 9 

angle but also the proper quadrant. Reference to Fig. 4 will 
show the following rule to be correct : — When the north point 
of the compass-box is toward the point whose bearing is desired, 
read the north end of the needle. When the south point of the 
box is toward the point, read the south end of the needle. If a 
bearing of the line is taken looking in the opposite direction it 
is called the reverse bearing. 

Since iron or steel near the instrument affects the position 
of the needle, great care should be taken that the chain, axe, or 
marking pins are not left near the compass. Small pieces of 
iron on the person, such as keys, iron buttons, or the iron wire in a 
stiff hat, also produce a noticeable effect on the needle. Electric 
currents are a great source of disturbance to the needle and in 
cities, where electricity is so common, the compass is practically 
useless. 

In reading the compass-needle, the surveyor should take 
care to read the farther end of the needle, always looking along 
the needle, not across it. By looking at the needle sidewise it is 
possible to make it appear to coincide with a graduation which is 
reaUy at one side of it. This error is called parallax. 

27. The Earth's Magnetism. — Dip of the Needle. — The 
earth is a great magnet. On account of its magnetic influence 
a permanent magnet, such as a compass-needle, when freely 
suspended will take a definite direction depending upon the 
direction of the lines of magnetic force at any given place and 
time. If the needle is perfectly balanced before it is magnetized 
it will, after being magnetized, dip toward the pole. In the 
northern hemisphere the end of the needle toward the north 
pole points downward, the inclination to the horizon being slight 
in low latitudes and great near the polar region. In order to 
counteract this dipping a small weight, usually a fine brass wire, 
is placed on the higher end of the needle at such a point that 
the needle assumes a horizontal position. 

28. DECLINATION OF THE NEEDLE. — The direction which 
the needle assumes after the counterweight is in position is 
called the magnetic meridian and rarely coincides with the true 
meridian. The angle which the needle makes with the true me- 
ridian is called the declination of the needle. When the north 



20 MEASUREMENT OF DIRECTION [Chap. IL 

end of the needle points east of the true, or geographical, north 
the declination is called east; when the north end of the needle 
points west of true north it has a west declination. 

29. Variations in Declination. — The needle does not con- 
stantly point in the same direction. Changes in the value of the 
declination are called variations of the declination,'* The prin- 
cipal variations are known as the Secular^ Daily^ Annualy and 
Irregular. 

The Secular Variation is a long, extremely slow swing. It 
is probably periodic in character but its period covers so many 
years that the nature of it is not thoroughly understood. The 
following table shows the amount of secular variation as observed 
in Massachusetts during two centuries. 



1750 


r 13' w. 


1800 


e*' 28' w. 


1850 


9*» 10' w. 


1900 


12'' 00' w. 



In the United States all east declinations are now gradually 
decreasing and all west declinations are gradually increasing, at 
an average rate of about 2 minutes per year. 

The Daily Variation consists of a swing which averages 
about 7 minutes of arc from its extreme easterly position at 
about 8 A.M. to its most westerly position at about 1.30 P.M. 
It is in its mean position at about 10 A.M. and at 5 or 6 P.M. 
The amount of daily variation is from 3 to 12 minutes according 
to the season and the locality. 

The Annual Variation is a periodic variation so small (about 
one minute a year) that it need not be considered in surveying 
work. 

*The angle called Declination by surveyors is usually called Variation by 
navigators. 

t See p. 107 of U.S. Coast and Geodetic Survey special publication entitled 
<«U. S. Magnetic Declination Tables and Isogenic Chart for 1902, and Principal 
Facts Relating to the Earth's Magnetism," by L. A. Bauer, issued in 1^2. 



TABLE 2. I 

Observed Declinations of Needle in Eastern MASSACHusBTTS.t ' 
Year. Declination. 

1700 10° 31' w. ! 




Fig. 5> IsoGONic Chart of the Vsn 



(From the U. S. Coast and Geodetic Surrey tpedal pablication entitled " U. S. Magnetic Da 

by L. A. Bi 




► States for the Epoch January, 1902 

in Tables and laogonic Chart for 1909, and Principal Facts Relating to the Earth's Magnetism," 
i l iMi tJ in 19M.) 



DECLINATION OF THE NEEDLE 2$ 

Irregular Variations in the declination are due chiefly to 
magnetic storms. They are uncertain in character and cannot 
be predicted. They are, however, usually observed whenever 
there is a display of the Aurora Borealis. Such storms often 
cause variations of from ten to twenty minutes in the United 
States and even more in higher latitudes. 

30. Isogenic Chart. — If lines are drawn on a map so 
as to join all places where the declination of the needle is the 
same at a given time, the result will be what is called an isogenic 
chart, (See Fig. 5.) Such charts have been constructed by 
the United States Coast and Geodetic Survey. While they do 
not give results at any place with great precision they are very 
useful in finding approximate values of the declination in differ- 
ent localities. 

An examination of the isogenic chart of the United States 
shows that in the Eastern States the needle points west of north 
while in the Western States it points east of north. The line of 
no declination, or the agonic line, paisses at the present time (1906) 
through the Carolinas, Ohio and Michigan. 

31. OBSERVATIONS FOR DECLINATION. — For any survey 
where the value of the present declination is important, it should 
be found by special observations. The value found at one place 
may be considerably different from that of a place only a few 
miles distant. The method of finding the declination by ob- 
servation on the Pole-Star (Polaris) is described in Art. 2 10, p. 187. 

ADJUSTMENTS OF THE COMPASS. 

32. The three adjustments which need to be most frequently 
made are (i) adjusting the bubbles, (2) straightening the needle, 
(3) centering the pivot-point. 

33. Adjustment of the bubbles. — To make the Plane 
of the Bubbles Perpendicular to the Vertical Axis. — Level the in- 
strument in any position. Turn 180° about the vertical axis 
and, if the bubbles move from the center, bring each half-way 
back by means of the adjusting screws; and repeat the process 
until the desired fineness of adjustment is secured. 

34. detecting errors in adjustment of the Needle. 
— If the readings of the two ends of the needle are not 180® 



26 



MEASUREMENT OF DIRECTION 



[Chap, n 




Fig. 6. 



Bent Compass- 
Needle. 



apart, this may be due to the needle being bent, to the pivot- 
point not being in the center of the graduated circle, or to both. 
If the difference of the two readings is the same in whatever 

direction the compass is turned, it 
follows that the needle is bent but 
the pivot-point is in the center of the 
circle. (See Fig. 6.) The bent needle 
is represented by the line AOB and 
the position of a straight needle shown 
by the line A OC. In the two positions 
shown it is seen that the difference in 
readings will be the same, i.e., arc 
CB = arc C'B\ If the difference of 
the readings varies as the compass 
is turned around it follows that the 
pivot-point is not in the center, and the needle may or may 
not be bent. Suppose the needle is straight but the pivot is 
not in the center, then the effect in different parts of the circle 
is shown in Fig. 7. When the needle is in the position ADy 
perpendicular to CO, (where C is 
the true center and C is the 
position of the pivot-point), then 
the error is a maximum. If B is 
a point 180® from A then the dif- 
ference of the two readings is BD, 
When the needle is at A^D' the 
error is less than before and equals 
B'ly. When the needle is in the 
line CC, i.e., in the position yi"Z>", 
the ends read alike. 

In making these adjustments 
it is better to first straighten the 
needle, because the error due to 
the needle being bent can be detected independently of the 
error of the pivot. 

35. TO STRAIGHTEN THE COMPASS-HEEDLE. — Level the 
instrument and let the needle down on the pivot. Remove the 
glass cover. By means of a brass wire or alight stick of wood 




Fig. 7. 



Pivot-Point 
Center. 



OUT of 



ADJUSTMENTS OF THE COMPASS 2/ 

Steady the needle so that one end of it, say the south end, is 
opposite some graduation on the circle as A in Fig. 8. Note 
the position of the north end of the needle C Now, without mov- 
ing the compass itself, turn the needle around so that the north 
end is at the graduation A. Hold it in this position with the 
brass wire and read the position of the south end C\ One- 
half the difference of the readings, or, the distance C^D is the 
^ c' amount by which the needle is bent. 

^ ^^^ Carefully remove the needle from the 

^^ pivot and bend it by the amount CD 
\ in the direction which will move the 
\ south end half-way back from C 
Q I toward C. It is better not to touch 

\ T j the needle with the hands more than 

\ I / is absolutely necessary as this weakens 

\ I y the magnetism. Instrument makers 

V^^^^ \ ^^ usually leave the central part of the 

^ needle quite soft so that it can be 

Fig. 8. Straightening the easily bent in making this adjustment. 
Compass-Needle. Since the amount by which the 

needle is bent is a matter of estimation it should be replaced 
on the pivot and the test repeated until it is found that reversing 
the needle does not change the readings. 

36, To CEHTER the PiVOT-POnrr. — If the difference of 
readings of the two ends of the needle varies in different parts 
of the circle it is due to the pivot-point being out of center. 
Take readings of the two ends of the needle in various positions 
of the compass and find the position of the needle in which the 
difference of the two readings is greatest (Art. 34, p. 25). The 
pivot is to be bent at right angles to this direction an amount 
equal to half this difference. Remove the needle and bend the 
pivot by means of a pair of small flat pliers. Replace the needle 
and see if the difference of end readings is zero. If not, 
the pivot must be bent until this condition is fulfilled. As the 
pivot may become bent somewhat in a direction other than that 
intended, a complete test for adjustment must be made again, 
and the process continued until the difference in the readings of the 
ends of the needle is zero in all positions of the compass. The 



28 



MEASUREMENT OF DIRECTION 



[Chap. n. 



metal at the base of the pivot is left soft so that it can be easily 
bent. 

37. To REMAGNETIZE THE NEEDLE. — Rub each end of 
the needle from the center toward the end several times with a 
bar-magnet, using the N end of the magnet for the S end of the 
needle and vice versa. (The N end of the magnet attracts the 
S end of the needle and repels its N end.) When the magnet 
is drawn along the needle it should move in a straight line, 
parallel to the axis of the needle. When returning the bar from 
the end of the needle toward the center, lift it several inches 
above the needle as indicated in Fig. 9. 




Fig. 9. ReMagnetizing the Compass-Needle. 
38. CcmMON SOURCES OF ERROR IN COMPASS WORK* — 

1. Iron or steel near compass. 

2. Parallax in reading needle, 

39* COlfMON MISTAKES.— 

1. Reading wrong end of needle. 

2. Not letting needle down on pivot. 

3. Reading the wrong side of the loth degree, 

viz., reading 61° instead of 59°. 



DETECTING LOCAL ATTRACTION 29 

40. DETECTING LOCAL ATTRACTION OF THE NEEDLE. — As 
the needle is always affected by masses of iron near the compass 
it is important that the bearings in any survey should be checked. 
This is most readily done by taking the bearing of any line from 
both its ends or from intermediate points on the line. If the two 
bearings agree it is probable that there is no local magnetic dis- 
turbance. If the two do not agree it remains to discover which 
is correct. 

In Fig. 1 1 suppose that the compass is at A and that the 




Fig. 11. Diagram Illustrating Local Attraction at A. 

bearing of AB is N 50^^ E, and with the compass at B the 
bearing BA is found to be S 49® W. It is evident that there is 
local attraction at one or both points. In order to ascertain the 
correct magnetic bearing, turn the compass toward a point C which 
is apparently free from magnetic disturbance, and observe the 
bearing of BQ which is, say, S 72° E. Now move the compass 
to Cand observe the bearing CB. If this is N 72^ W it indi- 
cates that there is no local attraction at C or B, hence S 59° W 
is the correct bearing of line BA, and there is i°J error in all 
bearings taken at A, If the bearings of BC and CB had not 
agreed it would have been necessary to take the bearing and 
reverse bearing of a new line CD. This process is continued 
until a line is found whose bearing and reverse bearing differ by 
exactly 180**. 



30 



MEASUREMENT OF DIRECTION 



[Chap. IL 



41. CALCULATING ANGLES FROM BEARINGS.— In calcu- 
lating the angle between two lines it is necessary only to 
remember that the bearing is in all cases reckoned from the 
meridian, either N or S, toward the E and W points. In Fig. lo, 



AOB =s difference of bearings. 
AOC = 1 80® — sum of bearings. 
AOD = 180® — difference of bear- w 

ings. 
AOF = sum of bearings. 




Fig. 10. 



PROBLEMS. 

I. Compute the angle AOB from the given bearings in each of the following 



(a) OA. N39«iE. (c) 
OB, N76<»iE. 

(b) OA, N 3S« IS' E. (d) 
OB, S 88° 00' W. 



OA, N 15° E. 
OB, S 36° E. 
OA, N 40° 15' E. 
OB, N 66° 45' W. 



. 2. The bearing of one side of a field in the shape of a regular hexagon is 
S \<f\ E. Find the bearings of the other sides taken around the field in order. 

3. (a) In 1859a certain line had a bearing of N 21° W. The declination of 
the needle at that place in 1859 was 8° 39' W. In 1902 the declination was 
10° 58' W. What was the bearing of the line in 1902 ? 

(b) In 1877 a line had a bearing of N 89° 30' E. The declination was 
0° 13' E. In 1902 the declination was 1° 39' W. Find the bearing of the line 
in 2902. 

(c) At a certain place the declination was 4° 25' W in 1700, 1° 39' W in 1750, 
0° 2x' E in 1800, 1° 03' W in 1850, 4° 00' W in 1900. If a line had a bearing of 
S 65°i W in 1900, what was its bearing in 1700, 1750, 1800, and 1850"? 

4. The following bearings were observed with a compass : AB, N 27°} E ; 
BA, S 25°i W ; ^C. S 88° W ; C^, N 87°! E ; CZ?, N 47°i W ; DC, S 47°^ E. 
Find the true bearing of AB. Where is the local attraction ? Which way is the 
needle deflected at each point, and how much ? 



CHAPTER III. 

MBASUREMIINT OF ANaiiBS. 

THE TRANSIT. 

42. GENERAL DESCRIPTION OF THE TRANSIT. — The en- 
gineer's transit is an instrument for measuring horizontal and 
vertical angles. A section of the transit is shown in Fig. 1 2. 




Fig. 12. Section of Transit. 

A^ inner spindle ; By outer spindle ; C, upper plate ; Z>, lower plate ; Ey graduated 
circle ; Fy vernier ; Gy magnetic needle ; Hy upper clamp (turned 90° from its nor- 
mal position so as to show in section, corresponding tangent screw not shown) ; 
Jy lower clamp ; Ky lower tangent screw ; Z, leveling screws ; My ball-and-socket 
joint ; Ny shifting head ; O, base of transit. 

Two spindles, one inside the other, are each attached to a hori- 
zontal circular plate, the outer spindle being attached to the 
lower plate and the inner one to the upper plate. Except in 
some older instruments, the lower plate carries a graduated 
circle and the upper plate carries the verniers for reading the 
circle. On this upper plate are two uprights or standards 

31 



32 MEASUREMENT OF ANGLES [Chap. m. 

supporting a horizontal axis. The length of the telescope and 
the height of the standards are commonly such as to allow the 
telescope to make a complete rotation on its horizontal axis. 
The motion of this axis is usually controlled by a clamp and a 
slow-motion screw called a tangent screw. In older instruments 
this often consisted of two opposing screws ; in modern instru- 
ments it usually consists of a single screw with an opposing 
spring. At the center of the horizontal axis is a telescope 
attached at right angles to it. 

For leveling the instrument, there are two spirit levels on the 
upper plate, one parallel and the other at right angles to the hori- 
zontal axis. The spirit level which is parallel to the axis is the 
more important one because it controls the position of the hori- 
zontal axis of the telescope ; it should be and generally is made 
more sensitive than the other. In the transit, the leveling is 
done by means of four (sometimes three) leveling screws. 

The upper plate is usually provided with a magnetic needle 
and a graduated circle so that the transit may be used as a com- 
pass. The lower spindle is attached to the base of the instru- 
ment by means of a ball-and-socket joint the same as in the 
compass. Both the upper and lower plates are provided with 
clamps for holding them in any desired position and with tan- 
gent screws for making exact settings. 

At the center of the ball-and-socket joint is a ring to which the 
plumb-line may be attached. The plumb-bob used with the tran- 
sit is generally heavier than that used in taking tape measure- 
ments. Modern transits are so made that the entire head of the 
instrument can be shifted laterally \vith reference to the tripod 
and can thus be readily placed exactly over a point on the 
ground. 

The horizontal circle is usually graduated either to half-degrees 
or to 20-minute spaces. The graduations are often numbered 
from o° to 360° by two rows of figures running in opposite direc- 
tions. In some transits they are numbered from 0° to 360° in a 
right-hand direction and, by a second row of figures, from o*^ 
each way to 180®; and still others (older types) are numbered 
from o® to 90"^ in opposite directions, like a compass circle. 
Transits are all provided with two opposite verniers. 



THE TRANSIT 



33 




Fig. 18. Engineer's Transit. 



34 



MEASUREMENT OF ANGLES 



[Chap. IIL 



M 

o 
u 

u 
.J 

H 



c/) 
55 

H 



O 

o 

H 
O 
» 

< 

Q 
D 
H 

o 
o 






43. The normal or ^/V^?^/ position of 
the transit is with the upper clamp and its 
tangent screw nearest the observer and the 
focusing screw of the telescope on the right- 
hand side (in some instruments, on top) of 
the telescope. When the instrument is 
turned 180° in azimuth from the direct 
position and the telescope is inverted (turned 
over about the horizontal axis) it is said to 
be in the reversed position. 

44. .If the telescope is provided with a 
long level tube and a vertical circle, or arc, 
it is called an Engineer's Transit^ or Sur- 
veyor's Transit. (Fig. 13.) If it does not 
have these attachments it is called a Plain 
Transit. 

45. The TELESCOPE. — The essential 
parts of the telescope are the objective^ the 
cross-hairs, and the eyepiece. (See Fig. 14.) 

The line of sight, or line of collimation, 
is the straight line drawn through the op- 
tical center of the objective and the point 
of intersection of the cross-hairs. When 
light from any point A falls on the objective, 
the rays from A are bent and brought to a 
focus at a single point B called the image. 
The only exception to this is in the case 
when A is on the optical axis ; the ray 
which coincides with the optical axis is not 
bent. The cross-hairs are placed in the 
telescope tube near where the image is 
formed, as shown in Fig. 14. The objective 
is screwed into a tube, which is inside the 
main tube and which can be moved by means 
of a rack-and-pinion screw so as to bring the 
plane of the image of the object into coinci- 
dence with the plane of the cross-hairs. The 
instrument is so constructed that the motion 



THE TRANSIT TELESCOPE 35 

of this tube is parallel to the line of sight. The eyepiece is 
simply a microscope for viewing the image and the cross-hairs. 
When the plane of the image coincides with the plane of the 
cross-hairs, both can be viewed at the same instant by means of 
the eyepiece. The adjustment of the eyepiece and the objective, 
to enable the cross-hairs and the image to be clearly seen at 
the same time, is QsH^&di focusing. 

In focusing, first the eye-piece tube is moved in or out until the 
cross-hairs appear distinct ; then the objective is moved until the 
image is distinct. If it is found that the cross-hairs are no longer 
distinct after moving the objective the above process is repeated 
until both image and cross-hairs are clearly seen at the same 
instant. The focus should be tested for i)arallax by moving the 
eye slightly from one side to the other ; if the cross-hairs appear to 
move over the image the focus is imperfect. In focusing on ob- 
jects at different distances it should be remembered that the 
nearer the object is to the telescope, the farther the objective must 
be from the cross-hairs ; "and that for points near the instrument 
the focus changes rapidly, i.e., the objective is moved consider- 
ably in changing from a focus on a point lo ft. away to one 
20 ft. away, whereas for distant objects the focus changes very 
slowly, the focus for 200 ft. being nearly the same as that for 
20CXD ft. An instrument can be quickly focused on a distant 
object if the objective is first moved in as far as it will go and 
then turned out slowly until the image is distinct. The objec- 
tive should not be turned too rapidly as it may pass the correct 
position before the eye can detect the distinct image. If an in- 
strument is badly out of focus it may be pointing directly at an 
object and yet the image may not be visible. 

46. The Objective. — The objective might consist of a simple 
bi-convex lens, like that shown in Fig. 15, which is formed by 
the intersection of two spheres. The line 0(y joining the 
centers of the two spheres is called the optical axis. If rays 
parallel to the optical axis fall on the lens those near the edge of 
the lens are bent, or refracted, more than those near the center, 
so that all the rays are brought to a focus (nearly) at a point F 
on the optical axis called ^^ principal focus. If light falls on 
the lens from any direction there is one of the rays such as 



36 



MEASUREMENT OF ANGLES 



[Chap. HI. 



AC or BD which passes through the lens without permanent 
deviation, i.e., it emerges from the other side of the lens parallel 
to its original direction. All such rays intersect at a point JTon 
the optical axis which is called the optical center, 

A simple bi-convex lens does not make the best objective 
because the rays do not all come to a focus at exactly the same 
point. This causes indistinctness and also color in the field of 




Fig. 15. Bi-CoNVEX Lens. 

view, particularly near the edges. This difficulty is overcome 
by using a combination of lenses, consisting of " crown ** and 
"flint** glass as shown in Fig. 14, which very nearly corrects 
these imperfections. 

The position of the image of any point is located on a straight 
line (nearly) through the 
point and the optical center ; 
hence it will be seen that 
the image formed by the 
objective is inverted. 

47. Cross-Hairs. — The 

cross-hairs consist of two 

very fine spider threads 

stretched across a metallic 

ring at right angles to each ^ ,« ^ 

t. f r J i_ Fig. 16. Cross-Hair Ring. 

other and fastened by means 

of shellac. The cross-hair ring (Fig. 16) is held in place by 

four capstan-headed screws which permit of its being moved 




MAGNIFYING POWER 37 

vertically or horizontally in the telescope tube. The holes in the 
tube through which the screws pass are large enough to allow 
some motion of the ring in adjusting. 

48. Eyepiece. — The eyepiece of the ordinary transit tele- 
scope may be either of two kinds, that which shows an inverted 
image or that which shows an erect image. An erecting eyepiece 
requires two more lenses than the inverting eyepiece, which add 
to its length and also absorb light ; but in spite of these disad- 
vantages the erecting eyepiece is generally used on ordinary 
transits. It will be seen, however, that with the same length of 
telescope a greater magnifying power and a clearer, definition of 
the image can be obtained by the use of the inverting eyepiece. 
These advantages are so important and the disadvantage of see- 
ing objects inverted is so slight that inverting eyepieces should 
be used more generally than they are at present. 

49. Magnifying Power. — The magnifying power is the 
amount by which an object is increased in apparent size. It is 

tan ^ A A 

equal to - — ^ — ,(or nearly equal to — ), A being the angle sub- 
tended by an object as seen through the telescope and a the 
angle as seen by the unaided eye. 

50. The magnifying power may be measured in two ways, 
(i) The dimensions on a graduated rod will appear magnified 
when viewed through a telescope. If, with one eye at the tele- 
scope, the rod is viewed directly with the other eye it will be 
noticed that one space as viewed through the telescope will 
appear to cover a certain number of spaces as seen with the naked 
eye. This number is approximately the magnifying power of the 
telescope. 

(2) Viewed through a telescope wrong-end-to, an object is re- 
duced in apparent size in the same ratio that it is magnified when 
seen through the telescope in the usual manner. Measure with 
a transit some small angle A between distant points and then 
place the telescope to be tested in front of the transit, with its 
objective next the objective of the transit. Measure the angle a 
between the same points ; this new angle will be smaller. Then 

tan -i A 
the Magnifying Power =——t—- The magnifying power 



38 



MEASUREMENT OF ANGLES 



[Chap. m. 



of the ordinary transit telescope is between twenty and thirty 
diameters. 

51. Field of View. — The field of view is the angular space 
that can be seen at one time through the telescope. It is the 
angle subtended at the optical center of the objective by the 
opening in the eyepiece. In the ordinary transit this angle is 
about one degree, but in some instruments it is considerably 
more. 



_— 9 



—7 






-K) 



& 



—4 

—3 

-I 







c^ 











— 10 




—9 




—6 




—7 


i : 


-5- 




-*'' 




—3 

— ^2 






5- 


(BUUO 







6- 



—9 

■—8 

■7 



3 
5- 



5- 



—3 
—Z 

I 



Fig. 17. 



Fig. 18. 



Fig. 19. 



52. THE VERlflER. — The vernier is a device for determin- 
ing the subdivision of the smallest division of a scale more accu- 



THE VERNIER 39 

lately than can be done by simply estimating the fractional part. 
It depends upon the fact that the eye can judge much more 
exactly when two lines coincide than it can estimate a fractional 
part of a space. 

A simple form of vernier, shown in Fig. 17, is constructed 
by taking a length equal to 9 divisions on the scale and dividing 
this length into 10 equal parts. One space on the vernier is 
then equal to -^^ of a space on the scale, i.e., it is -^ part 
shorter than a space on the scale, hence ad = -j^ of a space 
on the scale, cd = -j^^ of a space, etc. Now if the vernier is 
raised until a coincides with 6, i.e., until the first line on. the 
vernier coincides with the next higher line on the scale, then 
the index line has moved over ^ of a space and the reading 
will be 501. If the vernier is moved -^ space higher then 
line 2 coincides with the next higher line on the scale and 
the reading is 502, as shown in Fig. 18. Similarly Fig. 19 
shows reading 526. Thus it is seen that the number of the line 
on the vernier which coincides with a line on the scale is the 
number of tenths of the smallest division of the scale that the 
index point (zero) lies above the next lower division on the scale. 
Furthermore it will be seen from its construction that it is im- 
possible to have more than one coincidence at a time on a single 
vernier. The type of vernier just described is used on leveling 
rods. 

53. Verniers used on Transits. — In transits, since angles 

,may be measured in either direction, the verniers are usually 

double, Le., there is a single vernier on each side of the index 

point, one of which is to be used in reading angles to the right, 

and the other in reading angles to the left. 

The vernier most commonly found on the transit reads to 
one minute of arc (Fig. 20). When this vernier is used the 
circle is divided into degrees and half-degrees. The vernier 
scale is made by taking a length equal to 29 of the half-degree 
spaces and subdividing it into 30 equal parts. Each space on 
the vernier is then equal to f^ X 30' = 29'. Therefore the differ- 
ence in length of one division on the circle and one division on 
the vernier is equal to the difference between the 30' on the 
circle and the 29' on the vernier, or one minute of arc. In 



40 



MEASUREMENT OF ANGLES 



[Chap. III. 



Fig. 20 the zero of the vernier coincides with the 0° mark on 
the circle. The first graduation on the vernier to the left of the 
zero fails to coincide with the o*^ 30' line by just i' of arc. 
The second line on the vernier falls 2' short of the i*^ mark, 
the third line 3' short of the i*" 30' mark, etc. If the vernier 
should be moved one minute to the left the first line would coin- 



30 








VERNIER 






^ 


^ 


\0 ^ /O 


* 


fc 


1 ' 1 


Yiwv 










360 










CIRCLE 






Fig. 


ao. 


One-Minute Vernier Set 


AT 


o^ 



3? 




cide and the reading would be o*^ 01'. If the vernier were 
moved one minute more the second line would coincide and 
the reading would be o°02', etc. Therefore the number of the 
line on the vernier which coincides with some line on the circle 
is the number of minutes to be added to o*^. After the vernier 
has moved beyond the point where the 30' line coincides, it 
begins subdividing the next space of the circle, and we must 
then add the vernier reading to o® 30'. 



The following figures show various types of vernier com- 
monly used on transits. 




TRANSIT VERNIERS 4 1 

Fig. 21. — Double vernier reading id i\ Circle divided 
into 30' spaces. 29 divisions of the circle divided into 30 parts 
to make one division of the vernier. 

Reading, inner row of figures, 9® 16'. 
Reading, outer row of figures, 350° 44'. 

Since the vernier moves with the telescope, read the angle 
on th3 circb in the same direction that the teldScop3 has moved- 
Read the number of degrees and half-degrees the index has 
passed over and estimate roughly the number of minutes beyond 
the last half-degree mark. Then follow along the vernier in the 
same direction and find the coincidence. The number of this 
line is the number of minutes to be added to the degrees and 
half-degrees which were read from the circle. An estimate of 
the number of minutes should always be made as a check against 
large mistakes in reading the vernier or in reading the wrong 
vernier. 



Fig. 22. — Double vernier reading to 30". Circle divided 




Fig. 22. 

into 20' spaces. 39 divisions of the circle divided into 40 parts 
to make one division of the vernier. 

Reading, inner row of figures, 31® 17' 30". 
Reading, outer row of figures, 328° 42' 30". 



42 



MEASUREMENT OF ANGLES 



[Chap. IIL 



Fig. 23. — Single vernier reading to 20". Circle divided 
into 20' spaces. 59 divisions of the circle divided into 60 parts 
to make one division of the vernier. 

Reading, 73° 48' 40". 




CIRCLE 

Fig. 28. 

On account of the length of this vernier it is impracticable 
to use a double vernier. Where it is desirable to read the 
angles in either direction the circle has two rows of figures as 
shown in Fig. 24. 



Fig. 24 — Reading, inner row of figures, 73*" 48' 40". 
Reading, outer row of figures, 266"* 31' 20". 




CIRCLE 

Fig. 24. 

It is evident that if angles are to be read " clockwise " the 
index at the right end of this vernier should be set at o^ If 



TRANSIT VERNIERS 



43 



angles are to be measured in the opposite direction the index at 
the left end should be set at o^ To avoid this inconvenience of 
resetting, some surveyors set the middle line (ic/ line) of the 
vernier on o*^ and disregard the numbering on the vernier, 
reading it as explained under Fig. 26. 



Fig. 25. — Single vernier reading to 10". Circle divided 




into i& spaces. 59 divisions of the circle divided into 60 parts 
to make one division of the vernier. 

Readmg, 59° 15' 50". 



Fig. 26. — Single vernier reading in either direction to i'. 
CIRCLE 5 




5 



ijrm]r\ 




^ 9 



VERNIER 

Fig. 26. 



Circle divided into $<y spaces. 29 divisions of the circle 
divided into 30 parts to make one division of the vernier. 

Reading, 2° 23'. 



44 



MEASUREMENT OF ANGLES 



[Chap. HI. 



This vernier is read like the ordinary i' vernier except that 
if a coincidence is not reached by passing along the vernier in 
the direction in which the circle is numbered, it is necessary to 
go to the other end of the vernier and continue in the same 
direction, toward the center, until the coincidence is found. 
This vernier is used on the vertical circle of transits when the 
space is too small for a double vernier. 

There is another type of transit vernier, which is occasionally 
used, in which the degree is divided into hundredths instead of 
minutes. 

54. Eccentricity. — : If the two opposite verniers of a tran- 
sit do not read exactly alike it is usually due to a combination 
of two causes, (i) because the center of the vernier plate does 
not coincide with the center of the graduated circle, (2) because 
the vernier zeros have not been set exactly 180*^ apart. The 
first cause produces a variable difference while the second 
produces a constant difference. 

It will be noticed that the effect of these errors is similar 
to that described in Art. 34, p. 25, on Adjustments of the Com- 
pass ; the eccentricity of the circles of the transit corresponding 

to the bent pivot of the compass 
and the error in the position of 
the verniers of the transit corre- 
sponding to the bent needle of the 
compass. 

With reference to the eccen- 
tricity of the plates, let C in 
Fig. 27 be the center of the 
vernier plate and C the center of 
the circle. Let GF be a line 
through the two centers. When 
one vernier is at F and the 
other is at G the vernier readings 
will be the same as though C 
and C were coincident, since the displacement of the center of 
the circle occurs in the direction of the lines of graduation at F 
and G. If the telescope is then turned at right angles to its 
former position, the verniers then being at D and E^ the readings 



g D 



Fig. 27. 



Eccentricity of 
Circle. 



ECCENTRICITY OF CIRCLES 45 

of opposite verniers will differ by the maximum amount. Suppose 
that the graduations are numbered from o** right-handed to 360°. 
When the vernier is at an intermediate position, as at Ay it will 
be seen that it reads too much by the amount AA\ The opposite 
vernier at B reads too little by the amount BB\ Since AB and 
A'ff are parallel, BB* and AA' are equal. Consequently the 
mean of the two vernier readings will be the true reading and 
the eccentricity is in this way eliminated. Since the effect of 
eccentricity is never more than a very few minutes it is cus- 
tomary to read the degrees and minutes on one vernier and the 
minutes only on the other. 

55. In spite of the fact that the two verniers are not 180° 
apart no error is introduced provided; (i) that the same vernier 
is always used, or (2) that the mean of the two vernier readings 
is always taken. But if vernier A is set and the angle is read 
on vernier B an error does enter. Where only one vernier is 
read always read the vernier that was set at O^. 

In good instruments both of these errors are very small, 
usually smaller than the finest reading of the vernier. 

USE OF THE TRAIVSrr. 

56. SETTIlfG UP THE TRANSIT. — In setting the transit 
over a point, place one leg of the tripod in nearly the right posi- 
tion on the ground, then g^sp the other two and move the in- 
strument in such a way as to bring the head over the point and 
at the same time keep the plates of the instrument approximately 
level, giving the tripod sufficient spread to insure steadiness. 
The tripod legs should be pressed firmly into the ground. The 
nuts at the top of the tripod legs should be tight enough so that 
the legs are just on the point of falling of their own weight 
when raised from the ground. If they are loose the instrument 
is not rigid ; if they are too tight it is not in a stable condition 
and may shift at any moment. 

If the point is on sloping ground it is often convenient, and 
usually insures greater stability, to set two legs on the down- 
hill side and one leg uphill. When the center of the instrument 
is over the point but the tripod head is not nearly level it can be 



46 MEASUREMENT OF ANGLES [Chap. in. 

leveled approximately without moving the instrument away from 
the point by moving, one, sometimes two, of the tripod legs in an 
arc of a circle about the point. Nothing but practice will make 
one expert in setting up the transit. 

It is desirable to bring the instrument very nearly level by 
means of the tripod ; this is really a saving of time because under 
ordinary conditions it takes longer to level up by the leveling 
screws than by the tripod. It also saves time on the next set- 
up to have the leveling screws nearly in their mid position. If 
the transit is set by means of the tripod, say, within o.oi or 0.02 
ft. of the point, the exact position can be readily reached by 
means of the shifting head^ which may be moved freely after 
any two adjacent leveling screws are loosened. When the tran- 
sit has been brought directly over the point, the leveling screws 
should be brought back to a bearing. In the first (rough) setting 
the plumb-bob should hang, say, an inch above the point, but 
when the shifting head is used it should be lowered to within 
about \ inch or less of the point. 

57. In leveling the instrument, first turn the plates so that 
each plate level is parallel to a pair of opposite leveling screws. 




Fig. 28. Cut Showing How Fingers Move in Leveling. 

Each level is therefore controlled by the pair of leveling screws 
which is parallel to it. Great care should be used in leveling. 
The screws must not be loose as this will cause the plates to tip 
and perhaps to move horizontally which would change the posi- 
tion of the plumb-bob over the point. On the other hand they 



SETTING UP THE TRANSIT 47 

must not be too tight as this will not only injure the instrument 
but will cause errors due to strains in the metal. To level 
the instrument, grasp one pair of opposite screws between the 
thumbs and forefingers and turn so that the thumbs move either 
toward each other or away from each other, as illustrated in 
Fig. 28. In this way one screw is tightened as much as the 
other is loosened. The motion of both screws must be uniform ; 
if they bind, the one which is being loosened should be turned 
faster. If this does not appear to remedy matters then the 
other pair of screws is binding and should be loosened slightly. 
Only experience will teach one to level an instrument quickly 
and correctly. It may be convenient for beginners to remember 
that in leveling the instrument the bubble will move in the same 
direction as the left thumb moves. After one bubble has been 
brought nearly to the center of its tube the other bubble is 
centered in a similar manner by its pair of leveling screws. In- 
stead of tr)dng to cent'er one bubble exactly before beginning on 
the second one it is better to get both of them approximately 
level, after which first one bubble and then the other may be 
brought exactly to the centfer. After the instrument is leveled 
the plumb-bob should be examined to see that it has not been 
moved from over the point during the process of leveling. 

58. To MEASURE A HORIZONTAL ANGLE. — After setting 
the instrument up over the point, first set the zero of one of the 
verniers opposite the zero of the circle. This is done by turning 
the two plates until the two zeros are nearly opposite, clamping 
the plates firmly together with the upper clampi and then bring- 
ing the two into exact coincidence by means of the tangent screw 
which goes with the upper clamp. If a line on the vernier is 
coincident with a line on the circle then the two adjacent hnes 
on the vernier will fail to coincide with the corresponding lines 
on the circle by equal amounts (Art. 53, p. 39). Hence the coin- 
cidence of any line on the vernier with a line on the circle can 
be more accurately judged by examining also the adjacent di- 
visions and noting that they are symmetrical with respect to 
the coincident lines. A pocket magnifier, or "reading glass,'* 
is generally used for setting and reading the vernier. Never 
touch the clamp after a setting has been made by means of the 



48 MEASUREMENT OF ANGLES [Chap. HI. 

tangent screw. In setting with the tangent screw it is better 
to do this by a right-hand turn, i.e., by turning the screw in 
the direction which compresses the spring against which it 
works. If the screw needs to be turned back, instead of turn- 
ing it to the exact setting turn it back too far and then bring 
it up to the accurate setting with a right-hand motion, thereby 
insuring a firm bearing of the spring against the screw. The 
two plates which are now clamped in proper position are free to 
turn together about the vertical axis. Turn to the first object 
and point the telescope at it approximately by looking over the 
top of the telescope. When turning the instrument so as to 
sight the first point it is good practice to touch the lower plate 
only. Focus the telescope by moving the eyepiece until the 
cross-hairs are distinct and then moving the objective until the 
image is distinct. It is sometimes convenient to point the tele- 
scope at the object when focusing the cross-hairs so that they 
can be readily seen.* Test for parallax by moving the eye 
slightly from one side to the other. Move the telescope until 
the vertical cross-hair is very nearly on the point. It is better 
to use that part of the cross-hair which is near the center of the 
field of view. Clamp the lower plate by means of the lower 
clamp, and set exactly on the point by the lower tangent 5crew. 
The line of sight is now fixed on the first object. To measure 
the angle loosen the upper clamp, turn the telescope to the 
second point, and focus the objective if necessary. Set nearly 
on the point, clamp the upper plate, and set the vertical cross- 
hair exactly on the point by means of the upper tangent screw. 
The angle is then read on the vernier which was set at 0^. 

The tangent screws should not be used to move the plates 
over large angles. Acquire the habit of setting closely by hand 
and using the tangent screw for slight motions only. 

59. TO MEASURE AN ANGLE BY REPETITION. — The 
eyepiece magnifies the image so much thnt it is possible to set 
the cross-hair on a point much more closely than the vernier will 

* If the eyepiece is focused on the cross-hairs with the telescope pointing at 
the sky, as is frequently done, they will be found to be approximately in locos 
when looking at the object ; but for accurate work the eyepiece should be focused 
on the cross-hairs when the objective is in focus on the object. 



MEASURING ANGLES BY REPETITION 49 

read. The graduation of the circle is very accurate and can be 
depended upon closer than the vernier can be read, consequently 
the full value of the instrument is not utilized by single readings 
of an angle. To obtain the value of an angle more accurately 
proceed as follows. After the first angle has been measured 
leave the two plates clamped together, loosen the lower clamp 
and turn back to the first point. Set on the first point, using 
the lower clamp and its tangent screw. Then loosen the upper 
clamp and set on the second point, using the upper clamp 
and its tangent screw, thus adding another angle, equal to the 
first one, to the reading on the circle. Repeat this operation, 
say, six times. The total angle divided by six will give a more 
precise result than the first reading. Suppose that the angle is 
actually 18® 12' 08' ; if a "one-minute " instrument is being used 
it is impossible to read the 08" on the vernier, so the reading 
will be 18° 12'. Each repetition will add 08" (nearly) and after 
the 6th repetition, the amount wiU be 48" which will be read as 
i'. After the 6th pointing the total angle will then be read 109° 
13' which divided by 6 gives 18° 12' 10", a result in this case 
correct to the nearest 10". To eliminate errors in the adjust- 
ment of the transit the above process should be repeated with 
the instrument reversed and the mean of the two values used. 
(See Art. 79, p. 61.) It is customary to take only the ist and 
6th readings, but as a check against mistakes it is well for the 
beginner to examine the vernier reading after each repetition 
and see that ^ the second reading, J the third, etc., nearly 
equals the first reading. 

Repetition has also the advantage of eliminating, to a great 
extent, errors of graduation. If an angle is about 60® and is 
repeated 6 times it will cover a whole circumference. If there 
are systematic errors in the graduations the result is nearly free 
from them. The effect of accidental, or irregular, errors of 
graduation is decreased in proportion to the number of repeti- 
tions. In the best modern instruments the errors of graduation 
seldom exceed a few seconds. 

Little is gained by making a very large number of repetitions 
as there are systematic errors introduced by the action of the 
clamps, and the accuracy apparently gained is really lost on this 



50 MEASUREMENT OF ANGLES [Chap. HI. 

account. Three repetitions with the telescope normal and three 
with the telescope inverted are sufficient for anything but very 
exact work. 

It is desirable that as little time as possible should elapse 
between pointings, as the instrument cannot be relied upon to 
remain perfectly still. As a matter of fact it is vibrating and 
"creeping" nearly all the time from numerous causes. For 
example, when the instrument is set up on frozen ground, it will 
quickly change its position on account of the unequal settlement 
of the tripod legs. Changes of temperature, causing expansion 
or contraction of the metal of the instrument, and the effect of 
wind introduce errors. The more rapidly the measurements can 
be made, consistent with careful manipulation, the better the 
results will be. If the transit is set up on shaky ground 
the transitman should avoid walking around his instrument. 

60. Repetition is useful not only to secure precision, but 
also as a check against mistakes. If a mistake is made on the 
first reading of an angle the vernier, on the second reading, 
falls in a new place on the circle so that the mistake is not 
likely to be repeated. It is common practice to repeat, or 
"double," all important angles and divide the second reading 
by 2 simply as a check on the first reading. 

61. TO LAY OFF AN ANGLE BY REPETITION. — There is 
no direct method of laying off an angle by repetition as in the 
case of measuring an angle, therefore the following indirect 
method is used. With the vernier set at 0° and the telescope 
sighted on the first point the angle is carefully laid off on the 
circle and the second point set in line with the new position of 
the telescope. Then this angle which has been laid off is 
measured by repetition as precisely as is desired as described in 
Art. 59. The resulting angle obtained by repetition is a more 
precise value than the angle first set on the vernier. The 
difference between this value and the angle desired is the cor- 
rection which should be made at the second point. This can be 
readily done by measuring approximately the distance from the 
instrument to the second point, and computing the perpendicular 
offset to be laid off at the second point. (The offset for an 
angle of one minute at a distance of 100 ft. is nearly 0.03 ft.) 



LAYING OFF AN ANGLE BY REPETITION 5 1 

62. RUNNING A STRAIGHT LINE — One Point Visible from 
the Other. — There are several ways in which a straight line 
may be fixed on the ground, depending upon the existing con- 
ditions. If the line is fixed by the two end points one of which 
is visible from the other, the method of setting intermediate 
points would be to set the transit over one point, take a " fore- 
sight " on the other and place points in line. For very exact 
work the instrument should be used in both the direct and re- 
versed positions (Art. 79, p. 61). This will eliminate errors of 
adjustment such as failure of the telescope to revolve in a true 
vertical plane, or failure of the objective tube to travel parallel 
to the line of sight. 

63. RUNNING A STRAIGHT LINE — Neither Point Visible 
from the Other. — If neither point can be seen from the other 
then it is necessary to find some point, by trial, from which the 
terminal points can be seen. The transit is set up at some point 
estimated to be on the line, a "backsight" is taken on one of 
the points and the instrument clamped. The telescope is then 
reversed on its horizontal axis. If the vertical cross-hair strikes 
the second point the instrument is in line ; if not, then the error 
in the position of the instrument must be estimated (or meas- 
ured) and a second approximation made. In this way, by suc- 
cessive trials, the true point is attained. The final tests should 
be made with the instrument in direct and reversed positions to 
eliminate errors of adjustment of the line of sight and the hor- 
izontal axis. To eliminate errors in the adjustment of the plate 
bubbles the plate level which is perpendicular to the line should 
be relevded just before making the second backsight and while 
the telescope is pointing in that direction. This can be more 
readily done if, when the transit is set up, one pair of opposite 
leveling screws is turned so as to be in the direction of the line ; 
then the other pair will control the level which is perpendicular 
to the line of sight. After one point has* been found by this 
method other points may be set as described in the previous 
article. 

Another method of running a line between two points one of 
which is not visible from the other would be to run what is called 
a random line as described in Art. 191, p. 169. 



52 MEASUREMENT OF ANGLES [Chap. m. 

64. Prolonging a Straight Line. — If a line is fixed by two 
points A ^nd B and it is desired to prolong this line in the direc- 
tion AB, the instrument should be set up at A, a sight taken 
on B and other points set in line beyond B. When it is not 
possible to see beyond B from Ay the transit should be set up at 
B and points ahead should be set by the method of backsighting 
and foresighting as follows. With the transit at -5 a backsight 
is taken on A and the instrument clamped. The telescope is 
inverted and a point set ahead in line. The process is repeated, 
the backsight being taken with the telescope in the inverted 
position. The mean of the two results is a point on the line AB 
produced. The transit is then moved to the new point, a back- 
sight is taken on B, and another point set ahead as before. 

In this last case, if a line is prolonged several times its own 
length by backsighting and foresighting, there is likely to be a 
constantly increasing error. In the first case, where the line is 
run continually toward a point known to be correct, the errors 
are not accumulating. 

65. Methods of Showing Sights. — If the point sighted is 
within a few hundred feet of the instrument, a pencil may be used 
and held vertically in showing a point for the transitman to sight 
on. Sighting-rods are used on long distances.* Where only 
the top of the rod or pole is visible a considerable error is intro- 
duced if it is not held plumb. A plumb-line is much more 
accurate for such work but cannot be easily seen on long sights. 
Under conditions where the plumb-line cannot be readily seen 
some surveyors use for a sight an ordinary white card held with 
one edge against the string or held so that the center of the 
card is directly behind the string. If the edge of the card is 
held against the string, the transitman must be extremely careful 
that he is sighting on the proper edge.f 

* It is desirable that the foresight should be of a color such that the cross- 
hair is clearly seen, and of a width such that the cross-hair nearly (but not quite) 
covers it. 

t It is common among some surveyors to use a two-foot rule for a sight. 
The rule is opened so that it forms an inverted V (A). The plumb-string is 
jammed into the angle of the A by pressing the two arms of the rule together. 
The rule is then held so that the plumb-string as it hangs from the rule appears to 
bisect the angle of the A. 

Another device b to attach to the plumb-line an ordinary fish-line float (shaped 



RUNNING A STRAIGHT LINE 53 

Whenever the instrument is sighted along a line which is to 
be frequently used or along which the transit is to remain sifted 
for any considerable time the transitman should if possible select 
some well-defined point which is in the line of sight, called a 
"foresight." If no definite point can be found one may be 
placed in line for his use. By means of this " foresight " the 
transitman can detect if his instrument moves off the line, and 
can set the telescope exactly " on line " at any time without re- 
quiring the aid of another man to show him a point on the line. 

66, Signals. — In surveying work the distances are frequently 
so g^eat that it is necessary to use hand signals. The following 
are in common use. 

''Right'' or ''Left:' — The arm is extended in the direc- 
tion of the motion desired, the right arm being used for a motion 
to the right and the left arm for a motion to the left. A slow 
motion is used to indicate a long distance and a quick motion a 
short distance. 

"Plumb the Pole.'* — The hand is extended vertically above 
the head and moved slowly in the direction it is desired to have 
the pole plumbed. 

" All Right," — Both arms are extended horizontally and 
moved vertically. 

" Give a Foresight." — The transitman, desiring a foresight, 
motions to the rodman, by holding one arm vertically above his 
head. 

" Take a Foresight." — The rodman desiring the transit- 
man to sight on a point, motions the transitman by holding one 
arm vertically above his head and then he holds his lining-pole 
vertically on the point. 

" Give Line." — When the rodman desires to be placed "on 
line " he holds his lining-pole horizontally with both hands over 
his head and then brings it down to the ground in a vertical 
position. If the point is to be set carefully, as a transit point, 

like a plumb-bob). This may be fastened so that its axis coincides with the 
string and so that it can be raised and lowered on the string. It should be painted 
with such colots that it can be seen against any background. 

The man showing the sight for the transitman should always try to stand so 
that the sun will shine on the object he is holding ; on long sights it is difficult 
(sometimes imposaUe) to see an object in a shadow. 



54 MEASUREMENT OF ANGLES [Chap. III. 

the rodman waves the top end of pole in a circle before bringing 
it to the vertical position. 

''Pickup the Transit:' -^V^en the chief of the party de- 
sires to have the instrument set at another point he signals to 
the transitman by extending both arms downward and outward 
and then raising them quickly. 

All signals should be distinct so as to leave no doubt as to 
their meaning. Care should be taken to stand so that the back- 
ground will not prevent the signals being distinctly seen. The 
palms of the hands should be shown in making the signals, and 
for distant signals a white handkerchief is often used Where 
much distant signaling is to be done flags are attached to the 
lining-poles. Special signals may be devised for different kinds 
of work and conditions. 

67. TO MEASURE A VERTICAL ANGLE. — In measuring a 
vertical angle with a transit, first point the vertical cross-hair 
approximately at the object, then set the horizontal cross-hair 
exactly on the point by means of the clamp and tangent screw 
controlling the vertical motion. Next read the vertical arc or 
circle. Then, without disturbing the rest of the transit, unclamp 
the vertical arc, and bring the telescope to the horizontal position 
by means of the level attached to the telescope, and the clamp 
and tangent screw of the vertical arc. When the telescope 
bubble is in the center read the vertical arc again. This gives 
the index correctiony to be added or subtracted according to 
whether the two readings are on opposite or on the same side of 
zero. In some forms of transit the vernier is on a separate arm 
which also carries a level. By bringing this level to the center 
of the tube by means of its tangent screw the index correction 
is reduced to zero each time and the true angle read directly. 
Instruments provided with this form of level have no level 
attached to the telescope. 

If the transit has a complete vertical circle errors in the ad- 
justment of the bubble and the horizontal cross-hair may be eli- 
minated by inverting the telescope, turning it through 180° 
azimuth, and remeasuring the angle. The mean of the two 
results is free from such errors. If the transit is provided with 
only a portion of a circle the vernier will be off the arc when 



PRECAUTIONS IN USE OF TRANSIT 55 

the telescope is inverted, consequently with a transit of this type 
the elimination cannot he effected. 

68. PRECAUTIONS IN THE USE OF THE TRANSIT. — In 
the preceding text several sources of error and also precautions 
against mistakes have been mentioned, but in order that the be- 
ginner may appreciate the importance of handling the instru- 
ment carefully he should make the following simple tests. 

1. Set the transit up with the three points of the tripod 
rather near together so that the instrument will be high and un- 
stable. Sight the cross-hair on some definite object, such as the 
tip of a church spire, so that the slightest motion can be seen. 
Take one tripod leg between the thumb and forefinger and twist 
it strongly; at the same time look through the telescope and 
observe the eflfect. 

2. Press' the tripod leg laterally and observe the effect on 
the level attached to the telescope ; center the bubble before 
testing. 

3. Step on the ground about i or 2 inches from the foot of 
one of the tripod legs and observe the effect on the line of sight. 

4. Breathe on one end of the level vial and observe the mo- 
tion of the bubble. 

5. Press laterally on the eyepiece and observe the effect on 
the line of sight. 

These motions, plainly seen in such tests, are really going on 
all the time, even if they are not readily apparent to the observer, 
and show the necessity for careful and skillful manipulation. 
The overcoat dragging over the tripod, or a hand carelessly rest- 
ing on the tripod, are common sources of error in transit work. 

Before picking up the transit center the movable head bring, 
the leveling screws back to their mid position, loosen the lower 
dampi and turn the telescope either up or down. 



56 



MEASUREMENT OF ANGLES 



[Chap. HI. 



ADJUSTMENTS OF THE TRANSIT. 

69. If an instrument is badly out of adjustment in all re- 
spects, it is better not to try to completely adjust one part at a 
time but to bring the instrument as a whole gradually into ad- 
justment. If this is done, any one process of adjusting will not 
disturb the preceding adjustments, the parts are not subjected 
to strains, and the instrument will be found to remain in adjust- 
ment much longer than it would if each adjustment were com- 
pleted separately. 

Nearly all adjustments of the transit, in fact of nearly all 
surveying instruments, are made to depend on the principle of 
reversion. By reversing the position of the instrument the effect 
of an error is doubled. 

70. ADJUSTMENT OF THE PLATE BUBBLES. — To adjust 
the Plate Levels so that Each lies in a Plane Perpendicular to the 
Vertical Axis of the Instrument. Set up the transit and bring 



— il 




Fig. 29. Adjustment of the Plate Bubbles. 



the bubbles to the center of their respective tubes. Turn the 
plate 180° about its vertical axis and see if the bubbles remain 
in the center. If they move from the center, half this distance 
is the error in the adjustment of the tube. (See Fig. 29.) The 
adjustment is made by turning the capstan-headed screws on the 



ADJUSTMENTS OF TRANSIT $7 

bubble tube until the bubble moves half-way back to the center 
as nearly as this can be estimated Each bubble must be ad- 
justed independently. The adjustment should be tested again 
by releveling and reversing as before, and the process continued 
imtil the bubbles remain in the center when reversed. When 
both levels are adjusted the bubbles should remain in the 
centers during an entire revolution about the vertical axis. 

7z. Adjustmeut of the Cross-Hairs. — ist. To put the 
Vertical Cross-Hair in a Plane Perpendicular to the Horizontal 
Axis. Sight the vertical hair on some well-defined point, and, 
leaving both plates clamped, rotate the telescope slightly about 
the horizontal axis (see Fig. 30). 

The point should appear to travel on the vertical cross-hair 
throughout its entire length. If it does not, loosen the screws 




Fig. 80* Adjustment of the Cross-Hairs (First Part). 

holding the cross-hair ring, and by tapping lightly on one 
of the screws, rotate the ring until the above condition is 
satisfied. Tighten the screws and proceed with the next 
adjustment* 



58 



MEASUREMENT OF ANGLES 



[Chap. III. 



7a. 2nd. To make the Line of Sight Perpendicular to the 
Horizontal Axis.* (See Fig. 31.) Set the transit over a point 






Fig. 81. Adjustment of the Cross-Hatrs (Second Part). 

A. Level up, clamp both plates, and sight accurately on a 
point B which is approximately at the same level as A. 
Reverse the telescope and set C in line with the vertical cross- 
hair. By A, and C should be in a straight line. To test this, 
turn the instrument about the vertical axis until B is again 
sighted. Clamp the plate, reverse the telescope, and observe if 
point Cis in line. If not, set point D in line just to one side 
of C and then the cross-hair ring must be moved until the 
vertical hair appears to have moved to poinf E, one-fourth the 
distance from D toward C, since, in this case, a double reversal 
has been made. 

The cross-hair ring is moved by loosening the screw on one 
side of the telescope tube and tightening the opposite screw. 
If D falls to the right of C then the cross-hair ring should be 
moved to the left ; but if the transit has an erecting eyepiece 
the cross-hair will appear to move to the right when viewed 
through the telescope. If the transit has an inverting eyepiece 
the cross-hair appears to move in the same direction in which 
the cross-hair is actually moved. 

The process of reversal should be repeated until no further 
adjustment is required. When finally adjusted, the screws 
should hold the ring firmly but without straining it. 

* In making the adjustment in the shop with collimators instrument makers 
seldom level the transit carefully. In field adjustments it is desirable, although 
not necessary, to level the instrument. The essential condition is that the veiticaJ 
axis shall not alter its position. 



ADJUSTMENTS OF TRANSIT 



59 



73. Abjustmeut of the STAHDARDS. — To make the Hori- 
zontal Axis of the Telescope Perpendicular to the Vertical Axis of 
the Instrument. (See Fig. 32.) Set up the transit and sight 
the vertical cross-hair on a high point A, 
such as the top of a church steeple. 
Lower the telescope and set a point B in 
line, on the same level as the telescope. 
Reverse the telescope, turn the instru- 
ment about its vertical axis, and sight 
on B, Raise the telescope until the point 
A is visible and see if the cross-hair comes 
on A. If not, note point C in line and 
at same height as A. Then half the dis- 
tance from C to -4 is the error of adjust- 
ment. Loosen the screws in the pivot 
cap and raise or lower the adjustable end 
of the horizontal axis by means of the 
capstan-headed screw under the end of 
the axis. Repeat the test until the high 
and the low points are both on the cross- 
hair in either the direct or reversed posi- 
tions of the transit. The adjusting screw 
should be brought into position by a right- 
hand turn, otherwise the block on which 
the horizontal axis rests may stick and 
not follow the screw. The cap screws 
should then be tightened just enough to 
avoid looseness of the bearing. 

74. Adjustment of the Telescope Bubble. — This is adjusted 
by the "/^g"" mefAod, or direct method^ as explained in Art. 128, 
p. 91. This consists in first determining a level line by using 
the instrument in such a way as to eliminate the error of the 
bubble, and then centering the bubble while the line of sight is 
horizontal. 

75* Adjustment of the Auxiliary Level on the Vernier of the 
Vertical Arc— (See Art. (yy^ p. 54.) To adjust the Level 
80 that it is in the Center of the Tube when the Line of Sight 
18 Level and the Vernier reads 0^ This is adjusted by the " peg 




Fig. 32. Adjustment 
OF THE Standards. 



6o MEASUREMENT OF ANGLES [Chap. IIL 

method" (Art. 128, p. 91). The bubble is first brought to 
the center of the tube by means of its tangent screw. Then the 
telescope is moved until the vernier of the vertical arc reads o^ 
The instrument is then in condition to be used as a leveling 
instrument and is adjusted by the "peg method." 

If the telescope is provided with an attached level the auxiliary 
level could be adjusted by comparing it with the telescope level 
as follows. Level the telescope by means of its attached level, 
make the vernier read o by means of the tangent screw of the 
vernier, and then bring the bubble of the auxiliary level to the 
center by means of its adjusting screws. 

76. Adjustment of the Vernier of the Vertical Circle. — To 
make the Vernier read 0^ when the Telescope Bubble is in the 
Center of the Tube. If there is any index error (Art. (>y, p. 54) 
bring the bubble to the center, loosen the screws holding the 
vernier, and tap lightly until the zeros coincide. Tighten the 
screws and test again. In some instruments the vernier is con- 
trolled by a slow-motion screw for setting the index at the zero 
of the circle. 

77. Adjustment of the Objective Slide. — To make the 
Objective Slide move Parallel to the Line of Sight. If the tube 
holding the objective.is adjustable it must be placed so that the 
direction of the line of sight will not be disturbed when the 
telescope is focused. The adjustment may be made as follows. 
Adjust tlie line of sight as in Art. 72, using very distant 
points. This will require the objective to be drawn in nearly 
as far as it will go and hence the position of the objective will 
be changed but little by any subsequent lateral adjustment of 
the tube. Next repeat the test for the adjustment of the line of 
sight by using two points which are very near the instrument. 
In sighting on these points the objective must be run out and 
any error in its adjustment will change the direction of the line 
of sight so that it is no longer perpendicular to the horizontal axis 
of the instrument. In case the instrument fails to stand this 
test the objective slide does not move parallel to the line of 
sight. The adjustment is made by moving the adjustment 
screws of the objective slide so as to apparently increase the 
error making, by estimation, one-quarter the correction required. 



HOW TO ELIMINATE ERRORS 6l 

The adjustment of the line of sight should be again tested on 
two distant points and the cross-hairs moved in case the second 
adjustment appears to have disturbed the first. 

78. Shop Adjustments. — The adjustment of the objective 
slide and other adjustments such as centering the eyepiece tube 
and centering the circles are usually made by the instrument 
maker. 

79. HOW TO ELIMINATE THE EFFECT OF ERRORS OF 
ADJUSTMEUT in THE TRANSIT. — Errors of adjustment in the 
plate bubble may be avoided by leveling up and reversing as 
when adjusting. Then, instead of altering the adjustment, 
simply move the bubble half-way back by means of the leveling 
screws. This makes the vertical axis truly vertical. Then the 
bubbles should remain in the same parts of their respective 
tubes as the instrument revolves about its vertical axis. 

Errors of the line of sight and errors of the horizontal axis 
are eliminated by using the instrument with the telescope in the 
direct and then in the reversed position and taking the mean of 
the results whether the work is measuring angles or running 
straight lines. 

Errors of eccentricity of the circle are completely eliminated 
by reading the two opposite verniers and taking the mean. 

Errors of graduation of the circle are nearly eliminated by 
reading the angle in different parts of the circle or by measur- 
ing the angle by repetition. 

80. Care of Instruments. — A delicate instrument like the 
transit requires constant care in order that the various parts 
may not become loose or strained. Care should be taken that 
the tripod legs do not move too freely, and that the metal shoes 
on the feet of the tripod do not become loose. The transit 
should be securely screwed to the tripod. In caring for the 
lenses a camel's hair brush should be used for dusting them and 
soft linen with alcohol for cleaning them. The objective should 
not be unscrewed except when absolutely necessary, and when 
replaced it should be screwed in to the reference mark on the 
barrel of the telescope. Grease should never be used on exposed 
parts of an instrument, as it collects dust. Care should be 
taken not to strain the adjusting screws in making adjustments. 



62 MEASUREMENT OF ANGLES [Chap. HI. 

The instrument should be protected as much as possible from 
the sun, rain, and dust. If the instrument is carried in the box 
it is less likely to get out of adjustment than when carried on 
the shoulder, but the former is often inconvenient. It is cus- 
tomary in traveling by carriage or rail to carry the transit in its 
box. While being carried on the shoulder the lower clamp 
should be left undamped so that in case the instrument strikes 
against anything, some parts can give easily and save the instru- 
ment from a severe shock. When the transit is in use, be care- 
ful not to clamp it too hard, but clamp it firmly enough to in- 
sure a positive working of the tangent screws and so that no 
slipping can occur. 

8i. Common sources of error in Transit work. — 

1. Nonadjustment, eccentricity of circle, and errors of gradu- 
ation. « 

2. Changes due to temperature and wind. 

3. Uneven settling of tripod. 

4. Poor focusing (parallax). 

5. Inaccurate setting over point. 

6. Irregular refraction of atmosphere. 



82. Common MISTAKES IN TRANSIT WORK. — 

1. Reading in the wrong direction from the index on^ a 
double vernier. 

2. Reading the opposite vernier from the one which was set 

3. Reading the circle wrong, e.g., reading 59° for 61°. If 
the angle is nearly 90^ reading the wrong side of the 90 
point, e.g., 88° for 92°. 

4. Using the wrong tangent screw. 



64 



SOLAR ATTACHMENT 



[Chap. UI. 




Fig. 33. Solar Attachment to Transit. 

(The authors are indebted to C. L. Berger & Son 
for the photograph from which this cut was made.) 



SOLAR ATTACHMENT 65 



THE SOLAR ATTACHMBIIT. 

83. DESCRIPTIOH OF SOLAR ATTACHMEHT. — One of the 
most important auxiliaries to the engineer's transit is the solar 
attachment, one form of which is shown in Fig. 33. This is a 
small instrument which may be attached to the telescope and by 
means of which a true meridian line can be found by an observa- 
tion on the sun. In the form here shown the principal parts are 
the polar axis^ which is attached to the telescope perpendicular 
to the line of sight and to the horizontal axis, and a small 
telescope which is mounted on the polar axis. This telescope 
can be revolved about the polar axis and can be inclined to it 
at any desired angle. The polar axis is provided with four ad- 
justing screws for making it perpendicular to the line of sight 
and to the horizontal axis. 

Another form of attachment has the solar telescope replaced 
by a lens and a screen on which the sun's image can be thrown. 
This defines a line of sight and is in reality the equivalent of a 
telescope. This instrument is provided with the arc of a circle 
known as the declination arc, the use of which will be explained 
later. 

Still another form consists of a combination of mirrors (simi- 
lar to those of a sextant) which can be placed in front of the 
objective. In this form the telescope of the transit serves as the 
polar axis. 

While these various solar attachments differ in the details 
of construction, they all depend upon the same general prin- 
ciples. 

84. THE Celestial sphere. — In order to understand the 
theory of this instrument it will be necessary to define a few 
astronomical terms. Fig. 34 represents that half of the celes- 
tial sphere which is visible at one time to an observer on the 
surface of the earth. For the purposes of this problem the 
celestial sphere may be regarded as one having its center at 
the center of the earth and a radius equal to the distance of the 
sun from the earth. The sun in its apparent daily motion 
would then move around in a circle on the surface of this 



66 



MEASUREMENT OF ANGLES 



[Chap. HI. 



sphere. The circle NES IV is the observer's horizon and is the 
boundary between the visible and invisible parts of the celestial 
sphere. The point Z is the zenith and is the point where a 
plumb-line produced would pierce the celestial sphere. The 
circle SZPN is the observer's meridian and is a vertical circle 
through the pole. The circle EQ W is the celestial equator. 
The circle AMB^ parallel to the equator, is a parallel of decli- 
nation^ or the path described by the sun in its apparent daily 




Fig. 34. Diagram of the Celestial Hemisphere. 



motion from east to west. The sutis declination is its angular 
distance from the equator, or the arc OT. The declination is 
considered positive when north and negative when south. The 
polar distance of the sun is the complement of the declination 
represented by the arc OP 

85. OBSERVATION ON THE SUN FOR- MERIDIAN WITH 
SOLAR Attachment. — If the polar axis of the instrument is 
made to point to the celestial pole, i.e., made parallel to the 
earth's axis, then the small telescope can be made ' to follow 
the sun in its daily path by simply giving it an inclination to the 
polar axis equal to the sun!s polar distance and revolving it 
about the polar axis. 



OBSERVATION FOR MERIDIAN WITH SOLAR 6/ 

(i) To find the true meridian by an observation on the sun 
first make the angle between the polar axis and the solar tele- 
scope equal to the sun's polar distance at the time of the obser- 
vation. This is done by turning the solar telescope into the 
same plane ss the main telescope by sighting both on some dis- 
tant object, and then making the angle between the two telescopes 
equal to the sun's declination. Some instruments are provided 
with a declination arc upon which the declination angle can be 
laid off directly. Others have a small spirit level attached to 
the small telescope, in which case the vertical circle of the tran- 
sit is used for laying off the declination angle. Incline the main 
telescope until the reading of the vertical circle equals the de- 
clination, and clamp ; then level the solar telescope by means of 
the attached level. The angle between the polar axis and the 
solar telescope is then 90° plus or minus the reading of the 
vertical circle. 

(2) By means of the vertical cfa-cle of the transit incline the 
polar axis to the vertical by an angle equal to the co-latitude of 
the place, which is 90° minus the latitude. The polar axis now 
has the same angle of the elevation as the celestial pole. 

(3) If the observation is in the forenoon, place the solar 
telescope on the left of the main telescope (on the right if in the 
afternoon); then, by moving the whole instrument about the 
vertical axis and the solar telescope about the polar axis, point 
the solar telescope at the sun. The sun's image is brought to 
the center of the square formed by foifr cross-hairs, or ruled 
lines, in the solar telescope. The final setting is made by the 
tangent screw controlling the horizontal motion of the transit 
and the one controlling the motion of the solar about the polar 
axis. Only one position can be found where the solar telescope 
will point to the sun. In this position the vertical axis points 
to the zenith, the polar axis to the pole, and the solar telescope 
to the sun. The instrument has thus solved mechanically the 
spherical triangle having these three points (Z, /*, O) as vertices. 
The horizontal angle between the two telescopes is equal to the 
sun's true bearing. Since the solar telescope is pointing to the 
sun the main telescope must be in the plane of the meridian. 
If all of the work has been correctly done it will be observed 



68 MEASUREMENT OF ANGLES [Chap.HI. 

that the sun's image will remain between the cross-hairs set 
parallel to the equator, and therefore the sun can be followed 
in its path by a motion of the solar telescope alone. If it is 
necessary to move the instrument about the vertical axis to 
point the solar telescope again at the sun this shows that the 
main telescope was not truly in the meridian. 

After the meridian has been determined the main telescope 
may then be lowered and a point set which will be due north or 
due south of the instrument. 

86. Computation of Declination Settings. — The sun's polar 
distance may be obtained from the " American Ephemeris and 
Nautical Almanac/' published by the Government. The polar 
distance is not given directly, but its complement, the sun's 
apparent declination^ is given for each day and for the instant 
of Greenwich Mean Noon, The rate of change of the declina- 
tion, or the difference for I hour^ is also given. In order to use 
this for any given locality, it is first necessary to find the local 
or the standard time corresponding to mean noon of Greenwich. 
In the United States, where standard time is used, the relation 
to Greenwich time is very simple. In the Eastern time belt 
the time is exactly 5 hours earlier than at Greenwich ; in the 
Central^ 6 hours earlier ; in the Mountain, 7 hours earlier ; in 
the Pacific, 8 hours earlier. If a certain declination corresponds 
to Greenwich mean noon, then the same declination corresponds 
to 7 A.M. in the Eastern belt or 6 A.M. in the Central belt, 
etc. The declinatioif for any subsequent hour of the day may 
be found by adding (algebraically) the difference for i hour mul- 
tiplied by the number of hours elapsed. Declinations marked 
North must be regarded as positive and those marked South as 
negative. An examination of the values of the declination for 
successive days will show which way the correction is to be 
applied. It will be useful also to remember that the declination 
is 0° about March 21, and increases until about June 22, when 
it is approximately 23*^ 27' North ; it then decreases, passing the 
o*^ point about September 22, until about December 21 when it 
is approximately 23° 27' South; it then goes North until 
March 2 1 when it is o® again. 

After the correct declination is found it has still to be cor- 



OBSERVATION FOR MERIDIAN WITH SOLAR 69 

rected for refraction of the atmosphere. The effect of refrac- 
tion is to make the sun appear higher up in the sky than it 
actually is. In the northern hemisphere, when the declination 
is North this correction must be added, when South, subtracted ; 
or algebraically it is always added. 

The refraction correction may be taken from Table VII, 

p. 507. 

The co-latitude which must be set off on the vertical circle 
may be obtained from a map or may be determimed by an ob- 
servation which is made as follows. Set off the sun's .declination 
for noon, as for any other observation, the two telescopes being 
in the same vertical plane, and point the small telescope at the 
sun. By varying the angle of elevation of the main telescope, 
keep the solar telescope pomting at the sun until the maximum 
altitude is reached. The angle read on the vertical circle is the 
co-latitude (see also Art. 217, p. 195). 

Example. 

Latitude 40*»N. Longitude 4b 45m W. 

Jan. 10, 1900. 
Declination for Greenwich mean noon 21^ 59' 04^' 
Difference for ih + 22''.25 

TiMB. DBCX.INATION. RbPRACTION. SbTTINO. 

7h. A.M. 21° 59' 04" 

8 58 42 S'40" 21° 53' 02" 

9 58 20 2 51 21 55 29 

10 57 57 2 07 21 55 50 

11 57 35 I 51 21 55 44 

12 M. 57 13 (I 47) (21 55 26) 

1 P.M. 56 51 I 51 21 55 00 

2 56 28 2 07 21 54 21 

3 56 06 2 51 ^ 21 53 15 

4 55 44 5 40 21 50 04 

87. "^ Comstpck's Method of finding the Refraction. — Set 

the vertical cross-hair on one edge (or limi) of the sun and note 
the instant by a watch. Set the vernier of the plate 10' ahead 
and note the time when the limb again touches the cross-hair. 

* See Bulletin of the University of Wisconsin, Science Series, VoL I, No. 3. 



70 MEASUREMENT OF ANGLES [Chap. m. 

Call the number of seconds between these observations n. Read 
the altitude A. Then the refraction in minutes will be nearly 

, ^ 2000 

equal to — ; 

An 

88. Observation for meridian should not be made when the 
sun's altitude is less than about lo^ because the refraction cor- 
rection will be unreliable. Observations near noon are to be 
avoided because a slight error in altitude produces a large error 
in the resulting meridian. For good results therefore the obser- 
vation should be made neither within an hour of noon nor near 
sunrise or sunset. 

89. MISTAKES m USING THE SOLAR ATTACHMEHT. — 

1. Solar on wrong side of main telescope. 

2. Refraction correction appUed wrong way. 

ADJUSTMENTS OF THE SOLAR ATTACHMENT. 

go. Adjustment of polar axis. — To make the Polar 
Axis Perpendicular to the Plane of the Line of Sight and the Hori- 
zontal Axis. Level the transit and the main telescope. Bring the 
bubble of the solar telescope to the center of its tube while it is 
parallel to a pair of opposite adjusting screws which are at the 
foot of the polar axis. Reverse the solar telescope 1 80° about 
the polar axis. If the bubble moves from the center position, 
bring it half-way back by means of the adjusting screws just 
mentioned and the other half by means of the tangent screw 
controlling the vertical motion of the solar. This should be 
done over each pair of opposite adjusting screws and repeated 
until the bubble remains central in all positions. 

91. ADJUSTMENT OF THE CROSS-HAIRS. — To make the 
Vertical Cross-Hair truly Vertical. Sight on some distant point 
with all the clamps tightened and, by means of the tangent 
screw controlling the vertical motion of the solar, revolve the 
solar telescope about its horizontal axis and see if the vertical 
cross-hair remains on the point. If not, adjust by rotating the 
cross-hair ring, as described in Art. 71, p. 57. 



ADJUSTMENTS OF SOLAR ATTACHMENT J I 

92. adjustmeut of Telescope bubble. — To make the 
Axis of the Bubble Parallel to the Line of Sight. Level the main 
telescope and mark a point about 200 ft. from the instrument in 
line with the horizontal cross-hair. Measure the distance be- 
tween the two telescopes and lay this off above the first point 
which will give a point on a level with the center of the solar 
telescope. Sight the solar at this point and clamp. Bring the 
bubble to the center by means of the adjusting screws on the 
bubble tube. 



PROBLEMS. 

1. Is ^ necessary that the adjustments of the transit should be made in the 
order given in this chapter ? Give your reasons. 

2. A transit is sighting toward B from a point A, In setting up the transit at 
A it was carelessly set 0.0 1 ft. directly to one side of A, as at A'. What would 
be the resulting error, i.e., the difference in direction (^in seconds) between AS and 
A'B, (i) when AB — 40 ft., (2) when i^^ — 1000 ft.? 

3. An angle of 90° is laid ofif with a ** one minute " transit, and the angle then 
determined by six repetitions, the final reading being 179° 58'+ 360®. The point 
sighted is 185 feet from the transit. Compute the offset to be laid off in order to 
correct the first angle. Express the result in feet and also in inches. 

4. An angle measured with a transit is 10° 15' 41'^^ The telescope of a level- 
ing instrument is placed in front of the transit (with its objective toward the 
transit) and the angle again measured and found to be o^ 18' 22^'. What is the 
magnifying power of this level telescope ? 

5. Compute the declination setting for every hour when observations on the 
sun for meridian can be made at Boston ( Lat. 42^ 21' N, Long. 71^ 04' 30'' W) 
on each of the following dates. 

January i, 1906. 

DecL S 23° 03' 27".9 

Dift. for I hour, +ii ''.70 
April 16, 1906. 

Decl. N 9*> sy 34".2 

Diff. for I hour, + 53''.44 
July 2, 1906. 

Decl. N 23° 05' 49''.5 

Diff. for 1 hour, — io''.39 

Sept. 25, 1906. 

Decl. S o*> 35' 49".4 
Diff. for I hour, -58''.5i 



CHAPTER IV. 

MBASURBMBNT OF DIFFERIINCB OF BLBVATION. 

93, LEVEL SURFACE. — A level surface is a curved surface 
which at every point is perpendicular to the direction of gravity 
at that point, such, for example, as the surface of still water. 
Any line of sight which is perpendicular to the direction of 
gravity at a given point is therefore tangent to the level sur- 
face at that point and is called a horizontal line. 

94. The Spirit LeveL — In nearly all instruments the direc- 
tion of gravity is determined by means of either a plumb-line or 
a spirit level. A spirit level is a glass tube, the inside of which 
is ground to a circular curve longitudinally, and nearly filled 
with a liquid such as alcohol or ether, leaving enough space to 
form a bubble. The grinding is usually done only on the inside 
upper surface of the tube. The radius of the curve varies accord- 
ing to the use which is to be made of the level ; a very short ra- 
dius makes a slow moving bubble while a long radius makes a 
very sensitive bubble. It is important that the curve should be 
exactly circular so that equal distances on the tube should sub- 
tend equal angles at the center. The level is provided with a 
scale of equal parts, which may be either a metallic scale screwed 
to the brass case holding the glass bubble tube, or it may con- 
sist of lines etched on the glass itself. A point near the middle 
of the tube is selected as the zero point and the graduations are 
numbered both ways from that point. The straight line tangent 
to the curve at the zero point of the scale is called the axis of 
the bubble. The position of the bubble in the tube is deter- 
mined by noting the positions of both ends. The bubble will 
change its length with changes in temperature, consequently the 
reading of one end is not sufficient to determine the position of 
the bubble. On account of the action of gravity the bubble will 
always move toward the higher end of the tube ; hence, when 
the bubble is central the axis of the tube is horizontal. 

72 



THE WYE LEVEL 73 

95. Angular Value of One Division of the Level Tube. — The 

angular value of one division of a level tube is the angle, usually 
expressed in seconds, through which the axis of the tube must 
be tilted to cause the bubble to move over the length of one di- 
vision on the scale. The simplest way of finding this in the 
field consists in moving the bubble over several divisions on the 
scale by means of the leveling screws and observing the space on 
a rod passed over by the horizontal cross-hair, the rod being 
placed at a known distance from the instrument. The space on 
the rod divided by the distance to the rod gives the natural tan- 
gent of the angle through which the line of sight has moved. 
Since the angle is very small its value in seconds of arc may be 
obtained by dividing its tangent by the tangent of one second, 
(log tan i" = 4.6855749 — 10). Dividing the angle found by 
the number of divisions of the scale passed over on the bubble 
tube, gives a result which is the average number of seconds 
corresponding to a single division. 

In a properly constructed leveling instrument the value of 
one division of the level should have a definite relation to the 
magnifying power of the telescope. The smallest angular move- 
ment that can be detected by the level bubble should correspond 
to the smallest movement of the cross-hairs that can be detected 
by means of the telescope. 

THE LEVEL. 

96. The instruments chiefly used for the direct determina- 
tion of differences of elevation are known as the Wye Levels the 
Dumpy Level, and the Hand Level, The Precise Level differs 
in its details from the others but does not really constitute a 
diflferent type ; it is essentially a wye level or a dumpy level, 
according to the principle of its construction. The engineer's 
transit, which has the long level attached to the telescope, is 
frequently used for direct leveling. All of these instruments 
are so constructed that the line of sight is horizontal when the 
bubble of the attached spirit level is in the middle of its tube. 

97. THE WYE LEVEL. — In the wye level (Figs. 35 and 36) 
the spirit level is attached to the telescope tube which rests in 



74 



MEASUREMENT OF ELEVATION 



[Chap. IV. 




two Y shaped bearings from 
which it derives its name. Those 
parts of the telescope which bear 
on the wyes are made cylin- 
drical and are called fi'n^s or 
pivots. The telescope is held in 
the wyes by means of two clips. 
The level is attached to the 
telescope by means of screws 
which allow vertical and lateral 
adjustments. The two wye sup- 
ports are secured, by means of 
adjusting screws, to a horizontal 
bar which is attached rigidly at 
right angles to a spindle, or 
vertical axis, similar to that of 
a transit. The instrument is 
provided with leveling screws, 
clamp, and tangent screw, but 
has no shifting head nor plumb- 
line attachment. The whole 
upper portion of the instrument 
is screwed to a tripod in the 
same manner as a transit. The 
characteristic feature of the wye 
level is that the telescope can 
be lifted out of its supports, 
turned end for end and replaced, 
each ring then resting in the 
opposite wye. 

98. THE DUMPY LEVEL.— 
In the dumpy level (Fig. 37) 
the telescope, the vertical sup- 
ports, the horizontal bar and the 
vertical spindle are all made in 
one casting or else the parts are 
fastened together rigidly so as to 
be essentially one piece. The 



75 



3 

5 



H 

X 



w 

r 
w 

< 
w 




76 




Q 



c 



THE DUMPY LEVEL jy 

spirit level is fastened to the horizontal bar and can be adjusted 
in the vertical plane ; there is no other adjustable part except 
the cross-hair ring. 

99. Comparison of Wye and Dumpy Levels. — The wye level 
has long been a favorite in this country, chiefly on account of 
the ease with which it can be adjusted, which depends upon the 
fact that when the telescope is reversed in the wye supports the 
line through the centers of the pivots is exactly coincident with its 
first position. While this feature of the wye level is of practical 
advantage in adjusting the instrument it is based on the assump- 
tion that both pivots are circular and of exactly the same diame- 
ter, which may or may not be true. For, even supposing the 
pivots to be perfect when new, they" soon wear, and perhaps 
unevenly, and consequently the method of adjusting by reversal 
will then fail and the '^peg'' adj'tistmenty or direct method, 
must be used. (See Art. 128, p. 91.) It is not uncommon to find 
a wye level of excellent manufacture which, after being adjusted 
by reversals, jfails to stand the test by the direct method, but 
which is capable of excellent work when adjusted by the latter 
method. 

The dumpy level has very few movable parts, and conse- 
quently it does not easily get out of adjustment even when 
subjected to rough usage.* Furthermore the recent work of the 
United States Coast and Geodetic Survey with a new precise 
level, which is really a dumpy level with certain refinements, 
indicates the superiority of the dumpy form for the most precise 
work. 




Fig. 88. The Locke Hand Level. 

100. The Locke HAITO level. — The hand level (Fig. 
38) has no telescope, but is simply a metal tube with plain glass 

• See Reports of the Superintendent of the U. S. Coast and Geodetic Sur- 
vey for the year i89S-99» p. 351, and the year 1900, p. 525. 



78 MEASUREMENT OF ELEVATION [Chap. IV. 

covers at the ends and with a spirit level on top. When look- 
ing through the tube one sees the level bubble on one side 
of the tube in a mirror set at 45° with the line of sight, 
and the landscape on the other side. In order that the eye may 
see the bubble and the distant object at the same instant the 
instrument is focused on the bubble by means of a lens placed 
in a sliding tube. The level line is marked by a horizontal wire, 
which can be adjusted by means of two screws. The instrument 
is held at the eye and the farther end is raised or lowered until 
the bubble is in the center of the tube. At this instant a point 
in line with the horizontal wire is noted. In this way approxi- 
mate levels may be obtained. 



LEVELING RODS. 

loi. According to their construction rods are either Self- 
reading or Target rods, or a combination of the two. Self -read- 
ing rods are those which can be read directly from the instrument 
by the levelman whereas target rods can be read only by the 
rodman. The commonest forms of leveling rods are known as 
the Bostofty the New Yorky and the Philadelphia rods. (See 

Fig. 39.) 

102. BOSTON ROD. — The Boston rod (Fig. 39) is a target 
rod of well seasoned wood about 6 J ft. long, made in two strips, 
one of which slides in a groove in the other. A target is 
fastened rigidly to one of these strips about 0.3 ft. from one 
end. Clamps are provided for holding the two parts in any 
desired position. There is a scale on each side of the rod, one 
starting from either end, graduated to hundredths of a foot and 
each with a vernier placed about the height of the eye and read- 
ing to thousandths of a foot. When the rod-reading is less than 
5.8 ft. the rod is first placed on the ground with the target near 
the bottom. Then the strip carrying the target is raised to the 
proper height while the bottom of the other strip rests on the 
ground, as shown in Fig. 39. For readings over 5.8 ft. the rod 
is turned end for end so that the target is at the top and can be 
moved from 5.8 to 11. 4 ft., the limit of the rod. The terms 



■4 

i 

6 

i 

4 




1 



7 



^ 






J 



h 



e 



« 



I i 



Fig. 89. Leveling Rods. 



8o 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



^^ short rod'' and '^ long rod'' are used to distinguish these two 
positions. 

The common form of target used on the Boston rod is shown 
in Fig. 40. Instead of this target one of a design similar to 
that in Fig. 41 is sometimes used, in which the white strip in 
the center may be bisected by the horizontal cross-hair. 
Bisection is more precise under all conditions than setting on a 





Fig. 40. Boston Rod Target. 



Fig. 41. Bisection Target. 



single line or on the division line between two surfaces of 
different color. 

A serious objection to the Boston rod is that in reversing it 
(changing from long to short rod) any error in the position of 
the target with reference to the scale is doubled by the reversal, 
and such an error is not readily eliminated. 

103. NEW YORK ROD. — The New York rod (Fig. 39) 
consists of two strips of wood, arranged similarly to those of the 
Boston rod. Unlike the latter the target on the New York 
rod is movable. For "short rod" the target is moved up or 
down on the rod until the proper height is reached. The face 
of the rod is graduated to hundredths of a foot. The vernier is 
on the target itself and reads to thousandths of a foot. The 
graduations on the rod cannot be read from the instrument except 
at short distances. For "long rod*' the target is set at the 
highest graduation, usually 6.5 ft., and clamped to one of the 
sliding strips which is then raised until the target is in the right 
position. A clamp is provided for holding the two strips 
together. The reading for "long rod'* is found on the side of 
the strip that is raised, and opposite the vernier which is on the 



LEVELING RODS "* 8 1 

Other strip, the scale reading downward In this case the rod 
cannot be read directly from the instrument. 

104. PHILADELPHIA ROD. — This rod has the graduations 
plainly painted on its face so that it can be used as a self-reading 
rod (Fig. 39). It has also a target the arrangement of which is 
similar to that of the New York rod. The target has no vernier 
but is graduated directly to 0.005 ft- 5 ^^^ thousandths can be 
readily estimated. The rod is extended in the same manner as 
the New York rod, and it can be read to 0.005 ft. and estimated 
to o.ooi ft. by means of a scale fastened on the back of the rod. 
When the rod is fully extended, the graduations on the front 
face are continuous and the readings can be made directly by 
the levelman if desired. 

105. SPECIAL SELF-READING RODS. — There are a large 
number of self-reading rods of special design. One of the com- 
monest types shown in Fig. 39, is similar to the Philadelphia 
rod except that it has no target and is not graduated closer than 
tenths. The figures on the face of the rod are made of definite 
height (0.06 or 0.08 ft.) and of definite thickness (0.0 1 or 0.02 
ft.) so that it is easy for the levelman to estimate the readings 
to hundredths of a foot. These rods are usually constructed 
so that they can be extended for "long rod " readings. 

106. Tape Rod.* — The tape rod (Fig. 39) is a self-reading 
rod of decidedly different design from the Philadelphia rod. It 
is a wooden rod made in one piece with a metal roller set in it 
near each end. Passing over these rollers is a continuous steel 
band 20 ft. long and o. i ft. wide, on the outside of which for 
its entire length is painted a scale graduated to feet, tenths, and 
half-tenths, with the details of the numbers so designed that 
readings to the nearest 0.0 1 ft. can readily be made. Unlike the 
other rods mentioned the scale reads down on the face of the 
rod instead of up. It is provided with a clamp so that the metal 
band, or tape, can be set at any desired reading and held firmly 
in that position. The use of this type of rod is limited to cer- 



• This rod was invented by Thomas F. Richardson and is used extensively 
by the Metropolitan Water and Sewerage Board of Boston, Mass. 



82 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



tain kinds of work, its advantagjB being the time saved in calcu- 
lations as explained in Art. 228, p. 206. 

107. Precise Level Rod — The self-reading rod used by 
the U. S. Coast and Geodetic Survey is made of a single piece 
of wood, soaked in paraffin to prevent changes in length due to 
moisture. Metal plugs are inserted at equal distances so that 
changes in length can be accurately determined. It is divided 
into centimeters, painted alternately black and white. The bot- 
tom of the rod carries a foot-plate. The meters and centimeters 
are read directly and the millimeters estimated. This rod has 
attached to it a thermometer, and a level for plumbing. 

108. Advantages of the Self-Reading Rod. — While the ad- 
vantage in the speed with which leveling can be accomplished 
by use of the self-reading rod is well understood, it is also true 





Fig. 42. Rod Levels. 



although not so generally recognized that very accurate results 
can be obtained. For any single reading the error may be 
larger than with the target rod, but the errors of estimating 
fractional parts are compensating, so that in the long run the 
results are found to be very accurate. Precise leveling carried 
on by the U. S. Coast and Geodetic Survey and by European 
surveys has demonstrated the superiority of such rods. The 



SELF-READING RODS 83 

self-reading rod might to advantage be more generally used than 
it is at present. 

109. Attachments to the Rod for Plumbing. — In accurate 
work it will be convenient to use some device for holding the 
rod plumb. Spirit levels attached to brass ** angles*' which may 
be secured to a comer of the rod are very convenient. Two 
patterns are shown in Fig. 42. In some rods the levels are set 
permanently into the rod itself. 

no. Effect of Heat and Moisture. — Changes of temperature 
do not have a serious effect on rods since the coefficient of ex- 
pansion of wood is small. The effect of moisture is greater, 
however, and consequently if very accurate leveling is to be 
done the rod should be compared frequently with a standard. 
Rods soaked in paraffin are less affected by moisture than those 
which have not been so treated. 

USE OF THE LEVEL AlTD ROD. 

111. In order to obtain the difference in elevation between 
two points, hold the rod at the first point and, while the instru- 
ment is level, take a rod-reading. This is the distance that the 
bottom of the rod is below the line of sight of the level. Then 
take a rod-reading on the second point and the difference 
between the two rod-readings is the difference in elevation of 
the two points. 

112. To LEVEL THE INSTRUMENT. — Set up the instru- 
ment in such a position that the rod can be seen when held on 
either point and at such height that the horizontal cross- 
hair will strike somewhere on the rod. In setting up the level, 
time will be saved if the habit is formed of doing nearly all of 
the leveling by means of the tripod legs, using the leveling 
screws only for slight, motions of the bubble in bringing it to the 
middle of the tube. Turn the telescope so that it is directly 
over two opposite leveling screws. Bring the bubble to the 
center of the tube approximately; then turn the telescope until 
it is over the other pair of leveling screws and bring the bubble 
exactly to the center. Move the telescope back to the first 
position and level carefully, and again to the second position if 



84 MEASUREMENl OF ELEVATION [Chap. IV. 

necessary. If the instrument is in adjustment and is properly 
leveled in both directions, then the bubble will remain in the 
center during an entire revolution of the telescope about the 
vertical axis. The instrument should not be clamped ordinarily, 
but this may bie necessary -under some circumstances, for ex- 
ample, in a strong wind. 

113. To TAKE A ROD-READING. — The rodman holds the 
rod on the first point, taking pains to keep it as nearly plumb as 
possible. The levelman focuses the telescope on the rod, and 
brings the bubble to the center while the telescope is point- 
ing at the rod, because leveling over both sets of screws will not 
make the bubble remain in the center in all positions unless the 
adjustment is perfect. If a target rod is used, the target should 
be set so that the horizontal cross-hair bisects it while the bubble 
is in the center of the tube. It is not sufficient to trust the 
bubble to remain in the center; it should be examined just 
before setting the target and immediately afterward, at every read- 
ing. The levelman signals the rodman to move the target up or 
down. When the center of the target coincides with the hori- 
zontal cross-hair the levelman signals the rodman, " All right " 
(Art. 115), and the rodman clamps the target and reads the 
rod. This reading is then recorded in the note-book. In ac- 
curate work the levelman should check the position of the target 
after it has been clamped to make sure that it has not slipped 
in clamping. For readings to hundredths of a foot it is not neces- 
sary to clamp the target ; the rodman can hold the two parts of 
the rod firmly together while he reads it. 

While the levelman is sighting the target, the rodman should 
stand beside the rod so that he can hold it as nearly vertical as 
possible in the direction of the line of sight. The levelman can 
tell by means of the vertical cross-hair whether it is plumb in . 
the direction at right angles to the Ime of sight. It is extremely 
important that the rod be held plumb. Vertical lines on build- 
ings are a great aid to the rodman in judging when his rod is 
plumb. If the wind is not blowing the rodman can tell when 
the rod is plumb by balancing it on the point. 

114. Waving the Rod. — In careful work when the "long 
rod '* is used it may be plumbed in the direction of the line of 



USE OF LEVEL AND ROD . 85 

sight by *^'waving the rod'' To do this the rodman stands 
directly behind the rod and inclines it toward the instrument so 
that the target will drop below the line of sight. He then slowly 
draws it back, causing the target to rise. It will be highest 
when the rod is plumb. If at any point the target appears 
above the cross-hair it should be lowered. If, while the rod is 
being waved, the target does not reach the cross-hair the target 
must be raised and the process repeated until as the rod is 
waved there appears to be just one place where the target coin- 
cides with the horizontal line of sight. Whenever close results 
are desired it will be well to take several readings on each point 
and use the mean. 

115. Signals. — While the rodman is seldom very far away 
from the levelman in this work still it is often convenient to use 
hand signals. The following are commonly used in leveling. 

" Up'' or ^^ Down." — The levelman motions to the rodman 
by raising his arm above his shoulder for an upward motion and 
dropping his arm below his waist for a downward motion. A 
slow motion indicates that the target should be moved a consid- 
erable amount and a quick motion indicates a short distance. 

"-4// Right." — The levelman extends both hands hori- 
zontally and waves them up and down. 

** Plumb the Rod." — The hand is extended vertically above 
the head and moved slowly in the direction it is desired to have 
the rod plumbed. 

^^ Take a Turning Point." — The arm is swung slowly in a 
circle above the head. 

^^ Pick up the Level." — When a new set-up of the level is 
desired the chief of party signals the levelman by extending 
both arms downward and outward and then raising them quickly. 

Some surveyors use a system of signals for communicating 
the rod-readings, but mistakes are liable to be made unless great 
care is used. 

116. DIFFERENTIAL LEVELING. — Differential leveling is the 
name given to the process of finding the difference in eleva. 
tion of any two points. In Art. 1 1 1 the simplest case of differ- 
ential leveling is described. When the points are far apart the 
instrument is set up and a rod-reading is taken on the first point. 



86 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



This is called a backsight ox plus sight and is usually written B, 5. 
or + 5. Next the rod is taken to some well-defined point which 
will not change in elevation (such as the top of a firm rock) 
and held upon it and a reading taken. This is called ?l foresight 
or minus sight and is written F. S, or — 5. The difference be- 
tween the two readings gives the difference in elevation between, 
this new point and the first point. This second point is called a 
turning point and is written T, P. The level is next set up in 
a new position and a backsight taken on the turning point. A 




Diagram Illustrating Differential Leveling. 



new turning point is then selected and a foresight taken upon it. 
This process is continued until the foresight is taken on the final 
point. The elevation of the last point above the first is equal to 
the sum of all the backsights minus the sum of all the foresights. 
If the result is negative, i.e., if the sum of the foresights is the 
greater, then the last point is below the first. The form of notes 
for this work is shown below, and the fieldwork is illustrated by 
Fig. 43- 



DIFFERENTIAL LEVELING 



87 



POIMT 


B. S. 


F.S. 


Remarks 


A. 
T.P. 
T. P. 
T.P. 

B. 


8.160 
7.901 
9.446 
8.005 


2.404 
3.070 
6.906 
2.107 


Highest point on stone bound, S. W. cor. X and Y Sts. 
N. £. cor. stone step No. 64 M St. 




33.512 
14.487 


14.487 





Diff. 19.025 B above A. 

117. The Proper Length of Sight. — The proper length of 
sight will depend upon the distance at which the rod appears 
distinct and steady to the levelman, upon the variations in read- 
ings taken on the same point, and upon the degree of precision 
required. Under ordinary conditions the length of sight should 
not exceed about 300 ft. where elevations to the nearest o.oi ft. are 
desired. " Boiling " of the air due to irregular refraction is fre- 
quently so troublesome that long sights cannot be taken accurately. 

If the level is out of adjustment the resulting error in the rod- 
reading is proportional to the distance from the instrument to the 
rod. If the level is at equal distances from the rod the errors are 
equal and since it is the difference of the rod-readings that gives 
the difference in elevation, the error is eliminated from the final 
result if the rodman makes the distance to the point where the 
foresight is taken equal to the distance to the backsight by count- 
ing his paces as he goes from one point to the other. 

118. Effect of the Earth's Curvature and of Refraction on 
Leveling. — Since the surface of the earth is very nearly spherical, 




Fig. 44. Diagram Illustrating Effect of Earth's 
AND OF Refraction. 



Curvature 



any line on it made by the intersection of a vertical plane with the 
earth's surface is practically circular. In Fig. 44 the distance 
AA' varies nearly as AJV (see foot-note, p. 339). The effect of 



88 MEASXmEMENT OF ELEVATION [Chap. IV. 

the refraction of the atmosphere is to make this offset from the 
tangent appear to be A'B which is about one-seventh part smaller 
than A' A, This offset, corrected for refraction, is about 0.57 
ft. in one mile ; for 300 ft. it is 0,002 ft. ; for 500 ft., 0.005 ft. ; 
for 1000 ft., 0.020 ft. If the rod is equally distant from the 
instrument on the foresight and backsight the effect of curva- 
ture and refraction is eliminated from the result. 

119. PRECAUTIONS IN LEVEL WORK. — Nearly all of the 
precautions mentioned in Art. 68, p. 55, for the transit instru- 
ment, are also applicable to the level. Care should be taken 
not to strike the rod on the ground after it has been clamped 
and before it has been read. 



ADJUSTMENTS OF THE LEVEL. 
I. ADJUSTMENTS OF THE WYE LEVEL. 

120. ADJUSTMENT OF THE CROSS-HAIRS. — (a) To make the 
Horizontal Cross-Hair truly Horizontal when the Instrument is 
Leveled. This may be done by rotating the cross-hair ring as in 
the case of the transit (Art. 71, p. 57), if the instrument is so 
constructed that the telescope cannot be rotated in the wyes. 
In many instruments the telescope can be rotated in the wyes. 
In some levels the telescope is always free to rotate in the 
wyes, while others are provided with a stop regulated by an ad- 
justing screw, which prevents the telescope from rotating beyond 
a certain point. 

The instrument is leveled and some point found which is 
covered by the horizontal cross-hair. The telescope is turned 
slowly about the vertical axis so that the point appears to traverse 
the field of view. If the point remains on the cross-hair the ad- 
justment is perfect. If it does not, then an adjustment must be 
made, the manner of doing this depending upon the construction 
of the instrument. If the telescope cannot be rotated in the 
wyes the adjustment is made by rotating the cross-hair ring, 
similar to the adjustment described in Art. 71, p, 57. If the tele- 
scope has a stop-screw this must be moved until the instrument 



ADJUSTMENTS OF THE WYE LEVEL 



89 



satisfies this test. If the telescope can rotate freely in the wyes 
it can be turned by hand until it satisfies the test. Since there 
is nothing to hold the telescope in this position the adjustment 
in the last case is likely to be disturbed at any time. 

121. (b) When the above adjustment is completed the Line 
of Sight should be made to Coincide with the Axis of Pivots, or 
Parallel to it. (See Fig. 45.) Pull out the pins which hold the 
clips on the telescope and turn the clips back so that the telescope 
is free to turn in the wyes. Sight the intersection of the cross- 
hairs at some well-defined point, using the leveling screws for the 
vertical motion and the clamp and tangent screw for the hori- 
zontal motion. Then rotate the 

telescope 180° in the wyes, so that 
the level tube is above the tele- 
scope. The intersection of the 
cross-hairs should still be on the 
point. If not, move the horizontal 
cross-hair half-way back to its first 
position by means of the upper and 
lower adjusting screws of the cross- 
hair ring. Then move the vertical 
cross-hair half-way back to its first 
position by the other pair of screws. 
Repeat the test until the adjust- 
ment is perfect. 

122. Adjustment of the level Tube. — To make the 
Line of Sight and the Level Tube Parallel to Each Other. Two 
methods are used, — the direct ^ or '^peg,'* method and the in- 
direct method. While the former is the only one applicable to 
the dumpy level either one can be used for the wye level, 
although the indirect method is the simpler. 

123. ADJUSTMENT OF THE LEVEL TUBE BY INDIRECT 
HETHOD. — (a) To put the Axis of the Bubble Tube in the Same 
Plane with the Line of Sight. Bring the bubble to the center of 
the tube and rotate the telescope in the wyes for a few de- 
grees (very little is necessary) ; if the bubble moves toward one 
end of the tube that end must be the higher, which indicates 
the direction in which the adjustment should be made. Move 



/ £nd.ta 


|WI_ 


Cro 


k-holr \ 


1 ThifPb, 


»ton 


Ciw 


i-Mr 1 


I let PbtJi 


on 


Crw 


W 1 


\ 




■~7 


*• POINT 1 


X t 




J 


v^ 


lJ 




^ 



Fig. 46. Adjustment of 
THE Cross-Hairs (Sec- 
ond. Part). 



90 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



the screws controlling the lateral movement of the tube until the 
bubble returns to the center. Test the adjustment by rotating 
the telescope each way. 

124* (b) To make the Axis of the Bubble Tube and the Line of 
Sight Parallel to Each Other. First damp the instrument (over a 
pair of leveling screws), then bring the bubble to the center of 
the tube, lift the telescope out of the wyes, turn it end for end 
and set it down in the wyes, the eye end now being where the 
objective was originally. (See Fig. 46.) This operation must 
be performed with the greatest care, as the slightest jar of the 
instrument will vitiate the result. If the bubble returns to the 
center of the tube, the axis of the tube is in the correct position. 
If it does not return to the center, the end of the tube provided 



LIHE or SIGHT 




Fig. 46. Adjustment of the Bubble Tube by Indirect Method. 

with the vertical adjustment should be moved until the bubble 
moves half-way back to the center. This test must be repeated 
to make sure that the movement is due to defective adjustment 
and not to the jarring of the instrument. 

125. ADJUSTMENT OF THE WYES. — To make the Axis of 







Fig. 47. Adjustment of the Wyes. 



ADJUSTMENTS OF THE DUMPY LEVEL 9 1 

the Level Tube Perpendicular to the Vertical Axis of the Instrument. 
Bring the two dips down over the telescope and fasten them. 
Level the instrument, bring the bubble precisely to the middle 
of the tube over one set of leveling screws, and then turn the 
telescope 180° about the vertical axis. If the bubble moves 
from the center bring it half-way back by means of the adjusting 
screws at the foot of one of the wye supports. (See Fig. 47.) 
Since the bubble is brought to the center of the tube each 
time a rod-reading is taken this last adjustment in no^ way affects 
the accuracy of the leveling work but is a convenience and a 
saving of time. 



ZZ. ADJUSTMEIVTS OF THE DUMPY LEVEL. 

126. ADJUSTMENT OF THE CROSS-HAIR. — If the horizon- 
tal cross-hair is not truly horizontal when the mstrument is level 
it should be made so by rotating the cross-hair ring as described 
in the adjustment of the transit, Art. 71, p. 57. 

227. Adjustment of the bubble Tube. — To make the 
Axis of the Bubble Tube Perpendicular to the Vertical Axis. Owing 
to the construction of the dumpy level it is necessary to make 
this adjustment before making the line of sight parallel to the 
bubble tube. It is done by centering the bubble over one pair 
of leveling screws, and turning the instrument 180° about the 
vertical axis. If the bubble does not remain in the center of the 
tube, move it half-way back to the center by means of the adjust- 
ing screws on the level tube. 

128. The direct, or « Peg,»» Adjustment. —To make the 
Line of Sight Parallel to the Axis of the Bubble. (See Fig. 48.) 
Select two points A and B, say, 200 ft. or more apart. Set up 
the level close to A so that when a rod is held upon it the eye- 
piece wDl be only about a quarter of an inch from the rod. 
Look through the telescope wrong end to at the rod and find the 
reading opposite the center of the field. After a little experience 
it will be found that this can be done very accurately. From 
the fact that only a small portion of the rod is visible it will be 
foimd convenient to set a pencil-point on the rod at the center of 



92 MEASUREMENT OF ELEVATION [Chap. IV. 

the small field of view. Turn the telescope toward B and take 
a rod-reading on it in the usual way, being certain that the bub- 
ble is in the middle of the tube. The difference between these 
two rod-readings is the difference of elevation of the two points 
plus or minus the error of adjustment. The level is next taken to 
B and the above operation is repeated. The result is the differ- 
ence in elevation minus or plus the same error of adjustment. 
The mean of the two results is the true difference in elevation of 
points A and J5. Knowing the difference in elevation between 
the two points and the height of the instrument above B the rod- 
reading at A which will bring the target on the same level as 
the instrument may be computed. The bubble is brought to the 
center of the tube and the horizontal cross-hair raised or lowered 
by means of the adjusting screws on the cross-hair ring until the 
line of sight strikes the target. In this method the small error 
due to curvature of the earth (nearly o.ooi ft. for a 200-ft. sight) 
has been neglected. 

Example. (See Fig. 48.) 

Instrument at A. 

Rod-reading on A a 4.062 

Rod-reading on B— 5.129 

Diff . in elev. of A and B » i .067 
Instrument at B. 

Rod-reading on B « 5.076 

Rod-reading on A « 4.127 

Diff. in elev. of B and A « 0.949 

Mean of two diff. in elev. = — '-^^ — 1.008 true diff. in elev. 

2 

Instrument is now 5.076 above B. 

Rod-reading at A should be 5.076 — 1.008 » 4.068 to give a level sight. 

The peg method may be used for adjusting the wye level or 
the transit, the difference being that in the dumpy level the axis 
of the bubble tube is first made horizontal and then the line of 
sight is brought parallel to it, while in the wye level and in the 
transit the line of sight is first made horizontal and then the axis 
of the bubble tube is made parallel to it. Consequently, in the 
former case the cross-hair ring is moved in adjusting whereas in 
the latter case the adjustment is made in the bubble tube. This • 



THE PEG ADJUSTMENT 



93 



adjustment in its simplest form is described in the following 
article. 




Fig. 48. Peg Adjustment. 



129. Adjustment of the Locke haitd level. — In ad- 
justing the hand level the principle of the peg adjustment is 
used. The level is placed at a mark A (Fig. 49) and another 
mark B in line with the cross-hair is made, say, 100 ft. away. 




51?^-^ 




Fig. 49. Peg Adjustment for Hand Level. 

when the bubble is in the middle. The level is then taken to 
By held so that its center is at the height of this mark, and 
sighted toward the first point. A third point C is marked in 



94 MEASUREMENT OF ELEVATION [Chap. IV. 

line with the cross-hair when the bubble is in the middle. The 
point midway between A and C is at the same level as B. The 
adjustment is made by screws which move the horizontal wire. 

130. COMMON SOURCES OF ERROR IN LEVELING.— 

1. Improper focusing (parallax). 

2. Bubble not in middle of tube at instant of sighting. 

3. Rod not held plumb. 

4. Foresights and corresponding backsights on turning 

points not equally distant from the instrument. 

5. Poor turning points selected. (See Art. 224, p. 202.) 

131. COMMON MISTAKES.— 

1. Foresight and Backsight not taken on exactly the same 

point. 

2. Neglecting to set target accurately when " long rod " is 

used. 

3. In the use of the self-reading rod neglecting to clamp the 

rod at the proper place when "long rod '* is used. 

4. Reading the wrong foot-mark or tenth-mark. 

5. In keeping notes, — getting F. S. in B. S. column or 

vice versa, 

6. In working up notes, adding F. S. or subtracting B. S. 

PROBLEMS. 

1. A wye level was tested for the sensitiveness of the bubble, as follows : 
the rod was held on a point 200 ft. away ; the bubble was moved over 13.6 divi- 
sions of the scale ; the rod-readings at the two extreme positions of the bubble 
were 4.360 and 4.578. Compute the average angular value of one division of the 
level. 

2. A dumpy level was tested by the peg method with the following results. 

Instrument 2X A\ — Instrument at i9 : — 

B. S. on Ay 4.139 B. S. on B^ 3.900 

F. S. on B, 4.589 F. S. on A^ 3.250 

Find the rod-reading on A to give a level line of sight, the instrument remaining 
3.900 above B, Was the line of sight inclined upward or downward ? How 
much ? 



ERRORS AND MISTAOIS IN LEVELING 95 

3. The target on a Boston rod has been disturbed and it is desired to find oat 
if the target is in the correct position with reference to the scale. Describe a 
method by which the amount of this error can be determined. 

4. A New York rod is found to be 0.002 ft. short, due to wear on the brass 
foot-plate. Explain what effect this will have in finding the difference in eleva- 
tion between two points. 

5. (a). A level is set up and a B. S. of 5.0^8 is taken on a point 400 ft. away, 
then a F. S. of 3.260 is taken on a point 900 ft. away. What is the curvature and 
refraction correction ? What is the difference in elevation of the two points ? 

(b). In another case a B. S. of 8.266 was taken on a point 100 ft. away 
and a F. S. of 6.405 taken on a point 600 ft. away. What is the curvature and 
refraction correction ? What is the difference in elevation of the two points ? 



PART II. 
SURVEYING METHODS. 



PART II. 

SURVEYING METHODS. 

CHAPTER V. 

LAND SXTRVBTINO. 

132. SURVEYING FOR AREA. — In surveying a field for the 
purpose of finding its area the instruments and methods 
used will be determined largely by the degree of accuracy 
required. If it is permissible to have an error in the area of, 
say, 0.5 per cent then the compass and chain may be used. If 
accuracy much greater than this is required it will be necessary 
to use the transit and the steel tape. At the present time, 
however, in nearly all work except surveys of farms and wood- 
lands, the transit is used even under conditions where the 
compass would give the required accuracy. 

In surveying a field all the angles and lengths of the sides 
are determined consecutively, the survey ending at the point 
from which it was started. Then by trigonometry the position of 
the final point or of any other point with relation to the starting 
point can be readily calculated. If the survey were absolutely 
accurate the last point as calculated would coincide with' the 
first, but this condition is never attained in practice. The 
calculated distance between the two, divided by the perimeter 
of the field, is usually called the error of closure ;* it is often 
expressed in the form of a fraction in which the numerator is 
unity. In surveying with a compass and chain the error of 

closure expected is about i part in 500, expressed as 

133. SURVEYING FOR AREA WITH COMPASS AND CHADI. — 
If the area alone is desired the surveyor's 4-rod chain will be 

* The term error of closure more properly applies to the actual distance by 
which the survey fails to close, but as this is generally expressed in the form of a 
fraction the term has commonly been applied to the latter. 

99 



lOO 



LAND SURVEYING 



[Chap. V. 



convenient on account of the simple relation existing between 
the square chain and the acre (Art. 4, p. 3). In making a survey 
enclosing an area it is customary to begin at some convenient 
comer and to take the bearings and the distances in order 
around the field. As the measurements are made they are 
recorded in a field note-book. It is not necessary to take the 
sides in order, but since they must be arranged in order for the 
purpose of computing the area it will be convenient to have them 
so arranged in the original notes. If the length and bearing 
of any side are omitted the area is nevertheless completely 
determined (Art. 397, p. 366), but as these two measurements 
furnish a valuable check on the accuracy of all the measurements 



(Lbpt-Hand Pagb) 



(Right-hand Pagb) 



r Sry^cfmKxtLarofJB^Snm, Abrmttm^Akes. 


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F?™r:T7?l^-^HlE±rJ 






Avaorits 


A 


Om£ 


MXlkW 


17.1s 






$t^9i^Shms atrM^mM,,8Jmifm m^ l.^khardlutr. 


B 


MS^£ 


AfSfirr 


asj 






Pir—Stvmp 


C 


Nl'^E 


MTiw 


XJ6 






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snw 


/axT 






Cedar 6M.S'S£ «• 4«y* oa*. 


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6fon€ beoai/, £. skM Pfim S^. 


F 


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Fig. 60. Notes op Chain and Compass Survey. 



they* never should be omitted if they can be taken. It is of the 
utmost importance in every survey that check measurements 
should be taken. Even a few rough checks taken in the field 
which will require only a little extra time often prove to be of 
great value in detecting mistakes. Both a forward bearing and 
a back (or reversed) bearing should be taken at each comer ; 
from these the angle at a corner can be obtained free from error 
due to any local attraction of the needle. The' above process 
gives a series of connected straight lines and their bearings 
(or the angles between them), which is called a traverse. 

It is often impossible to set the compass up at the comers 
of the property, and in such cases assumed lines running 
parallel or approximately parallel to the property lines can be 



SURVEY BY A TRAVERSE lOI 

surveyed as described in Art. 134, and its area determined 
In some cases the compass can be set on the property line at 
an intermediate point and the bearing obtained, but the sur- 
veyor must be sure that there is no local attraction of the needle 
at this point. All points where the compass is set should 
be marked and described so that they can be found again. If 
any instrument point is not otherwise defined it may be tem- 
porarily marked by a small stake and several reference 
measurements made from this stake to prominent objects 
nearby, so that its position can be relocated if the stake is lost. 
These measurements are called ties. 

Notes of the traverse are usually recorded as shown in 
Fig. 50. 



SURVEY OF FIELD WITH TRABSIT AND TAPE. 

134. SURVEY OF A FIELD BY A TRAVERSE. — Surveying 
a field for area can usually be done in one of the three following 
ways. 

(i). By setting up the transit at the corners of the property 
and measuring the angles directly ; the distances being measured 
directly along the property lines. 

(2). When the property lines are so occupied by buildings 
or fences that the transit cannot be set up at the corners, but 
the distances can still be measured along the property lines, then 
the angles at the comers are obtained by measuring the angles 
between lines which are parallel to the property lines. 

(3). If the boundaries of the property are such that it is not 
practicable to set the transit up at the comers nor to measure 
the distance directly on the property lines, a traverse is run 
approximately parallel to the property lines and these lines con- 
nected with the traverse by means of angles and distances. 

135. In the case (2) the parallel lines are established in 
the following manner. Set the transit up at some point E 



I02 LAND SURVEYING [Chap. V. 

(Fig. 51) within 2 or 3 ft. of the comer A. Establish the line EF 
parallel to AD by making DF = AH by trial. Point H cannot 
be seen through the telescope, but it is so near the instrument 
that by means of the plumb-line on the transit it can be accu- 
rately sighted in by eye. Similarly EG is established parallel to 
AB. Then the angle FEG is measured ; and this is the property 



f7K 



« H r 

Fig. 61. Transit Lines Parallel to the Sides of Field. 

angle at^^. It is evident that the values of Alfsnd DFznd of 
/4/and BG are of no permanent use and are therefore not recorded 
in the notes. When practicable it is advisable to choose the 
transit point, K for example, on one of the property lines or its 
prolongation. Fig. 52 is a set of notes illustrating either 
case (I) or (2). 

136. In the case (3) the transit can be set up at an arbi- 
trary point marked by a stake and chosen far enough from one 
of the corners so that the telescope can be focused on it. In 
this way all the comers of the traverse are chosen so that the 
traverse will be approximately parallel to the sides of the field. 
The angles and distances of this traverse are then measured. 
To connect the property lines with this traverse, angles and dis- 
tances are measured to the respective corners of the property 
before the instrument is moved to the next point. Fig. 53 is a 
set of notes illustrating this case. Time can be saved in the 
computations and a good check on the work may be obtained if 
the property lines are also measured when possible. These are 
not only useful as checks on the accuracy of the survey, but the 



SURVEY BY A TRAVERSE 



103 



Kitkr , 

J.H.Brtiolley£skrie<tinhn,/V.Y ^nty ^ 

Bearing cfCF iromphn ofFbrk Com.CiSvtMericlian) 
Z^oe .oosiooshorf: 
IboihjH ^ 

^^ 207,42(f> " 



■/ 




/ihwYorkfbrkCom. 



t 




L.FBrown 




L.F.Brown 



Fig. 62. Notes of Survey with Transit and Tape. 



ro4 



LAND SURVEYING 



[Chap. V. 



Survey o/Land o/S/'/as Cokm/yr, Ai/rcrt//^ Af///Sj Ak. 



kVeJ/s 




yiUloyN 



Fig. 68. Notes of Survey with Transit and Tape. 



SURVEY BY SINGLE SET-UP IO5 

length of the sides will be needed in giving a description of the 
property. 

These three methods which have been described may be 
combined in any survey according to circumstances. 

137. Irregular Curved Boundaries. — When a tract of land is 
bounded by an irregular curved line such as a brook it is custom- 
ary to run the traverse line near it, sometimes crossing it several 
times, and to take perpendicular offsets to the brook. If it is a 
winding brook with no distinct turns in it, offsets at regular in- 
tervals are measured from the transit line as in the portion near 
point A of Fig. 53. Near point B in this figure the brook has 
practically a direct course between its turns, in which case the 
proper measurements to make are the offsets to those points 
where the course of the brook changes and the distances 
along the transit line between these offset lines. Since they 
are usually short the right-angle offset lines are laid oflf by 
eye. 

138. SURVEY OF A FIELD BY A SINGLE SET-UP OF 
TBE TRANSIT. — When it is necessary to economize time in the 
field at the expense of accuracy and of the time required to calculate 
the survey the following method may be used. If possible set up 
at a point within the field, preferably near the middle, from which 
all the corners can be seen, and measure the angles and distances to 
each corner. In this way the field is divided into several oblique 
triangles in each of which two sides and the included angle have 
been measured and from these the area and third side (property 
line) can be computed. As a check on the measured angles 
their sum should be 360° ; there is no check on the property 
lines unless they are measured directly. 

This method of surveying a field may be employed as a check 
on one of the other methods which have already been described, 
but is not recommended as a method to be used by itself except 
in emergencies. The weak point in it is the low degree of pre- 
cision with which the angles are usually measured. Here the 
effect of an error of, say, 30 seconds in an angle may often be 
much larger than the errors in the measured distances (Art. 352, 
P- 325). The additional measurement of the property line gives 
the length of all three sides of the various triangles into which 



I06 LAND SURVEYING [Chap. V. 

the field is divided. If the area is calculated from the three 
sides of the triangles, using the measured angles as checks only, 
an accurate result may be obtained, but at the expense of con- 
siderable office work. 

139. SURVEY OF A FIELD WITH A TAPE ONLY. — Some- 
times it may be necessary to survey a field when a transit is not 
at hand. This can be done by dividing the field into several tri- 
angles and measuring all their sides. To insure accuracy of re- 
sults the triangles should be so chosen that there are no angles 
in them less than 30*^ or greater than 1 50*^. This method will 
require a large amount of computation if the angles as well as 
the area of the field are desired. Lining in by eye will give ac- 
curate results in distances along the line, but only approximate 
side measurements can be obtained from such a line. 

140. Selecting the Comers. — If a corner is marked by a 
stone bound the exact point may be easily found ; but where it is 
simply defined as the intersection of stone walls or fences the 
surveyor will have to examine all evidence as to its position and 
use his judgment in deciding where the true corner is located 
(Art. 151, p. 116). When the property is bounded by a public 
way or a town boundary such data relating to the location of these 
lines must be obtained from the proper local authorities. After 
determining the position of the corner points, the surveyor should 
use precisely the same points in all distance or angle measure- 
ments. If stakes are used the exact point is marked by a small 
tack driven into the top of the stake. 

In deciding upon the location of the boundary lines from an 
examination of artificial features it should be borne in mind that 
it is customary to build fences or walls along highways entirely 
on private property so that the face of the wall or fence is on the 
side line of the highway. In cities the base-board of a fence is 
usually built so that its face is on the street line, but the location 
of the fences has no weight when the street line is defined by 
stone bounds or other permanent marks (Art. 253, p. 227). For 
boundaries between private lands the legal line is, in the case of 
a stream, the thread (not necessarily the center) of the stream ; 
the center of the stone wall or Virginia rail fence ; the line be- 
tween the bottom stringer and the boarding or pickets of an 



TIES 107 

ordinary fence, the fence-posts being entirely on one side of the 
boundary line. Not infrequently woodland is marked off by 
blazing the trees on one or both ^des of the boundary line, the 
blazing being done on the side of the tree nearest the boundary 
line. If a tree comes directly on the line it is blazed on both 
sides where the line strikes it. A small pile of stones, sometimes 
with a stake in the center of the pile, is often used to mark the 
corners of such land. 

141. Method of Procedtire. — In deciding where the traverse 
shall be run the surveyor should keep in mind both convenience 
in fieldwork and economy in office work. Frequently a method 
of procedure which shortens the time spent in the field will 
greatly increase the amount of the labor in the office. Circum- 
stances will determine which method should be used. If there 
is no special reason why the time in the field should be shortened, 
the best arrangement of the traverse will be th e one that will 
make the computation simple, and hence mistakes will be less 
liable to occur. If the lines of the traverse coincide with the 
boundary, as in cases (i) and (2), the amount of office work will 
be the least. If in case (3) the traverse lines are approximately 
parallel and near to the boundaries of the property the computa- 
tion of the small areas to be added to or subtracted from the 
area enclosed by the traverse is simplified to some extent. 

142. Ties. — All important points temporarily marked by 
stakes should be " tied in," i.e., measurements should be so taken 
that the point may be readily 
found or replaced in the future. 

There should be at least three \ $c^.6'Wrch 

horizontal ties which intersect at ^ .t^^^ 

angles not less than 30®. They 
should be taken from easily rec- 
ognized definite points, such as 
blazed trees, stone bounds, fence 

posts, or buildings. All such . 

measurements should be care- Xu^ujj, 

fuDy recorded, usually by means p,^ ^ approximate Ties. 
of a sketch Fig. 54 shows a 

stake located by ties measured to tenth of a foot ; these are taken 





io8 LAND SURVEYING [Chap, V. 







2^^^ simply to aid in finding the 

stake. 

It is often desired to take the 
ties so that the exact point can 
be replaced. In such cases the 
^\^3,^»^ surveyor should mark carefully 

"^•"•^ • by tack or crow-foot the exact 

Fig. 66. Exact Ties. points from which measure- 

ments (taken to -j-Jtt ^^0 ^^^ 
made, and record the entire information in the notes as shown 
in Fig. 55. 

143. Meastirement of the Angles of the Traverse. — The ' 
angles of the traverse may be measured in any one of three ways; 
by measuring the interior angle ^ by measuring the deflection angle^ 
which is the difference between the intierior angle and i8o^ or 
by measuring the azimuth angle. 

In practice the deflection angle is measured directly by sight- 
ing back on the previous point with the vernier at o® and the 
telescope inverted, then revolving the telescope about its hori- 
zontal axis to the direct position and turning the upper limb to 
the right or left until the next point is sighted. The deflection 
angle as recorded in the notes is marked /? or Z to indicate 
whether the telescope was turned to the right or left. It is evi- 
dent that a single measurement of the deflection angle is affected 
by any error in the adjustment of the line of sight as well as of 
the standards. If the deflection angle is "doubled " by turning 
to the backsight with the instrument direct and the angle re- 
peated a check on the angle is obtained and the errors of adjust- 
ment are also eliminated (Art. 79, p. 61). Where this procedure 
is followed it will be convenient to make the first backsight with 
the instrument direct so that when the second foresight is taken 
the instrument will again be in the direct position and ready for 
lining in. 

144. Measurement of Azimuth Angles. — By the azi- 
muth method the angles are measured as follows. The transit 
is set up at a point A (Fig. 56), the vernier set at o^ the tele- 
scope turned until it points to the south, and the lower plate 
clamped. Either the true or the magnetic south may be used. 



MEASURING ANGLES OF TRAVERSE 



109 



but if neither is known any arbitrary direction may be assumed. 
The upper clamp is loosened and the telescope sighted on B, 
The angle read on the vernier is the azimuth of ABy the circle 
being read in a clockwise direction (Art. 24, p. 16). The tran- 
sit is next moved to B, 

The azimuth of BC may be obtained in one of two ways. 
( I ) Invert the telescope and backsight on Ay the vernier remain- 
ing at the reading it had at A ; then clamp the lower plate, turn 
the telescope to its direct 
position, and sight on C. 
The angle on the vernier is 
the azimuth of BC referred 
to the same meridian as the 
azimuth of AB, The disad- 
vantage of this method is 
that the error of collimation 
enters the azimuth angle each 
time. (2) (Add 180° to the 
azimuth of AB^ set this off 
on the vernier, and sight on 
A, The telescope may then 
be turned directly to C (with- 
out inverting) and the azi- 
muth of BC can be read directly on the vernier. The disad- 
vantages of this method as compared with the former are that 
the error of eccentricity of the circle enters, that time is con- 
sumed in setting the veniier at each set-up of the instrument, 
and that there is an opportunity for mistakes in calculating and 
in making the setting on the vernier. 

In the azimuth method the angles of the traverse are checked 
by the fieldwork. After point D has been occupied, the transit 
is again set up at A and the azimuth of AB determined from a 
backsight on D. This azimuth of AB should agree with the 
original azimuth of this line. In ordinary land surveying the 
azimuth method of measuring the angles is little used. 

145. Checking the Fieldwork. — The transit is set over the 
selected points and the angles between the adjacent lines 
measured. If the work is not to be of unusual precision a 




Fig. 66. Azimuth Angles. 



I lO LAND SURVEYING [Chap. V. 

transit reading to one minute will be sufficient. A single 
measurement will give the angle with sufficient precision, but 
as it is important in all cases to have a check on the work it is 
advisable to "double the angle** (Art 60, p. 50), even though 
it is not necessary to use this method for the purpose of pre- 
cision. Referring to Fig. 52, p. 103, it will be seen that the 
angles were quadrupled where the sides were long, and doubled 
where they were short. In this case the angles were repeated 
to obtain greater precision. 

As an additional check against large errors in the angles, the 
magnetic bearing of each line should be read, thus enabling one 
to detect mistakes greater than a quarter of a degree and to 
guard against reading Right for Left in deflection angles. These 
bearings also show the approximate directions of the lines of the 
survey. This check should always be applied in the field so that 
any mistake in reading the angles can be rectified before leaving 
the work. This may be done by calculating each angle from the 
observed bearings of the adjacent sides ; or by starting with one 
observed bearing (assumed to be correct), calculating the other 
bearings in succession by means of the measured angles, and 
noting whether the observed bearings agree approximately with 
the calculated bearings. 

After the angles have been measured, the accuracy of the 
transit work may be tested by adding them together. The sum 
of the interior angles of the field should equal («-2) X 180°, 
where n is the number of sides in the field. If the deflection 
angles are used the sum of all the rigbt deflections should differ 
from the sum of all the left deflections by 360**, or in other 
words, the algebraic sum of the deflection angles should be 
3600. 

It is frequently important to check the distances before 
leaving the field. If there is any doubt as regards the correctness 
of the measurement of a line it should be remeasured, preferably 
in the opposite direction, so that the same mistake will not be re- 
peated. (See line AB in Fig. 52, p. 103.) If the traverse lines 
do not coincide with the boundaries, an independent check is 
obtained by measuring along the boundaries as well as on the 
traverse line, as in Fig. 53, p. 104. This furnishes at once a 



CHECKING FIELD WORK III 

rough check on the distances in the field and a close check after 
the survey has been calculated. It is often advisable to run a 
line across the traverse, especially when there are many sides 
to the field, thus dividing the field into two parts, as in 
Fig. 52, p. 103. If any mistake has been made it is then 
possible to tell in which portion of the traverse it occurred 
(Art. 407, p. 370- 

146. Accuracy Required. — In order that the accuracy m the 
measurement of distances shall be consistent with that of the angles 
it is necessary that great care should be exercised in holding the 
tape horizontal, in the plumbing, in the aligning, and in securing 
the proper tension. 

If the angles are measured to the nearest minute and the dis- 
tances to the nearest tenth of a foot, it will be sufficiently accu- 
rate to use sighting-rods in "giving line." The error of closure 
of such a survey should be not greater than j^j^^, but would sel- 
dom be less than j-^^^-f^ (Art. 132, p. 99). 

If the property is very valuable, as in the case of city build- 
ing lots, it is well to use a transit reading to 30" or 20". The 
angles should be repeated, not only as a check against mistakes, 
but to increase the precision of the measurement (Art. 59, p. 48). 
The tape measurements should be made with special care, and 
should be taken to the nearest hundredth of a foot. In the best 
work the temperature correction should be applied, a spring bal- 
ance should be used to give the right pull on the tape, the cor- 
rection to the standard distance should be determined (Art. 241, 
p. 216), the alignment given with the transit, and great care 
taken in plumbing. Sights are given by holding a pencil vertically 
on top of the tack on the stake or by plumb-line (Art. 65, p. 52). 
In this work it is important that the property line should be fol- 
lowed, when possible, to insure the most accurate results. In 
such work an error of closure of -^-qI-q^ or better is expected 
It is customary on most city work to neglect the effect of tempera- 
ture and to omit the use of the spring balance, the pull being 
carefully judged. This sort of work should give results as close 
as ^-a^^rs* ^^^ ^^ accuracy of :j^^^j is sometimes reached. 

147. Oi'ganization of Transit Party. — Transit surveys can be 
readily carried on by a party of three men. The note keeper 



112 LAND SURVEYING [Chap. V. 

who is in charge of the party directs the entire work ; the transit- 
man who has the instrument always in his care sets It up where 
directed by the note keeper, reads the angles and gives line when 
desired; the chainman generally acting as head-chainman and the 
note keeper as rear-chainman, measure all distances. 

148. Note Keeping. — All measurements should be recorded 
in a special note-book as soon as they are made and never left to 
be filled in from memory. The notes should be neat and in clear 
form so that there will be no doubt as to their meaning. Great 
care should be taken so that they shall not be susceptible of any 
interpretation except the right one. They are generally re- 
corded in pencil, but they should always be regarded as 
permanent records and not as temporary memoranda. As other 
persons who are not familiar with the locality will probably use 
the notes and will depend entirely on what is recorded, it is 
very important that the notes should contain all necessary data 
without any superfluous information. If the note keeper will 
bear in mind constantly how the survey is to be calculated or 
plotted it will aid him greatly in judging which measurements 
must be taken and which ones are unnecessary. Clearness is of 
utmost importance in note keeping, and to attain it the usual 
custom is not to attempt to sketch to scale ; and yet in surveys 
where considerable detail is desired it is sometimes well to 
carry out the sketches in the note-book approximately to scale. 
Care should be taken not to crowd the notes, — paper is 
cheap, — and an extra page of the note-book devoted to a 
survey may save hours of time in the office consumed in trying 
to interpret a page of crowded data. Too much stress cannot 
be laid on the importance of being careful not to lose the note- 
book ; not infrequently a note-book contains data which thou- 
sands of dollars could not replace. 

Although sufficient fulness to make the notes clear is 
desirable, it is customary to abbreviate the names of the 
artificial features most commonly met with by the surveyor. 
To properly understand a set of notes one must be familiar 
with these abbreviations, some of the more common of which 
are enumerated. 



NOTE KEEPING II3 

9M, Stone bound. 

M«n- Monument. 

-^ Triangulation Station. 

SHU Stake. 

Ik Tack. 

«*• Nail. 

•fk. Spike. 

^ Drill-hole. 

cii Crow-foot (a mark like this \| or >^). 

cciS Cut crow-foot (cut into wood or stone). 

^ Center. 

* Center line. 

ci^ Curb. 

C8. Catch basin. 

MX Manhole. 

Tki Telegraph pole. 

Fence. 

Fence, showing on which side the posts are. 

Base-board of fence. 

Line of building; the outside line is the base- 
board, the cross-hatched part is the line of the 
stone or brick underpinning. 

Distances should always be recorded in such a way as 
to indicate the precision with which they were taken. For 
example, if they were taken to hundredths of a foot and a 
measurement happened to be just 124 ft. it should be re- 
corded as 124.00, not as 124. The two zeros are of as much 
consequence as any other two digits which might have come in 
their places. Angles which have been read to the nearest half- 
minute, however, are recorded as follows : 6® 47' 30". It will 
be seen that this is not consistent with the foregoing. A more 




114 LAND SURVEYING [Chap. V. 

proper way of reading this angle would be 6® 47 Y, but this is not 
common practice. 

In addition to the measurements every set of notes should 
contain the following information: — the kind of work, the 
locality, the date, and the names of members of the field party. 
It is well to also state the names or numbers of the instruments 
used and their errors. Where a survey is continued for several 
pages the date may be placed at the top of every page ; other 
data need not be repeated. Fig. 50, p. 100, Fig. 52, p. 103, and 
Fig. 53, p. 104, are good examples of field notes. 

149. SURVEY OF A FIELD FOR A DEED. — In this case 
the lengths and bearings of all the boundaries are desired. The 
traverse lines should therefore follow the property lines, if 
possible. The bearings desired are not the observed magnetic 
bearings, but are those calculated by means of the transit angles 
as explained in Art. 145, p. no, and therefore are relatively as 
accurate as the angles themselves. In case a true meridian is 
found by observation (Chapter VII) the bearings should be re- 
ferred to this and marked true bearings by a note on the plan, 
and this information should also be contained in the deed. 

A plan which is to accompany a deed should show such 
features as watercourses, highways, buildings, and adjoining 
property lines, as well as stone bounds, stakes, fences, walls, or 
other artificial objects which mark the boundaries of the property. 

This plan should contain the following information. 

(i) Lengths of all property lines together with their calcu- 
lated bearings or the angles at the corners.* 

(2) Location and description of corner bounds. 

(3) Conventional sign or name on walls, fences, etc. 

* It is customary with many surveyors to omit from the plan certain data 
such as the angles or bearings, so that, while it may answer the purpose for which 
it was made, it does not contain all the data and frequently not enough to enable 
another surveyor to relocate the property by means of it. This is done, of course, 
so that when the tract is to be resurveyed or plotted it will be necessary to em- 
ploy the same surveyor who has in his possession data for which the owner has 
paid and which the surveyor should have turned over to him. For a valuable 
paper on this subject see " The Ownership of Surveys, and what Constitutes a 
Survey and Map," by Professor William G. Raymond, published in TAe Polytechnic^ 
the student journal of the Rensselaer Polytechnic Institute, Troy, N. Y., January. 
1894. 



DEED DESCRIPTION II5 

(4) Names of highways, streams or ponds, and names of 
adjacent property owners. 

(5) Scale of drawing and direction of the meridian used (true 
or magnetic). It is better to refer all bearings to the true 
meridian when possible, and in such a case the direction of the 
magnetic needle should also be shown.* 

(6) The title should include a simple and complete state- 
ment giving the name of owner, place, date, and name of sur- 
veyor. An explanatory note such as a statement as to whether 
bearings refer to true or magnetic meridian may also be neces- 
sary. (See Art. 468, p. 415.) 

150. Deed Description. — The written description of the prop- 
erty which is recorded in the deed should be given by bearings 
(or angles) and distances, stating in every case how the sides of 
the property are marked and whether bounded by a highway, 
stream, or private property, giving the name of- the present 
owner of the adjacent property. The following is an example 
of a deed description of the property shown in the form of notes 
in Fig, 53, p, 104. 

" Beginning at a point in the northerly line of Willow Road 
in the town of Bancroft Mills, Maine, at an iron pipe sunk in 
the ground at the S.E. corner of land now or formerly belonging 
to Nathan H. Barnes, and running along the said northerly line 
N 85** 34' E a distance of two hundred ninety-seven and seven- 
tenths (297.7) feet to the thread of channel of Stony Brook at 
land now or formerly belonging to James F. Hall; thence turn- 
ing and running in a northerly direction, by thread of channel of 
said Stony Brook and land of said Hall, a distance of about three 
hundred and eight (308 ±) feet to a stone wall at land now or 
formerly belonging to Hiram Cole; thence turning and running 
along the middle of said stone wall and by land of said Cole 

* As magnetic bearings are unreliable (Art. 28, p. 19) true bearings should be 
used wherever their adoption does not entail too much additional expense. In 
those parts of the country which have been subdivided by the U. S. General Land 
Office true meridians can be readily obtained from the government surveys ; in 
many of the older (Eastern) states true meridians have been established by local 
authorities. If the survey can be connected with any triangulation system such 
as that of the United States or state surveys then, since the true bearings of all 
of the triangulation lines are known, the bearings of the traverse lines can be 
obtained. 



Il6 LAND SURVEYING [Chap. V. 

N 86® 45' W a distance of two hundred and five and eight-tenths 
(205.8) feet to the middle of another stone wall at land of said 
Barnes; thence turning and running by latter stoiie wall and land 
of said Barnes S o** 53' E a distance of one hundred and seventy- 
seven and two-tenths (177.2) feet to a fence ; thence turning and 
running by said fence and land of said Barnes N 87** 09' W a 
distance of ninety-three and three-tenths (93.3) feet to an iron 
pipe sunk in the ground ; thence turning and running by a fence 
and land of said Barnes S i*^ 51' W a distance of one hundred 
and sixty-nine and four-tenths (169.4) feet to the point of begin- 
ning ; all the bearings being magnetic and the parcel containing 
a calculated area of 79,270 square feet more or less.** 

It is unfortunate that the description of the property in deeds 
in the vast majority of cases, does not define the property in 
such a manner that it can be plotted from the description. Some 
deeds are so loosely written as to contain only the names of the 
owners of adja^cent property, no bearings or distances being given. 

151. JUDICIAL FUNCTIONS OF THE SURVEYOR. — In rerun- 
ning old property lines which have been obliterated, the surveyor 
is called upon to set aside temporarily his strict adherence to the 
mathematical side of surveying and must endeavor to find if pos- 
sible where the lines originally ran. He should therefore be fa- 
miliar with the relative importance of various evidence regarding 
the location of the property lines, as determined by court deci- 
sions. It is distinctly his duty to find the position of the original 
boundaries of the property and not attempt to correct the original 
survey even though he may be sure that an error exists in it. 
Very often it is true that, owing to the cheapness of land, the 
original survey was roughly made with little thought of the effect 
it would have when the land became valuable. 

The surveyor therefore must first of all hunt for all physical 
evidence of the location of the boundaries * and failing in this he 



* It must not be assumed that a boundary is missing because it is not at once 
visible. Stone bounds are often buried two or three feet deep ; the top of a 
stake soon rots off, but evidences of the existence of the stake are often found 
many years after the top has disappeared, and the supposed location should be 
carefully dug over to find traces of the old stake. The shovel and common sense 
are of as much use as the transit and tape in relocating an old corner. 



..'"^^^'^ 



DISPUTED BOUNDARIES 11/ 

will base his judgment on any other reliable evidence such as 
occupancy or the word of competent witnesses. It is obvious 
that this is along equitable lines, since the property was originally 
purchased with reference to the actual or visible bounds which 
vest the owner with rights to the property bounded by these 
lines. 

If there is a dispute between adjoining owners over the loca- 
tion of a boundary line this presents a question which must be 
settled by the courts unless the parties can come to an agree- 
ment themselves. In such cases the surveyor acts simply as an 
expert in judging where the line originally ran and has no power 
to establish a new line. He can, however, be employed by the 
disputing parties as an arbitrator to decide on the equitable line, 
but they are not necessarily obliged to accept his judgment. 

If they come to an agreement between themselves, however, 
regarding the location of the line and occupy to that line, this 
agreement is binding even though no court has intervened in the 
matter. 

It is to be assumed that the deed was drawn by the grantor 
with honest intent to convey the property to the grantee. It is 
intended then that it shall be interpreted if possible so as to 
make it effectual rather than void. The deed should also be 
construed in the light of what was known at the time when the 
title was transferred. 

In the interpretation of a deed it is assumed that it was in- 
tended to convey property the boundaries of which will form a 
closed traverse. Therefore it is within the jurisdiction of the 
surveyor to reject any evident mistake in the description when 
running out the property line, e.g., a bearing may have been re- 
corded in the opposite direction or an entire side omitted. Where 
artificial features are mentioned as boundaries, these always take 
precedence over the recorded measurements or angles, but these 
marks must be mentioned in the deed in order to have the force 
or authority of monuments. When the area does not agree with 
the boundaries as described in the deed the boundaries control. 
All distances unless otherwise specified are to be taken as straight 
lines ; but distances given as so many feet along a wall or high- 
way are supposed to follow these lines even if they are not 



Il8 LAND SURVEYING [Chap. V. 

Straight. When a deed refers to a plan the dimensions on this 
plan become a part of the description of the property. 

Where property is bounded by a highway the abutters usu- 
ally own to the center line, but where it is an accepted street 
each abutter yields his portion of the street for public use ; if, 
however, the street is abandoned the land reverts to the original 
owners. If a street has been opened and used for a long period 
bounded by walls or fences, and there has been no protest re- 
garding them, these lines hold as legal boundaries. In the case 
of a line between private owners acquiescence in the location of 
the boundary will, in general, make it the legal line. But if 
there is a mistake in its location and it has not been brought to 
the attention of the interested parties or the question of its po- 
sition raised, then occupancy for many years does not make it a 
legal line. 

Where property is bounded by a non-navigable stream it ex- 
tends to the thread of the stream. If the property is described 
as running to the bank of a river it is interpreted to mean to the 
low water mark unless otherwise stated. Where original owner- 
ship ran to the shore line of a navigable river and the water has 
subsequently receded the proper subdivision is one that gives to 
each owner along the shore his proportional share of the channel 
of the river. These lines will therefore run, in general, perpen- 
dicular to the channel of the stream from the original intersec- 
tion of division lines and shore lines. 

A more complete statement of the principles mentioned above 
particularly with reference to the U. S. Public Land Surveys 
will be found in an address on " The Judicial Functions of Sur- 
veyors," by Chief-Justice Cooley of the Michigan Supreme Court, 
read before the Michigan Association of Engineers and Survey- 
ors, and published in the proceedings of the society for 1882, 
pp. 1 12-122. 

152. RERUNNING OLD SURVEYS FROM A DEED. — The vis- 
ible marks which are mentioned in a deed are of primary impor- 
tance in determining the extent of a piece of property ; the 
lengths of the sides and the bearings (or angles), which should 
agree with the boundaries, are of secondary importance. It 
sometimes occurs, however, that all evidences of artificial bound- 



RERUNNING OLD SURVEYS 1 19 

aries of the property or of portions of it are missing, and the 
surveyor must then fall back on the dimensions given in the 
deed as the best informatipn available (Art. 150, p. 115). Fur- 
thermore it is sometimes necessary to "run out '' an old deed 
to determine which of two lines is the correct boundary, or in 
some cases to find how close the actual boundaries of a property 
agree with the original deed. 

If the directions of the boundaries are defined in the deed by 
the magnetic bearings, as was formerly the usual custom, it is 
necessary first to find the declination of the needle at the date of 
the original survey as well as the present declination of the needle 
and to correct all the bearings accordingly (Art. 29, p. 20). 
The declination of the needle should appear on the original deed 
or plan ; but unfortunately it seldom does, and the year the sur- 
vey was made must then be obtained either from the deed, the 
old plan, or from witnesses, and the declination of the needle at 
that time computed. Observations at different places arid times 
have been compiled by the U. S. Coast and Geodetic Survey, 
and these results may be found in convenient form for calculation 
in the annual Reports of the Superintendent, particularly the 1886 
report.* From these observations the approximate change in 
declination may be obtained. In this way the magnetic bearings, 
corrected to date, can be determined as closely probably as the 
original bearings were taken. It is evident that the change in the 
declination of the needle between the date of the original survey 
and the present time is what is desired. If there exists there- 
fore one well-defined line which is known to be one of the original 
boundary lines, a bearing taken on this Une and compared with 
that given in the deed will determine directly the change in 
declination. There may be more than one well-defined line 
whose bearings can be obtained and a comparison of the results 
on these different lines will give an idea of the reliability of the 
original survey as well as a more accurate determination of the 
change in declination. 



* In 1902 the U. S. Coast and Geodetic Survey issued a special publication 
entitled, '* Magnetic Declination Tables and Isogenic Charts for 1902/' in which 
IS given a very complete list of declinations for various places in the United States. 



I20 LAND SURVEYING [Chap. V. 

Not infrequently in attempting to rerun old compass surveys 
it is found that the traverse as described in the deed does not 
•* close/' i.e., the last point does not coincide with the first. If 
this error of closure is small it may be due to the difference in 
length between the chain used for the original survey and the 
one being used. Before any attempt is made to run out the old 
survey this difference should be determined by measuring one or 
more of the well-defined lines of the property, if any can be found, 
and comparing the measurements obtained with the recorded 
distances. 

Occasionally it is found that the traverse will not close by a 
large amount owing to a mistake in the original survey. Often 
in such cases the deeds of adjacent property will show what 
the mistake was, and in such cases it is allowable to make a cor- 
rection if it will give a description that is consistent. For ex- 
ample, it occasionally happens that a bearing has been recorded in 
the reverse direction so that no area is enclosed by the boundaries* 
Sometimes an entire chain-length has been omitted in one of 
the lines and by supplying this the description is made consistent. 
Other inconsistencies are to be dealt with in the same general 
manner, or as suggested in the preceding article. 

153. How to Look Up a Recorded Deed. — In all the states of 
the Union the transfer of real property must be recorded in the 
respective county Registry of Deeds or in the office of the city 
or town clerk. At the Registry of Deeds is kept an exact copy 
of the deed, which can be examined by any one. It is fre- 
quently necessary for the surveyor to make use of these copies 
when it is not convenient to obtain the deed from the owner of 
the property or when it is necessary to look up the deed of ad- 
jacent property or previous transfers of any of them. 

In every Registry of Deeds an index of the deeds is kept, 
which is divided into two parts, th^ grantor index and the £^rantee 
index ; the grantor being the party who sells the land and the 
grantee the one who buys it. These indexes are frequently 
divided by years and for this reason the surveyor should know 
not only the name of the party who bought or sold the property 
(both if convenient to get them), but also the approximate date 
of the transaction. With this information he can readily find 



SURVEY OF PUBLIC LANDS 121 

in the proper index the name of the party, opposite which will 
appear the date of the transaction and the number of the deed 
book and page on which the copy of the deed is recorded. He 
then finds the deed book, from which he can copy whatever data 
he desires from the deed ; usually the description of the property 
is all that concerns the surveyor. In the deed book is usually a 
reference number in the margin or in the text of the deed which 
refers to the next preceding transfer of the same property or 
to any attachments, assignments, and the like which may have 
been made on it. This method of indexing and filing deeds is 
used in the New England States and in many of the other states ; 
in fact the general principles are the same throughout the coun- 
try although the details may differ to some extent. 

THE UAITED STATES SYSTEM OF SURVEYING THE PUBLIC LANDS.* 

154, THE SYSTEM. — The United States System of Sur- 
veying the Public Lands, which was inaugurated in 1784, and 
modified since by various acts of Congress, requires that the 
public lands "shall be divided by north and south lines run 
according to the true meridian, and by others crossing them at 
right angles so as to form townships six miles square," and that 
the comers of the townships thus surveyed "must be marked 
with progressive numbers from the beginning.** Also, that the 
townships shall be subdivided into thirty-six sections, each of 
which shall contain six hundred and forty acres, as nearly as 
may be, by a system of two sets of parallel lines, one governed 
by true meridians and the other by parallels of latitude, the 
latter intersecting the former at right angles, at intervals of a 
mile. 

Since the meridians converge it is evident that the require- 



* The vork of surveying the government lands is carried on under the direc- 
tion of the Commissioner of the General Land Office. In each of the districts 
where such surveys are made is a Surveyor General, appointed by the President. 
The work is usually done under contract by experienced surveyors, called Depu- 
ties. The Deputies are paid by the mile, according to classified rates. All surveys, 
before being accepted, are inspected by a corps of Examiners of Surveys, who are 
appointed especially for this duty. 



122 



LAND SURVEYING 



[Chap. V. 



ment that the lines shall conform to true meridians and also 
that townships shall be six miles square, is mathematically im- 
possible. 

In order to overcome this difficulty the subdivision is carried 
on as follows: — (See Fig. 57.) 

J 
LlSTANDAROI ' ' "* " 



T4H 
Kit 



T3N 



TEH 
RIE 



TIN 
RIE 



T4M 
REE 



T3N 
RZE 



TEN 
REE 



TIN 
REE 



T4N 
R3E 



T3H 
R3E 



TEN 
R3E 



TIN 
R3E 



Znd. 



1st. 



T4N 
R4E 



T3H 
R4E 



T2.N 
R4E 



TIM 
R4E 



5 

SrAMDARD 



FV^R/ 



LLEL^RH 



W^RALULNOmH 



It 
"B" 



J 



UJ 



J. 



Fig. 67. 



BASE-UNt 

Diagram Illustratinq Meridian, Base-Line, Standard 
Parallels, Ranges, and Townships. 



First. The establishment of a principal meridian conform- 
ing to the true meridian, and at right angles to it, a base-line 
conforming to a parallel of latitude, as is described in Art. 1 56, 
p. 124, and Art. 157, p. 126. 

Second. The establishment of standard parallels conform- 
ing to parallels of latitude, initiated from the principal meridian 
at intervals of 24 miles and extended east and west of the same. 

Third. The establishment of guide meridians conforming 
to true meridians, initiated upon the base-line and successive 
standard parallels at intervals of 24 miles, resulting in tracts of 
land 24 miles square, as nearly as may be, which shall be subse- 
quently divided into tracts of land 6 miles square by two sets of 
lines, one conforming to true meridians, crossed by others con- 
forming to parallels of latitude at intervals of 6 miles, containing 
23,040 acres, as nearly as may be, and designated townships. 



METHOD OF SUBDIVISION 



123 



Such townships are divided into 36 tracts, called sections^ 
each of which contains 640 acres, as nearly as may be, by two 
sets of parallel lines, one set parallel to a true meridian and the 
other conforming to parallels of latitude, intersecting at intervals 
of I mile, and at right angles, as nearly as may be, as shown 
in Fig. 57. 

Any series of contiguous townships or sections situated 
north and south of each other constitutes a range, while such 
a series situated in an east and west direction constitutes 
a tier. 

Section lines are surveyed from south to north, and from east to 
west, in order to place the excess or deficiency, according to the 
requirement of the law, on the north and west sides of the town- 
ships. # 

The tiers of townships are numbered, to the north or south, 
commencing with No. i at the base-line ; and the ranges or 
townships, to the east or west, beginning with No. i at the prin- 
cipal meridian of the system. 

The thirty-six sections into which a township is subdivided 
are numberec^ commencing with No, i at the northeast angle of 
the township, and proceed- 
ing west to No. 6, and then 
proceeding east to No. 12, 
and so on, alternately, to No. 
36, in the southeast angle as 
illustrated by Fig. 58. In all 
cases of surveys of frac- 
tional townships the sections 
will bear the same numbers 
they would have if the town- 
ship were complete. 

Standard parallels (form- 
erly called correction lines) 
are established at intervals of 
24 miles, north and south of 
the base line, and guide 
meridians at intervals of 24 
miles, east and west of the principal meridians ; thus confining 



6 


5 


4 


3 


2 


1 


♦7 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


??> 


24 


30 


29 


28 


27 


26 


25 


31 


3Z 


33 


34 


a.") 


36 



Fig. 58. Diagram of a Town- 
ship Illustrating Method of 
Numbering the Sections. 



124 LAND SURVEYING tCHAP. V. 

the errors resulting from convergence of meridians and inaccu- 
racies in measurement within comparatively small areas. 

" 155. Initial Points.* — Initial points from which the lines 
of the public surveys are to be extended will be established 
whenever necessary, under such special instructions as may be 
prescribed in each case by the Commissioner of the General 
Land Office. The locus of such initial points will be selected 
with great care and due consideration for their prominence and 
easy identification, and must be established astronomically. 

"An initial point should have a conspicuous location, visible 
from distant points on lines ; it should be perpetuated by an in- 
destructible monument, preferably a copper bolt firmly set in a 
rock edge ; and it should be witnessed by rock bearings, without 
relying on anything perishable like wood. 

" The initial point having been established the lines of public- 
land surveys will be extended therefrom. They are classified as 
follows : 

"Class I. Base lines and standard parallels. 

"Class 2. Principal. and guide meridians. 

" Class 3. Township exteriors (or meridional and latitudinal 
township boundaries). 

" Class 4. Subdivision and meander lines. 

" Only the base line and principal meridian can pass through 
the initial point. 

" 156. Base Line. — From the initial point the base line will 
be extended east and west on a true parallel of latitude, [Art. 
168, p. 148,] by the use of transit or solar instruments, as may 
be directed by the surveyor general in his written special instruc- 
tions. The transit will be used for the alinement of all import- 
ant lines. 

"The direction of base lines will conform to parallels of 
latitude and will be controlled by true meridians ; consequently 
the correct determination of true meridians by observations on 
Polaris at elongation is a matter of prime importance. 

" Certain reference lines, called tangents and secants, having 

* These instructions are taken from the *' Manual of Surveying Instructions 
for the Survey of the Public Lands of the United States," prepared by the Com- 
missioner of the General Land Office in 1902. 



INITIAL POINTS 12$ 

a known position and relation to the required parallel of latitude, 
will be prolonged as straight lines. Two back and two fore sights 
are taken at each setting of the instrument, the horizontal limb 
being revolved i8o^ in azimuth between the observations, in one 
method, taking the mean of observations. Another method, 
called double back and fore sights, is still more exact, and there- 
fore preferable. In this process the vertical cross-wire is fixed 
upon two transit points at some distance apart, in the rear, and 
then reversed to set one or two new points in advance. This 
not only insures a straight line, if the transit is leveled, but also 
detects the least error of collimation. 

" Where solar apparatus is used in connection with a transit, 
the deputy will test the instrument, whenever practicable, by 
comparing its indications with a meridian determined by Polaris 
observations ; and in all cases where error is discovered he will 
make the necessary corrections of his line before proceeding 
with the survey. All operations will be fully described in the 
field notes. 

"The proper township, section, and quarter-section corners 
will be established at lawful intervals, and meander corners at 
the intersection of the line with all meanderable streams, lakes, 
or bayous. 

" In order to detect errors and insure accuracy in measure- 
ment, two sets of chainmen will be employed ; one to note dis- 
tances to intermediate points and to locate topographical features, 
the other to act as a check. Each will measure 40 chains, and 
in case the difference is inconsiderable, the proper corner will 
be placed midway between the ending points of the two measure- 
ments ; but if the discrepancy exceed 8 links on even ground, or 
25 links on mountainous surface, the true distance will be found 
by careful re-chaining by one party or both. 

" The deputy will be present when each corner is thus estab- 
lished, and will record in the body of his field notes the distances to 
the same, according to the measurement by each set of chainmen. 

"To obviate collusion between the sets of chainmen, the 
second set should commence at a point in advance of the begin- 
ning comer of the first set, the initial difference in measurement 
thus obtained being known only to the deputy. 



126 LAND SURVEYING [Chap. V. 

Z57. '* Principal Meridian. — This line shall conform to a 
true meridian [Chapter VII] and will be extended from the initial 
point, either north or south, or in both directions, as the condi- 
tions may require, by the use of transit or solar instruments, as 
may be directed by the surveyor general in his special written 
instructions. The methods used for determination of directions, 
and the precautions to be observed to secure accuracy in measure- 
ment, are fully stated above under the title " Base Line," and will 
be complied, with in every particular. 

"In addition to the above general instructions, it is required 
that in all cases where the establishment of a new principal meri- 
dian seems to be necessary to the surveyor general, he shall 
submit the matter, together with his reasons therefor, to the 
Commissioner of the General Land Office, and the survey of 
such principal meridian shall not be commenced until written 
authority, together with such special instructions as he may deem 
necessary, shall have been received from the Commissioner. 

158. " Standard Parallels. — Standard parallels, which are also 
called correction lines, shall be extended east and west from the 
principal meridian, at intervals of 24 miles north and south of 
the base line, in the manner prescribed for running said line, 
and all requirements under the title * Base Line ' will be care- 
fully observed. 

" Where standard parallels have been placed at intervals of 30 
or 36 miles, regardless of existing instructions, and where gross 
irregularities require additional standard lines, from which to 
initiate new, or upon which to close old surveys, an intermediate 
correction line should be established to which a local name may 
be given, e.g., ' Cedar Creek Correction Line ' ; and the same 
will be run, in all respects, like the regular standard parallels. 

159. " Guide Meridians. — Guide meridians shall be extended 
north from the base line, or standard parallels, at intervals of 24 
miles east and west from the principal meridian, in the manner 
prescribed for running the principal meridian, and all the pro- 
visions for securing accuracy of alignment and measurement, 
found or referred to under the titles Base Line and Principal 
Meridian, will apply to the survey of said guide meridians. 

" When existing conditions require that such guide meridians 



PRINCIPAL MERIDIAN 12/ 

shall be run south from the base or correction lines, they will 
be initiated at properly established comers on such lines, 
marked as closing comers. 

"Where guide meridians have been improperly placed at 
intervals greatly exceeding the authorized distance of 24 miles, 
and standard lines are required to limit errors of old, or govern 
new surveys, a new guide merijdian may be run from a standard, 
or properly established closing corner, and a local name may be 
assigned to the same, e.g., 'Grass Valley Guide Meridian.' 
These additional guide meridians will be surveyed in all respects 
like regular guide meridians. 

160. " Township Exteriors. — Whenever practicable, the town- 
ship exteriors in a block of land 24 miles square, bounded by 
standard lines, will be surveyed successively through the block, 
beginning with those of the southwestem township. 

" The meridional boundaries of townships will have prece- 
dence in the order of survey and will be run from south to 
north on true meridians, with permanent comers at lawful 
distances ; the latitudinal boundaries will be run from east to 
west on random or trial lines, and corrected back on true lines. 

" The falling of a random, north or south of the township 
corner to be closed upon, will be carefully measured, and, with 
the resulting tme return course, will be duly recorded in the 
field notes. 

"Should it happen, however, that such random intersects 
the meridian of the objective comer, north or south of said 
comer, or falls short of, or overruns the length of the south 
boundary of the township by more than three chains (due 
allowance being made for convergency), said random, and, if 
necessary, all the exterior boundaries of the township, will be 
retraced and remeasured to discover and correct the error. 

" When running random lines from east to west, temporary 
corners will be set at intervals of 40.00 chains, and proper 
permanent comers will be established upon the true line, cor- 
rected back in accordance with these instructions, thereby 
throwing the excess or deficiency against the west boundary of 
the township, as required by law. 

"Whenever practicable, the exterior boundaries of town- 



128 LAND SURVEYING [Chap. V. 

ships belonging to the west range, in a tract or block 24 miles 
square, will first be surveyed in succession, through the range, 
from south to north ; and in a similar manner, the other three 
ranges will be surveyed in regular sequence. 

" In cases where impassable obstacles occur and the fore- 
going rules cannot be complied with, township corners will be 
established as follows : 

" In extending the south or north boundaries of a township 
to the west, where the southwest or northwest corners cannot 
be established in the regular way by running a north and south 
line, such boundaries will be run west on a true line, allowing 
for convergency on the west half mile ; and from the township 
corner established at the end of such boundary, the west 
boundary will be run north or south, as the case may be. In 
extending south or north boundaries of a township to the east, 
where the southeast or northeast comer cannot be established 
in the regular way, the same rule will be observed, except that 
such boundaries will be run east on a true line, and the east 
boundary run north or south, as the case may be. Allowance 
for the convergency of meridians will be made whenever 
necessary. 

161. " Method of Subdividing. — The exterior boundaries of 
a full township having been properly established so far as pos- 
sible, the subdivision thereof will be made as follows : 

"At or near the southeast corner of the township, a true 
meridian will be determined by Polaris or solar observations, 
and the deputy's instrument will be tested thereon ; then from 
said corner the first mile of the east and south boundaries will 
be retraced, if subdivisions and survey of the exteriors have 
been provided for in separate contracts ; but, if the survey of 
the exterior and subdivisional lines are included in the same 
contract, the retracements referred to will be omitted. All dis- 
crepancies resulting from disagreement of bearings or measure- 
ments will be carefully stated in the field notes. 

" The meridional sectional lines will be made parallel to the 
range line or east boundary of the township, by applying to the 
bearing of the latter a small correction, dependent on the lati- 
tude, taken from the following table, which gives, to the 



METHOD OF SUBDIVIDING 



129 



nearest whole minute, the convergency of two meridians 6 
miles long and from i to 5 miles apart ; and supplies directly 
the deviation of meridional section lines west of north, when 
the range line is a true meridian. Add the corrfection to the 
bearing of the range line, if the same is west of north, but 
subtract when it bears east of north. 



TABLE 3. 
Corrections for Convergency within a Township. 



Latitude. 



30 to 35 
35 to 40 
40 to 45 

45 to 50 
50 to 55 
55 to 60 
60 to 65 
65 to 70 



Correction to be applied to bearing of range lines 
at a distance of — 



I mile, a miles. 3 miles. 4 miles. 5 miles. 



3 
3 
4 

I 

7 

8 

10 



^'Example. — Latitude, 47®. Range Ime bears N. 0° 2' E. 
then parallel meridional section lines will be run as follows: 

From the corner for sections — 

35 and 36, N.o^ i' E. 
34 and 35, north. 
33 and 34, N. o® i' W. 
32 and 33, N. o^ 2' W. 
31 and 32, N. 0° 3' W. 

" After testing his instrument on the true meridian thus deter- 
mined, the deputy will commence at the corner to sections 35 
and 36, on the south boundary, and run a line parallel to the 
range line, establishing at 40.00 chains, the quarter-section corner 
between sections 35 and 36, and at 8Q.00 chains the corner for 
sections 25, 26, 35, and 36. 



I30 LAND SURVEYING [Chap. V- 

" From the last-named corner, a random line wDl be run east- 
ward, without blazing, parallel to the' south boundary of section 
36, to its intersection with the east boundary of the township, 
placing at 40.00 chains from the point of beginning, a post for 
temporary quarter-section corner. If the random line intersects 
said township boundary exactly at the corner for sections 25 and 
36, it will be blazed back and established as the true line, the 
permanent quarter-section comer being established theron, mid- 
way between the initial and terminal section comers. 

"When the objective corner is in sight from the starting 
comer, or the deputy has evidence of its location to prove that a 
different random course would fall closer to the corner, he may 
use such changed course for his random. A line may be run as 
a "randoni for distance only," when the course is certain. 

"If the random intersects said township boundary to the 
north or south of said corner, the falling will be carefully mea- 
sured, and from the data thus obtained, the tme return course 
will be calculated, and the true line blazed and established and 
the position of the quarter-section corner determined, as directed 
above. 

The details of the entire operation will be recorded in the 
field notes. 

"Having thus established the line between sections 25 and 
36, from the comer for sections 25, 26, 35, and 36, the west and 
north boundaries of sections 25, 24, 13, and 12, will be established 
as directed for those of section 36 ; with the exception that the 
random lines of said north boundaries will be run parallel to the 
established south boundary of section 36 ; e.g., the random line 
between sections 24 and 25 will be run parallel to the established 
south boundary of section 25, etc. 

"Then, from the last established section corner, i.e., the 
corner of sections i, 2, 11, and 12, the line between sections i 
and 2 will be projected northward, on a random line, parallel 
to the east boundary of the township, setting a post for tempo- 
rary quarter-section corner at 40.00 chains, to its intersection 
with the north boundary of the township. If the random inter- 
sects said north boundary exactly at comer for sections i and 2, 
it will be blazed back and established as the true line, the tem- 



SUBDIVISION OF TOWNSHIP 131 

porary quarter-section comer being established permanently in its 
original position, and the fractional measurement thrown into 
that portion of the line between said comer and the north boun- 
dary of the township. 

" If, however, said random intersects the north boundary of 
the township, to the east or west of the corner for sections i 
and 2, the consequent falling will be carefully measured, and 
from the data thus obtained the true return course will be calcu- 
lated and the true line established, the permanent quarter-section 
corner being placed upon the same at 40.00 chains from the 
initial corner of the random line, thereby throwing the fractional- 
measurement in that portion lying between the quarter-section 
comer and the north boundary of the township. 

"When the north boundary of a township is a base line or 
standard parallel, the line between sections i and 2 will be nm 
parallel to the range line as a true line, the quarter-section cor- 
ner will be placed at 40.00 chains, and a closing corner will be 
established at the point of intersection with such base or stand- 
ard line ; and in such case, the distance from said closing corner, 
to the nearest standard comer on such base or standard line, will 
be carefully measured and noted as a connection line. 

" Each successive range of sections progressing to the west, 
until the fifth range is retained, will be surveyed in a similar 
manner ; then, from the section corners established on the west 
boundary of said range of sections, random lines will be projected 
to their intersection with the west boundary of. the township, 
and the true return lines established as prescribed for the survey 
of the first or most eastern range of sections, with the exception 
that on the true lines thus established the quarter-section corners 
will be established at 40.00 chains from the initial comers of 
randoms, the fractional measurements being thereby thrown 
into those portions of the lines situated between said quarter- 
section corners and the west boundary of the township. 

" The following general requirements are reiterated for em- 
phasis : 

** The random of a latitudinal section line will alwa3rs be nm par- 
allel to the south boundary of the section to which it belongs, and 



132 



LAND SURVEYING 



[Chap. V. 



with the true bearing of said boundary; and when a section has 
no linear south boundary, the random will be run parallel to the 
south boundary of the range of sections in which it is situated, 
and fractional true lines will be run in a similar manner. 



H«f^^ 



>4 6Sr53W 



H.S3.SSW. 




S.e?3BL. 



5B5'saE 



Fig. 69. Portion of Township Illustrating Subdivision of 

Sections. 



" The deputy is not required to complete the survey of the 
first range of sections from the south to north before commenc- 
ing the survey of the second or any subsequent range of sections, 
but the corner on which any random line closes shall have been 
previously established by running the line which determines its 



SUBDIVISION OF SECTION 1 33 

position, except as follows : Where it is impracticable to estab- 
lish such section comer in the regular manner, it will be estab- 
lished by running the latitudinal section line as a true Une, with 
a true bearing, determined as above directed for random lines, 
setting the quarter-section comer at 40.00 chains and the section 
corner at 8o,oo chains. 

" Quarter-section comers, both upon meridional and latitudi- 
nal section lines, will be established at points equidistant from 
the corresponding section comers, except upon the lines closing 
on the north and west boundaries of the township, and in those 
"situations the quarter-section corners will always be established 
at precisely forty chains to the north or west (as the case may 
be) of the respective section corners from which those lines re- 
spectively start, by which procedure the excess or deficiency in 
the measurements will be thrown, according to law, on the 
extreme tier or range of quarter sections, as the case may be. 
(See Fig. 59.) 

" Where by reason of impassable objects only a portion of the 
south boundary of a township can be established, an auxiliary base 
line (or lines, as the case may require) will be run through the 
portion which has no linear south boundary, first random, then 
corrected, connecting properly established corresponding section 
comers (either interior or exterior) and as far south as possible ; 
and from such line or lines, the section lines will be extended 
northwardly in the usual manner, and any fraction south of said 
line will be surveyed in the opposite direction from the section 
comers on the auxiliary base thus established. 

"Where by re^on of impassable objects or other reasons no 
part of the south boundary of a township can be regularly es- 
tablished, the subdivision thereof will proceed from north to 
south and from east to west, thereby throwing all fractional 
measurements and areas against the west boundary, and the 
meanderable stream or other boundary limiting the township 
on the south. 

"If the east boundary is without regular section comers 
and the north boundary has been run eastwardly as a true line, 
with section corners at regular intervals of 80.00 chains, the 
subdivision of the township will be made from west to east, and 



1 34 LAND SURVEYING [Chap. V. 

fractional measurements and areas will be thrown against the 
irregular east boundary. 

"When the proper point for the establishment of a town- 
ship or section comer is inaccessible, and a witness comer can 
be erected upon each of the two lines which approach the same, 
at distances not exceeding twenty chains therefrom, said witness 
comers will be properly established, and the half miles, upon 
which they stand will be recognized as surveyed lines. 

" The witness comer will be marked as conspicuously as a 
section comer, and bearing trees will be used wherever possible. 

" The deputy will be required to furnish good evidence that 
the section corner is actually inaccessible. 

" Where impassable precipices, deep canyons, or lands other- 
wise quite unsurveyable, prevent the extension of regular lines, 
deputies are not authorized to set meander comers, nor to 
meander the line separating lands that can be traversed from 
those that cannot. In place of meandering, they are to set 
witness corners on line, near the intersection of section lines with 
the brink or foot of the impassable cliffs, or at the margin 
of the impracticable marsh, to represent an inaccessible regular 
section or quarter-section corner if within twenty chains. Such 
quarter sections thus marked may be platted as surveyed. 

" Where a large or desirable track is found to have its acces- 
sible section lines too short to justify the erection of such witness 
comers, and to render it regularly surveyed, offset lines may 
be run on lines of legal subdivision, far enough to show, by 
necessary witness comers, the 40-acre tracts that would other- 
wise have been excluded from survey. 

" The topographic sketches of mesas and impassable canyon 
regions, returned by deputies, will show as nearly as practicable 
the location of these features and their margins ; and where 
possible the comers on opposite sides of a canyon should be 
connected by triangulation at least once in each township. 

162. ''Meandering. — The running of meander lines has 
always been authorized in the survey of public lands fronting on 
large streams and other bodies of water, but does not appear to 
have been proper in other cases. The mere fact that an irregular 
or sinuous line must be mn, as in case of a reservation bound- 



MEANDER LINES 135 

ary, does not entitle it to be called a meander line except where 
it closely follows a stream or lake shore. The legal riparian 
rights connected with meandered lines do not apply in case 
of other irregular lines, as the latter are strict boundaries. 

" Lands bounded by waters are to be meandered at mean 
high-water mart This term has been defined in a State deci- 
sion (47 Iowa, 370) in substance as follows: High water mark 
in the Mississippi River is to be determined from the river-bed ; 
and that only is river-bed which the river occupies long enough 
to wrest it from vegetation. 

"In another case (14 Penn. St. 59) a bank is defined as the 
continuous margin where vegetation ceases, and the shore is the 
sandy space between it and low-water mark. 

" Numerous decisions in State and U. S. Supreme Courts, 
assert the principle that meander lines are not boundaries defin- 
ing the area of ownership of tracts adjacent to waters. The 
general rule is well set forth (10 Iowa, 549) by saying that in a 
navigable stream, as the Des Moines River in Iowa, high-water 
mark is the boundary line. When by actibn of the water the 
river bed changes, high-water mark changes and ownership of 
adjoining land changes with it. The location of meander lines 
does not aflfect the question. 

" Inasmuch as it is not practicable in public land surveys to 
meander in such a way as to follow and reproduce all the minute 
windings of the high-water line, the U. S. Supreme Court has 
given the principles governing the use and purpose of meander- 
ing shores, in its decision in a noted case (R. R. Co. v, Schurmeier, 
7 Wallace, 286-7) as follows : 

" In cases where the deputy finds it impossible to carry his 
meander line along mean high-water mark, his notes should state 



"Meander lines are ran in surveying fractional portions of the public lands 
bordering on navigable rivers, not as boundaries of the tract, but for the purpose 
of defining the sinuosities of the banks of the stream, and as the means of ascer- 
taining the quantity of land in the fraction subject to sale, which is to be paid for 
by the purchaser. In preparing the official plat from the field notes, the meander 
line is represented as the border line of the stream, and shows to a demonstration 
that the water-course, and not the meander line as actually ran on the land, is 
the boundary. 



I 



136 LAND SURVEYING [Chap. V. 

the distance therefrom, and the obstacles which justify the 
deviation. 

"Proceeding down stream, the bank on the left hand is 
termed the left bank and that on the right hand the right bank. 
These terms will be universally used to distinguish the two 
banks of a river or stream. 

" Navigable rivers, as well as all rivers not embraced in the 
class denominated 'navigable,' the right-angle width of which 
is three chains and upwards, will be meandered on both banks, 
at the ordinary mean high-water mark, by taking the general 
courses and distances of their sinuosities, and the same will be 
entered in the field book. Rivers not classed as navigable will 
not be meandered above the point where the average right-angle 
width is less than three chains, except that streams which are 
less than three chains wide and which are so deep, swift, and 
dangerous as to be impassable through the agricultural season, 
may be meandered, where good agricultural lands along the 
shores require their separation into fractional lots for the benefit 
of settlers. But such meander surveys shall be subject to re- 
jection if proved unnecessary by field inspection. 

" Shallow streams, without any well-defined channel or per- 
manent banks, will not be meandered ; except tide-water streams, 
whether more or less than three chains wide, which should 
be meandered at ordinary high-water mark, as far as tide-water 
extends. 

"At every point where either standard, township, or section 
lines intersect the bank of a navigable stream, or any meandera- 
ble shore, corners will be established at the time of running 
these lines. Such corners are called meander corners, and the 
deputy will commence at one of these corners, follow the bank 
or boundary line, and take the bearing and measure the length 
of each course, from the beginning corner to the next meander 
corner. 

" All courses reported are to be compass courses, taken or 
counted from the meridian, and not from a latitudinal line ; and 
* transit angles ' showing only the amount of deviation from the 
preceding course, are not allowed in field notes of meanders. 

"For convenience of testing by traverse, the courses of 



MEANDER LINES 1 37 

meander lines should be given by the nearest quarter degree. 
As meandered lines are not strict boundaries, this method will 
give results with approximate accuracy for good closings within 
the limits of a section. Meander lines will be examined in the 
field as well as rectangular lines, before acceptance. 

" All meanders should be traversed before leaving the vicin- 
ity, and if misclosure is found, indicating error in measurement 
or in reading courses, the lines must be re-meandered. 

"The crossing distance between meander corners on same 
line, and the true bearing and distance between corresponding 
meander corners, will be ascertained by triangulation or direct 
measurement, in order that both shores may be protracted. The 
particulars will be given in the field notes. 

" For convenience of platting and computation, the deputy 
is required to use in meanders distances having whole chains, 
or multiples of ten links, with odd links only in closing distances. 

" The meanders of all lakes, navigable bayous, and deep ponds 
of the area of twenty-five acres and upwards, will be commenced 
at a meander corner and continued, as above directed for navi- 
gable streams; from said corner, the courses and distances of 
the entire margin of the same, and the intersections with all 
meander corners established thereon, will be noted. 

" All streams falling into the river, lake, or bayou will be 
noted, and the width at their mouths stated ; also, the position, 
size, and depth of springs, whether the water be pure or mineral ; 
also, the heads and mouths of all bayous ; all islands, rapids, and 
bars will be noted, with intersections to their upper and lower ends, 
to establish their exact situation. The elevation of the banks of 
lakes, bayous, and streams, the height of falls and cascades, and 
the length and fall of rapids will be recorded in the field notes. 

" To meander a lake or deep pond lying entirely within the 
boundaries of a section, two lines will be run from the two 
nearest corners on different sides of such lake or pond, the 
courses and lengths of which will be recorded, and if coincident 
with unsurveyed lines of legal subdivisions, that fact will also be 
stated in the field notes, and at each of the points where said 
lines intersect the margin of the pond or lake, a special meander 
comer will be established as above directed. 



138 LAND SURVEYING [Chap. V. 

"A special meander comer is one established on a line of 
legal subdivision, not a standard, township, or section line. 

"The relative position of these points being thus definitely 
fixed in the section, the meandering will commence at one of 
them and be continued to the other, noting the intersection, and 
thence to the beginning. The proceedings are to be fully en- 
tered in the field notes. 

"Meander lines will not be established at the segregation 
line between dry and swamp or overflowed land, but at the ordi- 
nary high-water mark of the actual margin of the rivers or lakes 
on which such swamp or overflowed lands border. 

"The precise relative position of an island, in a township 
made fractional by a river or lake in which the island is situated, 
will be determined by triangulation from a special and carefully 
measured base line, initiated upon the surveyed lines, on or near 
the lake or river bank on the mainland, so as to connect by 
course and distance on a direct line, the meander comer on the 
mainland with the corresponding point on the island, where the 
proper meander corner will be established. 

" In making the connection of an island lying entirely within 
a section, with the mainland, a special base will be measured 
from the most convenient meander comer, and from such base, 
the location of an auxiliary meander comer (that is, one not on 
a line belonging to the system of rectangular surveying) will 
be determined by triangulation, at which the meanders of the 
island will be initiated. 

" In the survey of lands bordering on tide waters, meander 
corners may be temporarily set at the intersection of the sur- 
veyed lines with the line of mean high tide, but no monument 
should be placed in a position exposed to the beating of waves 
and the action of ice in severe weather. In all such cases, the 
rule given in section 90 must be observed, by establishing a wit- 
ness corner on line at a secure point near the true point for the 
meander comer. 

" The field notes of meanders will show the dates on which 

the work was performed 

The field notes of meanders will state and describe 

the corner from which the meanders commenced, and upon 



DATA TO BE NOTED 1 39 

which they closed, and will exhibit the meanders of each frac- 
tional section separately; following, and composing a part of 
such notes, will be given a description of the land, timber, depth 
of inundation to which the bottom is subject, and the banks, 
current, and bottom of the stream or body of water meandered. 
The utmost care will be taken to pass no object of topography, 
or change therein, without giving a particular description thereof 
in its proper place in the notes of the meanders. 

163. <^ Summary of objects and data intersected by the line 
or in its vicinity, to be noted. — i. The precise course and 
length of every line run, noting all necessary offsets therefrom, 
with the reason for making them, and method employed. 

" 2. The kind and diameter of all bearing trees, with the 
course and distance of the same from their respective corners ; 
and the precise relative position of witness comers to the true 
corners. 

" 3. The kind of materials of which corners are constructed. 

"4. Trees on line. The name, diameter, and distance on 
line to all trees which it intersects. 

" 5. Intersections by line of land objects. The distance at 
which the line intersects the boundary lines of every reservation, 
town site, donation claim, Indian allotment, settler's claim, im- 
provement, or rancho ; prairie, bottom land, swamp, marsh, grove, 
and windfall, with the course of the same at all points of inter- 
section ; also, the distances at which the line begins to ascend, 
arrives at the top, begins to descend, and reaches the foot of all 
remarkable hills and ridges, with their courses, and estimated 
height in feet, above the level land of the surrounding country, 
or above the bottom lands, ravines, or waters near which they 
are situated. Also, distance to and across large ravines, their 
depth and course. 

" 6. Intersections by line of water objects. All rivers, creeks, 
and smaller streams of water which the line crosses ; the distances 
measured on the true line to the bank first arrived at, the course 
down stream at points of intersection, and their widths on line. 
In cases of navigable streams, their width will be ascertained 
between the meander comers, as set forth under the proper 
head. 



I40 LAND SURVEYING [Chap. V. 

" 7. The land's surface — whether level, rolling, broken, hilly, 
or mountainous. 

" 8. The soil — whether rocky, stony, sandy, clay, etc., and 
also whether first, second, third, or fourth rate. 

" 9. Timber — the several kinds of timber and undergrowth, 
in the order in which they predominate. 

" 10. Bottom lands — to be described as wet or dry, and if 
subject to inundation, state to what depth. 

"II. Springs of water — whether fresh, saline, or mineral, 
with the course of the streams flowing from them. 

"12. Lakes and ponds — describing their banks and giving 
their height, and whether it be pure or stagnant, deep or shallow. 
• "13. Improvements. Towns and villages ; houses or cabins, 
fields, or other improvements with owners' names; mill sites, 
forges, and factories, U. S. mineral monuments, and all corners 
not belonging to the system of rectangular surveying ; will be 
located by bearing and distance, or by intersecting bearings from 
given points. 

" 14. Coal banks or beds ; peat or turf grounds ; minerals and 
ores ; with particular description of the same as to quality and ex- 
tent, and all diggings therefor ; also salt springs and licks. All 
reliable information that can be obtained respecting these objects, 
whether they be on the line or not, will appear, in the general 
description. 

"15. Roads and trails, wifh their directions, whence and 
whither. 

" 16. Rapids, cataracts, cascades, or falls of water, with the 
estimated height of their fall in feet. 

" 1 7. Precipices, eaves, sink holes, ravines, remarkable crags, 
stone quarries, ledges of rocks, with the kind of stone they 
afford. 

" 18. Natural curiosities, interesting fossils, petrifactions, 
organic remains, etc. ; also all ancient works of art, such as 
mounds, fortifications, embankments, ditches or objects of like 
nature. 

" 19. The magnetic declination will be incidentally noted 
at all points of the lines being surveyed, where any material change 
in the same indicates the probable presencie of iron ores ; and 



LIMITS FOR CLOSING I4I 

the position of such j)oints will be perfectly identified in the 
field notes. 

164. ** Prescribed Limits for Closings and Lengths of Lines. 
— If in running a random township exterior, such random 
exceeds or falls short of its proper length by more than three 
chains, allowing for convergency, or falls more than three chains 
to the right or left of the objective point (or shows a proportion- 
ate error for lines of greater or less length than six miles), 
it will be re-run, and if found correctly run, so much of the 
remaining boundaries of the township will be retraced, or 
resurveyed, as may be found necessary to locate cause of 
misclosure. 

"Every meridional section line, except those which ter- 
minate upon a fractional side of a township, will be 80 chains 
in length, without allowance of 50 links per mile for difference 
of measure, or any other allowance beyond a small reasonable 
discrepancy according to the nature of the surface, to be deter- 
mined after examination. 

"The random meridional or latitudinal lines through a tier 
or range of fractional sections shall fall within 50 links of the 
objective comers, and a greater falling will indicate negligence 
or error. 

" The actual lengths of meridional section lines through a 
fractional north or south tier of sections shall be within 1 50 links 
of their theoretical length. The latter will be determined from 
the given lengths of meridional boundaries on the east and the 
west range lines. 

" Each latitudinal section line, except in a fractional east or 
west range of sections, shall be within 50 links of the actual dis- 
tance established on the governing north or south boundary of 
the township for the width of the same range of sections. 

" The north boundary and the south boundary of any section, 
except in a fractional range, shall be within 50 links of equal 
length. 

"The meanders within each fractional section or between 
any two successive meander corners, or of an island or lake in 
the interior of a section, should close by traverse within a limit 
to be determined by allowing five-eighths of a link for each chain 



142 LAND SURVEYING [Chap. V. 

of such meander line. This rule does not apply to irregular bound- 
aries of reservations or private claims, except as far as the same 
are natural water boundaries. The total misclosure of meanders 
will not be permitted to exceed 150 links, except in large pri- 
vate land claims, which are governed by a different rule and 
limit. 

"In closing upon accepted surveys, when irregularities be- 
yond the allowable limits are developed, either in the length or 
direction of the closing lines, closing comers will be set, with 
quarter-section corners at 40 chains from the last interior sec- 
tion corner ; 

" And, in general, when conditions are met which result in a 
random line being defective, either in length or direction, such 
procedure will be adopted as will secure the greatest number of 
new rectangular legal subdivisions, without disturbing the con- 
dition of accepted surveys. 

165. ** Field Notes. — The proper blank books for original 
field notes will be furnished by the surveyor general, and in 
such books the deputy surveyor will make a faithful, distinct, 
and minute record of everything done and observed by himself 
and his assistants, pursuant to instructions, in relation to run- 
ning, measuring, and marking lines, establishing corners, etc., 
and present, as far as possible, full and complete topographical 
sketches of all standard and exterior lines, drawn to the usual 
scale of township exteriors. These 'original field notes' are 
not necessarily the entries made in the field, in the deputy's 
pocket note books called tablets ; but they are to be fully and 
correctly written out in ink, from such tablets, for the permanent 
record of the work. Tablets should be so fully written as to 
verify the original field notes whenever the surveyor general 
requires them for inspection. 

" A full description of all comers belonging to old surveys, 
from which the lines of new surveys start, or upon which they 
close, will in all cases be furnished the deputy from the sur- 
veyor general's office, when authority is given for commencing 
work; then, if the old comers are found to agree with said de- 
scriptions, the deputy will describe any one of them in this form, 
'which is a firmly set, marked and witnessed as described 



FIELD NOTES 1 43 

by the surveyor general ; ' but, should a comer, not answer the 
description supplied, the deputy will give a full description of 
such corner and its accessories, following the proper approved 
form given in these instructions. 

" A full description of each comer established under any one 
contract will be given once only ; subsequent reference to such 
comer will be made in the form, 'heretofore described,' or *the 
comer of sections 2, 3, 10, and 11,' as the case may require. 

" In all cases where a comer is reestablished, the field notes 
will describe fully the manner in which it is done. 

" The field notes of the survey of base, standard, and meri- 
dian lines will describe all corners established thereon, how es- 
tablished, the crossings of streams, ravines, hills, and mountains ; 
character of soil, timber, minerals, etc. ; and after the description 
of each township comer established in running such lines, the 
deputy will note particularly in the * general description' the 
character of townships on each side of the lines run. 

"The field notes of the survey of exterior boundaries of 
townships will describe the corners and topography, as above 
required, and the 'general description' at the end of such 
notes will describe the townships as fully as possible, and also 
state whether or not they should be subdivided. 

♦'The field notes of the subdivisional survey of townships 
will describe the comers and topography as above required, 
and the 'general description' at the end of such notes will 
state minutely the character of the land, soil, timber, etc., found 
in such townships. 

'* The topography will be given on the true line in all cases, 
and will be taken correctly, not estimated or approximated. 

" With the field notes of the survey of base lines and stand- 
ard parallels, and principal and guide meridians forming a tract 
24 miles square, including those of the township exteriors therein, 
the deputy will submit a diagram of the lines surveyed, drawn to 
a scale of half an inch to one mile, upon which will be written 
the true bearings and lengths of all surveyed lines, except 
the lengths of those which are actually 40.00 or 80.00 chains. 
These diagrams will exhibit all water courses, with the direction 
of each indicated by an arrow head pointing down stream ; also. 



144 LAND SURVEYING [Chap. V. 

the intersection of the lines with all prairies, marshes, swamps, 
ravines, lakes, ponds, mountains, hills, and all other natural or 
artificial topographical features mentioned in the field notes, to 
the fullest extent possible. 

" With the special instructions for making subdivisional sur- 
veys of townships into sections, the deputy will be furnished by 
the surveyor general with blank township diagrams drawn to a 
scale of one inch to forty chains, upon which the true bearings 
and lengths of the township and section lines, from which the 
surveys are to be projected, or upon which they are to close, will 
be carefully marked ; and on such diagrams the deputy who sub- 
divides will make appropriate sketches of the various objects 
of topography as they occur on his lines, so as to exhibit not 
only the points of intersection therewith, but also the direc- 
tions and relative positions of such objects between the lines, 
or within each section, as far as practicable, so that every 
topographical feature may be properly completed and connected 
in the showing. 

" Triangulations, offsets, or traverses, made to determine dis- 
tances that cannot be directly measured, such as those over 
deep streams, lakes, impassable swamps, caftons, etc., will be 
made on the random lines, when random lines are run. All 
particulars will be fully stated in the field notes. 

*'The exhibition of every mile of surveying, whether on 
standard, township, or subdivision lines, and the meanders in 
each section, will be complete in itself, and will be separated 
from other records by a bkck line drawn across that part of the 
page containing the body of notes. The description of the sur- 
face, soil, minerals, timber, undergrowth, etc., on each mile of line 
will follow the notes of survey of such line, and not be mingled 
with them. 

" Particular care will be taken to record at the end of each 
mile the number of chains of mountainous land, heavily timbered 
land, or land covered with dense undergrowth. 

" The date of each day's work will immediately follow the 
notes thereof. 

" Near the end of the field notes of exteriors and immediately 
before the 'general description,* the deputy surveyor will add, 



FIELD NOTES 145 

in the form shown in the specimen field notes, a tabular state- 
ment of the latitude and departure of all boundary lines of the 
township, derived from a traverse table, and will give the totals, 
and the errors in latitude and departure ; said errors shall in no 
case exceed three chains, the prescribed limit for the falling of 
the random north boundary of a township. If a part or the 
whole of one or more boundaries is made up of meander lines, 
the northings, southings, eastings, and westings of the full 
section lines, nearest said meanders, will replace the missing 
N., S., E., or W. township lines, as the case may require, 
thereby presenting the errors of said boundaries of a closed 
survey. 

" If all the exterior lines have been surveyed by the deputy, 
the bearings and distances for the table will be taken from 
his own notes. In a case where some of the boundaries have 
been surveyed under another contract, the deputy will use the 
bearings and distances supplied by the surveyor general, in con- 
nection with those of his own lines ; and, if errors exceed the 
allowance of three chains, specified in paragraph i of the " Pre- 
scribed Limits," the deputy will determine by retracement where 
the error occurs, correct the same before he leaves the field, and 
place the table in his original field notes. 

" Besides the ordinary notes taken on line (and which will 
always be written down on the spot, leaving nothing to be sup- 
plied by memory), the deputy will subjoin, at the conclusion of 
his book, such further description or information touching any 
matter or thing connected with the township (or other) survey 
which he may be able to afford, and may deem useful or neces- 
sary to be known — with a general description of the town- 
ship in the aggregate, as respects the face of the country, its 
soil and geological features, timber, minerals, waters, settle- 
ments, etc. 

'* Following the general description of the township will be 
placed * A list of the names of the individuals employed to 
assist in running, measuring, and marking the lines and corners 

described in the foregoing field notes of township No. of 

the base line of range No. of the meridian, show- 
ing the respective capacities in which they acted." 



146 



LAND SURVEYING 



[Chap. V. 



Specimen of Field Notes. 

Subdivision of T. 15 N., R. 20 E. 

Chains. I commence at the cor. of sees, i, 2, 35, and 36, on the S. bdy. of the 
Tp., which is a sandstone, 6 X 8 x 5 ins. above ground, fiimly set, and 
marked and witnessed as described by the surveyor general. 
Thence I run 

N. 0° 01' W., bet. sees. 35 and 36. 
Over level bottom land. 
4 .50 Wire fence, bears £. and W. 

20.00 Enter scattering cottonwood timber, bears £. and W. F. G. Alexan- 
der's house bears N. 28** W. 
29 .30 Leave scattering cottonwoods, bearing £. and W.; enter road, bears N. 
30.00 S£. cor. of F. G. Alexander's field ; thence along west side of road. 
39.50 To crossroads, bears £. to Mound City; N. to Lake City. F. G. 
Alexander's house bears S. 40® W. The } sec. cor. point will 
fall in road ; therefore 
Set a cedar post, 3 ft. long, 3 ins. sq., with quart of charcoal, 24 ins. 
in the groundi, for witness cor. to \ sec. cor., marked W C ^ S 35 
on W. and 36 on £. face ; dig pits, 18 x 18 X 12 ins. N. and S. of 
post, 3 ft. dist. ; and raise a mound of earth, 3) ft. base, i) ft. high. 
W. of cor. 
40.00 Point for \ sec. cor. in road. 

Deposit a marked stone, 24 ins. in the ground, for i sec. cor. 
The SE. cor. of Pat. Curran's field bears W., 5 Iks. dist. 
40.50 Set a limestone, 15x8x6 ins. 10 ins. in the ground, for witness cor. 
to i sec. cor., marked W C } S on W. face; dig pits, 18 x 18 x 12 
ins. N. and S. of stone, 3 ft. dist. ; and raise a mound of earth, 3} ft. 
base, i^ ft. high, W. of cor. 
Thence along £. side of field. 

50 . 50 NE. cor. of Pat. Curran's field, bears W. 4 Iks. dist. 

51 . 50 Leave road ; which turns to N. 70° W., leads to ferry on Yellowstone 
River ; thence to Lake City. 

57 . 50 Enter dense cottonwood and willow undergrowth, bears N. 54° E. and 

S. 54** W. 
72 . 50 Leave undergrowth, enter scattering timber, bears N. 60^ E. and S. 

60° W. 
80.00 Set a locust post, 3 ft. long, 4 ins. sq., 24 ins. in the ground, for cor. of 
sees. 25, 26, 35 and 36, marked 
T 15 N S 25 on NE., 
R 20 E S 36 on SE., 
S 35 on SW., and 

S 20 on N W. face ; with i notch on S. and E. faces ; from which 
An ash, 13 ins. diam., bears N. 22^ E., 26 Iks. dist., marked T 15 

N R 20 E S 25 B T. 
A sycamore, 23 ins. diam., bears S. 71}° E., 37 Iks. dist., marked 

T15N R20ES36BT. 
A walnut, 17 ins. diam., bears S. 64° W., 41 Ucs. dist., marked 

T 15N R20ES35 BT. 
A cottonwood, 13 ins. diam., bears N. 21 J® W., 36 Iks. dist. 
marked T 15 N R 20 E S 26 B T. 
Last 20.00 chs. of this mile subject to overflow, 2 to 4 ft. deep. 
Land, level bottom. 

Soil, alluvial ; ist rate. No stones were obtainable. 
Timber, scattering cottonwood, sycamore, ash, and walnut ; under- 
growth, cottonwood and willow. 
Dense undergrowth, 15.00 chs. 



MARKING THE CORNERS I47 

x66. Marking the Comers. — After the positions of the cor- 
ners are determined they are marked according to instructions 
issued by the Land Office. The character of the monuments 
set will depend upon the kind of comer to be marked, the charac- 
ter of the country, and the existing conditions. There are four- 
teen different classes of comers, as follows : 

1. Standard township comers. 

2. Closing township corners. 

3. Comers common to four townships. 

4. Comers common to two townships only. 

5. Comers referring to one township only. 

6. Standard section comers. 

7. Closing section corners. 

8. Corners common to four sections. 

9. Comers common to two sections only. 

10. Comers referring to one section only. 

1 1. Quarter-section corners. 

12. Standard quarter-section corners. 

13. Meandered comers. 

14. Comers on reservation or other boundaries not conforming 

to the regular system. 

There are eight different classes of monuments allowed, de- 
pending upon the character of the country and the difficulty of 
transportation. 

1. Stone, with pits and mounds of earth. 

2. Stone, with mounds of stone. 

3. Stone, with bearing trees. 

4. Post, with pits and mounds of earth. 

5. Post, with bearing trees. 

6. Mound of earth, with deposit, and stake in pit. 

7. Tree comer, with pits and mounds of earth. 

8. Tree corner, with bearing trees. 

There are many details in regard to the proper marking of 
corners which can only be learned by experience in this kind of 
surveying. 



148 LAND SURVEYING [Chap. V. 

167. To ESTABLISH A PARALLEL OF LATITUDE.— A paral- 
lel of latitude on the surface of a sphere is a curved line. This 
may be understood from the facts that the meridians converge 
toward the pole, and that a parallel is at every point at right 
angles to the meridian at that point. If vertical lines are drawn 
through every point on a parallel of latitude they will form a 
conical surface, the apex of the cone being at the center of the 
sphere. In the case of a straight line all of the verticals would 
lie in the same plane, and this plane would intersect the sphere 
in a great circle. 

A parallel of latitude may be run out by means of the solar 
attachment to the transit, since by means of this instrument the 
direction of the meridian may be quickly found whenever the sun 
is visible (Art. 85, p. 66). A line which at every point is at 
right angles to the meridian will be a true parallel of latitude. 
This method, however, is found to give results less accurate than 
are required, chiefly on account of the errors in the adjustment 
of the solar attachment. 

A better method of establishing a parallel is by taking offsets 
from a straight line. Two methods of doing this, known as the 
Secant Method and the Tangent Method^ are used in the Public 
Land Surveys. 

168. The Secant Method. — (Fig. 60.)' " This method consists 
of running a connected series of straight lin^s, each six miles long, 



T.13N,lt21E. 

3* »5 



^:::sA^ 7,.^<L .^^Z ,S^CAff r line __ — Y: 

' 5 I s s ? » ! 5 * . 



^ 71 ^ * [Offsers-To^ 4] 

Fig. 60. Secant Method for Establishing a Parallel of 

Latitude. 

on such courses that any one of the lines will intersect the curve of 
the parallel of latitude in two points, separated by an interval of 
four miles ; and from this line thus established, measuring north 





TABLE 4. 
Azimuths op the Secant, and Offsets, in Feet, to the Parallel. 

Ladtode in le£t4iaod colnmn and distance from starting point at top or bottom of the table. 



Lati- 
tude 



30 
31 
3a 
33 
34 
35 
36 
37 
38 
39 
40 
41 
43 
43 
44 
45 
46 
47 
48 
49 



Azlmnths and oflfsets at — 



80^ W A 
X.93 N. 

89»58'.4 
a.oz N. 

a.og N. 

890 68'.3 
3.Z7 N. 

89«68'.2 
3.35 N. 

89<»68'.2 
a.33 N. 

a8«68M 

3.43 N. 

89«58'.0 
3.51 H. 

89o58'.0 
3.61 H. 

«9«»57'.9 
3.70 N. 

89» 57'.8 
3.79 N. 

890 67'.7 
a.89 N. 

89«67'.7 
3.00 N. 

89° 67'.6 
3.ZZ N. 

89*67'.6 
3<33 N. 

89« 57'.4 
3.33 H. 

89<»67'.3 

3.44 N. 

89«B7'.2 
3.57 N. 

89» 57M 

3.70 N. 

89* 67'.0 
3.8a N. 

890 68'.9 
3.96 N. 



lAti. 



6 miles. 



imUe. 



89»68'.7 
0.87 N. 

89»68'.6 
0.91 N. 

89059.6 
0.94 N. 

89°68JS 
0.97 N. 

89«68'J5 
z.oz N. 

89<»58'J> 
Z.05 N. 

89*68'.4 
Z.09N. 

89°68'.3 
I.Z3 N. 

890 68'.3 
X.17 N. 

89° 68' .2 
Z.3Z H. 

89«58'.l 
Z.35N. 

89« 58^.0 
1.30 N. 

89«68'.0 
X.35 N. 

89»68'.0 
Z.40N. 

89*67'.9 
1.4s N. 

89« 67'.8 
Z.50N. 

89«67'.7 
X.55 N. 

890 57'.6 
i.6z N. 

89» 67'.5 
1.66 N. 

89«67'J5 
1.73 N. 

89» 67'.4 
Z.78N. 



similes. 



89«09'.O 
0.00 

89*58'.9 
0.00 

89*68'.9 
0.00 

89«58'.8 
0.00 

89°68'.8 
0.00 

89«»68'.8 
0.00 

890 58'.7 
0.00 

89°68'.6 
0.00 

89«»58'.6 
0.00 

89° 58' .6 
0.00 

89«»68'.5 
0.00 

89« 68^.4 
0.00 

89° 58' .4 
0.00 

89° 68'.4 
0.00 

89°58'.d 
0.00 

89°58'.3 
0.00 

89°58'.2 
0.00 

89°68'.l 
0.00 

89°58'.0 
0.00 

89°68'.0 
0.00 

89° 67'.9 
0.00 



zi miles. 3 iziiles. 3^ miles. 



89° 59' .2 
0.67 S. 

89° 59' .2 
0.70 S. 

89°69'.2 
0.73 s. 

89° 59'.! 
0.76 S. 

89°59M 
0.79 S. 

89° 59'.! 
0.83 S. 

89°89'.0 
0.8s S. 

89°68'.9 
0.88 S. 

89°58'.9 
0.9Z S. 

89°58'.9 
0.94 s. 

89°58'.9 
0.98 S. 

89°58'.8 
Z.03S. 

89° 68'.8 
z.05 s. 

89°58'.8 
Z.08S. 

89° 58'.7 

Z.Z3S. 

89°58'.7 
z.z6 S. 

89° 58'.6 
Z.3Z S. 

89°58'.6 
Z.35 S. 

89°58'J> 
Z.30S. 

89°58'.5 
Z.34 S. 

89°58'.4 
Z.39S. 



5 miles. 



4i miles. 



89°59'J{ 
Z.Z5S. 

89°59'.5 
Z.30 S. 

89°69'.5 
z.35 s. 

89°59'.4 
z.30 S. 

89°69'.4 
z.35 s. 

89°59'.4 
Z.40 S. 

89°59'.4 
Z.46 S. 

89°69'.3 
Z.5Z S. 

89° 59^.8 
Z.56S. 

89°69'.3 
Z.63S. 

89°59'.3 
1.68 S. 

89°69'.2 
Z.74S. 

89»59'.2 
Z.80 8. 

89° 69' .2 
Z.86S. 

89°69'.2 
Z.93 s. 

89°69'.l 
3.00 S. 

89° 59'.! 
3.07 S. 

89° 69'.! 
3.Z4S. 

89°59'.0 
3.33 S. 

89°59'.0 
3.30 S. 

89°69'.0 
3.38 S. 



4 miles. 



89°59'.7 
Z.44S. 

89° 59'.7 
Z.50 S. 

89°69'.7 
Z.56 S. 

89°69'.7 
Z.63 S. 

89° 69' .7 
Z.69 S. 

89° 59' .7 
Z.75 s. 

89° 69' .7 
Z.83 S. 

89°59'.7 
Z.89 S. 

89°59'.7 
1.95 s. 

89°59'.7 
3.03 S. 

89° 59' .7 
3.Z0S. 

89°59'.6 
3.Z7 S. 

89°59'.6 
3.35 S. 

89°59'.6 
a.J3S. 

89°69'.6 
3.4Z S. 

89°59'.6 
3.49 S. 

89° 59'.5 
3.59 s. 

89° 69^.5 
3.67 S. 

89°59'.6 
3.78 S. 

89°59'.5 
3.87 S. 

89° 59'.5 

3.97 S. 



similes. 



3 miles. 



90° (E. or W.) 
Z.54S. 

90°(E.orW.) 
Z.60 S. 

90° (E. or W.) 
Z.67 S. 

90° (E. or W.) 
Z.73S. 

90° (E. or W. 
Z.80S. 

90° (E. or W. 
Z.87S. 

90° (E. or W. 
Z.94S. 

90° (E. or W.) 
3.0Z S 

90° (E. or W. 
3.08 S.) 

90° (E. or W.) 
3.z6 S. 

90° (E. or W.) 
3.34 S. 

90° (E. or W.) 
3.33 S. 

90° (E. or W.) 
3.40 S. 

90° (E. or W.) 
3.48 S. 

90° (E. or W.) 
3.57 S. 

90° (E. or W.) 
3.6^ S 

90° (E orW.) 
3.76 S. 

90° (E. or W.) 
3.86 S. 

90° (E. or W.) 
3.96 S. 

90° (E. or W.) 
3.06 S 

90° (E. or W.) 

3.17 S 



Defleo- 

tion Angle 

and nat. 

tan. to 
Rad.eeft. 



3' 00".2 
0.69 i~s. 

3' 07".4 
0.73 ins. 

3'16".0 
0.75 ins. 

3' 22".6 
0.78 ins. 

3'30".4 
0.8 z ins. 

3' 88".4 
0.84 ins. 

3'4e".4 
0.87 ins. 

3' 66".0 
0.90 ins. 

4' 08".6 
0.93 ins. 

4' 12".6 
0.97 ins. 

4' 21 ".6 
z.ooins. 

4' 31". 2 
Z.04 ins. 

4'40".8 
z.o8ins. 

4^60" .8 
Z.X3 ins. 

6' 01".0 
z.z6 ins. 

5' 11" .8 
z. 30 ins. 

6' 22" .8 
z.34 i-^s. 

6' 34".2 
Z.38 ins. 

5' 4«".2 
Z.33 ins. 

5' 58".6 
z.38 ins. 

8' 11".4 
Z.43 ins. 



3 miles. 



Azimuths and offsets at- 



Deflec- 
tion Angle 
and nat. 

tan. to 
Rad. 66 ft. 



ISO LAND SURVEYING [Chap. V. 

or south, as the case may be, to attain other required points on 
the latitude curve." The o and 6 mile points of a parallel will 
be north of the secant, and the 2, 3, and 4 mile points will be 
south of the secant. 

The instrument is set up south of the township corner where 
the survey is to begin, the distance from the comer being found 
in Table 4 in the column headed "o miles.*' For example, in 
latitude 40° the transit would be set 2.79 ft. south of the comer. 
The direction of the first secant at its initial point is found by 
observing on Polaris (Chapter VII) to obtain the true meridian 
and then laying off the azimuth angle found in Table 4 under 
"o miles." (See Fig. 60.) This angle should be repeated sev- 
eral times to determine accurately the direction of the secant. 
This direction is then prolonged 6 miles. At each mile and half- 
mile point an offset is measured to establish a point on the curve, 
the distance and direction of the offset being shown in Table 4. 
When the 6-milfe point is reached the direction of a new secant 
is found by turning off to the north the deflection angle given in 
the right-hand column of Table 4. The offsets are then measured 
from this line as from the preceding one. The chief advantage 



3. * 

^ _ V an >< E^»*""^ ', I 

t'. *'. ^ 

^«-™^ lit 

[0ffaet3--nibte 6] • ^' 

Fig. 61. Tangent Method for Establishing a Parallel of 

Latitude. 




M 



of this method is that the offsets are short and hence much cut- 
ting is saved in wooded regions. 

169. "Tangent Method. — This method consists in laying 
off from a tme meridian, established by observations on Polaris 
at elongation, an , angle of 90®, producing the direction thus 



TANGENT METHOD 151 

determined, a distance of 6 miles, in a straight line, and 
measuring north therefrom, at half-mile intervals, distances of 
correct length, taken from Table 6 (interpolated if necessary), 
for the given latitude, to attain other points on the latitude 
curve passing through the tangential or initial points. 

"The azimuth or bearing of the tangent at successive mile 
points will be taken from Table 5 to the nearest whole minute 
only, and will be inserted in the field notes, no interpolation 
being required, except when test sights are taken. The true 
bearing between two points on a standard parallel will be, 
derived from Table 5 by taking it in the column headed with 
one-half the distance between said points. The offsets at 
intervals of one mile are inserted in Table 6 ; to obtain the 
length of offsets at the half-mile points, take one-fourth of the 
oflFset corresponding to twice the distance of the half-mile point 
from the tangential point. 

"This method is suitable for running standard parallels and 
latitudinal township Unes in a level open coimtry, where no 
intersections with topographical features will be required ; but, 
in all cases the secant method will be found most convenient." 



IS2 



LAND SURVEYING 



[Chap. V. 



TABLE 5. 
Azimuths op the Tangent to the Parallel. 

[The azimuth is the smaller angle the tangent makes with the true meridian and always 
measured from the north andtowards the tangential pomts.] 



Lati- 
tude. 


I mile. 


a 


mUes. 


3 


miles. 


4 


miles. 


S 


miles. 


6 miles. 








/ 


ff 





/ 


// 





, 


// 





/ 


f* 





/ 


// 





/ 


// 


90 

3a 


88 
89 
89 


69 
69 

69 


30.0 
28.8 
27.6 


89 
89 
89 


68 
68 
68 


69.9 
67.6 
66.0 


89 
89 
89 


68 
68 
68 


29.9 
26.3 
22.6 


89 
89 
89 


67 
67 
67 


69.9 
66.0 
60.0 


89 
89 
89 


57 
67 
67 


29.9 
23.8 
17.5 


89 
89 
89 


56 
66 
66 


698 
62 6 
45.0 


33 

34 
35 


89 
89 
89 


69 
60 
69 


28.2 
24.9 
23.6 


89 
89 
89 


68 
68 
68 


62.6 
49.9 
47.2 


89 
89 
89 


68 
68 
68 


18.7 
14.8 
10.8 


89 
89 
89 


67 
67 
67 


44.9 
39.7 
34.4 


89 
89 
89 


67 
67 
66 


11.2 
01.6 
58.0 


89 
80 
89 


66 
66 
66 


37.4 
296 
21.6 


36 


89 
89 
89 


69 
69 
69 


22.2 
20.8 
19.4 


89 
89 
89 


68 
68 
68 


44.4 

41.6 
38.8 


89 
89 
89 


68 
68 
67 


06.8 
02.6 
68.2 


89 
89 
89 


67 
67 
57 


28.9 
23.3 
17.5 


89 
89 
89 


66 
66 
56 


61.1 
44.1 
36.9 


89 
89 
89 


66 
66 
66 


13.4 
06.0 
66.3 


39 
40 
4« 


89 
89 
89 


69 
69 
69 


17.9 
18.4 
14.8 


89 
89 
89 


68 
68 
68 


36.8 
32.8 
29.6 


89 
89 
89 


67 
67 
67 


63.7 
49.2 
44.4 


89 
89 
89 


67 
67 
56 


11.6 
05.6 
60.3 


89 
89 
89 


56 
56 
56 


29.6 
21.9 
14.1 


89 
89 
89 


66 
55 
66 


47.6 
38.3 
28.9 


4a 
43 
44 


89 
89 
89 


69 
69 
69 


13.2 
11.6 
00.8 


89 
89 
89 


68 
68 
68 


26.4 
23.1 
19.6 


89 
89 
89 


67 
67 
67 


39.6 
34.6 
29.5 


89 
89 
89 


66 
66 
66 


62.8 
46.2 
39.3 


89 
89 
89 


56 
66 
65 


06.0 
67.7 
49.1 


89 
89 
89 


66 
66 

54 


19.2 
09.2 
68.9 


47 


89 
89 
89 


60 
69 
69 


06.0 
06.2 
04.3 


89 
89 
89 


68 
68 
68 


16.1 
12.4 
08.6 


89 
89 
89 


57 
67 
67 


24.1 
18.6 
12.9 


89 
89 
89 


66 
66 
66 


32.1 
24.8 
17.1 


89 

i 


65 

66 
66 


40.2 
31.0 
21.4 


89 
89 
89 


64 
64 
64 


48.2 
37.2 
26.7 


4S 

49 
50 


89 
89 
89 


60 
69 
68 


02.3 
00.2 
68.1 


89 
89 
89 


68 
68 
67 


04.6 
00.6 
66.2 


89 
89 
89 


67 
57 
66 


06.9 
00.7 
64.3 


89 
89 
89 


66 
66 
66 


09.2 
00.9 
62.6 


89 
89 
89 


66 
65 
54 


11.5 
01.2 
60.5 


89 
89 
89 


64 
64 
63 


13.8 
01.4 
48.6 



LaU- 
tade. 


7 


miles. 


8 miles. 


9 


miles. 


10 miles. 


I] 


miles. 


xa 


miles. 








/ 


„ 





, 


„ 





, 


ft 





/ 


„ 





f 


// 





/ 


/» 


30 
31 
3» 


89 
89 
89 


66 
56 
56 


29.8 
21.3 
12.6 


89 
89 
89 


55 
55 
66 


68.8 
50.0 
40.0 


88 
88 
88 


56 
55 
55 


29.8 
18.8 
07.6 


89 
89 
89 


64 
54 

64 


68.7 
47.6 
36.1 


89 
89 
89 


64 
64 
64 


29.7 
16.3 
02.6 


89 
89 
89 


63 
53 
63 


69.7 
45.1 
90.1 


33 
34 
35 


89 
89 
89 


66 
66 

66 


03.6 
54.5 
46.2 


89 
89 
89 


66 

56 
66 


28.8 
18.4 
08.8 


88 
88 
88 


54 
54 
54 


56.1 
44.4 
32.3 


89 
88 
89 


64 
54 
53 


22.3 
08.3 
65.8 


89 
89 
89 


63 
63 
53 


48.5 
34.2 
19.6 


89 
89 
88 


53 
52 
62 


14.8 
69.1 
43.1 


36 


89 
89 
89 


65 
65 

55 


36.6 
25.8 
15.7 


89 
89 
89 


54 
64 
54 


67.8 
46.6 
35.1 


88 
88 
89 


64 
54 
63 


20.0 
07.4 
54.5 


89 
89 
89 


53 
53 
53 


42.3 
28.2 
13.8 


89 
89 
89 


53 
52 
52 


04.5 
49.1 
33.2 


89 
89 
89 


62 
52 
61 


26.7 
09.9 
62.6 


39 
40 
41 


89 
89 
89 


55 

54 
54 


05.4 
54.7 
43.7 


89 
88 
89 


54 
54 
53 


23.3 
11.1 
68.5 


88 
88 
88 


53 
53 
53 


41.2 
27.5 
13.4 


89 
89 
89 


52 
62 
52 


58.1 
43.8 
28.2 


89 
89 
89 


52 
52 
51 


17.0 
00.2 
43.0 


89 
89 
89 


51 
61 
60 


34.9 
16.6 
57.8 


4a 
43 
44 


89 
89 
89 


54 
54 
54 


32.4 
20.8 
08.7 


89 
88 
89 


53 
53 
53 


45.6 
32.3 
18.5 


88 
88 
88 


52 
52 
52 


58.8 
43.8 
28.4 


89 
89 
89 


62 
61 
51 


12.0 
65.4 
38.2 


89 
89 
89 


51 
51 
60 


25.2 
06.8 
48.0 


89 
89 
89 


50 
60 
49 


38.4 
18.5 
57.8 


47 


89 
89 
89 


53 
63 
53 


56.3 
43.4 
30.0 


89 
88 
88 


63 
52 
52 


04.3 
49.5 
34.3 


88 
88 
88 


52 
51 
51 


12.3 
66.7 
38.6 


89 
89 
89 


51 
51 
60 


20.4 
01.8 
42.8 


89 
89 
89 


60 

60 
48 


28.4 
06.1 
47.2 


89 
89 
89 


49 
49 
48 


96.4 
14.3 
51.4 


48 
49 
50 


89 
89 
89 


53 
63 
62 


16.1 
01.7 
46.6 


88 
88 
88 


62 
52 
51 


18.4 
01.8 
44.7 


89 
89 
89 


51 
51 
60 


20.7 
02.1 
42.8 


89 
88 
89 


60 
60 
48 


23.0 
02.4 
40.8 


89 
89 
89 


49 
49 
48 


26.3 
02.6 
38.0 


89 
89 
89 


48 
48 
47 


27.6 
02.8 
37.1 



TANGENT METHOD 

TABLE a 
Offsets, in Chains, from Tangent to Parallel. 



»S3 



Lati- 
tnde. 


imile. 


a miles. 


3 miles. 


4 miles. 


smiles. 


6 miles. 


o 


CAaim. 


CJkaiMs. 


CAanu. 


C/kaiMS. 


Chains, 


ChaiMs. 


30 
3a 


0.006 
0006 
0.006 


0.023 
0.024 
0.025 


0.063 
0.065 
0.067 


0.09 
0.10 
0.10 


0.14 
0.16 
0.16 


0.21 
0.22 
0.23 


33 
34 
35 


0.007 
0.007 
0.007 


0.026 
0.027 
0.028 


0069 
0.061 
0.064 


0.10 
0.11 
0.11 


0.16 
0.17 
0.18 


0.24 
0.26 
0.26 


36 

12 


0.007 
0.008 
0.008 


0.029 
0.031 
0.032 


0.066 
0.068 
0.071 


0.12 
0.12 
0.13 


0.18 
0.19 
0.20 


0.28 
0.27 
0.28 


39 
40 
4X 


0.008 
0.008 
0.009 


0.033 
0.034 
0.036 


0.074 
0.076 
0.079 


0.13 
0.13 
0.14 


0.20 
0.21 
0.22 


0.29 
0.30 
0.32 


4a 
43 
44 


0.009 
0.009 
0.010 


0.036 
0.038 
0.039 


0.082 
0.086 
0.088 


0.14 
0.16 
0.16 


0.23 
0.24 
0.24 


0.38 
0.34 
0.36 


47 


0.010 
0.010 
0.011 


0.040 
0.042 
0.044 


0.091 
0.094 
0.097 


0.16 
0.17 
0.17 


0.26 
0.26 
0.27 


0.36 
0.37 
0.39 


48 
49 
50 


0.011 
0.012 
0.012 


0.046 
0.046 
0.048 


0.101 
0.104 
0.108 


0.18 
0.19 
0.19 


0.28 
0.29 
0.30 


0.40 
0.42 
0.43 



Lati- 
tude. 


7 miles. 


Smiles. 


9 miles. 


xo miles. 


xz miles. 


xa miles. 





CkaiTU, 


Chains, 


Chains. 


Chains. 




Chains, 


3D 
31 
38 


0.29 
0.30 
0.31 


0.37 
0.39 
0.40 


0.47 
0.49 
0.51 


0.68 
0.60 
0.63 


0.71 
0.74 
0.76 


0.84 
0.88 
0.91 


33 
34 
35 


0.32 
0.33 
0.36 


0.42 
0.43 
0.45 


0.53 
0.66 
0.57 


0.66 
0.68 
0.70 


0.79 
0.82 
0.86 


0.96 . 

0.98 

1.02 


36 

12 


0.36 
0.37 
0.38 


0.47 
0.48 
0.50 


0.69 
0.61 
0.64 


0.73 
0.76 
0.78 


0.89 
0.91 
0.95 


1.06 
1.10 
1.14 


39 
40 
41 


0.40 
0.41 
0.43 


0.62 
0.64 
0.56 


0.66 
0.68 
0.70 


0.81 
0.84 
0.87 


0.99 
1.02 
1.06 


1.18 
1.22 
1.26 


4a 
43 
44 


0.44 
0.46 
0.48 


0.58 
0.60 
0.62 


0.73 
0.75 
0.79 


0.90 
0.93 
0.97 


1.09 
1.14 
1.18 


1.31 
1.36 
1.40 


47 


0.49 
0.51 
0.63 


0.64 
0.66 
0.68 


0.81 
0.84 
0.87 


1.00 
1.04 
1.07 


1.22 
1.26 
1.31 


1.45 
1.60 
1.66 


48 
49 
30 


0.56 
0.67 
0.69 


0.71 
0.74 
0.77 


0.91 
0.93 
0.97 


1.12 
1.16 

,20 


1.36 
1.40 
1.46 


1.61 
1.67 
1.73 



154 



LAND SURVEYING 



[Chap. V. 




170. CONVERGENCE OF THE 
MERIDIANS. — The angular con- 
vergence of the meridians, given 
in Table 3, may be computed as 
follows. In Fig. 62 AB is an 
arc of a parallel of latitude and 
EQ the arc of the equator 
intercepted by the meridians 
through 4 and B. ATand BT 
are lines tangent to the meri- 
dians at A and B, meeting the 
earth's axis, prolonged, at T. It 
will be seen that the angle 
BTO equals the angle BOQ, 
which is the latitude of points 
A and B. The angle AO'B 
is the difference in longitude 
of points A and B, The angle 



Fig. 62. 
between the meridians at A and B is the angle A TB 

In the triangle B00\ 

B(y = BO cos O'BO 

= BO cos BOQ 

In the sector AO'By 



In the sector A TB, 
AB 



-^^= angle A TB (approximately) 
BT 



But 



BT= 



BO' 



BC 



sin BTO sin BOQ 



.'. angle ATB =4^, sin BOQ 
BU 

=^ngle AO'B sin BOQ, 

i.e., the angular convergence equals the difference in longitude 

times the sine of the latitude. 



CONVERGENCE OF MERIDIANS 



^SS 



Example. — To find the angular convergence between two 
meridians 6 miles apart in latitude 37®. The length of 1° of 
longitude in latitude 37° is 55.30 miles (Table 7). 



55.30 



X sin 37^ X 6o = 3'.9. 



TABLE 7. 
Length of a Degree in Longitude. 





Dqree of Longi- 




Degree of Longi- 




Dqnree of Longi- 


Lat. 


tude 


Lat. 


tude. 


Lat. 


tude. 




Statute MUtt. 




Statute MUee. 




Statute Miles. 





69.160 


30 


59-944 


60 


34.666 


I 


.150 


31 


0-334 


61 


33-615 


2 


•"? 


32 


58.706 


62 


32.553 
31 .481 


3 


.066 


33 


.060 


f3 


4 


68.992 


34 


57.396 


64 


30.399 


1 


"^■^ 


35 


56.715 


65 


29.308 


36 


.016 


66 


28.208 


- 1 


.647 


37 


55- 300 


67 


27.100 


.491 


38 


54.568 
53-819 


68 


25 983 


9 


•3M 


39 


69 


24.857 


10 


68.116 


40 


53-053 


70 


23 723 


II 


67.898 


41 


52.271 


71 


22.582 


12 


.659 


42 


51 -473 


72 


21 .435 
20.282 


13 


.400 


43 


50.659 


73 


14 


.120 


44 


49 -830 


74 


19.122 


•;i 


66.820 
•499 


t 


48.986 
.126 


M 


:^:?i! 


17 


.158 


47 


47.251 


77 


15.607 


18 


65.797 


48 


46 .362 


78 


14.425 


19 


.416 


49 


45 -459 


79 


13.238 


20 


65.015 


50 


44 542 


80 


12 .047 


21 


64.594 


51 


43 .611 


81 


10.853 


22 


, .'54 


52 


42 .667 


82 


9.656 


23 


"'■% 


53 


41.710 


l^ 


8.456 


24 


54 


40.740 


84 


7.253 


^1 


62.718 


55 


39-758 


85 


6.048 


.201 


56 


38.763 
37.756 


86 


4.841 


27 


61.665 


57 


87 


3-632 


28 


.110 


58 


36.737 


88 


2.422 


29 


60.536 


59 


35 -707 


89 


I .2X1 



CHAPTER VI. 

TRAVERSE IJNB8. — LOCATION OF BUILDINaS. — MISCEL- 

LANBons suRVxnriNa problems. 

TRAVERSE LINES. 

171. TRAVERSES WHICH DO NOT FORM CLOSED FIGURES. — 

A great many surveys, such, for example, as the preliminary sur- 
veys for railroads or pipe lines, call for traverses which do not 
return to the starting point. In this work the line is usually 
measured continuously from one end to the other, and the form 
of notes is commonly as follows. The starting point of the tra- 
verse is called ** Station o," the next station 100 ft. away is 
"Station i,'* the next "Station 2," etc. Every loo^ft. length is 
2^ full station and any fractional distance is called Ih^plus. The 
distance from Station o to any point, measured along the traverse 
line, is the station of that point and is recorded always by the 
number of the last station with the plus station in addition, e.g., 
the station of a point at 872.4 ft. from Station o is 8 + 72.4, 

At the angle points it is customary to measure the deflection 
angles rather than the interior angles because the former are 
usually the smaller. These should be checked in the field by 
"doubling*' the angles (Art. 143, p. 108). 

The notes are kept so as to read up the page. The left-hand 
page is for the traverse notes and the right-hand page for the 
sketch, the stations in the sketch being opposite the same station 
in the notes. Fig. 63 is a set of notes illustrating this type of 
traverse. Frequently no notes are kept in tabular form, all of 
the data being recorded on the sketch. 

172. METHODS OF CHECKING TRAVERSES WHICH DO NOT 
FORM CLOSED FIGURES. — Checking by Astronomical Methods. — 
The angles of any traverse can be checked by determining the 
azimuth of the first and last lines by astronomical methods. 
(See Chapter VII.) But since the meridians converge it is neces- 

156 



CHECKING TRAVERSES 



157 



sary to make proper allowance for this convergence, the amount 
of which can be obtained from Table 3, p. 1 29. 

173. Checking by Cut-Off Lines. — The angles may also be 
checked in some cases by cutting across from one point on the 
traverse to another at a considerable distance ahead, and measur- 
ing the angles from the traverse line at each end of this cut-off 

(Lbpt-Hano Pack.) (Right-Hand Paob.) 



PMimnary5uryefFcrX^YfUt,Knmhci^m 


^.W'^ 


^ 


Sfa. 


fhi/rt 


DeH.Aryli 


%!^ 


OoMTtns 


1 


9 


•♦0«2 


4f/7L 


Nif£ 


NO*06£ 


{ 


) 




6 

*4^ 










« -^tto, 


IviffSrSt 








7 


• t^Uf 


kMl 


Af$0*9f 


AfJoVw 


f < 


1 




6 

s 

4, 


•M?7 


MTrr/r 


/fz/iw 


Af/n»W 


~"^^i^-^ 


1 




3 












^ 




-!"« 










ft$^ 


^ 




z 










.^ 


Kf\ 




1 








MifXw 


HifUsn 




•- 


- 


v 














J 



Fig. 63. Traverse Notes. 

line, thereby obtaining all the angles of a closed traverse in which 
the length of one side only (the cut-off line) is missing. Some- 
times the angle at only one end of the cut-off line can be meas- 
ured, in which case the calculations for checking are not so simple 
as in the former case. When both angles have been measured 
the check consists in simply obtaining the algebraic sum of the de- 
flection angles, while in the latter case the traverse must be 
computed. 



1 58 TRAVERSE LINES [Chap. VI. 

174. Checking by Angles to a Distant Object. — A practical 
and very useful method of checking the azimuth of any line of the 
traverse is as follows. At intervals along the line, measure carefully 
the angle from the traverse line to some well-defined distant ob- 
ject, such as a distinct tree on a hill or the steeple of a church. 
If the survey is plotted and it is found by laying off the angles 
taken to the distant object that these lines do not meet at one 
point on the plan there is a mistake in the angles, and a study of 
the plot will show the approximate location of the mistake. If 
convenient, an angle to the distant object should be taken at 
every transit point. When plotted, if these lines meet at the 
same point in one section of the traverse and in another section 
meet at another point, then there is a mistake in the line which 
connects these two parts of the traverse. Frequently this dis- 
tant point is so far away that it cannot be plotted on the plan. 
In this case as well as when it is desired to check more accu. 
rately than by plotting, the location of the distant point with 
reference to the traverse hne can be computed by using these 

-measured angles, as explained in Art. 408, p. 372. Plotting will' 
not disclose minor errors of a few minutes only. 

175. Checking by Connecting with Triangulation Points. — 
An accurate and practical method of checking both the angles 
and distances of a traverse is to connect the traverse with reli- 
able triangulation points which can be easily identified. (See 
Art. 283, p. 255.) The latitude and longitude of these triangula- 
tion points and the distances between them can be obtained from 
the proper authorities. Sometimes the distances between them 
are not known but they can be computed. Then by connecting 
the traverse lines with these triangulation points by angles and 
distances a closed traverse is obtained, which serves as a good 
check. 

Many surveyors fail to appreciate the value of this method of 
checking and do not realize how many such points are available. 
The information concerning such triangulation points can be 
obtained from The U. S. Coast and Geodetic Survey, The U. S. 
Geological Survey, State surveys, and frequently from City or 
Town surveys. 



GENERAL PRINCIPLES 159 

LOCATION OF BUILDINGS FROM TRANSIT LINE. 

176. METHODS OF LOCATINO BUILDINGS. — Many objects, 
such as buildings, are plotted directly from the survey line. In 
this case the measurements taken should be such as will permit 
the most accurate and rapid plotting. Sometimes where it is 
desirable to shorten the amount of fieldwork, the methods used 
are such as to gain time at the expense of accuracy or of sim- 
plicity in plotting. The accuracy with which such locations are 
made will depend upon the purpose of the survey. In city plans 
the accurate location of buildings is of g^eat importance, while 
in topog^phic maps a rough location is often sufficient. There 
are so many different cases which will arise that this work 
requires considerable skill and judgment on the part of the 
surveyor. 

177. GEOMETRIC PRINCIPLES, — Whether the locations are 
accurate or only rough, the principles involved are the same. 
In order to make clear the various methods used in the location 
of buildings it will be well to enumerate the geometric principles 
involved before giving particular cases occurring in practice. 

A point may be located : — 

(i) By rectangular coordinates, i.e., by its station 
and perpendicular offset. 

(2) By two ties from known points. 

(3) By an angle and a distance from a known point. 

(4) By an angle at each of two known points. 

(5) By a perpendicular swing offset from a known 

line and a tie from a known point. 

(6) By perpendicular swing offsets from two known 

lines. 

A line may be located : — 

(i) By two points on the line. 
(2) By one point on the line and the direction of 
the line. 

178. Ties, offsets, swing Offsets, and range lines. — 

In the above, the word tie is used as meaning a direct horizon- 
tal measurement between two points. 



l60 LOCATION OF BUILDINGS [Chap. VI. 

An offset is the distance from a line, usually at right angles. 

A swing offset is the perpendicular distance to a line and is 
found by trial. The zero end of the tape is held at the point to 
be located and the tape is swung in a short arc about the point 
as a center, the tape being pulled taut and kept horizontal. 
The tape is read from the transit in various positions, and the 
shortest reading obtainable is the perpendicular distance desired. 

A range line is a line produced to intersect the transit line 
or some other line. 

179. GENERAL SUGGESnoilS. — By whatever method the 
buildings are located the following suggestions should be carried 
out. 

fi) All the sides of the building should be measured and 
checked by comparing the lengths of opposite sides. 

(2) Other things being equal, a long side of a building 
should be located in preference to a short side. 

(3) Taes should intersect at an angle as near 90° as practi- 
cable, and never less than 30°. 

(4) One or more clieck measurements should be taken in 
every case. 

(5) In order to secure the best location the surveyor should 
keep constantly in mind how the building or other object which 
is being located is to be plotted. 

In most work of this character it is customary to record the 
measurements to tenths of a foot. How precisely the measure- 
ments should be taken, however, depends upon the scale to 
which they are to be plotted. 

180. Typical Cases. — Although each case will have to 
be dealt with according to circumstances there are certain typi- 
cal cases which will ser\^e as guides. These are illustrated by 
the following examples. 

181. Example I. Building Near Transit Line and Nearly 
Parallel to it. — As will be seen in Fig. 64 swing offsets are 
taken at the two front comers which, together with the tie from 
A to station 1 and the length of the front of the building locate 
points A and B. Then the general dimensions of the building 
are sufficient to plot and check the remaining sides. It is as- 
sumed that the corners of the building are square unless it is 



TYPICAL CASES l6l 

obvious that they are not. The tie from C to station J? is a 
check against an error in the other measurements. 

Plotting. — This building would be plotted thus : — scale 
the distance AX perpendicular (estimated) to the transit line 




and draw a line with triangles parallel to the transit line ; then 
scale Al from station 1 to this parallel line. Point A is then 
located. Point B is located in the same way, AB being used as 
the tie from A, Then by means of triangles and scale the 
building is completed and the distance C2 scaled and compared 
with the notes. Another way to plot point A would be to set 
on the compass the distance lA and swing an arc about i as a 
center ; then, keeping the scale perpendicular to the transit line, 
find where the distance XA will cut this arc, thus locating point 
A, Point B can be similarly located after A has been plotted. 
For the same degree of accuracy distances can be measured 
more rapidly with a scale than they can be laid off with a com- 
pass, therefore the former method is usually more practicable. 

This building might have been located by four ties AO^ Al, 
Bl, and B^. The plotting in this case would be slow because 
at least two of the ties must be swung by use of a compass, and 
inaccurate because the intersections would be bad. 

182. Example II. Building Near Transit Line and Making 
a Slight Angle with it. — Fig. 65 illustrates two ways of locating 
, a building in such a position that the intersection of the transit 
1 line by the long side (produced) can be readily obtained. 

The left-hand building is located by the method of Example I. 
The tie Bl could have been taken instead of BS. It would 
have given a better intersection at B, but since it is a longer tie 
than B2 the fieldwork necessary is slightly greater. If B2 is 



1 62 LOCATION OF BUILDINGS [Chap. VL 

taken Bl might be measured as a check tie although Al would 
make a better check tie since it will also check the measurement 
of the side AB. 

The right-hand figiu-e illustrates another method of locating 
such a building. The front and side of the building are ranged 
out by eye, a method which is thoroughly practical and suffi- 
ciently precise for all ordinary purposes, and the plus station of 
points E and /'are measured. The range lines CE and DFare 
also measured and the check tie C3. C2 could have been taken 
as a check tie ; it would have given a better intersection at C 
than the tie C3^ but it is much longer. 

6j 




TRANSfT LINE ^ v 




Stal Z C 

Fig. 66. 

Plotting. — The left-hand building is plotted as described 
in Example I. In plotting the right-hand building the plus sta- 
tions on the transit line are first scaled. Then with the com- 
pass set at the distance EC an arc is swung from £ as a center. 
From F the distance FC is scaled to intersect the arc, which 
locates point C and the direction of the side CD, The building 
is then plotted with triangles and scale. The check tie C3 
should scale to agree with the notes and the line GC produced 
should strike point E, 

There is little difference between these two methods in the 
amount of fieldwork, there being only one more measurement in 
the right-hand than in the left-hand figures, but one extra check 
is thereby obtained. In plotting, the method used in the right- 
hand figure is shorter. 

183. Example III. Building Located Entirely by Direct 
Ties. — Any building not far from the transit line can be located 
and checked by four ties as in Fig. 66. This method has the 
advantage of being very simple and direct, especially in the field, 
but the plotting of the building calls for the use of the compass 
in two of the ties and hence is less rapid and accurate than 
where swing offsets or ranges can be used. 



TYPICAL CASES 1 63 

Plotting. — The plotting of this building is done by swing- 
ing the tie from one station to a corner of the building and scal- 
ing from the other station the tie to the same comer. Then the 




9ta2 

Fig. 66. 

other comer is plotted in the same way or by using the side of the 
building as one of the ties in case it gives a better intersection. 

184. Example IV. Building Located at a Considerable Skew 
to the Transit Line. — A building which is at a considerable skew 
to the transit line can best be located by range ties as illus- 
trated in Fig. 67. The range ties through A are sufficient to 




Fig. 67. 

locate the buflding, provided AE and AF are not too short in 
comparison with the sides of the building. If the seranges are 
long enough, then BS is a check tie ; but if the ranges are short, 
B3 must be depended upon to determine the position of point 
B and in this event one of the range ties becomes a check. But 
if A is within two or three feet of the transit line it will be well 
to omit one of the ranges and take the additional tie ^C or the 
range tie DC produced. 



1 64 LOCATION OF BUILDINGS [Chap. VI. 

Plotting. — If the ranges are of fair length the building is 
plotted as explained for the right-hand building in Art. 182, but 
if the range ties are short point B is located either by swinging 
the arc with radius EB and scaling B3 or by arc SB and scaling 
EB. Then the direction of AB is determined and the building 
is plotted. CA produced should strik^ at F^ and AF should scale 
the measured distance. 

185. Example V. Buildings at a Long Distance from the 
Transit Line. — It is evident that in this case (Fig, 68) the tape 



A. 



T1W<5IT /" %^ UNE,, 

SlAl 2 3 4. 5 6 

Fig. 68. 

is not long enough to allow the use of swing offsets. Range 
ties may be used provided the building is not so far away that 
the eye cannot judge the range line with reasonable accuracy. 
Sometimes the only methods available are long ties or angles or 
a combination of the two. In any specific case there may be 
some objections to any of these methods, and the surveyor will 
have to decide according to circumstances which method he will 
use. For example, where there are obstacles to the measure- 
ment of ties, the corners of the building may have to be located 
entirely by angles from two points on the transit line. Loca- 
tion by angles is objectionable because it is difficult to plot an 
angle quickly and at the same time accurately. It often hap- 
pens, however, that when a building is at a considerable distance 
from the transit line its accurate position is not required, since 
as a rule the features near the transit line are the important 
ones. This method of "cutting in " the comers of the building 
by angle is often used in rough topographic surveying and is 
decidedly the quickest of all methods so far as the fieldwork is 
concerned. 

Plotting. — The angles are laid off from the transit line 



TYPICAL CASES 165 

with a protractor and the proper intersections determine the 
corners of the buildings. If the building is measured the side 
between the comers located will be a check tie. 

In some cases, e.g., in making a topographic map on a small 
scale, the buildings are not measured at all, their corners being 
simply " cut in " by several angles from different transit points, 
and the shape of the building sketched in the notes. 

186. Example VI. Buiidymgs Located from Other Buildings. — 
Buildings which cannot be conveniently located from the transit 
line on account of intervening buildings may be defined by ties 
from the ones already located. Fig. 69 shows several ways 







I. 
f 



"-y^^ 



.^i 



TWAN srr ^^ ^i j)r^ ^'^'^ '-'^t 



Slu4 



Fig. 69. 



in which such buildings may be located. Any of the preceding 
methods are applicable, using the side of the house as a base-line, 
but it will be found that range ties are almost always prefer- 
able. For example, the bam is located by the distance BK, the 
range tie KC and the tie BC, and checked by the tie BE, An- 
other location of the barn is the distance AK or BK, the range 
tie KC, and the two range ties A/ and CJ, By this latter method 
the directions of both sides of the barn are checked. Still an- 
other location of the point C would be to substitute in the place 
of the range tie CK a swing offset from C to the house. The 
shed is located by the range ties AFzxid FG and by the tie AG, 
The check tie HD in general checks the location of both the 
bam and the shed. If the side HL is ranged out instead of 
the opposite side it will be seen that the tie AL will give a 



i66 



LOCATION OF BUILDINGS 



[Chap. VI. 



poorer intersection at Z. If convenient a tie from Z to ^ or 
the range GF continued to the transit line may be measured as 
a check. 

187. Example Vn. Buildings of Irregular Shape Occa- 
sionally a building of irregular shape has to be located. For 
example, the shop in Fig. 70 is located on the front by ties and 




swing offsets like Example I ; then the direction of AB is deter- 
mined by the range tie BC, The back corner E is determined 
by the ranges FH^nd EH, and by the dimensions of the build- 
ing; FA is assumed parallel to GB. If the angle F \sk right 
angle the tie EF may be taken instead of the range ties FH and 
EH^ but even when F is a right angle it will be well if time will 
permit to take these range distances as they give valuable checks 
on the other measurements which the single tie EF does not 
furnish. ED is scaled along HE produced and the rest of the 
building plotted by its dimensions and checked by AD. 

The ties shown on Fig. 70 to locate the factory chimney 
will locate its sides even if they do not form a regular polygon. 
If such a structure is situated at a considerable distance from 
the transit line probably the best way to locate it is by angles 
and 'distances to the corners, by the measurements of the sides, 
together with a few such ranges as NJ or KJ. 

188. Example Vm. Large City Buildings. — Fig. 71 illus- 
trates the location of several buildings in a city block where 
the transit line runs around the block. The fronts of the build- 
ings are located from the transit line and the rear comers are 
tied together. The range ties are shown by dotted lines and 
other ties by dashes. The angles measured are marked by 



TYPICAL CASES 



167 



arcs. At the curve ^B, the side lines of the building are ranged 
out to point C which is located from the transit line by an angle 




TRANSIT LINE 

Fig. 71. 



and distance and checked by a swing offset ; CD is also measured 
to locate point D on the curve. 

Frequently large buildings have their walls reinforced by 
pilasters, and care should be taken in such cases not to confuse 
the neat line of the wall with the line of the pilasters. 

189. Example IX. Locatioii of Buildings by Angles and 
Distances. — It will be seen from Figs. 71 and 72 that some of 
the buildings have been located by angles and distances from 
transit points. Any of the buildings in the above examples 
could be located by this method, and on account of the rapidity 
with which the work can be done in the field many surveyors 
prefer to use it almost exclusively. 

190. Location of Buildings and Fences from Transit Line. — 
Fig. 72 is a sample page from a note-book illustrating the above 
principles. It will be noticed that in the field notes the letter 
R appears where the lines are ranges. 



1 68 



LOCATION OF BUILDINGS 



[Chap. VI. 




Fig 72. 



RANDOM LINE I69 

MISCELLANEOUS SURVEYING PROBLEMS*. 

191. Random line. — Not infrequently in attempting to 
run a straight line between two points A and B (Fig. 73) it is 
impossible to see one point from the other or to see both points 
A and B from an intermediate set-up on a straight line between 
them. When this condition exists it is necessary to start at one 
point, e.g., Af and run what is called a trial, or random, line AC 
by the method explained in Art. 64, p. 52, in the direction of 
the other end of the line as nearly as can be judged. 

Where the random line passes the point B the perpendicular 
offset YB is measured and also the distance to point Y along AC. 
Unless the random line is very close, say, within about two feet of 
the line AB^ the point Y where a perpendicular to i4C will pass 
through B cannot be accurately chosen by eye. The method 
resorted to in this case is one which has very general application 
in all kinds of surveying work, and is as follows. 




Fig. 73. 

With the transit at A point X is set carefully on the line 
AC and as nearly opposite point B as possible. Then the instru- 
ment is set up at X and 90° turned off in the direction XZ. If 
this line does not strike B (and it seldom will exactly) the dis- 
tance BZ is carefully measured by a swing offset as described in 
Art. 178, p. 159. The distance BZ is equal to the distance XY 
which is added to AX giving the length of the long leg i4 F of the 
right triangle A YB. The distance YB is then measured, and 
AB and angle YAB are easily calculated. 

Angle DA Y has been measured from some previous course 



I70 MISCELLANEOUS SURVEYING PROBLEM [Chap. VL 

such zsAD and the addition of the angle YAB together with 

the known distance AB makes the traverse complete to the 

point B without any further fieldwork. If the transit is now 

moved to B with a view to carrying on the survey it will be 

found that, since A cannot be seen from B^ there is no point on the 

line BA to use as a backsight. But any point such as E can be 

AM 
readOy set on the line AB by making the offset ME = ^^ -j—. 

Another point can be similarly set on AB as a check on the back- 
sight. 




Fig. 73. 

This random line method is sometimes employed when AB 
is a boundary which is covered with shrubs. In such cases, al- 
though the view from AtoB may not be obstructed, it may be so 
difficult to measure the line AB that its length can be more 
easily obtained by the use of the random line while the angle 
DAB may be measured directly at i4. If it is desired to mark 
the line ABhy several intermediate points these may be estab- 
lished by means of perpendicular offsets calculated as described 
above. 

192. Obstacles on line. — When an obstacle of limited 
extent, such as a building or a small pond, lies on the transit 
line various methods are resorted to for prolonging the line 
through such obstructions; the most useful of these methods 
will be explained. 

193. Offsetting Transit Line. — This method is illustrated by 
Fig. 74. It is desired to produce the line AB beyond the house. 
Point B is set on line and as near as is practicable to the house. 



OFFSETTING AROUND BUILDING I71 

The instrument is then set up at B and a right angle ABF laid 
off with the transit. BFv& made any convenient distance which 
will bring the auxiliary line beyond the building. Similarly 
point E is set opposite point A^ and sometimes a second point E^ 
opposite^', points A and -4' being exactly on the transit line. 
These points E and E' need not be set by means of a transit 
set up at A and at A' unless AE is quite long. 

The instrument is then set up at F and backsighted on -£", the 
sight is checked on E\ the telescope inverted, and points G, IT, 
and H set on line. Leaving the telescope inverted, another 
backsight is taken on Ay and the process repeated as described 
in Art. 64, p. 52. Then the transit is moved to point G, and a 
right angle turned off, and point C set on the right angle line, 
the distance GC being made equal to BF, 




Fig. 74. 

Then by setting up at Cand sighting ahead on D, {DH= GC), 
and checking on point i?', {lyiT =^GC), the transit line is again 
run forward in its original location. The distance FG is care- 
fully measured which gives the distance BC, and thus it appears 
why it is so necessary that the lines BF and GC shall be laid off 
at rigth angles by means of the transit. The other offsets AE, 
A'E\ DHy and D'H' are not in any way connected with the 
measurement along the line; they simply define the direction of 
the line so that if convenient it is often only necessary to show 
these distances as swing offsets for the transitman to sight on. 
From what has been said it will be seen that offsets AE' and 
VIF are not absolutely necessary, but they serve as desirable 
checks on the work and in first-class surveying they should not 
be omitted. For obvious reasons the offsets AE^xrid DH should 
be taken as far back from the obstacle as is practicable. 

Should the house be in a hollow so that it is possible to see 
over it with the instrument at A^ the point D, or a foresight of 
some sort (Art. 64, p. 52) should be set on line beyond the house 



1 7^ MISCELLANEOUS SUrVeVING fROBLEiA^ fCHAP. Vl. 

to be used as a foresight when the transit is set up again on the 
original line. The distance may be obtained by an offset line 
around the house or by slope measurements to the ridgepole. 
Sometimes it is possible to place exactly on line on the ridgepole 
of the house a nail or a larger wooden sight which gives an 
excellent backsight when extending the line on the other side of 
the building. 

If the building has a flat roof it may not be out of the ques- 
tion to set a point on the roof exactly on line, move the instru- 
ment to this point on the roof, and prolong the line in this way. 
Under these conditions the transitman will have to be extremely 
careful in the use of his instrument as it will be set up on an in- 
secure foundation. If he walks around the transit he will find that 
it affects the level bubbles and the position of the line of sight ; 
it is therefore well for him if possible to stand in the same tracks 
while he backsights and foresights. Sometimes two men, one in 
front and one behind the transit, can carry on the work under these 
conditions more accurately and conveniently. This method in- 
sures an accurate prolongation of the line, but the distance 
through the building must be measured by an offset method, 
unless it can be done by plumbing from the edge of the flat roof. 

194. Short Transit Sights. — Sometimes the offset BF 
(Fig. 74) does not need to be more than 2 or 3 feet. The 
shorter this offset line can be made, and still clear the build- 
ing, the better. But to lay off the short line BF will require a 
method somewhat different from any that has been hereto- 
fore explained. As the ordinary transit instrument cannot be 
focused on a point much less than about 5 ft. distant it is im- 
possible to set point F directly. The method employed is to 
set a temporary point, say 10 ft. distant, on which the transit 
can be focused, and on a line perpendicular to the original 
transit line. From the transit point to this auxiliary point a 
piece of string may be stretched and the point F set at the re- 
quired distance from B and directly under the string. 

195, Bisection Method. — A method which is economical in 
field work but not very accurate is the following. In Fig. 75 
the instrument is set up at A, backsighted on the transit line, 
and equal angles turned off on each side of the transit line pro- 



MEASURING AROUND OBSTACLES 



173 



duced. Points B' and C are carefully set on one of these lines 
and at convenient distances from Ay and on the other line points 




Fig. 75. 

B' and L" are set at the same distances from A, Then point B 
is placed midway between W and B"y and similarly point C is 
set midway between C and C. The line EC is the prolonga- 
tion of the transit line. Of course the distance B^C should be 
made as long as practicable. The inaccuracy in this method 
lies entirely in laying off the two angles. (See Art. 61, p. 50.) 
In this case the distance AB can be computed from the formula 

~BB*^ 

(approximately). (See foot-note, p. 339.) 



AF ^AB= 
196. 



2AB 
Measuring Around a Small Obstacle. 



In Fig. 76 the 




line AB runs through a tree. Point D is set with the transit at 
A^ and DE is made equal to some convenient short distance and 
laid off at right angles to the transit line by eye. Then AE 
and EB are measured. The distance 



AB = AE 



DE" 



+ EB- 



DE" 



(See foot-note, p. 339.) 



2AE • 2EB 

When DE is taken as some whole number of feet the compu- 
tation of the above is extremely simple. 

This method of measuring around a small obstacle might be 
applied much more generally than it is at present if its accuracy 
and its simplicity were more fully realized by surveyors. 



174 MISCELLANEOUS SURVEYING PROBLEM [Chap. VI. 

197. Equilateral Triangle Method. — While this method re- 
quires much less fieldwork than the offset method described 
above it is at the same time less accurate. Point B (Fig. ^y) is set 
on the transit line as near the building as practicable but so that a 
line BC at 60° with the transit line can be run out. The instru- 
ment is set up at B, backsighted on A, and an angle of 120® laid 
off ; the line BC is made long enough so that wh^n the instru- 
ment is set up at Cand 60® is laid off from it, CjD will fall outside 
the building. BC is measured and CD is made equal to BC 
If the instrument is set up at D and angle CDE laid ofif equal to 
1 20** the line D£ is the continuation of the original transit line, 




and the line BD^BC> This method is subject in three places 
to the errors incident to laying off angles and, when BC and CD 
are small, it has in two of its intermediate steps the disadvan- 
tages due to producing a short line. 

108. INACCESSIBLE DISTANCES. — If the obstruction is a 
pond, points on the far side of it can be set and these should be 
used in producing the transit line. When the line can be pro- 
duced across the obstacles the following methods may be used. 

igg. Inaccessible Distance by Right Triangle Method. — 
In Fig. 78 the line AB is made any convenient length and at 
any convenient angle to the transit line. The line BC is 
laid off at 90° to BA and is intersected with the transit line and 
the distance BC measured, which distance should check with its 
computed value. Also the angle CAB can be measured which 
will check the transit work. 



INACCESSIBLE DISTANCES 



175 




Fig. 78. 

200. Intersecting Transit Lines. — In many kinds of 
surveying work it is necessary to put in points at the intersection 
of two transit lines. It would be an easy matter to set the 
point if two transits could be used, one on each line, and the 
sight simultaneously given by each transitman. As it is seldom 
practicable to use more than one transit in a surveying party the 
following method is resorted to. 

An estimate is made by eye where the lines will cross each 
other and temporary points not more than 10 ft. apart are set on 
one of the transit lines by means of the instrument, enough points 
being marked to make sure that the second line will cross some- 
where among this set of temporary points. A string is then 
used to connect two of these temporary points and the transit 
is set up on the other transit line and the point where the 
second line cuts the string is the intersection point. Some- 
times when the lines cross each other at nearly 90° the intersec- 
tion point can be estimated so closely that only two temporary 
points need be placed on the first line. In other cases, where 
the two transit lines cross at a very small angle, it is impossible 
to tell by eye within several feet where the lines will intersect 
and a number of points must be used because in practice the 
stretching line is seldom applicable for distances much over 
15 ft. For short distances the plumb-line can be used as a 
stretching line. 

201. Inaccessible Distance by Swing Offset Method, — If the 
distance across a pond or river is not great the following method 



1/6 MISCELLANEOUS SURVEYING PROBLEM [Chap. VL 

may be used. It has the advantage of requiring the min- 
imum amount of fieldwork. With the instrument at A 
(Fig. 79) point C is set on the transit line on the far side of the 
river. The instrument is then set up at C and the angle ACB 
measured between the transit line and a loo-ft. swing offset 
from point A. 

A pencil is held vertically at the lOO-ft. mark of the tape 
and while the zero point is held firmly at A the tape, which 
is constantly kept horizontal and taut, is swimg slowly in 
an arc ad. The transitman, using the tangent screw, can fol- 
low the pencil with the vertical cross-hair of the transit, stop- 
ping the cross-hair when the pencil is in its farthest position 




Fig. 79. 

from A. Then as the tape is swung the second time he can 
check his setting and when this is established the angle ACB is 
read. The distance AC then is very easily calculated. It 
should be noted, however, that if AC is several times as long as 
AB the resulting error in ACmzy be so great as to prohibit the 
use of this method where very precise results are required. 
There is no reason why the swing offset could not be made at 
C with the instrument at /I if more convenient. 

202. Inaccessible Distance by Tangent Offset Method. — In the 
method described above the distance across the pond may be so 
great that 100 ft. will be too short a base to use, or point A may be 
situated on ground sloping upward towards B so that a swing offset 



INACCESSIBLE DISTANCES 



177 



cannot be made. In such cases the line AB (Fig. 80) can be laid 
off at right angles to the transit line and of any convenient length. 



TRANSfT i 


\[fr m 


UNE 


\ 


A i 


11 lit 


^y^^ 


^"%. 


1 


& i 




^•^H 


N 


1^ 


' 



Fig. 80. 

Then the angle ACB is measured and the line AC computed. 
By another set-up of the instrument the angle B can be meas- 
ured as a check, and if the line BC does not cut across the pond 
its length can also be measured as a further check. 

203. Inaccessible Distance by Oblique Triangle Method. — 
Often the shores of a stream are covered with trees so that none 




of the above methods are applicable. It may be convenient to 
measure a line AB (Fig. 81) in but one direction along the shore. 
In this case the point C is first carefully set on the opposite side, 
the line AB measured along the shore, and the angles at A and 



178 MISCELLANEOUS SURVEYING PROBLEM [Chap. VI. 

• 
at C are measured. The distance AC can then be computed. 
It will be well also to set up at B and measure the angle £ as a 
check on the work. At the time when point C is set it is also 
good practice to set a point further ahead on the line, to use as a 
foresight to check the transit line when the instrument is moved 
across the river. 

204. To Obtain the Distance Between Two Inaccessible Points 
by Observation from Two Accessible Points. — In Fig. 82 the pomts 
A and B are inaccessible and it is desired to obtain the distance 
AB and the angle that AB makes with the transit line. From 
the point D the distance DC and the angles BDA and ADC are 
measured, and similarly at C the angles ACB and BCD are meas- 
ured. AB can then be calculated as follows : — in the triangle 
CBD compute CB\ in triangle ACD compute AC\ and in the 
triangle ACB calculate AB^ the inaccessible distance. In the tri- 



FlG. 82. 

angle ACBi angle ABC can be computed, which, together with 
the measured angle BCD^ will give the difference in direction 
between AB and CD. It is not at all necessary that DC should 
have, been measured as one straight line in the traverse ; the trav- 
erse might have run as indicated by the dotted lines, but in such 
an event the distance CD and the necessary angles could have 
been easily figured so that it could be reduced to the above prob- 
lem. 



INACCESSIBLE DISTANCES 1 79 

This problem occurs when the distance between two triangu- 
lation stations, A and Bt and the azimuth of AB are desired and 
when it is inconvenient or impossible to measure the line AB or 
to occupy the points with the transit. 

20$. To Obtain the Inaccessible Distance Between Two Ac- 
cessible Points by Observations on Two Inaccessible Points of 
Known Distance Apart. — In this case (Fig. 82) i4 and B are the 
two accessible points and C and D are the two inaccessible points 
but the distance DC is known ; the distance i45 is required. 
With the transit at A, the angles CAD and DAB are measured ; 
at B the angle CBD and ABC are measured. The length of 
the line CD is 'known. While it is simple to obtain CD in terms 
of i4J5, it is not easy to directly determine AB in terms of CD \ 
it will be well therefore to use an indirect method. Assume 
i4£ as unity. Then by the same process as described in the 
preceding problem the length of CD can be readily found. 
This establishes a ratio between the lengths of the lines AB and 
CA and the actual length of CD being known the distance AB 
can be computed. 

A problem of this sort would occur under the following 
circumstances. If the distance CD between two church spires 
were accurately known (from a triangulation system) and it is 
desired to use this line CD as a base-line for a survey, two points 
A and B could be assumed, and the distance between them and 
the azimuth ot AB could be found by this method. 



CHAPTER VII. 



OBBSRVATIONB FOR MBRIDIAN AND LATXTX7DB. 



O^SERVATIOHS FOR HERXDIAN. 






LMHni Clony#ien 



3o6. To ESTABLISH A TRUE MERIDIAlf LIRE BY OBSERVA- 
TION ON POLARIS WITH THE TRANSIT. — On account of the 
earth's daily rotation on its axis all heavenly bodies appear to 
revolve once a day around the earth. Stars in the south appear to 

revolve in large circles parallel 
to the daily path of the sun. 
As we look farther north the 
apparent size of the circles 
grows smaller. The center of 
these circles is the nortA pole of 
the celestial sphere^ a point in 
the sky in the prolongation of 
the earth's axis. The pole-star 
(Polaris) revolves about the 
pole in a small circle whose 
radius is less than a degree and 
a quarter (Fig. 83). This an- 
gular distance from the pole to 
a star is called its polar distance. 
When the star is directly 
above the pole its bearing is 
the same as that of the pole 
itself and the star is said to be 
at upper culmination. At this 
instant it is in the true meri- 
dian. About twelve hours later 
it will be below the pole at 
lower culmination and will be 
again in the true meridian. About half-way between these 
two positions the star reaches its greatest east or west bearing, 

i5q 




Fig. 88. 



MERIDIAN OBSERVATIONS ON POLARIS 



I8l 



and at such times is said to be at its greatest elongation. At 
either eastern or western elongation the star's bearing is not 
changing perceptibly because it is moving almost vertically, a 



»T3 
O 
to 

H 

o 

o 

g 

n 

o 



H 
S 

CO 

S 

n 
2 



► 
w 

en 
H 
W 
I« 

W 

r 
o 
5: 
o 
> 

H 

o 
a; 



NOIiVNIWiriD H3AV01 iv sisiHvaoj 
NaHM SNOixviiaxsNOO anx ^o NOixisod 




2; 
o 

H 

< 
o 

o 
2; 

CCS 

M 
H 

CO 

< 

H 






O 
H 

CO 

O 

u 

M 
S 
H 

O 



Fig. 84. 



Ursa Major ^ 

* I 

Position of the Constellations when 
Polaris is at UPPER CULMINATION. 

Relative Position of the Constellations near the 
North Pole. 



o 



1 82 OBSERVATIONS FOR MERIDIAN [Chap. VLL 

condition which is most favorable for an accurate observation. 
At cuhnination the star is changing its bearing at the maximum 
rate, and therefore this is not as good a time to make an accurate 
observation as at elongation. This star moves so slowly, how- 
ever, that even at culmination its bearing can be obtained with 
sufficient accuracy for determining the decUnation of the needle. 
Polaris can be easily found by means of two conspicuous con- 
stellations near it, Cassiopeia and Ursa Major, The seven most 
conspicuous stars of the latter form what is commonly known as 
the " Great Dipper'' (Fig. 84). The two stars forming the part 
of the bowl of the Dipper farthest from the handle are called the 
^^ pointers " because a line through them points almost directly 
at the pole. On the opposite side of Polaris is Cassiopeia, shaped 
like the letter W. A line drawn from h * Cassiopeia^ the lower 
left-hand star of the W, to f Ursa Majoris, the middle star of the 
Dipper handle, passes very dose to Polaris and also to the pole 
itself. 

207. OBSERVATION FOR MERIDIAN ON POLARIS AT 
Elongation. — When the Dipper is on the right and Cassiopeia 
on the left, Polaris is near its western elongation ; when the dipper is 
on the left Polaris is near eastern elongation. When the constella- 
tions are approaching one of these positions the transit should be 
set over a stake and leveled, and the telescope focused upon the 
star.f Unless the observation occurs at about sunrise or sunset 
it will be necessary to use an artificial light to make the cross-hairs 
visible. If the transit is not provided with a special reflector for 
throwing light down the tube a good substitute may be made by 
cutting a small hole in a piece of tracing cloth or oiled paper and 
then fastening it over the end of the telescope tube by a rubber 
band. If a lantern is then held in front and a little to one 
side of the telescope the cross-hairs can be plainly seen. The 
star should be bisected by the vertical wire and followed by 
means of the tangent screw in its horizontal motion until it no 

• The Greek Alphabet will be found on p. 516. 

t It is difficult to find a star in the field of view unless the telescope is focused 
for a very distant object. The surveyor will find it a convenience if he marks on 
the telescope lube the position of the objective tube when it is focused for a distant 
object. 



OBSERVATIONS AT ELONGATION 1 83 

longer changes its bearing but moves vertically. (It will be seen 
from Fig. 83 that when the star is approaching eastern elonga- 
tion it is moving eastward and upward ; when approaching western 
elongation it is moving westward and downward.) As soon as 
this position is reached the telescope should be lowered and a 
point set in line with the vertical cross-hair at a distance of sev- 
eral hundred feet from the transit. Everything should be ar- 
ranged beforehand so that this can be done quickly. Immediately 
after setting this point the instrument should be reversed and 
again pointed on the star. A second point is then set at one 
side of the first. The mean of these two points is free from the 
errors of adjustment of the transit. If the instrument is in ad- 
justment, of course, the first and second points coincide. On 
account of the great difference in altitude between the star and 
the mark the elimination of instrumental errors is of unusual 
importance (Art. 79, p. 61). For 10 minutes of time on either 
side of elongation the bearing of the star does not change more 
than 5 seconds of arc and therefore there is sufficient time to 
make these two pointings accurately. 

After the direction of the star at elongation has been found, 
the meridian may be established by laying off an angle equal to 
the azimuth, or true bearing of the star. Since this angle to be 
laid off is the horizontal angle between the star and the pole, it 
is not equal to the polar distance but may be found from the 
equation : — 

o- o^ » T- -D • Sin Polar Distance of Star * 

Sm Star s True Beanng = p; — = — : — 

Cos Latitude 

The mean polar distances for the years 1906 to 1920 m^y be 

* This equation may be derived as follows ; in Fig. 83, let P represent the 
pole, Z the zenith, and £ the position of the star at elongation. Then by spherical 
trigonometry, 

sin PZE sin P£ 

sin ZEP^^n ZP' 

But PZE is the angle between the two vertical circles and equals the bearing. 
ZEP^go^ because ZE is tangent to the circle WUEL^ which represents the path 
of Polaris. PE is the polar distance and ZP may be shown to be equal to 
90^ - latitude. 

Hence, sin /'Z-ff-?!?^. 

cos lat. 



1 84 



OBSERVATIONS FOR MERIDIAN [Chap. VII. 



TABLE 8. 
Mean Polar Distances of Polaris.* 



Ytttf. 


Mean Polar Distance. 


Year. 


Mean Polar Dbtance. 




* // 




/ /. 


1906 


I II 41.05 


1914 


I 09 12.07 


1907 


I II 22.37 


1915 
1916 


1 08 53.51 


1908 


I II 03.71 


I 08 34.97 


1909 


I 10 45.07 
I 10 26.44 


1917 


I 08 16.45 


1910 


1918 


I 07 57.94 


1911 


I 10 07.82 


1919 


I 07 39.45 
I 07 20.98 


1912 


I 09 49.22 


1920 


1913 


I 09 30.64 


. . 





■; 



found in Table 8. The latitude may be obtained from a reliable 

map or by observation (Arts. 216-17, p. 196). 

When the transit is set up at the south end of the line the 

^^^^_^ angle thus computed must be laid off to the right if 
the elongation is west, to the left if the elongation is 
east. A convenient and accurate way of laying off 
the angle is by measuring the distance between the 
two stakes A and B (Fig. 8$), and calculating the per- 
pendicular distance BC which must be laid off at the 
north stake B to give a meridian AC. 



* The above table was derived from data furnished by the Sup> 
erintendent of the United States Coast and Geodetic Survey. The 
Mean Polar Distance is the average value of the polar distance for 
the entire year. 

In taking the polar distance from the table for the purpose of 
looking up its sine the student should keep in mind the degree of 
precision desired in the computed azimuth. If the azimuth is to be 
within about one minute of the true value the polar disunce need be 
taken only to the nearest minute, but if the azimuth is to be correct 
within a few seconds the polar distance should be taken to the nearest 
second. It should be noted however that since the values ^ven in 
the table are only the average values for the year there will in general 
be an error of a few seconds due to neglecting the variation of the 
polar distance during the year. The exact value for every day in the 
year may be found in the "American Ephemeris and Nautical 
Almanac," published by the Bureau of Equipment, Navy Depart- 
FlG. 86. ment. 



OBSERVATIONS AT CULMINATION 1 8$ 

208. OBSERVATION FOR MERIDIAN ON POLARIS AT CULMI- 
NATION. — At the instant when Polaris is above the pole the 
star f Ursce Majoris will be almost exactly underneath Polaris. 
When Polaris is below the pole h Cassiopeia will be almost di- 
• rectly below Polaris (Fig. 84). In order to know the instant 
when Polaris is exactly on the meridian it is necessary first to 
observe the instant when one of these two stars is vertically 
below Polaris. From this the time when Polaris will be on the 
meridian can be calculated by adding a certain interval of time, 
and the meridian line can thus be directly established. This 
interval of time was, for f Ursa Majoris^ about 2"36» in the year 
1900, and it increases about 2 1 » per year. The intervals computed 
by this rule are only approximate, but are sufficiently accurate for 
many purposes and, as the change is very slow, the rule is good for 
many years. It may also be used for any latitude in the United 
States. When f Ursa Majoris cannot be used, as is the case in 
the spring of the year, especially in northern latitudes, a sim- 
ilar observation can be made on h Cassiopeia, The interval 
for this star was 3"24' for 1900, with an annual increase of 
about 20». 

The observation to determine when the two stars are in the 
same vertical plane is at best only approximate, since the instru- 
ment must be pointed first at one star and then at the other ; 
but since Polaris changes its azimuth only about i minute of 
angle in 2 minutes of time, there is no difficulty in getting fair 
results by this method. The vertical hair should first be set 
on Polaris, then the telescope lowered to the approximate 
altitude of the other star to be used. As soon as this star 
comes into the field the vertical hair is again set carefully on 
Polaris. As it will take the other star about 2 minutes to reach 
the center of the field there will be ample time for this pointing. 
Then the telescope is lowered and the instant when the star 
passes the vertical hair is observed by a watch. This will be the 
time desired, with an error of only a very few seconds. The time 
of culmination should then be computed as described above and 
the vertical hair set on Polaris when this computed time arrives. 
The telescope is then in the meridian which may be marked on 
the ground. 



1 86 



OBSERVATIONS FOR MERIDIAN 



[Chap, VII. 



It will be seen that in this method the actual error of the 
watch has no effect on the result since it is used only for meas- 
uring the interval of a few minutes. The error in the meridian 
obtained by this method will seldom exceed one minute of angle. 

209. To Find the Standard Time of Culminatioii and Elonga- 
tion. — The approximate times of culmination and elongation of 
Polaris for the ist and isth of each month in the year 1907 may 
be found in Table 9. 

TABLE 9. 

Approximate Times of Culmination and Elongation of Polaris 

Computed for the 90TH Meridian West of Greenwich, 

for the Year 1907. 



Date. 



7907 
Jan. I . 

" IS • 
Feb. I . 

". IS . 
Mar. I . 

a" '5- 
Apr. I . 

*• 15 . 
May I . 

" 15 . 
Jun. I . 

" IS . 
Jul. I . 

" 15. 
Aug. I . 

Sep. I . 

" IS . 
Oct. I . 

" 15 . 
Nov. I . , 

Dec. I . 

" 15 • 



ulminatioa. 


Western 


Lower 


Eastern 


Elongation. 


Culmination. 


Elongation. 


A m 


h m 


h m 


k m 


6 44 


12 39 


18 42 


49 


5 49 


II 44 


17 47 


23 50 


4 41 


10 36 


16 39 


22 42 


3 .46 


2 ^l 


15 44 


21 47 


2 51 


8 46 


14 49 


20 52 


I 56 


7 51 


13 54 


19 57 


49 


6 44 


12 47 


18 50 


23 SO 


5 40 


II 52 


17 55 


22 47 


4 46 


10 49 


16 52 


21 52 


3 51 


9 54 


15 57 


20 45 


2 44 


8 47 


14 50 


'2 51 


I so 


7 53 


13 56 


18 48 


47 


6 50 


12 53 
II 58 


17 53 


23 48 


5 55 


16 47 


22 42 


4 49 


10 52 


15 52 


21 47 


3 •?4 


I 57 


14 45 


20 40 


2 47 


8 50 


13 SO 


19 45 


I 52 


• 7 55 


12 47 


18 42 


49 


6 52 


II 53 


17 48 


23 51 


5 58 


10 46 


16 41 


22 44 


4 51 


2 51 


15 46 


21 49 


3 56 


8 47 


14 42 


20 45 


2 52 


7 52 


13 47 


19 50 


I 57 



To find the time for any other date interpolate between the 
values given in the table, the daily change being about 4 min- 
utes. 



STANDARD TIME 1 8/ 

In order to find the exact time of culmination or elongation 
for any observation it would be necessary to take into account 
the latitude and longitude of the place and the exact date of the 
observation. The times given in Table 9 are only approximate 
in any case and are to be regarded merely as a guide so that the 
surveyor may know when to prepare for his observations. 

The times are computed for mean local astronomical time at 
the 90th meridian west of Greenwich and for the year 1907. 
These numbers increase about J minute each year so that this 
table may be used to obtain approximate results for subsequent 
years. Astronomical time begins at noon of the civil day of the* 
same date and is reckoned from Oh to 24^ e.g., 18** would mean 
6^ A.M. The tabular numbers are nearly correct for the Stand- 
ard Meridians, i.e., the 7Sth, 90th, 105th, and 120th west of 
Greenwich. All watches keeping "railroad time,'* or "standard 
time," are set to the local mean time of one of these four meri- 
dians (Art. 86, p. 6S). To find the watch time of culmination 
or elongation for any other meridian, first find the difference 
in longitude in degrees between the place of observation and 
the standard meridian, and then convert this into minutes and 
seconds of time by dividing by 15, since 15° of longitude are 
equivalent to one hour of time. The standard, or watch, time 
of the observation is then obtained by adding this correction to 
the time taken from the table if the place is west or by sub- 
tracttng it if the place is east of the standard meridian. 

210. MERIDIAN OBSERVATIONS ON POLARIS WITH THE 
Compass. — In determining a meridian with the compass the 
observations are made as described for the transit except that 
the following modifications will be necessary. Suspend a long 
plumb-line a few feet away from the point where the instru- 
ment is to be set. Since the rear sight is the only part of the 
compass to be used in the observation it may be unscrewed 
from the compass and fastened to a piece of board. This board 
should be placed on a table. The compass sight may then be 
shifted to the right or left to bring it in line with the star and 
the plumb-line. The plumb-line should be illuminated by means 
of a lantern. ' The direction of the star may be marked by set- 
ting stakes in line. If the observation is made at elongation the 



1 88 OBSERVATIONS FOR MERIDIAN [Chap. VII. 

meridian should be laid out as described in Art. 207. In finding 
the declination of the needle the compass is set up over one of 
the meridian stakes and sighted at the other, when the declina- 
tion can be read off directly. In order to obtain as nearly as 
possible the mean value, of the declination this should be done 
at about 10 A.M. or 5 to 6 P.M. because at these times the 
needle is in its mean position for the day. 

211. Meridian Observation on Polaris at any Time 

WITH THE Transit. — in order to make this observation, it is necessary to 
know the local time very closely. As in most cases the time which the surveyor 
canies is ** standard time *' it is assumed that such is the case here. The observa- 
tion itself consists in either marking the direction of the star, as previously de- 
scribed, and noting the time by the watch when the star is sighted ; or in repeating 
the angle between the star and some reference mark, the time of each pointing on 
the star being noted. In the latter case, take the average of the observed 
times and assume that it corresponds to the average angle. This is very nearly 
true if the observations extend over a few minutes of time only. 

After finding the standard time of the observation, the next step is to compute 
the hour angle of the star at the time of the observation. Take from the Nautical 
Almanac : (i) the right ascension of Polaris for the date; (2) the right ascension 
of the ** mean sun " for the date ; (3) the increase in the sur^s right ascension since 
Greenwich noon, which is found in Table III in the Appendix to the Nautical 
Almanac. Remember that the dates in the Almanac are in Astronomical time 
(Art. 209, p. 186). Reduce the standard time to local time by adding or subtract- 
ing the difference in longitude expressed in hours, minutes, and seconds, remem- 
bering that if the place is west of the standard meridian the local time is earlier 
than standard time and vice versa. To the local time add the sun's right ascen- 
sion and the correction from Table III, Appendix, Nautical Almanac. The result 
is the sidereal time. From this subtract the star's right ascension, and the result 
is the hour angle of the star reckoned from the meridian from oh to 24h in the di- 
rection of the star's apparent motion. Convert this angle into degrees, minutes, 
and seconds. The azimuth of the star may now be computed from the formula, 

^ „ sin / 

• tan Z =- 



cos L tan Z> — sin Z cos / 



where Z ^ the azimuth, or true bearing ; / — the hour angle ; Z i>« the latitude ; 
D <-i the declination » go'' — the polar distance. If the hour angle is between oh 
and I2h the star is west of the meridian ; if between I2h and 24h it is east of the 
meridian (see Example below). 

In the " Manual of Surveying Instruction " issued by the General Land Office 
a set of tables is given which will enable the surveyor to perform all of the above 
work by simple inspection and without the aid of the Nautical Almanac. 



* See Hayford's Geodetic Astronomy, p> an, Art. 193. 



AZIMUTH OF POLARIS AT ANY HOUR 1 89 



Example. 

Observation on Polaris for azimath April 15, 1908. Latitude 38^ 58^. Longi. 
tude 92® 25^. Angle between a mark (approximately N.W.) and Polaris is 
repeated 6 times* Watch im 138 fast. The times are 





8h 35m 
8 37 
8 38 
8 39 
8 41 

8 43 


40» 
20 
50 
59 
30 
00 


Mean of 6 readings 8 39 
Watch fast i 
True Central time 8 38 
Longitude of Standard Meridian 


26.2 

13 

»3 
6 



Greenwich time 14^ 38™ 13' 

From Nautical Almanac, Right Ascension of " Mean Sun ** at Greenwich Mean 
Noon a- ih 32™ 57S.82 ; Right Ascension of Polaris =» ih 25m ois.47 ; Declination 
of Polaris — + 88** 48' 52'' ; Correction from Table III (Nautical Almanac) for 
Greenwich Time = I4h ^S^ =» 2^ 24s. 2 



92° 


25' = 


6h ogm ^ffi 


.'. longitude correction » 


09m 40* 


Mean of observed times 




8h 38m 13s 


Longitude correction 




9 40 


Local time 




8 28 33 


Right Ascension " Mean Sun 


>» 


I 32 58 


Correction (Table III) 




2 24 


Sidereal time 




10 03 55 


Right Ascension Polaris 




I 2$ 01 


Hour Ang^e Polaris 


r» 


8h 38^548 




/- 


129^3' 30" 


log cos Z = 9.89071 


log! 


sin Z « 9.79856 


log tan Z> » 1.68413 


log< 


cos/- 9.80558 (n)* 


1.57484 




9.60414 (n) 


37.570 




- .4019 


.402 






37.972 






log sin / s- 


9.88600 


log denominator — 


1.57946 


log tan Z B 


8.30654 


Z« 


i<>09' 


37" W. of N. 



* The n after the logarithm indicates that the number corresponding is negative 



190 OBSERVATIONS FOR MERIDIAN [Chap. VU. 

212. Solar OBSERVATIOIVS. — Where great accuracy is not 
required many surveyors prefer solar observations because 
they can be made without much additional work, while star 
observations have to be made at night and require special 
arrangements for illuminating the field of view and the mark. 
If it is sufficient for the purpose in view to obtain the azimuth 
within i minute of angle solar observations will answer. In 
making these observations with the ordinary transit it is neces- 
sary to have some means of cutting down the sun's light so 
that it will not be too bright for the eye while making point- 
ings. This is usually effected by placing a dark glass over the 
eyepiece. A dark glass in front of the objective wiU introduce 
error into the pointings unless the faces of this glass have been 
made plane and exactly parallel. If the instrument is not pro- 
vided with a dark glass the observation may be made by hold- 
ing a white card back of the eyepiece while the telescope is 
pointing at the sun. If the eyepiece tube is drawn out the 
sun's disc and the cross-hairs can both be sharply focused on the 
card. By this means pointings can be made almost as well as 
by direct observation. It is also well to cut down the amount 
of light entering the objective by having a cap with a hole in 
the center or by using a piece of tracing cloth as explained in 
Art. 207, p. 182. 

213. OBSERVATION FOR MERIDIAN BY EQUAL ALTITUDES 
OF THE SUN IN THE FORENOON AND AFTERNOON. — This obser- 
vation consists in measuring in the forenoon the horizontal angle 
betweeii the sun and some reference mark at the instant when 
the sun has a certain altitude, and again measuring the angle 
when the sun has an equal altitude in the afternoon. If the 
distance of the sun from the equator were the same in the two 
cases the horizontal angles between the sun and the meridian 
would be the same in both observations, hence the mean of the 
two readings of the horizontal circle would be the reading for 
the meridian. But since the sun is changing its distance from the 
equator the measured angles must be corrected accordingly. 
The correction is computed by the equation 

COS L sin / 



MERIDIAN BY SOLAR OBSERVATIONS I9I 

in which X = the correction to the mean vernier reading, 
d^ the hourly change in declination of the sun taken from 
Table 10 and multiplied by half the number of hours between 
the two observations, L = the latitude, and / = half the elapsed 
time converted into degrees, minutes, and seconds. Since the 
hourly change for any given day is nearly the same year after 
year an almanac is not necessary but the table given below is 
sufficient. 

TABLE 10. 

Hourly Change in the Sun's Declination. 

ist. loth. 20tb. 30th. 

January +12" + 22" +32" +41" 

February +43 +49 +54 

March +57 +59 +59 +58 

April +58 +54 +49 +46 

May +45 +39 +39 +23 

June +21 4-12 +02 —09 

July —10 —19 —28 —36 

August -38 —44 -49 -54 

September —54 —57 —58 —59 

October -58 -57 -54 -49 

November —48 —42 —34 —25 

December —23 —14 —02 +10 

The observation is made as follows : — * at some time in the 
forenoon, preferably not later than 9 o'clock, the instrument is 
set up at one end of the line the azimuth of which is to be 
found, and one vernier is set at 0°. The vertical cross-hair is 
then sighted at the other end of the line and the lower plate 
clamped. The upper clamp is loosened and the telescope turned 
until the sun can be seen in the field of view. The horizontal 
cross-hair is to be set on the lower edge of the sun and the 
vertical cross-hair on the left edge. Since the sun is rising and 
also changing its bearing it is difficult to set both of the cross- 
hairs at once and it will be found easier to set the horizontal 
hair so that it will cut across the sun's disc leaving it clamped 
in this position while the vertical hair is kept tangent to the left 
edge of the sun by means of the upper tangent screw. When 
the sun has risen until the lower edge is on the horizontal hair 

* The nearer the sun is due East or due West, the better the result. 



192 OBSERVATIONS FOR MERIDIAN [Chap. VIL 

the instrument is in the desired position and after this position 
is reached the upper tangent screw should not be moved. As 
soon as this position is reached the time is noted. Both the 
vertical and the horizontal circles should now be read and the 
angles recorded. 

In the afternoon, when the sun is found to be nearly at the 
same altitude as at the forenoon observation, the instrument 
should be set up at the same point and again sighted on the 
mark. The observation described above is repeated, the point- 
ings now being made on the lower and right edges of the disc. 
The telescope is inclined until the vernier of the vertical circle 
reads the same as it did at the forenoon observation. When the 
sun comes into the field the vertical hair is set on the right edge 
and kept there until the lower edge is in contact with the hori- 
zontal hair. The time is again noted and the verniers are read. 
If desired, the accuracy may be increased by taking several pairs 
of observations. The mean of the two circle readings (suppos- 
ing the graduations to be numbered from o° to 360® in a clock- 
wise direction) is now to be corrected for the sun's change in 
declination. The correction as obtained by the formula given on 
p. 190 is to be added to the mean vernier reading ii d is minus, 
and subtracted if d is plus, i.e., if the sun is going south the mean 
vernier reading is east of the south point, and vice versa. 
When the circle reading of the south point is known the true 
bearing of the mark becomes known and the bearings of other 
points may be found (see Example below). 

The disadvantage of this method is that it is necessary to be 
at the same place both in the forenoon and afternoon, whereas in 
many cases the surveyor might in the afternoon be a long distance 
from where he was working in the forenoon. 

Example. 

Latitude 42® 18^ N. April 19, 1906. 

A.M. Observation. P.M. Observation. 

Reading on Mark, o^oc/oc/' Reading on Mark, o^oq'oc/' 

Pointings on Upper and Left Limbs. Pointings on Upper and Right Limbs. 

Vertical Arc, 24^*58' Vertical Arc, 24**58' 

Horizontal Circle, 357**i4'i5" Horizontal Circle, i62®28'oo'' 

Time 7hi9ni3oe Time 4I&1 21^158 



MERIDIAN BY SOLAR OBSERVATIONS 



193 



it elapsed time <- 41^26111228 
- 66°3S'3o- 
log sin / 9.96270 
log cos L 9.86902 

983172 
log 230".9 2.36342 

2.53170 
correction 340". 2 — 5'4o''.2 



Increase in declination in 4li26°^22B«i 
52'' X 444 - 230"-9 



Mean circle reading ■- 79^51 '08'' 
5 40 



S 79<»45'28'' E 



Azimuth of mark » 280^14^32'^ 

' 214. OBSERVATION FOR MERIDIAN BY A SINGLE ALTI- 
TUDE OF THE SUN. — The azimuth of a line may be obtained 
by measuring a single altitude of the sun with the transit and 
computing the azimuth by spherical trigonometry. The instru- 
ment is set at 0° and pointed at a mark. The upper clamp is 
loosened and pointings made as follows. First, the cross-hairs 
are set on the left and lower limbs of the sun and both circles 
are read ; the time is also noted. If desired several sets of 
observations may be made. Second, the cross-hairs are set on 
the right and upper limbs, and the reading of the circles and 
the time are again recorded. The mean of the vertical circle 
readings is taken, and corrected for atmospheric refraction by 
subtracting the correction given in Table 11. This corrected 
mean is called A in the formula given below. 



TABLE 11. 

Refraction Correction. 



Altitude. 


Refraction. 






10° 


_/ 


19" 






II 




51 






12 




27 






13 




07 






14 




49 






15 
16 




34 
20 






17 
18 

»9 


2 

2 


08 
57 
48 





Altitude. 


Refraction. 


20*» 


2' 39'' 


25 


2 04 


30 


I 41 


35 


1 23 


40 


I 09 


45 


58 


50 


49 


60 


34 


70 


21 


So 


10 



194 OBSERVATIONS FOR MERIDIAN [Chap. Vn. 

In order to compute the azimuth it is necessary to know the 
latitude of the place. This may be obtained from a reliable map 
or from an observation as described in Art. 2x6, p. 196. It is 
also necessary to know the declination of the sun at the instant 
of the observation ; this is found as described in Art. 86, p. 68. 
If Z represents the azimuth of the sun's center from the south; 
Z, the latitude ; A, the altitude ; /, the distance from the north 
pole to the sun (or 90°-declination) ; and j = |^ (Z + A +/) ; 
then 

cot' \Z ^ si" (-y - L) sin {s - h) 
^ cos s cos {s — /) 

Five place logarithms will give the value of Z within 10 seconds 
of angle, which is amply accurate for this observation. 

When the true bearing of the sun is known the bearing of 
the mark from the instrument can be foimd. 



OBSERVATION FOR AZIMUTH 



195 



Example. 

Observation on Sun for Azimuth. 

Latitude 42*' 21' N. Longitude 4^ 44m i8« W 

Time, Nov. a8, 1905, A.M. 



Horizontal Circle 


Vertical Circle 


Watch 


Vernier A B 






Mark 238^ 14' 14' 




A.M. 


Right and Lower Limbs 311 48 48.5 


14*^41' 


8h 39m 428 


M It u .. J, 2 20 20 


IS 00 


8 42 19 


The inst. reversed 






Left and Upper Limbe 312 27 26.5 


15 55 


8 45 34 


« « u « 3,2 52 51.5 


16 08 


8 47 34 


Mark 238 14 14 







Mean reading on Mark » 238^ 14^0 
« « •* San — 312 21 .7 

Mark N. of Sun — 74. 07'.7 

Observed Altitude 15° 26'.o 



Mean - 15° 26' 



Mean -i 8h 4311 
5 



4/« 



Greenwich Time — 13^ 43m 478 
Sun's apparent declination at 
Greenwich Mean Noon — — 21® 14' 54''.4 



Refraction 3.5 


Difference for i hour — — 26".8i 


True Altitude 15® 22'.5 « h 


- 26''. 81 X 13^.73 - - 6'o8'' I 




Declination — — 21® 2i'o2".5 




Polar Distance — 1 1 1 ® 2 1 ' 02''.5 


Z- 420 2I'.0 


log sin (j - Z) - 9.82708 


h - 15'' 22^5 


log sin (j — A) — 9.97062 


P^ III*>2l'.0 


log sec J — 1. 02132 


t - 84^32'.2 


log sec (j — />) — 04940 


*-Z - 42*> Il'.2 


2) 0.86842 


s^h -69«09'.7 


cot } Z » 0.43421 


/ - -P - - 260 48^.8 


1 Z - 20*> I2'.4 




Z -40° 24'.8 East of South 




Mark N. of Sun 74° o7'-7 




ii4^32'-5 




MarkN 65** 27^.5 E 



215. OBSERVATION FOR MERIDIAN BY MEANS OF THE 
SOLAR ATTACHMENT. — This observation has been described 
in detail in Art. 85, p. 66. 



196 OBSERVATIONS FOR LATITUDE [Chap. VII. 



OBSERVATIOIIS FOR LATXTUDE. 

216. (I) BY THE ALTITUDE OF POLARIS AT UPPER OR LOWER 
CULMINATION. — When Polaris is approaching either culmination 
(see Art. 206, p. 180, and Fig. 83) set up the transit and point the 
horizontal hair on the star. Keep the cross-hair pointed on the 
star until the culmination is reached. Read the vertical arc and 
determine the index correction. The altitude is to be corrected 
for refraction by Table 1 1, p. 193. This gives the true altitude. 
If Polaris is at upper culmination subtract from the true altitude 
the polar distance of the star at the date of the observation (Table 
8, p. 184). If the star is at lower culmination the polar distance 
is to be added. The result is the latitude of the place of obser- 
vation. 

217. (2). BY THE ALTITUDE OF THE SUN AT NOON.— The ob- 
servation consists in finding the greatest altitude of the sun's 
lower limb. This will occur when the sun is on the meridian 
(very nearly). Begin the observation a little before apparent 
noon, remembering that this differs sometimes as much as 17"" 
from mean noon.* Furthermore it should be remembered that 
standard time may differ a half hour or so from inean time. 
When the maximum altitude is found the following corrections 
are to be made: first, the refraction correction is to be sub- 
tracted, (Table II, p. 193); second, the sun's semi-diameter 
(found in the Nautical Almanac) is to be added; third, the 
sun's declination is to be subtracted if plus or added if minus. 
The result, subtracted from 90°, is the latitude. 



* Apparent noon occurs when the sun is on the meridian. Mean nqon is the 
instant when the sun would be on the meridian if it moved at a uniform rate along 
the equator. The difference between the two is known as the Equation of time 
and may be found in the Nautical Almanac. For example, on November ist, the 
sun ]>asses the meridian i6m i8s before mean noon, i.e., when it is izh com cos 
apparent time it is i ih 43m 42s mean time. 



OBSERVATIONS FOR LATITUDE I97 



Example. 

Observed 'mazimiim altitude of the sun's lower limb on 



Jan. 8, 1906. « 
Observed altitude 
Index Correction 

Refraction 


25** 06' 

2S« od'.o 

i'.o 

25« 0/.0 

2'.0 

2500^0 

i6'.3 
25^ 2I'.3 
22« l8'.0 


Index Correction — + 

Declination of sun at 
Greenwich mean noon 


l' 


■ 22"i9'35'' 
+ 1 33 


(S) 


Sun's semi-diameter 
Altitude of son's center ~ 
Declination - 




220 l8'02'' 


(S) 


LaUtude 


47" 
42" 


39'.3 




Diff. ih - 4. 19 
+ i9"-58 X 4^. 


r. 58 

74 - + 1' 


33" 






PROBLElfS. 











1. (a) What was the azimuth of Polaris at its greatest western elongation at 
Boston when the polar distance of the star was i^ 14' 1 2'^ ? The latitude of Boston 
is42*'2i'N. 

(b) In making an observation for meridian two stakes were set 329 feet apart, 
marking the direction of the star at elongation. Compute the length of the per- 
pendicular offset to be laid off at one end of the line to obtain the true meridian. 

2. What is the approximate Eastern Standard Time of the eastern elongation 
of Polaris on August loth at a place in longitude 72^ 56' West ? 

3. Observation on May 15, 1906, for determining the azimuth of a line from 
an altitude of the sun. Reading of vernier A of the horizontal circle while point- 
ing on the azimuth mark « 0° 00^. At first pointing on sun, lower and right limbs, 
vernier A, horizontal circle read 168^ 59' ; vertical arc read 43® 36'; the Eastern 
Standard Time was 2h 52m 45s P.M. At second pointing on the sun, upper and 
left limbs, vernier A, read 168** 52'; vertical arc, 42® 33'; time, 2h 55m 379 P.M. 
The second pointing on the mark -> o® 00', the mark being to the left of the sun. 
The sun's declination at Greenwich Mean Noon was +18® 42' 43".6 (North). The 
change for i hour was -f- 35^'.94 (sun going north). The latitude of the place was 
42** 1/ N. ; The longitude was 71^ 05' W. Find the azimuth of the mark. 

4. Observation for latitude. The observed altitude of Polaris at upper cul- 
mination was 43° 27'. The polar distance of the star was i^ 12''. What was the 
latitude of the place ? 

5. Observation for latitude. The observed maximum altitude of the sun's 
lower limb on August loth, 1906, was 66^ 29'. The Eastern Standard Time was 
approximately iil^ 50m A.M. The semi-diameter of the sun was 15' 48^^7. The 
declination of the sun at Greenwich Mean Noon was North 15^46' ^3" '3 ( + ). The 
difference for i hour was— 43''.46 (sun going south). What was the latitude of 
the place ? 



CHAPTER VIII. 

LEVELZNO. 

2i8. DEFnfinoilS. — Leveling consists in ascertaining differ- 
ences in elevation; there are two kinds, Direct Leveling^ and 
Trigonometric Leveling, The former alone will be considered 
in this book, as trigonometric leveling is used only in advanced 
surveying work. 

Wherever extensive leveling operations are to be carried on 
it is necessary to have a system of reference points called bench 
marks (B.Ms.), the relative heights of which are accurately known. 
These heights are usually referred to some definite zero plane, 
such, for instance, as mean sea-level or mean low water^ and the 
height of a point above this plane is called its elevation. This 
plane is called the datum. (See Art. 237, p. 211, and Art. 250, 
p. 226.) Strictly speaking it is not a plane but a level surface, ie., 
it is at every point perpendicular to the direction of gravity. If 
mean sea-level is not known a datum can be arbitrarily assumed. 

219. LEVELING TO ESTABLISH BENCH MARKS. — When it is 
necessary to run a line of levels to establish new bench marks 
the rod is first held on some bench mark the elevation of which 
is accurately known, and a backsight taken (Art. 1 16, p. 85). If 
this backsight is added to the known elevation of the bench 
mark it gives the height of the instrument {H. /.) above the 
datum. A turning point is then selected ahead on the route (to 
be traversed), and a foresight taken on it. (See Art. 224, p. 202.) 
If the foresight is subtracted from the height of the instrument 
the elevation of the turning point is obtained. When a target 
rod is used it is customary to take readings on bench marks and 
turning points to thousandths of a foot, and in this case often 
more than one rod-reading is taken on each point. If the first 
and second readings agree within 0.002 ft. it is unnecessary to 
take more readings ; if they differ by a greater amount it may 
be necessary to take three or four or even more readings to 
properly determine the correct value. The object of taking 

198 



BENCH MARK LEVELING I99 

more than one reading is not so much to increase the precision 
as to check the former readings. 

When it is desired to establish a bench mark a suitable point 
is selected and used as a turning point. The elevation of this 
bench mark could be obtained by simply taking a foresight upon 
it and not using it as a turning point, but by making the bench 
mark also a turning point it becomes a part of the line of levels 
and if the levels check, the elevation of the bench mark is also 
checked. Each bench mark established should be carefully re- 
corded by a description or a sketch, or both. The elevations 
of the remaining turning points are as accurate as the elevations 
of the bench marks themselves, so that any of the turning 
points might be used as a bench mark. Consequently it is 
advisable to describe those turning points which can be readily 
identified so that they may be used when it is not convenient or 
possible to use one of the established bench marks. 

In leveling up or down slopes the levelman should be able to 
judge quickly where to set his instrument in order to have it the 
desired height above the turning point. In going downhill the 
rod-reading of the backsight should be as small as possible in order 
to overcome the height with the minimum number of set-ups of 
the level. But while the levelman may waste much time by 
having large backsights necessitating additional set-ups, it is 
also possible for him to waste quite as much time in attempting 
to place his instrument so as to get very small backsights. The 
proper way to handle the instrument is as follows. Set up 
roughly (without pressing the tripod legs into the ground), turn 
the telescope toward the rod and then level it, approximately, 
in that direction. By sighting along the outside of the telescope, 
the approximate place where the line of sight will strike the rod 
can be noted and the distance the instrument should be moved 
up or down the slope can readily be estimated. Then move to 
the new position, level up carefully, and proceed to take the back- 
sight. This general procedure should be followed whether level- 
ing up or down a slope. 

220. In this work it is very important to eliminate as far as 
possible errors of adjustment in the instrument. If at every set- 
up of the level the foresight and its corresponding backsight are 



200 



LEVELING 



[Chap. VIII. 



taken at points which are equally distant from the instrument such 
errors will be eliminated. If the level is not in perfect adjustment 
the resulting error in any reading is proportional to the distance. 
At equal distances from the instrument the errors are equal, and, 
since it is the difference of the rod-readings that gives the dif- 
ference in elevation, the error is eliminated from the final result 
by this method. By making the length of foresights and back- 
sights equal on turning points it is possible to eliminate not only 
the error due to non-adjustment of the bubble but also any error 
due to non-adjustment of the objective tube, since this will occupy 
the same position in the telescope in each sight. The distance 
to the backsight is determined by the place where the instrument 
is set up, and the rodman, as he passes from one turning point to 
the next, can by pacing make the foresight distance approximately 
equal to that of the backsight. The line of levels should be "closed" 
by continuing the leveling until the original bench mark, or some 
other bench mark whose elevation is well established, is reached. 
221. The notes for this work may consist of five columns, as 
shown in Fig. 86. The height of instrument is obtained by add- 
ing the backsight to the elevation of the point on which it is taken. 
The elevation of any point is found by subtracting the foresight 
for that point from the height of the instrument. Notice that the 



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Fig. 86. Bench Mark Level Notes. 



DOUBLE RODDED LINES 



201 



calculations maybe checked by adding the foresights and the back- 
sights. The difference of these sums should be the same as the 
difference in elevation between the first and last points. 

222. Double Rodded Lines. — A good check on the line of 
levels may be secured by running a double line of turning points. 
Instead of taking a foresight on a single turning point, foresights 
may be taken on two different points near together, from the 
same set-up of the instrument. When the level is set up again 
a backsight is taken on each turning point and two independent 
values of the new height of mstrument are obtained. In 
ordinary bench mark leveling these two values should not differ 
by more than 0.002 or 0.003 ft. from the previous difference, 
i.e., if the two heights of instrument differed by 0.013 at a 
certain set-up they should not differ by more than 0.016 nor 
less than 0.0 10 at the next set-up. If the two turning points 
of a pair are so chosen that their diflference in elevation is more 
than a foot then any mistake of a foot in the computations or 
in reading the rod will be immediately detected. 

In this way, by little additional work the accuracy of the levels 
may be checked as the work progresses. This method of using 
double turning points is particularly useful in running long lines of 
levels where no established bench marks are available for checking. 

223. A set of notes illustrating double turning points is 
shown in Fig. 87. It will be noticed that the higher and lower 

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Fig. 87. Bench Mark Level Notes, Double Rodded Lines. 



202 • LEVELING [Chap. Vni. 

turning points of a pair are arranged in a systematic order. The 
readings in this case have been taken on the lower turning point 
first at each set-up. It is very important that some definite 
system shall be followed so that the two lines of levels will not 
be confused. 

224. Bench Marks and Turning Points. — Both the bench 
marks and the turning points should be such that their elevations 
will not change during the time they are needed. The only 
difference between the two is that turning points may be of use 
for only a few minutes while bench marks may be needed for 
many years. Bench marks should be very carefully and accu- 
rately described, and their heights should be checked before 
being accepted as correct. They are frequently taken on such 
points as these : — stone bounds, tops of boulders, spikes in 
trees, and on sills, stone steps, or underpinning of buildings. 
Curb stones or tops of hydrants are also used but are not so 
permanent. As it is often impossible in a new country to find 
existing points where bench marks can be established, it is usual 
in such cases to set stone monuments or iron rods and to care- 
fully determine their elevation. The U. S. Geological Survey, 
for example, sets an iron pipe with a cap on the top of it ; or in 
some cases a plate with a horizontal line across it in the masonry 
wall of a building. Some of the bench marks of the U. S. 
Coast and Geodetic Survey and of the Missouri River Com- 
mission consist of stones buried 3 or 4 ft. under ground. The , 
exact bench is the top of a spherical headed bolt set in the top 
of the stone. This is reached by lowering the rod through an 
iron pipe which extends to the surface of the ground. 

Bench marks should be established at frequent intervals for 
convenience in dependent work. Some surveyors consider it 
advisable to have two bench marks in the same locality to ser\'e 
as checks on each other. In choosing a bench or a turning 
point it is best to select a point which is slightly raised so that 
the rod will always rest on exactly the same point. A rounded 
surface is better than a sharp point, especially when it is on a 
rock, as the rod may chip off a small piece and alter the eleva- 
tion. If a turning point is taken on a flat surface it is difficult 
to get the rod at exactly the same height each time. Bench 



BENCH MARKS AND TURNING POINTS 203 

marks are, however, sometimes established on flat level surfaces 
such as the coping stone of a masonry structure, because per- 
manence is of more importance than great precision. Bench 
marks are not only described in the notes, but are themselves fre- 
quently marked by red chalk, by chisel marks, or drill-holes. 

225. LEVELINO FOR PROFILE. — Profile leveling is for the 
purpose of determining the changes in elevation of the surface of 
the ground along some definite line. The line is first " stationed,*' 
i.e., marked at every hundred feet or such other interval as is 
desired. The level is set up and a backsight taken on a bench 
mark to determine the height of the instrument. Foresights 
are then read on as many station points on the line as can be 
conveniently taken from the position of the instrument. Inter- 
mediate sights are taken at any points where marked changes 
of slope occur, and the plus stations of these intermediate points 
are recorded with the rod-readings. It will be noticed that here 
the terms foresight and backsight do not refer to the forward 
and backward directions. A backsight is a reading taken on a 
point of known elevation for the purpose of obtaining the height 
of the instrument. A foresight is a reading taken on a new point 
to determine its elevation. For this reason backsights are fre- 
quently called //m^ sights ( + 5), and foresights are called mintis 
sights (— 5). When it is necessary to move the level to a new 
position in order to take readings on stations ahead, a turning 
point is selected and its elevation determined. The level is 
then taken forward and its new height of instrument determined 
by taking a backsight on the turning point. This general pro- 
cess is continued until the end of the line is reached. 

A line of levels should be checked by connecting with some 
reliable bench mark if possible. If there are any bench marks 
along the line of levels they should be used as turning points if 
convenient, or at least check readings should be taken on them 
in order to detect mistakes. In such a case it is evident that 
the reading taken on the bench mark is really a foresight since 
its elevation is being found anew from the height of instru- 
ment. Readings on bench marks and turning points should be 
taken to thousandths or to hundredths of a foot, depending upon 
the accuracy desired. If the elevations of the profile are de- 



204 



LEVELING 



(Chap. VUI. 



sired to the nearest hundredth of a foot, as in the case of a 
raikoad track, the turning points should be taken to thousandths 
of a foot. Elevations on the surface of the ground will not usu- 
ally be needed closer than to tenths in which case the T. Ps. are 
taken only to hundredths. In calculating the elevations the re- 
sults should not be carried to more decimal places than the rod- 
readings themselves, otherwise the results will appear to be more 
accurate than they really are. 





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Fig. 88. Profile Level Notes. 

226. Profile notes are kept as shown in Fig. 88. In this 
case also the heights of instrument and the elevations of turn- 
ing points may be checked by means of the sums of the fore- 
sights and backsights, provided only the sights on turning points 
and the initial and final benches are included. If it seems 
desirable the elevations of stations may be checked by means of 



PROFILE LEVELING 



20S 



differences in foresights. The difference between the eleva- 
tions of any two points, which are obtained at the same set-up 
of the instrument, is equal to the difference between the fore- 
sights taken on these points. For example, if the difference be- 
tween the foresights on stations 4 and 5 is 3 ft. this should also 
be the difference between their elevations. In these notes the 
elevations of B. Ms. and T. Ps. are put in a different column 
from the surface elevations simply for the sake of clearness, but 
many surveyors prefer to put all the elevations in the same 
column. Another arrangement of columns which will be found 
convenient when plotting the notes is to place the station col- 
umn immediately to the right of the elevation column. 



T^i 




PLAN 




SIKO 



PROFILL 
Fig. 89. 



Fig. 89 represents a rough plan and profile of the line of 
levels shown by the notes in Fig. 88. Angle points in the 
transit line are shown in the plan, but they do not appear in the 
profile of the line. It will be noticed that the T. Ps. and B. Ms. 
are not on the transit line in plan, and that they consequently 
do not appear on the profile. It is not customary to introduce 



206 



LEVELING 



[Chap. VIII. 



any sketches into the profile notes except those used in describ- 
ing bench marks or turning points. 

227. Cross-Sectioning. — If it is desired to know the shape 
of the surface of a piece of ground, the area may be divided 
into squares and the elevation taken at each corner of these 
squares and at as many intermediate points as seem necessary 
to determine the changes of slope. These surface elevations 
are obtained to tenths of a foot. The squares which may be 
anywhere from 10 ft. to 100 ft. on a side are laid out with the 
transit and tape, stakes being driven at the comers. It is well 
to choose some long line of the traverse as the primary line 
from which the cross-section system is to be laid out. The 
points are usually designated by a system of rectangular coor- 
dinates, one set of parallel lines being marked by letters and the 
other by numbers, as shown in Fig. 90. For example, the 





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point / would be called (C, 7) ; the point j, {D, 5) ; the point 
r, {B f 80, 4 -h 35) ; etc. The notes are kept as in profile leveling 
except as to designation of points. 

228. Use of the Tape Rod in Cross-Section Work. — In this 
work, where there are a large number of elevations to be cal- 
culated, it will save much time to use a tape rod (Art. ic6, 
p. 81), which is so arranged that no elaborate figuring is re- 
quired In this rod the numbers increase from the top toward 
the bottom, the opposite way from ordinary rods. The level is 



CROSS-SECTION LEVELS 20/ 

set up at a convenient point and the rod held on a bench mark. 
The tape, or band, on the rod is then moved up or down as di- 
rected by the levelman until he reads the feet, tenths, and hun- 
dredths which are the same as those of the elevation of the bench 
mark, e.g., if the elevation of the B. M. is 195.62, the tape will be 
moved until it reads 5.62. If the rod is then held on a point 
1.61 ft. lower than the bench, the rod-reading will be 4.01, since 
with this rod the readings decrease as the rod is lowered. The 
elevation of the point is then 194.01 ft., or sufficiently precise for 
topographic work, 194.0 ft. In this way the elevations are read 
directly on the rod to feet and decimals of feet, the tens and hun- 
dreds of feet being supplied mentaUy. Obviously the only notes 
kept are the columns of stations and elevations. 

229. CROSS-SECTIONING FOR EARTHWORK. — Whenever it 
is desired to ascertain the quantity of earthwork in an excava- 
tion or an embankment, it is necessary to take levels to determine 
the vertical dimensions, and to obtain the horizontal dimensions 
by means of the transit and tape. The three general cases where 
the quantity of earthwork is to be estimated by the engineer 
are : (i) an excavation or embankment having a known base and 
side slopes as in the construction of a railroad or a highway, (2) 
an irregular excavation from a bank of earth called a borrow-pit^ 
(3) a trench excavation such as is used for sewer construction. 

230. ( i) Road Cross-Sections. — Cross-sections for estimating 
the earthwork in highways or railroads are usually taken at full 
station points (sometimes oftener) and at right angles to the 
center line of the road.* By this method is obtained a section 
of the general shape shown in Figs. 91 and 92. These cross- 
sections are taken in the field before the construction begins so 
that a proper record of the surface heights can be obtained be- 
fore the ground is disturbed. 

From the plan of the proposed road its alignment is staked 
out and a profile is taken along the center line, which is subse- 
quently plotted (Art. 225, p. 203). On this profile the grade line 
is drawn, which corresponds to the finished surface of the road. 
Roads are usually first finished to sub-grade^ which is below the 

* For a more complete treatment of this snbject see ** Railroad Carves and 
Earthwork," by Professor C. F. Allen, published by Spon & Chamberlain, 
New York. 



208 LEVELING [Chap. VIIL 

completed surface by an amount equal to the thickness of the 
road covering, i.e., the pavement of a highway or the ballast in 
the case of a railroad. The width of the base of the road and 
the inclination of the side slopes are known. For ordinary 
gravel the slope is usually i^ ft. horizontal to i ft. vertical, called 
"a slope of I J to i." 

For construction work the engineer sets grade stakes at 
every full station or oftener on the center line and at both sides 
where the finished slope intersects the surface of the ground, 
e.g., at points A, B and C on Figs. 91 and 92. All of these 



Fig. 91. Excavation. Fig. 92. Embankment. 

stakes are marked, giving the amount of "cut" or "fill" to be 
made at these points. The cut or fill marked on the stakes at 
B and C is the vertical distance from the base of the road to the 
surface of the ground at these points, e.g., the distance 6C 

These cuts and fills are determined in the field by the follow- 
ing method. The level is set up and the height of instru- 
ment obtained from some convenient bench mark. Then, the 
elevation of the finished grade being known (from the profile 
prepared in the office), the difference between the height of 
instrument and the elevation of the finished road will give what 
is called the rod-reading for grade^ i.e., the rod-reading which 
would be obtained if the foot of the rod could be held on the 
finished surface of the road. Then the rod is held on the surface 
of the ground at the center stake and a reading is taken (to the 
nearest tenth of a foot), and the difference between the rod- 
reading for grade and the rod-reading on the surface will give 
the cut or fill at that point, and this is marked on the center 
grade stake thus, C5.2 or F4.7. 

231. Setting Slope Stakes. — The points where the 
side slopes intersect the surface of the ground are found by 
trial as follows. Hold the rod at a point where it is estimated 
that the side slope will cut the surface, and take a rod-reading. 
The difference between this rod-reading and the rod-reading for 



ROAD CROSS-SECTIONS 



209 



grade will give the cut or fill at this point, from which the 
distance out from the center of the section to the point on the 
side slope having this cut can be computed. This distance out 
equals (J base -f cut x slope). Then the distance is measured 
from the center to the rod, and if the measured distance 
equals the computed distance the rod was held at the right 
place and the stake should be driven and marked with the cut 
or fill at that point (distance bC^ Fig. 92). If the measured 
distance does not agree with the calculated distance a second 
trial must be made by holding the rod at another point and 
repeating the operation. The difference between the measured 
and calculated distances is an aid in judging where the rod 
should be held at the second trial. After a little practice it 
will be possible to set the slope stake at the second or third trial. 
232. Earthwork Notes for Road Cross-Sections. — 
The notes for this work will contain the cut or fill at the 
center, the cut or fill at either side, and the corresponding 
distances out. A cut is usually written in the notes as a plus 
( + ) height and a fill as a minus (— ) height; but the stakes 

Lbft-Hand Pagb. 



Cross-SecHon fordbmssfomffoad. 




Sta, 



Surface 



/2 

// 
/O 



aas 

37S 






96ZS 



M&^^3^oJ^£i^ 



96fio 
3XS> 



Cross-Ssc/jbns, 6cBe4o'-5k^^l 



i€,0 



-MiO *L6 



M^ t4'jO ^^ t. 



H.S 



23X) 



f^ZX) 






Fig. 93. Cross-Section Notes for a Road. 

are marked C or F rather than + or — . If the surface is 
irregular levels are taken at intermediate points and are recorded 
as shown opposite Sta. 11 + 50, and Sta. 12 in the notes, 
Fig. 93. Where the surface of the ground is parallel to the 



210 LEVELING [Chap. Vm. 

base of the road, as in Sta. lo, the section is called a Level 
Section. Where the surface of the ground is not parallel to 
the base and where three cuts or fills only are recorded, as at 
Sta. 1 1, the section is called a Three Level Section. If, besides 
the three readings which are taken for a three level section, two 
more intermediate readings are taken one directly over each end 
of the base, as at Sta. 1 1 + 50, the section is called a Five 
Level Section. If intermediate readings (one or more of them) 
are taken anywhere except over the ends of the base, as in Sta. 
12, the section is called an Irregular Section, For methods of 
computing the amount of earthwork see Chapter XII. 

It will be noticed that in the column of the notes headed 
" Cross-Sections " the distances out appear above and the corre- 
sponding cuts below the lines. Besides this set of notes there is 
a simple set of level notes similar to Fig. 86, p. 200, from which 
the height of instrument is determined. This is conveniently kept 
in another part of the note-book, often at the back of the book. 

233. (2) Cross-Sections for Borrow-Pits. — The ground is 
first staked out in squares or rectangles and the elevation at each 
corner and at every change in slope is determined as explained 
in Art. 227, p. 206. Then the work of excavating is carried on, 
and when it is desired to determine the amount that has been 
excavated, the same system of cross-sections is again run out 
and the new elevations at the corners and at the necessary 
intermediate points are determined. 

The notes are kept as shown in Fig. 90, p. 206. For methods 
of computing the earthwork in borrow-pits see Art. 373, p. 342. 

234. (3) Cross-Sections for Trench Excavation. — The sur- 
face elevations are determined by making a profile of the 
line. The grade of the bottom of the trench is obtained either 
from the plan or by direct leveling. The width of the trench is 
measured wherever it changes and the stations of these places 
noted. For methods of computing the quantity of earthwork 
see Chapter XII. 

235. LEVELING TO ESTABLISH A GRADE LINE. — The level 
may be used for setting points at desired elevations as, for 
example, in establishing the grade line of a sewer. To set any 
point at a given elevation, set up the level and take a backsight 



ESTABLISmNG A GRADE LINE 211 

on a bench mark, thus determining the height of instrument. 
Subtract the given elevation from the height of instrument and 
the result is the rod-reading for grade. Raise or lower the rod 
until the horizontal cross-hair indicates this reading. The foot 
of the rod is then at grade. This is usually set for construction 
work to hundredths of a foot ; for some purposes tenths of a 
foot will be sufficiently exact. If a target rod is used the target 
is set at the proper reading, and the bottom of the rod is at 
grade when the cross-hair bisects the target: 

If the grade line comes beneath the surface of the ground 
and cannot be reached a point may be set a convenient whole 
number of feet above grade and the depth marked on a stake> 
or vice versa if the grade line comes far above the surface. 

236. " Shooting in " a Grade Line. — To save time and to 
diminish the liability of mistakes, grades are often set by a 
method known as " shooting in '* the grade. First set a point 
at the proper elevation at each end of the straight grade line. 
The instrument (usually a transit with a telescope bubble) is 
set up 6 or 8 inches to one side of the first point, and the dis- 
tance from the top of the first stake to the axis of the telescope 
is measured with the tape or rod.* Then the rod, which is set at 
this reading, is carried to the last point on the straight grade line, 
and, whDe it is held vertical on this point, the instrument man 
raises or lowers the telescope until the horizontal cross-hair is on 
the target, clamping the instrument in this position. If. a level 
is used the horizontal cross-hair is set by means of the leveling 
screws ; but if the transit is used the cross-hair is set by means 
of the clamp and tangent screw of the vertical motion. The 
line of sight is then along an inclined line parallel to the grade 
line. All intermediate points on the grade line are then set by 
raising or lowering the rod until the target coincides with the 
horizontal cross-hair. 

237. To ESTABLISH A DATUM PLAITE BY MEANS OF TTOAL 
OBSERVATIONS. — Whenever it is necessary to establish a datum 
from tidal observations it may be determined as follows. Set up 

* Where the grade is flat some surveyors prefer to set the instrument just 
behind the point instead of to one side of it. 



212 



LEVELING 



[Chap. vm. 



a vertical staff, graduated to feet and tenths, in such a manner 
that the high and low water can be read. Read the positions 
of high and low water for each day for as long a period as prac- 
ticable. The mean value obtained from an equal number of 
high and low water observations will give the approximate value 
of mean sea-level. If the observations extend over just one lunar 
month the result will be fairly good, whereas in less than one 
month a satisfactory result cannot be obtained ; to determine this 
accurately will require observations extending over several years. 
The proper location of the gauge is an important factor in 
obtaining the true mean sea-level. The place chosen for setting 
up the gauge should be near the open sea, so that local conditions 
will not influence the tide. It should be somewhat sheltered 
against bad weather. The water should be deep so that at the 
lowest tide the water will stand at same height on the gauge. 

At the beginning of the series the zero of 
the staff and some permanent bench marks 
should be connected by a line of levels. This 
should be tested occasionally to see if the staff 
is moved. After the reading of the rod for 
mean sea-level is found the elevation of the 
bench mark can be computed. 

238. The Staff Gauge. — This is a form 
of gauge (Fig. 94) which can be easily con- 

/l ^H structed, and which is sufficient where only a 
I ^^1 short series of observations is to be made. If 
!l (5% niade in sections not over 3 feet long, as de- 
I ^N scribed below, it can easily be packed in a box 
^, ^^ for transportation. Each section consists of 
I two strips of wood about i^ inches square, 

■ and 3 feet long, fastened together at the 

I ends by strips of brass, leaving a space 

I between them of about i inch. In this 

B space is placed a glass tube of about J inch 

I 4(ll!l!ii diameter and held in place by brass hooks. 
M^l| On one side of the tube is a red strip blovm 
into the glass. When the gauge is set up 
for observations the sections are screwed to 




Fig. 



94. Staff 
Gauge. 



LEVELING ACROSS A RIVER 213 

a long vertical piece of joist. The ends of the tube are nearly 
closed by corks, in which small glass tubes of approximately i 
mm. (inside) diameter have been inserted. When the water rises 
in the main tube, the red strip appears to be much wider than it 
really is on account of the refraction of light by the water. Above 
the water surface the strip appears its true width. By observing 
the position of the wide strip the height of the water surface can be 
read within a hundredth of a foot. The heights are read on a scale 
of feet painted on the wooden strips. If the size of the small 
glass tube is properly chosen, the fluctuations of the water sur- 
face outside will not disturb the water in the tube, so that the 
reading is a fair average of the water surface. A gauge of this 
sort may be read by means of a transit telescope or field glass 
at a distance of several hundred feet. 

When a long series of observations is to be made a self-regis- 
tering tide gauge should be used. A description of such a gauge 
may be found in the Reports of the U. S. Coast and Geodetic 
Survey.* 

239. LEVELING ACROSS A RIVER. — While the effect of cur- 
vature and refraction (Art. 118, p. 87) is usually negligible in 
leveling operations, it may in certain special cases become of 
great importance to eliminate this error. For example, it is 
sometimes necessary to carry a line of levels across a river of 
considerable width, say, half a mile. In this distance the correc- 
tion for curvature and refraction amounts to about 0.143 ft. un- 
der normal conditions, which in a line of bench levels is too large 
a quantity to neglect. If the correction as derived from formu- 
las could be depended upon under all circumstances it would be 
sufficient to compute and apply it to the rod-reading. But the 
amount of the refraction correction is so variable that the actual 
value often differs considerably from the computed value. 

If it is desired to obtain the difference in elevation between 
two distant points with great accuracy it will be necessary to use 
a method which will eliminate the effects of curvature and re- 
fraction no matter what their actual amount may be. In Fig. 
95 suppose a backsight were taken on T. P.j with the instrument 

• Report for 1897, pp. 315-320 and pp. 480-489. 
Report for 1853, pp. 94-96. 



214 LEVELING [Chap. Vm. 

at A and then a foresight taken on T. P., The elevation of T. P., 
as computed from T. P.^ will be too low by the amount ab^ 
since the foresight on T. P., is too great by this amount. If 
the difference in elevation is determined by the instrument at 
B the backsight on T. P.j is too large by the amount cd. Hence 
the H. I. of the instrument at ^ is too great, and consequently 



Fig. 96. Leveling Across a River. 

the elevation of T. P., too great by the amount cd. The mean 
of the two determinations would give the true elevation of T. P., 
if ab^cd, but this occurs only when the two sights are taken 
under the same atmospheric conditions. Therefore it will be 
seen that the two sights must be taken simtdtaneously. In 
order to eliminate the errors of adjustment * in the instrument 
it is necessary to use the same instrument at both ends of the 
line. To accomplish both of these results at once it is neces- 
sary to take simultaneous readings with two instruments and then 
to repeat the operation with the instruments interchanged. The 
magnifying powers of the two telescopes and the sensitiveness of 
the two spirit levels should be about equal in order to g^ve the 
best results. It will be noticed that this process is similar to 
that of the peg adjustment (Art. 128, p. 91). 

* Errors due to non-adjustment are of unusual importance because the sight is 
much longer than that used in adjusting the instrument. 



LEVELING PROBLEMS 



215 



PROBLEMS. 
X. Compute fhe following set of level notes. 



Sta. 


B.S. 


H. L 


F. S. 


Elev. 


B.M., 
B. M., 
T. P.j 
B.M.. 
T.P.. 
B. M., 


4.702 
11.846 
7.276 
8.760 
0.687 
1.607 




6.727 
9.689 
4.726 
11.000 
8.496 


16^27 



2. Compute the elevations in the following set of level notes. 



Sta. 


B.S. 


H. L 


F. S. 


Elev. 


B.M.^ 


6.427 




4.273 


62.473 


20 






6.2 




21 






7.4 




+4* 






5-2 




22 






4.7 




T. P.„ 


4.724 




9.976 




23 






11.2 




+63 






10.4 




B. M.]! 


oj^09 




7.482 




24 






11.2 





3. Compute the elevations in the following set of level notes. 



Sta. 


+s. 


H.L 


-s. 


Elev. 


B.M... 


6.214 






84.238 


T. P., L. 


3-515 




9.280 




T. P.j H. 


2.152 




7.919 




T. P., L. 


2.971 




8.263 




B.M«,H. 


2.338 




7.629 




T. P., L. 


4.278 




7.529 




T. P.,H. 


2.646 




5894 




B.M.^L. 


5721 




6.072 




T. P.4 H. 


4.837 




5.187 




B.M.„ 






5817 





4. Make up a set of cross section notes for road construction which shall be 
consistent with the following data: width of road, 50 ft., slopes i J to i ; grade 
elevation of Sta. o » 107.20; grade, I- i 4. Show complete notes from Sta. o to 
Sta. 3 inclusive as follows: Sta. o« a level section; Sta. i, a three level section; 
Sta. 2, a five level section ; Sta. 3, an inegnlar section. 



CHAPTER IX. 

CITY SURVBYINO. 

240. INSTRUMENTS USED. — Owing to the comparatively 
high value of land in cities and to the fact that a large propor- 
tion of city surveying is the establishing of lines and grades for 
construction work, the chain and compass are discarded entirely 
and the steel tape and transit are used. 

241. Tapes and Tape Measurements. — The tape most com- 
monly employed is the light loo-ft. steel tape, graduated to 
hundredths of a foot, described in Art. 7, p. 5. All ordinary 
measurements are taken in the usual manner, the pull and the 
horizontal position of the tape being judged by the men taking 
the measurements. But frequently it is necessary to obtain 
results with a greater degree of accuracy than is afforded by 
the ordinary method of measurement. For example, in measur- 
ing the base-line for triangulation work or in the survey of the 
valuable portions of large cities, there is call for an accuracy of 
measiu-ements which can only be obtained by using a method 
which will insure a uniform pull on the tape, a careful align- 
ment, little or no sag in the tape, and some means by which the 
temperature of the tape can be taken and its correction applied 
to the results. In such cases the pull is measured by use of a 
tension handle (ordinary spring balance) which can be attached 
by a clamp to any part of the tape, the alignment is given with 
the transit, and, where feasible, just enough pull is given so that 
the stretch in the tape equals the shortage due to sag. The 
correction for temperature can be computed from the difference 
between the temperature of the tape taken in the field and the 
temperature at which it is standardized /"Art. 19, p. 13). The 
tape should be compared with the City Standard (Art. 243, p. 
21 8 \ at a definite tension, and the temperature noted at the 
time. From this information all of the field measurements can 

216 



INSTRUMENTS USED 21/ 

be reduced to agree with the City Standard and very accurate 
results may be obtained. 

Where the ground is not level and there is call for frequent 
plumbing it is impossible to obtain accurate results unless the 
plumbing is carefully done by experienced tapemen. For very 
accurate work it may be desirable to entirely eliminate the 
plumbing. This is sometimes done by measuring directly on 
the surface (on the slope) from point to point, and by means of 
the level instrument and rod the relative elevations of these 
points are obtained and the horizontal projection of the slope 
distances computed. Instead of measuring the difference in 
elevation between the two ends of the line, the angle of inclina- 
tion of the slope line is often measured on the vertical arc of a 
transit which is set up over one of the end points. 

The government Bureau of Standards at Washington will, 
for a nominal charge, standardize tapes ; and city and private 
engineers frequently avail themselves of this opportunity. This 
Bureau will give the exact length of the tape at a given temper- 
ature or the temperature at which the tape is of standard length, 
whichever is desired by the engineer. It is well to have the 
tape also tested at a few intermediate points, e.g., the 25 ft., 50 
ft., and 75 ft. marks. One tape which has been standardized 
should be kept in reserve, with which tapes in service can be 
compared both when new and after being mended. 

Besides the ordinary steel tape, steel or metallic tapes 
reading to tenths of a foot are used in taking measurements for 
making approximate estimates of construction and for measuring 
earthwork, paving, and the like. 

242. Transits and Levels. — The transits usually employed 
in city work read to 30" or to 20" ; and for most city work no 
finer graduation is necessary. With these instruments the re- 
quired precision in reading angles on triangulation work 
can be obtained by repeating the angles as explained in Art. 
59, p. 48. In such work, however, it will be of advantage to 
have an instrument reading to 10". It is well also to have 
one or more transits equipped with stadia hairs for use on 
rough surveys. 

Much of the city work, such as the staking out of new streets, 



2l8 CITY SURVEYING [Chap. IX. 

paving, sewers, or curbs, requires the establishment of both lines 
and grades. Since this class of work does not as a rule call for 
very precise results, the measurements and rod-readings are usually 
taken to hundreidths of a foot. It is not convenient, for the ordi- 
nary surveying party of three men, to carry both a transit and a 
level instrument in addition to the ordinary equipment of sight- 
ing-rods, level-rod, stakes, tape, etc., so the engineer's transit, 
>yith a level attached to the telescope, is extensively used in set- 
ting grades as well as in establishing lines. For this reason sev- 
eral of the transits in a city office ,should be equipped with telescope 
levels and some of them with vertical arcs. The degree of pre- 
cision possible with an engineer's transit is entirely satisfactory 
for all ordinary leveling. 

Where leveling work alone is to be done the ordinary wye or 
dumpy level instrument is used together with target or self -reading 
rods. (See Chapter IV.) For bench leveling it is customary, in 
large cities at least, to use a precise levels an instrument which is 
similar in principle to the ordinary level but which has a more 
delicate bubble and a telescope of higher power, and is therefore 
capable of yielding more accurate results. 

243. CITY STANDARD.* — It is customary in all large cities to 
have a standard of length, usually 100 ft. long, established in 
some convenient place, often near the office of the City Engineer. 
It sometimes consists of two brass plugs set in a stone pavement, 
or it may be a long steel rod supported on rollers on the side of 
a wall or building in such a way that the rod can expand or con- 
tract freely. The end points and the so-ft. points are so marked 
that they can be readily found and used by any surveyor who 
desires to test his tape. 

A city standard is often established by carefully transferring 
the length of some other standard, by means of different tapes 
and under different weather conditions ; or it can be established 
by means of a tape which has been standardized by the U. S. 
Bureau of Standards (Art. 241, p. 216). The City Standard is 



• See a paper entitled "The 100 -foot Standard of Length of the Boston 
'Water Works at Chestnut Hill Reservoir," by Charles W. Sherman, published in 
the Jour. Assoc. Eng. Soc., Vol. XVIII, Ko. 4, April, 1897. 



STANr)ARD OF LENGTH 219 

generally placed where it will not be exposed to the direct rays 
of the sun, and with this end in view it is sometimes covered with 
a wooden box. 

When a tape is tested it should be stretched out at full length 
beside the standard and left there until it acquires the same 
temperature as the standard before the comparison is made, to 
avoid the necessity of applying a temperature correction. 



CITY LAYOUTS. 

244. In la)ring out or extending a city it is the duty of the 
surveyor to consider the future needs of its population and to 
design the general plan of the city accordingly. Nearly all of 
our large cities show examples of lack of forethought relative 
to future growth, which have necessitated the outlay of millions 
of dollars for revision of street lines, sewer systems, water 
works, and the like. 

Occasionally the engineer is called upon to plan a new city 
or to' design the general layout of the suburbs of an existing 
city. The basis for such work should be a top)ographic map of 
the entire area, for the topographic features of a locality will 
influence its development to a marked degree. 

245. STREETS. — In planning the arrangement of the streets 
for a city such features as a water front, a river or lake, the loca- 
tion of an existing railroad, or the probable location of some 
projected railroad line will determine to a large degree where 
the business section of the city will be located. This section 
should then be so divided as to yield the greatest convenience 
for business purposes. Other sections will be reserved for 
residential districts, and their design will be of a different 
character. Easy access should be provided from the business 

"to the residential districts and to outlying towns or adjacent 
cities. 

The streets must be of the proper width to accommodate 
the traffic they are to carry, and their alignment and grades 
must be carefully studied with the topographic map as a guide. 
Adequate drainage' of the streets is, of course, one. of the most 



220 CITY SURVEYING [Chap. DC 

important features, for which ample provision must be made 
in establishing the alignments and grades. 

In the business section the traffic will move in certain 
directions, e.g., to and from important points such as a river, 
railroad station, or freight yard, and this traffic must be pro- 
vided for by wide streets with easy grades. In the residential 
portions, narrower streets and steeper grades are permissible 
when made necessary by the topography of the district. 

246. Location of Streets. — In establishing the location of 
city streets in hilly districts it is probable that to obtain the 
essential requisites of easy grades and good drainage the topog- 
raphy will govern the street layout. Whereas in a practically 
level country, with no steep grades in any direction, the street 
layout can be such that the most direct communication between 
diflferent parts of the city is secured. 

Fig. 96 shows the location of a rectangular system of streets 
laid out without reference to the topographic features. The 
lower portion is on rolling ground where this system may be 
properly applied ; but from a study of the contours it will be 
seen that in the upper portion this method introduces very steep 
grades on all of the streets which cross the valley and also leaves 
a hollow in these streets which is difficult to drain. Fig. 97 
shows a layout which will obviate this difficulty to some extent, 
the diagonal streets being located in the valleys to take the sur- 
face drainage of surrounding property. It is obviotis that the 
construction of a sewer through these diagonal streets will be 
much more economical than through the streets as laid out in 
Fig. 96, for a sewer must have a continual drop toward its out- 
let, and cannot be laid uphill and downhill like a water pipe. 

With reference to directness of communication between 
different parts of a city the two general systems which have 
been used in this country are the rectangular block system and 
a combination of rectangular blocks with diagonal streets, run- 
ning in the direction of the greatest traffic. 

The rectangular system gives the maximum area for private 
occupation and is consistent with the general style of rectangu- 
lar building construction. Where the topography admits of it, 
this system of streets is advisable. Many of our large cities, 



LOCATION OF STREETS 




Fig. 06. Layout of Streets without Regard to Topography. 




Fig. 97. Layout of Streets with Regard to Topography. 



222 CITY SURVEYING [Chap. IX. 

like Philadelphia, for example, have been laid out in this manner. 
The streets frequently run parallel and perpendicular to the 
shore of a lake or river. More often, however, they are laid out 
in north and south, and east and west directions. When diagonal 
streets also are introduced they should connect the points be- 
tween which the traffic is the heaviest. Indianapolis is planned 
in this manner, having four broad diagonal avenues running from 
a central park ; but the city of Washington (Fig. 98) is the best 
example of this system in the United States. 

247. Size of Blocks and Lots. — No definite size of blocks 
and lots can be prescribed which will fit all conditions. Experi- 
ence has shown that the depth of lot most convenient for both 
business and residential districts is from 100 to 150 feet. In 
business districts particularly, it is well to provide an alley from 
15 to 25 ft. wide running lengthwise through the block. This 
makes the width of blocks from 215 to 325 feet, which is about 
the range in existing cities. 

The length of the blocks should be in the direction of great- 
est travel, and this dimension will therefore depend upon the 
necessity for cross-streets to accommodate the traffic which 
moves at right angles to the principal line of traffic. In busi- 
ness districts then the cross-streets should be much more fre- 
quent than in residential p)ortions of the same city.. The length 
of blocks therefore varies considerably in different cities and in 
different parts of the same city ; ranging all the way from 400 
to 900 feet. In New York the typical blocks are 200 X 900 
ft., and 200 X 400 ft. ; in Boston they vary in width from 125 
to 252 ft. and in length from 200 to 700 ft., depending upon the 
locality. 

The frontage of lots is frequently 25 ft. in business and 
congested residential districts and 50 feet or more in sub- 
urban districts, but these dimensions are by no means 
universal. 

248. Width of Streets. — The widest streets should in gen- 
eral be the ones which have the greatest traffic. Important 
business streets should be from 100 to 150 ft. in width, while 
streets of secondary importance in business districts may be 
from 60 to 80 ft. wide. In residential districts the main streets 



223 




224 CITY SURVEYING [Chap. DC 

should be 60 to 80 ft. wide, but those of lesser importance are 
often made 50 ft. These widths, however, are more liberal than 
have been used in many of our older cities, e.g., such cities as 
Boston, Baltimore, and New York which are especially afflicted 
with narrow streets. 

The alleys which are run through the middle of city blocks 
should be made from 15 to 20 ft. wide. If they are made nar- 
rower than 1 5 ft. two teams cannot pass each other unless cer- 
tain parts Qi the alley are widened for this purpose. Alleys 
furnish a convenient place for the location of water pipes and 
sewers. 

The width of sidewalks varies greatly with the locality. In 
business districts, where there is usually a necessity for ample 
width, some cities devote two-fifths of the entire width of the 
street to sidewalks; while in residential districts, the sidewalks 
are frequently much narrower in proportion to the width of the 
street. In Boston the general rule is to make each sidewalk 
one-sixth the width of the street. Sidewalks 8 ft. wide are 
ample for most residential districts. In some localities walks 
as narrow as 4 ft. are laid out with a liberal grass-plot between 
the sidewalk and the roadway, which not only gives a pleasing 
appearance to the street, but also lessens the width of sidewalk 
and of roadway to be paved and maintained, thereby decreasing 
the burden of taxation and leaving room for an increase in width 
of roading if afterwards needed. 

249. STREET Grades. — In connection with the layout of 
anew city or suburb the grade of the streets is of quite as 
much importance as the street alignment. While, in the 
residential districts of some cities, street grades as steep as 10 
and 15 per cent, are not uncommon, still it is considered 
advisable, if possible without excessive cost, to keep the grades 
down to about 5 or 6 per cent., especially those which extend 
for any considerable distance. In business districts, where 
heavy loads are to be hauled, it is desirable that the grades 
should not exceed 3.5 or 4 per cent. In any case where one 
street crosses another the grade should be flattened between 
curb lines to 3 or 4 per cent, if the grade of either street is 
greater than this amount. 



STREET GRADES 225 

On account of drainage it is well to build a street with a 
slight grade rather than level. A grade of 6 inches in loo feet 
is a good working minimum for proper drainage, and if the 
street does not have this gradient the gutters must be made of 
varying depth so as to properly carry ofiF the water. Other 
elements which govern the rate of grades are the cost of earth- 
work and the proper balancing of the excavation and embank- 
ment in the construction, the effect on abutting property, and 
the general appearance of the street. 

At points where there is a decided change in grade it is 
customary to introduce a parabolic vertical curve. (Art.. 268, 
p. 242.) 

For the purpose of establishing the grades, profiles are 
made of each street. Levels taken for the purpose of making 
a profile should include elevations at the center of the street and 
along both side lines, and it is often desirable to have a cross- 
section plan of the entire area of the vicinity where the street is 
to be located. A description of the street grade is written up 
for acceptance by the proper municipal authorities. When this 
description has been formally accepted by an order of the City 
Government the grade is said to have been "established." 
Such an order may refer to the profile by title or recorded 
number, instead of a description of the grade. The profile of 
each street should contain one or more cross-sections on which 
is indicated to what part of the cross-section the profile refers, 
i.e., whether the profile grade is the grade of the center of the 
street, the curb, or the sidewalk at the property line. 

The following is an example of a description of an estab- 
lished street grade : — 

"Beginning at Station 146 (Maple St.) at the junction of 
the center lines of Maple St. and Ocean Ave., at grade * 52.00, 
the grade line falls 0.50 per 100 for 726 ft. to grade 48.37— 
thence rises 0.82 per 100 for 322 ft. to grade 51.01 — thence 



• The word grade U frequently used to mean the elevation of a point. In 
such a case care should be taken not to confuse the meaning of grade with rate of 
grade. The latter is sometimes called gradients a word which has some advan- 
tages but is not entirely satisfactory. 



226 CITY SURVEYING [Chap. IX 

falls 0.50 per 100 for 122 ft. to grade 50.40 — thence falls by a 
vertical curve for 100 ft. as follows : 

Sta. EleT. 

157+60 S<MO 

157 + 85 4990 

158+ 10 49-30 

158 + 35 48.55 

158 + 60 47.70 

thence falls 3.60 per 100 for 239 ft. to Station 160 + 99 
(Maple St.), grade 39.10." 

250. The Datum Plane. — One of the first tasks of the 
surveyor in laying out a town site is to establish a datum plane to 
which all elevations may be referred. It is customary to choose 
a datum that bears an intimate relation to the topography of the 
locality. For example, if the town is located on the seashore a 
series of tidal observations may be taken to determine the mean 
sea-level or mean low water either of which is often used as a 
datum (Art. 237, p. 211). The mean level of lakes is used as 
a datum for many inland cities. Frequently the elevation of 
some point not far from the town site has been established by 
the U. S. Geological Survey, the U. S. Coast and Geodetic Sur- 
vey, or by the line of levels of a railroad ; and by careful level- 
ing the elevation of some permanent point in the town site can 
be established which will serve as the starting point for all the 
elevations in the town. Where nothing of this sort is available, 
the elevation of some point is found by barometer so that the 
recorded elevation may approximate the actual height above sea- 
level. 

251. ESTABLISHING BENCH MARKS. — When the datum has 
been determined, bench marks are established by the method 
explained in Art. 219, p. 198. The establishment, at the start, 
of a reliable system of bench marks is of utmost importance, in 
order that the elevations of all parts of the city shall refer to the 
same datum. In laying out construction work it is absolutely 
necessary that bench marks which can be relied upon shall be 
available and sufficiently numerous to be of use in any section 
of the city without requiring several set-ups of the level to con- 
nect a bench mark with the level work that is to be done. 



WATER AND SEWER SYSTEMS 22/ 

Another advantage in having them close together is that they may 
serve as ready checks on each other as well as on the work at 
hand. It is not uncommon for a bench mark to be disturbed, 
and, if the level work is not occasionally checked on some other 
bench mark, an error will surely enter into all of the level work 
which was started from that bench. 

252. WATER AND SEWER SYSTEMS. — The water and sewer 
systems of any community are of vital importance and provis- 
ion for them must be made in the layout of every town site. 
The location of the water supply and the storage and distributing 
reservoirs is a matter of such magnitude that it cannot be dis- 
cussed in this short treatise.* The conditions essential to an 
economical water or sewer system will sometimes radically affect 
the alignment and grades of many of the streets. The gradi- 
ents of water pipes are of little importance since the water is 
working under pressure, and the pipes can be laid uphill and 
downhill so long as there is sufficient " head " to force the water 
through the pipes. 

In a sewer system the problem is far different ; every sewer 
must have proper g^dients, and the entire system must fall 
gradually from the most remote points to the main sewer outlet. 
The topographic map therefore is of utmost importance as a basis 
for a study of this problem, f 

STAKING OUT CITY WORK. 

253. STAKING OUT A NEW DISTRICT. — In staking out 
a new district the information at hand is usually a plan of the 
proposed layout of the streets which has been studied out in 
the office from a map of the district. If this layout has been 
approved by the municipal authorities the street lines as they 
appear on the plan are the "established lines." 

It is the surveyor's duty to stake out these lines on the 
ground, connecting them properly with the street lines of the 

* See Public Water Supply, by Toumeaure and Russell, published by John 
Wiley & Sons, New York. 

t See Sewerage, by Professor A. P. Folwell, published by John Wiley & 
Sons, New York. 



228 CITY SURVEYING [Chap. DC 

older portion of the city, and in short, to produce on the ground 
a layout exactly like that on the plan. Sometimes the angles 
and distances necessary for the layout have been computed m 
the office, but more frequently these are not determined until 
the lines are laid out on the ground. In reproducing these lines 
on the ground the surveyor will often find that the exact dimen- 
sions given on the plan do not correspond with his fieldwork 
owing probably to the fact that' his tape differs in length from 
that used by the surveyor who made the original plan. In such 
a case he must distribute the discrepancies (unless ihey are large 
enough to indicate that a mistake has been made) in the proper 
manner in his work. 

Not infrequently the entire work is staked out from a plan 
which has been made in the office, and the exact angles and 
distances as determined in the field are recorded on this plan 
which then goes to the proper authorities to be put in the form 
of a city order. As soon as the plan is accepted the street lines 
should be marked by monuments (Art. 254), so that there may 
be no difficulty in retracing the lines as they were originally laid 
out and accepted. If considerable grading work is to be done 
in building the new streets it may not be practicable to set 
many of the corner bounds at first on account of the likelihood 
of their being disturbed. In such cases it is the duty of the 
surveyor to properly reference the points by cross transit lines 
or otherwise before construction work begins ; for it is important 
that the layout, as recorded in the city order, shall be accurately 
and definitely defined so that when the streets are brought to 
the proper grade and the monuments are finally set they will 
mark the exact position of the original layout. 

254. MONUMENTS. — It is important and at the same time 
customary to define street lines by setting stone bounds, often 
called monuments^ at the street corners and at angles in the 
street lines. The bounds are set sometimes on the side lines, 
sometimes on the center lines, and sometimes in the sidewalks. 

At street intersections, one monument at the intersection of 
the center lines will suffice to mark both street lines, but since 
this point will come in the center of the road pavement where it 
is likely to be disturbed by traffic or by street repairing it is sel- 



MONUMENTS 22$ 

dom placed there. The more practicable method is to defini 
the street lines by marking the side lines at the angles or, in 
the case of rounded corners, at the beginning and end of the 
curves. It is not necessary that all four corners of a street 
intersection shall be marked, as a bound on one corner will define 
the side lines of the two streets and, the width of the streets 
being known, the other sides can easily be determined. Nor 
is it necessary to place a bound at one of the corners of every 
street intersection, provided a street is straight for several blocks, 
although it is good practice to do so. On account of the liability 
of bounds which are placed on the side lines of the street being 
disturbed by building operations, some surveyors prefer to place 
them on an offset line, say 2 ft. from the street line. All mon- 
uments should be placed with extreme care as regards both 
their accuracy of position and their stability. If any bounds 
are set with more care than others, they should be the ones 
which occur at angle points in the street lines rather than the 
intermediate bounds which are set along a straight line. 

Monuments are usually roughly squared stone posts about 4 
to 8 inches square and 3 to 4 feet long, the length depending 
upon the severity of the climate, e.g., in New England a monu- 
ment less than 4 ft. long is likely to be disturbed by frost action. 
They are carefully squared on top and a drill-hole in this end 
marks the exact point. This drill-hole may be made before the 
stone is set in place, or after it has been placed so that its 
center is about in position the exact point may be defined by 
drilling a hole in the top of the bound. Frequently the hole is 
filled with lead and a copper nail set in the lead is used to mark 
the exact point. For nice definition of the point, a copper bolt 
is inserted and two lines scratched across it ; the intersection 
marks the exact point. When the stone bound is placed at the 
intersection of the side lines of the streets it is sometimes 
located entirely in the sidewalk in such a way that its inside 
corner is exactly on the intersection of the street lines. In such 
a case the three other corners of the bound are usually chipped 
off so that there may be no mistake as to which corner defines 
the line, but the line comer frequently becomes worn oflf and 
this practice is therefore not recommended. Some surveyors 



230 



CITY SURVEYING 



[Crap. DC 



use, in the place of stone bounds, a piece of iron pipe or iron 
plug with a punch-hole in the top of it, driven into the ground 
or embedded in cement concrete. Long heavy stakes are 
employed to temporarily define intermediate points or points of 
secondary importance. 

255. Setting Stone Bounds. ^— When the street lines are 
laid out the corners are marked by tacks in the top of ordinary 
wooden stakes. The monuments which are to take the place of 
the stakes should be set before the frost has entered the ground 
or before any other disturbance of the stakes has taken place. 
When the bound is ready to be set the first thing to do is to 
drive four temporary stakes around the corner stake about two 
feet from it and in such a way that a line stretched from two 
opposite stakes will pass over the tack in the head of the corner 
stake (Fig. 99). Then tacks are carefully set in the tops of 

these temporary stakes in 
such positions that a stretch- 
ing line running from the 
tack on one stake to the 
tack on the opposite stake 
will pass exactly over the 
tack in the corner stake. 

Then the comer stake is 
removed and the hole dug 
for the stone bound. Care 
should be taken not to 
dig the hole any deeper 
than is necessary so that the 
bound may be set on firm 
earth. As to the position of the top of the bound with refer- 
ence to the surrounding ground, surveyors disagree. Some 
prefer that the monument should stick out of the ground so 
that it can be readily found ; while others claim that if it pro- 
jects above the surface the bound is likely to become misplaced 
by traffic, and therefore that it is better to set it just flush with 
the ground or slightly below the natural surface. If any grad- 
ing is to be done in the vicinity the bound should be set so that 
it will conform to the proposed grade. When the hole for the 



SHi-B- 



Shntrching 



I 



B 



E3r 



Line 



-BSHu 



Stone Dound 



Hk. 



Fig. 99. Setting a Stone Bound. 



SETTING STONE BOUNDS 



231 



w:^A 






bound has been dug to the proper depth it is well to stretch the 
strings across between the temporary stakes and plumb down 
roughly into the hole to determine where the center of the 
bound will come, so that when the monument is dropped into 
the hole it can be placed so that it will set plumb. 

The bound having been set in the hole, the next operation is 
to fill around it. This should be done with considerable care, 
the material being properly rammed as the filling proceeds and 
the bound kept in such a position that the drill-hole in the top 
of it, if there is one, shall be exactly under the intersection of 
the strings. It is sometimes desirable to put in a foundation 
of concrete and to fill with concrete around the monument to 
within a foot of the surface, as shown in Fig. 100, where a 
very substantial bound is re- 
quired, or where the ground is 
so soft as to furnish an insecure 
foundation. If the top of the 
bound is plain and the hole is 
to be drilled after the bound 
is in place, care should be taken 
to place the monument so that 
this hole will come practically 
in the center of the top in order 
that it may present a workman- 
like appearance. After the 
bound is set exactly in place the 
temporary stakes are removed. 

Some surveyors prefer to use only two opposite stakes and 
one stretching line, the position of the monument being deter- 
mined by a measurement along the stretching line from one or 
both of the temporary stakes. Still another method of tempo- 
rarily tying in the stone bound, and one which many surveyors 
use, is to set two stakes such as A and B in Fig. 99, and 
either measure the distance from them to the bound or set 
them at some even distance from the bound. This process 
of using temporary stakes and the stretching line is employed 
also in setting other types of bounds such as gas pipes or 
iron rods. . 



W3^: 



i»l 



W^M 



Fig. 100. Stone Bound with 
Concrete Foundation. 



232 



CITY SURVEYING 



[Chap. DC 



In the construction of buildings or fences, monuments axe 
frequently distCirbed and too often they are reset by the owner 
of the property without the services of a surveyor. In rerunning 
a street line, therefore, a surveyor should be on the lookout 
for such conditions, and he should be cautious in the use of 
any monument which he has any reason to suspect may have 
been misplaced. 

256. CURVED LAYOUTS. — It is not unusual for streets to 
be laid out with curved lines. In the design of boulevards, 
parks, and residential sections a landscape architect is often 
called in and the plan he presents is sometimes almost devoid of 
any straight street lines. (See Fig. loi.) The surveyor must 




Fig. 101. Curved Layout for REsmENXiAL Part of a City. 

take this plan and from the design there given stake out the 
layout and obtain the necessary dimensions to definitely locate 
all parts of it. 

As a rule the landscape architect simply draws on the topo- 
graphic map his scheme of layout with very few dimensions and 
leaves the rest to be worked out by the surveyor. Occasionally 



CIRCULAR CURVES 233 

the radii of the curves are noted on the plan, but the street 
widths are often the only dimensions given. If the radii are 
not given the surveyor must determine from the plan either 
these radii or some other distances, such as the tangent lengths, 
so that he can go into the field, and, beginning with some known 
street line, run out the new street lines in such a way that when 
the data he determines are plotted the lines will coincide with 
those on the plan prepared by the landscape architect. As a 
rule these curved lines can be made up of a combmation of cir- 
cular curves. 

257. ELEMENTS OF A CIRCULAR CURVE.— Before consider- 
ing how to stake out a curve it will be well first to refer to the 
elements of a simple circular curve. In Fig. 102 which repre- 
sents a simple circular curve 

05=- Radius =/? 

A HB = Length of Arc = L^ 

AB^ Long Chord =C 

VA = F5 = Tangent Distance = T 

VH = External Distance = E 

HF == Middle Ordinate = M 

I = Intersection Angle, or 

Central Angle 
V = Vertex 

P.C. = Point of Curvature 
P.r. = Point of Tangency fig. 102. Circular Curve. 

From simple geometric and trigonometric relations, 




2 K 


T = i?tan - 
2 


Exsec ^ = £ 
2 R 


E = R exsec - 

2 


Vers^ = ^ 
2 R 


M = R vers- 
2 


Sinl=£. 

2 2R 


C= 2/2 sin - 
2 




L^—RX Circular measure of /.* 



• The curves used in railroad engineering are usually very flat, so that there is 
little difference between the chords and their corresponding arcs. This fact 



234 



CITY SURVEYING 



[Chap. DC 



258. STAKING OUT CIRCULAR CURVES In Fig. I02 the 

two lines BC and EA are produced in the field and a point is set 
at their intersection V, as described in Art. 200, p. 175. The 

instrument is then set up at V 
and the central angle / carefully 
measured, or if point / is inacces- 
sible other angles such as VEC 
and VCE may be measured from 
which / can be easily computed. 
Then the radius R which is deter- 
mined from the plan being known, 
the tangent distance T is obtained 
by the formula, T' = i? tan ^ /. 
Points P.T. and P,C. are then set 
and the curve is usually laid out 

by the method of deflection angles as explained in the following 

article. 

259. DEFLECTION ANGLEa — A deflection angle is usually 
referred to as an angle between a tangent and a chord, e.g., in 
Fig. 103 angles VAh, VAc, etc., are deflection angles. Since 




Fig. 102. Circular Curve. 




Fig. 103. Deflection Angles. 



makes it possible to compute the length 0/ curve by a simple approximate method, 
which, however, is sufficiently exact for most railroad work. 

The Degree of Curve^ which is the angle at the center subtended by a chord of 
100 ft., 18 an element of the circular curve which is used extensively in railroad en- 



DEFLECTION ANGLES 235 

the angle between a tangent and a chord is measured by half the 
included arc these deflection angles must be equal to half the 
angle at the center subtended by the same chord or arc. 

If the total length of the curve is divided into an even num- 
ber of parts, «, the angle at the center under each of these arcs 

will be -, and the deflection angle for one chord will be — , 
n 2n 

which in Fig. 103 is the angle VAh, Angle hAc = angle VAh^ 

both being measured by one-half of equal arcs. It follows then 

that the deflection angle to point 

/ / 

^ = 2 X — = - 

2n n 

2n 2n 

I 2/ 
^ = 4 X — = — 
2« n 

etc. 

Evidently, after the first deflection VAb is found, the 
other ddSections can be obtained by simply adding the incre- 
ment — to the preceding deflection angle, and this is the 

2H 

method which should be used. The deflection angle from the 

P. C. to the P. T, should be equal to - , and this check should 

always be applied to the computations before they are used in 
laying out the curve. 

The chords Ab, be, cdy etc. are equal since their arcs are 

equal. With the radius and the central angle [ - for one chord] 

given, the chord length can readily be found from the formula, 

gineering. The central angle divided by the degree of curve will give the number 
of loo-ft. chords in the length of the curve, i-c.-^ = Z ( in loo-ft. stations). 

Therefore L (in feet) -■ -^ • For a complete discusaon of railroad curves see 

** Railroad Carves and Earthwork/' by Professor C. F. Allen, published by Spon 
& Chamberlain, New York. 



^26 CITY SURVEYING [Chap. DC 

c = 2 sin — Since the angle at the center is usually small 

and the radius large the angle will have to be carried out in 

some instances much closer than to the nearest minute in order 

that the length of the chord may be obtained to hundredths of 

a foot (Art. 371, p. 341). An approximate value for the chord 

length corresponding to a given arc may be obtained by the 

approximate formula, 

^ C * 

, or = 



24i?«' 24/e* 

in which 4 is the length of the arc, c is the chord length, and 
R the radius. 

The fieldbooks in use by most surveyors contain tables of 
chords and corresponding arcs for curves of different radii, 
which assist greatly in shortening these computations. 

When the deflection angles have been computed and checked 
and the chord length found, the instrument is set up at A, 
(Fig. 103) a foresight taken on the vertex with the vernier 
reading o^ and the point 6 set by measuring Ab and placing d 
on line by means of the transit on which the first deflection 
angle VAb has been laid off. Point c is set by measuring 6c 
and placing c on line with the transit on which the second 
deflection angle has been laid off, and so on, until the last point 
(P. T.) has been set. 

It is evident that with the transit at the P.C. the curve 
could have been laid out just as well by taking the measurements 
from the P. T, end, and some surveyors prefer to do it this way. 
Similarly the instrument might just as well have been set up at 
the P.T. instead of the P.C. and the measurements started from 
the P.C. if it were found to be more convenient. 

* The following will give some idea of the accaracy of this formula. 
With R -■ 100 and /• — 25, the formula gives c — 25.065, (correct value 

is 25.066). 
With R « 100 and /• — 50, the formu a gives c — 50.521, (correct value 

is 50.536). 
With R « 1000 and /« » 100, the formula gives c « 100.042, (correct value 

is 100.042). 
. . .This formula will be found very useful if a slide rule is employed for the 
computation. 



LAYING OUT CIRCULAR CURVES 



^37 



It is sometimes necessary to set definite station points on the 
curve rather than to cut the curve up into several equal parts as 
suggested above. The principle is exactly the same as described 
above ; but in figuring the deflection angles and the chord lengths 
to be used the computations are not quite so simple. No trouble 
will be experienced, however, if it is borne in mind that the total 
deflection angle to any point is equal to half the central angle to 
that point from the P.C.y and that the central angle for any arc 
bears the same relation to the entire central angle that the arc 
does to the entire length of curve. 

260, Keeping the Notes. — In a curved street the notes of 
alignment generally refer to the center line, the two side lines 
being parallel to the center line. All three of these lines have 
to be nm out by the use of chords and deflection angles ; Fig. 
104 is an example of a concise form of notes for this work. In 



^CMTWI 



Station 



Oishmee 
(An:) 



Chords 



UA- I Ufiher I ^y>»/ 



DthbcfiMt 



Ait^MS 



nuKfrK 



ft*200 
T'96j66 



t8*SZj50 

l7t22M 
/6^72AZ 



30jOB 
Jif,oo 
SOjOO 
S>jOO 



S&59 

.S3J9 
SSS9 



3(kQS 
49,87 

49.87 



70f€t. 



24J9 
4/.I4 



4StS7 



4IM 
4/.f4. 



2SU7'-4Cr 
7'09^40 



i?7 



/?C. 



Fig. 104. Notes of a Circular Curve. 



the first column is a description of the curve, which refers to 
the center line of the street. This particular curve is marked 
" To Right " meaning that it deflects to the right while passing 
around it in the direction in which the stations run! In the 
third column are the distances measured on the actual arc along 
the center line. The next three columns headed " Chords " are 
the chord measurements across the curve from station to station 
on the left side line, the center line, and the right side line of 
the street, the terms left and right meanmg left and right look- 
ing in the direction in which the stations run. In the column 
headed"*' Deflection Angles '* are the total deflections to be laid 
off with the instrument set up at theP.C. These same deflection 



238 



CITY SURVEYING 



[Chap. DC 



angles are used in running out the side lines for the chords which 
have been computed for the side lines run between points which 
are radially opposite the corresponding points on the center line. 
The computation of these notes will be found in Art. 371, p. 341. 

261. When the Entire Curve Cannot be Laid Out from One 
End. — It is often impossible to see from the P.C* to the P, T, 
of a curve on account of intervening obstructions. In such a 
case the curve is run from the P.C. as far as is practicable and a 
point is carefully set on the curve ; then the transit is brought 
forward and set up at the point thus fixed, and the curve extended 
beyond. There are two different methods employed in this case. 

262. First Method. — Assume the circular curve in Fig. 
105 to be laid out from A to d as described above. Point d is 




Fig. 106. Intermediate Set-up on Curve, 

carefully set and the instrument then taken to that point and 
set up. The vernier is turned back to o*^ and beyond 0° by the 
value of the deflection angle VAd, Then by using the lower 
clamp and tangent screw the telescope is sighted on point A. 
The upper plate is then undamped and, if the telescope is turned 
so that the arc reads 0° the instrument will be pointing along 
the direction of an auxiliary tangent df, for angles VAd and Adf 
are equal. It is well to note whether the instrument appears 
to point in the direction of the tangent. Then reverse the tele- 
scope, set off on the vernier the angle gde = — , and lay out the 

2H 



INTERMEDIATE SET-UPS. 



239 



curve from dtoB just as though it were an independent curve 
beginning at d and ending at B. 

263. Second Method. — When the transit has been set 
up at dy the vernier is set at 0° and a backsight taken on A, 
Then an angle equal to the deflection angle VAe is laid off on 
the arc; this will cause the telescope to point in some such 
direction as dA. The line of sight is reversed and point e set on 
//^/ produced, making the chord de of the proper length. Then 
point B is set by laying off on the vernier an angle equal to VAB 
and measuring the chord eB. This method is correct for 

VAe^ VAd+dAe 

^fdA + kdft being measured by half of equal arcs. 

This second method is sometimes to be preferred since the 
original deflection angles figured can be used throughout the 
curve. The first method calls for the calculation of a few more 
angles ; but this is so simple a process that there is probably 
little choice between the two methods. 

264. CURVED STREET CORNERS. — It is the practice in many 
cities to curve the corners of the streets by introducing a cir- 
cular curve of short radius. Where both street lines are straight 
the problem is handled as explained in Art. 258, p. 234. 

265. * One Street 
Line Straight, the Other 
Curved. — In Fig. 106 
the curved street line 
DEF intersects the 
straight street line 
A V and at this point 
the circular curve whose 
center is C and with a 
given radius r is to be in- 
troduced ta round off the 
corner. It is required 
to stake out the curve 
GE on the ground. In 




Fig. 



• The authors are indebted to I. T. Famham» City Engineer of Newton, 
Mass., for the solution of the problems given in Arts. 265-6, 



240 



CITY SURVEYING 



[Chap. DC 



the field any tangent line, such as FV, is run ofF from some 
known point on the curve and intersected with A V, and the 
angle fi and the distance FV^lvq measured. In the right triangle 
CT^Fin which R and T'are known, compute angle a and distance 
CV. In the right triangle CAVyC Fand 7 = ^ — a being known, 
compute CA and AV\ CA' = CA -r; CC ^ R^ r. In the 
right triangle CA'C\ CA' and CC being known, compute A'C 

and A'CC = GCE. 
Angle ilCF - 1 80° — p. 
Angle ECF^ACF — 
A'CC\ from which the 
length of the arc FE 
can be readily com- 
puted, which locates the 
point E. VG^ A F— 
A'C J which locates point 
G of the curve GE^ and 
any intermediate points 
can be located as ex- 
plained in the previous 
articles. 

As the radius OE is often quite short the center of the 
curve can be located from either its P,C, or P.T. or both, and 
any intermediate points on the curve can be easily swung in 
from its center. 

266. Both Street Lines Curved. — In Fig. 107 the two curved 
street lines ABD and A^B'T/ intersect each other and the curve 
whose center is E and with a g^ven radius r is introduced at the 
intersection of the two street lines. It is required to locate the 
curve B*B on the ground. In the field the tangent DV 
is run off from some known point D on the curve ABD and 
intersected with a tangent jy Ffrom the curve A'SU and angle 
a and distances T and T' are measured. In the right triangle 
CDVy R and T being known, compute angle C7Z) and distance 
CV. Similarly in the triangle CVD' compute angle C'VIV and 
distance C^. In the oblique triangle CYC, C7* CV and 
angle CVC = 360° — (a + CVD + CVD') being known, com- 
pute CC and the angle CC'F and CCV, In the oblique tri- 




FiG. 106. 



ROUNDING STREET CORNERS 



241 



angle CCE, CE -:- R + r, CE ^ R' -- r, and CC being known, 
compute the angle CCE, CCE and CEd which is the sup- 
plement of the central angle of the curve BB. Angle 
DCB = DCV+ VCC — CCE, from which arc DB can be com- 




Fig. 107. 

puted. Similarly angle lyCW = D'CV + VCC - CCE, from 
which arc D'B' is computed. These locate the P.C. and P.T. 
of the small curve whose center is E. 

267. STAKING OUT STREET GRADES. — The fieldwork neces- 
sary in setting grade stakes is explained in Arts. 235-6, p. 210. 
When new streets are constructed the excavation or embankment 
is first brought to sub-grade, i.e., to the grade of the bottom of the 
road covering or pavement. The grade stakes set for this work are 
usually the center and the two side slope stakes, properly marked 
with the cut or fill, as described in Arts. 230-2, pp. 207-10. 

As the work progresses the center stake is dug out or covered 
up and when the construction has progressed nearly to the sub- 



242 



CITY SURVEYING 



[Chap. DC. 



grade it is customary to set stakes at the elevation of the sub- 
grade along the center line and on each side line of the street. 

268. Vertical Curves. — Where the rate of grade of a street 
changes, in order to avoid an abrupt transition from one grade to 
the other, a vertical curve is introduced which is tangent to both 
grade lines. The simplest curve to locate for this purpose is the 
parabola. 

In Fig. 108 L f^and KAf represent two grade lines intersect- 
ing at V. The parabola AHB is tangent to these lines at A and 
B. It is often customary to set the grade stakes on a vertical 




Fig. 108. 



Vertical Parabolic Curve. 



curve at every 25-ft. station ; in such a case, then, Fig. 108 rep- 
resents a vertical curve 200 ft. long on which the elevation of 
nine points must be determined. The equation of the parabola is 



^2 _ 4 p^^ or j' = (constant) Xy 



(0 



the jr dimensions being parallel to VK (vertical) and the^ dimen- 
sions being along A V, From the equation it is readily seen that 
the offsets from the tangent vary as the squares of the distances 
along the tangent, oxx^'.x^^^ y'^ y}. The lines VP and NQ are 
vertical and AQ is horizontal. Since the curve extends an equal 
distance each side of V, AP =^ PQ ; and therefore AK =^ KB. 
NB^aVH; VIf=4CC; CC = 4DD'; etc. (from equation i.) 
Let ^ and g-^ represent the rate of grade of LV and VM, 
and n the number of 25-ft. stations (in this case 4) on each side 
of the vertex V, then 

NB^-U-hg,) 

4 



KV = 



NB 



(from similar triangles) 



VERTICAL CURVES 243 

but NB = 4 VH (from above) 

therefore KV = 2HV, 

or point ff is midway between V and K. 

The elevation of f^is determined from the established grade. 
The number of 25-ft. stations will determine the distance VA 
and VB. The elevation of A and of B can be readily computed 
along their respective straight grade lines. 

Elev./r = ?l?X-^+Elev.B 



Elev. If = 



2 
Elev. V+ Elev. /i: 



2 
Fif == Elev. /T- Elev. F. 

Elevations of all the other intermediate points along the curve can 
be computed by finding the elevation of the points D\ C\ £', 
Fy G', and J/ and by adding to these elevations the ordinates 
D'A CCET £, etc. 

VH 



lyn = rj = 

CC = GG = 
EE =^ FF = 



16 
VH 

4 
16 



269. CROSS-SECTION OF STREET. — On account of the ne- 
cessity for draining the surface of a road the center is raised or 
" crowned" above the grade of the gutters by an amount depend- 
ing on various conditions. The shape of the road surface is 
sometimes two planes, running straight from the gutter to a 
summit or ridge in the center of the street, this ridge being 
rounded off by rolling ; but more frequently it is a curved sur- 
face in the form of a parabola or a circle. The ordinary width 
and crown of streets are such that the parabola and the circle 
are practically coincident. 

When a street is to be paved the curbstones are first set to 
proper line and grade, then stakes are set for the finished grade 
of the roadway. The center grade stake is frequently the only 



244 CITY SURVEYING [Chap. DC 

grade given and va templet, or form, which can be set on the 
curbs and on this center stake is used to give the form of the 
cross-section. The form of the teftiplet for this work is laid 
out by the surveyor. If no templet is used he should put 
in intermediate grade stakes betVeen the center and the curb 
lines. In either case the surveyor, triust compute the necessary 
ordinates to give the proper shape to the surface. 

Usually the mean transverse slope of the pavement is given 
either in the form of a ratio thus : 

•hM T CI Crown I 

Mean Transverse Slope = 



Half the Width of Carriageway 30 
or. Mean Transverse Slope = f ' per ft. 

270. Gutters at Same Elevation. — Fig. 109 represents the 




"< w > 

Fig. 109. Cross-Section of Pavement ; Gutters at same Elevation 

cross-section of a pavement and sidewalks. The crown AB is 
computed from the mean transverse slope and the width of the 
pavement. 

The ordinate Diy at any other point on the parabola 
= CC X -TTTzr since in a parabola the offsets from a tangent 

Q 

vary as the square of the distance out along the tangent (Art. 
268, p. 242). But CC = AB ; hence, if D' is half-way from 

AB 

the center to the curb, DD^ = . 

4 

271. One Gutter Higher than the Other. — When one gutter 
is higher than the other the following application of the parabola 



CROSS-SECTION OF PAVEMENT 245 

can be used In Fig. 1 10 the maximum ordinate .r is at a dis- 



FiG. 110. Cross-Section of Pavement: One Gutter Higher 

THAN THE OTHER. 

tance a from the lower gutter. The first step is to find this 
distance a and then x is readily found from the mean transverse 

X 

slope since - = Mean transverse slope. When x is found the 
other offsets can be computed as explained in the previous article. 

At A the offset = - 
4 
at 5 = JT— * 

^ x-b 

at C= 

4 

The width of pavement, the difference in elevation of the 
gutters, and the mean transverse slope being given, the formula 
for a is derived as follows. 

In Fig. no, W ^ width of pavement. 

R -■ radius of the circular curve DACB. 
a ^ distance from the line of the lower gutter to the highest 

point of the pavement. 
/ i« distance from the line of the highest gutter to the highest 

point of the pavement 
b mm difference in elevation between the two gutters. 
/ » mean transverse slope, expressed as a ratio of crown to half 

the width of pavement. 
X i* difference in elevation between the lower gutter and the 

highest point on the pavement. 

x - £_ (i) (See (I) in foot-note, p. 339.) 

2 R 

and X ^b ^ — (2) (See (i) in foot-note, p. 339.) 

2R 

2R 



246 



CITY SURVEYING 



[Chap. IX. 



Combiiiing (i) and (2),a> 


-/»- 


zRb 


{a+/)(a- 


-/)- 


zRb 


a 


-/- 


2Rb 


But a 


+ /- 


W 


.*. a 


-/- 


zRb 
W 


(tf + /) + (a . 


-/)- 


'^+'^ 




2a « 


^+'^ 




a « 


IV Rb 




T'^ W 


From (i), 


2-ff- 


a* 



(3) 



But 



- i- Mean transverse slope ■■ j 

a 



From (3), 



.-. aJ? - eL . « 

as s 



ab 

W Ys 

a ^ — + _ 

2 ^ 



\ 2Ws] 



2 



2 



2lVs 



272. If, instead of assuming the mean transverse slope of 
the pavement, the elevation of the center of the pavement D 
(Fig. Ill) with respect to the elevation of A and B is assumed. 



CROSS-SECTION OF PAVEMENT 



247 



then DG is readily found and the elevation of such points as E 
or F, which are midway between D and the gutters, are com- 
puted from the method explained in Art. 270, eE and fF both 

being equal to -—1 . 
4 



Similarly, Elevation E = Elevation G + 



Elevation F = Elevation H -h 



Elevation J = Elevation K + 



Elevation L == Elevation M + 



4 

4 

7DC 

16 

tdc 

16 



etc. 



373. IRREGULAR SHAPED BLOCKa — There is a wide vari- 
ance of practice in the method of cutting up irregular shaped 
blocks into lots. One good general rule in such cases is to give 



J' 


60 


eo 


60 


60 


60 


60 


./>^ 


8 


^ 


^ 

k 




r>J 


^^ 


U^ - 


s X> 


> 


y^^ 






l^ 


i^ 















Fig. 112. Arrangebcent of Lots in a Wedge-Shaped Block. 



each lot as much street front as is possible consistent with mak- 
ing the side lines of the lots at right angles to the street lines. 
If the side lines do not run at right angles to the street there 
will be portions of the lot which are not available for the custom- 
ary rectangular style of building construction and which are 
therefore not so desirable for business purposes. This is not of 



248 



CITY SURVEYING 



[Chap. DC 



SO much importance in residential districts where the rectangu- 
lar system is often purposely avoided to some extent, to obtain 
a layout which has an attractive appearance, as illustrated by 
Fig. loi, p. 232. 

Fig. 112 is an example of an irregular shaped block in which 
rectangular lots have been planned, the wedge-shaped remnants 
being thrown into the comer lots. 

274. STAKING OUT City lots.— In staking out the lots of 
a rectangular block, the comers of which have been established, 
the most direct method is as follows. The transit is set up on 
the S. B. at A, (Fig. 1 1 3)> a sight is taken on B, and the front 



_ji 



J 





9 






















'» 


f 




90' 




JO 


M 


• 


ao 

9- 




SO 

■ 


ae 

M 


M 
tt 


M 

n 


JO 

n 


90 




V 


t 


i 


4 


9 


« 


7 


• 


» 


» 


II 


«c 




A 




«> 


SO 




M 


»•_ 


» 


*» 


ae 


ao 


30 


-iUS 


t 



~i r 



Fig. 113. Rectangular City Block. 



comer stakes of lots i, 2, 3, 4, etc., are set, with a tack, exactly 
on line, in the top of each stake. All such work should be done 
to the nearest 0.0 1 ft. It will be well first to measure the line 
ABf to see that it is just 600 ft. long. Since it is assumed that 
considerable care was used in setting the S. Bs. exactly in the 
correct position, if it. is to be found to be a few hundredths over 
or under 600 ft., it is probable that this discrepancy is due to 
the difference between the length of the tape on the present 
.work and that, used in the original layout. In such a case the 
twelve lots m\ist be laid out with equal frontages. For example. 



STAKING OUT CITY LOTS 249 

it may be a hot day when the lots are to be staked out and the 
tape may give a distance from ^4 to 5 of 599.88 ft. In this case 
each lot should measure 49.99 f^- wide. 

With the instrument still at A and sighted on C, point D is 
set by measuring 66 ft. from C, and then point E v% placed mid- 
way between A and D. Whatever slight discrepancy there may 
be in the distance between the S. B. at A and that at C is thrown 
into the depth of the lots rather than the width of the street. 

By setting up the instrument at B and sighting on H^ points 
F and G are set. Then by setting up at F and sighting on D 
the front comers of lots 1 3, 14, 1 5, etc., are determined. Another 
set-up of the transit at G with the line of sight on E will allow 
the "back bone" to be run out and the back corners of all the 
lots established. The check on the lines ABy EG^ and DF is 
their total length. The depth of the lots can easily be checked 
by taking direct measurements from the front to their rear corner 
stakes. If a further check is desired the transit can be set up 
at each of the front corner stakes of the lots on one street and 
a right angle turned off to check the position of the rear corner 
stakes and the front corner stakes of the lots on the street be- 
yond. 

By the method suggested above the street lines are made 
straight and the slight inaccuracies which may occur in the field- 
work are put into the back and side lines of the lots. 

Some surveyors prefer, after the front stakes on both streets 
are located, to set up the instrument at each front comer and 
locate the back corner stakes by tuming a 90® angle and laying 
out the depth of the lot, at the same time checking the position 
of the front stakes on the street on the other side of the block. 
Then the distances along EG are measured to check this field- 
work. 

275. STAKING OUT CURB* LINES AND GRADES. — If the 
line stakes which are set for the curbstones are placed directly 
on the line of the curb they will be disturbed when the trench 
is excavated. For this reason they are usually set in the side- 
walk on an offset line, say, 3 ft. from the outside edge of the 

* Called edgestones in some localities. 



250 CITY SURVEYING [Chap. IX. 

curb, arid at intervals of about 25 ft. The grade stakes are set at 
about the sarae interval, with their tops at grade or at some even 
distance (6 inches or i foot) above or below the grade of the 
curb. Sometimes the grade stakes are not driven so that their 
tops bear any relation to the finished grade, but a horizontal 
chalkmark is made on the side of the stake marking the proper 
grade. A stake can be marked much more quickly than the 
top can be driven to the exact grade. 

When new curbstones are being set in an old street, stakes 
cannot as a rule be used. The sidewalks are too hard to permit 
the driving of stakes, and even if they could be driven those 
projecting above the surface of the sidewalk would be a source 
of danger to pedestrians. In such cases it is customary to use 
heavy spikes about 6" long. These are driven into the side- 
walk on the offset line and the elevation of their tops determined 
by leveling. The difference between the elevation of each spike 
and the grade of the curb opposite it is calculated. A list of the 
stations and the distances the spikes are above or below the curb 
is given to the foreman in charge of the work. These distances 
should always be transposed into feet and inches (to the nearest 
J") before being given to the foreman, as it is seldom that the 
men employed to lay the curbstones have any conception of the 
meaning of tenths and hundredths of a foot. (See Art. 7, 

p. 5.) 

Where there are trees growing in line with the curbs, a nail 
can sometimes be set in the side of a tree on the line of the curb 
as well as at its grade. Points like these, of course, should be set 
in preference to offset stakes or spikes wherever possible, as 
there is little liability of the workmen misinterpreting such 
marks. They can fasten their string directly to the nail and set 
the curb to agree with it. 

Before the curbstones are ordered the surveyor usually meas- 
ures the distances between trees and locates driveways, and 
then makes out a list of the lengths of straight, of curved, and of 
chamfered stones (opposite driveways) to be used on the job. 
This list is used in ordering the stones, and when they are de- 
livered they should be found to fit the conditions without the 
necessity of cutting any of them. 



STAKING OUT CURBS AND SEWERS 25 1 

276. STAKING OUT SEWERS. — The lines and grades of 
sewers are sometimes run out in the same way as those described 
for curbstones. The stakes or spikes (in bard paving) are set on 
an offset line and the grades figured as described in Art. 275. 

Another method which is extensively used is to spike out the 
center line of the sewer and, from the profile of the street, 
determine the depth of digging. When the excavation is com- 
pleted the surveyor again runs out the center line and places batter- 
boards at the proper grade and line. This eliminates the errors 
which are likely to creep in during the leveling over from the 
ofifset spikes as is done in the previous method. 

277. STAKING OUT STREET RAILWAY TRACKS. — The 
lines and gfrades for street railway tracks are given usually by the 
use of an offset line of spikes. The spikes are frequently placed 
on an offset line 5 ft. from the center, or on a line 3 ft. from the 
gauge of the nearer rail, and at every 50-ft. station or oftener. 
The diflFerences between the desired elevation of the track and 
the spikes is calculated, and this information is given to the fore- 
man in charge, usually in the form of printed "grade sheets." 

278. RERUNNING STREET LINES AND GRADES. — There is 
a constant call for lines and grades of streets. All kinds of 
work, such as the construction of fences, buildings, and street 
improvements, call for rerunning the street lines and grades. 

The work of running out the line is simple enough if the 
original S. Bs. are in place. It is not uncommon, however, to 
find that in excavating a cellar on a corner lot the corner bound 
has been disturbed or that it has been removed entirely ; and 
before the line can be properly staked out it may be necessary 
to begin at some reliable S. B. farther down the street or even 
on some other nearby street line. 

When the line has finally been rerun it is customary to take 
and record swing offsets from the corners of the underpining of 
several of the buildings located along the street and near to the 
line. By this record of offsets, then, this street line can very 
easily and quickly be run out at any future time, and any dis- 
turbance of the S. Bs, at the comers can readily be detected. 
Several offsets to substantial buildings are often of more perma- 
nent valne than stone bounds. In some offices these offsets to 



252 CITY SURVEYING [Chap. DC 

buildings are recorded directly on the street plans. Whenever 
a street line or grade is rerun full note should be made showing 
all measurements taken for determining the lines or grades. 

Sometimes the original street lines have been so completely 
obliterated that it is necessary to resurvey them and make a new 
record plan and description of them and have these new lines 
"established" by a city ordinance. Such work, for example, 
has been done by the City of Providence since 1857 when a state 
law was passed requiring that accurate street lines be marked 
where the adjacent land was about to be built upon. To 
properly carry out this law the resurvey of a number of the 
principal streets was required and the policy then originated has 
been continued. 

When a new building is to be constructed the owner generally 
requests the City Engineer to define the street grade in front of 
his property. The surveyor who has charge of this work goes 
to the place and levels from the nearest B. M. to the site of the 
new building. He has in his possession the established grade 
of the street and its cross-section. From these he can compute 
the elevation of the sidewalk grade at those points along the 
street line where the grades are desired. On the fence or on 
stakes set on the side line of the street he marks the grade of 
the sidewalk at the property line, usually to a hundredth of a foot. 

279. REVISHfG STREET LUfES. — In older cities much is 
being done toward straightening some of the crooked streets, 
and widening the narrow streets. A survey of existing struc- 
tures is made and plotted, and the new street lines are then 
studied with reference to existing conditions. Several proposed 
lines are sometimes considered and run out on the ground. The 
line finally selected is carefully run out and offsets to existing 
structures determined so that it may be definitely located, and 
the areas of all property taken from each abutter are then 
surveyed, computed, and described. This layout is then accepted 
by city ordinance and the necessary construction is made in 
accordance with the revision. 

280. REYISIHG STREET GRADES. — Sometimes the estab. 
lished grades of city streets have been laid down in the early 
days of the city, and it is subsequently found that these grades 



BATTER-BOARDS FOR BUILDINGS 253 

need revision. In such a case the surveyor will make a profile 
of the center line of the street, of each curb (if there are any) 
and sometimes along the side lines of the street. He will also 
take all necessary elevations on the steps of buildings which lie 
near the street lines, and a few levels in the front yards of abut- 
ting property. From a study of these grades together with a 
plan of the street the new grade line is laid out so as to affect 
existing property as little as possible. When this grade line has 
been accepted it is run out in the usual manner and the street 
regraded. Stakes for final grading are set to hundredths of a foot. 

281. SETTING Batter-Boards for a BuiLDmG. — One 

of the most common tasks of the surveyor is to set the batter- 
boards for the excavation and construction of the cellar of a new 
building. The dimensions of the building and the elevation at 
which to set it are usually obtained from the architect, although 
sometimes the elevation of the ground floor of the building is 
recorded on the plan itself. In a brick or stone building the 
lines to be defined are the outside neat lines of the building, and 
the elevation desired is usually the top of the first floor. In the 
case of a wooden building the line usually given is the outside 
line of the brick or stone imderpinning and the elevation given 
is the top of this underpinning on which the sill of the house is 
to rest. Sometimes the outside line of the sill is desired instead 
of the outside line of the underpinning. There should be a 
definite understanding in regard to these points before the work 
of staking out is begun. 

Generally there is no elevation marked on the plan and the 
surveyor is simply told to set the top of underpining a certain 
distance above the sidewalk or above the surface of some portion 
of the lot. If there is an elevation referred to City Datum 
marked on the plan, he should level from the nearest B. M. and 
set the batter-boards at the grade given. 

The location of the building on the lot is given either by 
plan or by orders from the architect or owner. Not infre- 
quently the surveyor receives the directions to place the build- 
ing so tliat its front line is on line with the other buildings on 
the street and so that it will stand a certain number of feet from 
one of the side lines of the lot. 



254 



CITY SURVEYING 



[Chap. DC 



His first work is to stake out the location of the building by 
accurately setting temporary stakes at all of the corners of the 
building, e.g., in Fig. 114, at A^ B, C, A ^> and F, A stake 

should be set at G also 
so that the entire work 
can be checked by meas- 
uring the diagonals AG 
and FB, and GD Ind EC. 
These checks should 
always be applied where 
possible. Then the posts 
for the batter-boards are 
driven into the ground 3 
or 4 ft. outside the line of 
the cellar so that they 
will not be disturbed 
when the-walls are being 
constructed. On these 
posts, which are usually 
of 2" X 4" scantling, i" 
boards are nailed. These 
boards are set by the sur- 
veyor so that their top 
edges are level with the 
grade of the top of the 
underpinning or for whatever other part of the building he is 
giving grades. After the batter-boards are all in place they 
should be checked roughly by sighting across them ; they should 
all appear at the same level. Sometimes, however, on account 
of the slope of the ground some of them have to be set a definite 
number of feet above or below grade. 

Then the lines are to be marked by nails driven in the top 
of these batter-boards. The transit is set up on one of the 
comer stakes of the house at A (Fig. 1 14), for example, and a 
sight is taken on F. This line is then marked on the batter- 
board beyond (at/) and on the one near the transit (at a). If 
the batter-board is so near the transit that the telescope cannot 
be focused on it, then point a can be set within a hundredth 




Setting Batter-Boards for 
A Building. 



BATTER-BOARDS FOR BUILDINGS 255 

of a foot by eye if the surveyor will stand outside of the 
batter-board and sight point ^i in a line determined by point 
/and the plumb-line on the instrument. Then a sight is taken 
along AB and this line is produced both ways and nails set on 
the batter-boards at h and/. In a similar manner all of the lines 
are marked on the batters. These points should be marked 
with nails driven in the top edges of the batter-boards and 
there should be some lettering on the boards to make clear 
which lines have been given. It is well for the surveyor also 
to show these marks to the builder or inspector and have it 
clearly understood just what parts of the structure these lines 
and grades govern. 

It is customary to set batters for the jogs in the building as 
well as for the main corners ; but small bay windows of dwell- 
ings are not usually staked out, but are constructed from 
wooden patterns made and set by the builder. 

As soon as the excavation is begun the corner stakes are 
dug out and the building lines are then obtained by stretching 
Unes between the nails in the opposite batter-boards. These 
batter-boards are preserved until the sills or first floor are in 
place, when they may be removed. 

382. dry Plans AIID records. — Every city has a large 
number of valuable plans and records in its possession. Too 
frequently these are not kept with anything like the care con- 
sistent with the amount of money that has been expended to 
obtain them. For suggestions regarding the filing and index- 
ing of pkns and records see Arts. 483-7, pp. 431-3. 

RECTANGULAR COORDHf ATE SYSTEM OF SURVEYING CITIES. 

283. GENERAL DESCRIPTION. — It is customary to disregard 
the effect of curvature of the earth in the survey of a city on 
account of its limited extent, and to use a system of rectangular 
coordinates based upon plane surveying. In a coordinate system 
two arbitrary lines are chosen for coordinate axes, one usually 
coinciding with some meridian and the other at right angles to it. 
All points in the city are located by distances from these two 
axes, these distances being known as X's and Y's, or sometimes 



B56 CITY SURVEYING [Chap. K. 

as latitudes and longitudes. The axes are sometimes chosen 
entirely outside the area to be surveyed, and where they meet 
(their origin) is designated as (o, o.). Sometimes they are taken 
through some conspicuous point, such as the tower of the city 
hall, and are considered as being certain distances from the zero 
lines as (lo ooo, lo ooo). By either of these arrangements 
negative values for coordinates are avoided. The coordinates are 
usually considered positive toward the north and the east, in 
accordance with the custom of analytic geometry, as is the case 
in ordinary land surveying. The convergence of the meridians 
is neglected and all points having the same X coordinate there- 
fore lie on a straight line parallel to the initial meridian and 
are not all on the same true meridian line. 

In the survey of the city of Baltimore (Fig. 115) the origin 
of coordinates was taken through the Washington Monument 
in the central part of the city, and the map divided into squares 
1000 feet on a side. Each square mile is shown on a separate 
page of the atlas of the city and these squares are designated 
by their number north or south, and east or west of the origin, 
as 1S2W, 3N4E, etc. Any point is designated by the distance 
in feet north or south, and east or west, as (icxx) E, 2000 N). 

One of the chief advantages of any coordinate system is that 
if any point is lost it can be exactly replaced by means of the 
known coordinates. This would be especially true in case a 
large section of the city were destroyed by fire. 

284. TriaNGULATION scheme. — The principal points of 
the survey are usually located by a system of triangulation. 
Prominent points are selected in such positions that the lines 
joining them form well shaped triangles, i.e., preferably triangles 
which are not far from jequilateral. These points may be signals 
on tops of hills, church spires, and the like. If the cupola of the 
city hall, or some such point is chosen as the origin of coordi- 
nates it should also be one of the triangulation points. Points 
which can be occupied by an instrument are in general to be 
preferred. Such points as steeples or flag poles are definite 
enough, but where no definite object exists on which to sight 
the instrument signals are erected for this purpose. Such a 
signal usually consists of a pole placed carefully over the exact 




FlO. 116. TrI ANGULATION SCHEHB f 
(Printed by pennission of Major Joseph W. Shirley, Chief E&g 




Survey op the city of Baltimore. 

tr of the Topographical Survey Commission, Baltimore, Md.) 



MEASUREMENT OF BASE-LINE 26 1 

point and braced in a vertical position by other poles forming 
a tripod. 

The system of triangles should cover the entire area but 
should not contain more lines than are necessary to establish 
a sufficient number of points to control the subsequent work 
of the survey. 

285. MEASUREMENT OF BASE-LINE. — At least one line in 
the system must be chosen where its length can be very ac- 
curately measured; this is called the base-line. The lengths of 
all the other lines are to be computed from this line by means 
of the measured angles, hence it will be seen how important 
it is that this line should be measured with great accuracy, 
and that it should also form a well shaped triangle with the 
connecting triangulation stations. 

It should be chosen if possible in some level spot where there 
are no serious obstacles to the measurement. It is sometimes 
an advantage to have the ends of the base-line slightly elevated 
above the general level. The base should be measured with a 
steel tape the exact length of which is known. The tension 
should be kept constant by means of a spring balance, and the 
temperature carefully taken. If the work is done on a cloudy or 
rainy day the thermometer readings will represent the tempera- 
ture of the tape much more nearly than when taken in sunshine. 
The points should be lined in with a transit and the tape held 
horizontal, or, if the measurements are taken directly from stake 
to stake, the slope should be determined, by means of a leveling 
instrument. There should be at least two independent measure- 
ments of the line. 

286. MEASUREMENT OF ANGLES. — If possible all of the 
angles of each triangle should be measured by repetition. An 
"inverting" instrument reading to 20" or to 10" is to be pre- 
ferred for this work. The angles are repeated at least six times 
with the telescope direct and the same number of times with 
the telescope inverted. Several of these sets of readings are 
made beginning each time with a different initial setting on the 
circle. For example, if the first setting was at 0° and four sets are 
to be taken the second would begin with a setting of 90®, and 
so on. In each case both verniers should be read and the mean 



262 CITY SURVEYING [Chap. DC. 

of the two taken. Sometimes the direction of the measurement 
is changed during the set, the first six repetitions being taken 
from left to right, and the second six from right to left. In this 
work it is important that the instrument should be carefully 
centered over the point, and that the signals are also carefully 
centered It is also important to keep the instrument carefully 
leveled, especially if there is great difference in the angular ele- 
vation of the points sighted. 

287. Adjustment of the Angles. — The test of the accuracy 
of the angle measurements is in the " closure " of the triangles. 
In good work the sum of the angles of a triangle should not 
differ from 180® by more than about 5 seconds, under fair con- 
ditions. After the angles have been measured the errors in the 
closure of the triangles should be distributed equally among the 
angles, thus making the sum of the angles in each triangle ex- 
actly equal to i8o<*. If the best results are desired all of the 
discrepancies due to errors of measurement can be removed by 
adjusting the system in accordance with the " Method of Least 
Squares." In ordinary work, however, where the errors have 
been kept small, the expense of such a computation is not war- 
ranted. After all of the angles have been corrected the sides 
of the triangles may be computed. 

288. Azimuth. — If the coordinate lines are to run N and 
S and E and W it is necessary to know the astronomical azimuth 
of at least one line of the triangulation system before the 
coordinates can be computed. This may be determined by 
observation on Polaris as described in Chapter VII, or, in case 
there are other triangulation points already established in the 
vicirifity, the new system can be connected with them and the 
azimuths computed from one of these lines. Azimuths are 
reckoned in this work from lines parallel to the initial meri- 
dian, from the south point right-handed, i.e., in the direction 
S-W-N-E, and from 0° to 360®. When the azimuth of one 
line is known all of the others may be computed. With the 
azimuth and length of each line known the difference of the 
latitudes and departures, i.e., the difference of the Xs and Fs of 
the ends can be found, and with the coordinates of some one 



SECONDARY TRIANGXJLATION 263 

point gi>^en, or assumed, the coordinates of all of the other 
points can be computed as explained in Art. 410, p. 373. 

289. SECONDARY AND TERTIARY TRIANGULATION.— After 
the prmcipal triangles have been completed, forming a system of 
control, smaller triangles are selected, locating a system of points 
of lesser importance so far as the survey is concerned. This is 
called the secondary system. Sometimes a third {or tertiary) 
system is introduced, the triangles being still smaller. The ter- 
tiary triangles are the ones that would be used for locating the 
city boundaries, street comers, and important monuments. 

It frequently happens that, owing to the large number of 
angle measurements and the consequent accumulated error, the 
lengths of the sides of the small triangles become much less 
accurate than they would be if measured directly; and since 
many of these lines naturally lie in places where the distance 
can easily be measured, this measurement should be made 
as a check, in which case this line becomes a secondary 
base-line. It is a good plan to introduce these measurements 
frequently, where it can be conveniently done without great 
expense, in order to prevent the errors of the survey from 
accumulating unnecessarily. 

290. Traverses. — After all of the triangulation is com- 
pleted the system is extended by running traverses with the 
transit and tape, from one known point to another. The tri- 
angulation points are regarded as fixed and the errors of closure 
of the traverses are assumed to be entirely in the traverse sur- 
veys, the traverses being made to fit in exactly between the 
triangulation points. 

All street lines, or parallel offset lines, are connected with 
the coordinate system so that the azimuth of every street line in 
the city may be known, and the coordinates of all important 
points, such as street corners and lot comers, are computed. 

291. METHOD OF LOCATraO PROPERTY LINES AND BUILD- 
INGS. — Since the coordinates of the property cqmers are to be 
computed it is advisable to locate them by angle and distance 
from the transit points, for with these data the calculation of 
the coordinates is simple. The buildings are located from the 
transit line by methods explained in Chapter VI. 



CHAPTER X. 

TOPOaRAPHICAL S UKVl S YJma . 

292. In making a survey for a topographical map the methods 
used will depend upon the purpose for which the map is made 
and the degree of accuracy which is required. But whatever the 
purpose of the map may be it is not necessary to locate points 
in the field more accurately than they can be represented on 
paper, whereas in surveying for an area measurements are made 
with far greater precision than would be necessary for the pur- 
pose of plotting. 

While most of the details of topographical surveying can be 
filled in more economically by the use of the transit and stadia 
or by the plane table it is thought best to describe here only 
the more elementary methods, and to reserve the complete 
treatment of the stadia and plane table for an advanced work. 

293. TRIANGULATION for Control. — In all cases where the 
area is large it will be advisable to use a system of triangulation 
to control the survey, as this is the cheapest method of accurately 
determining the relative position of a few points whfch are a con- 
siderable distance apart. The details of this triangulation work 
have already been described under the head of " RectanguIarCo- 
ordinate System of Surveying Cities," Chapter IX. One line 
of the survey, the base-line, must be carefully measured. The 
precision with which the angles of all the triangles must be 
measured depends upon the use to be made of the map. After 
the principal triangulation points have been established then 
positions are plotted on the map. This may be done conven- 
iently by the method of rectangular coordinates described in 
Art. 283, p. 255. The extension of the system to smaller systems 
of triangles, called secondary and tertiary, may be made if neces- 
sary. After the triangulation system has been extended far 

264 



TRIANGULATION 265 

enough to furnish a sufficient number of points for controlling 
the accuracy of the map, traverses, may be run wherever con- 
venient or necessary for locating topographic details. In all 
cases the traverses should be connected with the triangulation 
points at frequent intervals in order that the relative positions 
of all points may be kept as nearly correct as possible. Where 
a high degree of accuracy is necessary these traverses should 
be run with a transit and tape ; if, however, errors of a foot or 
two would not be appreciable on the map it will be sufficiently 
accurate to use the stadia method of measuring the distances 
and thus save time. 

294. LOCATION OF POnVTS FROM THE TRANSIT LINE.— 

Where a tape is used for measuring the distances, such objects 
as fences, walls, and buildings may be located as described in 
Chapter VI, but it will not be necessary to make the measure- 
ments with as great precision. Fig. 116 is a sample page of 
notes of a topographical survey where the transit and tape were 
used. On city plans, which are frequently drawn to a scale of 
40 feet to an inch, a fraction of a foot can easily be shown. On 
a topographic map the scale is often such that an error of a frac- 
tion of a foot becomes insignificant in the side measurements 
from the transit line, where such errors cannot accumulate. In 
some cases it may be sufficient to obtain the distances by pacing, 
and the angles or directions by means of a pocket compass. Lo-. 
cations may frequently be checked by noting where range lines 
intersect the transit line. In making a series of measurements 
it is well to take each measurement with a little more precision 
than is actually needed for plotting, in order to be sure that the 
accumulated errors do not become too large. 

In taking measurements the surveyor should constantly keep 
in mind how the notes can be plotted ; this will often prevent 
the omission of necessary measurements. No matter whether 
an accurate or only a rough survey is desired check measure- 
ments should be taken on all important lines. 

29s. CONTOUR LINES. — There are two general systems of 
representing on paper the form of the surface of the ground. 



266 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 



]Jr- 




Fig. lie. Field Notes of a Portion of Topographical Survey 
WITH Transit and Tape. , 



HACHURES AND CONTOURS 



267 



In one of these systems (Fig. 117) slopes are represented by 
hachure lines, i.e., lines which always run in the direction of the 
steepest slope of the ground. In the other system (Fig. 118) 
contour lines, lines joining points of equal elevation, are used. 
In the latter system elevations may be read directly from the 
map, and for this reason it is much more used by surveyors. 




Fig. 117. Hachure Lines. 



Fig. 118. Contour Lines. 



A contour line is the intersection of a level surface with the 
surface of the ground. A clearer conception of a contour line 
may be obtained from the following. Imagine a valley, or de- 
pression in the surface of the ground, partly filled with water. 
The shore line of this body of water will then be a contour line, 
since it is the intersection of a level surface with the surface of 
the ground. If the water stands at an elevation of 50 feet the 
shore line is the 50-ft. contour. If the surface of the water 
were raised 5 feet the new shore line would then be the S5-ft. 
contour. Contour lines if extended far enough will therefore 
be closed curves, and all of the points on any one contour will 
have the same elevation above the datum. It is customary to 
take contours a whole number of feet above the datum, spacing 
them in regard to height, so as to make the contour intervals 
equal, e.g., a contour may be taken at every 5 feet or every 10 
feet of elevation. Since the contours are equidistant in a verti- 
cal direction their distance apart in a horizontal direction shows 
the steepness of the slope. 



268 TOPOGRAPHICAL SURVEYING [Chap. X. 

Fig. 1 19 illustrates contour maps of simple solids. 

CLEVAHON 




Fig. 119. 



cornouR map 
Contour Maps of Simple Solids. 



296. Characteristics of (Contours. — The chief characteristics 
of contours are illustrated in Fig. 121, and may be summed up 
as follows. 

1. All points on anyone contour have the same elevation, as 
at A. 

2. Every contour closes on itself, either within or beyond 
the limits of the map. In the latter case the contour line will 
not end within the limits of the map but will run to the edge of 
the map, as at B- 

3. A contour which closes within the limits of the map in- 
dicates either a summit or a depression. In depressions there 
will usually be found a pond or a lake; but where there is.no 
water the contours are usually marked in some way to indicate 
a depression, as at C. 

4. Contours can never cross each other except where there 
is an overhanging cliff, iij which case there must be two inter- 
sections, as at D. Such cases as this seldom occur. 



CHARACTERISTICS OF CONTOURS 269 

5. On a uniform slope contours are spaced equally, as at E. 

6. On a plane surface they are straight and parallel to each 
other, as at F. 

7. In crossing a valley the contours run up the valley on one 
side and, turning at the stream, run back on the other side, as 
at G. Since the contours are always at right angles to the lines 
of steepest slope they are at right angles to the thread of the 
stream at the point of crossing. 

8. Contours cross the ridge lines (watersheds) at right 
angles, as at H. 




Fio. 121. Illustrating Characteristics op Contours. 



270 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 




Fig. 120, Contour Crossing a 
Street. 



Fig. 1 20 shows a contour across an ordinary city street with 

sidewalks and curbstones, 
the street being located on 
a steep grade. In order to 
trace out the position of a 
contour it is necessary to 
keep in mind that it is a 
line all points on which are 
at the same elevation. It 
will be noticed that the con- 
tour from A to B crosses 
the sidewalk in a straight 
line but not perpendicular 
to the street line because the 
sidewalk is sloped toward 
the gutter. Turning at B 
it runs straight along the face of the curbstone until it strikes 
the gutter at Z?, and returns on the other side of the gutter 
along the surface of the road, the point E being where it swings 
around and travels back toward the other gutter. The other 
half of the street is similar. If the center of the road is at 
the same elevation as the top of the curb opposite, then E will 
be opposite B. This illustrates how contours run around val- 
leys (gutters) and ridges (crown of street). 

If the side of the street to the right {HF) were at a lower 
elevation than the left side then the contour at the point where 
it crosses the gutter, /s would be farther up the road irom E^ 
le., the contour would be unsymmetrical, EF being longer 
than DE, 

297. RELATION BETWEEN CONTOUR MAP AND PROFILE. — 
If a line is drawn across a contour map the profile of the surface 
along that line may be constructed, since the points where the 
contours are cut by the line are points of known elevation and 
the horizontal distances between these points can be scaled or 
projected from the map. The profile shown in Fig. 122 is con- 
structed by first drawing, as a basis for the profile, equidistant 
lines, corresponding to the contour interval, and parallel to AB. 
From the points where AB cuts the contours lines are projected 



PROFILE FROM A MAP 



271 



to the corresponding line on the profile. Conversely, if the pro- 
files of a sufficient number of lines on the map are given it is pos- 
sible to plot these lines on the map, mark the elevations, and 
from these points to sketch the contours as described in Art. 
301, p. 276. 




PROFILE ON.LINLAB 
Fig. 122. Profh^e Constructed from a Contour Map. 



298. RELATION BETWEEN CONTOUR MAP AND SIDE ELE- 
VATION OR PROJECTION. — A photograph of a landscape 
represents approximately a side elevation of the country. To 



272 TOPOGRAPHICAL SURVEYING [Chap. X. 

construct such a projection from a contour map (Fig. 123), lines 




projection on, plane ad 
Fig. 123. Side Elevation Constructed prom a Contour Map. 

are drawn perpendicular to AB, the plane of projection, and tan- 
gent to the contours. These tangent points show the limits 
between the visible and invisible portions of the landscape, the 
observer being assumed to stand on the line AB and to look in 
a direction perpendicular to ^45. 



DRAINAGE AREAS 273 

299. DRAINAGE AREAS. — The drainage area that supplies a 
stream or pond is limited by the divide line which is a line 
drawn on the ridges surrounding a depression as indicated by 
the dotted line on Fig. 124. Since the perpendicular to the 
contout at any point is the direction of steepest slope the direc- 
tion in which water will flow at any point can be determined at 
once by examining the contours. On the ridge there is a line 
(its summit) on one side of which water will flow down one of 
the slopes and on the other side of .which it will flow down the 
other slope. This line is the divide line or watershed line. 

If a dam were built as shown in Fig. 124, its elevation being 
960 ft., the area actually flooded by the water at full height of 
dam is the area included within the 960 ft. contour, which is in- 
dicated by the shaded section. The drainage area for the por- 
tion of the stream above the dam is the area included within the 
heavy dotted line, which follows the line of the divide. 

300. SKETCHING CONTOURS FROM STREAMS AND SUM- 
MITS. — The present topography of some parts of the country 
is due almost entirely to erosion by streams. Consequently the 
position and fall of the streams give more information regarding 
the position of the contours than any other topographic features. 
If a definite position of the contours is desired it will be neces- 
sary to obtain the elevation of a few governing points on the 
ridges as well as the location and elevation of the streams, as 
shown in Fig. 126. ' 

In sketching in contours from these data it should be borne 
in mind that the contours cross the stream at right angles to its 
thread and that they curve around from the hill on either side so 
as to represent the valley of the stream. The contours are 
farther apart at the top and bottom of the slope of an eroded 
hill than near the middle, because in these portions the slope 
is somewhat flatter. A stream is usually steeper near its source 
than in the lower portion and therefore the contours are closer 
together near the source. This is true of most cases but the 
shape of the contours in any particular case will depend upon the 
geological formation. Fig. 127 represents the same country as 
Fig. 126 but with the contours sketched on it, following out the 
general suggestions which have just been mentioned. 



274 



TOPOGRAPHICAL SURVEYING [Chap. X. 




Fig. 124. Illustrating Flooded Area and Drainage Area. 



CONTOUR SKETCHING 



275 




Fig. 126. Map Showing the Location and Elevation of Streams 
AND Summits. 




Fig. 127. Contours Sketched from the Data given in thk 

m.ap above. 



2y6 



TOPOGRAPHICAL SURVEYING. 



3ai. SKETCHING CONTOURS FROM KNOWN ELEVATIONS.^ 

— A portion of the country can be cross-sectioned as described 
in Art. 227, p. 206, or profiles can be run on any desired lines 
as explained in Art. 225, p. 203. From these known elevations 
contours can be sketched by interpolation. This is usually done 
by estimation and the principle involved is the same whether the 
elevations were obtained by cross-sectioning or by profiles. 

Fig. 125 illustrates how contours can be sketched from cross- 




TRAVERSE UNC 

Fig. 126. Contour Sketched for Cross-Section Notes. 



section notes. The points at which elevations are taken in the 
field should be so chosen that the slope of the ground is practi- 
cally uniform between any two adjacent points. Then by simple 
interpolation the contours may be accurately sketched. This 
interpolation may be done by geometric construction, but for 
most topographic work it is accurate enough to interpolate by 
eye. 

302. MISTAKES IN SKETCHING CONTOURS. — Fig. 1 28 
shows several examples of impossible and incorrectly sketched 
contours ; the streams are assumed to be correctly located. The 
numbers on the figure refer to the tabulation made in Art. 296, 
p. 268, and will assist in detecting the type of error present. 



SKETCHING CONTOURS FROM CROSS-SECTIONS 277 




Fig. 128. Contours INCORRECTLY Sketched. 

303. Locating CoirroURS. — Contours are often most eco- 
nomically located by means of the transit and stadia or by an 
instrument called the plane table* In this chapter, however, 
only those methods will be considered which call for the use of 
the transit and tape. 



* A discussion of the Stadia and the Plane Table does not come within the 
province of this book. 



278 TOPOGRAPHICAL SURVEYING. [Chap. X. 

304. Locating Contours by Cross-Sections. — A very com- 
mon as well as expensive method of locating contours is that of 
taking cross-sections. Elevations on the surface of the ground 
are usually taken to tenths of a foot. From these elevations 
the contours may be sketched by interpolating between these 
known elevations as explained in Art. 301. The accuracy may 
be increased by taking a larger number of intermediate points. 
The size of the squares used should depend upon the roughness 
of the surface. 

305. Locating Contours by Profiles. — In some cases where 
the ground is fairly smooth it is sufficient to take a few profiles 
on known lines, not necessarily at right angles to each other. 
These lines are stationed and elevations are taken at every full 
station and at the points of marked change in slope. From 
these data the contours are sketched on the map by interpola- 
tion as described in Art. 301. 

306. Locating Points on the Contours. — Where the contour 
interval is small, say one or two feet, and the topography is to 
be determined with considerable accuracy, it is advisable to find, 
in the field, points actually on the contours and thus avoid the 
errors of interpolation. The rodman moves up or down the 
slope until the rod-reading indicates that the foot of the rod is 
on a contour. The position of the rod may then be located by 
an angle and a distance from some known line, the distance 
being taken with a tape. 

307. Locating Contours by the Hand LeveL— A more rapid 
but less accurate way of putting in contours is by means of the 
hand level. The work is done by making profiles of lines whose 
positions on the map are known. A point on some contour is 
found in the following manner. 

The first step to take is to measure to the nearest tenth of 
a foot the distance from the ground to the eye of the leveler, 
which may be, say, 5.4 ft. If the B. M. is at elevation 143.43 
and it is desired to locate a point on the 140-ft. contour, the 
rodman holds the rod (or a tape) on the B. M. while the leveler 
attempts to place himself on the 140ft. contour. When he is 
on the 140-ft. contour the elevation of his eye (H.I.) is 145.4 



LOCATING CONTOURS BY THE HAND LEVEL 279 

and the rod-reading at the B. M. must be 145.4 — 143*43 "= 
1.97, or 2.0 to the nearest tenth of a foot. The leveler there- 
fore travels along the line on which the point is to be located 
until he reads 1.97 on the rod. His feet are then on the 140- 
ft. contour, the position of which is located from some known 
point on the line. Sometimes this is done by measurement and 
sometimes by pacing. A point on the 145-ft. contour could 
have been located first by applying the same principle, but if 
the 140-ft. contour is established it is very easy to locate a 
point on the i4S-ft. contour as follows. The distance from 
the leveler's feet to his eye being 5.4 ft., if he stands on the 
140-ft. contour and reads 0.4 ft. on the rod, the bottom of the 
rod must be on the 145-ft. contour. By trial then the point is 
found where the rod reads 0.4 ft.* Then the leveler walks up 
the hill and, standing on the point just found, places the rodman 
on the next higher contour by the same process. 

In working down the hill to locate the 135-ft. contour, if the 
leveler is standing on the 140-ft. contour, the rod will be on the 
135-ft. contour when it reads 10.4 ft. Or, when the 140-ft. 
contour has been found by the leveler the rodman comes forward 
and holds the rod on this spot and the leveler backs down the 
hill until he reads 0.4 ft. on the rod ; he is then standing on the 
135-ft. contour. Some surveyors prefer to cut a stick just 5 ft. 
long and hold the hand level on the top of it in taking sights. 

The points thus found at regular contour elevations are then 
plotted on the corresponding lines and the contours sketched by 
joining points of equal elevation. Where the lines which are 
profiled are far apart or where the country is very rough it is 
frequently necessary to obtain the correct position of the con- 
tours, to locate extra points on them between these profiled lines. 
The extra points are located by right-angle offsets from the 
lines. Most oi this work is plotted in the field upon paper ruled 
in small squares to facilitate sketching. Where practicable it is 
always well to sketch the contours in the field rather than in the 
office. 

* For very rough work sometimes the rod is not used, the leveler simply es- 
timating where the rod-reading will come on the rodman*s body and placing him 
so that his feet will be on the proper contour. 



280 TOPOGRAPHICAL SURVEYING [Chap. X. 

308. LOCATION OF STREAMS AND SHORE LINES. — Streams 
or shore lines of ponds may be very rapidly located by stadia 
measurements. If the shore lines are to be located by tape 
measurements, however, a convenient way is to run a transit 
line aproximately parallel to the general direction of the shore 
line, and to take perpendicular offsets at regular intervals and 
at all points where there is a marked change in the direction of 
the shore line, as was done in the notes in Fig. 53, p. 104. 

309. Contour problems. — There are many surveying 
problems involving earthwork which can be worked out approxi- 
mately by use of a contour map. As a rule the smaller the 
contour interval, the more accurate will be the result of such 
work. Contour studies occur in a variety of problems, so 
numerous that it would be useless to attempt to cover the sub- 
ject fully. Three typical problems, however, are illustrated and 
explained ; and these contain the essential principles applicable 
to practically all contour studies. 

310. Example i. — (Fig. 129). Given a contour map, the 
surface being represented by contours shown by full lines, a 
plane (extended indefinitely) is passed through the straight lines 
AB and CD, which are level and parallel, AB being at elevation 
12.5 and CD being at elevation 40. It is required to find where 
this plane intersects the surface, and to shade the portion which 
is above the plane. 

Since the proposed surface is a plane, contours on it will be 
parallel to AB and CD, The elevations of AB and CD being 
known, other contours, such as ef and gh, can be interpolated 
between AB and CD. Their interval is made 5 ft. the same as 
the contour interval for the original surface. Evidently the 
point where any of these parallel lines crosses an original con- 
tour of the same elevation, as 7, k, /, my or «, is a point on the 
intersection of the plane with the surface. Joining these points 
gives the line of intersection of the plane with the original sur- 
face, which is indicated by the heavy full line on the figure. 
Such points as y, j, or / are determined by interpolation. Inter- 
mediate contours are drawn at one-foot intervals between the 
original surface contours ; corresponding lines are interpolated 
between the straight contours which show the plane ; additional 



CONTOUR PROBLEMS 



281 



intersections obtained, and in this way the point / is determined. 
Again it will be seen that point /, with reference to the parallel 
straight contours, is at about 18.5; with reference to the original 




Fig. 129. 



contours, it will be seen that wt is about three-tenths of wr^ the 
distance between contours, and this makes the elevation of point 
/ equal to 18.5. 

311. Example 2. — (Fig. 130.) Given a contour map which 
ncludes a road, and on which the original contours are represented 
by full lines. It is desired that all of the road between A and B 
shall be visible from the ground at point C. Sketch on the map 
and shade the portions which will have to be cut down to fulfill 
this requirement. * 

The general method of solving this problem is to sketch a 
new set of contours on the map, which will represent a uniform 



282 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 



slope from C to the nearer edge of the road. Everything that 
is above the surface represented by these new contours must be 
cut away. 

First draw lines, such as Cay Cb^ and Cr, the points a, by and 
c being points on the upper side of the road between which it 
may be assumed that the slope is uniform (Art. 301, p. 276). 
Along these lines interpolate points which will lie on the uniform 
slope from C to the road and also on the regular 5 ft. intervals 
which correspond to the contours. For example along the line Ca 




Fig. 130. 



from the summit which is at elevation 89 to the road at ^ which 
is at elevation 55, there is a drop of 34 ft., or a little less than 
7 contour intervals. Points ^, /, gy //, etc. are therefore plotted so 
as to divide Ca into 9 equal parts. Similarly points /, 7, ky etc. 
are plotted along the line C6, but the point by being at elevation 
56, is plotted so that the distance ib is four-fifths of the other 
distances ijyjky etc. When these points have been plotted on all 
of the necessary diagonal lines, the contours representing a uni- 
form slope from C to the road are sketched on the map as shown 
by the dotted lines on the figure. The points, such as w, «, or 
Ty where the new contours cut the old contours of equal elevation, 
are points of " no cut and no fill.'* A line connecting these 



CONTOUR PROBLEMS 



283 



points encloses portions of either cut or fill. The shaded por- 
tions or the figure, where the new contours are nearer C than 
the corresponding old ones, represent the portions where it 
will be necessary to excavate to the surface represented by the 
dotted contours. In the central portion of the figure, from point 
c to /, the road can already be seen. 

312. Example 3. — (Fig. 131.) Given a contour map on 
which are shown the two side lines of a road, the contours being 
represented by full lines. The road is to be built on a 4% 
down grade starting at ^4 at elevation 55. Scale i inch =150 




Fig. 131. 



feet Side slopes of road to be i^ horizontal to i vertical. It 
is desired to sketch the new contours on the slopes of the road, 
to sketch on the map the top and foot of slopes, and to designate 
the portion in embankment and the portion in excavation. 

First, the new contours which are to cross the road are 
plotted at abf cd, ef, gh. These will be 125 ft. apart, as a 4 % 
grade falls 5 ft. in a distance of 125 ft. If the road is assumed 
to be level on top, then these lines will cross the road at right 
angles to its general direction as shown in the figure. From 
points a and by on either edge of the road, the new contour 



284 



TOPOGRAPHICAL SURVEYING 



[CttAP. x: 



lines will follow along the slope, e.g., the line ao represents the 
new 50 ft. contour. Where this contour ao passes point c it is 
just 5 ft. above the road. Since the slope of the cut is ij to i, 
then the distance cut from c must be U x 5 = 7.5 ft.; opposite 
e it is 10 ft. below the road and similarly the distance out from 
e must be 15 ft. Where this new 50 ft. contour meets the old 




Fig. 131. 



50 ft. contour at o, is a point at the top of the slope. Similarly 
all of the new contour lines, which are represented on the 
figure by dash lines are plotted and their intersections with the 
corresponding contours of the original surface give points of " no 
cut" or "no fill,'* or top of slope (in excavated portions) and 
foot of slope (in embankment portions). These lines are shown 
in the figure by heavy dotted lines. Where this heavy dotted 
line crosses the road it marks a " no cut " and " no fill " line, 
i.e., the road bed cuts the surface of the ground. 



CHAPTER XL 

MININa STTRVETINa'.* 

313. GENERAL REMARKS. — In this chapter the limitations 
and difl&culties met with in surveying a mine will be pointed out 
and some of the instruments and methods generally used will 
be described. As rocky and precipitous mountain regions are 
more the home of metal mining than of any other industry, the 
special diflSculties of surface' surveying in such localities will also 
be considered. Lastly, the methods of establishing the bound- 
aries of mining claims in United States territory will be briefly 
described. 

Two of the principal objects to be accomplished in accurate 
mine surveying are the locating of the ownership boundaries 
underground and the laying out of passageways so as to connect 
with one another, thereby facilitating the working of the mine. 
Such passages are usually highly inclined and while under con- 
struction are called connections. 

314. DEFINITIONS OF MINING TERMS. — The following 
terms are in common use in mining surveying. 

Adit. A horizontal underground passageway running from the 
surface and used only for drainage and ventilation. 

Apex. The trace of the intersection of the vein with the sur- 
face of the undisturbed rock formation. 

Compartment. One of the smaller passageways of a large shaft, 
divided by timber partitions. 

Connections. Passageways which are being driven from one ac- 
cessible part of a mine to another. 

Cross-cut. A horizontal passageway at right angles to or across 
the direction of the deposit. 

* This chapter was written by Blarney Stevens, M. Sc, Mining Engineer, 
Ellamar, Alaska. 

285 



286 MINING SURVEYING [Chap. XI. 

Dip. The inclination of the plane of the deposit to the 
horizon. 

Drift. A horizontal passageway along, or parallel to, the trend 
of the deposit. 

Heading. Any preliminary passageway driven to explore the 
mine or to facilitate future operations. 

Levels. Horizontal passageways run at regular intervals (verti- 
cally) along the deposit for working the mine. 

Manhole. A small passage from one level into the next level 
above or below, or into stopes. 

Mill-hole. A passage between a stope and a level through which 
the ore is conveyed. 

Outcrop. The portion of the vein where it intersects the surface 
of the ground. 

Pitch. The direction of an ore body (called a chimney or 
chute) in an ore bearing body, sometimes expressed as an 
azimuth. 

Raise. A passage leading upwards from any portion of the 
mine. 

Shaft. A vertical or steeply inclined passage used in working the 
mine. 

Stopes. Rooms excavated, within the walls of the deposit and 
above or below the leyels, for exploiting the mines. 

Strike. The direction (bearing) of a horizontal line in the plane 
of the deposit. The strike is always at right angles to the 
dip. 

SPuU. Timber running crosswise between the side walls of a 
passageway. 

Tunnel. A horizontal passageway from the surface to the mine. 

Wall. The boundary between a highly inclined vein and the 
rock each side of it. The upper wall is called the " hang- 
ing wall " and the lower one the " foot wall." 

Winze. A subsidiary shaft not starting from the surface. 



MINING TRANSITS 287 

MmuiO mSTRUMEinS. 

Owing to the confined nature and steep inclination of many 
of the passages through which survey lines have to be carried, 
specially constructed instruments are necessary. 

315. Mining transits. — in modem mining, all the 
accurate angle measurements are taken with a transit, the details 
being filled in with a miner^s dial or other light compass instru- 
ment. Several forms of transit are designed for mining and 
mountain work. The essentials are lightness and capability of 
measuring accurate azimuths of nearly vertical or of very short 
sights. 

With an ordinary transit one cannot take a downward sight 
more steeply inclined than 55® or 60° to the horizon. For 
taking highly inclined sights various devices have been used by 
which telescopic sights may be taken over the edge of the horizon- 
tal circle of the instrument. This is commonly done by attach- 
ing an auxiliary telescope, usually smaller than the main tele- 
scope, to the side or to the top of the ordinary engineer's transit 
so that the instrument will afford all the advantages of the ordi- 
nary transit and also allow vertical sights to be taken. 

316. SIDE TELESCOPE. — Fig. 132 shows a mining transit 
in which the auxiliary telescope is attached to an end extension 
of the horizontal axis. When this instrument is used the azi- 
muths which are measiu-ed by means of the side telescope have 
to be corrected for the eccentricity of this telescope. A striding 
level is used to adjust the horizontal axis. This is a sensitive 
spirit level having two V-shaped bearings so that it can be set on 
top of the horizontal axis; it can be lifted and turned end for end. 

317. Top Telescope. — in this type of mining transit the 
auxiliary telescope is mounted on top of the main telescope. 
Since this telescope is directly over the main telescope, azimuths 
measured with the auxiliary telescope will be the same as though 
they were measured by the use of the main telescope. But if 
vertical angles are measured by means of the top telescope it will 
be necessary to allow for the distance between the two telescopes. 

318. Adjustments of Side Telescope. — It is assumed that 
all ordinary adjustments of the transit have been made; in 



288 



MINING SURVEYING 



[Chap. XL 




Fig. 132. Mining Transit with Side Telescope. 

(From the catalogue of C. L. Berger & Sons, by permission.) 



ADJUSTMENT OF SIDE TELESCOPE 289 

mining work the adjustment of the objective slide (Art. 77, p. 60) 
is of unusual importance. The side telescope is generally ad- 
justed by first making the line of sight parallel to the axis of the 
telescope tube. This is done by the cross-hair adjustment and 
the aid of a pair of fixed wyes in which the tube is rotated; it is 
the same adjustment as for the level, Art. 121, p. 89. It is as- 
sumed that the instrument maker has made the optical axis 
parallel to the axis of the tube. 

Secondly, the line of sight is made parallel to that of the main 
telescope. It is first brought into a vertical plane parallel to the 
vertical plane of the main telescope by means of the adjusting 
screws on the vertical trivet plate of the side telescope, the sight of 
each telescope being taken to the same very distant object. If it 
is not convenient to sight on a distant object, a piece of paper 
with two vertical marks connected by a horizontal line may be 
used, the distance between the marks being equal to the distance 
between the telescopes. This piece of paper should be set at 
right angles to the line of sight and not too near the instrument. 
The vertical cross-hair of the main telescope is sighted at one 
point by means of the clamp and tangent screw of the plates and 
then the vertical cross-hair of the side telescope is sighted at the 
other point by means of the trivet plate adjustment on the side 
telescope. 

The side telescope and main telescope are then brought into 
the same plane at right angles to the vertical plane as follows: — 
the horizontal cross-hair of the main telescope is sighted at some 
point, preferably a distant one; then the horizontal cross- hair of 
the side telescope is sighted at the same point by means of the 
tangent screws on the side telescope. 

With this form of attachment a correction for eccentricity 
of the side telescope is necessary in both azimuth and altitude 
readings. The necessity for correction in azimuth may be elimi- 
nated by using the instrument in both the direct and reversed 
positions; reversing brings the side telescope to the other side of 
the main telescope, thereby eliminating the errors of the line of 
sight. The striding level should be used in both of its positions, 
i.e., with the main telescope direct the striding level is used in 
both positions and two azimuths are read, and with the telescope 



290 MINING SURVEYING [Chap. XI. 

inverted the striding level is again used in both positions and two 
more azimuths are read. The mean of the two readings of each 
pair gives two mean lines of sight which are symmetrically related 
to the vertical plane passing through the two station points and 
the correct azimuth reading is therefore the mean of these two 
azimuths. There is a small correction to be applied to the mean 
of the altitude readings but this is not usually of any importance. 

319. Adjustment of Top Telescope. — The top telescope is 
adjusted in much the same manner as the side telescope. No 
corrections have to be apphed for single azimuths readings, if the 
instrument is in adjustment; but the altitude readings require a 
correction on account of eccentricity of the telescope. This 
form of instrument will not reverse so that errors cannot be elimi- 
nated in that way, and it is not so well adapted to the use of the 
striding level. 

320. IWTERCHANGEABLE SIDE AND TOP TELESCOPE.— 
Some instruments are made with an interchangeable telescope 
which can be attached at either the top or the side of the main tele- 
scope, according to whether horizontal or vertical angles are being 
measured. In such an instrument no correction for eccentricity 
of the auxiliary telescope is necessary, and it is arranged so as 
not to require readjustment when changed from side to top or 
vice versa, 

321. Eccentric bearing telescope. — Another form 
of instrument (Fig. 133) known as the "eccentric bearing" 
transit, is regarded by many as the most accurate type. This 
instrument has an extra pair of supports for the horizontal axis 
of the telescope, which are so arranged that the axis can be dis- 
placed horizontally by a fixed amount along the course sighted. 
When the telescope is set in the eccentric supports vertical sights 
can be taken. A striding level is used to adjust the horizontal 
axis. With this instrument the foresight and backsight should 
both be taken with the horizontal axis in the same pair of bear- 
ings, as these two pairs of bearings are not so adjusted that the 
horizontal axis is exactly parallel in the two positions. All 
errors of adjustment may be eliminated by taking four readings, 
two with the horizontal axis in one position and two with it 
tinned end fd|^ end in the eccentric supports, leveling up with the 



291 



ts? 



6 i 



H 

> 

P g 

r H 

» O 



H 
w 
r 

CD 

rs 
o 

P3 




-^^ 



J I 



292 MINING SURVEYING [Chap. XI. 

striding level each time the sight is taken, the striding level being 
used in both its positions in each of the two positions of the hori- 
zontal axis. 

322. COMBINED SOLAR ATTACHMEIfT AND TOP TELESCOPE. 
— A special top telescope is sometimes made to do the duty of 
a solar attachment; but it is now generally admitted that better 
meridian determinations can be made by direct, single observations 
with the main telescope, and the surveyor is advised not to get any 
such complex attachment for mining work. 

323. In comparing the relative merits of the various forms of 
attachment it must be remembered that the object to be accom- 
plished is to transfer the meridian accurately from one station to 
another, these stations being close together in plan and distant 
in elevation. All other virtues of any attachment are of minor 
importance. Therefore, in addition to the ordinary adjust- 
ments of the transit, special care must be taken to get the hori- 
zontal axis of the telescope truly horizontal and the line of sight 
exactly perpendicular to it. A high power telescope is more 
necessary than for ordinary surveying, as a small variation of the 
line of sight means a large error in the azimuth. 

324. USE OF THE ORDINARY TRANSIT IN MINING SUR- 
VEYING. — Where a special attachment is not to be obtained, 
or when the auxiliary telescope is too small for accurate work, 
the ordinary transit can be used in such a manner as to ac- 
complish the same result as the eccentric bearing instrument 
(Art. 321). The instrument, firmly screwed on to the tripod, is 
inclined over the shaft at an angle just sufficient for the line of 
sight to clear the horizontal plate. It is then braced in position 
by such rigid supports as the circumstances afford, and the head 
of the instrument is rotated so that the horizontal axis of the tele- 
scope becomes truly horizontal, as determined by a striding level, 
while the telescope is sighting in the desired azimuth. One or 
more station points are then set out down the mine and one each 
way on the surface, all in the same azimuth, and these are re- 
spectively connected with the mine and surface surveys. All 
errors of adjustment may be eliminated by repeating sights with 
the telescope in the direct and the reversed positions and by re- 



COMPASSES USED IN MINES 293 

« 

versing the striding level each time and taking the mean posi- 
tion of the four points so set. 

An attachment which is very necessary in performing some 
of the work required of mining transits is the reflecting or pris- 
matic eyepiece. This makes it possible to take any sight what- 
ever above the horizon, and being a handy instrument to use and 
not requiring any adjustment, it shotild be carried by every mine 
surveyor. 

325. Compasses used IW mines. — The transit has taken, 
to a great extent, the place of the old miner's dial in which the 
compass was the mam feature. This is partly because, in mod- 
ern mines, so much heavy machinery is used that the compass 
needle cannot be depended upon, even to its ordinary degree of 
accuracy. 

Compasses, however, serve a useful purpose in general min- 
ing work. They are made in many sizes and of different design. 
A compass with a plain needle is to be preferred to one with a 
swinging card, since the former can be brought to a central posi- 
tion more quickly and is more accurate by reason of the lesser 
amount of weight on the center bearing. Compasses may be 
used for reconnoissance surveys and also for filling in the details 
of a mine from the main stations. A mining compass should be 
capable of sighting fairly high altitudes above or below the hori- 
zon, and a sighting clinometer* attachment for measuring 
vertical angles is very convenient as it obviates the use of any 
other instrument. A small modem mining dial mounted on a 
light tripod fulfills all these conditions. The hanging compass 
and clinometer is made so as to be hung from a wire stretched 
between two station points thus rendering sighting unnecessary, 
but it is not much used. 

A mounted compass is more accurate than one simply held 

in the hand, but any hand compass may always be mounted when 

\ 

* The Abney hand level and clinometer consists of a modification of the hand 
level described in Art. 100, p. 77. On top of the instrument is a level tobe pivoted 
ai the center of a graduated arc, and seen by reflection in a mirror placed inside 
the telescope tube. The instrument is pointed along the line whose inclination is 
desired and the level turned until the bubble is in its mid position, when the angle 
of inclination may be read on the graduated arc. 



294 MINING SURVEYING [Chap. XI 

the conditions permit. Perhaps the best form of hand compass 
is one in which the observer looks down on the instrument and 
the line of sight is reflected upward towards him by a hinged 
mirror so that the object and the compass box are seen simul- 
taneously. 

Ore of a magnetic nature has often been discovered by local 
variations of the compass needle and by the dipping needle, a 
special self- plumbing form of which is made for the use of miners. 

A combination compass, or clinometer of special form, is 
also useful in taking local strikes and dips of formation. One 
of the straight edges of the instrument is f)ut against the ledge 
of rock and turned in contact with it until the level line is reached 
as shown by an attached spirit level. The instrument is then 
folded up or down about this edge as a hinge until the compass 
needle is horizontal and the strike is read. The dip is always 
at right angles to the strike, but it is not of great importance to 
set out this right angle accurately. 



UirDERGROUND SURVEYING. 

326. TRANSFERRING A MERIDIAN INTO A MINE BY USE 
OF THE Transit. — Only a moment's thought will convince 
the student that some difficulty must be experienced in ac- 
curately transferring the meridian to the bottom of a narrow 
shaft several hundred feet in depth. The ordinary method of 
transferring a meridian into a mine is to set up the transit at a 
station fixed at the mouth of the shaft and, after taking a back- 
sight on the previous station on the surface, to take a foresight 
down the shaft, the line of sight being made as much inclined to 
the vertical as possible. Having ascertained the intervening 
distance, the transit is set up at the bottom station, a backsight 
taken on the top station, and the survey then carried into the gal- 
leries of the mine. The top and bottom stations are not always 
the surface and bottom of the shaft, although for simplicity, 
they may be referred to as such in this chapter. 

In sighting from both ends of the same highly inclined line it 
will be found that errors due to the line of sight not being per- 



UNDERGROUND SURVEYING 295 

pendicular to the horizontal axis are eliminated if the readings 
are made with the telescope in the same position at both sights, 
whereas errors due to inclination of the horizontal axis are elim- 
inated if the readings are made with the telescope direct when at 
the top and reversed when at the bottom of the shaft, or 
vice versa. 

When it is impossible to sight up a shaft on account of its 
being too wet, two or more points can be set in line at the bot- 
tom of the shaft by means of the instrument when at the top, 
and these will determine a line of known azimuth at the bottom 
of the shaft. 

In some cases a wire is stretched horizontally across the 
bottom of the shaft and as far back into the workings as possible, 
the wire being carefully aligned by the instrument at the top. 
This method may admit of even more accuracy than that of 
taking a backsight to the surface from a station established on 
the bottom of the mine. Errors due to a slight inclination of 
the horizontal axis are not important when this method is used 
and for that reason it is also useful in cases where a sensitive 
striding level is not to be had. The effect of a slight inclination 
of the horizontal axis is simply to shift the line slightly to one 
side but parallel to the true position. 

When no extra telescope or eccentric bearings are to be had, 
an ordinary transit with a prismatic eyepiece attached may be 
used to drop the meridian down a vertical or highly inclined 
shaft, provided it is not so wet as to prevent sighting upward 
from below. To accomplish this a thin wire is stretched hori- 
zontally across the top of the shaft at a known azimuth; the wire 
should be prolonged one or both ways in order to give a good 
base-line. Two points may be fixed at the top of the shaft if 
preferred. The transit is then set up on the bottom and it is 
brought by trial into the same vertical plane as the wire. The 
striding level is used in both positions and the transit is used in 
both the direct and reversed positions to eliminate errors*. 

It is to be noted that in mining and mountain work slight 
errors occur in sighting up steep inclines owing to the refraction 
of the atmosphere, but this is so slight that it does not aflFect the 
transfer of the meridian and is never taken account of. 



296 MINING SURVEYING [Chap. XI. 

327. PLUMBING THE MERIDIAN DOWN A SHAFT.— To 

the mine surveyor the plumb-line is an instrument of precision, 
excelling even the transit, and under most conditions, the work 
of transferring the meridian down a mine can be accomplished 
more accurately by means of the plumb-line than by any other 
method accessible to the surveyor. 

The method usually followed is to suspend two bobs from the 
staging above the mine so that a horizontal line in their plane can 
be sighted both from above and from below. The transit is 
set up both above and below on this line and thus an azimuth 
connection is established between the surface and the workings. 
Sometimes a much longer base-line than can be directly sighted 
can be obtained by plumbing down at the comers of a shaft as 
shown in Fig. 134. Points A and B have been plumbed down 

and, by the triangulation method 
there indicated, a connection with 
the underground traverse can be 
established. In this triangle the 
angles shoidd be chosen so as 
to give good intersections. 

All kinds of drafts in the shaft 
should be avoided during the 
alignment at the bottom. No 
cages or skips should be run and 
Fig. 134. Triangulating at the the passages leading to the shaft 
Bottom of Shaft. u * u j j -^l 

may have to be damped with 

sheets of canvas. No lateral streams of water should impinge 
on the plumb-lines; in fact it is desirable that no water at all 
should drop in their vicinity. 

The best plumb-line for this work is one made of wire. An- 
nealed copper wire is most flexible, but soft steel or piano wire 
being thinner will be less afi^ected by drafts and will also stretch 
less. The plumb-bob should not weigh less than five pounds 
and should be heavier for a deep shaft. A good working weight 
is one-third of the load at which the wire will break. 

The plumb-bob is hung in a bucket or a barrel of viscous 
liquid so as to bring it to a standstill in the shortest possible 
time. The shape of the plumb- bob is of importance in this respect 




PLUMBING DOWN A SHAFT 297 



r%l 



and the form shown in Fig. 135 is a good one, 
since it prevents rotary as well as lateral oscilla- 
tions. It should hang near the top of the ves- 
sel as the wire will be in a high state of tension 
and will stretch considerably. A mark should 
also be made on the wire showing how far the 
bob is above the bottom of the vessel. 

The liquid must be a true one (not a mud 
or slime) and it must be neither too limpid nor 
too viscous; for in the former case it will not 
stop the oscillations within a reasonable period, 
and in the latter the bob may not reach the cen- 
tral position quickly enough. The amplitude fig. 136. Type 
of the vibrations of the plumb-bob decreases in of Plumb-Bob 
a fixed ratio with equal increments of time, and j^^^ MERmil^ 
the viscosity of the fluid should be such as to Down a Shaft. 
make each oscillation, say, about one-quarter 
of the preceding. The ratio of decrease during equal incre- 
ments of time is independent of the length of the plumb-Une 
and of the amplitude of the oscillations if the resistance is 
purely viscous. This law makes it possible to select the fluid 
above ground, with the aid of a short length of wire attached to 
the bob; it applies only when the bob swings through a very small 
arc so that the resistance is wholly viscous. It may be noted 
that the period of oscillation varies approximately as the square 
root of the length of the plumb-hne, the same as for a pendulum 
swinging in air. 

If the shaft is wet the vessel should be covered with a sloping 
lid having a hole in it of an inch or so in diameter so that the 
wire can swing freely. In order to obtain as long a base-line as 
possible the wire should be hung as near to the casing of the shaft 
as is consistent with the precaution that it shall be perfectly 
plumb. It should be carefully examined along all its length to 
make sure that there are no obstacles to interfere with it. In 
some cases it may be sufiicient to pass a lighted candle around 
the wire at the bottom and observe any obstacles by sighting 
from the top.- The distance between the wires at the bottom 
and top of the shaft should always be measured and compared. 



298 MINING SURVEYING [Chap. XL 

as this gives the best test of the accuracy of the plumbing opera- 
tion. If four lines one in each comer of the shaft are hung 
instead of two an accurate check or measure of the errors is 
possible. 

When once the plumb-lines are hung the meridim may be 
transferred to all the levels of the mine once and for all time, 
so that a little extra precaution and time given to this operation 
are worth while. The surveyor should always keep in mind the 
fact that in plumbing the meridian down the mine the direction 
of the meridian is of much more importance than the actual posi- 
tion of the points themselves, because an error due to an incorrect 
direction of the meridian may be multiplied many hundreds of 
times in carrying the traverse through the mine (Art. 348, p. 316). 

328. TRANSFERRING A MERIDIAN INTO A MINE WHEN 
THERE ARE TWO SHAFTS. — The above methods presuppose 
that the mine has so far been opened only by one shaft. If 
there is a second shaft or an adit, it is, of course, only necessary 
to plumb or otherwise transfer the position down each shaft; the 
computed distance between these points then becomes a base-line 
of substantial length. In Fig. 136 the traverse ^4 J5 C-D is run 

B TRAVERSE 



^^ 


\. 


A ^^ - 


_Closinq Line >^ -. 


A' """■- -^G' 


/o* 

UNDERGROUND TRAVERSE E 




PLAN 




Fig. 136. 



out on the surface to connect the two shafts at A and D. The 
points A and D are plumbed down the shafts and the correspond- 
ing points A' and D' established at the bottom. A.i underground 
traverse A'OF'E'U is then run out. In the surface traverse 
the length and azimuth of AD and in the underground traverse 
the length and azimuth of A'D' are missing. The horizontal 



UNDERGROUND TRAVERSES 299 

length and azimuth of each of these lines can be determined from 
their respective traverses as explained in Art. 397, p. 366. The sur- 
face traverse is referred to the true meridian, and, since nothing is 
yet known in regard to the direction of the meridian in the mine, 
the underground traverse is referred to an assumed meridian. 
The true azimuth of A' D' is the same as the azimuth of AD^ pro- 
vided the plumbing down the shaft has been accurately done. 
The diflference between the true and assumed azimuths of A'ly is 
a correction to be applied to the azimuths of all of the lines of this 
underground traverse. 

329. UNDERGROUKD TRAVERSES. — Surveying in a mine 
is necessarily a process of traversing, for only the working passages 
are available for lines of survey. The line of traverse is not always 
in the center of the passage but is often varied from it in order 
that the longest possible sight may be taken. In the tortuous 
passages of a mine it is frequently necessary to take very short 
sights on the main traverse and since the azimuth is transferred to 
distant connections through these short lines great care should 
be exercised. The positions of the walls of the passages are noted 
as the work proceeds and are sketched in approximately on 
the plot. After the main traverses have been run, the surface 
boimdaries, if touched, may be accurately estabUshed and the 
stopes and working places surveyed by more convenient and less 
accurate methods, from the stations already established. 

It is often very convenient in underground work to take the 
azimuth from an estimated general direction (or strike) of the vein ; 
for the diriection of the meridian is of no importance in the actual 
working of a mine, while the direction of most of the passages will 
usually vary only a few degrees from the strike, and thus all trav- 
erse calculations are simplified. 

A speedy and convenient manner of running an underground 
traverse is to use three tripods having leveling heads and centering 
plates like those of the transit. The transit fits on to any of 
these heads and while it is attached to one of them the other two 
are surmounted by lamp targets in* which the sighting center has 
exactly the same position as the sighting center of the transit 
would have if set on the same tripod. These tripods are placed 
vertically over or under the stations and the transit is attached to 



300 MINING SURVEYING [Chap. XI. 

the middle one. When the transit head is moved from the 
middle to the foremost tripod a target takes its former place and 
the hindmost tripod is brought ahead of the transit and set up on 
the new forward station. The lamp behind the plumb-target or 
plumb-line should give a diflFused illumination of considerable 
area so that it may be easily found with the telescope and so that 
it may render the cross-hairs of the telescope plainly visible. In 
cases where the illumination of the object is such that the hairs 
cannot be distinguished, a light is thrown obliquely into the 
telescope tube in front of the hairs, preferably by a tube reflector 
(Fig. 133) in front of the object glass. 

Sometimes a brass lamp with a small central flame, called a 
plummet-lamp, is suspended in place of a plumb-line and the 
flame is sighted at, but this is too small a target for quick work 
and the surveyor may also mistake other lights, such as miner's 
lamps or candles, for it when sighting through the telescope. 

330. Establishing Station Points. — The station point is es- 
tablished either on the floor or the roof, according to the character 
and condition of the mine: the chief object sought is permanence 
of position rather than convenience in getting at the point for 
future use, which is of secondary importance. In a vein mine a 
timber in the roof, especially a stuU, is often more permanent than 
the floor or rock roof, but any timber is likely to be moved by the 
miners. The hanging wall- is a good place for the station, but if 
the inclination is small, as in a coal vein, the foot wall or floor is 
best. 

To establish a station, get a miner to make a drill hole about 
six inches deep, more or less, according to the hardness of the 
rock. Cut a wooden plug to fit this hole tightly when ham- 
mered in dry, and do not let any more of the plug project than is 
necessary. Small screw eyes make good roof station points 
from which to suspend the plumb-line, but where the lines are 
short a finishing nail bent to a sharp angle is better as the plumb- 
line will then always hang in exactly the same position. For 
measuring between stations a hundred-foot steel ribbon tape, 
divided to hundredths of a foot is used; but for long straight 
tunnels and shaft work, a longer steel wire tape is more con- 
venient. 



NOTES OF A MINE SURVEY 



301 



Survey of Bear Creek Mine, West Boulder, Montana. 



Sta. 


Bearing. 


Distance. 


Vert. 
Angle. 


Back, 
•ighton 


May 17, 1906. 
Pvty : Keene, Chase, Holbrook. 





NSS^lCE 


650.8 


- 1*17' 


San 


To top of sdr shaft extending 
to I at level ; C, of S. edge 
of air shaft, 4' x 4'. Sta. 2. 





N 2°lo'W 


"74 


-8o*>io' 


Sun 


To Sta. 1 01 at ist level. 
Line runs 3' from S. side 
and 4.5' from IV. side of 
shaft; shaft 8' X 8'. 


1 01 


N 87^45' E 


230-8 


+ o«45' 





To Sta. 102 in ist level. 


lOl 


N 2°io'W 


112. 6 


-8o*>io' 





To Sta. 201 in 2nd level. 


102 


N89*»io'E 


750 


+ o-sy 


lOI 


To top of center of raise ex- 
tending to 2nd level, raise 
4' X 4'- Sta. 107. 












102 


N89*»io'E 


153 5 


+ o°53' 


lOI 


To Sta. 103 in 1st level. 


103 


S 89^15' E 


105 7 


+ o*'39' 


102 


To Sta. I04 in ist level. 


104 


S 88°i2' E 


162. 1 


+ o*'48' 


103 


To Sta. 105 at foot of S, 
side of air shaft extending 
to surface. 


lOS 


S 9°55'W 


92.3 


+88*25' 


104 


To top of air shaft, Sta. 2. 


105 


S 88«i2' E 


15.9 


level 


Compass 


To Sta. 106 at breast of ist 
level. 


201 


N 2^10' W 


iiS.S 


-80*10' 


lOI 


To Sta. 301 in 3rd level. 


201 


N 85*»52' E 


167.4 


+ 0*50' 


lOI 


To Sta. 202 in 2nd level. 


201 


S 85<'46' W 


196.0 


+ 0*47' 


lOI 


To Sta. 205 in 2nd level. 


202 


N 88^20' E 


138.0 


+ 0*44' 


201 


To C. of raise extending to 
1st and 3rd levels, 4' x 4' 
Sta. 208. 


202 


N 88^20' E 


106.3 


+ 0*44' 


201 


To Sta. 203 in 2nd level. 


203 


S 89*»05' E 


176.9 


+ 0*42' 


202 


To Sta. 204 at breast of 2nd 
level. 


208 


S 3^14' E 


113.7 


+ 77^19' 


202 


To Sta. 107. 


205 


S 86*10' W 


216.8 


+ 0*48' 


201 


To Sta. 206 in 2nd level. 



302 



MINING SURVEYING 



[Chap. XI. 



Survey of Bear Creek Mine, West Boulder, Montana. (Cont'd.) 



su. 


Bearing. 


Distance. 


Vert. 
Angle. 


Back- 
sight on 


^ 


206 


S 87°I4' W 


118.0 


+ 0°4I' 


205 


To top center of winze ex- 
tending to 3rd level, 4' X 
4'. Sta. 209. 


206 


S 87^14' w 


152.0 


+ o°4l' 


205 


To Sta. 207 at breast of 2nd 
level. 


301 


N 86«2o' E 


304.0 


+ 0^46' 


201 


To Sta. 302 at C. of raise ex- 
tending to 2nd and 4th 
levels, 4' X ^\ 


301 


N 86^20' E 


316.0 


+ 0^46' 


201 


To Su. 303 in 3Td level. 


301 


S 86°4o' W 


195.0 


+ 0^50' 


201 


To Sta. 305 in 3rd level. 


301 


N 2°io'W 


116. 8 


- 8o°io' 


201 


To Sta. 401 at 4th level. 


302 


S 5^35' E 


116. 5 


+ 78^29' 


301 


To Sta. 208. 


303 


S 89^07' E 


289.0 


+ 0O39' 


301 


To Sta. 304 at breast of 3rd 
level. 


305 


S 88^52' W 


186.2 


+ 0^46' 


301 


To Sta, 306 in 3rd level. 


306 


S 89^48' W 


150.0 


+ o°43' 


305 


To Sta, 307 at C. of bottom 
of winze extending to 2nd 
level, 4' X 4'. 


307 


S 2°4i'E 


120.5 


+ 7i°ii' 


306 


To Sta. 209. 


307 


S 89^48' W 


10.9 


level 


Compass 


To Sta. 308 at breast of 3rd 
level. 


401 


N 85^48' E 


219.7 


+ 0^48' 


301 


To Sta. 402 in 4th level. 


401 


S 88«io' W 


116. 4 


+ 0^52' 


301 


To Sta. 406 at breast of 4th 
level. 


401 


N 2°io'W 


49-7 


^88°io' 


301 


To bottom of shaft, 3' from 
S, side and 4' from E side. 
Shaft 8' X 8'. 


402 


N 89^56' E 


85.0 


+ 0O45' 


401 


To Sta. 403 to C of raise ex- 
tending to 3rd level, 4' X 4'. 


402 


N 89'»s6' E 


92.6 


+ o°45' 


401 


To Sta. 404 in 4th level. 


403 


S o°o6'E 


116.2 


+ 8i°46' 


402 


To. Sta. 302. 


404 S 87*'2o' E 


217.6 


+ 0O43' 


402 


To Sta. 405 at breast of 4th 
level. 



lOb 



MAP OF 

BEAR CREEK MINE 

WEST BOWLDER. MONT. 

Scale. I in. "SO ffr. 

Scale of Reduced Plate » 
I in. « 150 ft. 




TRANSVERSE SECTION 

LOOKING WeST 



\ 



PLOTTING A MINE TRAVERSE 305 

331. Notes of a Mine Traverse. — As a rule the notes of mine 
surveys are kept in the form of sketches, especially the details, 
such as the location and extent of the stopes. These details are 
plotted on to the skeleton survey which is simply a traverse, the 
notes for which may be kept in the following form. 

The different station points of this survey are numbered for 
identification only, their numbers bearing no relation to the 
distances between them. For convenience the stations on the 
first level are numbered loi, 102, etc.; on the second level 201, 
202, and so on. In larger ard more complex mines the 
system of numbering and lettering stations is carried out still 
farther. 

332. Plotting a Mine Traverse. — This survey can be plotted 
by computing three sets of coordinates which give all the data 
needed for showing the mine in plan, longitudinal section, and 
transverse section. If the three coordinate planes are the meri- 
dian plane, the east and west plane, and the horizontal plane, then 
the vertical and horizontal distances from each slope measure- 
ment are first obtained by multiplying the measured distance by 
the sine and the cosine respectively of their vertical angles. The 
vertical distance is the difference in elevation between the two 
points. From the horizontal projection and the azimuth, or 
bearing, the latitude and departure of the course can be computed 
as usual (Art. 384, p. 352). A plot of these notes will be seen in 

Fig- 137- 

It is assumed in plotting these notes that all the transit lines 
in the galleries run 2 ft. below the roof and in the center of the 
galleries, which are 6 ft. high and 4 ft. wide; conditions which 
are more uniform than would occur in actual practice. The 
measurements which locate the walls of the galleries have been 
purposely omitted from the foregoing notes for the sake of 
simplic'ty. 

If it is desired to substitute for the meridian plane a vertical 
plane through the strike and for the east and west plane one 
which is at right angles to the strike, then all of the true bearings 
or azimuths must be corrected by an amount equal to the strike. 
After these bearings have been corrected the three coordinates 
are calculated in the same way as described above. The ad- 



306 MINING SURVEYING [Chap. XL 

vantage of this latter method is that the levels are shown in their 
full length in the longitudinal section and the shaft is shown in 
its true length in the transverse section. 

333. UlfDERGROUITD LEVELING. — The drainage of a 
mine is usually toward the shaft (Fig. 137), and the grade of the 
levels is such as will make the tractive force of a full car going 
towards the shaft equal to the pull required to move an empty 
car in the opposite direction. When connections, other than ver- 
tical ones, have to be made the grades must be taken into consid- 
eration. In this work an ordinary surveyor's level is generally 
used in conjunction with a short leveling rod about five or six 
feet high. 

334. MINE MAPS AND CHARTS. — The galleries of a mine 
are often so nearly over one another that confusion is liable to 
arise in charting, unless some special means of identifying them 
is employed. As these galleries or working passages appertain 
to definite levels or strata, a different color may be assigned to 
each level or strata and adhered to throughout. The lines of 
survey are in a colored ink and the passages or workings are of a 
fainter tint of the same color. These colors can also be adhered 
to on the elevations, of which there are usually two, one along the 
strike and the other at right angles to it. (See Fig. 137.) 

Some surveyors use large scale plots and simply mark the 
position of the stations on them so that when a course has to be 
set out its distance and direction can be scaled directly from the 
map. 

Another method is to use a small scale map and mark on it, 
in figures, the exact coordinates of every station point. The 
origin, or point of reference, is usually the plumb-line of the 
shaft, and the two vertical planes of reference may conveniently 
be taken through the estimated general strike and dip of the vein. 
The true course of the survey lines may also be marked and all 
the exact data can be clearly kept in a minimum space. 

The progress of work in the stopes or rooms of the mine is 
generally represented on different plans from those used to show 
the main headings. These working plots may be either vertical, 
horizontal, or parallel to the vein or seam. In any case, the 
thickness of the deposit is recorded at frequent intervals together 



MINE MAPS 307 

with other particulars, such as thickness of waste or value of ore. 
These thicknesses are all measured at right angles to the plane 
of the working plan, so that when multiphed by the area on the 
plot, the cubic capacity of any section is obtained. Where the 
ore occurs in irregular masses, not conforming particularly to 
any one plane, the above system does not apply and some other 
method must be devised by the surveyor. 

The best way of estimating amounts not mined is to sketch 
their probable extent on such a chart from the data available and 
to make use of the area and thickness method as suggested in the 
preceding paragraph. Ground explored by bore-holes but not 
opened by headings may be best shown by plotting in plan the 
positions of both of the walls, where struck in the bore-holes, 
marking the elevations in figures on the plan. Contours may 
then be sketched in colors, and a very good idea of the shape and 
trend of the ore body obtained, and the quantities of ore may 
also be calculated therefrom. (See Computation of Volume, 
Chapter XII.) 

335. LAYING OUT MUflHG WORK. — Drifts or cross-cuts are 
laid out by putting in two nails or hooks in the roof, not too 
near together, from which the miner can hang two plumb-lines 
and sight the center of the heading he is to rim. 

Vertical shafts are carefully plumbed on the inside of the 
frames, and frame by frame, as these are put in. It is best to 
hang the plumb-Une from several frames above the bottom one, 
as these upper ones are more likely to have ceased to move. 
Hang the line an even fraction of an inch each way from the true 
position of the comers and note any accidental variation in the 
last frame set, so that in future work, if it is desired to hang the 
plumb-line from this frame, its error of position can be allowed 
for. The dimensions of a shaft or drift are given either "in 
the clear," meaning net measurements inside all timbers, or 
"over all" meaning gross measurement outside all timber and 
lagging. 

336. UlfDERGROUND SURVEYING PROBLEMS. — In the prac- 
tice of mine surveying, problems are constantly arising which 
tax the ability and ingenuity of the surveyor, although the actual 



3o8 



MINING SURVEYING 



[Chap. XL 




solution of most of them is quite simple. A few of the common 
problems met with in such work are given below. 

337. Vertical Angle Correction 
for Eccentricity of the Top Tele- 
scope. — As has been stated in Art. 
317, all vertical angles taken by 
means of the top telescope must be 
corrected for the eccentricity of this 
attachment. In Fig. 138 the vertical 
angle has been taken to a point C in 
the bottom of a shaft. The distance 
AC was measured, A being the hori- 
zontal axis of the main telescope. 
Since the transit is set up over a siu*- 
face station at JE, the distances de- 
sired are EC and AD. HB and 
H'A are both horizontal, 
V - ACB. 

JK \ ' Am -- ^^ — Distance between telescopes 
AC Distance measured 

AD = AC sin F, 
and DC = AC cos 7'. 
The height of instrument above the datum being known the ele- 
vation of C can be readily calculated. 

Had the distance BC been measured instead oi AC then 
DC=CF + FD = BCcos V + .45 sin F. 
Similarly i4Z) = BC sin V - AB cos V. 

338. Vertical Angle Correction for 
Eccentric Bearing Telescope. — In 

Fig. 139, A is the central bearing for 
the telescope and B is the eccentric 
bearing in which the telescope rested 
when the vertical angle V and the 
distance BC were measured. 



Fig. 188. 
then V 



DC = FC '\- AB = BC cos V + AB 
iD ^ BF ^ BC sin V. 




Fig. 189. 



ESTABLISHING BOUNDARY UNDERGROUND 309 

339. To Establish a Boundary Line of the Claim Underground. 
— In Fig. 140 points A and B are on the boundary of the claim. 




The shaft is located at 5, and it is desired to prolong the 
underground working in the drift ilV to a point K' vertically 
under the boundary line. The surface traverse BADCS is run 
out, point 5 is plumbed down to 5', and the meridian trans- 
ferred into the mine. Then the underground traverse 
^EF'G'H'r is run out. The horizontal projections of all the 
measured lines on both traverses are computed (or measured), 
and the length of the level line AH' and its bearing can be cal- 
culated as described in Art. 398, p. 367. In the horizontal triangle 
AH'K'y AH' and all the angles being known, the line H'K' can 
readily be computed. If the drift H'J is not level the distance 
from H' along the drift to the boundary plane will be equal to 
the horizontal distanc : H'K divided by the cosine of the vertical 
angle. 

340. To Lay Out a Connection in a Mine. — Here the prob- 
lem is to determine the bearing (or azimuth) and the vertical 
angle and the distance to run from point -4 in a mine to point B 
in another portion of the mine. A traverse can be run from A 
to B through the passages already cut in the mine, and all the 
distances reduced to horizontal distances which, together T/vdth 
the azimuths, form a traverse in which the length of the closing 
line AB (horizontal projection) and its azimuth are missing. 
These can easily be computed by the method explained in Art. 
398, p. 367. The difference in elevation between the actual points 



3IO MINING SURVEYING [Chap. XI. 

A and B together with the length of the horizontal projection of 
AB will give the vertical angle; from these data the direct dis- 
tance between the points A and B can be computed. 

341. HYDRAULIC SURVEYING FOR MDfES. — The miner's 
unit for measuring water is the miner^s inch. By an inch of 
water was originally meant such continuous flow as will go 
through a one inch square hole, the head of water behind it 
being usually six to nine inches. This very loose definition has 
been done away with but the name still applies, being defined 
more exactly as ninety cubic feet of water per hour (ij cubic 
feet per minute). In spite of all criticism, the miner's inch has 
become by custom the standard unit for the flow of water in most 
mining districts. It no doubt retains its hold on the practical 
mind because no good definite time or capacity units are in gen- 
eral use, seconds, minutes, hours, and days, or gallons and cubic 
feet with their clumsy relations to one another, being used ac- 
cording to the whim of the individual. To get an idea of the 
magnitude of a standard miner's inch, it may be remembered 
that it is equivalent to a stream one inch square running at a 
uniform rate of 3.6 feet per second. This is about a medium 
speed for small mountain streams; and, with a little practice, 
the flow of such a stream in miner's inches may be calculated 
mentally, after rough measurements have been made of the cross- 
section of the stream and the speed of flow of the water at the 
surface. The accuracy of this process is within the ordinary 
limits of fluctuation of the stream from day to day. If the flow 
has to be recorded over a long period it is well to put in a weir. 

For estimating the flow of larger and more important moun- 
tain streams, a portion of the stream where the width and flow 
are comparatively uniform may be chosen and the length of this 
portion measured and marked by flags. A cross-section of the 
bottom of the stream is obtained at each flag and at intermediate 
points if necessary by measuring the depth at equal intervals 
across the stream; from these a mean cross-section is obtained. 
Floats are started at intervals across the stream opposite the up- 
stream flag, and timed with a stop watch while running to the 
down-stream flag; the speed of each float represents the velocity 
of the stream in its respective longitudinal strip. Each velocity 



SURFACE SURVEYING 3II 

is multiplied by the area of the corresponding portion of the 
cross-section of the stream, and from the total flow so computed 
a certain percentage is deducted for the excess of surface over 
mean flow; this, for ordinary mountain streams, is approximately 
twice the percentage of the grade of the channel. 

The surveying and staking out of mining ditches, flumes, and 
pipe lines follow the general practice for this work in other fields 
of engineering. 

342. Testing for Ore by Electric Currents. — Methods of 
testing the earth for ores by means of electrical currents and 
waves are being experimented upon, and the working out, record- 
ing, and plotting of the results are likely to become a part of the 
mine surveyor's work. 

SURFACE SURVEYIlfG. 

343. SURFACE SURVEYING Df RUGGED MOUITrAIN REGIONS. 

— In accurate work, such as the surveying of mining 
claims for patent,* the ordinary mining transit may be used. 
Measurements are made with a steel wire tape, 300 to 500 feet 
long and marked every 10 feet (or 20 feet) so as to be used with 
a short auxiliary steel ribbon tape which is divided to hun- 
dredths of a foot. The measurements are taken from the center 
of the instrument to the object at which it is pointed, care being 
taken not to overstretch the tape nor to kink it. The most 
accurate work is done by stretching the tape with a tension 
handle (a spring balance) which can be attached by a clamp to 
any part of the tape. Where it is feasible, just enough tension 
is given so that the stretch of the tape compensates for the short- 
age due to sag. In many cases assistants will have to hold the 
middle point or the points at one-third and two-thirds the length 
of the tape up to the line of sight, giving at the same time enough 
puU to make the sag equal in the different sections of the tape. 

There are several systems of traversing. The most common 
is to measure the height of the center of the instrument above the 

* By patent proceedings is meant the proceedings necessary to obtain from 
the government a fee simple deed to the mining claim. 



312 MINING SURVEYING [Chap. XI. 

station point, and then to sight an equal height on a graduated 
staff held on the back and forward stations, recording the azi- 
muth, vertical angle, and distance. Another method is to sight 
and measure to targets set at a fixed height above the stations, 
recording the vertical angle only at alternate stations. If the 
vertical angles are read at every station there wi.l be two sets of 
vertical angle and distance measurements. The three tripod 
method may also be used as described for underground work; 
and lastly two transits and instrument men may be em- 
ployed, each sighting to the other's telescope and measuring the 
distances between them. Each of these methods has its advan- 
tages and disadvantages, and the best one to use depends upon the 
conditions of the work to be done. In some cases there will be 
twice as many altitudes and in some cases twice as many dis- 
tance readings as are actually needed, but these extra readings 
may be used as a check available in the field. 

In making general maps of a mining district, only monu- 
ments and important locations need be accurately shown. This 
accurate work which is the first to be done forms a skeleton on 
which to make a general map. The topography can be filled in 
by a transit fitted with fixed stadia wires and a compass. 

The best topographical data in mountainous country are ob- 
tained by running traverses along the ridges and valleys; these 
are also usually the best places to travel. Much sketching is 
necessary and the work should be plotted by the surveyor him- 
self each day as the work proceeds. In this work a rough deter- 
mination of the topography is sufficient, since the plans are usu- 
ally plotted to the scale of i^^^^i^ or smaller, and therefore such 
instruments as the hand compass, clinometer, and aneroid baro- 
meter can be used. With such instruments one man can do the 
entire work. The plane table cannot be used to advantage in 
mountain or mine surveying, but photographic surveying may 
orten prove useful in filling in details of topography. 

344. MINE BOUNDARIES. — ^APPROPRIATIONS UNDER UNITED 
STATES LAWS.* — In most countries mineral rights are defined 

• For further information with regard to this subject see the Manual of 
Instructions for the Survey of the Mineral Land of the United States, issued in 
1895 by the Commissioner of the General Land Office, Washington, D. C. 



MINING CLAIMS 3^3 

by vertical planes through lines marked out on the surface. 
Title to metalliferous lands, however, as granted by the United 
States, conveys the right to all minerals included in the down- 
ward prolongation of the portions of veins cut off by the verti- 
cal end bounding planes, i.e., a vein can be worked in the 
dip indefinitely, but in the direction of the strike it is limited by 
the end bounding planes of the claim. This law has given rise to 
much litigation and there are still many unsettled points involved. 
The Federal law allows a claim to cover 1500 feet located 
along the direction of a vein and 300 feet of surface ground on 
each side of it. These dimensions which constitute the maxi- 
mum can be reduced by local laws. The ordinary method of 
locating a claim is shown in Fig. 141. The discovery being 

WW _____^.___ ^^ 

yi — k^ — £ — ^c 

sw : SE 

Fig. 141. Plan of Lode Claim. 

made at D the center line WC-D-EC is run and then the end 
lines SE-NE and SW-NW are put in, being made parallel with 
each other and straight. The side lines must not be 6ver 300 
feet apart, measured at right angles from the center line. 

A monument with explanations is placed at each of the seven 
points marked. If in a timbered country, the lines run should 
be blazed, and squared trees may be used as monuments. At 
D (Fig. 141) a location notice is posted, defining the boundaries 
of the claim and containing such explanation as would identify 
the claims in case of dispute. The miner usually makes the loca- 
tion survey himself, using approximate courses and distances. 
There is legally no objection to this work being done roughly, 
but when a patent survey comes to be made, neither the dimen- 
sions specified in the location notice nor the limits of the claim 
as marked off on the ground can be exceeded. So when the 
location survey is roughly made certain "fractions " of ground 



314 MINING SURVEYING [Chap. XL 

are not included, and these may cause much trouble, especially 
when "groups" of claims are located. 

In such preliminary surveying, traverses may be run along 
coiu'ses where the sights can be conveniently taken and the azi- 
muth taken from the direction of the vein; this may save much 
time and considerably simplify the work, (specially in thickly 
timbered regions (Art. 329, p. 299). In the description it is 
sufficient to state the approximate compass bearings of the 
boundaries. The center line and side lines need not be straight or 
parallel, but are assumed to be so unless marked with additional 
monuments. If, on account of the crookedness of the vein, it 
is advisable to make the center line of the claim a series of straight 
lines (like a traverse), this can be done, but the above condi- 
tions must be fulfilled with regard to the length and breadth of 
the claim and the two end lin6s must be parallel. In order to 
guard against troublesome litigation, an effort is sometimes 
made to surround a valuable claim with others, thus forming a 
"group." The more valuable claim is then protected as regards 
all " extralateral rights." 

Flat deposits, such as coal and placer, are subject only 
to vertical bounding planes, and, provided the boundaries are 
marked plainly on the ground and the legal dimensions are 
not exceeded, no difficulty need be encountered. The Federal 
law allows 20 acres to be taken for a placer claim but fixes no 
limits in regard to breadth or length. Local laws can regulate the 
size, provided the 20 acre limit per claim is not exceeded. The 
coal lands law is made subject to the general system of public 
land surveys for agricultural lands. 

345. SURVEYING FOR PATENT. — The surveying of claims 
for patent from the United States Government can only be 
obtained by those who have received appointment of United 
States Deputy Mineral Surveyor and they must have an order 
from the Surveyor General of the state or territory in which 
the claims are located before making any such survey. 

In survejdng for patent, much more accurate work has to be 
done than when merely locating a claim. After the shape of 
the claim as originally staked has been determined, the positions 
of the new corners and other boundary marks are computed and 



PATENT SURVEYING 315 

laid out on the ground. The original claim cannot anywhere be 
exceeded and usually has to be cut down so as to make the end 
lines parallel and bring the dimensions of the claim within 
statutory limits. All this must be done accurately, the limit of 
error allowed being one in. two thousand. Besides 'the marking 
of the boundaries on the ground, the position of at least one of 
the comers of each claim must be determined with reference to 
permanent monuments recognized by the government. The 
true meridian must also be determined by observations of the 
sun and all courses must be referred to it. The position of all 
buildings and surface improvement must be found and shown 
on the plot, and also the position of all comers of other claims 
for which a patent has already been applied. The surveyor 
must also make an estimate of the value of and describe all 
improvements, such as tunnels, shafts, open-cuts and other 
mining work done on the ground, and these should amount to 
not less than $500.00 worth per claim. The Manual of Instruc- 
tions describes in detail the character of the comers required to 
be established, and a great many other details which must be 
known to the Deputy Mineral Surveyor before his survey will 
be accepted, and defines the penalties attached to poor or dis- 
honest work. Patented claims may overlap, and in fact do, in 
all mining districts, but in making application for patents to 
claims which lap on ground previously patented, the exact rights 
desired on the area of intersection must be defined. 

Placer claims may be taken in twenty acre tracts, the bound- 
ing lines of which must conform with the general system of 
survey lines established by the Government, but if such survey 
has not been extended to the district, they must be bounded by 
true meridian and east and west lines. The survey of coal land 
is subject to somewhat similar rules. 

346. THE SURVEYING OF BOREHOLES. — Boreholes, 
whether made by a rotary or a percussion drill, are never per- 
fectly straight and unless the ground is remarkably homo- 
geneous, are not amenable to any mathematical law. Means 
have been devised, however, of measuring the strike and dip of 
a hole at any particular distance from its mouth. The trend of 
the borehole can thus be plotted with some degree of approxi- 



3l6 MINING SURVEYING [Chap. XL 

mation and the position of any particular body or strata struck 
in the borehole determined. One method depends in principle 
upon the conversion from liquid into jelly, by cooling, of a 
solution of gelatin, contained in a small vessel together with a 
compass needle and a plumb-bob and of such a shape as to 
align itself with any part of the hole in which it may be placed 
Another instrument takes a photographic record of the 
position of the compass needle and plumb-bob, after the lapse of 
such an interval of time as is necessary to place the instrument 
in proper position and allow the needle and plumb-bob to come 
to rest. The position of points in any plane stratum, as found 
by three boreholes, determines it. If, however, the angle at 
which a borehole cuts this stratum is known, only two boreholes 
are necessary and if the strike and dip of the stratum is known, 
one borehole is sufficient to determine it. 

347. STAKING OUT THE PROBABLE APEX OF A VEDf. — 
It is often required to prolong the course of an inclined vein on 
the rugged surface, either for exploration purposes or to locate 
a claim. This may be accomplished by setting up on the vein a 
transit fitted with a solar attachment, the main telescope being 
inclined at the angle of dip of the vein in altitude and pointed at 
right angles to the strike The solar attachment, when set for 
the zero declination, will sight points only in the plane of the 
vein. 

As veins are usually somewhat irregular, the survey need not 
necessarily be made with a transit. Instead, such instruments 
as the compass, clinometer, or a small improvised plane table may 
be used. 

348. ECONOMIC PRINCIPLES. — The surveying of mine 
workings is strictly an economic problem and the surveyor must 
study it as such. The accuracy attained must be such that the 
cost in obtaining it and the saving of expense in mining oper- 
ations through it together efTect the maximum of economy. The 
surveyor bearing this in mind will be neither too careless nor too 
exact. He will not, for example, close down the mine in order 
to carry a meridian into it when no important connections are 
needed, and there are no boundary disputes. On the other 
hand, in some cases accuracy of a survey is of such prime impor- 



ECONOMIC PRINCIPLES 



317 



tance that a temporary interference with the working of the mine 
may be warranted. 

In any given case the surveyor must make a scientific meas- 
ure of the accuracy re- 
quired. There is no 
better method of ob- 
taining accurate results 
than to go over the lines 
several times with vary- 
ing conditions, but this 
is not always good econ- 
omy, especially in such 
work as plumbing a 
shaft which necessitates 
a temporary stoppage of 
all hoisting operations. 
Often the controlling 
error will be the error in 
plumbing of the meri- 
dian. In Fig. 142, D is 
the horizontal distance 
in a straight line from 
the shaft to the connec- 
tion, d is the distance be- 
tween the plumb-lines, 
and e its error as ascertained by measuring or other means. 

The controlling error at the connection \s E = ^^and is in a 

direction perpendicular to that in which D was measured on the 
chart. This is obvious, for the surveys of the galleries are con- 
sidered accurate, the error being one of relative rotation around 
the shaft as an axis. Where there are many angular errors of 
the same degree of magnitude, such as occur when a number of 
short sights are included in the traverse, the distances may be 
measured from these short lines to the connection and their 
respective errors E^ E^, E^, etc. found at these localities. These 
are then resolved according to their respective latitudes and 
departures into S„ .S„ S,, etc., and W^, W,^, W^, etc. The greatest 




Shorr 



Fig. 142. 



3l8 MINING SURVEYING [Chap. XI. 

possible error is then 5i + 5, + 5, + etc. to the north or south 
and W^ + W^ + W^ -}- etc., to the east or west, these summa- 
tions being made without regard to any sign. 

Likewise the mean proba ble error (by method of least 
squares), is V5i^ + S^ + S^ + etc. to the north or south and 
VWi' 4- W,* + Wg^ + etc. to the east or west. Errors due to 
the measurement of distances, which are not likely to be great, 
may be divided into latitudes and departures directly and com- 
pounded with those due to angular error. 

Besides being of immediate service to the surveyor, the prac- 
tice of computing possible and probable errors gets him in the 
habit of thinking along the most business-like lines instead of 
drifting into a rut or losing interest in his work. 

The surveyor should keep his plans up to date and see that 
the men in charge of the mining operations fully understand 
their instructions; for many mining "bosses" of considerable 
experience get entirely wrong notions of the shape of their work- 
ings and are often too proud to ask for information. The sur- 
veyor, without assuming a "know-it-all '' attitude, can, from the 
specialized nature of his work, often make useful suggestions 
in regard to the exploration of a mine. By working always in 
harmony with the other officials of the mine, he can further the 
interests of all concerned, both employers and employees. 



PROBLEMS. 



1. From a monument at the mouth of a tunnel a line is run in the tunnel, azi- 
muth 37° 24', slope distance 424 ft., vertical angle 4- 2° 10'; thence azimuth 62° 42', 
slope distance 278.5 ft., vertical angle + 2° 18' to breast. From the same monu- 
ment a line is run on the surface, azimuth 98° 33', slope distance 318.5 ft., vertical 
angle — 3° 22'; thence azimuth 38° 02', slope distance 647 ft., vertical angle -f 14° 13' 
to the center of a vertical shaft. How deep must the shaft be to meet a connect- 
ing drift run on a grade of + 2.4 % from the breast of tunnel, and what is the 
slope length and azimuth of this drift? 

2. The strike of a certain vein at point of outcrop is N 43° E and the dip is 
71° 5o'» pitch S.E. From this point of outcrop a surface line is run, N 83° 15' E, 
slope distance 248 ft., vertical angle — 12° 34'; thence S 2^54' E, slope distance 
208.5, vertical angle — 14° 34' to a point from which the tunnel is to be driven in 
the direction N 71° W and with a grade of 4- 3.8^ until it intersects the vein. 



ECONOMIC PRINCIPLES 319 

(a) What would be the slope length of such a tunnel? 

(b) What would be the slope length and bearing of the shortest possible tun- 
nel run on a + 1.3% grade to intersect the vein ? 

3. A vein has a pitch of S 67® W and its dip is 55®. What is the azimuth of 
an incline on the vein having a slope of 44®? 

4. From the bottom of vertical shaft No. i a horizontal traverse was run in the 
mme to the bottom of vertical shaft No. 2 as follows: Assumed azimuth 0°, distance 
243 ft; thence azimuth 340°, distance 121 ft.; thence southeasterly a distance of 473 
ft. along a vein which shows a pitch of 60** (azimuth) and a dip of 35®; thence azi- 
muth 42°, distance 25 ft. to the center of shaft No. 2. From a point vertically 
above the last point a line is run on the surface with true azimuth 116^20', dis- 
tance 41 1 ft. (horizontal) to a point A from which the center of shaft No. i is sighted 
at azimuth 71® 3c/. 

(a) How much deeper will shaft No. 2 have to be sunk to reach the vein? 

(b) What is the true strike of the vein ? 

5. A vertical winze has been sunk below the level of a tunnel. It is desired 
to sink a vertical shaft from the surface to connect with the winze. The monu- 
ment X is established at the mouth of the tunnel and the monument Y is near the 
site of the proposed shaft. Y bears S 88*58' 56" W, 896.796 ft. from X, The 
following are the notes of the survey connecting X and the winze comers A, B, C, 
and D: — 

Station. Mean Deflection. Horizontal Distance. Station. 

X .. 0° oo' 896.796 Y 

403.080 I 

587.208 2 

3 32* 23' 43" ^ 67.000 3 

3 54** 43' 47" R 44.803 4 

4 39** 51' 57" ^ 41.075 '5 



45° 05' 


34" 


R 


74° OS' 


06" 


L 


32° 23' 


43" 


L 


54° 43' 


47" 


R 


39° 51' 


57" 


R 


31° 10' 


10" 


R 


31° 10' 


10" 


R 


31° 43' 


40" 


R 


24° 02' 


40" 


R 



5 31° 10' 10" R 19-573 Cor. A 

27.240 Cor. B 

21.477 Cor. C 

25773 Cor. D 

Required the location of the shaft comers on the surface. 

6. From a monument M at the mouth of a tunnel a traverse is run in the tunnel, 
azimuth 20® 35', distance 352 ft., vertical angle -f- 1° to point A ; thence azimuth 61®, 
distance 528 ft., vertical angle -h i® 40' to point B at the breast of the tunnel 
From M a surface traverse is run, azimuth 11® 10', distance 578 ft., vertical angle 
4- 4° 25' to point C; tbence azimuth 11**, distance 407 ft., vertical angle -f- 14** 20' 
to point D, which is the center of a vertical shaft 120 ft. deep. Find the length 
and grade of a connecting incline from the bottom of the shaft to the breast of the 
tunnel. 

7. The course of Tunnel il is N 34° 45' 10" W., the grade 0.42%, and the ele- 
vation of the mouth 2570 ft. The course of Tunnel B b N o® 45' 00" W, the 
grade 0.33%, and the elevation of the mouth 2608 ft. The following traverse con- 



320 MINING SURVEYING [Chap. XL 

nects the mouths of the two tunnels: — from mouth of Tunnel B, N o® 45' 10" W, 
100 ft.; thence N 19® if 30" E, 381.60 ft.; thence S 10® 21' 20" E, i 30.60 ft; 
thence N 74*14' 30" E, 3662.01 ft.; thence N ^s"* 4$' 30^' E, 1547.21 ft; thence 
N 73° 48' 00" E, 1455.00 ft.; thence S 12® oo' 00" E, 205.40 ft; thence 
S 70® oo' 10" E, 205.00 ft. to the mouth of Tunnel A . Where and how far must one 
upraise vertically in order to connect the tunnels? Do not consider the dimen- 
sions of the tunnels. 

8. Assuming the transit to be in perfect adjustment what is the error in hori- 
zontal angle in sighting down a 500-ft shaft, 5 ft in breadth, when the telescope 
cannot be sighted closer than 3 seconds along the inclined line? 



PART III. 
COMPUTATIONS. 



3*1 



PART III. 

COMPUTATIONS. 

CHAPTER. Xn. 

GXSNISRAI. PRINdPLES.— MISCBLZJLNXIOnS PROBLEMS.— 
EARTHWORK COMPX7TATIONS. 

349. GENERAL REMARKS. — The ultimate purpose of many 
surveys is to obtain certain numerical results to represent 
quantities such as areas or volumes. In the section on Survey- 
ing Methods it has been pointed out that in all surveys there 
should be a proper relation between the precision of measure- 
ment of the angles and distances. To secure final results to any 
given degree of precision, the measurements in the field must be 
taken with sufficient precision to yield such results. In com- 
puting from a given set of field notes the surveyor should first 
determine how many places of figures he should use in the com- 
putations, the aim being to obtain all the accuracy which the 
field measurements wiU )deld without wasting time by using 
more significant figures than are necessary. Professor Silas W. 
Holman* in the preface to his "Computation Rules and Loga- 
rithms" says: — "It would probably be within safe limits to 
assert that one-half of the time expended in computations is 
wasted through the use of an excessive number of places of fig- 
ures, and through failiu-e to employ logarithms." 

Final results should be carried to as many significant figures 
as the data will warrant and no more. In order to insure the 
desired precision in the last figure of the result it will usually be 
necessary to carry the intermediate work one place further than 
is required for the final result. 

350. The number of significant figures in the result of an 
observation is the number of digits which are known. For in- 
stance, if a distance is recorded as 24,000 ft. when its value was 

* See " Computation Rules and Logarithms," by Professor Silas W. Holman, 
published by Macmillan & Co., New York. 

323 



324 COMPUTATIONS [Chap XH. 

obtained to the nearest thousand feet only, it contains but two 
significant figures. The zeros are simply put in to show the 
place of the decimal point. If, however, the distance has been 
measured to the nearest foot and found to be 24,000 ft. there 
are five significant figures, for the zeros are here as significant as 
the 2 or 4. Similarly a measurement such as 0.00047 contains 
but two significant figures, the zeros simply designating the posi- 
tion of the decimal point, for, had this same value been recorded 
in a unit jj^-^^j^-^jj^ as large the result would have been 47. 

Again, if a series of rod-readings are taken on different points 
to thousandths of a foot and three of the readings are 4.876, 
5.106, and 4.000 it is evident that each of these readings contains 
four significant figures; if each of them is multiplied by 1.246 the 
respective results are 6.075, 6.362, and 4.984. But had the 
results been measured to the nearest tenth of a foot and found to 
be 4.9, 5.1, and 4.0 these values when multiplied by 1.246 should 
appear as 6.1, 6.4, and 5.0. This illustration indicates the proper 
use of significant figures. Since the rod- readings 4.9, 5.1, and 4.0 
are reliable only to about 1.5 to 2 per cent, the multiple 1.246 
should be used in this computation as 1.25. Similarly in the use 
of such a constant as 7r=3.i4i5927 it is a waste of time to use any 
more significant figures in the constant than exist in numbers with 
which the constant is to be combined in the computation. 

351. In deciding how many places of decimals to use in the 
trigonometric functions the student should examine the tabular 
differences and determine what percentage error is introduced 
by any error in an angle. For example, suppose an angle of a 
triangle to have been measured in the field to the nearest minute. 
There may be an error of 30 seconds in this angle, and it will be 
seen from the table of natural sines that the tabular difference 
for one minute in the fourth decimal place varies from 3 for a 
small angle to less than i for a large angle, and that the vari- 
ation is about the same for cosines, and for tangents and cotan- 
gents of angles under 45°. Then for half a minute the differ- 
ence will be, on an average, about i in the fourth place. There- 
fore, in general, four places will be sufficient when the angles 
have been measured to the nearest minute only. But if there 
are several steps in the computations it may be advisable to use 



SIGNIFICANT FIGURES $2$ 

five-place tables. Similarly it can be seen that five-place tables 
of functions will, in general, give angles to the nearest lo seconds, 
and six-place tables to the nearest second. These are only aver- 
age results and are intended to give the student a suggestion as 
to how to decide for himself whether to use four, five, or six- 
place tables. It is obviously a great saving of time to use four- 
place tables where foiir places are needed rather than to use six 
or seven-place tables and drop oflF the last two or three digits. 
The alnount of labor increases about as the square of the number 
of places in the tables, i.e., work with 6-place tables: work with 
4-place table = 36 : 16. 

352. The following simple examples illustrate the useless- 
ness of measuring the distances with a precision which is incon- 
sistent with that of the angles, when the angles are to be used in 
the computation of other distances. Given the measurements 
shown on Fig. 143. If the angle B was measured to the nearest 
minute only there may be an error of 
30 seconds in this angle and the tab- 
ular difference for 30 seconds for the 
sine and cosine of this angle in four- 
place tables is o.oooi; therefore use 
four-place tables. In this case it is 
evident that the 0.02 on the hypo- 
tenuse distance is of no value what- 
ever in determining the length of the 
other two sides a and 6, that the 0.6 
being the fourth significant figure 

should be retained, and that the resulting length of a or 6 will not 
be reliable to more than four significant figures. 

log 773.6 = 2.8885 log 773.6 = 2.8885 

log cos 44''i2' = 9.8555 log sin 44^12' = 9.8433 

log a = 2.7440 log b = 2.7318 

a = 554.6 b = 539.3 

If it is assumed, however, that the angle B is measured by 
repetition and found to be 44°i2'25'^ the error in the original 
angle then was about 25^^. By using the same value for the hypo- 




326 COMPUTATIONS [Chap. XIL 

tennse (773.6) and six-place tables to secure greater precision the 
value of a is 554.5 and of b 539.4. Comparing these results with 
those obtained above will give a good idea of the error in length 
of these lines due to reading the angle to the nearest minute only 
and also a proper conception of the fallacy of computing with 
tables of more than four places when the angles are read to the 
nearest minute only. The difference between the values of a 
and b obtained by use of the angle 44° 12' and similar results by 
use of 44° 12' 25^^ is due entirely to the 25^^ and not to the fact that 
four-place tables were used in the former case and six-place 
tables in the latter, for in both cases the result has been obtained 
to four significant figures only. 

It is also evident that when the angle B was measured to the 
nearest minute it was inconsistent to measure the hypotenuse 
closer than to the nearest tenth of a foot. But if angle B was 
measured to the nearest 10 seconds the line AB should have been 
measured to the nearest hundredth. It should not, however, 
be assumed that in all cases where 'angles are only measured to 
the nearest minute the sides should be recorded to tenths of a 
foot. It is the percentage error in the measurement of the sides 
which must be the same as the percentage error in the angles. 
If the sides are very short, they should be measured to hun- 
dredths of a foot to be consistent with angles to the nearest min- 
ute. In general, when the angles are read to nearest minute 
only, the sides should be measured to four significant figures; 
with angle to nearest 10 seconds they should be measured to 
five significant figures; and with angles measured to i second the 
sides should be measured to six significant figures. All the sides 
of a triangle of considerable size might be measured to hun- 
dredths of a foot, the angles being recorded to the nearest minute 
only, and the distances used for the computations, the angles 
serving merely as checks; this, of course, is practicable at times. 
353. In Fig. 144 the angle is measured to the nearest minute, 

but the distance is measured to him- 
dredths of a foot. In this case we 
are to determine the length of a long 
line from a short one and the error 
in the short Une is therefore miilti- 




LOGARITHMIC OR NATURAL FUNCTIONS 327 

plied several times. The same degree of precision should be 
secured in the measured line BC as is desired in the computed 
lines AC or ABy which, it is assumed in this case, is required 
to four significant figures. In order that the measurements of 
line BC and angle A may be consistent with the precision of 
the required result, BC should be taken to the nearest hundredth 
of a foot and angle A to the nearest minute. In this computa- 
tion four- place tables should be used and the value obtained for 
AC ox AB should be recorded only to four significant figures. 

log 12.34 = 1. 0913 
log tan 3°56' = 8.8373 
log AC = 2.2540 
AC = 179.5 

If i4C is desired to the nearest hundredth of a foot the angle 
A might be determined closely by repetition, but this will not 
give the length ACio the nearest hundredth unless BC has been 
measured closer than to the nearest hundredth; for, suppose 
there is an error of 0.005 ft. in the measurement of 5C, then the 
line AC being about 15 times as long as BC will have an error of 
0.075 ft- ^^ matter how exact the angle at A may be measured. 
In other words, if -4 C is desired correct to five significant figures 
BC should contain five significant figures. Evidently the prac- 
tical way of obtaining an exact value for the inaccessible dis- 
tance i4C is to measure AB io the nearest hundredth, and to 
compute AC from AB and BC, using the angle at -4 as a check 
on the measured distances. In both of the above examples it 
is assumed that the 90*^ is exact. 

354. LOGARITHMIC OR NATURAL FUNCTIONS. — The ques- 
tion as to whether logarithmic or natural functions shall be 
used will depend upon the computation in hand. Many sur- 
veyors have become so accustomed to using naturals that they 
will often use them when logarithms would require less work 
and offer fewer opportunities for mistakes. Each method has 
its proper place, and the computer must decide which will be the 
better in any given case. The use of logarithms saves consid- 
erable time spent in actual computation because the process is 



328 



COMPUTATIONS 



[Chap. XIL 



simpler, but, on the other hand, looking up the logarithms con- 
sumes time. The result is in many cases, however, a saving of 
time over that required to do the arithmetical work of multi- 
plying or dividing. While the multiplication of two numbers 
of three or four digits each can possibly be done directly more 
quickly than by logarithms, still it takes more mental effort and 
there is more opportunity for making mistakes ; but in case sev- 
eral such multiplications are to be made logarithms are almost 
always preferable. Furthermore when there are several multi- 
plications of the same number logarithms will save time since the 
logarithm of this common number has to be taken from the table 
but once. Frequently, however, the computation is so simple 
that the use of logarithms would be almost absurd, e.g., the multi- 
plication of any number by a simple number like 20, 25, 150, or 
500. If a function of an angle is to be multiplied or divided by 

any such number the natural 
function should of course be 
used. 

355. SHORTCUTS.— The so- 
lution of a right triangle, when 
one of the angles is small, involv- 
ing the use of the cosine of this 
small angle, can often be more 
easily obtained by the use of the 
versed sine or external secant of 
Fig. 146. the angle. In Fig. 145 




Vers A - 



CD 
AB 



ExsecA«^ 



AB = 207.42 
A = f 32' 
AC =207.42 cos 7^32' 
But AC = AB - CD 

= 207.42 — 207.42 vers ^^ 32' 
= 207.42— 207.42 X 0.00863 
(207.42 X 0.00863 = 1.79. by slide rule.) 
= 207.42 — 1.79 
- 205.63 



(I) 
(2) 



SHORT CUTS 329 

Obviously, when the angle is quite small, the result of the multi- 
plication indicated in ( 2) can be taken from the table to the near- 
est hundredth of a foot with much less efiort than is required for 
the computation called for in (i). In fact, the computation in 
(2) can often be done more quickly by the use of natural num- 
bers than by logarithms, and in most cases the slide rule will 
give results suflSciently exact (Art. 359, p. 330). 

Had AC been given (205.63) and the angle -4, (7° 32') then 

COS 7*^ 32 
But AB =- AE + EB 

= 205.63 + 205.63 exsec 7° 32' 
= 205.63 + 205.63 X 0.00871 
(205.63 X 0.00871 = 1.79, by slide rule.) 
= 205.63 + 1.79 
= 207.42 

356. There are many "short cuts" in arithmetical work 
which are of great value to the computer, and the student should 
endeavor to learn the most common and simple ones. The fol- 
lowing are a few illustrations. 

247 X 100 24700 

247X25 = -^^— = -^ 

4 4 

68200 
682 X so = 

694 X 150 = 69400 -f 34700 

927 X 62.5 = 92700 X I 

672 X 1002.3 = 672000 -f 1344 -h 201.6 

547 X .9968 = 547 (i - .0032) = 547 - 5.47 X .32 

-^ == -^ (reducing minutes to decimals of a degree) 
60 o 

-^ = 8.43 X 8 
12.5 ^^ 



330 COMPUTATIONS [Chap. Xn. 

The student should cultivate the habit of performing men- 
tally as much of the work as can be done without fatigue, delay, 
or danger of mistakes. No hard and fast rule can be laid down 
in this matter, as some persons have more aptitude than others 
for work of this kind. Such subtractions as i8o**— 36^47' iS*' 
should always be performed mentally. Also in taking the co- 
logarithm of a number from a table of logarithms the result 
should be written down directly. 

357. ARRANGEMEIfT OF COMPUTATIONS. — All survey- 
ing computations should be kept in a special computation 
book. At the head of the page should appear the title of the 
work, the number and page of the field note-book from which 
the data are copied, the names of the computer and checker, 
and the date. The work should be arranged neatly and system- 
atically so that every part of the computations can be traced 
by any one who is familiar with such work. Where possible the 
work should be so arranged that numbers will have to be written 
but once. Each important value, each column, etc. should be 
labeled so that it can be readily found. 

358. CHECKS. — It is very important that all calculations 
should be checked, not merely at the end of the computation 
but also at as many intermediate steps as possible. In this way 
a great waste of time may be prevented and serious mistakes 
avoided. One good method of checking is to perform the oper- 
ations when possible by two independent methods, for example, 
by the use of logarithms and by natural functions. Very often 
two men do the computing, one man's work acting as a check 
on that of the other. The two may each work by the same or 
by different methods, and the results may be compared at inter- 
vals. Every part of the work should be done independently, from 
the copjring of data out of the npte-book to the final results. It 
is not uncommon to find two men computing the same area 
where only one of them looks up the logarithms. In case a mis- 
take is made in looking up the logarithms the results may check 
but both are wrong. The computer should also check his work 
roughly by estimating approximately what the result should be. 

359. SLIDE RULE. — A valuable aid in checking calculations is 
an instrument known as the slide rule, which enables the computer 



SUDE RULE 331 

to multiply and divide numbers by logarithms by a purely mechan- 
ical process. It is really the equivalent of a table of logarithms. 
It consists of a wooden rule, usually about 10 inches long, having 
a groove in one side in which runs a small wooden strip called the 
slide. On one face of the rule are placed two scales, A and J9, 
Fig. 146, one above and one below the slide which is indicated by 



i HilmiDinN I [i [iti[ i [iti[iiii| i in[mi|im[ I [l[i[i[iDD[t 



) l ]l]l] l ]l3l]l] l ]l[ll l l|lill[ll l j|llll[lMl|lll l [ I [ I [I [I [l[s 



Fig. 146. 

B and C. These are constructed by plotting logarithms of num- 
bers by subdividing a unit of some convenient length, say 10 
inches. For example, the log of i is o, so this is taken as the 
left end of the scale and the number i placed at this point. The 
log of 2, to three significant figures, is 0.301, and a line is placed 
therefore at a distance equal to ^^ of the 10 inches, or 3.01 
inches, and marked with the number 2. Similarly at 4.77 (log 
3 = 0.477) a line is marked 3. In this way the logarithms of 
other numbers are plotted. The space between i and 2 is 
subdivided by plotting log i.i, log 1.2, etc. The subdivision is 
continued until the spaces are as small as will admit of rapid 
and accurate reading of the scale. 

It is customary to make the spacing on the upper scale just 
half that on the lower, i.e., if 10 inches is chosen as the unit for 
the lower scale, then the unit for the upper scale will be 5 inches. 
Since the length of this upper sca\e is only half the length of the 
rule there ^re usually two scales exactly alike marked on the upper 
part of the rule, the right end of one coinciding with the left end 
of the other. 

On the slide are two scales, B and C, exact duplicates of those 
on the rule and so placed that when the end line of the scale B 
on the slide is placed opposite the end line of the scale A on the 
rule, every line on the slide is exactly opposite its corresponding 
line on the rule. A runner is usually attached to the rule for 
convenience in setting and reading the scales. This runner is 
a small metal slide which fits over the face of the rule in such 



332 COMPUTATIONS [Chap. Xfl. 

a way that it can be slid along the rule and set at any reading of 
the scale. It is usually provided with a fine line running cross- 
wise of the rule which is used in marking the exact setting. 

Multiplication or division of numbers is performed by add- 
ing or subtracting the scale distances corresponding to these 
numbers. The scale distance is the logarithm of the number. 
Adding two scale distances is, in effect, adding two logarithms, 
and the resulting scale distance is the logarithm of the number 
marked opposite on the scale. For example, if the left end of 
scale C, Fig. 147, is set opposite the number 2 of the scale Z), 
then opposite the number 3 on scale C, is found the product, 6, 
on scale D. The distances which have been added are those 
corresponding to log 2 and log 3 respectively. The sum of 
these distances is the distance corresponding to log 6. Division 
is performed by placing the divisor on scale C over the di\adend 
on scale D and reading the result, opposite the end of the scale 
C on the scale D. 

Fig. 147 shows the position of the scales for dividing 6 by 3. 



I 



1I 1 I i lll i l l l i l i f . l.l .i . i.l I I .I I |*ii . iilM.nrM.M | ir.nr^ / 



iimfiitiliiiir I. f .l.f.ifj.f. 



^ ■ j . j.j,j . j . jg.j . >tniiiiiiiliinii i i i C.i.iihi.|, ' v ' C'va'V'Ova'v^^^ 



Fig. 147. 

The scales A and B may be used in a like manner. It is evident 
that, by setting the runner on the result of one operation and 
then moving the slide so that one of its ends coincides with the 
runner setting, continued multiplication and division can be 
performed without the necessity of reading intermediate results. 
Scale D may be used in connection with scale A for obtain- 
ing squares or extracting square roots. Since the spaces on scale 
A are one-half those on scale D the number 4 on scale A is oppo- 
site number 2 on scale D, 9 is opposite 3, and so on, every num- 
ber on scale A being the square of the corresponding number on 
scale D. Other scales, generally log sines and log tangents, are 
placed on the reverse side of the slide, so that trigonometric cal- 
culations can also be performed with this instrument. Results 



REDUCING FIELD NOTES 333 

obtained with the ordinary 10 inch slide rule are usually correct 
to 3 significant figures, so that this slide rule is the equivalent 
of three-place logarithm tables. 

360. Thacher Slide Rule. — The Thacher slide rule consists 
of a cylinder about four inches in diameter and eighteen inches 
long woi king within a framework of triangular bars. On these 
bars is fastened a scale corresponding to the scale on an ordi- 
nary slide rule, and on the cylinder is marked another scale like 
that on the bars. The cylinder is the slide and the triangular 
bars form the rule. This rule is operated in a manner similar 
to the one explained above. Results can be obtained with it 
which are correct to four and usually to five significant figures. 

361. REDUCIIIG THE FIELD NOTES FOR COMPUTATIONS. 
— Before any of the computations are made the measure- 
ments taken in the field frequently have to be corrected on 
account of erroneous length of tape. This correction can usu- 
ally be made mentally when the distar ces are transcribed into 
the computation book. The errors in the angles are balanced 
by altering the value of those angles which were taken from short 
sights since the angular errors are most Ukely to occm: in these. 
In some cases, where it has been found desirable to take meas- 
urements on a slope, these distances are reduced to horizontal 
distances by multiplying them by the versed sine of the vertical 
angle and subtracting the result from the corrected slope dis- 
tance; the correction for error in the tape being made before this 
is done. Sometimes instead of a vertical angle the slope dis- 
tance and the difiFerence in elevation between the points are the 
data contained in the field notes. In this case the formula given 
in Art. 20, p. 13, should ordinarily be used. 

362. Curved boundary by offsets, — The offsets to the 
brook (Fig. 53, p. 104) were taken at regular intervals in one 
portion of the survey and in another portion offsets were taken 
at the points where the direction of the brook changes. The 
offsets which were taken at regular intervals give a series of 
trapezoids with equal altitudes the area of which can be obtained 
by one computation. Although there are several approximate 
rules for this computation the two most common are what are 
known as the Trapezoidal Rule and Simpson^ s One-Third Rule. 



334 COMPUTATIONS [Chap. XII. 

363. Trapezoidal Rule. — If the figure is considered as made 
up of a series of trapezoids their area can be found by the follow- 
ing rule: — 



Area = d (^ + 2 A + ^) 



where d = common distance between offsets, 
Ae and h\ = end offsets of the series of trapezoids, 
and 2 A = sum of the intermediate offsets. 



364. Simpson's One-Third Rule. — In the development of 
this formula the curved line is assumed to be a parabolic curve. 
It is claimed by some that this afiFords results more nearly correct 
than the Trapezoidal Rule, although for most problems of this 
kind, where the offsets at best can give but an approximate 
location of the boundary, frequently a brook or crooked wall the 
center of which must be estimated, it is quite probable that the 
Trapezoidal Rule is sufficiently exact. Simpson's One-Third 
Rule is as follows: — 

Area = - (A. + 2S Aodd + 4S Aeren + A'e) 

where d = common distance between offsets, 
Ae and h\ = end offsets of the series, 
2 2 hodd = twice the sum of all the odd offsets 

(the 3d, 5th, 7th, etc., from the end) 
4 2 Aevcn = twice the sum of all the even offsets 

(the 2d, 4th, 6th, etc., from the end). 



For this rule to apply there must be an even number of 
trapezoids; if there is an odd number, an even number of them 
may be computed by this rule and the extra trapezoid must be 
computed separately. Or, if there is a triangle or trapezoid at 
the end of this series, which has a base greater or less than d, 
it must also be computed separately. 



TRAPEZOIDAL RULE 



335 



Fig. 148 shows the computation of a series by both methods 
and also the computation of several trapezoids and triangles at 
the ends of the series. The data are taken from the field notes 
in Fig. 53, p. 104. 




I 

z 



B K V 
-ByStmpsoi7''S Offe^/ref fii/M 

r 
m 

38.S 

3cs'.5* 



0/ Jftfp^zo/cfa/ /fuM 

if 



6jO 

AS 

4.0 



^f^MS^niit 









\0I9 

230 

/03 



383 
/03A- 



&6 

f? 

pArg 

/6,ZK.9Z7s/3j^^Z 
aA>.7xZ7Za Z9/ ¥S/XZU 

ZSowxB*^ 



Fig. 148. 



336 COMPUTATIONS [Chap. XII. 

365- STRAIGHTENING CROOKED BOUNDARY LINES. — In 
Fig. 149, AEFGH represents a curved boundary between two 



Fig. 149. Straightening a Crooked Boundary. 

tracts of land, and it is desired to run a line from A so as to make 
the boundary a straight line and to leave each tract of the same 
area as before. 

The trial line ABh first run, and the distance AB^ the angles 
at A and 5, and the necessary offsets to the curved boundary are 
measured in the field. Then the areas of the property between 
this trial line and the curved line are computed as explained in 
the previous articles. The sum of the fractional areas on one 
side of the trial line and the sum of the areas on the other side of 
it should be equal. If not made so by the trial line, the difference 
between these sums is the area of a correction triangle ABC 
which must be taken from one tract and added to the other. 
The area and the base AB being known the altitude dC can tc 
computed. Then in the triangle ABC, the lines BC and AC 
and the angle at A are calculated; and the line AC is staked out, 
its calculated length being checked by measuring the line AC 
in the field and the angle at A being checked by the measured 
distance BC. 

366. AREA BY TRIANGLES. — If the field has been sur- 
veyed by setting the transit in the middle of the field and taking 
angles between the corners (Art. 138, p. 105), the areas of the 
triangles may be foimd by the trigonometric formula: 

Area = J a 6 sin C, 

where C is the angle included between the sides a and h. 

If all three sides of any of the triangles have been measured 



STRAIGHTENING CROOKED BOUNDARY 



337 



or if the fidd has been surveyed with the tape alone (Art. 139, 
p. 106), the area of the triangles can be found by the trigono- 
metric formula: — 



Area = V5 (5 - a ) (^ - 6 ) ( j — c ) 

a + b + c 



where a, 6, and c are the sides and 5 = - 






367. AREA OF A QUADRILATERAL BY TRIANGLES.— 

Most city lots have four sides, and while the Double Meridian 
Distance Method (Art. 384, p. 352) is often employed in com- 
puting their areas, it is not at all uncommon in computing such 
quadrilateral lots to divide them into triangles, checking the field- 
work and computations, and computing the areas by triangles. 

In Fig. 150, A BCD represents an ordinary city lot in which 
all the sides and angles have 
been measured. It is evident 
that the diagonal BC can be 
computed either from BDy 
CD, and the angle Z), or from 
AB, AC, and the angle A. 
These two determinations of 
BC should check each other. 
Similariy two independent de- 
terminations oi AD can be 
found. These evidently check 
all the fieldwork and calcula- 
tions as far as they have gone. 
In computing these triangles 
the best way is to resolve all 
the work into right triangle 
calculations, as suggested by 
the dotted lines on the figure. 
Not only is this method more simple than to use the oblique 
triangle formulas, but it gives at the same time altitude distances 
which are useful in computing the area of the lot. The area 
can be obtained by calculating the area of one pair of triangles 
and readily checked by calculating the other pair. 





338 COMPUTATIONS [Chap. XII. 

368. AREA OF CURVBD 
CORNER LOT. — In Fig. 151, 
ABFHGDE is the boundary 
of a corner lot, all the angles 
and distances of which have 
been determined in the field. The 
area of ABCDE can be easily 
computed by the method ex- 
plained in Art. 384, p. 352. 
Then the area of FCGH must be 
subtracted from the traverse 
^^®- ^^^- area. The angle / is known and 

the radius KF of the curve is given or can be computed from data 
such as CH or CF obtained in the field (Art. 257, p. 233), 

KFHG = EmjiHK ^ P X O.OI74533* X HK ^^^^ 

2 2 

VI, p. 506.) 

KFCG ^ FC X FK 
FCGH = KFCG - KFHG 

The area of FCGH could have been calculated by comput- 
ing the area of the triangle FCG and then subtracting the area of 
the segment FHG from it. The area of this segment, however, 
cannot be calculated accurately by any short formula. An 
approximate formula for the area of a segment is 

2 
Area of Circular Segment = - MC (approximate), where M is 

3 
the middle ordinate and C is the chord length. 

M = 

Expressed in terms of C and R, 

Area of Circular Segment = (approximately). 

I2/v 

* The length of the arc of curve whose radius is i and whose central angle 
is I® is 0.0174533, which will give results to six significant figures, provided /and 
/^ are correct to six significant figures. 

t In Fig, 152, OB =» Radius of circular curve. 

C/f «= Middle Ordinate for chord AB, 
CD is drawn tangent to the curve. 



M = —— ' (approximately). 



AREA OF CURVED CORNER LOT 339 

These formulas are fairly accurate when M is very small a& 
compared with C. They are most useful, however, as a check 
on computations made by the prec ding method. 

369. ROUGH Checks on areas. — if the traverse has 
been plotted to scale, it can be easily divided into simple figures 
such as rectangles or triangles, their dimensions scaled from the 
plan, and their areas computed, thereby giving an independent 
rough check on the area. 

A piece of tracing cloth divided into small squares can be 
placed over the plan of the traverse and the number of squares 
counted and the fractional parts estimated, generally to tenths 
of a square, by inspection. Then the area of one square being 
known an approximate area of the traverse may be obtained. 

370. Plardme er. — One of the commonest ways of checking 
the area of a traverse is to obtain its area by means of an instru- 

DB « Tangent Offset for chord CB. 
OE is drawn perpendicular to CB, 
In the two similar triangles OEB and CBD^ 
DBiCB^BEi OB 

DB',CB ^ — lOB 

2 

DB^^-^ 
2 OB 

Offset from Tangent - J^^^^ 




a X Sadins 
But DB « CH, and AB ^zxCB (approximately) (1) 

\ 2 I AB* 

•• ^^" 2 OB " 8aS (approximately) 

Middle Ordinate - 3 ^^^^1^, (approximately) (2) 

The following will give some idea of the accuracy of this formula: 
When radius — 20 ft. and chord » 10 ft, M -s 0.625, (correct value is 0.635). 
When radius — 100 ft and chord » 25 ft., M » 0.781, (correct value is 0.784). 
When radius — 100 ft. and chord ■■ 100 ft., M ■■ 12.500, (correctvalue is 13.397). 
When radius ^ 1000 ft. and chord «» 100 ft., M a 1.250, (correct value is I.251). 

It is evident from the above that this formula will not give accurate results 
when the chord is large in comparison with the radius. 



34^ COMPUTATIONS [Chap. XIL 

ment called the planimeier^ which is in principle a mechanical 
integrator. It is a small instrument consisting of an arm, carry- 
ing a tracing point, which is fastened to the frame of the instru- 
ment; the arm can be adjusted to any desired length. The 
frame touches the paper at only two points; one, the anchor 
point, and the other, the circumference of a small wheel which is 
free to revolve. On the rim of this wheel is a scale which is read 
by means of a small vernier. The length of the arm can he 
regulated by setting it at the proper reading on a scale which is 
marked on the arm, so that a unit on the wheel scale will repre- 
sent any desired unit area such as a square inch or a square 
centimeter. 

In using the instrument the anchor point is set at some con- 
venient position on the drawing outside of the area to be measured 
and then the tracing point is run around the perimeter of the area 
to be determined. The reading on the wheel is recorded when 
the tracer is at the starting point. The tracer, in passing around 
the perimeter, should be kept as closely as possible on the bound- 
ary line and should return exactly to the starting point. Then 
the scale is again read, and the difference between the two read- 
ings is the area which has been traced out, expressed in some unit 
depending on the length of the arm. The result can be easily 
transposed into the unit of the scale of the map. 

Usually the settings for the scale on the arm are furnished by 
the maker for various units of area. It is safer to test this setting 
by running the instrument around a known area, such as 4 square 
inches and determining the interval passed over by the wheel by 
making several tests and by setting the anchor point at different 
positions. This interval divided by 4 will be the value of one 
square inch of plan area and this is equivalent to a certain number 
of square feet of surface, depending upon the scale of the map. 
It is important that the sides of the trial square should be laid off 
so that they agree with the present scale of the map which, owing 
to swelling or shrinking of the paper, is frequently not quite the 
same as when it was first drawn (Art. 479, p. 428).* 

♦ When areas are desired from U. S. Geological Survey maps on which are 
sho^-n parallels of latitude and longitude it is best to refer all planimetered areas 
to the areas of a quadrilateral, say, i^ on a side. The area of such quadrilateral 



DEFLECTION ANGLES AND CHORDS 



341 



371. DEFLECnON ANGLES AND CHORDS FOR A CIRCULAR 
CURVE. — The computations shown in Fig. 153 refer to the notes 
in Fig. 104, p. 237. In the discussion of the simple curve as 



6/VEN:-/f'20O, curve to Right, I'Si^-JS'-iO^^PC, ^/e^TZM 

Width of S treet 70^ 

T^/rtBfn. 2^^47*40" 



iZO0X.^6330«Se.66T 
Si* ^.B90H79 
SS'^.Oio/8// 






.9003360 K2oo^/eo.oaLc 

Def/ection Arrg/es, 

DefkcHanJL for Soft ^^ x ^S*^7Uo''^^xZS:79^ 

Log 7zssin:-^tto496 

DmfiecHonL for3ao6H:^33SiLyjkH.f6rSaf Log laoM '^^^ 

eo 



Log.60/6 • 




3/: 

.4*ie'3/'*defl.3a09it, 



43: 
T09'43''deff.Mff. 

/d^Z2MsiZi'Z9'/0 

4-ta-3o 
RT ia^SlS^^W-^r-^CCheck^ 



Chords 



SOthArc. 

Sln7"09'4o"^JZ467 

44?0 



i/Z47xZ^ s 



49868 Cent^Chd. 
&7Z7- 
S8SB Uft Chd, 
41.14 /fight Chd. 



30,08 fk Arc 
^in4^ie'3o^J07StZ. 

3a048(MChd. 
\0JS/XZK3Ss> S.2S7^ 

3S.3/ UftChd, 
24.79 Right Chd. 



Fig. 153. 



can be taken from a publication entitled Geological Tables and Formulas, by 
S. S. Gannett, Bulletin No. 232, U. S. Geological Survey, and by simple proper* 
tion the desired area found. 



34^ COMPUTATIONS [Chap. XIL 

applied to city surveying (Art. 259, p. 234) will be found the for- 
mulas which have been used in the computations in Fig. 153. 
The length of the curve L^ is found by taking from Table VI, 
("Lengths of Circular Arcs: Radius — i "), the length of an arc 
for 51^, for 35', and for 20^ successively and adding them, which 
gives the arc of a curve whose radius is i and whose central angle 
is 51^ 35' 20^. This is then multiplied by the radius (200) which 
gives the value of £c, which is added to the station of the P.C. to 
obtain the station of the P.T. 

372. COMPUTATION OF OBSERVATIONS. — The computations 
relating to observations for meridian and latitude will be found 
in Chapter VII. 

COMPUTATION OF VOLUME. 

373. BORROW-PITS.* — Fig. 154 is a plan of a portion 
of a borrow-pit, at the corners of which the depth of excavation 
is marked in feet and tenths. Each of the regular sections of 
earthwork is a truncated rectangular prism whose volume is 
equal to the average of the four comer heights multiplied by the 
area of the cross-section, or expressed as a formula. 

Volume Truncated Rectangular Prism = .4 X ^t + h» +^8+^4 

4 
where A is the area of the cross-section and A„ A„ A,, and h^ are 
the comer heights. 

For a truncated triangular prism such as dbcy using the same 
notation, 

h + h •{• h 
Volume Truncated Triangular Prism = i4 X ^• 

In computing a trapezoidal prism, such ssfdhg, the trapezoid 
is subdivided into a rectangle /sA^ and a triangle /de; or for jhds, 
into two triangles by diagonal lines, as jhs and kds and their 
volumes may be computed by the above formula. 

When there are several prisms with the same cross-section, as 
shown in Fig. 154, these rectangular prisms can be computed as 
one solid by assembling them as follows : — multiply each comer 

* For a complete discussion of the computation of Borrow-Pits see Railroad 
Curves and Earthwork by Professor C. F. Allen, published by Spon & Chamber- 
lain, New York. 



BORROW nrs 



343 



height by the number of rectangular prisms in which it occurs 
and then add these results and divide by 4. This is then multi- 
plied by the area of the cross-section of one prism. For example, 
in Fig. 154, the quantity bounded by amnrsja can be foimd by 



m 



^ 



^ 



•P 




Fig. 154. Plan of Portion of a Borrow-Pit. 



one computation because it is composed of a series of prisms 
having the same cross-section. Li the summation of the heights, 
those at a, m, n, f , and s are taken but once, those at such points 
as c, g, A, etc. are multiplied by 2, at j the height is multiplied by 
3, and at such points as k it is mutliplied by 4. 

Where the excavation is completed to a certain level, as in a 
cellar, it is a special case of above. The area of the cellar can be 



344 COMPUTATIONS [Chap. XIL 

divided into rectangles, their comer heights taken, and from these 
the volume can be computed. 

374. VOLUME OF PRISMOID. — The data obtained from 
field notes are usually in the form of cross-sections which are 
taken at right angles to some general line of the construction, 
thereby dividing the earthwork into prismoidal solids with their 
bases parallel and their sides either plane or warped surfaces. 
The bases of the solids are the cross-sections which are obtained 
by taking sections of trench excavation or of road construction 
(Figs. 91 and 92, p. 208). 

' 375. End Area Formula. — The simplest method of com- 
puting the volume of a prismoidal solid is to average the areas of 
the two bases and multiply by the distance between them, which, 
expressed as a formula, is 

V = di_±A« X / {End Area Formula) 

2 

in which A^ and A^ are the areas of the two end bases and / is 

the distance between them. This method is used to a very great 

extent throughout the country, although it does not give suflS- 

ciently accurate results for certain classes of work. 

376. Prismoidal Formula. — The correct volume of a pris- 
moid is expressed by the Prismoidal Formula: 

Volume of Prismoid = -(i4, + 4i4n» +-4,) 

in which /is the distance between the two bases, A^ and -4,; and 
ilm is the '* middle area,^^ i.e., the area half-way between the two 
bases, which is obtained by averaging the corresponding dimen- 
sions of the two end areas, A^ and -4,; it should not be taken as 
the mean of A^ and A^. 

377. The end areas can easily be computed by dividing them 
into triangles as shown in Fig. 155, the area of which can be 
found readily from the dimensions given in the field notes. 




VOLUME OF PRISMOID 345 

20.0 2I.S 

Notes of section: . . + 4-0 ", — 

+60 +1.0 

Area = 4 X (21.5 + 29) , 20 X (i + 6 ) 
2 2 

= 2 X 50.5 + 10 X 7 = 171. 

It is also the custom with some surveyors to plot each section 
carefully to scale and to obtain its area by use of the planimeter 
(Art. 370, p. 339). This is probably the most practical method 
when the sections are very irregular since the field work does 
not warrant the use of very accurate methods. 

There are several other methods employed in computing 
earthwork but the above are by far the most common. 

Several sets of Earthwork Tables and Diagrams have been 
published which reduce the work of computation very materi- 
ally. 

378. ESTIMATES FOR GRADING.— Estimates for grad- 
ing may be conveniently made by means of a topographic map. 
On this map will appear the contours of the original surface. 
The contours representing the finished surface are also sketched 
upon the map, and the smaller the interval between the contours 
the more accurate will be the result. In Fig. 156 the full lines 
represent the contours of the original surface which is to be al- 
tered so that when the necessary cutting and filling has been done 
the new surface will have the appearance indicated by the dash 
contours. At contour 20 and at contour 25 no grading is to be 
done. On the plan, first sketch the lines ABCDEF and 
AGHIJB which are lines of "no cut '* and "no fill,'* i.e., lines 
which enclose areas that are either to be excavated or filled. The 
amount of excavation and embankment must be computed sepa- 
rately. In sketching such lines the lines AB^ ED, and Hi, as 
will be seen, follow the intersection of the original contours with 
the new ones, since at these points there is no cut or fill. There 
are no direct data on the plan which define where the earthwork 
ends at C but the assumption is here made that the fill will run 
out to meet the original surface at about the next contour at C. 
In this example the fJl must run out somewhere between the 
24-ft. contom: and the 25-ft. contour, for if it ran beyond the 25-ft. 



34fi 



COMPUTATIONS 



[Chap. Xn. 



contour there would be another new 25-ft. contour shown on 
the plan. Therefore the line BCD has been sketched to repre- 
sent the limits of the fill in that vicinity; similarly EFA^ AGH, 
and IJB have been sketched. 

There are three general methods of computing the earth- 
work from the data given on the plan; (i) by computing directly 
the amount of cut or fill between successive contours, (2) by 




25 -."-'t '^'^ 




Scale 



Fig. 156. 

assuming a horizontal plane below the lowest part of the earth- 
work and computing the volume of the earth between this plane 
and the original surface, then computing the volume between 
the same plane and the finished surface; the difference between 
these two volumes will be the amount of earthwork, or j) by 
drawing on the plan a line of no cut or fill, a line representing, 



VOLUMES FROM CONTOURS S47 

say, 5 ft. cut or fill, a line representing lo ft. cut or fill and so om 
Then compute the volume between these successive 5-ft. 
layers. 

379. (i ) Referring to Fig. 154 and applying the first method, 
the volume of the solid AMPE is that of a solid having two paral- 
lel end planes AKEL (a plane at elevation 21) being the lower, 
and MNPO (a plane at elevation 22) being the upper pi ne. 
The altitude between these two end planes will be the difference 
in elevation between 21 and 22, or will be i ft. 

The areas of the horizontal planes AKEL, MNPO, QRST, 
and BUDV may be obtained by planimeter (Art. 370, p. 339) or 
otherwise, and the volume of the solid AKEL-MNPO may be 
obta.ned by the End Area Method (Art. 375, p. 344), its altitude 
being i ft. If it is desired to obtain the volume by the use of 
the Prismoidal Formula the volume of the solid AKEL-QRST 
may be found by using AKEL as one base, QRST as the other, 
and MNOP as the middle area, the altitude, or length, of the 
solid being the difference between 21 and 23, or 2 ft. The solid 
AKEL-F may be considered to be a pyramid with a base AKEL 
and an altitude equal to the vertical distance between the con- 
tour 21 and the point F which is in this case on contour 20, or 
a vertical distance here of i ft. 

Example. 

In Fig. 154 the amount of fiU on the area ABCDEFrs computed below. 
Area AELF — 900 sq. ft. 900 x } — 300 cu. ft. (Pyramid) 

" MNPO - 1000 _rlr_Z X I - 9SO. 

** QRST - 1020 looo -f 1020 ^ J _ j^j^ 

- BUVD - 680 '^^^-^ ^ X 1 - 850. 

2 

680 X § » 23a (Pyrandd) 

3) 3340- cu. ft. 
9)1113 

124. cu. yds. Total FiU. 

380, (2) Referring again to Fig. 156 and applying the sec- 
ond method, the area of ABCDEF is found (by planimeter) ; this 
is the area of a plane at, say, elevation 20, since none of the fill 



348 COMPUTATIONS [Chap. XIL 

extends below contour 20. Then the area of A BCD EL is found, 
which is the area of the plane cutting the original ground at ele- 
vation 21. Similarly the areas of MBCDPO, QBCDST, and 
BCDV are found. The volume of the solids between these 
planes may be computed by the End Area Method or by use of 
the Prismoidal Formula, in which case every other contour 
plane is used as a middle area as explained in the preceding 
paragraph. The volume of solid whose base is BCDV is a pyr- 
amid whose altitude is the vertical distance between the 24-ft. 
contour and point C, which in this case is i ft. 

By the same general method the areas of ABCDEK, 
MBCDPN, etc., which refer to the new surface of the ground, 
may be obtained, and the volume of the solids between succes- 
sive contour planes computed. The difference between this 
quantity and the quantity between a plane at elevation 20 and 
the original surface will give the amount of fill. 

While in this particular problem the first method is the 
shorter, still there are cases where the second method will be 
somewhat simpler. It is particularly useful when the actual 
amount of cut or fill is not desired but when it is required to know 
if the proposed alterations will require more or less earth than 
can be easily obtained on the premises and, if so, about how 
much the excess will be. In this case the portions of cut and 
fill will not have to be computed separately. A line is drawn 
around the limits of the entire area where the grading is to be 
done, the volume between an assumed plane and the original 
surface is found, and then the volume between the same plane 
and the proposed surface. The difference between the two 
values will give the amount of excess of earthwork. 

381. (3) Fig. 157 illustrates a third method of computing 
earthwork from the data given on a topographic map. The 
original contours are shov n in full lines and the contours of the 
proposed surface in dash Unes. Through the intersection of 
the new contours with the ori£inal ones is dra^vn the line of "no 
cut " (zero line), the line vhere the cut is just 5 ft. (marked 5), 
the line of 10 ft. cut (marked 10), etc. These dotted curves 
enclose areas which are the horizontal projections of irregular 
surfaces which are parallel to the final surface and at 5 ft., 10 ft., 



VOLUMES FROM CONTOURS 



349 




Ftg. 167. 



15 ft., etc., above the final surface. The solids included oe- 
tween these 5 ft. irregular surfaces are layers of earth each 5 ft. 
thick, and their volumes may be computed by either the End 
Area Method or by the Prismoidal Formula as explained in the 
preceding methods. The areas of these horizontal projections 
are obtained from the map and the vertical dimensions of the 
solids are the contour intervals. 

382. ROUGH ESTIMATES. — Rough estimates of the quantity 
of earthwork are often required for preliminary estimates of the 
cost of construction or for monthly estimates of the amount of 
work done. For preliminary estimates of road construction, very 



3 JO COMPUTATIONS [Chap. XH. 

frequently the notes of alignment and the profile of the center line 
are the only information at hand. From this profile the center 
cuts or fills can be obtained, and the cross-sections can be as- 
sumed to be level sections (Art. 232, p. 209) and computed by the 
End Area Method. The slight errors resulting will be corrected 
in the final estimate. 

In obtaining the required data from which to make an ap- 
proximate estimate of the quantity of earthwork, the engineer 
has an opportunity to exercise his judgment to an unusual degree. 
Rough estimates do not, as a rule, call for a large amount of field- 
work. It is important that as few measurements as possible 
should be taken and that these should also be at the proper 
places to give complete data and to allow simple computations. 
Too often engineers, as soon as they arrive on the work and 
before making a study of their problems, begin to take measure- 
ments, consequently they return to the office after hours of 
hard work with a mass of figures from which it will take several 
more hours to compute the quantities. Whereas, a few mo- 
ments' thought given to the choosing of the proper measure- 
ments to be taken in the field would give data which could be 
computed in a few moments by use of the slide rule, affording 
results sufficiently accurate for rough estimates. 



PROBLEMS. 

1. A series of perpendtcnlar offsets are taken from a straight line to a 
carved boundary line. The offsets are 1 5 ft. apart and were taken in the fol- 
lowing order: 6.8, 7.2, 4.6, 5.7, 7.1, 6.3, and 6.8. 

(a) Find the area between the straight and curved lines by the Trapezoidal 
Rule. 

(b) Find the same area by Simpson's One-Third Rule. 

2. It is desired to substitute for a curved boundary line a straight line which 
shall part off the same areas as the curved line. A trial straight line AB has 
been run; its bearing is S 10° 15' W, its length is 418.5 ft., and point ^ is on 
a boundary line CD which has a bearing S 80^ W. The sum of the areas between 
the trial line and the crooked boundary on the easterly side is 2657. ft.; on the 
westerly side it is 789 r. ft. It is required to determine the distance BX along 
CD such that AX shall be the strsught boundary line desired. Also find the 
length of the line AX, 



ROUGH ESTIMATES 35 1 

3. In the quadrilateral ACBD the distances and angles which were taken 
in the field are as follows: 

A ^—50.63 A B C— 105** 39' 00" 

B C- 163.78 B A Z>-89° zf 30" 

CZ>-93.8o 
D A^ 160.24 
/> ^-167.73 

Check the fieldwork by computations, and figure the area of the quadrilateral 
by using right triangles entirely. 

4. Two street lines intersect at an angle (deflection angle) of 48® \f yf\ 
The comer lot is rounded off by a drcular curve of 40-ft. radius. 

(a) Find the length of this curve to the nearest j^ ft. 

(b) Find the area of the land included between the curve and the two tan- 
gents to the curve (the two street lines produced). 

5. Find the quantity in cubic yards, in the borrow-pit shown in Fig. 154; the 
squares are 25 ft. on a side, and the line ast\& straight. 

6. At station 6 a rectangular trench was measured and found to be 3 ft 
wide and 4 ft. deep. At station 6+70 it was found to be 3.2 ft. wide and 8.6 ft. 
deep. 

(a) Find by use of the Prismoidal Formula the quantity of eaitliwork be- 
tween stations 6 and 6+70. Result in cubic yards. 

(b) Find the volume of the same by End Area Method. 

7. The following is a set of notes of the earthwork of a road embankment. 

« -J7f +^, JM 

+ 8.0 +5.6 

,, + 60 -3?:? 2^ ^^.o 2^ 2^ 
+ 10.0 +4.5 +7.5 +6.0 

Sta.1, 21± +6.0 2^1 

+ 40 +7.2 

The base of the road is 30 ft. and the slopes are i) to i. 

Tlnd by the End Area Method the quantity of earthwork from Sta. 11 to la. Re- 
sult in cubic yaxds. 



CHAPTER XIII. 

AREA BY DOUBLB MERIDIAN DISTANCES.— COORDINATES. 

383. COMPUTATION OF AREA. — The computation of the 
area of any piece of property which has been surveyed as a 
traverse will in general consist of (i) the computation of the area 
enclosed by the traverse and (2), where the traverse does not 
follow the property line, the computation of fractional areas to be 
added to or subtracted from the area of the traverse as the case 
may be. 

384. COMPUTATION OF AREA BY DOUBLE MERIDIAN 
DISTANCE METHOD. — In the field notes the length and the 
bearing of each line of the traverse are recorded. To obtain the 
area enclosed the points of the survey are referred to a system 
of rectangular coordinates. In Fig. 158 the coordinate axes 
chosen are the magnetic meridian through the most westerly 
point Fy and a line through F at right angles to the meridian. In 
compass surveys it is convenient to use the magnetic meridian 
for one of the axes; in transit surveys the true meridian is often 
used when its direction is known, but any arbitrary line may be 
used as an axis and some convenience results from choosing one 
of the lines of the survey as one of the axes. 

In computing the area, first find the length of the projection 
of each line on each of the coordinate axes, or in other words, 
find the northing or southing and the easting or westing of each 
line, or course, of the traverse. The projection of any line on the 
meridian is called its difference of laiittide or simply its latitude. 
The projection of a line on the other axis is called its difference 
of departure, or simply its departure,^ In Fig. 158 the latitude 
of FA is Fq; the departure of FA is qA. The latitude and de- 
parture of each course are computed by solving the right triangle 
formed by drawing lines through the extremities of this course 

* Some authors use the terms latitude difference and longitude difference. 

352 



DOUBLE MERIDIAN DISTANCES 



353 




Fig. 168. 



and parallel to the coordinate axes. It is evident from the 
figure that 

Latitude = Distance x cos Bearing, 
and Departure = Distance x sin Bearing. 

Latitu'les are called North or South and departures Edst or Westy 
depending upon the direction of the course as shown by its letters, 
e.g., if the 1 earing is N 30** E thi ; course has a North latitude 
and n East departure. North latitudes and East departures are 
considered as positive (+), South latitudes and West departures 



354 



COMPUTATIONS 



[Chap. xm. 



as negative ( — ). In the figure the courses are assumed to run 
from Fto A, from A to 5, etc. 

385. After all of the latitudes and departures have been 
computed (supposing for the present that the traverse is a closed 




Fig. 158 



figure) proceed to find the areas of all the trapezoids or trian- 
gles, such as DErSf EFr, etc., formed by (i) the courses, (2) 
their projections on the meridian, and (3) the perpendiculars 



DOUBLE MERIDIAN DISTANCES 355 

from the extremities of the courses to the meridian. It is evident 
in the figure shown that the area of the field is equal to 

{AB wq 4- BCtw 4- CDst) - {DErs + EFr + FAq)y 

that is, in this figure the sum of all the areas determined by lines 
running northward minus the sum of all the areas determined by 
lines running southward. These are known as north areas and 
south areas. ^ In computing the areas of these trapezoids it is 
convenient as well as customary to find the double areas and 
divide the final result by 2 instead of dividing by 2 in figuring 
each trapezoid. The area of any trapezoid equals the average 
distance of the extremities of the course from the meridian, 
multiplied by the length of the projection of the course on the 
meridian. This average distance of the ends of the line from 
the meridian is known as the meridian distance of the course, 
i.e., the meridian distance of the middle point of the course. In 
computing the double areas, tvrice this distance, or the double 
meridian distance (D.M.D.), is used, which is equal to the sum 
of the distances of the ends of the course from the meridian. 
In arranging the data for computing the double meridian dis- 
tances, the courses must be tabulated in consecutive order around 
the traverse, whether they were so taken in the field or not. The 
D.M.D. of the course FA is qA which is the departure of the 
course FA. The D.M.D. of AB is qA ^ wB == qA -\- qA -¥ 
zBy i.e., the D.M.D. of course FA + the dep. of FA + the dep. 
of AB. The D.M.D. of BC == tC + wB ^ iv + vC + qA + 
zB= qA -{-wB + zB+vC^ D.M.D. of AB 4- dep. of AB + 
dep. of BC. 

Hence the D.M.D. of all of the courses may be computed by 
the following rules : — 

(i) The D.M.D. of the first course (starting from the 
primary meridianf) equals the departure of the course itself. 



* If the traverse had been run around the field in the opposite direction these 
north areas would become south areas. The result would be the same, however, 
ixi either case since it is the algebraic sum of the areas which is obtained. 

t Any meridian could have been chosen as the primary meridian, but negative 
signs are avoided if the most westerly point is chosen as the starting point. 



3S6 COMPUTATIONS [Chap. XIIL 

(2) The D.M.D. of any other course equak the D.M.D. of 
the preceding course plus the departure of the preceding course 
plus the departure of the course itself. 

(3) The D.M.D. of the last course should be numerically 
equal to its departure, but with opposite sign. 

The double areas of all the trapezoids may now be found 
by simply multiplying the D.M.D. of each course by the latitude 
of the same course, North latitudes being regarded as plus and 
South latitudes as miniis. The sum of all the north double areas 
minus the sum of all the south double areas equals twice the area 
of the field. Be careful to divide by 2 after competing the other 
details of the computation. 

386. COMPUTATION FOR AREA OF COMPASS SURVEY BY 
D.M.D. METHOD. — The details of the above are illustrated 
in Fig. 159, which is the computation of the area of the traverse 
given in the compass notes in Fig. 50, p. 100. It will be seen 
from a study of the notes that there was local attraction of i° at 
station 5, and that in the following computations the corrected 
bearings are used (Art. 41, p. 30). 

In Fig. 159 the hearings, distances, latitudes, departures, and 
D.M.D 's, which are recorded on a line with station F are those 
corresponding to the course FA; those recorded on a line with 
station A refer to the course AB; etc. After the bearings and 
distances are entered in the table the places which are to be 
blank in the remaining columns are cancelled as shown; this is 
a check against putting the results of the computations in the 
wrong spaces. In computing the latitudes and departures the 
log distance is first entered; the log sin bearing is written below 
this and the log cos bearing is recorded above. To obtain the 
log latitude add the upper two logarithms; to obtain the log 
departure add the lower two logarithms. When the latitude and 
departure of a course have been obtained see if the results appear 
to be consistent with the given bearing and distance; when the 
bearing of a course, for example, is less than 45® its latitude is 
greater than its departure and vice versa. 



DOUBLE MERIDIAN DISTANCES 



357 



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Meridian Distance Method. 



358 COMPUTATIONS [Chap. XHI. 

387. Balancing a Chain and Compass Traverse. — Before 
the D.M.D. method can be properly appUed the errors of meas- 
urement of the traverse should be so distributed that the figure 
becomes a closed polygon. If the field is a closed polygon the 
sum of the north latitudes will equal the sum of the south lati- 
tudes, and the sum of the east departures will equal the sum of 
the west departures. As soon as the latitudes and departures 
are computed this test is applied. If the sums differ, the error 
is distributed in such a way as to make the sums exactly equal, 
and at the same time to give to each latitude and departure its 
most probable value. In the case of a compass survey the errors 
are fully as likely to be in the bearings, which have been read to the 
nearest quarter of a degree, as in the distances; hence if nothing 
definite is known in regard to the errors they are assumed to be 
proportional to the lengths of the lines and the survey is balanced 
by the following rule which alters not only the length of the lines 
but also their directions. 

388. The correction to be applied ^ ^'*^ j ^ >, / I ^/ ^^J' 
course is to the total error ^^ j ^ /, . | ^^ ^^^ length of the course 
is to the perimeter of the field. 

This rule is based upon purely mathematical considerations 
and should be applied only when nothing is known as to where 
the errors probably occurred. Usually the surveyor knows where 
the error is probably greatest and consequently in balancing the 
survey he will place the largest corrections where, in his judg- 
ment, they belong. In measuring with the chain, the recorded 
distances tend always to be too long, because the sag, poor align- 
ment, and poor plumbing, all serve to shorten the chain; conse- 
quently the probability is that the recorded measurements are 
too long, therefore in balancing it is more logical to subtract 
from the latitudes and departures in the columns whose sums 
are greater rather than to add anything to the latitudes and de- 
partures in the smaller columns. The corrections should of 
course be applied in such a way as to decrease the difference 



DOUBLE PARALLEL DISTANCES SS9 

between the two columns. In the example (Fig. 159) the total 
error in latitude is 0.08 and the total error in departure is 0.07. 
The perimeter of the traverse is 129.77. Hence the correction 
per chain-length is 0.062 links for latitudes, and 0.054 for depart- 
ures. The corrected values of the latitudes and departures are 
given in the columns headed balanced latitudes and balanced de- 
partures. 

389. From the balanced departures we then compute the 
D.M.D. of each course as shown in the next column. Observe 
that the last D.M.D. (point F), as computed from the preceding 
one, is exactly equal to the departure of the last course. This 
checks the computation of the D.M.D.'s. The D.M.D.'s are 
now multipUed by their corresponding latitudes and the prod- 
ucts placed in the double a ea columns, those having N latitudes 
being placed in the column of north ( + ) double areas and those 
having S latitudes in the column of south ( ~ ) double areas. The 
sums of these columns differ by 2245.8. One-half of this, or, 
1 1 23. is the area of the field in square chains, which equals 
1 1 2.3 acres. 

By proceeding around the field in the reverse direction the 
letters of all of the bearings would be changed, in which case the 
column of south double areas would be the larger. 

390. Double Parallel Distance. — There is no particular 
reason for using the trapezoids formed by projecting the courses 
on to the meridian rather than those formed by projecting them 
on to the other axis. In the latter case the Double Parallel Dis- 
tance {D.P.D,) should be computed, and the result multiplied 
by the departure for each course. 

In the D.M.D. method the computations have been checked 
at every step with the exception of the multiplication of the 
D.M.D.'s by the latitudes. A check on this part of the work 
can be obtained by figuring the area by use of the D.P.D.'s. 
This furnishes an example of a very desirable method of check- 
ing, as a different set of figures is used in computing the double 
areas, and the opportunity for repeating the same error is thus 
avoided. Fig. 160 shows the computation by the D.P.D. method 
of the area of the same survey as is calculated by the D.M.D. 
method in Fig. 159. 



86a 



COMPUTATIONS 



[Chap. xm. 



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391. Error of Closure. — An indication of the accuracy of 
the survey is found in the error of closure. If a complete traverse 
of the field has been made the final point, as computed, should 
coincide with the first. The amount by which they fail to coin- 
cide is the total error of the survey and may be found by the 
formula . 

where / is the error in latitude and d is the error in departure. 
If this distance E is divided by the perimeter of the field the 
resulting fraction is called the error of closure, which in this sur- 
vey is approximately i^cr^^ (see Art. 132, p. qq). 

392. COMPUTATION OF AREA OF A TRANSIT AND TAPE 
SURVEY. — The field notes show the lengths of the sides 
of the traverse, all of the angles and perhaps the magnetic bear- 



DOUBLE MERIDIAN DISTANCES 361 

ings of some or all of the courses. If an observation has been 
made for determining the direction of the meridian, this affords 
the means of computing the true bearings of all of the traverse 
lines. 

393. The first* step in reducing the notes (provided it has 
not already been done in the field) is to see if the difference 
between the sum of the right and left deflection angles equals 
360®. If interior angles have been measured, their sum should 
equal the number of sides of the field times two right angles, 
minus four right angles. If there is a small error in the sum 
of the angles this is usually adjusted by placing the errcr in the 
angles where it probably occurred. If nothing is known as to 
where it probably occurred the corrections should be made in 
the angles adjacent to the short lines, as any error in sighting or 
setting up the transit causes a greater angular error in a short 
line than in a long one. 

The transit survey is referred to a system of rectangular 
coordinates, as in case of the compass survey. If the direction 
of the true meridian is known (either from a special observation 
or by connection with some other survey referred to the meridian), 
it is advisable to use this meridian as one of the coordinate axes. 
If the direction of the true meridian is not known the magnetic 
meridian may be used. This of course is convenient in some 
respects because the bearings taken in the field already refer 
to this meridian. If not even the magnetic meridian is known it 
will then be advisable to choose some line of the survey (pre- 
ferably a long one) as the axis, for using one of the traverse lines 
as an axis saves computing the latitude and departure of one 
course. 

Whatever line is chosen as an axis, the bearings used for 
computing the latitudes and departures are to be obtained from 
the measured angles (after correction), and not from the observed 
bearings. For instance, if some line is selected and its mag- 
netic bearing used, then the bearings of all of the other lines 
should be computed from this one by^means of the (corrected) 
transit angles. In this way the bearings are relatively as accurate 
as the transit angles, even though the whole survey may be 
referred to an erroneous meridian due to the error of the magnetic 



362 



COMPUTATIONS 



[Chap. xm. 



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Fig. 161. Area of a Transit and Tape Survey by Double 
Meridian Distance Method. 



(The remainder of the computations is in Fig. 161 A.) 



DOUBLE MERIDIAN DISTANCES 



363 



LMCos.Bt(r« 
U9 $10. Bear 

UjLflt 



N 

1.7305U 
1.806500 

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Dep ^.10 

H 

Ug OMO 1.80650 

LogUrt- i.<iM80 

Log Area 5.71850 

Area J2Z8 



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LtAkna 3.36384 
Anca 2311 



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1.712437 
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2211414 

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20125 



161 A. 

(These compatations go with Fig. i6i.) 

bearing of the first line. In calculating these bearings the work 
should be checked by computing the bearing of each line from 
that preceding, the bearing of the last line being followed by the 
calculation of a new bearing of the first line of the traverse which 
must agree with the magnetic bearing assumed for it, provided 
the deflection angles have been adjusted so that their algebraic 
sum is 360®. The observed magnetic bearings of the different 
courses will serve as a check against large mistakes in this cal- 
culation. 

3g4. When all of the bearings have been figured the latitudes 
and departures are to be computed. In good transit surveys 
five places in the trigonometric functions will usually be necessary. 
If the angles are measured, by repetition, to a small fraction of 
a minute, seven-place logarithmic tables may profitably be em- 



364 COMPUTATIONS [Chap. XHI. 

ployed, as much interpolation is avoided by their use, but the 
logarithm! need not be taken out to more than five or six places. 
Seven places, of course, are more than are necessary so far as 
precision is concerned (Art. 351, p. 324). 

The computation of the latitudes and departures may be 
conveniently arranged as shown in Fig. 161 which is the com- 
putation of the survey in Fig. 52, p. IC3. After the latitudes and 
departures have been calculated they are arranged in tabular 
form. The columns of latitudes and the columns of departures 
are added and compared just as in a compass survey. 

395. Balancing a Transit and Tape Traverse. — In adjusting 

(balancing) a transit traverse a different rule is used from the 

one given in Art. 388. In the case of a transit survey the error is 

chiefly in the measurement of distances, as it is much easier to 

secure accurate results in the angular work than in the tape 

measurements. Hence the following rule for balancing the 

survey is applicable: — 

( laiituie ) 
The correction to he applied ^^^^ ) j j, 4 \^f ^^V course is 

,.,,.. . ( latitvde \ . ( latitude ) . .1 . 

is to the sum of all of the latitudes (without regard to algebraic 
sign). 

As in the case of a compass survey, the surveyor's knowledge 
of the circumstances should always take precedence over the rule, 
and it is probably more nearly correct to shorten the latitudes or 
departures in the larger columns than to lengthen them in the 
smaller columns. This is because distances are usually recorded 
longer than they actually are; the only cases where the distance 
is probably too short is when an excessive pull has been given 
to the tape or a mistake made in measurement. It will be 
observed in the original notes (Fig. 52, p. 103) that the distances 
BCf GH, and KA were all questioned, i.e., they were measured 
under such conditions that it is probable that there may be one 
or two hundredths error in them. In balancing the latitudes and 
departures then, this information is used. In Fig. 161 it will 
be seen that in balancing the survey the latitudes and departures 



DOUBLE MERIDIAN DISTANCES 



365 



of these questioned measurements have been changed in such a 
way as to reduce the length of £C, GjEZ", and KA each one hun- 
dredth of a foot. 

In balancing the angles^ in which there was an error of 15 
seconds, it will be noticed that the correction for this error, 
being small, was put into one angle, that at C, one of whose 
sides is the shortest Une in the traverse. The area is computed 
as explained in Art. 385, p. 354. 

396. Fractional Areas. — Fig. 162 is the computation of the 



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Fig. 162. 



Computation of Transit and Tape Survey, including 
Fractional Area. 



survey shown in Fig. 53, p. 104. The traverse was run with a 
transit and tape, the angles being measured to the nearest minute 



366 COMPUTATICM^S [Chap. Xm. 

and the sides to tenths of a foot. Nothing appears in the fidd 
notes to indicate that any of the lines were diflScult to measure, 
so it is assumed that any errors in measurement are as likely to 
occur in one line as another. Therefore, in balancing the lati- 
tudes and departures of this s vey, the rule given in Art. 395 is 
applied. In balancing the angles, in which there was an error 
of I minute, the entire error was placed in the angle at D where 
the side DE is short in comparison with the other sides. 

It will be noticed that the distances which appear on the 
sketches in the computation are slighdy diflFerent from those 
which appear in the field notes (Fig. 53) ; this is due to the fact that 
the distances have been corrected for erroneous length of tape 
before undertaking to calculate the area. The intermediate 
steps in the computation of this traverse do not appear in Fig. 
162, but they are the same as in the last traverse. The D.M.D.'s 
were computed from Fy the most westerly point. The compu- 
tation of the fractional areas is also given. 

397. SUPPLYING MISSING DATA, — If any two of the bear- 
ings or distances are omitted in the traverse of a field the miss- 
ing data can be supplied and the area obtained by computations 
based on the measurements taken. As has been shown in 
Art. 387, p. 358, the algebraic sum of all the latitudes in a closed 
survey must equal zero, and the algebraic sum of all the depart- 
ures must equal zero; or, to put it in the form of an equation. 



Zi cos A -{- Z2 cos B -h Z^ cos C + etc. = 
Zi sin A -f Z2 sin 5 + Z, sin C + etc. = 



o 



where Zj, Z„ Z,, etc., are the lengths of the corresponding courses. 
Therefore from these two equations any two unknown values in 
them can be computed. n 

The missing data could be any of the following combina- 
tions: — 

(1) The bearing and length of a line. 

(2) The length of a line and the bearing of another line. 

(3) The length of two lines. 

(4) The bearings of two lines. 



SUPPLYING MISSING DATA 367 

398* Case (i) where the bearing and length of a line are 
missing is by far the most common. Its solution is also more 
direct than that of the other cases. 

If the latitudes and departures of all of the measured sides 
are calculated, the sum of N and S latitudes will be found to 
differ, and the amount by which they differ is the latitude of 
the omitted side plus or minus the errors of latitudes. Similarly 
the amount by which the E and W departures differ is the 
departure of the course omitted plus or minus the errors cf 
departures. From the latitude and departure of a course its 
length and bearing may be readily found. 

A practical application of this case is found in the problems 
of subdividing a field by a line running from one known point to 
another, the direction and length of 'the dividing line not having 
been measured. The area of the portion cut off by this line can 
readily be computed by the above method. In case the angles 
were taken with the transit, the bearing of one line would be 
assumed to be correct and all other bearings computed to 
correspond. 

It is evident from the above that in supplying missing data 
the observed measurements must be assumed to be correct, as 
there is no way of proving this from the computations. For this 
reason it is never advisable, when it can possibly be avoided, to 
supply missing data derived from computations on which a field 
check has not been obtained. 

399. The solutions of the other three cases of missing data 
are not so simple, as they involve the use of simultaneous equa- 
tions; they will not be discussed here. 

400- Besides the four cases mentioned above there are some 
special cases which are capable of solution. In Fig. 163 the 
lines and angles measured are ^ 

shown by full lines. The ^rfX^<r^)^^^v^^ 

bearing of -45 is given. Here ^^^\ ^ — '*^*.^.,^^ 

one side and two angles are ^v*"^-^ ^^^^^ 

missing. The solution is as ^s^.,,-^"^"^"^-^ J:J 
follows. In the triangle EAB x\ "^--^/^ 

find £B, £5.4, and i4JE5. In ^ Fig"i63"" * 

the triangle EDC find JSC, 
DCE, and DEC. Then in the triangle £J5C, in which £C, 



36a COMPUTATIONS [Chap. XIH. 

£J5, and EBC are known, find ECB, CEB, and BC. All the 
angles and sides are then known. Other special cases may be 
solved in a similar manner. 

401. DETECTING MISTAKES. — Mistakes in fieldwork may 
often be detected by means of the calculations. One of the 
easiest mistakes to make in surve)ang is to omit a whole tape- 
length in counting. If such a mistake were made and the lati- 
tudes and departures were computed, the linear error of closure 
of the survey would prove to be about a tape-length. In order 
to find in which line this mistake probably occurred compute the 
bearing of this linear error of closure and examine the traverse to 
find a line having a bearing the same or nearly the same. The 
error in departure divided by the error in latitude equals the 
tangent of the bearing of the line which represents the error of 
closure of the traverse. The errors of the survey, of course, 
will prevent these bearings from agreeing exactly. If two 
mistakes have been made it may be difficult and sometimes im- 
possible to determine where they occurred. When an error of 
this sort is indicated by the computation the line should be re- 
measured. It is bad practice to change an observed measure- 
ment because it is found by calculation to disagree with other 
measured distances. 

It may, and frequently does, happen that there is more than 
one line in the traverse which has about the same bearing. In 
such a case it is impossible to tell in which of these lines the mis- 
take occurred. But if a cut-off line is measured as was sug- 
gested in Art. 145, p. 109, and one portion of the survey bal- 
ances, the other part will contain the mistake. By proceeding 
in this way the number of lines in which the mistake could occur 
is reduced so that its location can be determined and checked 
by field measurement. 

402. The Subdivision of land. — There are a great 
many different problems which may arise in the subdivision of 
land and which may be solved simply by the application of the 
principles of trigonometry. A few of these problems are so 
common and so frequently involved in the working out of more 
complicated cases that their solution will be given. 



SUBDIVISION OF LAND 



369 



403. To Cut Off from a Traverse a Given Area by a Straight 
Line starting from a. Known Point on the Traverse. — In Fig. 
164, ABCDE represents the traverse which has been plotted and. 
whose area has been computed. 
It is desired to cut off a cert in 
area by a line running from F 
which is at a known distance 
fiom A or E. The line FG is 
drawn on the plan so as to make 
the area FG'DE approximately 
equal to the desired area. The 
line DC is scaled ofiF and the 
scaled distance used as a trial 
length. Then the side FG' and 
its bearing can be found by the 
method explained in Art. 398, 
p. 367, and the area FEDG com- 
puted in the usual manner. The 
difference between the required 
area and the area of FEDG is the 
amount to be added to or sub- 
tracted from FEDG'. If this correction area is a minus area then 
the triangle FG'G will represent it. In this triangle the base FG 
and its area being known the altitude hG and the distances GG 
and FG can be readily computed. In the traverse FGDE^ 
which is the required area, the length of the missing side FG 
and its bearing can be supplied. 

Instead of using the trial line FG the line FD might have 
been first assumed and the correction triangle would then be 
FDG. This method has the advantage of containing one less 
side in the first trial area, but the correction triangle is large, 
whereas in the method explained above the correction triangJe is 
small which may be of advantage in that part of the computa- 
tion. 

404. To Cut Off from a Traverse a Given Area by a Line 
running in a Given Direction. — In Fig. 164, ABCDE repre- 
sents a closed traverse from which is to be cut off a given area by a 
line running at a given angle (BJK) with AB. On the plot of the 




370 



COMPUTATIONS 



[Chap. xm. 



traverse draw the line J'K' in the given direction cutting ofi 
J'BCK' which is, as nearly as can be judged, the required area. 
Scale the distance BJ' and use this trial distance in the compu- 
tations. Then compute the distance J'K' and the area of 
J'BCK' either by the method of Art. 398, p. 367, or by dividing 
J'BCK' into two oblique triangles. The difference between 

this area and the required area 
is then found, which is a cor- 
rection trapezoid to be added 
to or subtracted from J'BCK'. 
In this case it will be assumed 
that it is to be added to J'BCK', 
In this correction trapezoid 
the area and one base J'K' are 
known; also the base angles, J' 
and K'. From these data an 
approximate value for the a'ti- 
tude of the trapezoid can be 
obtained and the length of the 
other base K"J" of the trapezoid 
computed from this altitude 
and the length of J'K', Then 
the area of this trapezoid 
J'K'K^J" can be accuratdy de- 
termined; the difference between this and the required correction 
will be small and the dimensions of the second correction trape- 
zoid J^K^K'J' can probably be readily Computed from its area 
and the length of J^K" which are known. By successive trials, 
probably not more than two, the correct line JK can be found. 
If lines AB and CD are approximately parallel the trapezoid is 
nearly a parallelogram and its correct altitude can then be 
quickly determined. 

405. To Find the Area Cut Off from a Traverse by a Line 
running in a Given Direction from a Given Point in the Traverse. 
— This problem may be readily solved by drawing a line from 
the given point in the traverse to the comer which lies nearest 
the other extremity of the cut-off line. The area of the traverse 
thus formed is then computed, and this area corrected by means 
of a correction triangle. 




TRAVERSES WHICH DO NOT CLOSE 



371 




Fio. les. 



In Fig. 165, ABCDEFG repre- 
sents a plot of a field. It is desired 
to run the line from £ in a given 
direction. EjEZ" and to compute the 
area HEFGAB cut off by this line. 
The latitude and departure of points 
B and E being known the bearing 
and length of BE and the area of 
ABEFG can be computed. Then 
the area and the remaining sides of 
the triangle BEH can be obtained 
from BE and\he angles at B and E. 

It is obvious that the solution 
pf such problems as these is greatly facilitated by plotting the 
traverse before attempting the computations. 

CALCULATIONS RELATHI G TO TRAVERSES WHICH DO NOT CLOSE. 

406. To Calculate the 
Total distance bjetween 
End Points. — Fig. 166 repre- 
sents the traverse ABCDEF in 
which the distance AF and the 
angle BAF are desired. AB can 
be assumed as one of a pair of 
rectangular coordinate axes and 
the coordinates of point F {AH 
and HF) computed by the 
method explained in Art. 410, 
P- 373- ^^ ^^^ the angle BAF 
can then be easily found. This 
method is of service in check- 
ing traverse plots of this type. 
407. CUT-OFF LINES. — The calculation of cut-off lines, 
like the line CF\n Fig. 166, is the same problem as was explained 
in Art. 398, p. 367. The angles DCF and EFC have been 
measiu-ed in the field and the traverse CDEF is thus complete 
except that the^ength of the line CF is unknown. The length 
of CF and the angle it makes with AB can be readily computed 
since the coordinates of C and F can be found. 




Fig. lee. 



37^ 



COMPUTATIONS 



[Chap. xm. 



408. COMPUTATION OF AZIMUTHS WHEN CHECKING 
ANGLES TO A DISTANT OBJECT. — In this kind of problem 
the coordinates of all the points along the traverse can be com- 
puted with reference to some coordinate axes. At A and B 
(Fig. 167) an^es have been taken to 5, and from these angles 
the coordinates of point 5, referred io AB and a line perpen- 
dicular to AB as axes, can be computed (Art. 410, p. 373). Co- 
ordinates of S referred to the same axes 
should have the same value when 
figured from BC as a base as when 
calculated from the base CD and so on. 
If, however, when computed by means 
of angles at D and £, the point falls 
at 5', and angles E and F give its 
location also at 5' there is evidence 
of a mistake in the traverse at D. If 
the two locations of S and 5' are 
such that a line between them is par- 
allel to either CD or Z>£, the mis- 
take was probably made in the meas- 
urement of the line parallel to SS' 
and the distance 55' should be ap- 
proximately equal to the amoimt of 
the mistake in measurement. If, 
however, SS' is not parallel to either 
CD or DE the mistake probably lies in the angle at D. 




Fig. 167. 



409. CALCULATION OF TRIANGULATION. — In a triangu- 
lation system the base-line is the only line whose length is 
known at the start. The sides of any triangle are found from 
the law of sines, i.e.. 



sini4 __ a 
sin J3 b 
sin -4 __ a 
sinC c 



tf sing 

sin^l 

a sin C 

sin^l 



COORDINATES 373 

Assuming a to be the base and the angles A, J3, and C to 
have been measured the calculations are arranged as follows: 

(i) log a (1400.74) = 3-1463575 

(2) Colog Sin^l (57^42' 16'') = 0.0729874 

(3) log Sin B (61^ 17' 53^^) = 9-943o639 

(4) log Sin C (60^ 59' si'') = 9.9418088 

Sum of (i) (2) (3) log* = 3 1624088 
Sum of (1) (2) (4) log c = 3.1611537 

410. COORDINATES. — In many cities the coordinate sys- 
tem of surveying is used (see Chapter IX). In this system the 
position of each comer of the different lots is fixed by rectangu- 
lar coordinates measured from two lines at right angles to each 
other. 

Often the origin of coordinates O (Fig. 168) is so chosen that 
the whole city is in the first 
quadrant YO X. Distances 
measured parallel to XX^ 
are usually called abscissas 
and those parallel to YY^ 
ordinates. 

The advantage of this x* 

system of surveying lies in 
the fact that since all surveys 
refer to the same reference 
lines, they are therefore tied 
to each other; and also in the 
fact that a lot can be relo- 
cated from the coordinates of 
its comers even if all of the corner bounds have been destroyed. 

Generally the coordinate lines rim N and S, and E and W, 
but when city streets have been laid out at right angles to each 
other and not on N and S, and E and W lines, it may be more 
convenient to have the system of coordinates parallel to the street 
lines. 

The coordinates of any unknown point are usually computed 
from the coordinates of some other point to which the unknown 



V 



Fig. 168. 



374 



COMPUTATIONS 



[Chap. Xm. 



point is tied by an angle and distance. The difference in coor- 
dinates between the known and unknown points will be obtained 
as follows: — 

Difference in X = distance x sin azimuth angle. 
Difference in Y == distance x cos azimuth angle. 

Sometimes the unknown point is located by angles from two 
other known points, in which case the distance between the two 
points whose coJ'rdinates are known can be computed and then 
the distance from one of the known points to the unknown point. 
The problem is then in the form described in the previous para- 
graph. 



L..-x..^J 



-V 




— 1~ X5-r 



^ 

—L.- 



I 

I 
I 
I 



Fig. 160. 



4X1. TO DETERMINE THE AREA OF A FIELD BY REC- 
TANGULAR COORDINATES. — The area of the field i, 2, 3, 4 
(Fig. 169) is equal to the trapezoids 

(a, I, 2, b) + (6, 2, 3, c) - (a, i, 4, d) - (J, 4, 3» <^)- 

Expressed as an equation in terms of the coordinates the 
area is 



AREA BY COORDINATES 



375 



I, 2, 3, 4 - U -A)^^ +(J, - Js)^y^' 

From this equation is derived the following rule for obtaining 
the area of a closed field from the coordinates of its corners: — 

(i) Number the comers consecutively around the field. 

(2) Multiply each | ^^^^cissa | j^^ ^j^^ difference between the 

following and the preceding j i^ • { > always subtracting the 

preceding from the following, and take one-half of the sum of 
the products. 

412. Fig. 170 is the computation, by coordinates, of an area 



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^1534 


153A 






J 


N43*t£ 


a75 


3,37 


— 


B^ 


— 


f^3i3B 


'347 


aid 


6^ 


'6M7 




SKB 


4- 


NSS^I^ 


S.O0 


4^9 


— 


— 


e.fs 


^4^ 


~tB 


il75 


J6S? 


-M47 




3Z9£ 










f^.33 


IS.3S 


/i,M 


fZ.lf^ 












^7L1 


im 



Fig. 170. Computation of Compass Survey by Coordinates. 

from the field notes. The origin of coordinates is 10 chains W 
and 10 chains S of station 2. 

413. Equation (i ) may be developed into the following form: 

When this formula is to be used the coordinates may be arranged 
in the following simple manner: 






(4) 



376 



DOUBLE MERIDIAN DISTaMCES 



(Chap. XHL 



From equation (3) it will be seen that the area is equal to the 
sum of the products of the ordinates joined by full lines in (4) 
minus the sum of the products of the ordinates joined by broken 
lines. This formula involves the multiplications of larger num- 
bers than in (2), but does not require any intermediate sub- 
tractions. 



PROBLEMS. 



X. The latitude of a line of a traverse is + 106.42 ft; its departure is 
— a 73.62. What is its bearing ? 

2. From the following notes of a compass survey, compute by the double 
meridian distance method the area in acres. 



Station. 


Bearing. 


Distance 
(Chains). 


A 


N46«iW 


20.76 


B 


N5i°iE 


13 83 


C 


East 


21.35 


D 


S56«E 


2'. 60 


E 


S33nW 


18.80 


F 


N74°iW 


30. 8 



3. In the following notes of a compass survey the length and bearing of one 
of the courses were omitted. Substitute the correct values and compute the area 
(in acres) by the double meridian distance method. 



Station. 


Bearing. 


Distance 
(Chains). 


I 


S40*>W 


17-50 


2 


N45**W 


22.25 


3 


N 36*^1 E 


31-25 


4 


North 


13-50 


5 


(omitted) 


(omitted) 


6 


ssojw 


34.25 


7 


West 


32.50 



rkoMXhta 



177 



4. Rram the notes given in Fig. 52, p. 103, and Ffg. 161, p. 3*2, compute by 
the (k>uble meridian distance method the area of the traverse ABCDEK, 

5. In the following traverse there are two mistakes. Find where they occur 
and determine their amounts. 



Station. 


Obser\'ed 
Bearing. 


Deflection 
Angle. 


Distance 
(Feet). 


Calculated 
Bearings. 


Remarks. 


A 
B 
C 
D 
E 


N34^E 

S73**iE 

Sio'^lW 

N26oiW 

S52OW 


164** 14' R 
62«» 16' R 
84*» 22' R 

I42«49'R 
103** 41' L 


240.2 

163.7 
207.6 

273-1 
147-4 


N34*»oo'E 


C£=i88.i 
BC£= 34** 14' 
DEC-^ 81^25' 



6. The following b a set of notes of an irregular boundary of a lot of land. 
It is desired to straighten this crooked boundary line by substituting a straight line 
running from B to the line EF, Find the bearing of the new boundary line and 
its length; also the distance along EF from point E to the point where the new line 
cuts£F. 



Station. 


Bearing. 


Distance 
(Feet). 


A 


S 89* 14' E 


373.62 


B 


N 13'' 10' E 


100.27 


C 


N o*>i7'W 


91.26 


D 


N270 39'E 


112.48 


E 


N72«>i2'W 


346.07 


F 


S 5**o7'W 


272.42 




etc. 


etc. 



7. (a) In the lot of land, A BCD, the lines AB and DC both have a bearing 
of N 23** E; the bearing of i4Z> is due East; AD is 600 ft., i4B is 272.7 ft., and DC 
is 484.6 ft. Find the length of a line EF parallel toAB which will cut off an area 
ABFE equal to half an acre. Also find the length of the lines AE, and BF. (b) 
What is the area of EFCD? 

8. Given the notes of a traverse, which does not close, as follows: — 



Station. 


Deflection 
Angle. 





6+40 
9+20 

14+ 55 
17 + 18 
20+64 


60 17' L 
18043'L 

12047'R 
45^24'L 
680o6'R 


Find the length 
of a straight line 
from to 20+ 64 
and the angle it 
makes with the 
line from to 6 
+ 40. 



37* DOUBLE MERIDIAN DISTANCES [Chap. XHI. 

' 9. Compute the area of the foUomng traverse by coordinates. 



Station. 


Deflection 
Angle. 


Bearing. 


Distance 
(Feet). 


A 


780 10' oo* L 




208.64 


B 


88« 28' oo* L 




43<^-79 


C 


6^'' 02' IS*' L 




S6.48 


D 


33" 39' 15^ L 




98.80 


E 


•go^oo'oo-'R 




68.62 


F 


.220 5I'00*'L 

%68o 50' IS-' L 




95- 10 


G 


N36*>i4'oo»W 


207.41 


H 


89* 48' oo*' L 




103-75 


I 


SS" 00' IS- L 




96.75 


J 


530 49'oo«'R 




420.77 



PART IV. 
PLOTTING. 



PART IV. 

PLOTTING. 
CHAPTER XIV. 

DRAPTINa IK8TR02ffBNT8 AND MATERIALS. 

It is assumed in this section that the student is familiar with 
the ordinary drawing instruments such as the T-square, triangles, 
dividers, compasses, and scales, as well as with their use. 

ENGINEERING DRAFTING INSTRUMENTS. 

414. There are several drafting instruments which are used 
by engineers and surveyors but which are not so generally em- 
ployed in other kinds of drafting work. The most common of 
these are briefly described in the following articles. 

415. STRAIGHT-EDGE. — Engineering drawings are made 
with greater accuracy than much of the drafting work of other 
professions. In fact many engineering drawings are limited in 
precision only by the eyesight of the draftsman. It is evident, 
then, that to use a T-square which is run up and down the more 
or less uneven edge of a drawing board will not produce draw- 
ings of sufficient accuracy. For this reason in many classes of 
engineering work the edge of the drawing board is not relied 
upon. Furthermore, in most plots of surveying work the lines 
are not parallel or perpendicular to each other except by chance, 
but run at any angle which the notes require; and there is there- 
fore not so much call for the use of a T-square as there is in archi- 
tectural, machine, or structural drawings. All drawings are 
usually laid out starting from some straight line drawn on the 
paper by means of a straight-edge, which is simply a flat piece 
of steel or wood like the blade of a T-square. Steel straight- 
edges are more accurate and are more commonly used by engi- 
neering draftsmen than the wooden ones, the edges of which are 
likely to nick or warp and become imtrue. They can be ob- 

38' 



382 DRAFTING INSTRUMENTS [Chap. XIV. 

tained of almost any length and of any desired weight, the com- 
mon length being about 3 feet. 

416. ENGINEER'S SCALE. — Practically all engineering 
plans are made on a scale of 10, 20, 30, etc. feet to an inch. In 
the engineer's scale, therefore, the inch is divided into 10, 20, 30, 
etc. parts, instead of into eighths and sixteenths as in the archi- 
tect's scale. Engineer's scales are made 3, 6, 12, 18, and 24 
inches long. • One form is the flat wooden rule with both edges 
beveled and a scale marked on each bevel. Some flat rules are 
beveled on both faces and on both edges of each face, thereby 
giving four scales on one rule. Still another very common form 
is the triangular scale, made of wood or metal, and having six 
different scales, one on each edge of the three faces. In such 
rules the scales are usually 20, 30, 40, 50, 60, and 80 ft. or 10, 20, 
30, 40, 50, and 60 ft. to an inch. Scales are, however, often 
made having the inch divided into 100 parts, but in plotting a 
map which is on a scale of 100 ft. to an inch the work is probably 
more easily done by using a scale of 10, 20, or 50 divisions to an 
inch and estimating the fractional part of a division than by try- 
ing to plot with a loo-ft. scale which is so finely graduated as 
to be very hard to read without the aid of a magnifying glass. 
A 20-ft. or 50-ft. scale is more satisfactory for precision than a 
lo-ft. scale when it is desired to plot on a scale of 100 ft. to the 
inch. A plan on a 206-ft. scale is always plotted by using a 20- 
ft. scale, a 300-ft. plan by using a 30-ft. scale, etc. 

A map covering considerable area, like the map of a state, for 
example, must be plotted to a very small scale, and this is usually 
given in the form of a ratio such as i to 500, i to 2500, etc., mean- 
ing that one unit on the map is ^ J^, j^j^-^, etc. of the correspond- 
ing distance on the ground; this is sometimes called the natural 
scale. For plotting such maps specially constructed scales with 
decimal subdivisions are used. 

4x7. PROTRACTOR, — A protractor is a graduated arc 
made of metal, paper, celluloid, or horn, and is used in plotting 
angles. There are many varieties of protractor, most of them 
being either circular or semicircular. 

418. Semicircular Protractor. — Probably the most common 
is the semicircular protractor which is usually divided into de- 



SCALES AND PROTRACTORS 383 

grees, half-degrees, and sometimes into quarter-degrees. Fig. 
171 represents a semicircular protractor divided into degrees. 




C 

Fig. 171. Semicircular Protractor. 

In plotting an angle with this protractor the bottom line of the 
instrument is made to coincide with the line from which the angle 
is to be laid oflF, and the center of the protractor, point C, is 
made to coincide with the point on the line. On the outside of 
the arc a mark is made on the drawing at the desired reading. 
The protractor is then removed from the drawing and the line 
drawn on the plan. 




Fig. 172. Semicircular Protractor. 



Instead of having the 0° and 180^ of the protractor on its 
lower edge some instruments are made as shown in Fig. 172. 



1 



384 DRAFTING INSTRUMENTS [Chap. XIV. 

This form is claimed by some draftsmen to be more convenient, 
because in handling the protractor by placing the fingers on the 
base neither the graduations nor the line on the plan are covered 
by the hand. 

419. Full-Circle Protractor. — The full-circle protractor is 
of use particularly in stadia work or in plotting any notes where 
azimuth angles of over 180® have been taken. For such work 
as stadia plotting an ordinary paper protractor 8 to 12 inches 
in diameter is sufiiciently accurate, and, in fact, paper pro- 
tractors of this size will yield more accurate results than .the 
smaller metal ones. 

420. Some of the metal protractors are provided with an 
arm and vernier attachment. These, while giving more precise 
results, require more time for manipulation, and a plain metal 
protractor with a diameter of, say, 8 inches will give sufficiently 
close results for all ordinary work. As a matter of fact a pro- 
tractor with a vernier reading to minutes can be set much closer 
than the line can be drawn, and it is therefore a waste of time to 
attempt to lay oflF the angles on a drawing with any such accuracy. 
There is, however, a protractor of this type with a vernier read- 
ing to about 5 minutes which may be of use in precise plotting. 

421. Three-Armed Protractor. — The three-armed protrac- 
tor is used for plotting two angles which have been taken with an 
instrument (usually a sextant) between three known points, for 
the purpose of locating the position of the observer (the vertex 
of the two angles). The protractor has three arms, the beveled 
edges of which are radial lines. The middle arm is fixed at the 
o® mark and the other two arms, which are movable, can be 
laid oflF at any desired angle from the fixed arm by means of the 
graduations on the circle, which number each way from the 
fixed arm. The two movable arms having been set at the de- 
sired angles and clamped, the protractor is laid on the plan and 
shifted about until each of the three known points, (which have 
already been plotted on the plan), lies on a beveled edge of one 
of the three arms of the protractor. When the protractor is in 
this position its center locates the point desired which is then 
marked by a needle point Only one location of this center 
point can be obtained except in the case where the three known 



PROTRACTORS AND PANTOGRAPH 385 

points lie in the circumference of a circle which passes through 
the center. 

422. There are several other types of protractor made, but 
the principle and use of all of them are much the same as those 
of the simple types which have been explained. It is well in 
purchasing a protractor to test it to see that the center point lies 
on a straight line between the 0° and 180® marks, that the edge 
of the protractor is the arc of a true circle, and that the gradua- 
tions are uniform. 

423. PAMTOGRAPH. — This instrument is composed of several 
flat pieces of metal or wood joined in sjich a way as to form 
a parallelogram. One of the three points i4, By and C, (Fig. 
173) is fixed and the other two movable. The remaining bear* 




Fig. 178. The Pantograph. 

ing points are not essential except to support and steady the 
instrument. The two movable points are so attached to the 
instrument that they will trace out exactly similar figures. 
The instrument is used for copying a plan either to the same or 
to a different scale. There are several different forms of panto- 
graph varying considerably in appearance, but they are all based 
on the same principle. The essential condition in their design 
is that all three points A, B, and C, must lie in a straight line and 
each point must be on one of three different sides (or sides pro- 
duced) of a jointed parallelogram. Any one of the three points 
can be the fixed point. It is evident then that by changing the 
relative positions of these points, by moving them up or down the 
arms of the parallelogram, but always keeping the points on a 



386 DRAFTING INSTRUMENTS [Chap. XIV. 

Straight line, the scale of the copy can be made to bear any 
desired relation to the scale of the original drawing. These 
instruments are usually provided with scales marked on the arms 
indicating the proper settings for various reductions or enlarge- 
ments. With a pantograph very accurate results cannot as a 
rule be obtained because there is lost motion in the several joints 
of the instrument. Some of the expensive metal pantographs, 
however, will give fairly good results. 

424. Parallel ruler. — This is a beveled rule made 
of metal and mounted on two rollers of exactly the same diameter. 
It is used for drawing parallel lines. This instrument can be 
made to do accurate work, but it must be handled with a great 
deal of care to prevent the rollers from slipping. It is especially 
useful in drafting diagrams of graphical statics in connection 
mth structural design, in drawing the parallel sides of buildings, 
section lining, blocking out for titles, and in drafting large titles 
which require mechanical lettering. 

425. BEAM Compass. — This is an instrument used for 
drawing the arcs of circles whose radii are longer than can be 
set out with the ordinary compass drafting instrument. It is 
composed of a strip of wood or metal with two metal attach- 
ments which can be fastened to it. One of the§e attachments 
carries a needle point and the other, which is usually provided 
with a slow-motion screw for exact settings, carries a pencil or 
a pen. This instrument is particularly useful in laying out large 
rectangles such as are called for when surveys are plotted by 
coordinates (Art. 449, p. 401). 

426. COHTOUR PEN. — This pen is constructed very much 
like an ordinary right-line ruling pen except that it has a metal 
shaft, running through the entire length of the holder, to which 
the pen is attached. The shaft revolves inside of the holder, 
and the pen is so shaped that it drags behind taking a position 
ia the direction in which it is being moved. It is used for draw- 
ing irregular curved lines such as contours or shore lines. Not 
a litde practice is required before one can use a pen of this type 
accurately. When skill in its use is once acquired, however, a 
plan can be easily made on which the contours all have a uni- 
form weight of line giving a very satisfactory appearance. The 



BEAM COMPASS. — PROPORTIONAL DIVIDERS 387 

purpose of a contour line is to show the facts as to the surface, 
and this pen should not be used unless it is found by trial that 
it does the work in hand properly. Accuracy is more important 
than appearance. 

427. PROPORTIONAL DIVIDERS. — Proportional dividers are 
substantially an ordinary pair of dividers with both legs pro- 
longed through the pivot-point thereby forming another pair of 
legs above the pivot. The pivot is movable so that it can be 
pushed up and down in a slot in the legs and clamped in any 
desired position, thereby altering the relative lengths of the two 
pairs of legs. The sliding is accomplished in some dividers by 
a rack-and-pinion motion. When the pivot is in the middle 
position the legs are equal, and the space between the two points 
of one pair of legs is equal to the space between the other pair. 
There are marks on the legs showing the proper settings for the 
pivot so that the space between one pair of points will bear 
any desired ratio to the space between the other pair. The 
marks on the legs should not be accepted as correct, but should 
be tested by actual trial. One end of the proportional dividers 
is used to space off the distances from the original map and the 
other end used to plot that distance on the new map. Thus by 
means of this instrument a drawing can be enlarged or reduced 
to a definite scale without the use of the engineer's scale. 

A drawing which is to. be made two-thirds the size of the 
original can be readily reduced by scaling the distances from the 
original with a 20-ft. scale and plotting them on the new draw- 
ing by use of a 30-ft. scale. But when the reduction is some 
odd ratio which cannot be readily accomplished by means of the 
engineer's scale proportional dividers are very useful. 

428. RAILROAD CURVES, FRENCH CURVES, FLEXIBLE 
CURVE, AND SPLINE. — For drawing arcs of curves of long 
radiij such as occur on railroad plans and on plans of curved 
streets, in city work, curves made of wood, hard rubber, celluloid, 
or metal are used; these come in sets of about one hundred, with 
radii varying from about 2 inches to 300 inches. The metal 
curves are the most common and are made with the inside and 
outside edges of the same radii both edges being beveled. When 
a pencil line is drawn the beveled edges may be used against the 



388 DRAFTING INSTRUMENTS [Chap. XIV. 

paper, and when ink lines are drawn the curve can be turned 
over so that the beveled edges are up, thus preventing the ink 
from running in under the curve on the paper. Some curves for 
railroad work are made with a short straight edge tangent to 
the curve at one end and with the point where the curve begins 
marked by a line across it. 

429. Irregular curves, called French Curves^ are of a variety 
of shapes. They are made of wood, hard rubber, and celluloid, 
and are used to guide the pencil or pen in tracing out irregular 
curved lines on the map. 

430. A Flexible Curve consists of a strip of rubber fastened 
to a flexible metal back. This curve can be twisted to conform 
to any irregular curved line on the map and can then be used 
as a guide against which the pencil or pen is held in tracing out 
the curve. 

431. A Spline is a long thin flexible piece of wood, hard 
rubber, celluloid, or metal which can be bent so as to confcrm 
to a curve. It is usually held in position by specially designed 
weights with light metal arms which fit into a thin groove in the 
top edge of the spline. This instrument is used by naval 
architects for drawing long flat irregular curves such as occur 
in ship designs. In engineering drafting it is used in drawing 
the lines of arches, ^hich frequently are not circxdar. 

DRAWING PAPERS. 

432. The drawing papers used by surveyors may be divided 
into four general classes; (i) those used for plotting plans, (2) 
tracing paper or tracing cloth which is used for copying draw- 
ings, (3) cross-section and profile papers, and (4) process papers. 

433. DRAWING PAPER FOR PLANS. — There are numer- 
ous grades of drawing paper ranging from very cheap "detail " 
to heavy paper mounted on cloth, called "mounted paper." 
For rough plots which are to be copied later or which are for 
temporary use only, a manilla detail paper is frequently used; 
but where the drawing is to be of a more permanent character 
a heavy white or manilla paper is used. Still more permanent 



I 



DRAWING PAPERS 389 

plans, such as the plan of a survey of a city, should be plotted on 
heavy mounted paper. There is generally a right and a wrong 
side to all papers, which can be distinguished by the "water- 
mark "; this will read direct when the right side of the paper is 
toward the observer. A paper to be satisfactory for use should 
have a surface not too porous to take ink nicely, and of a fiber 
such that after scratching with a knife or rubbing with an ink 
eraser, the surface will still take ink effectively. No paper, 
however, after scratching can be expected to take bottle red ink, 
which permeates the fiber with extraordinary ease. 

434. TRACING PAPER AKD TRACING CLOTH. — In mak- 
ing copies of drawings, a thin transparent paper called tracing 
paper is often used. It is not tough enough to withstand rough 
handling and is used only for drawings of a temporary charac- 
ter. There are, however, certain kinds of transparent bond 
paper in use which will withstand considerable hard usage. 

435. For more permanent drawings a tracing cloth is used, 
made of a very uniform quality of linen coated with a prepara- 
tion to render it transparent. Most tracing cloth as it comes 
from the manufacturer will not readily take the ink, and it is 
necessary to rub powdered chalk or talc powder over the entire 
surface of the cloth before inking the drawing. After the sur- 
face chalk is brushed off, the tracing cloth is ready for use. Trac- 
ing linen generally has one side glazed and the other dull. Pencil 
lines can be drawn on the rough side, but the smooth side will not 
take even a very' soft pencil; either side may be used for ink draw- 
ings. Some draftsmen prefer to use the glazed side but the dull 
side is more commonly used. A tracing inked on the glazed side 
may be tinted on the dull side either by crayons or by a wash; 
the latter will cockle the cloth unless it is put on quite " dry." It 
is easier to erase from the glazed than from the dull side, but 
the dull side will stand more erasing,* and gives more uniform 
lines. 

* Erasure of ink lines from a tracing, as well as from any drawing paper, is a 
delicate undertaking. Success will result if the followmg suggestions are carefully 
observed : — with a smooth sharp knife pick off the ink from the paper ; this can be 
done almost without touching the paper. When practically aD of the ink is o&, 
rub the line with a pencil eraser. This will take o£f the rest of the line except 



390 DRAFTING INSTRUMENTS IChap. XIV. 

In making a tracing of another tracing it will be found that 
the lines can be more readily seen if a white paper is put under 
the lower tracing. It frequently happens that it is necessary 
to make a tracing of a blue-print. The white lines of the blue- 
print are not easily seen through the tracing linen. An arrange- 
ment which will assist greatly in such work is to have a piece of 
plate glass set into the top at one end of a drawing table in such 
a way that it forms part of the top of the table. The blue-print 
is placed over this glass and the light shining through from the 
under side of this glass and through the blue-print will make 
the white lines easily visible for copying. 

It is common practice, after a survey is made and before or 
during the computation of it, to plot the field notes accurately 
on detail paper and later to copy the plot on tracing cloth, which 
is the final drawing of the survey. 

From these tracing drawings any number of process prints 
can be made (Art. 438), the tracing taking the place of the 
negative used in photographic printing. 

436. CROSS-SECTION, AlID PROFILE PAPERS. — Paper di- 
vided into square inches which, in turn, are divided into small 
subdivisions is used to plot cross-sections of earthwork and the 
like. The inch squares are usually divided into J'^, ^"^ -j^, or 
^''. Cross-section paper can also be obtained divided accord- 
ing to the metric system, or with logarithmic divisions. Cross- 
section paper usually comes in sheets. 

437. Profile Paper which, as the name implies, is used for 
plotting profiles comes in rolls of 10 yds. or more. The vertical 
divisions are usually much smaller than the horizontal divi- 
sions, which makes it easier to plot the elevations accurately. 
The horizontal distances to be plotted occur mostly at full sta- 



perbaps a few specks of ink which can readily be removed by a sharp knife. This 
method of erasing takes more time than the ordinary method of rubbing with an 
ink eraser until the line has disappeared, but it leaves the paper in much better 
condition to take another line. It is impossible to obtain good results by this 
method unless the knife has an edge which is both smooth and sharp Where 
the surface of the tracing cloth has been damaged the applicaiion of a thin coat- 
ing of coUodioii on the damaged portion will produce a surface which will take 
the ink. 



PROCESS PAPERS 39I 

tion points, which are represented on the profile by the vertical 
rulings on the paper. 

Both the cross-section and the profile papers come in colors, 
(usually red, green, blue, orange, or burnt sienna) so that a black 
or a red ink line (the two most commonly used) will show up 
distinctly on the paper. These papers can be obtained also of 
very thin transparent material or in tracing cloth form, suitable 
for use in making process prints. Profile papers usually come 
in long rolls 20 inches wide. 

438. PROCESS PAPERS. — Blue-Prints. — The most com- 
mon process paper used in drafting oflSces is blue-print paper. 
It is a white paper coated on one side with a solution which is 
sensitive to light. After the solution is applied, the paper is 
dried and then rolled and sealed up for the market in light-proof 
rolls of 10 yds. or more. Fresh blue- print paper has a greenish- 
yellow color. The process of coating the paper and the gen- 
eral handling of the blue-print business is so well advanced and 
the price of the prepared paper is so low that surveyors now-a- 
days seldom coat their own paper. The process is a very simple 
one, however, and in emergencies, when commercial blue-print 
paper cannot be obtained, it may be very useful to know how to 
prepare it. A good formula for the solution is given below. 

Make the following two solutions separately (in the light 
if desirable) and mix, in subdued light or in a dark room, equal 
parts of each of them. 

Solution (i) 

Citrate of Iron and Ammonia, i part (by weight) 
Water, 5 parts C* '' ) 

Solution (2) 

Red Prussiate of Potash (re- 

crystalized), i part (by weight) 

Water, 5 parts (*' *' ) 

The mixed solution is applied to the paper by means of a 
camel's hair brush or a sponge; this is done in a dark room or 
in subdued light. The paper is coated by passing the sponge 
lightly over the surface three or four times, first lengthwise of 
the paper and then crosswise, giving the paper as dry a coating 



39^ DRAFTINO INSTRUMENTS [Chap. XIV. 

as possible consistent with having an even coating; it is then 
hung up to dry. The above coating will require about 5 minutes 
exposure in bright sunlight; for quick printing paper, use a 
larger proportion of citrate of iron and ammonia. 

The blue-print of a plan is generally made in a printing 
frame, which is merely a rectangular frame holding a piece of 
heavy glass, with a back to the frame which can be lifted from 
the glass. This back is padded so as to fit tight against the glass 
when the back is clamped into position. The process of taking 
a print is, briefly, to expose the tracing, with the blue-print paper 
under it, to the sunlight a proper length of time and then re- 
move the blue-print paper and wash it in water. 

•439. In detail, the process is as follows. First, turn the 
printing-frame over so that the glass is on the bottom, and remove 
the back of the frame. Then, after the tracing cloth has been 
rolled, if necessary, so that it will lie flat, place it with its face 
against the glass. Place the blue-print paper, which has been 
cut to the proper size, on top of the tracing with the sensitized 
side of the paper next to the tracing. The back of the frame is 
then clamped into position and the frame turned over so that 
the glass is up. It should then te examined to see that the trac- 
ing has been put into the frame with its ink lines against the 
glass, that the Hue-print paper is imder the entire tracing, and 
that mere are nq wrinkles in the tracing. All of the process to 
this stage should be done in subdued light, usually in a rocm 
with the shades drawn to keep out most of the sunlight. 

The frame is then moved out into the direct sunlight, placed 
as nearly as may be at right angles to the rays of sunlight, and 
left there a proper length of time, which will depend upon the 
sensitiveness of the coating of the paper and the intensity of the 
light. Some blue-print papers will print in 20 seconds, others 
require 5 or 6 m nutes in direct sunlight. In purchasing, then, 
it is necessary to ascertain from the dealer the "speed " of the 
paper and govern the exposure accord ngly. Blue-prints can 
be made in cloudy weather as well as when the sun is vis ble, the 
only difference being that it requires a much longer time for the 
exposure. In all cases where the time of exposure is doubtful 
the following simple test may be applied. Instead of taking a 



BLUE-PRINTING 393 

print of the entire tracing the first time, use only a small piece 
of the blue- print paper and put it in the frame as explained above 
and expose it a given t*me. Take it out and wash it, and from 
th s test judge the length of exposure necessary to give the print 
of the entire drawing. An under-exposed print, after it has 
been washed, will be light blue in color with white lines; an over- 
exposed print will be dark blue with bluish-white lines. The 
result desired is a dark or meditun blue background with white 
lines. It should be borne in mind, in judging the results, that 
all prints become a little darker when they are dry. 

In washing the print it should be entirely immersed in clear 
water at first; care should be taken that no part of the print is 
left dry. It should be washed by moving it back and forth in 
the water or by pouring water over it until the greenish solution is 
entirely washed off its face. The print should be left in the water 
for lo to 20 minutes, then it is hung up to dry. It will dry more 
quickly if hung so that one corner is lower than the others. It 
should not be hung where the sun will shine on it as the sun- 
light will fade it. 

In taking prints great care must be exercised not to get the 
tracing wet. When the prints are being washed the tracing 
should always be put in a safe place where the water will not 
spatter on it and it should never be handled with moist hands. 
It is practically impossible to eradicate the effect of a drop of 
water or even the marks made by damp fingers on tracing cloth; 
it is sure to show in every subsequent print which is taken from 
the tracing. 

440. Blue-print cloth is prepared in the same manner as 
the blue-print paper. Its advantage over the paper lies solely 
in the fact that it does not shrink as badly and is much more 
durable. Prints which are to be used on construction work 
where they are sure to get rough usage are sometimes made on 
cloth. 

441. Vandyke Solar Paper. — There has always been a call 
for a sensitive paper which will give positive prints, — a black, 
a brown, or a blue line on a white background. Such effect was 
secured by the old so-called "black print process," but its opera- 
tion was not altogether simple and good results were not reason- 



394 DRAFTING INSTRUMENTS [Chap. XIV. 

ably sure. The Vandyke paper has apparently solved this 
difficulty,* and in addition affords other advantages which the 
old "black process " paper did not possess. 

Vandyke paper is a sensitized paper which is printed in the 
same way as a blue-print, except that the tracing is put into the 
frame so that the ink lines will be against the Vandyke paper. 
The exposure is about 5 minutes in direct sunlight or, more defi- 
nitely, until the portion of the Vandyke paper which protrudes 
beyond the tracing is a rich dark tan color. Fresh Vandyke paper 
is light yellpw in color. The print is washed for about 5 minutes 
in clear water (where it grows lighter in color) and then it is put 
into a solution consisting of about one-half ounce of fixing salt 
(hyposulphite of soda) to one quart of water, where it turns dark 
brown. It is left in the fixing bath about 5 minutes, after which 
the print is again washed in water for 20 to 30 minutes and then 
hung up to dry. The fixing solution may be applied with a 
sponge or brush if only a few Vandykes are being made, but it is 
better to immerse them in a tank containing the solution. 

After the Vandyke print is washed the body is dark brown in 
color while the lines are white. This is not the final print to be 
sent out; it is simply the negative. 

This Vandyke print is then put into the printing-frame in 
place of the tracing, the face of the Vandyke being next to the 
sensitive side of the process paper, and from it as many prints as 
are desired are made on blue-print paper or on any kind of sen- 
sitized paper desired. These blue-prints made from Vandykes 
have a white background while the lines of the drawing appear 
in deep blue lines, for in this case the rays of the sun act only 
through the white parts of the Vandyke (the lines), whereas in 
making an ordinary blue-print from a tracing the sun's rays act on 
the paper through all parts of the tracing cloth except where the 
lines appear. Where brown lines on a white background are 
desired, the print is made by using a sensitized sheet of Vandyke 
paper, in place of the blue- print paper. 

One of the advantages of this process is that, as soon as a Van- 
dyke has been made from the tracing, the tracing can be filed 
away and kept in excellent condition, the Vandyke being used in 
making all prints. 



VANDYKE PRINTS 395 

Another advantage in the use of the blue-prints which hare 
been made by this process is that any additions made in pencil or 
ink show clearly on the white background of the print which is 
not true of the ordinary blue-print, on which corrections must be 
made with a bleaching fluid or water-color. 

442. Electrical Printing Frames. — The uncertainty of the 
sunlight for making prints has brought forward a printing frame 
in which an artificial light is used. 

One form of electrical printing frame is an apparatus con- 
sisting of a hollow glass cylinder, formed of two sections of glass, 
and resting on a circular base which is rotated by clock work. 
An electric light is suspended in the center line of the cylinder 
where it travels up and down by means of a clock work attach- 
ment. 

The tracing and paper are wrapped around the outer sur- 
face of the glass where they are tightly held against the glass by 
a canvas which is wound around the cylinder by means of a 
vertical roller operated by a handwheel. The cylinder can be 
rotated at any desired speed and the light which travels up and 
down the axis of the cylinder can be moved through any desired 
distance or at any desired speed. These motions are all made 
automatically when the apparatus is once adjusted. 

In another type of electrical machine several horizontal rollers 
are provided, with the light so arranged that as the tracing and 
blue-print paper passes from one roller to another the exposure 
is made. The speed of the machine is controllable and the length 
of the tracing that can be printed is limited only by the length of 
the roll of blue-print paper. With this machine, then, long 
plans or profiles can be printed without the necessity of frequent 
splicing which is required with other types of printing frame; 
furthermore the color of the print is also uniform throughout. 
The machine is driven by an electric motor. There are several 
machines of this general type on the market; some of them are 
provided with an apparatus for washing the prints as fast as they 
come from the machine. 

443. INKS AND WATER-COLORS. — Bottled ink, which is 
prepared in various colors, is used extensively on engineering 
drawings. The so-called "waterproof" inks differ from other 



396 DRAFTINfe INSTRUMENTS [Chap. XIV. 

inks in that a water-color wash can be put over the lines without 
causing them to "run." Bottled inks are satisfactory for most 
drawings, but when very sharp and fine hair-lines are required it 
is well to use the stick india ink. This is made by grinding the ink 
together with a little water in a saucer made for this purpose, 
until the ink is thick and black enough to be used. If the ink 
becomes dry it can be restored to as good condition as when first 
ground by adding water, a drop or two at a time, and rubbing it 
with a piece of cork or a pestle; if the water is added too rapidly 
the ink will flake. 

While the bottled black inks are fairly well prepared, the red 
inks are very unsatisfactory. They will sometimes run on paper 
where only very slight erasures have been made; in fact, on some 
of the cheaper papers red ink will always run. For tracing pur- 
poses red ink is wholly unsatisfactory, as it is impossible to ob- 
tain a good reproduction of a red ink line by any of the 
process prints. Where red lines are needed the use of scarlet 
vermilion water-color will te found to give not only a brilliant 
red line on the tracing, but also **body '' enough in the color 
so that the lines will print fully as well as the black ink lines. 
Scarlet vermilion water-color will give much better lines on any 
paper than the bottled red inks. Only enough water should be 
used to make the water-color flow well in the pen. Other water- 
colors are used in th ^ place of the bottled colored inks, such as 
P. ujsian blue i ste d of bottled blue i :k, or ^nt sienna instead 
of brow:i ink, and these give much better results. 

It is frequently necessary on blue-prints to represent additions 
in white, red, or yellow. A white line can easily be put on by 
using Chinese white water-color; but sometimes a bleaching fluid 
is used which bleaches out the blue leaving the white paper visible. 
The best color for a red line on blue-prints is scarlet vermilion 
water-color; and for a yellow line none of the ordinary yellow 
water-colors gives as brilliant lines as Schoenfeld & Co.'s lighi 
chrome yellow. 

For tinting drawings water-colors and dilute inks are used. 
Effective tinting may be done on tracings by using colored pencils 
on the rough side of the linen. 



CHAPTER XV. 

METBODS OF PLOTTING. 

444. LAYDf G OUT A PLAN. — Laying out a plan requires 
careful work. If a good-looking plan is to be obtained this part 
of the work must be done with not a little judgment. Besides the 
plan of the survey or property the drawing must have a title, and 
sometimes notes and a needle to show the direction of the merid- 
ian. These must all be arranged so that the entire drawing 
when completed will have a symmetrical appearance. Often 
the plot is of such awkward shape that it is very difficult to lay 
out the drawing so that it will look well, and the draftsman's 
artistic instincts are taxed to the utmost to produce a satis- 
factory result. 

445. Scale. — In many cases the scale of the plan as well as 
the general arrangement of its parts must be chosen by the 
engineer. Surveys of considerable extent which do not contain a 
great many details, such, for example, as the preliminary survey 
for a railroad, may be drawn to a scale of 400 ft. to an inch. A 
plan of a large piece of woodland or a topographical map of a 
section of a town may be represented on a scale of from ico ft. 
to 400 ft. to an inch. A plan of a city lot for a deed is represented 
on a 20- ft. to 80- ft. scale; and city streets, such as sewer plans 
and the like, are frequently drawn to a scale of 20 ft. to 40 ft. to 
an inch. Sometimes on plans of construction work drawings of 
different scale are made on the same sheet. The drawing for a 
conduit, for example, may be represented by a general plan on 
a scale of 80 ft. to an inch, while on the same sheet the conduit 
may be shown in section on a scale of 4 ft. to an inch. 

The field maps of the U. S. Coast and Geodetic Survey are 
usually plotted on a scale of TTriirTy, but some special maps are 
made on scales as large as ^uVtt. The field maps of the U. S. 
Geological Survey are mostly plotted to a scale of rahj^-s and 
reduced on the lithograph sheets to v-^iis-c or i-^-^vjf- 

397 



398 . METHODS OF PLOTTING [Chap. XV. 

These remarks in regard to scales are not to be con- 
sidered in any sense as hard and fast rules to govern all condi- 
tions They are suggested simply to give some idea of the exist- 
ing practice in this matter. 

METHODS OF PLOTTmO TRAVERSES. 

446. PLOTTING BY PROTRACTOR AHD SCALE. —The 

most common method of plotting angles is by use of the pro- 
tractor (Art. 417, p. 382), and of plotting distances, by use of the 
engineer's scale. Every traverse consists of a series of straight 
lines and angles, which can be plotted by a protractor in the fol- 
lowing manner. First, the survey to be mapped should be 
sketched out roughly to scale, in order to ascertain its extent and 
shape so as to decide the size of paper necessary for any given 
scale of drawing and to determine its general position on the 
sheet, which will fix the direction of the first line of the traverse, 
to be used as a starting line for the entire drawing. This having 
been done, the first line is drawn in the proper place on the paper, 
its length is scaled off by using the proper scale, and its two ex- 
tremities accurately marked by pencil dots or by means of a 
needle point, and surrounded by a light penciled circle. The line 
should be drawn so that it will extend beyond the next angle 
point a distance greater than the radius of the protractor, this 
extension of line being of use in the manipulation of the pro- 
tractor. 

The protractor is placed so that its center is exactly on the 
second angle point and so that both the 0° and i8c° marks of the 
protractor exactly coincide with the line. The traverse angle taken 
from the field notes is plotted, the protractor removed, the line 
drawn, and the length of the second course carefully scaled. 
Then the protractor is placed along this new line and opposite 
the third point, the angle at that point is laid off, the next line 
drawn, and the distance scaled. By this process the entire 
traverse is plotted. 

447. Checks. — On all plotting work, just as on all field- 
work and computations, frequent checks should be applied to 
insure accuracy. 



PROTRACTOR AND SCALE 399 

If the traverse is a closed traverse the plot, of course, should 
dose on the paper.* If it does not and the error of closure is in a 
direction parallel to any one of the lines, there is probably a mis- 
take in plotting the length of that line. If there is no indication 
of this sort the mistake .may be either in scaling, in laying oflF the 
angles, or in both. In such a case the entire plot should be 
checked unless there is some reason to think that a certain line 
may have been laid off at the wrong angle, in which event that 
questionable angle should be replotted. The bearings of all the 
lines of the traverse can be computed with reference to the mag- 
netic or to any assumed meridian ; any line can be produced to 
meet the meridian line, and this angle measured and checked. 
Similarly, the bearing of the last line of a traverse which dDes 
not close can be computed and the angle the last line makes with 
the meridian measured. If it checks the computed angle it is 
evident that no error has been made in the angles uiless mis- 
takes were made that exactly balance each other, which is not 
probable. In this way, by "cutting into" the drawing here 
and there, the angular error, if there is one, can be quickly **run 
down," without laying out all of the angles again and so possibly 
repeating the mistake that was originally made. The angles 
measured in applying this check have different values from the 
ones first laid out, and the chance of repeating the original mis- 
take is thereby eliminated. If no error is found to exist in the 
angles, the distances should next be checked. This can be done 
in two ways, and in some drawings both of these checks should 
be applied. 

First, scale each line separately setting down the results in- 
dependently upon a sheet of paper. After these are all recorded 
(and not before), compare the lengths with the lengths of lines as 
taken from the field notes. No error should be allowed to pass 
if it is large enough to be readily plotted by the use of the scale. 

* Instead of plotting every line of the traverse from its preceding line and 
returning, in the case of a closed traverse, to the other end of the starting line, it 
may be well to plot half the traverse from one end of the starting line and the 
other half from the other end ; the check will then come at a point about half-way 
around the traverse. The advantage of this method lies in the fact that accumu- 
lative errors are to some extent avoided since they are carried through only half 
a8 many courses. 



400 METHODS OF PLOTTING [Chap. XV. 

Second, take a long straight piece of paper, lay this on the 
drawing, and mark off tlie length of the first line on the edge of 
the paper; then mark off the length of the second Une starting 
from the mark which denotes the end of the first hne, and proceed 
in a similar way to the end of the traverse. Apply the scale to 
the strip of paper and read the station of each mark; record each 
of these independently and afterwards compare them with the 
field notes. The entire length of Une should check within a 
reasonable amount depending upon the scale; the allowable 
error can easily be determined by the principle explained in 
Art. 23, p. 14. 

By checking angles and distances by the above methods 
errors of any consequence can be avoided; in any case a drafts- 
man should not allow a drawing to leave his hands which has 
not been properly checked and known to be correct. 

When the traverse is not closed, such checks as have been 
described above must always be applied; otherwise there is no 
assurance whatever that the plan is correct. It is especially 
necessary to check the bearings of lines frequently, so that the 
accumulation of small errors may not become appreciable. 

448. Protractor and T-Square. — While the ordinary 
T-square is not much used in plotting engineering plans, there 
are some occasions where it is convenient to use it. Where a 
traverse has been run by bearings or by deflection angles the 
T-square with a shifting head can be conveniently used in con- 
nection with a protractor for plotting the angles by bearings. 

The piper is fastened to a drawing board having a metal edge, 
which insures one straight edge to the board. A meridian line 
is drawn on the paper, and the shifting head of the T-square 
is fastened so that the blade coincides with the meridian line. 
Then as the T-square is slid up and down the edge of the draw- 
ing board its blade always takes a direction parallel to the meri- 
dian. By means of the protractor shown in Fig. 172 the bear- 
ing of each line can be readily hid off or checked as illustrated 
by Fig. 174 and the distances laid off with the scale. In order 
to secure a satisfactory check, the deflection angles should be 
laid off directly from the previous line, and the bearings checked 
by means of the T-square and protractor. 



T-SQUARE AND PROTRACTOR 



40t 



It is evident that the bearings of the lines may be computed 
just as well from any assumed meridian as from the magnetic or 
true meridian ; and that the drawing can be fastened to the board 




Fig. 174. Laying off Bearings by Use of T-Square and 
Protractor. 



in such a way that the T-square can be conveniently used. 
This method is especially applicable to compass surveys as it ob- 
viates the necessity of drawing a new meridian line through each 
angle point. 

This method can be easily applied also as a means of check- 
ing any of the angles of a traverse which have been plotted by 
any of the ordinary methods. 

449. PLOTTING BY RECTANGULAR COORDINATES. — In plot- 
ting by this system all points in the traverse are referred to a 
pair of coordinate axes. For convenience thess axes are often 
the same as those used in calculating the area enclosed by the 
traverse. The advantages of this method are, (i) that all meas- 
urements are made by means of the scale only and (2) that the 
plotting may be readily checked. 

To plot a survey of a field by rectangular coordinates, first 
calculate the total latitude and the total departure^ that is, the 
ordinate and the abscissa, of eich point in the survey. If the 
meridian through the most westerly point and the perpendicular 
through the most southerly point are chosen as the axes negative 



402 



METHODS OF PLOTTING 



[Chap. XV. 



signs in the coordinates will be avoided. The coordinates of the 
transit points are computed by beginning with the most westerly 
point, whose total departure is zero, and adding successively the 
departure of each of the courses around the traverse. East 
departures are called positive and West departures negative. 
The total departure of the starting point as computed from 
that of the preceding point will be zero if no mistake is made in 
the computations. The total latitudes may be computed in a 
similar manner beginning, preferably, with the most southerly 
point as zero. 

450. For plotting the points on the plan, a convenient 
method of procedure is to construct a rectangle whose height 
equals the difference in latitude of the most northerly and the 
most southerly points and whose width equals the difference in 
departure of the most westerly and the most easterly points. If 
the most westerly and the most southerly points are taken as zero 
then the greatest ordinate and the greatest abscissa give the 
dimensions of the rectangle. The right angles should be laid 
off either by the use of a reliable straight-edge and a triangle or 
by the beam compass. 

451. The better method, however, is to construct the 
perpendiculars by means of a straight-edge and a triangle. It is 




Fig. 176. Erecting a Perpendicular with a Straight-edge and 
AN Inaccurate Triangle. 



RECTANGULAR COORDINATES 403 

not at all necessary, although it is always desirable, that the 
triangle shall be accurate. It should be used in the following 
manner. It is first placed against the straight-edge, as shown by 
the full lines in Fig. 175, and a point A, marked on the paper. 
Point C is also marked opposite a certain definite part of the 
triangle. Then the triangle is reversed to the dotted position 
and brought so that its edge coincides with point A, and then 
point B is marked opposite point C, as nearly as can be judged. 
A point D is plotted midway between B and C and the line AD 
is then drawn which is perpendicular to the straight-edge. If the 
triangle is accurate point B will fall on point C, so that this 
is a method of testing the accuracy of the right angle of any tri- 
angle. If it is found to be inaccurate it should be sent to an 
instrument maker and be "trued up.** A few cents spent in 
keeping drafting instruments in shape will save hours of time 
trying to locate small errors, which are often due to the inac- 
curacy of the instruments used. 

If the compass is used the right angle may be laid off by 
geometric construction. On account of the difficulty of judging 
the points of intersection of the arcs, very careful work is re- 
quired to obtain good results with the compass. 

Since the accuracy of all of the subsequent work of a coordi- 
nate plot depends upon the accuracy with which the rectangle 
is constructed, great care should be taken to check this part of 
the work. The opposite sides of the rectangle should be equal 
and the two diagonals should be equal, and these conditions 
should be tested by scaling or with a beam compass before con- 
tinuing with the plot. 

452. After the rectangle has been constructed, all points in 
the survey can be plotted by use of the scale and straight-edge. 
To plot any point, lay off its total latitude on both the easterly 
and the westerly of the two meridian lines of the rectangle, be- 
ginning at the southerly line of the rectangle. Draw a line 
through both of these points by means of a straight-edge.* 

* Accurate work, of course, cannot be obtained with a straight-edge that is 
not true. A straight-edge can easily be tested by drawing a fine pencil line on the 
paper along one edge of the straight-edge; then turn the straight-edge over on 
its other side, fit the same edge to the two ends of the pencil line, and see if the 
edge coincides with the line. 



404 



METHODS OF PLOTTING 



[Chap. XV. 



Then lay off along this line the total departure, beginning at the 
westerly side of the rectangle, thus obtaining the desired position 
of the point. 

The computations of the total latitudes and departures and 
the method of plotting a traverse by the coordinate method are 
shown in Fig. 176. This is the survey which is shown in the 



we.71 >G 




< 6IZ32 > 

Fig. 176. Computations and Plotting by Rectangular 
Coordinates. 

calculations in Fig. 161, p. 362, and in the form of notes in Fig. 
52, p. 103. 

453. Plotting by rectangular coordinates is the most accurate 
of all the methods usually employed. It is not very often ap- 
plied, however, to traverses which do not close, as there is seldom 
any other use for the coordinates of such a traverse, and the 



RECTANGULAR COORDINATES 405 

labor of computing them for this purpose alone is hardly war- 
ranted. For such traverses, therefore, either the protractor and 
scale, the Tangent Method, or the Chord Method (which are ex- 
plained in the following articles) may be employed. But for 
plans of a closed traverse, where the latitudes and departures 
have been computed in cpnnection with calculating its area, this 
coordinate system of plotting is frequently used. 

454. Checks. — When the transit points have been plotted, 
the scale distance between consecutive points should equal the 
distance measured in the field. It sometimes happens that some 
of the transit lines run so nearly parallel to one of the axes that 
the distances will scale the right amount even though a mistake 
has been made in laying off one of the coordinates. In such a 
case any appreciable error can be detected by testing the bear- 
ings of the lines by means of a protractor. These two tests, 
together with the scaled distances of any cut-off lines which may 
have been measured in the field, (Art. 145, p. 109), form a good 
check on the accuracy of the plotting. Since all of the points 
are plotted independently errors cannot accumulate. If it is 
found that any scaled distance fails to check with the measured 
distance it is probable that one of the two adjacent lines will also 
fail to check and that the point common to the two erroneous 
lines is in the wrong position. 

It should be remembered that everything depends upon the 
accuracy of the rectangle and that nothing should be plotted 
until it is certain that the right-angles have been accurately laid 
off. 

455. PLOTTING BY TANGENTS. — The traverse should first be 
plotted approximately on some convenient small scale by use of 
the protractor and scale, to ascertain its extent and shape. The 
importance of this little plot is often overlooked, with the result 
that when the plan is completed it is found to be too close to one 
edge of the paper or otherwise awkwardly located on the sheet. 
It takes only a few moments to draw such a sketch, and unless 
the draftsman is sure of the shape and extent of the plot he' 
should always determine it in some such manner before the plan 
is started. 

The directions of all the lines are referred to some meridian 



4o6 



METHODS OF PLOTTING 



[Chap. XV. 



and the bearings determined with an accuracy consistent with the 
measured angles. From the auxiliary plot it can be decided 
where to start the first course of the traverse on the paper and 
in what direction to draw the meridian, so that the lines of the 
completed traverse will be well balanced with the edges of the 
sheet, and so that the needle will be pointing, in a general way, 
toward the top of the drawing rather than toward the bottom. 
The bearing of the first line is plotted as follows (Fig. 177). 




Fig. 177. Plotting by Tangent Offsets. 

Lay off on the meridian line a length Aa of at least 10 inches 
and erect a perpendicular at a on the right-hand side of the 
meridian if the bearing of the first course is east, and on the 
left-hand side if it is west. Look up in the table of natural 
functions the tangent of the bearing of the first course and scale 
off this distance ai on the perpendicular.* Draw A6 which is 



* These distances and also the lo-inch base-lines are all laid off by use of the 
engineer*s scale. By using the lo-ft. or loo-ft. scale the tangents can be laid off 
without any computation, whereas with the other scales the tangent must be multi- 
plied by some number, e.g., by 2 if the 20-ft. scale is used, by 3 if the 30-ft. scale 
is used, etc., taking care in the pointing off. 

If it is deemed unnecessary to use a base as long as 10 inches, one can be laid 
off at the *' 10 ** mark on any engineer's scale and the tangent distances laid off by 
using the same scale, e.g., if a 20-ft. scale is used the ^ 10 " mark will give a base- 
line 5 inches long. 



PLOTTING BY TANGENTS 407 

the direction of the first course. On this line scale off ABy the 
length of the first course. On this line produced lay off Bg equal 
to 10 inches and erect a perpendicular, scaling off on the perpen- 
dicular the length gd equal to the tangent of the deflection angle 
at B, This determines the direction of ^Cfrom the first course. 
The remaining lines of the traverse are plotted in the same manner, 
using each time the deflection angle. 

456. Checks. — Unless the survey is a closed traverse checks 
must be occasionally applied. Every third or fourth course 
should be checked by finding the angle between it and the meri- 
dian line. This angle should be found by the same method 
(tangent offset method) and by using a base of 10 inches as in 
plotting the angles. In checking the course £>e, for example, a 
meridian is drawn through Z> parallel to Auy De is scaled off 10 
inches, and a perpendicular ef erected. The distance ef is scaled 
and from the table of tangents the angle /De is obtained. If the 
angle that the course makes with the meridian line disagrees 
with the calculated bearing of that course by any considerable 
amount, say, 10 minutes of angle or more, the previous courses 
should be replotted. If the error is less than 10 minutes the 
course which is being checked should be drawn in the correct 
direction so that even the slight error discovered may not be 
carried further along in the plot. Then after the plotting has 
proceeded for three or four more courses the check is again 
applied. 

The bearings of the lines can be checked by use of the pro- 
tractor and this will detect errors of any considerable size, but 
this method will not disclose any small errors ; moreover, if it is 
desired to have the plot when completed as accurate as could be 
expected from the precise method employed, it is entirely incon- 
sistent to check by use of a method which is far less accurate 
than the one used in making the plot. For this reason the 
checks on the direction of the lines are applied with the same 
care and by the same method as was used in the original layout 
of the angles. 

Occasionally it is more convenient to plot the complement of 
an angle rather than the angle itself, as was done in plotting the 
line EF, In this case the right angle erected at E must be laid 



4o8 



METHODS OF PLOTTING 



[Chap. XV. 



off with great care, preferably by the method explained in Art. 
45 1, p. 402. 

It is evident that the direction of each course could have 
been plotted by drawing a meridian line through the transit points 
and by laying off the bearings by the tangent method. But if 
such a method were used there would be no single check applied 
that would check all the previous courses, which is an important 
feature of the method explained above. 

If the traverse is not closed the lengths of the lines of the 
traverse should always be checked by the methods explained in 
Art. 447, p. 398. 

457- PLOTTING BY CHORDS. — This method, which is em- 
ployed by many draftsmen in plotting traverse lines, is fairly 
good although probably not so accurate as the Coordinate or 
as convenient as the Tangent Methods. 

Fig. 1 78 represents the traverse ABCDEF which has been 




Fig. 178. Plotting by Chords. 

plotted by chords. It is the same traverse that is shown in 
Fig. 177. 

On the meridian line the distance Aa is scaled off equal to 
10 inches and the arc ab swung from -4 as a center by use of 
the ordinary pencil compass. Then from a table of chords * the 

* Tables of chords can be found in Trautwine*s " Civil Engineer's Pocket 
Book," published by John Wiley & Sons, New York. 



PLOTTING BY CHORDS 409 

length of the chord ab is found for the angle aAb, The point b 
is sometimes located by setting the dividers at the distance ab 
and with ^z as a center intersecting the arc ^ at ^; but the 
more accurate method is to scale from point a the chord distance 
and mark the point b on the arc. Then the line Ab is drawn and 
AB scaled off on it. With 5 as a center the arc gd is drawn 
and the chord gd, corresponding to the deflection angle at 5, is 
scaled off. Bd is then drawn and BC scaled off on it. In the 
same way the entire traverse is plotted. 

458. Use of the Sine, — It is evident that the chord 

ab^2y. lox sm — • 
2' 

hence, if a table of chords is not available, a table of sines (always 
easily obtainable) can be used. The sine of half the angle can 
be taken from the tables and multiplied by 20 mentally. Some 
draftsmen use the table of sines and a radius of 5 inches to avoid 
the multiplication. This is not recommended because a base of 
5 inches is not long enough to insure a very accurate drawing. 
The necessity of multiplying by 2 can very easily be done aWay 
with by laying off the radius with a 2o-ft. scale and scaling off 
the sine of the angle with a lo-ft. scale. 

With dividers of the ordinary size it is impossible to lay out 
an arc with a lo-inch radius. In such a case either beam com- 
passes must be used or the radius employed must be shorter, so 
short, in fact, that it will frequently be better to resort to the 
Tangent Method. 

459, Checks. — Since this method is usually applied to 
traverses which do not close it is desirable to check every 
fourth or fifth course so that a mistake will not be carried too 
far before it is discovered and thereby cause a waste of time. 
In Fig. 178 it is desired to check the calculated bearing of De, 
The meridian Df is drawn through D parallel to Aa, the arc fe 
is swung with Z^ as a center and with a radius of 10 inches, and 
the chord ef\^ scaled. From the table of chords (or sines) the 
angle fDe (the bearing) can be found. It should agree reason- 
ably well with the calculated bearing. The degree of precision 
to be expected when plotting by chords is a little less than 



4IO METHODS OF PLOTTING [Chap. XV. 

that suggested for the Tangent Method in Art. 455, unless the 
beam compass is used. The Tangent Method, especially if the 
right angles are laid off by reversing the triangle, gives more 
accurate results than the Chord Method, for the use of the 
ordinary compass in the Chord Method is a fruitful source of 
error unless it is handled with the utmost care. 



METHOD OF PLOTTING DETAILS. 

460- BUILDINGS, FENCES, STREAMS, ETC. — The previous 
articles have dealt with the plotting of the traverse lines only, 
and these in many cases form merely the skeleton of the final 
plan. In the field the details of the survey are located from the 
transit line ; and, in a similar manner, the details are located on 
the plan from the traverse line which has already been plotted. 

Buildings, fences, shore-lines, streams, etc. are all plotted by 
means of the scale for distances and the protractor for the 
angles. Often a smaller protractor is used for this sort of work 
than for the traverse lines. This is permissible, for the lines 
which locate the details are usually short in comparison with 
the traverse lines and the resulting error is small in any case ; 
furthermore any slight error in the location of a detail will not 
as a rule affect the rest of the drawing, whereas an error in a 
transit line will, of course, have an effect on all of the rest of 
the drawing. The plotting of buildings has been taken up in 
connection with their location. (See Chapter VI.) 

In plotting a set of notes where several angles have been 
taken at one point, such as in stadia surveying, it is well to plot 
all of the angles first, marking them by number or by their 
value, and then to plot the distances with the scale. 

461. Contours. — Where contours are located by the cross- 
section method (Art. 304, p. 278), this cross-section system is 
laid out in soft penciled lines on the drawing. The elevations 
which were taken are written at their respective points on the 
plan and then the contours desired are sketched. The ground 
is assumed to slope uniformly between adjacent elevations, and, 
by interpolation between these points, the location of the con- 
tours on the plan can be made. When the contours have been 



PLOTTING DETAILS 4 1 1 

located, the cross-section lines and elevations are erased unless 
the plan is intended to be used as a working drawing. As a 
rule all useful data, such as construction lines and dimensions, 
are left on a working drawing. 

When the contours are located by any other means the 
principle is the same. The points whose elevations have been 
determined are plotted by scale and protractor, and the contours 
are interpolated between the elevatibns and sketched on the 
plan. 

462. CROSS-SECTIONS. — In plotting on cross-section paper, 
the rulings of the paper are used as the scale, and all the dimen- 
sions of the cross-section, which are to be plotted, are laid off by 
counting the number of squares on the cross-section paper. 

In highway, railroad, and dam construction it is often neces- 
sary to keep a record of the progress made on the earthwork by 
plotting the cross-section at each station, and, as the work goes 
on, to mark on each section in colored ink the progress of the 
work for each month. In this way monthly estimates can be 
readily made, and the cross-section sheets will also give a record 
of the progress of the work, each month being represented by a 
different colored line or by a different style of line. 

Where a series of cross-sections like this are to be plotted 
the station number and the elevation of the finished grade are 
recorded just under or ovier the section. To avoid mistakes in 
numbering the sections this should be done at the time of plot- 
ting the section. 

As these cross-section sheets rarely go outside the office they 
are usually considered in the same class with working drawings, 
and dimensions, such as the areas of sections or the quantities 
of earthwork, are usually recorded on them, together with any 
other data which may be of use in calculating the volumes. 

463. Profiles. — Profiles are almost always plotted on pro- 
file paper, although occasionally they are plotted on the same 
sheet with the plan so that the two can be readily compared. 

The profile is intended to show (graphically) relative eleva- 
tions. In most surveys the differences in elevation are so small 
in comparison with the horizontal distances that it is necessary 
to exaggerate the vertical scale of the profile so that the eleva- 



412 METHODS OF PLOTTING [Chap. XV. 

tions can be read from the profile With a reasonable degree of 
accuracy. The horizontal scale of the profile should be the same 
as the scale of the plan, but the vertical scale should be exagger- 
ated, say, 5 to 20 times the horizontal scale, depending upon 
how close it is desired to read the elevations from the drawing. 
If the horizontal scale of the profile is 80 ft. to an inch its ver- 
tical scale should probably be 20, 10, or 8 ft. to an inch. 

464. In plotting any 'profile the first step is to lay it out 
properly on the paper, i.e., to decide, from an examination of the 
range of the elevations, where to start it on the paper so that it 
will look well when completed, and so that any additions or 
studies which may subsequently be drawn on it will come within 
the limits of the paper. Station o of the profile should come on 
one of the heavy vertical lines, and the heavy horizontal lines 
should represent some even elevation such as 100, 125, 150, etc. 

The profile is plotted by using the rulings of the profile 
paper as a scale ; it is drawn in pencil first and afterward inked 
in. It will be found, if these profile papers are carefully meas- 
ured with a scale, that they are not as a rule very accurate. 
The rulings may be uniform, but owing to the shrinkage of the 
paper the divisions frequently do not scale as long as they should. 
In plotting a profile or section on such paper no attempt is made 
to use a scale ; the scale of the paper is assumed to be correct 
and the intermediate points are plotted by estimation, which can 
almost always be accurately done since the rulings of the paper 
are quite close together. 

The data for a profile of the ground generally consist of 
levels taken in the field at such points that the ground may be 
assumed to run straight between adjacent elevations. For this 
reason, in drawing the profile, the points where the slope of the 
ground changes should not be rounded off. On the other hand, 
however, the ground probably does not come to an actual angle 
at that point. The profile should be plotted therefore as a series 
of free-hand straight lines drawn so that the angles are not 
emphasized. When a profile is made from a contour map, the 
line should be a smooth, rather than an angular line. 

465. Profiles of the surface of the ground are generally 
made for the purpose of studying some proposed construction 



J 



PLOTTING PROFILES 413 

which is represented on the profile by a grade line, consisting 
usually of a series of straight lines. The points where the gra- 
dient changes are plotted and connected by straight ruled lines 
unless the proposed grade should happen to be a vertical curve 
(Art. 268, p. 242). Vertical lines are also drawn from the bot;tom 
of the profile to the grade line at these points. 

466. When the elevations are such that the profile, if con- 
tinued, will run off the top or bottom of the paper the entire 
surface line is lowered or raised some even number of feet, such 
as 20 or 50 ft., and the plotting continued : the number of feet 
represented between two heavy horizontal rulings of the pro- 
file paper should determine the drop or rise of the grade line. 
This change should be made, when convenient, on one of the 
heavy vertical rulings of the piaper or on one of the vertical lines 
where the gradient changes. 

467. Checks. — After plotting the surface aind grade eleva- 
tions in pencil, read off from the profile the station and elevation 
of each point as plotted and record both the station and eleva- 
tion on a piece of paper. Compare these readings with the data 
given and make the necessary corrections. Time can be saved 
if one man reads off the station and elevation from the profile 
while a second man compares the readings with the note-book. 
A quick method of plotting profiles is to have one man read the 
notes while the other man plots them, but when the profile is 
being checked this method should not be used ; the man, prefer- 
ably the one who did not do the plotting, should read from the 
profile as plotted and these readings should be compared with the 
note-book. 

PROBLEMS. 

1. Plot the surveys given in Fig. 50, p. 100, and in Fig. 53, p. 104, by Pro- 
tractor and Scale, Rectangular Coordinates, Tangents, or Chords. 

2. Plot by use of Scale and Protractor the notes given in Fig. 72, p. 168, and 
in Fig. 116, p. 266. 



Miw)Mt Fork Conmissioit 



- s 

M 




^^"^Z? 



CHAPTER XVL 

FINISHINa AND nUNG DRAWINGS. 

468, WHAT SHOULD APPEAR ON A DRAWING. — Drawings 
are made for a great variety of purposes, so that the data which 
a plan should contain depend entirely upon the use to which it 
is to be put. There are, however, several important things which 
should appear on every engineering drawing. In the first place, 
it should have a complete title which should be a brief descrip- 
tion of the drawing. The title should state whether the drawing 
is a plan, cross-section, profile, etc. ; what it represents, — a lot 
of land, a sewer, a railroad, etc. ; the name of the owner ; the 
place ; the date ; the scale ; and the name of the surveyor. Be- 
sides the title, some plans, such as land plans, always require the 
names of owners of abutting property, and a meridian. Notes 
are frequently added giving such information as is necessary to 
interpret the plan. All essential dimensions are lettered in their 
proper places. 

Besides these it is well to insert in some inconspicuous place 
(preferably near the border) the number of the note-book and the 
page from which the notes were plotted, and also the initials of 
the draftsman who made the drawing and of the man who 
checked it. 

Fig. 1 79 represents a land plan which contains all of the es- 
sentials ; it is a plot of the land shown in the form of notes in 
Fig- 52, p. 103; its computations are on p. 362; and its working 
plot is illustrated by Fig. 176, p. 404. 

469, Traverse lines. — The convenient use of a plan some- 
times requires the traverse line to be shown on the completed 
drawing. In such a case it is usually shown as a full colored line, 
each of the angle points being represented by a very small circle 
of the same color, the center of which marks the angle point. 
Sometimes the lines of the traverse are drawn to the angle points 

415 



4i6 



FINISHING AND HLING DRAWINGS [Chap. XVL 



which are marked by very short lines bisecting the angles. Fig. 
1 80 illustrates these two methods of marking transit points. 




Fig. 180. Methods of Marking Angle Points on Traverse Lines. 



Tnangulation stations are represented by a small equilateral 
triangle drawn around the station point. Fig. 115, p. 258, con- 
tains several examples of this. 

470. PHYSICAL FEATURES. — The boundaries of property 
and the physical features which are represented on a plan, such 
as streets, buildings, etc., are usually drawn in black ink. Any 
additions or proposed changes are frequently drawn in colored 
ink, usually in red, although water-color is much better for 
the reasons stated in Art. 443, p. 395. 

Shore lines and brooks are represented either in black or in 
Prussian blue. As a rule the shore line should be one of the 
heaviest, if not the heaviest line, on the drawing. Water-lining, 
shown in the topographical signs in Fig. 181, adds materially to 
the prominence and appearance of a shore line. 

471. TOPOGRAPHIC CONVENTIONAL SIGNS. — On topo- 
graphic maps certain physical features are shown by conventional 
signs which have come to be used so generally that they are 
practically standard throughout the country. A few of the 
more common of these symbols are shown in Fig. 181. The 
one representing " cultivated land " and the horizontal lines of 
the "salt marsh" and "fresh marsh" symbols are ruled; the 
rest are executed with an ordinary pen, Gillott's No. 303 being 
a good one for such work. 

It will be noticed that in the symbol for " grass " the indi- 
vidual lines of a group all radiate from a center below the group, 
and also that they end on a horizontal line at the bottom. This 



CONVENTIONAL SIGNS 



417 






^ 






fJ!» 






.fta.%^ 



^aiu^gtiA. 



Deciduous Trees (Oak). 



-•IM -'^ -«^ 




•Wl*. 


.*1X*. 


.^1//- «ilM, 


*»»/A. 


<iU^ 




^•^«^ *^ 




*iU<». 




M. 


.^■». 


ai 


^Mld. 




*»w*. 






^//. 




4M4» 


#/A» 




*•/«* *'*'* 






.«!//*. 


aiiK*. 


^MM 












-:£& 



Grass. 




Fresh Marsh. 




Waterlining. 



-r^X^\t 









JU "m OJ •*. • ,/, ^ * 



Deciduous Trees (Round Leaf), 



v?i.s K s \ \ .s \ \ sX X A 



Cultivated Land. 




Salt Marsh — Sand. 










Ledges — Evergreen Trees. 



Fig. 181. Topographic Conventional Signs. 



41 8 FINISHING AND FILING DRAWINGS [Chap. XVI. 

horizontal line, in the case of "grass" or "marsh" symbols, 
should always be parallel to the bottom of the map. 

In executing "water-lining " the first line outside the shore 
line should be a light full line drawn just as close to the shore 
line as possible, and should follow very carefully every irregular- 
ity of the shore line. The next water-line should be drawn 
parallel to the first but with a little more space between them 
than was left between the shore line and the first water-line. 
Then the third water-line should be spaced a little farther out, 
and so on ; five to ten lines are sufficient to represent this sym- 
bol properly. As the succession lines are added farther and 
farther from the shore line, the little irregularities of the shore 
gradually disappear until the outer water-line shows only a few 
irregularities opposite the most prominent ones of the shore. 

Wat^r-lining, as well as fresh marsh and salt marsh symbols, 
is often represented in Prussian blue. In fact, on some topo- 
graphic maps most of the signs are represented by colors, — 
the trees by green, the grass by a light green tint, water by a 
light blue tint, cultivated land by yellow ochre, and so on. 

Contour lines (shown in several of the cuts in Chapter X.) 
are almost always drawn in burnt sienna water-color. Every 
fifth or tenth contour is usually represented by a line slightly 
heavier and also a little darker in color. Gillott's No. 303 i>en 
will be found to give good results for this work ; but a contour 
pen, if it can be handled well, will give very uniform lines esj)e- 
cially where the contours have no sharp turns. In numbering 
the contours some prefer to break the lines and place the num- 
bers in the spaces, while others prefer to place the numbers 
just above or below the contours. Frequently a number is 
placed on every contour, but for most plans this is entirely un- 
necessary. If the contours are somewhat regular it is only 
necessary to number, say, every fifth contour. A good general 
rule to follow is to number only those lines which are necessary 
in order that the elevation of any contour may be found without 
appreciable mental effort. The numbers on the contours should 
be small plain figures in burnt sienna. 

The shape of the surface of the ground is sometimes repre- 
sented by hachure lines, which are illustrated in Fig. 182. The 



CONVENTIONAL SIGNS 



419 



contour lines are first sketched in pencil as a guide to the drafts- 
man in drawing the hachure lines, which should be drawn normal 
to the contours. The short 
lines are drawn from the summit 
downward in rows, each row 
just touching the next pre- 
ceding row. The steepness 
of the slope is represented by 
the weight and length of the 
lines,— the steeper the slope the 
heavier and shorter the lines. 
The individual lines are equally 
spaced, but on the flat slopes 
where the lines are lighter they 
have the appearance of being 
spaced farther apart. 

472. Such physical features as railroads, highways, buildings, 




Fig. 182. Hachure Lines. 




*■ -- 



1 1 I t I I I I I 



I I I I I I I I I 



,f- — 

□ 
O 
BMxl232 



Building. (On large scale maps.) 

Bam or Shed. (On large scale maps.) 

Buildings. (On small scale maps.) 

Fence. 

City or Town Boundary. 

Stone wall. 

Stone Retaining wall. 

Single Track Railroad. 

Double Track Railroad. 

Roads. 

Trail. 
Bridge. 

Triangulation Station. 

Stadia Station. 

Transit Point. Intersection Point 

Bench Mark. (- TnanguUHon.) 



Fig. 183. 



420 



D 




3 






QRST 


> 


QRST 


N 

>» 

X 

> 


Ph 


:3 


Q. 




00 ^ - 


O 


{» 


O 


(0 


«:l 


^oa 




Zc6 




^. n ? '^'^ 


^N 0. 


2N 


Ql 


^ cl, ^ •*. 


1 ^^^ 


c 


i_l>- 


O 

c 


1 ^ S^ ^ 




3 


2^ 


w 

J 

^ 


Sutmp] 

KJLM 

ijklmn 

3456 7 


ffi> 


::? 


I > 


:z> 


Af^ 


o 


Xi 


o 


X 


^t^ 


[9n 




L. 
U 





0^ ^ 


P 




O 


o 




o 


^ 


O 


SI 


2 s 


ffl 


CS 


m 


(D 


5 3 


<: 




< 




^ ^ 



421 




422 



FINISHING AND FILING DRAWINGS [Chap. XVL 



and boundaries are usually represented in black ink by the sym- 
bols shown in Fig. 183. 

473. LETTERUfG.* — The lettering on a drawing probably 
has more to do with its appearance than any other feature. To 
be able to do good lettering at first is a gift which but few 
men possess. It is an art that can be acquired by the most 
awkward draftsman, however, if he will study it carefully and 
devote a little time to systematic practice. 

Several di£ferent styles of lettering are shown in Figs. 184 
and 185. The general style to use in 'any given case depends 
on the type of drawing and on the use to which it is to be 
put. On plans which are to be sent from the office as com- 
pleted drawings such letters as the Roman or Gothic may be 
appropriate. Stump writing is a style of lettering which is 
difficult to execute but whose appearance, when well done, is 
very artistic. The ornate lettering in vogue a few years ago 
has been superseded by simpler styles which require much less 
time to produce. For construction drawings, like a plan of a 
bridge or a conduit, for example, the Reinhardt letters are used 




Fig. 186. Lettering on Slopes. 



* For a complete discussion and illustrations of lettering see any of the fol- 
lowing publications : "Plain Lettering," by Professor Henry S. Jacoby, pubHshed 
by the Engineering News Publishing Company ; ** Technic of Mechanical Draft- 
ing,** by Charles W. Reinhardt, published by the Engineering News Publishing 
Company ; " Letter Plates,** by Professor Charles L. Adams, Mass. Inst, of 
Technology, published by Professor Adams. 



LETTERING 423 

to a considerable extent. The title of such a plan looks well 
lettered in either erect or inclined Gothic. 

All plans should be lettered so as to read from the bottom. 
Unless a draftsman exercises considerable care he will find, 
when the plan is completed, that some of the lettering is upside 
down. Fig. 186 illustrates the proper lettering of lines of vari- 
ous slopes. 

474. Titles. — The design of the title of a plan gives the drafts- 
man an opportunity to exercise good taste. It should be so ar- 
ranged and the size of the letters so chosen that the most im- 
portant part of the title strikes the eye first. In general, each 
line of lettering should be centered, and the spacing between the 
lines should be so arranged that no part will either appear crowded 
or seem to be floating away from the rest of the title. The gen- 
eral outline of the title should be pleasing to the eye. In some 
of the larger offices, in order to save the time of the draftsman, 
titles are set up in type and printed on the map. 

Fig. 187 shows a set of titles which are well balanced and 
complete. Fig. 188 shows the style of lettering appropriate for 
a profile, a cross-section, or construction details. 



424 FINISHING AND FILING DRAWINGS [Chap. XVI. 



F^reLtnUnary Slltv^ ftor ct RtLtLnocLcL 
/yonv 

Crescent Beach, to Wbodlawn Centeteny. 

Oct€>ben, /802^ 
Sccvle '400 rieet to / irtch, 

COMMONWEALTH OF MASSACHUSETTS. 

METROPOLITAN WATER WORKS. 

WACHUSETT DAM 



UPPER GATE-CHAMBER. 



JULY 9, I900. 

UNITED STATES 
COAST AND GEODETIC SURVEY 

SKETCH OF GENERAL PROGRESS 

JUNE 30 1897 
Easteni Sheet 



Fig. 187. Titles of Plans. 



TITLES 4:^5 



TRACK ELEVATION. 

C. <Sc Vs/. L R. R. 

Cross- Section of Brid3e Showing 

Floor Construction. 
Scale i in.* Ift. 



HORIZONTAL SECTIONS 

thuouch upper through lower through lower 
sluice-gate sluice-gate valve well 

01t545«ft 



PineliminaiyRvfile 

for 9 Railroad from 

Bedford Junction to North Liberty 

Sta.0 to Sta498*68.7 
May/906 



Fig. 188. Titles of Profiles 



426 FINISHING AND FILING DRAWINGS [Chap. XVI. 

475. Notes. — Most drawings require notes of some sort. 
These are usually executed with a plain letter like the Reinhardt 
alphabet. In Fig. 189 are a few samples the general style of 
which is consistent with modern practice. 



Note:- Th/s re/nforcemenf is 8-0 "fongf, 
ancf comes direcf/y under each track. 
Leave amp/e room for brf'of^e-seat 

Note:-The datum plane used for con- 
tours and soundings on this map is 
"Boston City BaseV 
Boston City Base is 0.64 ft below 
base known as"Mean Low Water at 
Navy Yard" which is the datum used 
by the U.S. Coast Survey, the U.S. 
Engineers Office, and the Mass. 
Harbor and Land Commission. 

Soundings and Contours confirmed and ex- 
tended by data from map (1.-476) on file witti 
Massachusetts Hartwr and Land Commission. 



Fig. 180. Samples ok Notes. 



NOTES — BORDER LINES 427 

476. Border Lines. — The border line of a drawing should 
consist of a heavy single line or double lines closely spaced. It 
should neither be so heavy nor of such fancy design as to be con- 
spicuous. Plain clear drawings are the practice of to-day, and the 
border line should be in keeping with the rest of the drawing. 
For drawings 2 ft. long, the border should be about J" from the 
. edge of the sheet : for drawings 4 ft. long, i" to i-J" looks well. 
On some, particularly office drawings, the border is unnecessary 
and may be undesirable. Fig. 190 gives a few examples of sim- 
ple practical border lines. 



Fig. 190. Border Lines. 

477. Meridians. — On all land plans it is customary to draw 
either the true or the magnetic meridian ; often both of them are 
represented. To be in keeping with the rest of the drawing this 
should be simple in design. Too frequently, however, the drafts- 
man attempts to "lay himself out " on the needle with the result 
that it is so large and ornate that it is the first thing in the draw- 
ing that strikes the eye. The simple meridians shown in Fig. 
191 are suggested as suitable for ordinary land plans. 

The plan should always be drawn, if possible, so that the 



428 



FINISHING AND FILING DRAWINGS [Chap. XVI. 



meridian will point, in general, toward the top of the drawing 
rather than toward the bottom. Sometimes it is drawn with its 
upper part above and its tail below the drawing. In such a case 




Fig. 191. 



5 

MERroiANS. 



the line of the meridian must never cut any of the lines of the 
drawings: it should be interrupted far enough from the drawing 
so that it cannot be mistaken for one of the property lines. 

478. Scales. — On account of the shrinkage of drawing 
paper the scale is sometimes drawn on the plan itself at the time 
that the drawing is plotted. It is well to have it sufficiently 
long, say, 3 to 10 inches (depending upon the size of the draw- 
ing), so that it will be of use in detecting the amount of shrink- 
age. This, of course, will determine the shrinkage only in the 
direction of the scale. These scales are usually placed directly 
under the title or in one of the lower corners. Fig. 192 gives 
two examples of scales. 

In plotting a coordinate survey, the intersections of the north 
and south with the east and west lines should be marked on the 
finished drawing, as these are of great assistance in plotting addi- 
tions. Moreover the distances between these points give a 
reliable measure of the change in scale of the map due to 
shrinkage. 

479. SHRINKAGE OF DRAWING PAPERS. — All of the papers 
in use will shrink and swell more or less with variations of 



MERIDIANS — SCALES 429 

weather conditions. The heavy mounted papers are affected 
the least, but large drawings even on such paper will be found 
on examination to change in size perceptibly. The fact that they 
do not always shrink the same amount in different directions 





T 


SCALE ^ 

?....T «r •¥» *p 


_J5.r«r 




i * t 


"^ 






SCALe, 1 INCH • ZOO PtlT 




B» 




FiQ. 192. Scales. 


•00 ao 



makes it difficult to estimate the amount of the change and to 
allow for it. This effect can be estimated quite closely, how- 
ever, by testing the drawing by measuring accurately a few lines 
running in different directions when it is plotted and scaling the 
same lines at any other time and making allowance for the 
change. Scaled distances on tracing cloth are quite unreliable 
if it is not kept in a dry place, and blue-prints generally shrink 
in washing so that scale measurements taken from them usually 
contain considerable error. 

480, MAPS OF LARGE EXTENT. — Some maps, like the 
location map of a railroad or the map of a city, are so large that 
they must be made in sections. In such cases two slightly differ- 
ent methods are employed. One method is to plot the several 
sheets so that the drawing on one will extend to but not include 
any of the drawing on the adjacent sheet, the limits of the draw- 
ings being defined by straight lines. The other method is to 
have the drawing on each sheet lap over the drawings on the 
adjacent sheets a little. In this case marks are made on all 
drawings which make it possible to fit them to the correspond- 
ing marks on the adjacent drawings when they are being used 
jointly. 

In attempting to arrange the sheets of adjacent drawings after 
they have been in use for any considerable time, it is often found 
that they do not fit well on account of the unequal shrinking and 



430 FINISHING AND FILING DRAWINGS [Chap. XVL 

swelling of the paper. Moreover in plotting lines on separate 
sheets so that they will fit exactly, there are mechanical difficul- 
ties which can only be appreciated by the draftsman who has had 
experience with them. These objections, together with the fact 
that a comprehensive view of the whole situation cannot be taken 
in at one time, have led some engineers to prefer large and un- 
wieldy drawings to a system of separate sheets, but the latter are 
much more convenient when the plans are to be used in the 
field. 

481. Inking in a profile. — The surface line is usually 
shown as a full firm black line and the grade line as a full red 
line (Art. 443, p. 395). A horizontal base-line is sometimes drawn 
in red a short distance above the bottom of the paper and ver- 
tical red lines are drawn from this line to the grade line at every 
change of gradient and at both ends of the profile. On these 
vertical lines are recorded the grade elevations at these points 
and the " plus " if the place where the gradient changes is not 
at a full station. On the base-line between these red vertical 
lines is recorded the gradient of the grade line above. Under 
the base-line is the stationing, which is marked at every heavy 
vertical ruling of the profile paper, together with any other notes 
of alignment which may be desired. 

Information such as the names of streets, brooks, etc., is let- 
tered vertically above the profile and at the proper station. A 
title and the scale are sometimes placed on the face of the pro- 
file ; sometimes these are put on the back of the profile at one 
end of it (or both in the case of a long profile), so that the title 
can be read when it is rolled up. 

482. Cleaning drawings. — Every drawing, during its 
construction, collects more or less dirt. Often construction 
lines are drawn which must be erased when the plan is 
completed. In cleaning a drawing an ordinary soft pencil 
eraser is used for the pencil lines while a sponge eraser or stale 
bread crumbs will remove the dirt satisfactorily without aflfect- 
ing the ink lines. 

To take off the pencil lines and dirt from tracing cloth, 
wash the drawing with a cloth saturated with gasolene or 
benzine. This will remove pencil lines entirely and will clean 



FILING PLANS 43 1 

the tracing perfectly without any injurious effect on the tracing 
cloth. 

483. FILING DRAWIHGS. — While the particular method of 
filing plans varies considerably in different offices, there are 
a few general ideas carried out by all drafting offices in regard 
to the preservation as well as the systematic filing of drawings. 
There is no doubt that the best method of filing plans is to 
keep them flat, but this is not practicable with large plans 
which must usually be filed in rolls. In all systems of plan 
filing there appears to be a proper use of both flat and rolled 
plans. 

In large offices plans are, as a rule, made in several standard 
sizes prescribed by the rules of the office, and are filed flat in 
shallow drawers which are built to fit the different sizes of draw- 
ings. In some offices the adherence to standard sizes is very 
rigid, and considerable time is often spent to bring drawings 
within the limits of one of these sizes. When these sizes are 
exceeded the plans are either made in sections of standard size, 
as explained in Art. 480, or they are made as large plans which 
are rolled and filed away in pasteboard tubes. Sometimes very 
large plans are filed flat by hanging them from an overhead 
frame. 

Plans filed flat are marked each with its proper index 
number in one corner, preferably the lower right-hand corner, 
so that as the drawer is opened the numbers can be readily exam- 
ined. In some offices it is required" that in returning a draw- 
ing it shall be placed in its proper order in the drawer as well 
as in the proper drawer, while in other offices the plan drawers 
are made very shallow, so as to contain only about 15 or 20 
drawings, and when a plan is returned no attempt is made to 
put it in any particular place in the drawer, there being, at the 
most, only a very few drawings to handle to obtain the one 
desired. 

Rolled drawings are marked on the side of the rolls at each 
end so as to be easily read by one standing in front of the shelf 
on which the plans are stored. Another style of roll is closed 
at one end with a white label on the outside of the closed end. 
When the plan has been put into the tube it is so placed on the 



43^ FINISHING AND FILING DRAWINGS [Chap. XVI. 

shelf that the label on which the plan number is marked is at 
the front edge of the shelf where it can be conveniently read. 
When the plan is in use the empty tube is left on the shelf 
with its open end outward so that its number is in the back part 
of the shelf where it cannot be read. 

Large plans which are made in sections are often filed in 
large folios or books in such a way that they can be readily 
taken out and used separately. 

484. INDEXING. DRAWINGS. — There are so many systems of 
indexing plans that no attempt will be made to explain them other 
than to suggest a few of the essentials of any good system. 
Every system of numbering the plans should be such that one 
can tell from its number whether the drawing is a sketch, a 
working drawing, a finished drawing, a tracing, or a process 
print. The numbering also should suggest the type of draw- 
ing, as a land plan, a construction plan, etc. 

For offices where few plans are on file an index book may 
suffice for recording the plans, but in large drafting offices the 
card catalogue system is used extensively. By a judicious use 
of "markers" a card catalogue system can be so devised that 
it will be necessary to examine only a very few cards to find 
the one corresponding to any plan. Frequently it is necessary 
to index a plan by two or three different cards under different 
general headings.. 

485. FILING Note-Books. — Note-books should always be 
filed in vaults where they will be protected against fire. Too 
frequently through lack of forethought note-books containing 
information which it has cost thousands of dollars to collect are 
carelessly filed on a shelf in the drafting office. In some offices 
the rules require that every note-book and valuable plan shall 
be placed in the vault at the end of the day's work, and this 
appears to be the proper practice. 

Some offices go so far as to require that all notes shall be 
copied in ink and the original notes kept permanently filed in 
the vault to guard against their loss. Whether a copy is made 
or not, the original should be preserved as it has a value, in 
a lawsuit for instance, which any copy does not possess. When 
copies are made of the original notes they are sometimes made 



INDEXING — FILING 43 3 

in a loose-leaf book so that if any notes are taken from the office 
it is not necessary to take more than a very few leaves of the 
copy; the original notes never go from the office except in rare 
cases. 

486. Indexing Notes. — The notes contained in the field note- 
books are often indexed either in a book for this purpose or by 
means of a card catalogue. The method of indexing is similar 
to that used for plans. 

487. Other Records. — Other records, such as borings, sound- 
ings, estimates, computations, etc.^ are carefully filed and indexed 
so that it will be easy to refer to them. 



TABLES. 



43S 



437 
TABLE I.— LOGARITHMS OF NL'MBERS. 



N 


0123456789 


100 


00000 00043 00087 OOISO 00173 00217 00260 00303 00346 00389 




0432 0475 0518 0561 0604 0647 0689 0732 077i 0817 




0860 0903 0946 0988 1030 1072 1115 1157 1199 1242 




1284 1326 1368 1410 1452 1494 1536 1578 1620 1662 




1703 1745 1787 1828 1870 1912 1953 1995 2036 2078 




2119 2160 2202 2243 2284 2325 2366 2407 2449 2490 




2531 2572 2612 265i^ 2694 2735 2776 2816 2857 2898 




2938 2979 3019 3060 3100 3141 3181 3222 3262 3302 




3342 3383 3423 3463 3503 3543 3583 3623 8663 3703 




3743 3782 3822 3862 3902 3941 3981 4021 4060 4100 


110 


04139 04179 04218 04258 04297 04336 04376 04415 04454 04498 




4532 4571 4610 4650 4689 4727 4766 4805 4844 4883 




4922 4961 4999 5038 5077 5116 5154 5192 5231 5269 




5308 5346 5385 5423 5461 5500 5538 5576 6614 6652 




5690 6729 5767 5805 5843 5881 6918 6956 5994 6032 




6070 6108 6146 6183 6221 6258 6296 6333 6371 6408 




6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 




6819 6856 6893 6930 6967 7004 7041 7078 7115 7151 




7188 7225 7262 7298 7336 7372 7408 7445 7482 7518 




7555 7591 7628 7664 7700 7737 7773 7809 7846 7882 


120 


07918 07954 07990 08027 08063 08099 08135 08171 08207 08248 




8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 




8636 8672 8707 8743 8778 8814 8849 8884 8920 8956 




8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 




9342 9377 9412 9447 9482 9617 9552 9587 9621 9666 




9691 9726 9760 9795 9830 9864 9899 9934 996810003 




10037 10072 10106 10140 10175 10209 10243 10278 10312 0346 




0380 0415 0449 0483 0517 0551 0586 0619 0653 0687 




0721 0755 0789 0823 0857 0890 0924 0958 0992 1025 




1069 1093 1126 1160 1193 1227 1261 1294 1327 1361 


180 


11394 11428 11461 11494 11528 11661 11694 11628 11661 11694 




1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 




2057 2090 2123 2156 2189 2222 2254 2287 2320 2352 




2385 2418 2460 2483 2516 2548 2681 2613 2646 2678 




2710 2743 2775 2808 2840 2872 2905 2937 2969 3001 




3033 3066 3098 3130 3162 3194 3226 8258 3290 3322 




3354 3386 3418 3450 3481 3513 3545 3577 3609 3640 




3672 3704 3735 3767 3799 3830 3862 3893 3925 3956 




3988 4019 4051 4082 4114 4145 4176 4208 4239 4270 




4301 4333 4364 4396 4426 4457 4489 4520 4561 4582 


140 


14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 




4922 4953 4983 6014 5045 5076 6106 6137 6168 6198 




5229 6259 5290 5320 5351 6381 6412 6442 5473 6503 




6534 5564 6694 6625 5655 5686 6715 5746 6776 6806 




6836 5866 5897 5927 5957 5987 6017 6047 6077 6107 




6137 6167 6197 6227 6266 6286 6316 6346 6376 6406 




6435 6466 6495 6624 6554 6584 6613 6643 6673 6702 




6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 




7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 




7319 7348 7377 7406 7436 7464 7493 7522 7561 7680 


150 


17609 17638 17667 17696 17726 17764 17782 17811 17840 17869 



438 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


150 


17600 17688 17667 17606 17725 17764 17782 17811 17840 17869 




7898 7926 7956 7984 8013 8041 8070 8099 8127 8166 




8184 8218 8241 8270 8298 8827 8865 8884 8412 8441 




8469 8498 8626 8564 8688 8611 8689 8667 8696 8724 




8762 8780 8808 8887 8865 8893 8921 8949 8977 9006 




9088 9061 9089 9117 9146 9178 9201 9229 9267 9285 




9812 9840 9868 9896 9424 9461 9479 9607 9685 9562 




9590 9618 9646 9678 9700 9728 9766 9788 9811 9888 




9866 9898 9921 9948 9976 20008 20080 20068 20086 20112 




20140 20167 20194 20222 20249 0276 0808 0880 0868 0385 


160 


20412 20489 20466 20498 20620 20648 20575 20602 20629 20666 




0688 0710 0787 0768 0790 0817 0844 0871 0898 0925 




0952 0978 1006 1082 1069 1086 1112 1189 1166 1192 




1219 1246 1272 1299 1826 1852 1878 1405 1481 1458 




1484 1511 1687 1664 1590 1617 1648 1669 1696 1722 




1748 1775 1801 1827 1854 1880 1906 1982 1968 1985 




2011 2087 2063 2089 2115 2141 2167 2194 2220 2246 




2272 2298 2824 2850 2876 2401 2427 2458 2479 2505 




2681 2667 2588 2608 2684 2660 2686 2712 2787 2763 




2789 2814 2840 2866 2891 2917 2943 2968 2994 8019 


170 


23045 23070 23096 28121 28147 28172 23198 23223 23249 23274 




3300 3326 3350 3876 3401 8426 3452 3477 3502 3528 




3563 3578 3603 8629 8664 3679 3704 3729 3764 8779 




3805 3880 3865 3880 3905 3930 3955 8980 4005 4030 




4065 4080 4105 4180 4155 4180 4204 4229 4264 4279 




4304 4329 4368 4878 4403 4428 4462 4477 4602 4527 




4561 4576 4601 4626 4650 4674 4699 4724 4748 4773 




4797 4822 4846 4871 4895 4920 4944 4969 4993 6018 




6042 5066 6091 6115 5139 6164 6188 6212 5237 5261 




6286 5810 5334 6858 5382 5406 6431 5465 6479 5603 


180 


26627 25661 26676 26600 25624 25648 26672 26696 26720 25744 




5768 5792 5816 6840 6864 6888 6912 6985 5959 6983 




6007 6081 6065 6079 6102 6126 6160 6174 6198 6221 




6246 6269 6293 6816 6340 6864 6887 6411 6435 6468 




6482 6606 6629 6653 6576 6600 6628 6647 6670 6694 




6717 6741 6764 6788 6811 6884 6858 6881 6905 6928 




6961 6975 6998 7021 7045 7068 7091 7114 7138 7161 




7184 7207 7281 7264 7277 7300 7323 7346 7870 7393 




7416 7439 7462 7486 7608 7631 7664 7677 7600 7623 




7646 7669 7692 7716 7788 7761 7784 7807 7830 7862 


190 


27876 27898 27921 27944 27967 27989 28012 28a% 28058 28081 




8108 8126 8149 8171 8194 8217 8240 8262 8285 8307 




8830 8363 8375 8308 8421 8448 8466 8488 8511 8533 




8666 8578 8601 8628 8646 8668 8691 8713 8736 8758 




8780 8808 8825 8847 8870 8892 8914 8987 8969 8981 




9008 9026 9048 9070 9092 9115 9137 9159 9181 9203 




9226 9248 9270 9292 9314 9330 9368 9380 9403 9425 




9447 9469 9491 9513 9535 9567 9679 9601 9623 9645 




9667 9688 9710 9732 9754 9776 9798 9820 9842 9863 




9886 9907 9929 9961 9978 9994 30016 30038 80060 30081 


800 


80103 30125 30146 30168 30190 30211 30233 30265 30276 30298 



439 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


200 


30103 3012i 30146 30168 30190 30211 30233 3025^ 30276 30208 




0320 0341 0363 0384 0406 0428 0449 0471 0492 0614 




0636 0567 0578 0600 0621 0643 0664 0686 0707 0728 




0750 0771 0792 0814 0835 0866 0878 0899 0920 0942 




0963 0984 1006 1027 1048 1069 1091 1112 1133 1164 




1175 1197 1218 1239 1260 1281 1302 1323 1345 1366 




1387 1408 1429 1450 1471 1492 1513 1634 1656 1576 




1697 1618 1639 1660 1681 1702 1723 1744 1765 1786 




1806 1827 1848 1869 1890 1911 1931 1952 1973 1994 




2015 2036 2066 2077 2098 2118 2139 2160 2181 2201 


210 


32222 32243 32263 32284 32305 32326 32346 32366 32387 32408 




2428 2449 2469 2490 2510 2531 2562 2672 2693 2613 




2634 2654 2675 2695 2716 2736 2766 2777 2797 2818 




2838 2858 2879 2899 2919 2940 2960 2980 3001 3021 




3041 3062 3082 3102 3122 3143 3163 3183 3203 3224 




3244 3264 3284 3304 3325 3345 3365 3386 3406 3426 




3446 3465 3486 3506 3526 3646 3566 3686 3606 3626 




3646 3666 3686 3706 3726 3746 3766 3786 3806 3826 




3846 3866 3885 3906 3925 3946 3966 8985 4005 4025 




4044 4064 4084 4104 4124 4143 4163 4183 4208 4223 


220 


34242 34262 34282 34301 34321 84341 34361 34380 34400 34420 




4439 4469 4479 4498 4518 4537 4557 4677 4696 4616 




4636 4655 4674 4694 4713 4733 4763 4772 4792 4811 




4830 4850 4869 4889 4908 4928 4947 4967 4986 6005 




6025 6044 6064 5083 5102 5122 5141 6160 6180 6199 




6218 6238 5257 5276 5295 5315 5334 6363 6372 6392 




6411 5430 5449 5468 6488 5507 6626 6646 6664 5583 




5603 5622 5641 5660 5679 5698 6717 6736 6756 6774 




6793 6813 5832 5851 5870 6889 6908 6927 6946 6965 




6984 6003 6021 6040 6059 6078 6097 6116 6136 6154 


280 


36173 86192 36211 36229 36248 36267 36286 36305 36324 36342 




6361 6380 6309 6418 6436 6455 6474 6493 6611 6630 




6549 6568 6586 6605 6624 6642 6661 6680 6698 6717 




6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 




6922 6940 6959 6977 6996 7014 7033 7061 7070 7088 




7107 7126 7144 7102 7181 7199 7218 7236 7254 7273 




7291 7310 7328 7346 7365 738i5 7401 7420 7438 7467 




7475 7493 7511 7530 7548 7566 7585 7603 7621 7639 




7658 7676 7094 7712 7731 7749 7767 7786 7803 7822 




7840 7858 7876 7894 7912 7931 7949 7967 7985 8003 


240 


38021 38039 38057 38076 38093 38112 38130 38148 38166 38184 




8202 8220 82;i8 8256 8274 8292 8310 8328 8346 8364 




8382 8399 8417 8435 8453 8471 8489 8507 8525 8543 




8561 8578 8596 8014 8632 86^0 8668 8686 8703 8721 




8739 8757 8775 8792 8810 8828 8846 8863 8881 8899 




8917 8934 8952 8970 8987 9005 9023 9041 9068 9076 




9094 9111 9129 9146 9164 9182 9199 9217 9235 9262 




9270 9287 9305 9322 9840 9358 9376 9393 9410 9428 




9445 9463 9480 9498 9516 9533 9550 9568 9585 9602 




9620 9637 9655 9672 9690 9707 9724 9742 9769 9777 


250 


39794 39811 39829 39846 39863 39881 39898 39916 39933 39950 



440 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


01234567 8 9 


250 


39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 


1 


9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 


8 


40140 40157 0175 0192 0209 0226 0243 0261 0278 0295 


8 


0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 


4 


0483 0500 0518 0535 0552 0569 0586 0603 0620 0637 


5 


0654 0671 0688 0705 0722 0739 0756 0773 0790 0807 


6 


0824 0841 0858 0875 0892 0909 0926 0943 0960 0976 


7 


0993 1010 1027 1044 1061 1078 1095 IHI 1128 1145 


8 


1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 


9 


1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 


260 


41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 


1 


1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 


2 


1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 


8 


1996 2012 2029 2045 2062 2078 2095 2111 2127 2144 


4 


2160 2177 2193 2210 2226 2243 2259 2275 2292 2308 


6 


2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 


6 


2488 2504 2521 2537 2553 2570 2586 2602 2619 2635 


7 


2651 2667 2684 2700 2716 2732 2749 2765 2781 2797 


8 


2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 


9 


2975 2991 3008 3024 3040 3056 3072 3088 3104 3120 


270 


43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 


1 


3297 3313 3329 3345 3361 3377 3393 3409 34£5 3441 


2 


3457 3473 3489 3505 3521 3537 3553 3569 3584 3600 


8 


3616 3632 3648 3664 3680 3696 3712 3727 3743 3759 


4 


3775 3791 3807 3823 3838 3854 3870 3886 3902 3917 


6 


3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 


6 


4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 


7 


4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 


8 


4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 


9 


4560 4576 4592 4607 4623 4638 4054 4669 4685 4700 


280 


44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 


1 


4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 


2 


5025 5040 5056 5071 5086 5102 5117 5133 5148 6163 


8 


5179 5194 6209 6225 5240 5256 5271 5286 6301 6317 


4 


6332 6347 6362 6378 5393 6408 6423 6439 6464 6469 


5 


6484 6500 5515 6530 6646 6561 6576 5691 6606 6621 


6 


6637 6652 5667 5682 5697 6712 6728 6743 6768 6773 


7 


6788 6803 5818 6834 6849 6864 6879 6894 6909 6924 


8 


6939 6954 6969 6984 6000 6015 6030 6045 6060 6075 


9 


6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 


290 


46240 46265 46270 46285 46300 46315 46330 46345 46369 46374 


1 


6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 


2 


6538 6553 6568 6583 6598 6613 6627 6642 6657 6672 


8 


6687 6702 6716 6731 6746 6761 6776 6790 6806 6820 


4 


6835 6850 6864 6879 6894 6909 6923 6938 6963 6967 


6 


6982 6997 7012 7026 7041 7056 7070 7086 7100 7114 


6 


7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 


7 


7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 


8 


7422 7436 7461 7465 7480 7494 7609 7524 7538 7653 


9 


7667 7582 7596 7611 7625 7640 7654 7669 7683 7698 


800 


47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 



441 



TABLE I.— LOGARITHMS OF NXmBERS. 



N 


0123456789 


800 


47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 




7867 7871 7885 7900 7914 7929 7943 7958 7972 7986 




8001 8015 8029 8044 8068 8073 8087 8101 8116 8130 


3 


8144 8169 8173 8187 8202 8216 8230 8244 8259 8273 




8287 8302 8316 8330 8344 8359 8373 8387 8401 8416 




8430 8444 8468 8473 8487 8501 8516 8630 8644 8668 




8572 8686 8601 8615 8629 8643 8667 8671 8686 8700 




8714 8728 8742 8766 8770 8785 8799 8813 8827 8841 




8855 8869 8883 8897 8911 8926 8940 8964 8968 8982 




8996 9010 9024 9038 9062 9066 9080 9094 9108 9122 


810 


49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 




9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 




9415 9429 9443 9467 9471 9485 9499 9613 9627 9641 




9664 9668 9682 9696 9610 9624 9638 9661 9665 9679 




9693 9707 9721 9734 9748 9762 9776 9790 9803 9817 




9831 984i 9869 9872 9886 9900 9914 9927 9941 9965 




9969 9982 9996 50010 50024 60037 6005160065 60079 60092 




50106 50120 50183 0147 0161 0174 0188 0202 0215 0229 




0243 0256 0270 0284 0297 0811 0325 0338 0352 0366 




0379 0393 0406 0420 0433 0447 0461 0474 0488 0601 


820 


50515 50629 60542 50556 50669 50583 60696 50610 50623 50637 




0651 0664 0678 0691 0705 0718 0732 0746 0769 0772 




0786 0799 0813 0826 0840 0863 0866 0880 0893 0907 




0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 




1055 1068 1081 1095 1108 1121 1135 1148 1162 1175 




1188 1202 1215 1228 1242 1256 1268 1282 1295 1308 




1322 1335 1348 1362 1376 1388 1402 1415 1428 1441 




1465 1468 1481 1495 1508 1521 1634 1548 1561 1674 




1587 1601 1614 1627 1640 1664 1667 1680 1693 1706 




1720 1733 1746 1759 1772 1786 1799 1812 1825 1838 


880 


51861 51865 51878 61891 51904 61917 51930 51948 51967 61970 




1983 1996 2009 2022 2035 2048 2061 2075 2088 2101 




2114 2127 2140 2163 2166 2179 2192 2205 2218 2231 




2244 2267 2270 2284 2297 2310 2323 2336 2349 2362 




2375 2388 2401 2414 2427 2440 2463 2466 2479 2492 




2504 2517 2530 2643 2656 2569 2582 2696 2608 2621 




2634 2647 2660 2673 2686 2699 2711 2724 2737 2760 




2763 2776- 2789 2802 2815 2827 2840 2863 2866 2879 




2892 2905 2917 2930 2943 2966 2969 2982 2994 3007 




8020 3033 8046 8068 8071 3084 3097 3110 3122 3186 


840 


53148 58161 53173 53186 53199 63212 63224 53287 63250 63263 




8276 8288 8301 8814 3326 8339 3362 3364 8377 3:^90 




84a3 3415 3428 8441 8463 8466 8479 8491 8604 3617 




3629 3642 3665 8667 8580 8693 8605 8618 3631 3643 




3666 8668 8681 3694 8706 3719 3732 3744 3767 3769 




8782 3794 3807 3820 3832 8845 8867 3870 3882 3896 




8908 3920 3933 8945 8968 8970 3983 3996 4008 4020 




4033 4045 4068 4070 4083 4095 4108 4120 4133 4146 




4168 4170 4183 4196 4208 4220 4233 4246 4268 4270 




4283 4295 4307 4320 4832 4345 4367 4370 4382 4394 


360 


54407 64419 64432 54444 64460 64469 64481 64494 54506 54618 



442 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0128456789 


850 


64407 64419 64432 64444 64466 64469 64481 64494 64606 64618 




4631 4643 4656 4668 4680 4593 460^ 4617 4630 4642 




4664 4667 4679 4691 4704 4716 4728 4741 4763 4765 




4777 4790 4802 4814 4827 4839 4861 4864 4876 4888 




4900 4913 492i 4937 4949 4962 4974 4986 4998 6011 




6023 6035 6047 6060 6072 6084 5096 6108 6121 6133 




5145 5157 5169 6182 5194 6206 5218 5230 5242 6265 




5267 5279 5291 6303 5315 5328 5340 5352 5364 6376 




5388 5400 6413 6425 6437 6449 6461 6473 6486 6497 




6509 6622 6634 6546 5558 5570 6682 6594 6606 6618 


860 


56630 66642 66664 56666 65678 66691 65703 65715 65727 66739 




5751 5763 6775 5787 6799 5811 5823 6835 5847 6859 




6871 6883 6895 5907 6919 5931 6943 6955 6967 5979 




5991 6003 6015 6027 6038 6050 6062 6074 6086 6098 




6110 6122 6134 6146 6158 6170 6182 6194 6205 6217 




6229 6241 6253 6265 6277 6289 6301 6312 6324 6336 




6348 6360 6372 6384 6396 6407 6419 6431 6443 6455 




6467 6478 6490 6502 6514 6526 6538 6549 6561 6673 




6685 6697 6608 6620 6632 6644 6656 6667 6679 6691 




6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 


870 


66820 66832 66844 66866 66867 56879 66891 66902 66914 66926 




6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 




7064 7066 7078 7089 7101 7113 7124 7136 7148 7159 




7171 7183 7194 7206 7217 7229 7241 7262 7264 7276 




7287 7299 7310 7322 7334 7346 7357 7368 7380 7392 




7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 




7519 7530 7542 7553 7565 7576 7588 7600 7611 7623 




7634 7646 7657 7669 7680 7692 7703 7715 7726 7738 




7749 7761 7772 7784 7796 7807 7818 7830 7841 7862 




7864 7875 7887 7898 7910 7921 7933 7944 7966 7967 


880 


57978 67990 68001 68013 58024 68035 58047 68058 68070 68081 




8092 8104 8115 8127 8138 8149 8161 8172 8184 8195 




8206 8218 8229 8240 8252 8263 8274 8286 8297 8309 




8320 8331 8343 8354 8365 8377 8388 8399 8410 8422 




8433 8444 8456 8467 8478 8490 8501 8512 8524 8635 




8646 8657 8569 8580 8591 8602 8614 8625 8636 8647 




8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 




8771 8782 8794 8805 8816 8827 883& 8850 8861 8872 




8883 8894 8906 8917 8928 8939 8950 8961 8973 8984 




8995 9006 9017 9028 9040 9051 9062 9073 9084 9096 


890 


69106 69118 69129 59140 59151 69162 59173 69184 69195 69207 




9218 9229 9240 9251 9262 9273 9284 9296 9306 9318 




9329 9340 9351 9362 9373 9384 9396 9406 9417 9428 




9439 9450 9461 9472 9483 9494 9506 9517 9528 9539 




9650 9561 9572 9583 9594 9605 9616 9627 9638 9649 




9660 9671 9682 9693 9704 9715 9'726 9737 9748 9769 




9770 9780 9791 9802 9813 9824 9836 9846 9867 9868 




9879 9890 9901 9912 9923 9934 9945 9966 9966 9977 




9988 9999 60010 60021 60032 60043 60054 60065 60076 60086 




60097 60108 0119 0130 0141 0152 0163 0173 0184 0196 


400 


60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 



..J 



443 
TABLE I.— LOGARITHMS OF NUMBERS. 



■isr 


O 12^4:5 67 89 


400 e0206 60217 60228 60239 60249 60260 60271 60282 60293 60304 | 




0314 0325 0336 0347 0358 0369 0379 0390 0401 0412 




0423 0433 0444 0455 0466 0477 0487 0498 0509 0520 




0531 0541 0552 0563 0574 0584 0595 0606 0617 0627 




0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 




0746 0756 0767 0778 0788 0799 0810 0821 0831 0842 




0853 0863 0874 0885 0896 0906 0917 0927 0938 0949 




0959 0970 0981 0991 1002 1013 1023 1034 1045 1055 




1066 1077 1087 1098 1109 1119 1130 1140 1151 1162 




1172 1183 1194 1204 1215 1225 1236 1247 1257 1268 


410 


61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 




1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 




1490 1500 1611 1521 1532 1542 1553 1563 1674 1584 




1695 1606 1616 1627 1637 1648 1658 1669 1679 1690 




1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 




1805 1815 1826 1836 1847 1867 1868 1878 1888 1899 




1909 1920 1930 1941 1951 1962 1972 1982 1993 2003 




2014 2024 2034 2045 2055 2066 2076 2086 2097 2107 




2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 




2221 2232 2242 2252 2263 2273 2284 2294 2304 2315 


420 


62325 62335 62346 62356 62366 62377 62387 62897 62408 62418 




2428 2439 2449 2459 2469 2480 2490 2600 2511 2621 




2531 2542 2552 2562 2572 2583 2593 2603 2613 2624 




2634 2644 2655 2665 2676 2685 2696 2706 2716 2726 




2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 




2839 2849 2859 2870 2880 2890 2900 2910 2921 2931 




2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 




3043 3063 3063 3073 3083 3094 3104 8114 3124 3134 




3144 3165 3165 3175 3185 3196 3205 3215 3226 3236 




3246 3256 3266 3276 3286 3296 3306 3317 3327 3337 


480 


63347 63367 63367 63377 63387 63397 63407 63417 63428 63438 




3448 3458 3408 3478 3488 3498 3508 3518 3528 3638 




3548 3558 3568 3579 3689 3599 3609 3619 8629 3639 




3649 3659 3669 3679 3689 3699 3709 3719 3729 3739 




3749 3769 3769 3779 3789 3799 8809 3819 3829 3839 




3849 3859 3809 3879 3889 3899 3909 3919 3929 8939 




3949 3969 3969 3979 3988 3998 4008 4018 4028 4a38 




4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 




4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 




4246 4266 4266 4276 4286 4296 4306 4316 4326 4335 


440 


64345 64366 64365 64375 64385 64395 64404 64414 64424 64434 




4444 4454 4464 4473 4483 4493 4503 4513 4523 4532 




4542 4552 4562 4572 4582 4591 4601 4611 4621 4631 




4640 4650 4660 4670 4680 4089 4699 4709 4719 4729 




4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 




4836 4846 4856 4865 4876 4885 4895 4904 4914 4924 




4933 4943 4953 4963 4972 4982 4992 5002 6011 5021 




6031 5040 5050 5060 6070 6079 6089 6099 6108 6118 




6128 6137 6147 6157 5167 6176 6186 5196 6206 6215 




5225 6234 6244 5264 6263 5273 6283 5292 6302 5312 


460 


65321 65331 65341 65350 65360 65369 65379 66389 65398 65408 



444 
TABLE I.— LOGARITHMS OF NUMBERS. 



450 


0123456789 


66821 66331 65341 65360 66360 66369 66379 66389 66398 66408 


1 


5418 6427 6437 6447 6456 6466 6476 6486 649^ 6604 


2 


6614 6523 6633 6643 6552 6562 6671 6681 6691 6600 


8 


6610 6619 6629 6639 6648 6668 5667 6677 6686 6696 


4 


6706 6715 6725 6734 6744 6763 5763 6772 6782 5792 


5 


6801 6811 5820 6830 6839 6849 6858 6868 6877 6887 


6 


6896 6906 6916 5926 6935 6944 6954 6963 6973 6982 


7 


5992 6001 6011 6020 6030 6039 6049 6068 6068 6077 


8 


6087 6096 6106 6115 6124 6134 6143 6163 6162 6172 


9 


6181 6191 6200 6210 6219 6229 6238 6247 6267 6266 


460 


66276 66285 66295 66304 66314 66323 66332 66342 66361 66361 


1 


6370 6380 6389 6398 6408 6417 6427 6436 6446 6465 


2 


6464 6474 6483 6492 6502 6611 6621 6530 6639 6549 


8 


6658 6567 6677 6586 6596 6605 6614 6624 6633 6642 


4 


6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 


5 


6745 6755 6764 6773 6783 6792 6801 6811 6820 6829 


6 


6839 6848 6857 6867 6876 6886 6894 6904 6918 6922 


7 


6932 6941 6950 6960 6969 6978 6987 6997 7006 7015 


8 


7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 


9 


7117 7127 7136 7146 7164 7164 7173 7182 7191 7201 


470 


67210 67219 67228 67237 67247 67266 67266 67274 67284 67293 


1 


7302 7311 7321 7330 7339 7348 7367 7367 7376 7385 


9 


7394 7403 7413 7422 7431 7440 7449 7469 7468 7477 


8 


7486 7496 7504 7514 7623 7632 7641 7550 7660 7569 


4 


7578 7687 7596 7005 7614 7624 7633 7642 7661 7660 


5 


7669 7679 7688 7697 7706 7715 7724 7733 7742 7752 


6 


7761 7770 7779 7788 7797 7806 7815 7825 7834 7843 


7 


7852 7861 7870 7879 7888 7897 7906 7916 7925 7934 


8 


7943 7962 7961 7970 7979 7988 7997 8006 8016 8024 


9 


8034 8043 8052 8061 8070 8079 8088 8097 8106 8116 


480 


68124 68133 68142 68151 68160 68169 68178 68187 68196 68206 


1 


8215 8224 8233 8242 8251 8260 8269 8278 8287 8296 


2 


8305 8314 8323 8332 8341 8350 8359 8368 8377 8386 


8 


8395 8404 8413 8422 8431 8440 8449 8468 8467 8476 


4 


8485 8494 8502 8511 8520 8529 8538 8547 8566 8566 


6 


8574 8583 8592 8601 8610 8619 8628 8637 8646 8655 


6 


8664 8673 8681 8690 8699 8708 8717 8726 8736 8744 


7 


8753 8762 8771 8780 8789 8797 8806 8816 8824 8833 


8 


8842 8851 8860 8869 8878 8886 8896 8904 8913 8922 


9 


8931 8940 8949 8958 8966 8976 8984 8993 9002 9011 


490 


69020 69028 69037 69046 69056 69064 69073 69082 69090 69099 


1 


9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 


8 


9197 9205 9214 9223 9232 9241 9249 9268 9267 9276 


8 


928S 9294 9302 9311 9320 9329 9338 9346 9366 9364 


4 


9373 9381 9390 9399 9408 9417 9426 9434 9443 9452 


6 


9461 9469 9478 9487 9496 9604 9513 9622 9631 9639 


6 


9548 9557 9666 9574 9583 9592 9601 9609 9618 9627 


7 


9636 9644 9653 9662 9671 9679 9688 9697 9706 9714 


8 


9723 9732 9740 9749 9758 9767 9775 9784 9793 9801 


9 


9810 9819 9827 9836 9845 9854 9862 9871 9880 9888 


600 


69807 69906 69914 69923 69932 69940 69949 69968 .69966 69976 



445 

TABLE T.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


500 


69897 69906 69914 69023 69932 69940 69949 69958 69966 69975 


1 


9984 9992 70001 70010 70018 70027 70036 70044 70053 70062 


2 


70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 


8 


0157 0165 0174 0183 0191 0200 0209 0217 0226 0234 


4 


0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 


5 


0329 0338 0346 0365 0364 0372 0381 0389 0398 0406 


6 


0415 0424 0432 0441 0449 0458 0467 0475 0484 0492 


7 


0501 0509 0518 0626 0635 0544 0562 0661 0669 0578 


8 


0586 0595 0603 0612 0621 0629 0638 0646 0655 0663 


9 


0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 


510 


70767 70766 70774 70783 70791 70800 70808 70817 70825 70834 


1 


0842 0851 0859 0868 0876 0885 0893 0902 0910 0919 


2 


0927 0935 0944 0962 0961 0969 0978 0986 0995 1003 


8 


1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 


4 


1096 1105 1113 1122 1130 1139 1147 1165 1164 1172 


5 


1181 1189 1198 1206 1214 1223 1231 1240 1248 1257 


6 


1265 1273 1282 1290 1299 1307 1315 1324 1332 1341 


7 


1349 1357 1366 1374 1383 1391 1399 1408 1416 1425 


8 


1433 1441 1450 1458 1466 1475 1483 1492 1500 1508 


9 


1617 1625 1533 1642 1660 1669 1667 1575 1584 1692 


520 


71600 71609 71617 71626 71634 71642 71650 71669 71667 71675 


1 


1684 1692 1700 1709 1717 1725 1734 1742 1750 1759 


2 


1767 1775 1784 1792 1800 1809 1817 1825 1834 1842 


3 


1850 1858 1867 1875 1883 1892 1900 1908 1917 1925 


4 


1933 1941 1950 1968 1966 1975 1983 1991 1999 2008 


6 


2016 2024 2032 2041 2049 2067 2066 2074 2082 2090 


6 


2099 2107 2115 2123 2132 2140 2148 2156 2165 2173 


7 


2181 2189 2198 2206 2214 2222 2230 2239 2247 2265 


8 


22(53 2272 2280 2288 2296 2304 2313 2321 2329 2337 


9 


2346 2354 2362 2370 2378 2387 2395 2403 2411 2419 


680 


72428 72436 72444 72462 72460 72469 72477 72485 72493 72601 


1 


2509 2518 2526 2534 2542 2550 2668 2667 2675 2583 


2 


2591 2599 2607 2616 2624 2632 2640 2648 2656 2665 


8 


2673 2681 2689 2697 2706 2713 2722 2730 2738 2746 


4 


2754 2762 2770 2779 2787 2795 2803 2811 2819 2827 


« 


2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 


6 


2916 2925 2933 2941 2949 2957 2966 2973 2981 2989 


7 


2997 3006 3014 3022 3030 3038 3046 3064 3062 3070 


8 


3078 3086 3094 3102 3111 3119 3127 3135 3143 3151 


9 


3159 3167 3175 3183 3191 3199 3207 3215 3223 3231 


540 


73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 


1 


3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 


2 


3400 3408 3416 3424 3432 3440 3448 3466 3464 3472 


8 


3480 3488 3496 3504 3612 3520 3628 3536 3644 3652 


4 


3560 3568 3576 3584 3692 3600 3608 3616 8624 3632 


5 


3640 3648 3656 3664 3672 3679 3687 3696 3703 3711 


6 


3719 3727 3735 3743 3751 3759 3707 3775 3783 3791 


7 


3799 3807 3815 3823 3830 3838 3846 3854.3862 3870 


8 


3878 3886 3894 3002 3910 3918 3926 3933 3941 3949 


9 


3957 3965 3073 3981 3989 3997 4005 4013 4020 4028 


550 


74036 74044 74052 74060 74008 74070 74084 74092 74099 74107 



446 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


550 


74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 


1 


4115 4123 4131 4139 4147 4165 4162 4170 4178 4186 


2 


4194 4202 4210 4218 4225 4233 4241 4249 4257 4265 


8 


4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 


4 


4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 


6 


4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 


6 


4507 4515 4523 4531 4539 4547 4554 4562 4570 4578 


7 


4586 4593 4001 4009 4617 4624 4632 4640 4648 4666 


8 


4863 4671 4679 4687 4695 4702 4710 4718 4726 4733 


9 


4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 


560 


74819 74827 74834 74842 74850 74858 74866 74873 74881 74889 


1 


4896 4904 4912 4920 4927 4985 4943 4950 4958 4966 


2 


4974 4981 4989 4997 5005 5012 6020 5028 5036 6043 


8 


6051 6069 5066 5074 6082 5089 5097 6105 5113 6120 


4 


6128 6136 6143 5151 5159 5166 6174 6182 5189 5197 


5 


6205 5213 6220 6228 5236 5243 6251 6259 5266 5274 


6 


5282 6289 6297 5305 5312 6320 6328 6335 5343 6351 


7 


5358 5366 5374 5381 5389 6397 6404 5412 5420 6427 


8 


6435 6442 5450 6458 6405 6473 5481 6488 6496 6504 


9 


6611 5619 6526 5534 6542 6549 5667 6565 6672 6580 


570 


76587 75595 75603 76610 75618 75626 75633 75641 76648 76656 


1 


6664 5671 5679 5686 6694 5702 6709 5717 5724 5732 


2 


6740 5747 5755 6762 5770 5778 6786 6793 5800 6808 


8 


5815 6823 6831 6838 6846 6853 6861 6868 5876 6884 


4 


5891 6899 6906 6914 6921 5929 6937 6944 6962 5969 


5 


5967 6974 6982 5989 6997 6005 6012 6020 6027 6035 


6 


6042 6050 6057 6065 6072 6080 6087 6095 6103 6110 


7 


6118 6125 6133 6140 6148 6155 6163 6170 6178 6186 


8 


6193 6200 6208 6216 6223 6230 6238 6246 6253 6260 


9 


6268 6276 6283 6290 6298 6306 6313 6320 6328 6336 


580 


76343 76350 76358 76365 76373 76380 76388 76396 76403 76410 


1 


6418 6425 6433 6440 6448 6455 0402 6470 6477 6485 


2 


6492 6500 6507 6515 6522 6530 6537 6645 6552 6669 


8 


6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 


4 


6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 


5 


6716 6723 6730 6738 6745 6753 6760 6768 6775 6782 


6 


6790 6797 6805 6812 6819 6827 6834 6842 6849 6866 


7 


6864 6871 6879 6886 0893 6901 6908 6916 6923 6930 


8 


6938 6945 6963 6960 6967 6975 6982 6989 6997 7004 


9 


7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 


590 


77085 77093 77100 77107 77115 77122 77129 77137 77144 77161 


1 


7159 7166 7173 7181 7188 7195 7203 7210 7217 7225 


2 


7232 7240 7247 7254 7202 7209 7276 7283 7291 7298 


8 


7305 7313 7320 7327 7335 7342 7349 7357 7364 7371 


4 


7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 


5 


7452 7459 7466 7474 7481 7488 7495 7608 7610 7617 


6 


7525 7532 7539 7546 7554 7561 7568 7576 7683 7690 


7 


7597 7605 7612 7619 7627 7634 7641 7648 7656 7663 


8 


7670 7677 7685 7692 7699 7706 7714 7721 7728 7736 


9 


7743 7750 7767 7764 7772 7779 7786 7793 7801 7808 


600 


77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 



447 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


600 


77816 77822 77830 77837 77844 77851 77869 77866 77873 77880 




7887 789i 7902 7909 7916 7924 7931 7938 7946 7952 




7960 7967 7974 7981 7988 7996 8003 8010 8017 802i 




8032 8039 8046 8063 8061 8068 8075 8082 8089 8097 




8104 8111 8118 8126 8132 8140 8147 8154 8161 8168 




8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 




8247 8254 8262 8269 8276 8283 8290 8297 8305 8312 




8319 8326 8333 8340 8347 8355 8362 8369 8376 8383 




aSOO 8398 8405 8412 8419 8426 8433 8440 8447 8465 




8462 8469 8476 8483 8490 8497 8604 8612 8519 8626 


610 


78533 78540 78547 78654 78561 78669 78676 78683 78690 78697 




8604 8611 8618 8625 8633 8640 8647 8654 8661 8668 




8675 8682 8689 8696 8704 8711 8718 8725 8782 8739 




8746 8753 8760 8767 8774 8781 8789 8796 8803 8810 




8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 




8888 8895 8902 8909 8916 8923 8930 8937 8944 8951 




8958 8966 8972 8979 8986 8993 9000 9007 9014 9021 




9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 




9099 9106 9113 9120 9127 9134 9141 9148 9166 9162 




9169 9176 9183 9190 9197 9204 0211 9218 9226 9232 


620 


79239 79246 79263 79260 79267 79274 79281 79288 79296 79802 




9309 9316 9323 9330 9337 9344 9351 9358 9366 9372 




9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 




9449 9466 9463 9470 9477 9484 9491 9498 9505 9511 




9618 9525 9532 9539 9546 9563 9560 9567 9674 9581 




9588 9595 9602 9609 9616 9623 9630 9637 9644 9650 




9657 9664 9671 9678 9685 9692 9699 9706 9713 9720 




9727 9734 9741 9748 9754 9761 9768 9775 9782 9789 




9796 9803 9810 9817 9824 9831 9837 9844 9851 9858 




9865 9872 9879 9886 9893 9900 9906 9913 9920 9927 


680 


79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 




80003 80010 80017 80024 80030 80037 80044 80051 80058 80065 




0072 0079 0085 0092 0099 0106 0113 0120 0127 0134 




0140 0147 0154 0161 0168 0175 0182 0188 0196 0202 




0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 




0277 0284 0291 0298 0305 0312 0318 0325 0332 0339 




0346 0363 0359 0366 0373 0380 0387 0393 0400 0407 




0414 0421 0428 0434 0441 0448 0455 0462 0468 0476 




0482 0489 0496 0502 0509 0516 0523 0530 0536 0543 




0560 0657 0564 0570 0577 0584 0691 0598 0604 0611 


640 


80618 80625 80632 80638 80645 80652 80659 80665 80672 80679 




0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 




0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 




0821 0828 0835 0841 0848 0855 0862 0868 0875 0882 




0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 




0956 0963 0969 0976 0983 0990 0996 1003 1010 1017 




1023 1030 1037 1043 1050 1067 1064 1070 1077 1084 




1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 




1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 




1224 1231 1238 1245 1251 1268 1265 1271 1278 1285 


660 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81361 



448 





TABLE I.— LOGARITHMS OF NUMBERS*. \- 


N 


012 34567 89 


650 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81361 




1358 1365 1371 1378 1385 1391 1398 1405 1411 1418 




1425 1431 1438 1445 1461 1468 14(J5 1471 1478 1485 




1491 1498 1605 1611 1618 1525 1531 1538 1644 1551 




1668 1664 1571 1578 1584 1591 1698 1604 1611 1617 




1624 1631 1637 1644 1661 1657 1664 1671 1677 1684 




1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 




1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 




1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 




1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 


660 


81964 81961 81968 81974 81981 81987 81994 82000 82007 82014 




2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 




2086 2092 2099 2106 2112 2119 2126 2132 2138 2145 




2161 2168 2164 2171 2178 2184 2191 2197 2204 2210 




2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 




2282 2289 2296 2302 2308 2315 2321 2328 2334 2341 




2347 2354 2360 2367 2373 2380 2387 2393 2400 2406 




2413 2419 2426 2432 2439 2445 2452 2468 2465 2471 




2478 2484 2491 2497 2504 2610 2617 2523 2630 2636 




2643 2649 2666 2562 2669 2575 2582 2688 2695 2601 


670 


82607 82614 82620 82627 82633 82640 82646 82653 82669 82666 




2672 2679 2686 2692 2698 2705 2711 2718 2724 2730 




2737 2743 2750 2756 2763 2769 2776 2782 2789 2796 




2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 




2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 




2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 




2995 3001 3008 3014 3020 3027 3033 3040 3046 3052 




3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 




3123 3129 3136 3142 3149 3165 3161 3168 3174 3181 




3187 3193 3200 8206 3213 3219 8226 3232 3238 3245 


680 


83261 83267 83264 83270 83276 83283 83289 83296 83302 83308 




3315 3321 3327 3334 3340 3347 3353 8369 3366 3372 




3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 




3442 3448 3465 3461 3467 3474 3480 3487 3493 3499 




3606 3612 3618 3525 3531 3637 8644 3660 3656 3563 




3669 3575 3682 3588 3594 3601 3607 3613 3620 3626 




3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 




8696 3702 3708 3715 3721 3727 3734 3740 3746 3763 




8759 3766 3771 3778 3784 3790 3797 3803 3809 3816 




3822 3828 3835 3841 3847 3853 3860 3866 3872 3879 


690 


83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 




3948 3964 3960 3967 3973 3979 3985 3992 8998 4004 




4011 4017 4023 4029 4036 4042 4048 4065 4061 4067 




4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 




4136 4142 4148 4165 4161 4167 4173 4180 4186 4192 




4198 4205 4211 4217 4223 4230 4236 4242 4248 4255 




4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 




4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 




4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 




4448 4464 4460 4466 4473 4479 4485 4491 4497 4604 


700 


84610 84516 84522 84528 84535 84541 84547 84553 84559 84666 



449 





TABLE I.— LOGARITHMS OF NUMBERS. 


N 


0128456789 


700 


84510 84516 84522 84528 8453i 84541 84547 84553 84559 84566 




4572 4578 4584 4590 4597 4603 4609 4615 4621 4628 




4634 4640 4646 4652 4658 466^ 4671 4677 4688 4689 




4696 4702 4708 4714 4720 4726 4783 4789 4745 4751 




4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 




4819 4825 4831 4837 4844 48^0 4856 4862 4868 4874 




4880 4887 4893 4890 4905 4911 4917 4924 4930 4936 




4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 




5003 5009 5016 5022 5028 6034 5040 5046 5052 5058 




6065 5071 6077 6083 5089 6095 5101 5107 5114 5120 


710 


85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 




5187 5193 5199 5205 5211 5217 5224 5230 5236 5242 




5248 5254 5260 5266 5272 5278 5285 5291 5297 5303 




6309 5315 5321 5327 5333 5339 5345 6352 5358 5364 




5370 5376 5382 5388 5394 5400 5406 6412 5418 5425 




6431 6437 6443 5449 5455 6461 6467 6473 6479 5485 




6491 5497 5503 6509 6516 6522 5528 5534 6540 5546 




6552 6558 5664 6570 5576 5582 6688 6594 5600 6606 




6612 5618 5625 5631 6637 5643 5649 6655 6661 6667 




6673 6679 6685 6691 5697 5703 5709 5715 5721 5727 


720 


85788 85739 85746 85751 85757 85763 85769 85775 85781 85788 




6794 6800 5806 6812 5818 5824 6830 5836 5842 5848 




5854 5860 5866 6872 5878 5884 5890 5896 5902 6908 




6914 6920 5926 5932 5938 5944 5950 6956 5962 6968 




5974 6980 5986 5992 5998 6004 6010 6016 6022 6028 




6034 6040 6046 6052 6058 6064 6070 6076 6082 6088 




6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 




6153 6159 6165 6171 6177 6183 6189 6196 6201 6207 




6213 6219 6225 6231 6237 6243 6249 6255 6261 6267 




6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 


780 


86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 




6392 6398 6404 6410 6415 6421 6427 6433 6439 6445 




6451 6457 6463 6469 6475 6481 6487 6493 6499 6504 




6610 6616 6522 6528 6534 6540 6546 6552 6558 6564 




6670 6676 6581 6587 6593 6599 6605 6611 6617 6623 




66"?^ 6635 6641 6646 6652 6658 6664 6670 6676 6682 




6688 6694 6700 6705 6711 6717 6723 6729 6735 6741 




6747 6753 6759 6764 6770 6776 6782 6788 6794 6800 




6806 6812 6817 6823 6829 6836 6841 6847 6853 6859 




6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 


740 


86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 




6982 6988 6994 6999 7005 7011 7017 7023 7029 7035 




7040 7046 7052 7058 7064 7070 7075 7081 7087 7093 




7099 7105 7111 7116 7122 7128 7134 7140 7146 7151 




7157 7163 7169 7175 7181 7186 7192 7198 7204 7210 




7216 7221 7227- 7233 7239 7245 7251 7256 7262 7268 




7274 7280 7286 7291 7297 7303 7309 7315 7320 7326 




7332 7338 7344 7349 7355 7361 7367 7373 7379 7384 




7890 7896 7402 7408 7413 7419 7425 7431 7437 7442 




7448 7454 7460 7466 7471 7477 7483 7489 7495 7500 


750 


87606 87512 87518 87523 87529 87535 87541 87547 87662 87568 



450 

TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


750 


87S06 87612 87618 87623 87529 87635 87641 87647 87662 87668 




7564 7670 7576 7681 7687 7693 7699 7604 7610 7616 




7622 7628 7633 7639 7645 7651 7666 7662 7668 7674 




7679 7686 7691 7697 7703 7708 7714 7720 7726 7731 




7737 7743 7749 7764 7760 7766 7772 7777 7783 7789 




7795 7800 7806 7812 7818 7823 7829 7835 7841 7846 




7852 7858 7864 7869 7875 7881 7887 7892 7898 7904 




7910 7915 7921 7927 79;^ 7938 7944 7950 7966 7961 




7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 




8024 8030 8036 8041 8047 8063 8068 8064 8070 8076 


760 


88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 




8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 




8195 8201 8207 8213 8218 8224 8230 8236 8241 8247 




8262 8258 8264 8270 8276 8281 8287 8292 8298 8304 




8309 8315 8321 8326 8332 8338 8343 8349 8365 8360 




8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 




8423 8429 8434 8440 8446 8461 8467 8463 8468 8474 




8480 8486 8491 8497 8502 8508 8513 8519 8525 8630 




8536 8542 8547 8563 8559 8564 8570 8676 8581 8587 




8693 8698 8604 8610 8616 8621 8627 8632 8638 8643 


770 


88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 




8706 8711 8717 8722 8728 8734 8739 8745 8760 8766 




8762 8767 8773 8779 8784 8790 8795 8801 8807 8812 




8818 8824 8829 8835 8840 8846 8852 8867 8863 8868 




8874 8880 8885 8891 8897 8902 8908 8913 8919 8925 




8930 8936 8941 8947 8963 8968 8964 8969 8975 8981 




8986 8992 8997 9003 9009 9014 9020 9026 9031 9037 




9042 9048 9053 9059 9064 9070 9076 9081 9087 9092 




9098 9104 9109 9115 9120 9126 9131 9187 9143 9148 




9164 9159 9165 9170 9176 9182 9187 9193 9198 9204 


780 


89209 89215 89221 89226 89232 89237 89243 89248 89264 89260 




9265 9271 9276 9282 9287 9293 9298 9304 9310 9316 




9321 9326 9332 9337 9343 9348 9354 9360 9366 9371 




9376 9382 9387 9393 9398 9404 9409 9415 9421 9426 




9432 9437 9443 9448 9454 9459 9465 9470 9476 9481 




9487 9492 9498 9504 9509 9515 9620 9626 9631 9637 




9542 9548 9553 9559 9564 9570 9576 9681 9686 9592 




9697 9603 9609 9614 9620 9626 9631 9636 9642 9647 




9653 9658 9664 9669 9675 9680 9686 9691 9697 9702 




9708 9713 9719 9724 9730 9735 9741 9746 9762 9757 


790 


89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 




9818 9823 9829 9834 9840 9845 9861 9866 9862 9867 




9873 9878 9883 9889 9894 9900 9906 9911 9916 9922 




9927 9933 9938 9944 9949 9955 9960 9966 9971 9977 




9982 9988 9993 9998 90004 90009 90015 90020 90026 90031 




90037 90042 90048 90053 0059 0064 0069 0075 0080 0086 




0091 0097 0102 0108 0113 0119 0124 0129 0135 0140 




0146 0151 0167 0162 0168 0173 0179 0184 0189 0195 




0200 0206 0211 0217 0222 0227 0233 0238 0244 0249 




0255 0260 0266 0271 0276 0282 0287 0293 0296 0804 


800 


90309 90314 90820 90326 90331 90336 90342 90347 90362 90368 



451 

TABLE I.— LOGARITHMS OF NUMBERS. 



K 


0123456780 


800 


90300 00314 90320 90326 90331 90336 90342 90347 90352 90358 




0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 




0417 0423 0428 0434 0439 0445 04^0 0455 0461 0466 




0472 0477 0482 0488 0493 0499 0504 0509 0515 0520 




0526 0531 0536 0542 0547 0553 0558 0563 0569 0574 




0580 0585 0590 0596 0601 0607 0612 0617 0623 0628 




0634 0639 0644 0650 0655 0660 0666 0671 0677 0682 




0687 0693 0698 0703 0709 0714 0720 0725 0730 0736 




0741 0747 0752 0757 0768 0768 0773 0779 0784 0789 




0795 0800 0806 0811 0816 0822 0827 0832 0838 0843 


810 


90849 90864 90859 90865 90870 90875 90881 90886 90891 90897 




0902 0907 0913 0918 0924 0929 0934 0940 0945 0950 




0956 0961 0966 0972 0977 0982 0988 0993 0998 1004 




1009 1014 1020 1026 1030 1036 1041 1046 1062 1057 




1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 




1116 1121 1126 1132 1137 1142 1148 1153 1158 1164 




1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 




1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 




1275 1281 1286 1291 1297 1302 1307 1312 1318 1323 




1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 


820 


91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 




1434 1440 1445 1450 1455 1461 1466 1471 1477 1482 




1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 




1540 1546 1561 1566 1661 1666 1572 1577 1582 1587 




1693 1598 1603 1609 1614 1619 1624 1630 1635 1640 




1645 1651 1656 1661 1666 1672 1677 1682 1687 1693 




1698 1703 1709 1714 1719 1724 1730 1735 1740 1746 




1751 1766 1761 1766 1772 1777 1782 1787 1793 1798 




1803 1808 1814 1819 1824 1829 1834 1840 184^ 1850 




1866 1861 1866 1871 1876 1882 1887 1892 1897 1903 


880 


91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 




1960 1966 1971 1976 1981 1986 1991 1997 2002 2007 




2012 2018 2023 2028 2033 2038 2044 2049 2064 2059 




2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 




2117 2122 2127 2132 2137 2143 2148 2153 2158 2163 




2169 2174 2179 2184 2189 2195 2200 2205 2210 2215 




2221 2226 2231 2236 2241 2247 2252 2267 2262 2267 




2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 




2324 2330 2335 2340 2346 2360 2365 2361 2366 2371 




2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 


^TcV 


92428 92433 92438 92443 92449 92464 92469 92464 92469 92474 




2480 2485 2490 2496 2500 2606 2611 2616 2621 2526 




2631 2536 2642 2647 2562 2567 2662 2667 2572 2578 




2683 2588 2693 2698 2603 2609 2614 2619 2624 2629 




2634 2639 2645 2650 2665 2660 2666 2670 2675 2681 




2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 




2737 2742 2747 2762 2768 2763 2768 2773 2778 2783 




2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 




2840 2845 2850 2865 2860 2866 2870 2875 2881 2886 




2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 


850 


92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 



45^ 



TABLE L— LOGARITHMS OF NUMBERS. 



N 


0123456780 


850 


92042 02947 92952 92957 92962 92967 92978 92978 92983 92988 




2993 2998 3003 3008 3013 3018 3024 3029 3034 3039 




3044 3049 3054 3059 3064 3069 3075 3080 3085 3090 




3095 8100 310$ 3110 3115 3120 8125 3181 3136 3141 




3146 3151 3156 3161 3166 3171 3176 3181 3186 8192 




3197 3202 3207 3212 3217 3222 3227 3232 8237 8242 




3247 3252 3258 8263 3268 3273 3278 3283 3288 3293 




3298 3303 3308 3313 3318 3328 3328 3334 3339 3344 




3349 3354 8359 3364 3369 3374 3379 3384 3389 3394 




3399 3404 3409 3414 3420 3425 3430 3435 8440 3445 


860 


93450 93455 93460 93465 93470 93475 93480 93485 93490 93495 




3500 3505 3510 3516 8520 3526 3531 3536 3541 3546 




3551 3556 3561 3566 3571 3576 3581 3586 3591 3596 




3601 3606 3611 3616 3621 3626 8631 3636 3641 3646 




3651 3656 3661 3666 3671 3676 3682 3687 3692 3697 




3702 3707 3712 3717 3722 3727 3732 3737 3742 3747 




3752 3757 3762 8767 3772 8777 3782 3787 3792 3797 




3802 3807 3812 3817 3822 3827 3832 3837 3842 3847 




3852 3857 3862 3867 3872 3877 3882 3887 3892 3897 




3902 3907 3912 3917 3922 3927 3932 3937 3942 3947 


870 


93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 




4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 




4052 4057 4062 4067 4072 4077 4082 4086 4091 4096 




4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 




4151 4156 4161 4166 4171 4176 4181 4186 4191 4196 




4201 4206 4211 4216 4221 4226 4231 4236 4240 4245 




4250 4255 4260 4265 4270 4275 4280 4285 4290 4295 




4800 4305 4310 4315 4320 4325 4330 4335 4340 4345 




4349 4354 4359 4364 4369 4374 4379 4384 4389 4394 




4399 4404 4409 4414 4419 4424 4429 4433 4438 4443 


880 


94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 




4498 4603 4507 4512 4517 4522 4527 4532 4537 4542 




4547 4552 4557 4562 4567 4571 4576 4581 4586 4591 




4596 4601 4606 4611 4616 4621 4626 4630 4635 4640 




4645 4650 4655 4660 4665 4670 4675 4680 4685 4689 




4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 




4743 4748 4753 4768 4763 4768 4773 4778 4783 4787 




4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 




4841 4846 4851 4856 4861 4866 4871 4876 4880 4885 




4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 


890 


94939 94944 94949 94954 94969 94963 94968 94973 94978 94983 




4988 4993 4998 5002 5007 5012 5017 5022 5027 5032 




5036 6041 5046 5051 5056 5061 5066 5071 6075 5080 




5085 5090 5095 5100 6105 5109 5114 5119 5124 5129 




5134 6139 5143 5148 5153 5168 5163 5168 6173 5177 




5182 5187 6192 5197 5202 5207 5211 5216 5221 5226 




5231 5236 5240 5245 6250 6255 5260 5265 5270 5274 




6279 5284 5289 5294 6299 5303 5308 5313 5318 5323 




5328 5332 5337 5842 5347 5352 5357 5361 5366 5371 




5376 5381 5386 5390 5395 5400 5405 6410 5415 5419 


900 


95424 95429 95434 95439 95444 95448 95458 95458 95463 95468 



453 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456780 


900 


95424 95429 954:M 95439 96444 95448 96453 95458 95463 95468 




5472 5477 5482 5487 5492 5497 5501 5606 5511 5616 




5521 6525 65.S0 5535 5540 654^ 65^0 6664 6569 5664 




6669 6574 5578 5583 5688 5593 6598 5602 5607 6612 




6617 5022 5626 5631 5630 5641 5646 5650 5665 5660 




6665 6670 6674 5679 5684 6689 5694 6698 6703 5708 




5713 6718 6722 5727 6732 6737 6742 6746 6751 5756 




6761 6766 5770 5775 6780 5785 6789 5794 5799 5804 




5809 6813 5818 5823 5828 6832 6837 6842 5847 6862 




5856 6861 5866 5871 6876 6880 6885 6890 6895 6899 


910 


95904 95909 95914 96918 95923 96928 95933 95938 95942 96947 




5952 6957 5061 5966 6971 6976 6980 5985 5990 6995 




5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 




6047 6052 6057 6061 6066 6071 6076 6080 6085 6090 




6005 6099 6104 6109 6114 6118 6123 6128 6133 6137 




6142 6147 6152 6156 6161 6166 6171 6176 6180 6185 




6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 




6237 6242 6246 6251 6256 6261 6265 6270 6275 6280 




6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 




6332 6336 6341 6346 6350 6355 6360 6365 6369 6874 


920 


96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 




6426 6431 6435 6440 6445 6450 6454 6459 6464 6468 




6473 6478 6483 6487 6402 6497 6501 6506 6611 6616 




6620 6525 6530 6534 6539 6544 6548 6653 6668 6562 


4 


6567 6572 6577 6581 6586 6591 6696 6600 6605 6609 




6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 




6661 6066 6670 6676 6680 6685 6689 6694 6699 6703 




6708 6713 6717 6722 6727 6731 6736 6741 6746 6760 




6755 6759 6764 6769 6774 6778 6783 6788 6792 6797 




6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 


980 


96848 96853 96858 96802 96867 96872 96876 96881 96886 96890 




6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 




6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 




6988 6993 6997 7002 7007 7011 7016 7021 7025 7030 




7035 7039 7044 7049 7053 7058 7063 7067 7072 7077 




7081 7086 7090 7095 7100 7104 7109 7114 7118 7123 




7128 7132 7137 7142 7146 7151 7166 7160 7165 7169 




7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 




7220 7225 7230 7234 7239 7243 7248 7253 7257 7262 




7267 7271 7276 7280 7285 7290 7294 7299 7304 7308 


940 


97313 97317 97322 97327 97331 97336 97340 97345 97350 97364 




7359 7304 7368 7373 7377 7382 7387 7391 7396 7400 




7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 




7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 




7497 7502 7506 7511 7516 7520 7525 7629 7534 7639 




7543 7548 7552 7667 7562 7566 7571 7576 7580 7685 




7589 7594 7598 7603 7607 7612 7617 7621 7626 7630 




7635 7640 7644 7(549 7653 7658 7663 7667 7672 7676 




7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 




7727 7731 7736 7740 7745 7749 7754 7759 7763 7768 


960 


97772 97777 97782 97786 97791 97795 97800 97804 97809 978U 



454 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456 7'80 


960 


97772 97777 97782 97786 97791 97796 97800 97804 97809 97813 




7818 7823 7827 7832 7836 7841 7846 7860 7865 7869 




7864 7868 7873 7877 7882 7886 7891 7896 7000 7905 




7909 7914 7918 7923 7928 7932 7937 7941 7946 7980 




7965.7969 7964 7968 7973 7978 7982 7987 7991 7996 




8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 




8046 8060 8055 8069 8064 8068 8073 8078 8082 8087 




8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 




8137 8141 8146 8150 8155 8169 8164 8168 8173 8177 




8182 8186 8191 8196 8200 8204 8209 8214 8218 8223 


960 


98227 98232 98236 98241 98246 98250 98264 98269 98263 98268 




8272 8277 8281 8286 8290 8295 8299 8304 8308 8313 




8318 8322 8327 8331 8336 8340 8345 8349 8364 8:^68 




8363 8367 ^372 8376 8381 8386 8300 8394 8399 8403 




8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 




8463 8467 8462 8466 8471 8476 8480 8484 8489 8493 




8498 8602 8607 8511 8516 8620 8625 8629 8634 8638 




8543 8547 8552 8556 8561 8566 8670 8674 8579 8683 




8588 8592 8597 8601 8006 8610 8614 8619 8623 8628 




8632 8637 8641 8646 8660 8665 8669 8664 8668 8673 


970 


98677 98682 98686 98691 98696 98700 98704 98709 98713 98717 




8722 8726 8731 8736 8740 8744 8749 8763 8768 8762 




8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 




8811 8816 8820 8825 8829 8834 8838 8843 8847 8861 




8866 8860 8865 8869 8874 8878 8883 8887 8892 8896 




8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 




8945 8949 8964 8958 8963 8967 8972 8976 8981 8986 




8989 8994 8998 9003 9007 9012 9016 9021 9026 9029 




9034 90;38 9043 9047 9062 9056 9061 9065 9069 9074 




9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 


980 


99123 99127 99131 99136 99140 99145 99149 99154 99168 99162 




9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 




9211 9216 9220 9224 9229 9233 9238 9242 9247 9261 




9266 9260 9264 9269 9273 9277 9282 9286 9291 9296 




9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 




9344 9348 9352 9357 9301 9366 9370 9374 9879 9383 




9388 9392 9396 9401 9406 9410 9414 9419 9423 9427 




9432 9436 9441 9445 9449 9464 9468 9463 9467 9471 




9476 9480 9484 9489 9493 9498 9602 9606 9611 9516 




9620 9624 9628 9633 9537 9542 9646 9660 9665 9669 


990 


99664 99568 99572 99677 99581 99586 99690 99694 99599 99603 




9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 




9651 9656 9660 9664 9669 9673 9677 9682 9686 9691 




9695 9699 9704 9708 9712 9717 9721 9726 9730 9734 




9739 9743 9747 9752 9756 9760 9765 9769 9774 9778 




9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 




9826 9830 9835 9839 9843 9848 9862 9866 9861 9865 




9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 




9913 9917 9922 9926 9930 9935 9939 9944 9948 9962 




9957 9961 9966 9970 9974 9978 9983 9987 9991 9996 


1000 


00000 00004 00009 00013 00017 00022 00026 00030 00085 00a39 



455 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



t 




©• 


1 


• 


«• 


/ 


SiDe 


CkMdne 


Sine 


Oosine 


Sine 


Cosine 





—00 


10.00000 


8.84186 


9.99993 


6.54268 


9.99974 


60 


1 


6.46878 


00000 


84908 


:>9998 


54648 


99978 


60 


9 


78476 


00000 


85609 




54999 


99978 


58 


8 


04066 


00000 


86304 


99993 


55354 


99978 


57 


4 


7.00579 


00000 


26968 


99992 


56706 


99978 


56 


5 


16870 


00000 


27661 


99992 


56054 


69971 


65 





84188 


00000 


28324 


QOOOO 


56400 


99971 


54 


7 


80888 


00000 


28975 


99992 


56748 


99970 


58 


8 


88688 


00000 


29621 


99992 


57064 


99970 


58 


9 


41797 


00000 


80256 


99991 


57421 


99969 


61 


10 


7.46878 


10.00000 


8.30679 


9.99991 


8.57757 


9.99969 


60 


11 


50518 


00000 


81495 


99991 


58069 


99986 


49 


18 


54891 


00000 


82106 


99990 


56419 


99966 


48 


18 


57767 


00000 


82702 


99990 


58747 


99967 


47 


14 


60966 


00000 


88208 


99990 


58078 


99967 


46 


16 


68968 


00000 


88875 


99990 


50895 


99967 


46 


18 


66784 


00000 


81450 


99989 


59715 


99960 


44 


17 


69417 


9.99999 


S.^18 


99989 


60088 


99966 


48 


18 


71900 


OOQQQ 
WtfW 


85678 


99969 


60849 


99965 


48 


19 


74848 




86131 


99969 


60668 


VWwV 


41 


90 


7.76475 


9.99999 


8.36676 


9.99986 


8.60978 


9.99964 


40 


81 


78694 


OQOQO 
VWW 


37217 


99988 


61288 


99963 


89 


88 


80615 




87750 


99968 


61589 


99968 


88 


88 


88545 


99999 


88276 


99987 


61894 


99062 


87 


84 


84898 


99999 


88796 


99967 


68196 


99903 


86 


85 


86166 


00099 


89310 


99987 


(B497 


99961 


85 


26 


87870 


99999 


89618 


99986 


62796 


99961 


84 


37 


80509 


99999 


40890 


99986 


68091 


99980 


88 


88 


91088 


99999 


40816 


99966 


68886 


99980 


82 


89 


98618 


OMQR 


41807 


99085 


68876 


99959 


81 


80 


7.94064 


9.99996 


8.41798 


9.99966 


8.68968 


9.99959 


80 


81 


95506 


90998 


42272 


99983 


64256 


99958 


29 


88 


96697 


99998 


42746 


99964 


64548 


99968 


26 


88 


96283 


99908 


48216 


99984 


64827 


99957 


87 


84 


99520 


QQAQO 


48680 


99964 


65110 


99956 


26 


85 


8.00779 


09006 


44139 


99983 


65891 


99956 


86 


86 


02008 


QQQQO 


44594 


99988 


65670 


99955 


84 


87 


08198 


99997 


45044 


99963 


65947 


99965 


23 


88 


04350 


99997 


45489 


99982 


66228 


99954 


22 


89 


05478 


99997 


45930 


99962 


66497 


99954 


21 


40 


8.0657B 


9.99997 


8.46866 


9.99962 


8.66709 


9.99958 


20 


41 


07050 


99997 


46799 


99961 


67089 




19 


48 


06696 


99997 


47226 


99981 


67306 


99952 


18 


48 


09718 


99997 


47650 


99061 


6757B 


99951 


17 


44 


10717 


OOOCM 
WMID 


46069 


99960 


67841 


99951 


16 


45 


11698 


99996 


48486 


99980 


68104 


99960 


15 


46 


12647 


OQQOA 


48896 


99979 


68367 


99949 


14 


47 


18581 


99996 


49304 


99979 


68627 


99949 


13 


48 


14496 


OQQOA 


49706 


99979 


68866 


99948 


12 


49 


15891 


99996 


50106 


99976 


60144 


99918 


11 


60 


8.16868 


9.99996 


8.50504 


9.99978 


6.60400 


9.99947 


10 


51 


17126 


99996 


50807 


99977 


69654 


99946 





58 


17971 


99996 


51287 


99977 


609O7 


99946 


8 


68 


18796 


90995 


51678 


99977 


70159 


99045 


7 


54 


19610 


99995 


58055 


99976 


70409 


99944 


6 


66 


20407 


QQQOA 


52484 


99976 


70658 


90944 


5 


56 


81189 


99994 


52810 


99975 


70905 


99943 


4 


57 


819.58 


QQQQti 


53188 


99975 


' 71151 


99942 


8 


58 


22718 


90994 


53552 


991)74 


71396 


99942 


2 


50 


28156 


99994 


53019 


99974 


71636 


99941 


1 


60 


24186 


90003 


54282 


99974 


71880 


99940 





$ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 




89° 


88 


« 


87 


o 



4S6 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



9 




go 




4* 


«• 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cofdne 





8.71880 


9.99940 


8.84858 


9.99894 


8.94080 


9.99884 


60 


1 


72120 


99940 


84589 


0S8Q8 


94174 


99888 


59 


9 


Tum 


99988 


84n8 


99692 


94817 


99682 


58 


8 


78597 


99988 


84807 


99691 


94461 


99681 


57 


4 


78884 


99988 


85076 


99691 


94608 


99880 


66 


6 


78069 


99987 


85252 


99690 


94746 


99889 


55 


6 


78308 


99986 


86429 


99889 


94887 


99628 


54 


7 


78685 


99936 


85606 


99888 


96029 


99827 


53 


8 


78T67 


99985 


85780 


99887 


95170 


99625 


62 


9 


78997 


99934 


85966 


99886 


06810 


99624 


61 


10 


8.74826 


9.99984 


8.86128 


9.99686 


8.06460 


9.99688 


50 


11 


74454 


99988 


86301 


99684 


95569 


99622 


49 


18 


74680 


99982 


86474 


99888 


9.5?^ 


99821 


48 


18 


74906 


99982 


86646 


99682 


93867 


99620 


47 


14 


75130 


99981 


86816 


99681 


96005 


99819 


46 


15 


75868 


99980 


80967 


99880 


06148 


99817 


45 


16 


75575 


99929 


87156 


99679 


96280 


99616 


44 


17 


75795 


99929 


87825 


99679 


96417 


99815 


48 


18 


76015 


99928 


87494 


99678 


96568 


99814 


48 


19 


76234 


99927 


87661 


90877 


96689 


99618 


•41 


80 


8.76451 


9.99926 


8.87829 


9.99876 


8.96825 


9.99618 


40 


81 


76667 


99926 


87996 


99675 


96960 


99610 


89 


23 


76888 


99925 


88101 


99674 


97095 


99609 


88 


88 


77097 


99924 


88326 


99678 


97829 


99606 


87 


84 


77810 


99928 


88490 


998<2 


97868 


99807 


86 


85 


77522 


999-28 


88G54 


99671 


97496 


99606 


35 


26 


77738 


99922 


88817 


99870 


97629 


99804 


84 


27 


77948 


99921 


88080 


99869 


97762 


99808 


88 


88 


78162 


99920 


89142 


99868 


97894 


99F02 


82 


89 


78360 


99920 


89304 


90667 


98026 


99801 


31 


80 


8.78668 


9.99919 


8.89464 


9.99666 


8.98157 


9.99600 


80 


81 


78T74 


99918 


89683 


99665 


96288 


99798 


29 


38 


78979 


99917 


89784 


99864 


98419 


99797 


28 


88 


79188 


99917 


89948 


99668 


98549 


99796 


27 


84 


79386 


99916 


90102 


99662 


96679 


99795 


28 


86 


79588 


99916 


90260 


99661 


98806 


99798 


25 


86 


79789 


99914 


90417 


99860 


98937 


99792 


24 


87 


79990 


99918 


90574 


99650 


99066 


99791 


88 


88 


80189 


99918 


90780 


99658 


99194 


99790 


82 


89 


80388 


99912 


90886 


99657 


99822 


99788 


21 


40 


8.80585 


9.99911 


8.91040 


9.99656 


8.99450 


9.99787 


80 


41 


80782 


99910 


91195 


99855 


99577 


99786 


19 


42 


80978 


99909 


91849 


99854 


99704 


99785 


18 


48 


81178 


99909 


01502 


99853 


99630 


99783 


17 


44 


81367 


99906 


91655 


99868 


99056 


99782 


16 


45 


81560 


99907 


91807 


99651 


9.00082 


99781 


16 


46 


81752 


99906 


91959 


99850 


00207 


99780 


14 


47 


81944 


99905 


92110 


99648 


00:^32 


99r78 


18 


48 


82134 


99904 


92261 


99647 


00456 


99777 


18 


49 


82324 


99904 


92411 


99646 


00681 


99776 


11 


50 


8.82518 


9.99903 


8.98561 


9.99645 


9.00704 


9.99776 


10 


51 


82701 


99902 


92710 


99644 


00628 


99778 


9 


58 


R2888 


09901 


92859 


99848 


00951 


99772 


8 


58 


83075 


99900 


93007 


99842 


01074 


99771 


7 


54 


88261 


90699 


9:3154 


99841 


01196 


99769 


6 


55 


88446 


9'.«98 


93301 


99840 


01818 


99768 


6 


56 


8'i630 


99S98 


03448 


99839 


01440 


99767 


4 


57 


83818 


99897 


93594 


99838 


01561 




8 


58 


83996 


99896 


98740 


99887 


01682 


99764 


S 


59 


84177 


99895 


08885 


99836 


01803 




1 


60 


84868 


99894 


94030 


99884 


01923 


99761 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


9 




86« 




86« 




84- 



457 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



# 


6 


e 




7» 




%• 


/ 


Bine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.01988 


9.99761 


9.08589 


9.99675 


9.14856 


9.99575 


60 


1 


0^48 


99760 


06692 


99674 


14445 


99574 


59 


2 


02168 


99759 


08795 


99672 


14585 


9957% 


58 


8 


02388 


99W7 


06897 


99670 


14624 


99570 


67 


4 


0^402 


09756 


06999 


99669 


14714 


99568 


56 


6 


Oisao 


99766 


00101 


99667 


14808 


99566 


66 


6 


ami 


99758 


Qoao-i 


99660 


14891 


99666 


54 


7 


02757 


9am 


09S04 


99664 


14980 


99508 


58 


8 


0tf8r4 


09406 


99663 


16069 


99561 


52 


9 


0299;3 


90749 


09506 


99661 


15167 


99559 


51 


10 


9.03109 


9.99748 


0.09606 


0.99659 


9.15245 


0.99657 


50 


11 


08326 


99747 


09707 


99658 


15388 


99656 


49 


19 


03342 


99745 


09807 


99656 


15421 


99564 


48 


18 


03458 


99744 


0^907 


99055 


15608 


99662 


47 


14 


08574 


99748 


10006 


99653 


16696 


90650 


40 


15 


aMS90 


99741 


10106 


99G51 


15688 


99548 


45 


16 


03S03 


99740 


10205 


99650 


16770 


99546 


44 


17 


mio 


99788 


10304 


99648 


15657 


99545 


48 


18 


01034 


99787 


10402 


99647 


15944 


99548 


42 


19 


01149 


99786 


10501 


99645 


16060 


99541 


41 


20 


9.04262 


9.99784 


9.10609 


9.99648 


0.16116 


9.99689 


40 


21 


04876 


99783 


10697 


99642 


16203 


99537 


88 


23 


04490 


99781 


10795 


99640 


16269 


09585 


88 


23 


04608 


99780 


10603 


99688 


16874 


09633 


87 


24 


04715 


99^ 


10990 


99637 


164GO 


99582 


86 


25 


04828 


99727 


11087 


99635 


16545 


99580 


35 


26 


04940 


99726 


11184 


99638 


16631 


99628 


84 


27 


06052 


99724 


11281 


99682 


16716 


99526 


88 


28 


05164 


99723 


iiarr 


99630 


16801 


99524 


82 


29 


05276 


99721 


11474 


99629 


16886 


99522 


31 


80 


9.05886 


9.99720 


0.11670 


0.99627 


0.16970 


9.99520 


80 


81 


06497 


99718 


11G66 


99625 


17055 


99518 


29 


88 


05607 


99717 


11761 


99624 


17139 


99517 


28 


88 


05717 


99716 


lia57 


99622 


17228 


99515 


27 


84 


05827 


99714 


11952 


99620 


17807 


99518 


26 


85 


05037 


90718 


12047 


99618 


17391 


99511 


25 


86 


06046 


99711 


12142 


99617 


17474 


99509 


24 


87 


06155 


99710 


12286 


99615 


17568 


99507 


28 


88 


06.264 


99706 


1233i 


99618 


17641 


99605 


22 


89 


0687:2 


99707 


12425 


9U612 


17724 


99508 


21 


40 


9.06481 


9.99706 


9.12619 


&.99610 


9.17807 


9.99501 


20 


41 


06580 


99704 


12612 


99608 


17890 


99499 


19 


42 


06696 


99702 


12706 


99607 


17978 


99497 


18 


43 


06804 


99701 


12799 


99605 


18055 


99496 


17 


44 


06911 


QQAQQ 


12892 


99608 


18187 


99494 


10 


45 


07018 


99608 


12985 


99601 


18220 


99492 


15 


40 


07124 


QQMIA 


18078 


99600 


18902 


99490 


14 


47 


07231 


99605 


18171 


99596 


18388 


99488 


18 


48 


07837 


99693 


13268 


99596 


18465 


99486 


12 


49 


07442 


99692 


18365 


99696 


18547 


99484 


11 


60 


9.07548 


9.99690 


9.18447 


0.99693 


9.16628 


9.99482 


10 


51 


07653 


99689 


18589 


99591 


18709 


99480 





62 


07768 


99687 


13030 


90589 


18790 


99478 


8 


68 


07868 


99686 


18722 


99588 


18871 


99476 


7 


64 


07968 


90684 


18818 


99586 


18952 


99474 


6 


66 


08072 


99683 


18904 


99584 


19088 


99472 


5 


66 


06176 


99681 


i:i994 


99582 


19118 


99470 


4 


67 


08^80 


99680 


14085 


99581 


19193 


99468 


8 


68 


06888 


99678 


14175 


99579 


19273 


99J66 


8 


80 


06480 


99677 


14266 


99577 


19358 


99464 


1 


60 


08589 


99675 


14366 


99575 


19483 


99462 


• 


# 


' Cosine 


Sine 


Cosirift 


Sine 


Cosine 


Sine 


/ 




8r» 




82» 




81» 



4S8 



TABLE 11.— LOGARITHMIC SINES AND COSINES. 



t 




BO 


10« 


IP 1 


/ 


Sine 


Cosine 


Sine 


Coeine 


Sine 


Cosine 




9.19488 


9.99482 


9.28967 


9.99886 


9.28060 


9.99195 


60 




19618 


09460 


24089 


99888 


28185 


99192 


69 




19602 


99468 


24110 


99831 


28190 


99190 


66 




19672 


90456 


24181 


99888 


28854 


99187 


67 




19751 


99454 


24258 


99396 


28819 


99186 


66 




19880 


99462 


24824 


99684 


28884 


99182 


65 




19909 


99450 


24805 


09822 


28448 


99180 


64 




19088 


09448 


24466 


99319 


28512 


99177 


68 




20067 


99446 


24536 


99317 


28677 


99175 


69 




20146 


99444 


24607 


99315 


28641 


99172 


61 




0.20828 


9.99442 


9.24677 


9.99818 


9.88706 


9.99170 


60 




80802 


09440 


24748 


99810 


28769 


99167 


49 




90880 


09438 


24818 


99308 


28888 


99165 


48 




90468 


99486 


24888 


99806 


88896 


99162 


47 




90685 


99434 


24968 


99804 


28960 


99160 


46 




90618 


99482 


2SQ86 


99801 


29024 


99157 


46 




90601 


99429 


26098 


99899 


29087 


99156 


44 




<0768 


99427 


25168 


99897 


29150 


99162 


48 




90646 


99426 


95287 


99894 


29814 


99160 


42 




90022 


99428 


25807 


99898 


29877 


99147 


41 


90 


9.90999 


9.99421 


0.26876 


0.99890 


9.89340 


9.99146 


40 


21 


91076 


99419 


25445 


99888 


29408 


99148 


89 


23 


91168 


99417 


25514 


99885 


29466 


99140 


88 


28 


8!«» 


99416 


25688 


99888 


29529 


99187 


87 


24 


91806 


99418 


25652 


99881 


29591 


99185 


86 


25 


21862 


90411 


25721 


99278 


29654 


99188 


35 


23 


21466 


99400 


25700 


99876 


29716 


99180 


84 


27 


21984 


09407 


85858 


99874 


29779 


99127 


38 


28 


21010 


99404 


26927 


99871 


29841 


99124 


88 


29 


21686 


99402 


26995 


99269 


29908 


99122 


31 


80 


0.21761 


9.99400 


9.26068 


9.99867 


9.29966 


9.99119 


80 


81 




99396 


26181 


99864 


80088 


99117 


29 


82 


219B7 


99896 


26199 


99862 


80090 


99114 


26 


88 


99894 


26267 


99860 


80161 


99112 


27 


84 


29062 


99392 


26336 


99857 


80818 


99109 


26 


86 


22187 


99390 


96408 


99856 


80875 


09106 


25 


80 


22211 


99388 


26470 


99252 


80886 


90104 


24 


37 


22S86 


99.385 


26688 


OfiStfO 


80898 


90001 


28 


88 


22861 


99388 


26605 


99848 


80450 


99009 


22 


89 


22485 


99881 


26672 


99845 


80621 


99096 


21 


40 


9.29609 


9.99379 


9.86789 


9.99848 


9.80688 


0.99096 


20 


41 


22588 


99377 


26806 


99241 


80648 


99091 


19 


42 


22667 


99375 


26878 


99888 


807t)4 


99088 


18 


48 


22781 


99378 


26040 


99886 


80766 


99086 


17 


44 


28806 


99870 


27007 


99288 


80826 


99068 


16 


45 


22878 


99866 


27078 


90881 


80887 


99060 


16 


46 


22962 


99366 


27140 


99829 


80947 


99078 


14 


47 


28025 


99864 


97206 


99226 


81006 


99075 


18 


48 


28096 


99362 


27878 


99284 


81068 


99072 


12 


49 


28171 


99859 


27889 


99281 


81129 


99070 


11 


50 


9.28244 


9.99867 


0.27405 


9.99819 


9.81180 


0.99067 


10 


51 


28317 


99256 


27471 


99217 


81250 


99064 


9 


62 


83890 


99868 


27fS87 


99814 


81810 


99062 


8 


68 


23462 


99361 


27602 


99812 


81870 


99060 


7 


64 


28585 


99348 


27666 


99809 


81480 


09066 


6 


66 


28607 


99346 


27734 


99807 


81490 


99054 


6 


50 


23879 


90844 


27799 


99804 


81540 


99061 


4 


67 


28763 


9934? 


27864 


99808 


81609 


99048 


8 


66 


28823 


99340 


27980 


99800 


81669 


99046 


2 


SO 


23895 


993:J7 


£7995 


91>197 


317S8 


90043 


1 


m 


wm 


B9335 


280ft0 


HOI 95 


3178ft 


990*0 





# 


COBlJie 


Sine 


Cosine 


Sine 


Cosine 


Sine 


* 




W 




7t* 




yg- 



459 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



/ 


!«• 


IS* 


14- 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.81788 


9.99040 


9.85200 


9.06872 


9.88868 


9.96690 


60 


1 


81847 


99088 


85263 


06869 


88418 


96687 


50 


2 


31907 


99066 


85318 


96867 


38469 


96684 


58 


8 


81966 


99082 


85878 


06864 


88519 


96681 


57 


4 


880:25 


99080 


35427 


98861 


88570 


96678 


56 


5 


82084 


99027 


85481 


08868 


88620 


98675 


56 


6 


88143 


99024 


86536 


96865 


88670 


98671 


54 


7 


82202 


99028 


85590 


96862 


88721 


96668 


58 


8 


82261 


99019 


85644 


98649 


88771 


96665 


52 


9 


88819 


99016 


86696 


08846 


88821 


98662 


51 


10 


9.82878 


9.99018 


9.85752 


9.96848 


9.88871 


0.98659 


60 


11 


38487 


99011 


85606 


96840 


38921 


96666 


49 


18 


32495 


99006 


85860 


08837 


38971 


96652 


48 


18 


82553 


99005 


85914 


98684 


89021 


98649 


47 


14 


38612 


99002 


85968 


86881 


80071 


96646 


46 


15 


88670 


99000 


86022 


96828 


89121 


06648 


45 


16 


82728 


96997 


86075 


96625 


89170 


96640 


44 


17 


32786 


96004 


86129 


98822 


89220 


98636 


48 


18 


82844 


98991 


86182 


98819 


39270 


96688 


42 


19 


88908 


96969 


36286 


06816 


89819 


96(380 


41 


80 


9.82960 


9.96986 


9.86280 


0.06818 


9.89660 


0.086^ 


40 


81 


88018 


96088 


86842 


98810 


88418 


96623 


89 


88 


88075 


96060 


86895 


06807 


39467 


96620 


38 


88 


83183 


96978 


86449 


86804 


89517 


06617 


.97 


84 


33190 


96975 


86502 


98601 


89566 


98614 


86 


85 


83248 


96972 


86655 


96708 


89615 


98610 


85 


86 


83305 


98969 


86608 


96796 


89664 


98607 


84 


87 


83362 


98967 


86660 


98798 


89718 


98604 


S3 


88 


88420 


98964 


86718 


96789 


89762 


96601 


82 


80 


83477 


98961 


86766 


96786 


89611 


96597 


31 


80 


0.33584 


9.98958 


9.86819 


9.98788 


9.89660 


0.98594 


30 


81 


38591 


96955 


86871 


96780 


89909 


98591 


29 


88 


38647 


98853 


86924 


96777 


39958 


98588 


26 


S3 


33704 


96950 


36976 


98774 


40006 


98584 


27 


84 


88761 


96947 


87028 


98771 


40055 


98581 


26 


85 


88818 


98944 


87081 


96768 


40103 


98578 


25 


86 


88874 


98941 


87138 


98765 


40152 


96574 


24 


87 


83931 


96038 


37185 


96708 


40200 


96571 


88 


88 


88987 


98936 


87287 


96750 


40249 


98568 


22 


89 


84043 


98988 


37288 


96756 


40297 


96565 


81 


40 


9.84100 


9.98900 


9.87841 


0.96758 


9.40846 


9.98561 


20 


41 


84156 


98927 


87398 


98750 


40894 


96558 


19 


48 


84212 


96924 


87445 


98746 


40442 


96566 


18 


43 


84268 


96921 


87497 


98743 


40490 


98561 


17 


44 


84324 


98919 


37549 


96740 


40538 


98548 


16 


45 


84380 


98916 


87600 


98787 


40586 


98546 


15 


46 


34436 


98918 


87652 


98734 


40684 


98541 


14 


47 


34491 


98910 


37708 


98781 


40682 


98538 


18 


48 


34547 


98907 


87755 


96728 


40780 


98535 


12 


49 


84602 


96904 


87806 


98725 


40778 


96531 


11 


60 


9.34658 


9.98901 


9.87858 


9.96722 


0.40625 


9.98528 


10 


51 


34713 


98898 


87909 


96ri9 


40878 


96525 


9 


68 


84769 


98896 


37960 


96715 


40921 


98521 


8. 


53 


84824 


98893 


38011 


96712 


40968 


98518 


7 


54 


34879 


98890 


38062 


96700 


41016 


96515 


6 


65 


84984 


98887 


88118 


08706 


41068 


9a-)ii 


5 


56 


34989 


98884 


88164 


96708 


41111 


96508 


4 


57 


85044 


98881 


38215 


98700 


41158 


98505 


8 


68 


35099 


9W78 


mm 


9^97 


41305 


98501 


2 


59 


SB154 


Qf^Tr. 


S6:i37 


U*i*i 


41i^J 


9S49H 


1 


m 


mm9 


&S8::J 


3im)^ 


98COU 


41300 


HfUM 





t 


Cofrtno 


sine 


CkrislDe 


Sine 


Cofltne 


Sfne 


f 




77- 




7«* 




76' 



460 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



9 


16« 


16«» 


II* 





Sine 


CkNiine 


Sine 


Cosine 


Sine 


Cosine 







9.41800 


9.96494 


9.44084 


9.98884 


9.46504 


9.98060 


60 


1 


41847 


98191 


44078 


98281 


46635 


98066 


60 


2 


41804 


96488 


44122 


98277 


46676 


96068 


66 


8 


41441 


98484 


44166 


98878 


46717 


98048 


67 


4 


41488 


96481 


44210 


98270 


46758 


08044 


66 


5 


41585 


98477 


44258 


98266 


46800 


98040 


66 





41588 


98474 


44297 


96262 


46841 


98086 


54 


7 


410;® 


98471 


44341 


98259 


46888 


98088 


68 


8 


41675 


98467 


44385 


98255 


46023 


9800 


58 


9 


41722 


98464 


44428 


98251 


46964 


98ffii5 


51 


10 


9.41768 


9.98460 


9.44472 


9.98248 


9.47006 


9.96081 


60 


11 


41815 


98457 


44516 


98244 


47045 


98017 


49 


12 


41861 


98458 


44559 


96240 


47066 


08018 


48 


18 


41908 


98450 


44G02 


98237 


47187 


98009 


47 


H 


41954 


98447 


44646 


98283 


47168 


96006 


46 


15 


42001 


96448 


446K9 


98229 


47209 


98001 


45 


16 


42047 


9B440 


447^ 


96226 


47^249 


97997 


44 


17 


42093 


98436 


44776 


98222 


47290 


97998 


48 


18 


42140 


98483 


44810 


96218 


47380 


97969 


42 


19 


42186 


98429 


44862 


98215 


47371 


97986 


41 


20 


9.42232 


9.96426 


9.44906 


9.96211 


9.47411 


9.97988 


40 


21 


42278 


98422 


44948 


96207 


47452 


97978 


80 


82 


42324 


98419 


44998 


98204 


47498 


97974 


88 


23 


42370 


9W15 


4.'W)3» 


98200 


47583 


97970 


87 


24 


42416 


98412 


45077 


98196 


47573 


97966 


86 


25 


42461 


98409 


46120 


98192 


47618 


97962 


85 


26 


42507 


96406 


45168 


96189 


47654 


97958 


84 


27 


42553 


98402 


45206 


96185 


47694 


97954 


83 


28 


42599 


98396 


45249 


96181 


47734 


97960 


8< 


29 


42644 


06895 


45292 


98m 


47774 


97946 


81 


80 


9.42690 


9.98891 


9.45834 


9.98174 


9.47814 


9.97948 


80 


81 


42735 


98.388 


45377 


98170 


47854 


97988 


29 


82 


42781 


98384 


45419 


98166 


47894 


97984 


28 


83 


42826 


96381 


45463 


96168 


47934 


97980 


27 


84 


42872 


96377 


45504 


98159 


47974 


97926 


26 


85 


42917 


96878 


45547 


96155 


4.-014 


97928 


85 


38 


42968 


96370 


45589 


96151 


48054 


97918 


84 


87 


43008 


98366 


4.^632 


98147 


48094 


97914 


88 


88 


48058 


9a363 


4.'>674 


98144 


48188 


97910 


88 


89 


43098 


96359 


45716 


96140 


48178 


97906 


21 


40 


0.48148 


9.96856 


9.45758 


9.96186 


9.48818 


9.97908 


80 


41 


43188 


98352 


45801 


96188 


^ 48252 


97898 


19 


42 


48283 


98^9 


45848 


96129 


48298 


97894 


18 


43 


48278 


98345 


45885 


98125 


48382 


97890 


17 


44 


48328 


9a342 


45927 


98181 


48371 


97686 


16 


45 


48367 


98338 


45969 


98117 


48411 


97882 


15 


40 


43412 


98334 


46011 


98118 


48450 


97878 


14 


47 


48157 


98331 


46058 


96110 


48490 


97874 


18 


48 


43502 


98327 


46095 


98106 


48629 


97870 


12 


49 


48546 


98324 


46186 


96108 


48568 


07866 


11 


50 


9.43591 


9.98320 


9.46178 


9.98098 


9.48007 


9.97861 


10 


51 




98317 


462-iO 


98094 


48647 


97857 


9 


52 


43680 


0^813 


46J68 


98090 


48686 


97858 


8 


58 


43724 


98:309 


46.308 


9S087 


48?25 


97849 


7 


54 


43769 


98:W6 


46.345 


98083 


48784 


97845 


6 


55 


43813 


98302 


46:^ 


98079 


48808 


97841 


5 


56 


43857 


96299 


46428 


98075 


4aS42 


97887 


4 


57 


43901 


98295 


46469 


9S071 


48881 


97838 


8 


58 


43946 


98291 


46511 


98067 


48920 


97829 


8 


59 


48090 


98388 


46558 


98063 


48959 


97885 


1 


60 


44034 


96284 


46594 


98060 


48996 


97821 





# 


CoRlne 


Sine 


Cosine 


Sine 


Cosine 


Sine 


f 




74« 




78<» 




720 



46i 



TABLE 11.— LOGARITHMIC SINES AND COSINES. 



/ 


IS'* 


!»• 


20O 


/ 


Sine 


Cosine 


Bine 


Coeine 


Bine 


Cosine 





9.4899R 


9.97821 


9.61864 


9.97507 


9.58406 


9.97299 


60 


1 


49037 


97817 


51801 


97563 


68440 


97294 


69 


2 


49076 


97812 


51838 


97658 


68475 


97889 


66 


8 


49113 


97808 


61374 


97554 


68609 


97886 


67 


4 


49153 


97804 


61411 


97550 


68644 


97280 


66 


6 


49192 


97800 


61447 


97545 


68678 




66« 


6 


49231 


97796 


61484 


97541 


68618 


97871 


64 


7 


49269 


97792 


51520 


97586 


68647 


97866 


68 


8 


49308 


97788 


61557 


97582 


68682 


97268 


68 


9 


49847 


97784 


51598 


97528 


68716 


9Rtt7 


51 


10 


9.49885 


9.97779 


9.61689 


9.97K8 


9.58751 


9.97868 


60 


11 


49424 


97776 


61666 


97519 


58786 


97248 


49 


n 


49462 


97771 


61702 


97516 


63819 


97848 


48 


18 


49500 


97767 


51788 


97510 


63854 


97888 


47 


14 


49539 


97763 


51774 


97506 


53888 


vnoi 


46 


15 


49577 


97759 


51811 


97601 


53922 


97829 


46 


16 


49615 


97754 


51847 


97497 


63967 


97884 


44 


17 


49654 


97750 


61883 


97498 


68991 


97220 


48 


18 


49692 


97746 


61019 


97488 


54085 


97216 


48 


19 


49730 


97742 


61966 


97484 


f059 


9?210 


41 


20 


9.49768 


9.97738 


9.51991 


9.97479 


9.64098 


9.97806 


40 


21 


49806 


97784 


62027 


97476 


54127 


9?201 


89 


22 


49844 


9r729 


52063 


97470 


64161 


97196 


88 


23 


49882 


97725 


62099 


97466 


54196 


97192 


87 


24 


499*^0 


97721 


52135 


97461 


64229 


97187 


86 


25 


49958 


97717 


52171 


97457 


54288 


97188 


85 


26 


40996 


97718 


52207 


97458 


54297 


97178 


84 


27 


50084 


9T708 


62242 


97448 


64881 


9n78 


88 


28 


60072 


97704 


62278 


97444 


54365 


97168 


88 


29 


50110 


97700 


52314 


97489 


54399 


97168 


81 


80 


9.50148 


9.97696 


9.52350 


9.97435 


9.54488 


9.97169 


80 


81 


50185 


97691 


52385 


97480 


54466 


97154 


89 


82 


60228 


97687 


62421 


97426 


54500 


97149 


28 


83 


50261 


97688 


524.56 


97421 


64584 


97145 


87 


34 


50298 


97679 


52498 


97417 


54567 


97140 


86 


85 


50836 


97674 


52527 


97412 


54601 


97186 


85 


86 


50^74 


97670 


525G3 


97408 


64685 


97180 


84 


87 


50411 


97666 


52598 


97403 


54668 


97126 


88 


88 


50449 


97662 


52634 


97899 


54702 


97121 


88 


89 


50486 


97657 


52669 


97394 


54736 


97116 


81 


40 


9.50523 


9.97658 


9.52706 


9.97890 


9.54769 


9.97111 


80 


41 


60561 


97649 


5t.'740 


97:«5 


54802 


97107 




42 


50599 


97645 


527T5 


97381 


64836 


97108 




48 


50C35 


97640 


52811 


97876 


54869 


97097 




44 


50673 


97636 


52846 


97372 


54908 


97098 




45 


60710 


97632 


5'J881 


97867 


54936 


97087 




46 


50747 


97628 


52916 


97368 


64960 


97088 




47 


50784 


97623 


62951 


97858 


65008 


97078 




48 


508J1 


97619 


52986 


97358 


65036 


97078 




49 


50a58 


97615 


53021 


97349 


66060 


97068 




SO 


9.50896 


9.97610 


9.68056 


9.97344 


9.55108 


9.97068 




51 


509-3;3 


97606 


53092 


97340 


65186 


97059 




52 


60970 


97602 


63126 


97835 


55169 


97054 




58 


51007 


97597 


53161 


97881 


65208 


97049 




54 


510^ 


97598 


58196 


97826 


65285 


97044 




56 


51080 


97589 


63231 


97828 


65.?68 


97039 




56 


51117 


97584 


53266 


97317 


55301 


97086 




57 


51154 


97580 


53301 


97312 


5.5384 


97080 




58 


61191 


97576 


53.336 


97808 


55367 


97025 




69 


51227 


97571 


63370 


97803 


55400 


97020 




60 


51264 


97567 


53405 


97299 


56433 


9J016 




f 


Ck>8lne 


Sine 


Cosine 


Sine 


Cosine 


Sine 


i 




710 




70* 




«9» 



462 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



/ 


«!• 


220 


w 1 


§ 


Sine 


CkMine 


Sine 


Cosine 


Sine 


Cosine 





9.56488 


9.97016 


9.57368 


9.96717 


9.59168 


9.96408 


00 


1 


OOvOO 


97010 


67889 


96711 


59218 


96897 


60 


s 


66499 


97005 


67490 


96706 


69247 


96898 


68 


8 


666ftS 


97001 


67451 


96701 


59877 


968^7 


57 


4 


66564 


96906 


67488 


96696 


69807 


96881 


66 


6 


65697 


96991 


67514 


96691 


59386 


96876 


65 


6 


56680 


96986 


67546 


96686 


5986« 


96870 


54 


7 


56668 


96961 


57576 


96681 


59896 


96865 


53 


8 


56695 


96976 




96676 


59485 


96860 


62 


9 


66788 


96971 


67688 


96670 


69456 


96854 


61 


10 


9.65781 


9.96966 


9.57660 


9.96666 


9.59484 


9.96849 


60 


11 


56798 


9G008 


67700 


96660 


59514 


96848 


49 


18 


6o8<!6 


96957 


67731 


96656 


50548 


96336 


48 


18 


66858 


96958 


67763 


96650 


69578 


96883 


47 


14 


66691 


96947 


57798 


96646 


69608 


96827 


46 


16 


55988 


96948 


57884 


96640 


69682 


96882 


45 


16 


65956 


96987 


67856 


96684 


69661 


96316 


44 


17 


55968 


96988 


67885 


96629 


69690 


96311 


48 


18 


56081 


96987 


67916 


96684 


69780 


96806 


48 


19 


66068 


96928 


67947 


96619 


60749 


96800 


41 


90 


9.56065 


9.96917 


9.67978 


9.96614 


9.69778 


9.96294 


40 


81 


66118 


96918 


68008 


96608 


69808 


96269 


89 


83 


66160 


9(3907 


58089 


96608 


50687 


96284 


38 


88 


66188 


96908 


68070 


96698 


59666 


96278 


37 


84 


56215 


96898 


68101 


96596 


59896 


96273 


86 


85 


66•^47 


96898 


58181 


96588 


50924 


96267 


86 


86 


66879 


96888 


68168 


96588 


69954 


96262 


84 


87 


56311 


96888 


58198 


96577 


69088 


96256 


83 


88 




96878 


68288 


96578 


60018 


96251 


83 


89 


56376 


96878 


58258 


96567 


60041 


96246 


81 


80 


9.60408 


9.96868 


9.58884 


9.96568 


9.60070 


9.06840 


80 


81 


56440 


96863 


58814 


96666 


60099 


06234 


29 


38 


66478 


96858 


58845 


96561 


60126 


96829 


88 


83 


66504 


96S53 


68376 


90546 


60157 


9e8;'8 


27 


84 


56686 


96848 


58406 


96541 


60186 


9«UI8 


S6 


85 


66566 


96848 


68486 


96585 


60815 


96818 


85 


86 


KitKOQ 


96838 


58467 


96530 


60844 


96207 


84 


87 


56631 


96838 


68497 


96525 


60278 


96801 


83 


88 


66668 


96828 


58527 


96580 


60808 


96196 


98 


89 


66695 


96828 


58557 


96514 


60831 


96190 


81 


40 


9.56727 


9.96818 


9.58588 


9.96509 


9.60359 


9.96186 


80 


41 


56759 


96818 


68618 


96504. 


ws^ 


96179 




43 


66790 


96808 


58648 


96496 


60417 


96174 




48 


56882 


96803 


68678 


96498 


60446 


96168 




44 


56854 


96708 


68709 


96486 


60474 


96188 




45 


56886 


96798 


68739 


96488 


60508 


96157 




46 


56917 


96788 


68769 


96477 


60638 


96151 




47 


56949 


96788 


58799 


96478 


60561 


96146 




48 


56980 


96778 


58889 


96467 


60589 


96140 




49 


67013 


96778 


68859 


96461 


60618 


96185 




60 


9.57044 


9.96767 


9.58889 


9.96456 


9.60646 


9.96189 




61 


57075 


96768 


58019 


96451 


60676 


96188 




68 


57107 


96757 


68949 


96445 


60704 


96118 




58 


67188 


96758 


58979 


96440 


60788 


96118 




54 


67169 


96747 


69009 


96435 


60761" 


96107 




66 


57201 


96748 


59OT9 


96489 


60r89 


96101 




66 


67-i82 


i»787 


59069 


96484 


60618 


96095 




57 


57264 


9G738 


59098 


96419 


60646 


96090 




58 


67295 


96727 


69188 


96418 


60875 


960H4 




50 


67826 


96728 


59156 


96406 


60908 


96079 




60 


67858 


96717 


59188 


96403 


60981 


96078 




i 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


» 




68* 




«?• 




W 



463 



TALBE n.— LOGARITHMIC SINES AND COSINES. 



/ 


24* 


26* 


«6» 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Coffine 




9.60981 


9.96078 


9.62695 


9.96788 


9.64184 


9.96866 


60 




60900 


96067 


62022 


95722 


64210 


95360 


69 




00988 


96068 


62649 


95716 


64286 


95854 


58 




01010 


96056 


02676 


95710 


64262 


95348 


67 




01045 


96060 


62708 


95704 


64288 


96841 


66 




OlOTB 


96045 


62730 


95696 


64813 


95886 


65 




01101 


96039 


62757 


05692 


64889 


95829 


54 




01129 


90034 


62784 


95686 


64866 


95828 


68 




01158 


96028 


62811 


95680 


64891 


95817 


52 




01180 


96022 


62888 


95674 


64417 


95810 


51 




9.01214 


9.90017 


9.62805 


9.96668 


9.64442 


9.95804 


60 




01212 


96011 


02892 


95663 


64468 


96286 


49 




01270 


90005 


62918 


95657 


64494 


95292 


48 




01:298 


9600O 


62915 


95661 


64510 


95286 


47 




01328 


96091 


02972 


95646 


64545 


95279 


46 




01354 


95988 


02990 


95689 


64571 


96278 


45 




01882 


95962 




95038 


64696 


95267 


44 




01411 


95977 


03052 


95627 


64622 


95261 


48 


18 


01488 


95971 


08079 


95621 


64647 


96254 


42 


19 


61460 


95905 


03100 


95615 


64678 


96248 


41 


SO 


9.01494 


9.96900 


9.63133 


9.95609 


9.64608 


9.96242 


40 


21 


01522 


95954 


08159 


95608 


64724 


95286 


89 


33 


01560 


95948 


03186 


95597 


64749 


95229 


88 


28 


01578 


96942 


68218 


95591 


64775 


95228 


87 


24 


01600 


959.S7 


63230 


95585 


64800 


95217 


86 


25 


01681 


95981 


03260 


93579 


64826 


95211 


85 


20 


01062 


95925 


68292 


95578 


64851 


96201 


34 


27 


61689 


95920 


63319 


95567 


64877 


95196 


88 


28 


61717 


95914 


68345 


95561 


64902 


96192 


82 


89 


61746 


96008 


68372 


95655 


64927 


95185 


31 


80 


9.61778 


9.95902 


9.68398 


9.95549 


9.64953 


9.95179 


80 


81 


61800 


95897 


63425 


95548 


64978 


96178 


29 


32 


618-J8 


95891 


63451 


95587 


65003 


96167 


28 


83 


61856 


968S5 


03478 


95531 


65029 


95160 


27 


34 


61883 


95879 


C3504 


95525 


65054 


95154 


26 


85 


61911 


95873 


03581 


96519 


65070 


95148 


25 


86 


61989 


95868 


08567 


95518 


65104 


95141 


24 


87 


61966 


95862 


0.3583 


95507 


65180 


95135 


28 


88 


61994 


95830 


08610 


95500 


65155 


95189 


28 


89 


62021 


95850 


63686 


95494 


65180 


95122 


21 


40 


9.63049 


9.96844 


9.68662 


9.95488 


9.65305 


9.95116 


SO 


41 


62076 


95839 


63689 


95482 


65280 


• 95110 


19 


42 


62104 


95838 


63715 


95476 


66255 


95108 


18 


48 


62131 


95827 


63741 


95170 


65281 


95097 


17 


41 


62159 


96821 


63767 


95464 


65806 


95000 


16 


45 


6S186 


95815 


63794 


95458 


65881 


95081 


15 


46 


62214 


95810 


63820 


95452 


65356 


95078 


14 


47 


62241 


95804 


63846 


95446 


65881 


95071 


18 


48 


62268 


95706 


63R72 


05440 


65406 


95065 


18 


49 


62296 


95793 


63S98 


95484 


65431 


95059 


11 


50 


9.62328 


9.95780 


9.03924 


9.95427 


9.65456 


9.95058 


10 


51 


62850 


95780 


68950 


95421 


65481 


95046 


9 


52 


62877 


95775 


63976 


95416 


65506 


05039 


8 


58 


62405 


96709 


04002 


95409 


ftWtt 


95033 


7 


54 


62432 


95757 


64028 


95408 


65566 


95027 


6 


55 


62459 


04054 


95897 


65580 


95020 


5 


60 


62486 


95751 


64060 


95391 


06605 


96014 


4 


67 


62513 


95745 


04100 


95384 


06680 


95007 


8 


68 


62541 


95739 


04182 


95378 


06056 


95001 


2 


59 


012568 


95738 


04158 


95872 


66G80 


94995 


1 


00 


62596 




04181 


95866 


65705 


94968 





r 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




66* 




«4* 




68* 



464 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



/ 


«7« 


28«» 


290 


i 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.66705 


9.94968 


9.6n61 


9.94598 


9.68557 


9.94188 


60 


1 


65789 


94988 


67186 


94687 


68580 


94176 


60 


i 


66764 


94975 


67208 


94580 


68608 


94168 


58 


8 


66779 


94969 


67888 


94573 


68625 


94161 


57 


4 


66804 


94968 


678S6 


M567 


68648 


94164 


66 


5 


65888 


94956 


67280 


94560 


68671 


94147 


66 


6 


65858 


94949 


67808 


94568 


68C94 


94140 


64 


7 


66878 


91948 


67887 


94546 


68716 


94133 


68 


8 


65008 


94936 


67850 


94540 


68739 


94186 


68 


9 


65987 


94980 


67874 


94588 


68768 


94119 


51 


10, 


9.66068 


9.94988 


9.67398 


9.94526 


9.68784 


9.94118 


60 


11 


66976 


94917 


67421 


94519 


68807 


94106 


40 


12 


66001 


94911 


67445 


94513 


68829 


94098 


48 


;8 


66085 


94904 


67468 


U4506 


68852 


iM090 


47 


14 


660S0 


94898 


67492 


94499 


66875 


94063 


46 


16 


66075 


94891 


67516 


94493 


68897 


94076 


45 


16 


66099 


94885 


67539 


94485 


68920 


94069 


44 


17 


66184 


94878 


67562 


94479 


68948 


94068 


48 


18 


66148 


94871 


67586 


94472 


68965 


94055 


48 


19 


66173 


04865 


67609 


94465 


68967 


94046 


41 


80 


9.66197 


9.94868 


9.67688 


9.94458 


9.69010 


9.94041 


40 


21 


66881 


94858 


67656 


94451 




94034 


89 


82 


66846 


94845 


67680 


94445 


600r>5 


94027 


38 


88 


66270 


94889 


67708 


94438 


69077 


94080 


87 


84 


66895 


94882 


67726 


94431 


69100 


94018 


86 


85 


66819 


94826 


67750 


91484 


69128 


94006 


85 


86 


66348 


94819 


67778 


94417 


69144 


98996 


84 


87 


66868 


94818 


67796 


94410 


60167 


93991 


83 


88 


66898 


94806 


67KJ0 


94404 


69189 


93984 


82 


89 


66416 


94799 


67848 


94397 


69818 


98977 


81 


80 


9.66441 


9.94798 


9.67866 


9.94890 


9.69284 


9.98970 


80 


81 


66405 


94786 


67890 


94883 


69866 


98968 


89 


88 


66489 


94780 


67918 


94376 


60879 


98956 


26 


83 


66518 


94778 


67986 


94369 


69801 


98948 


27 


84 


66587 


94767 


67959 


94868 


69823 


93941 


86 


85 


6G568 


94760 


67982 


94355 


69346 


93984 


85 


86 


66586 


94768 


68006 


94349 


69368 


03987 


84 


87 


66610 


94747 


68029 


94343 


69390 


98980 


88 


88 


66684 


94740 


68052 


94385 


69418 


98918 


88 


89 


66668 


94734 


68075 


94828 


69434 


93906 


81 


40 


9.66688 


9.94787 


9.68096 


9.94821 


9.69456 


9.93898 


80 


41 


66706 


• 94720 


68121 


94314 


69479 


98891 


19 


48 


66781 


94714 


68144 


94307 


60ri01 


93884 


18 


43 


66765 


94707 


68167 


94300 


69628 


98876 


17 


44 


66T79 


94700 


68190 


94293 


69546 


<93869 


16 


45 


66803 


94694 


68218 


94286 


69567 


9S862 


15 


46 


66887 


94687 


68237 


94279 


60589 


93856 


14 


47 


66851 


94680 


68260 


94278 


69611 


93847 


18 


48 


66875 


94674 


68283 


94266 


69638 


98840 


18 


49 


66899 


94667 


68805 


94269 


69055 


98838 


11 


60 


9.66988 


9.94660 


9.68828 


9.94258 


9.69677 


9.93886 


10 


61 


66946 


94654 


68S51 


94245 


, 69699 


93819 


9 


63 


66970 


94647 


68874 


94238 


69781 


98811 


6 


63 


66994 


94640 


68397 


94231 


69748 


93804 


7 


54 


67018 


94684 


66420 


94284 


69766 


98797 


6 


55 


67048 


94627 


68448 


94217 


69787 


93789 


6 


66 


67066 


94620 


68466 


94210 


69809 


93788 


4 


67 


67090 


94614 


68489 


94208 


69881 


98776 


8 


68 


67113 


94607 


68518 


94196 


69858 


98768 


8 


69 


67187 


94600 


68534 


94189 


69876 


93760 


1 


60 


67161 


94598 


68567 


94188 


69897 


98758 





# 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


«ao 




6P 




60» 



465 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



f 


$0<» 


81» 


82«» 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.69697 


9.98758 


9.71184 


9.98807 


9.79421 


9.92842 


60 


1 


69919 


9S746 


71205 


86299 


72441 


92884 


69 


8 


69941 


98788 


71226 


98991 


72481 


92826 


68 


8 


69968 


98731 


71247 


93284 


72482 


92818 


67 


4 


69964 


03794 


71268 


98276 


72502 


92810 


56 


5 


70006 


93717 


71289 


98269 


72529 


92803 


65 


6 




98700 


71810 


98261 


72542 


92795 


54 


7 


70090 


93709 


71881 


98258 


72fi62 


9-2787 


68 


8 


70072 


98696 


71852 


98946 


72588 


92779 


69 


9 


70008 


93687 


71878 


93238 


72609 


92771 


61 


10 


9.70115 


9.93680 


9.71398 


9.98280 


9.72622 


9.92768 


60 


11 


70187 


93673 


TI414 


93228 


72648 


92765 


49 


18 


70159 


98665 


71486 


98216 


72668 


92747 


48 


18 


70180 


93658 


71456 


93207 


72883 


92739 


47 


14 


70902 


98650 


71477 


98-200 


T/m 


92781 


46 


16 


70224 


93648 


71498 


98192 


72798 


92728 


46 


18 


70245 


96686 


71619 


98184 


72748 


92715 


44 


17 


70267 


93698 


71589 


98177 


72768 


92707 


48 


18 


70288 


93621 


71660 


98169 


72788 


92G99 


49 


19 


70810 


93614 


71581 


98161 


72808 


92691 


41 


90 


9.70382 


9.98606 


9.71608 


9.98154 


9.72828 


9.92683 


40 


21 


70858 


08599 


71629 


98146 


72848 


92676 


39 


82 


70875 


98591 


71648 


93138 


72868 


92667 


88 


28 


70896 


98584 


71664 


93181 


72888 


92659 


87 


24 


70418 


98577 


71686 


98128 


79902 


9-2651 


36 


25 


70489 


98569 


71706 


93U5 


72922 


99648 


.35 


26 


70461 


93.569 


71726 


98108 


72942 


92685 


84 


97 


70489 


98554 


71747 


96100 


72962 


92697 


88 


28 


70604 


98547 


71767 


98099 


72988 


92619 


82 


29 


70525 


96589 


71788 


98084 


78002 


99611 


81 


80 


9.70547 


9.93588 


9.71809 


9.98077 


9.78088 


9.92608 


80 


81 


70508 


98595 


71899 


98069 


78041 


92595 


99 


82 


70690 


98517 


71850 


98061 


78061 


92587 


26 


88 


70611 


93510 


71870 


93058 


78081 


92579 


97 


84 


70638 


03509 


71891 


98046 


78101 


92571 


26 


85 


70654 


98496 


71911 


1»8088 


78121 


92568 


85 


86 


70675 


98487 


71939 


98080 


78140 


92555 


94 


87 


70697 


98480 


71952 


93029 


78160 


92646 


98 


88 


70718 


98479 


71978 


93014 


73180 




22 


89 


70739 


93465 


71994 


98007 


7820O 


99580 


91 


40 


9.70761 


9.93457 


9.72014 


9.92999 


9.7S819 


9.99529 


80 


41 


70782 


93450 


72034 


92991 


78289 


92514 


19 


42 


70808 


98442 


72055 


92088 


78269 


92506 


18 


48 


70624 


98436 


78076 


92976 


78278 


92496 


17 


44 


70846 


98427 


72096 


92968 


78298 


92490 


16 


45 


70867 


93420 


72116 


92960 


78318 


09488 


16 


46 


70888 


98412 


721.37 


92959 


78337 


99478 


14 


47 


70909 


93405 


72157 


92944 


78a'i7 


92465 


18 


48 


70931 


93397 


72m 


92986 


78877 


99457 


19 


49 


70939 


96390 


72198 


92929 


73896 


99449 


11 


60 


9.70978 


9.93882 


9.72918 


9.92921 


9.78416 


9.99441 


10 


61 


70994 


93375 


72288 


92918 


78485 


92438 


« 


52 


71015 


98367 


72259 


92906 


78455 


92425 


8 


58 


71036 


93360 


72279 


92897 


78474 


. 92416 


7 


64 


71068 


93352 


72299 


92889 


78494 


92408 


6 


56 


71079 


0^344 


72.320 


92881 


78518 


92400 


6 


56 


71100 


M387 


72340 


92874 


73588 


92399 


4 


67 


71121 


98329 


72860 


92866 


78562 


92884 


8 


66 


71149 


98392 


72381 


9-2858 


78579 


92876 


9 


69 


71168 


93314 


72401 


92850 


78691 


92367 


1 


60 


71184 


93807 


72421 


92842 


78611 


9-2869 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




69« 




68« 




67» 



466 



TABLE II.— LOGARl'l'HMIC SINES AND COSINES. 


/ 


88» 


S4* 


S6- 1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.78611 


9.98869 


9.74766 


9.91857 


9.76859 


9.91886 


60 


1 


78680 


98851 


74776 


91849 


75877 


91828 


69 


3 


78650 


93848 


74794 


91840 


76806 


91319 


68 


8 


78669 


98885 


74813 


91882 


75918 


91810 


67 


4 


78689 


98888 


74881 


91828 


75931 


91801 


56 


6 




92818 


74860 


91815 


75949 


91898 


66 


6 


78727 


92810 


74868 


91806 


75967 


91288 


54 


7 


78747 


98808 


74887 


91798 


75985 


91874 


58 


8 


78766 


92298 


74906 


91789 


76008 


91266 


62 


9 


78T85 


92285 


74924 


91781 


76021 


91867 


61 


10 


9.78805 


9.92277 


9.74948 


9.91772 


9.76039 


9.91248 


60 


11 


78824 


92269 


74961 


91763 


76057 


91289 


49 


18 


73848 


92260 


74960 


91756 


76076 


91280 


48 


18 


78868 


92268 


74999 


91746 


76098 


91881 


47 


14 


^T«88 


98844 


76017 


91788 


76111 


91818 


46 


16 


78901 


92286 


7S086 


91729 


76129 


91208 


45 


16 


78921 


92227 


75054 


91780 


76146 


91194 


44 


17 


78940 


92219 


75078 


91712 


76164 


91186 


48 


18 


78959 


92811 


75091 


91703 


76188 


91176 


48 


19 


78978 


98802 


75110 


91695 


76200 


91167 


41 


20 


9.78997 


9.92194 


9.75188 


9.9ir86 


9.76218 


9.91158 


40 


81 


74017 


92186 


75147 


91677 


76286 


91149 


89 


28 


74086 


92177 


75166 


91669 


76268 


91141 


88 


8S 


74065 


98169 


75184 


91660 


76271 


91188 


87 


24 


74074 


92161 


76208 


91651 


76289 


91128 


86 


25 


74098 


92162 


75221 


91648 


76307 


91114 


35 


80 


74118 


92144 


76289 


91684 


76324 


91105 


84 


27 


74182 


92186 


76258 


91625 


76342 


91096 


38 


28 


74151 


92127 


75276 


91617 


76860 


91087 


82 


29 


74170 


92119 


76294 


91608 


76878 


91078 


31 


90 


9.74189 


9.92111 


9.76818 


9.91509 


9.76895 


9.91069 


80 


81 


74208 


92102 


76381 


91591 


76418 


91060 


89 


88 


74227 


92094 


76860 


91582 


76481 


91061 


88 


88 


74246 


92086 


75368 


91578 


76448 


91048 


27 


84 


'i4885 


92077 


75886 


91566 


76466 


91088 


86 


85 


74284 


92069 


75405 


91556 


76484 


91028 


85 


86 


74808 


92060 


75423 


91547 


78501 


91014 


84 


8/ 


74828 


98062 


75441 


915S8 


76619 


91006 


28 


88 


74841 


92044 


75459 


91680 


76537 


90996 


28 


89 


.74360 


98086 


76478 


91521 


76554 


90987 


21 


40 


9.74879 


9.98087 


9.75496 


9.91512 


9.76572 


9.90978 


90 


41 


74898 


92018 


75514 


91504 


76590 


00969 


19 


42 


74417 


92010 


76583 


91495 


76607 


90960 




48 


74486 


92002 


75561 


91486 


76625 


90961 




44 


74455 


91998 


75569 


91477 


76642 


90949 




45 


74474 


91985 


75587 


91469 


76660 


90988 




46 


74496 


91976 


76605 


91460 


76677 


90924 




47 


74518 


91968 


76624 


91451 


76696 


90916 




48 


74581 


91959 


76642 


91442 


76718 


90906 




49 


74549 


91961 


76660 


91488 


V6780 


90696 




60 


9.74!M» 


9.91943 


9.75678 


9.91425 


9.76747 


9.90887 




61 


74587 


91964 


75696 


91416 


76766 


90878 




68 


74606 


91926 


75714 


91407 


767B8 


90809 




58 


74625 


91917 




91898 


76800 


90860 




64 


74644 


91908 


75751 


91389 


76817 


00861 




66 


74668 


91900 


75769 


91881 


76836 


90849 




66 


74681 


91891 


75787 


91872 


768.'$8 


90888 




67 


74700 


91888 


75805 


91368 


76870 


90828 




68 


74719 


91874 


75828 


91354 


76887 


90814 




69 


74787 


91866 


75841 


91346 


76904 


00806 




60 


74756 


91867 


75860 


91836 


76028 


90796 




9 


Ck)8iiie 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 






66* 




fiS' 




ft4* 



467 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



# 


se* ' 




S?' 


88« 


/ 


Gttne 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.76099 


9.90796 


9.77946 


9.90985 


9.780S4 


9.69658 


60 


1 


76989 


90787 


77968 


90996 


78950 


89648 


59 


» 


76957 


90777 


7;t«o 


90916 


78967 


89638 


58 


8 


76974 


90768 


77997 


90206 


78983 


89624 


57 


4 


76091 


90758 


78018 


90197 


78999 


89614 


56 


5 


77009 


90750 


78000 


90187 


79015 


89604 


55 


8 


770« 


90741 


78047 


90178 


7O081 


80594 


54 


7 


77048 


00731 


78068 


90168 


79047 


89584 


58 


8 


77061 


90799 


78080 


• 90159 


79068 


89574 


59 


9 


77078 


90718 


78097 


90149 


79079 


80564 


61 


10 


9.77096 


9.90704 


9.78118 


9.90130 


9.79095 


9.89564 


50 


11 


77119 


90694 


78180 


90180 


79111 


89544 


49 


18 


77130 


90685 


78147 


90190 


79198 


89684 


48 


18 


77147 


90676 


78183 


90111 


79144 


89524 


47 


14 


77164 


90667 


78180 


90101 


79160 


89514 


46 


15 


77181 


90657 


78197 


90091 


79176 


89604 


46 


16 


77199 


90648 


78218 


90089 


79199 


89495 


44 


17 


77216 


90689 


78230 


90079 


79906 


89486 


48 


18 


77988 


90630 


78246 


90068 


79224 


89475 


49 


19 


77250 


90620 


7B263 


90058 


79240 


89465 


41 


90 


9.77968 


9.90611 


9.7S280 


9.90048 
90084 


9.79956 


9.89456 


40 


91 


77285 


90609 


78296 


79279 


89445 


80 


99 


77302 


90592 


78813 


90024 


79988 


89485 


88 


28 


77319 


90583 


78829 


90014 


79804 


89425 


87 


94 


77386 


90574 


78340 


90005 


70819 


89415 


86 


95 


77858 


90565 


78869 


89995 


79385 


89405 


85 


98 


77870 


90555 


78379 


89965 


79851 


80896 


84 


97 


77387 


90546 


78396 


89976 


79867 


89386 


88 


98 


77406 


90587 


78419 


89966 


79388 


89375 


89 


99 


77429 


90627 


78428 


80950 


79899 


89364 


81 


80 


9.77489 


9.90618 


9.78445 


9.89947 


9.79415 


9.89854 


80 


81 


77456 


90509 


78461 


89937 


79481 


89844 


99 


89 


77478 


90499 


78478 


89927 


79447 


89334 


28 


88 


77490 


90490 


78494 


89918 


79468 


89824 


97 


81 


77507 


90480 


78510 


89906 


79478 


89314 


96 


85 


77524 


90471 


78527 


89808 


79494 


89304 


95 


86 


77541 


90469 


78548 


ovooo 


79510 


89294 


94 


87 


77558 


90152 


78560 


80879 


79596 


89284 


93 


88 


77575 


90448 


78576 


80869 


79549 


89274 


29 


89 


77598 


90434 


78599 


89859 


79558 


89964 


91 


40 


9.77609 


9.90424 


9.78609 


9.80849 


9.79678 


9.89254 


90 


41 


77696 


90415 


78626 


89840 


79580 


89944 


19 


49 


77648 


90406 


78649 


80830 




89288 


18 


48 


T?«60 


90896 


78658 


898,»0 


78621 


89228 


17 


41 


77677 


90386 


78674 


89810 


79686 


89918 


16 


45 


77694 


90377 


78691 


89801 


79659 


80908 


16 


46 


77711 


9030S 


78707 


89791 


79668 


89198 


14 


47 


77728 


90358 


78728 


89781 


79684 


89188 


18 


48 


77744 


90349 


78789 


89771 


79699 


89178 


19 


49 


77761 


90389 


78756 


89761 


79715 


89169 


11 


SO 


9.77778 


9.90830 


9.78772 


9.89759 


9.79731 


9.80159 


10 


51 


77795 


90820 


78788 


89749 


79746 


89149 


9 


59 


77819 


90311 


78805 


89739 


79769 


89139 


8 


58 


7?»J9 


90301 


78821 


80729 


79778 


89129 


7 


54 


77846 


90-.^2 


78837 


89719 


79798 


89119 


6 


65 


77889 


90282 


78853 


89709 


79809 


89101 


6 


56 


77879 


90278 


7S8C9 


89693 


79825 


89091 


4 


57 


77896 


90963 


78886 


89688 


79640 


80061 


8 


58 


77918 


90254 


78002 


89078 


79856 


89071 


9 


50 


r79:» 


90244 


78918 


89663 


79872 


89060 


1 


(SO 


T7946 


90235 


78984 


89653 


79887 


89050 





/ 


Ckwlne 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


680 




62« 


51- 1 



468 



TABLE 11.— LOGARITHMIC SINES AND COSINES. 



/ 


89* 


40« 


41» 


/ 


Sine 


Cosine 


Sine 


Oodne 


Sine 


Cosine 





9.79887 


9.89060 


9.80007 


9.88426 


9.81694 


••2X32 

87767 


60 


1 


70908 


89040 


80882 


88415 


81709 


69 


2 


79918 


89060 


80887 


88404 


81728 


87756 


68 


8 


79934 


89020 


80862 


86894 


81788 


87745 


67 


4 


79950 


89009 


80667 


88888 


81782 


87784 


66 


5 


79966 


88999 


80882 


88872 


81767 


87788 


66 


6 


79981 


88989 


80897 


88808 


81781 


87718 


54 


7 


79996 


88078 


80912 


88851 


81796 


erroi 


68 


8 


80012 


88968 


80927 


88840 


81810 


87090 


68 


9 


800(27 


QQQeO 
OOWUO 


80942 


86880 


81826 


87679 


51 


10 


9.80048 


9.88948 


9.80967 


9.88819 


9.81889 


9.87668 


60 


11 


80068 


88067 


80972 


88808 


81664 


87667 


49 


12 


80074 


88927 


80987 


88208 


81868 


87646 


48 


18 


80089 


88917 


81002 


8B287 


81882 


87686 


47 


14 


80106 


88906 


81017 


88276 


81897 


87624 


46 


15 


80120 


88806 


81082 


88266 


81911 


87818 


45 


16 


80186 


88886 


81047 


88266 


81926 


87601 


44 


17 


80161 


88875 


81061 


88244 


81940 


87690 


48 


18 


80166 


88865 


81076 


88284 


81956 


87879 


42 


19 


80182 


88855 


81091 


88S28 


61909 


87688 


41 


80 


9.80197 


9.88844 


9.81106 


9.88212 


9.81968 


9.8;i»7 


40 


21 


80218 


88884 


81121 


88201 


81998 


87646 


80 


22 


80228 


88824 


81186 


88191 


88019 


87586 


88 


28 


80244 


88818 


81161 


88180 


8S026 


87584 


87 


24 


80250 


8S808 


81166 


88169 


82041 


87518 


86 


25 


80274 


88798 


81180 


88168 


82066 


87501 


86 


26 


80290 


88782 


81195 


88148 


82069 


87490 


84 


27 


80805 




81210 


88187 


880S4 


87479 


88 


28 


80820 


88761 


81225 


88126 


88006 


87468 


88 


29 


80386 


88751 


81240 


88115 


82112 


87457 


81 


80 


9.80851 


9.88741 


9.81254 


9.88105 


9.82186 


9.87446 


80 


81 


80866 


88780 


81269 


88094 


82141 


87484 


89 


82 


80882 


88720 


81284 


88088 


82166 


87488 


88 


88 


80897 


88709 


81299 


88072 


82169 


87412 


27 


84 


80412 


88699 


81314 


88061 


82184 


87401 


86 


85 


80428 


88688 


81888 


88051 


82198 


87880 


86 


86 


80448 


88678 


81848 


88040 


82212 




24 


87 


80458 


88668 


81858 


68089 


82226 


87867 


28 


88 


80478 


88657 


81872 


88018 


82240 


87866 


22 


89 


80489 


88647 


81887 


88007 


82256 


87846 


21 


40 


9.80604 


9.88686 


9.81402 


9.87996 


9.88269 


9.87884 


90 


41 


80619 


88626 


81417 


87986 


82288 


8788S 


19 


42 


80534 


88616 


81481 


87976 


88297 


87811 


18 


48 


80660 


88605 


81446 


87964 


88811 


87800 


17 


44 


80565 


88594 


81461 


87958 


88820 


87888 


16 


45 


80580 


88584 


81475 


87942 


82840 


87877 


15 


46 


80595 


8S578 


81490 


87931 


82854 


87866 


14 


47 


80610 


88663 


81606 


87920 


82868 


87855 


18 


48 


60625 


88562 


81519 




»2882 


67248 


12 


49 


80641 


88542 


81584 


87898 


88896 


87282 


11 


60 


9.80656 


9.88581 


9.81549 


9.87887 


9.82410 


0.87S21 


10 


61 


80671 


88521 


81663 


87877 


82424 




9 


52 


80686 


88510 


81578 


87866 


82489 


87196 


8 


58 


80701 


88499 


81592 


87866 


82468 


87187. 


7 


64 


80716 


88489 


81607 


87844 


82467 


87176' 


6 


55 


80781 


88478 


81632 


87838 


88481 


87164 


6 


66 


80746 


88468 


81686 


87882 


82496 


67168 


4 


57 


80762 


88457 


81651 


87811 


88609 


67141 


8 


58 


80777 


88447 


81666 


67800 


68523 


87180 


2 


50 


80792 


88486 


81680 


87789 


88587 


87119 


1 


60 


80807 


88425 


81694 




82551 


87107 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


i 


W 




490 




48* 



469 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



/ 


4t* 


48- 


44«» 


/ 


Sine 


Coeiue 


Sine 


Coaine 


Sine 


Cowne 




9.82561 


9.87107 


9.88878 


9.86418 


9.84177 


9.85693 


60 




82566 


87096 


83392 


86401 


84190 


85681 


59 




62679 


870S5 


88405 


8G880 


64208 


65669 


58 




8^596 


8T073 


88419 


86877 


84216 


a5657 


67 




82607 


87062 


83482 


86866 


84229 


86645 


66 




82621 


87050 


83446 


86354 


64242 


66682 


55 




82685 


87039 


83469 


86842 


64255 


85020 


54 




82649 


87028 


83478 


8G330 


84260 


65606 


58 




82668 


87016 


83486 


80318 


84282 


65506 


52 




82677 


87005 


88500 


86806 


84206 


65588 


51 




9.82691 


9.86993 


9.88518 


9.86295 


9.84306 


9.65571 


60 




82T05 


86982 


88527 


86283 


84381 


85569 


49 




82719 


86970 


88540 


86271 


84384 


85647 


46 




82788 


86059 


83554 


86250 


84347 


65584 


47 




82747 


86947 


88567 


86247 


64860 


65622 


46 




82r61 


86936 


88381 


86235 


64878 


85610 


46 




82775 


86924 


8S594 


86228 


84885 


85497 


44 




82788 


86918 


83608 


86211 


84396 


65485 


48 


18 


82802 


86902 


83621 


86200 


64411 


86478 


48 


19 


82816 


86890 


83684 


86186 


64424 


85460 


41 


90 


9.82880 


9.86879 


9.88648 


9.86176 


9.84487 


0.86448 


40 


21 


8-2814 


86867 


88661 


86164 


84450 


85436 


89 


S2 


828S6 


86855 


83674 


86152 


84468 


65428 


86 


fa 


82872 


86844 


83688 


86140 


84476 


85411 


87 


S4 


82885 


86832 


88701 


86128 


64489 


65899 


86 


25 


82899 


86821 


83715 


86116 


84508 


65886 


.35 


28 


8;>918 


86809 


83728 


86104 


64515 


85874 


84 


27 


82927 


86798 


88741 




84526 


85361 


83 


28 


82941 


86786 


83755 


86080 


84540 


65349 


82 


29 


82956 


86775 


88768 


86008 


64553 


86837 


31 


80 


9.»^y68 


9.86768 


9.88781 


9.86056 


9.84566 


9.86384 


30 


81 


8J982 


867S2 


83795 


86044 


64579 


86312 


29 


88 


82996 


86740 


83806 


860ffi 


64592 


65299 


88 


88 


sw\o 


86728 


88821 


86020 


64605 


65287 


27 


84 


83028 


86717 


83884 


86008 


84618 


65274 


26 


«i 


88037 


86705 


83848 


85996 


84680 


65262 


85 


86 


83051 


86694 


88861 


86964 


64643 


65250 


24 


87 


8:^063 


86682 


88874 


85972 


64656 


85237 


28 


88 


R3078 


8C670 


83887 


86060 


64660 


85225 


22 


89 


88092 


86659 


83901 


85948 


84682 


85212 


81 


40 


9.83106 


9.86647 


9.83914 


9.85966 


9.84694 


9.a5200 


80 


41 


88120 


86635 


88927 


85924 


64707 


65187 


19 


42 


83133 


86624 


88940 


85912 


64?20 


85175 


16 


48 


83147 


86612 


83954 


85900 


64788 


86162 


17 


44 


83161 


86600 


83967 


a5888 


84745 


85150 


16 


45 


83174 


86589 


83960 


85876 


84758 


85187 


16 


46 


83i88 


80577 


a3TO3 


85881 


84771 


85125 


14 


47 


88202 


86566 


84006 


85861 


84784 


65112 


18 


48 


88il5 


86564 


g405JO 


85830 


64796 


65100 


12 


49 


832;!9 


86542 


84a» 


86827 


64809 


85087 


11 


80 


9.83242 


9.80530 


9.84046 


9.85815 


9.R4R22 


9.86074 


10 


61 


83256 


86518 


84059 


85H03 


84835 


85002 


9 


62 


83v>70 


86507 


84072 


85791 


84847 


A5(»49 


8 


58 


83-283 


86495 


840«S 


85779 


84860 


86037 


7 


54 


83297 


86488 


84098 


86766 


64873 


86024 


6 


55 


83:^10 


86472 


84112 


857.54 


84885 


85012 


5 


56 


83324 


86460 


84125 


86742 


64S98 


64999 


4 


57 


88338 


8644S 


84188 


85730 


84011 


64986 


8 


58 


88ail 


86436 


84151 


85718 


84928 


64974 


8 


59 


83865 


86425 


84164 


85706 


81936 


84961 


1 


60 


83378 


86413 


84177 


85698 


64949 


64949 





/ 


Ooeine 


Sine 


C!o8iiie 


Slne_ 


Cosine 


Sine 


/ 




470 




46« 




46* 



470 



TABLE m 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 




©• 




!• 


9 


• 


/ 


Tta 


CJotan 


Tan 


Ootan 


Tan 


Cotan 




— 00 


00 


8.94199 


11.75808 


8.54808 


11.45699 


60 




6.46878 


18.58637 


94910 


76090 


54669 


45881 


59 




76476 


28584 


95616 


74884 


66097 


44973 


68 




94066 


05916 


96819 


73668 


66882 


44618 


57 




7.06579 


12.93421 


96996 


78004 


66784 


44966 


66 




16270 


88780 


97669 


72331 


66063 


48917 


65 




84188 


75819 


98839 


71668 


66429 


48571 


54 




80688 


69118 


28986 


71014 


66778 


48927 


53 




86688 


68818 


99629 


70871 


6ni4 


49886 


68 




41T97 


58208 


80263 


69737 


67469 


49548 


61 




7.46878 


19.58687 


8.80688 


11.69119 


8.67788 


11.49919 


60 




60618 


49488 


81505 


68496 


68121 


418W 


49 




64S91 


45709 


S2119 


67888 


66461 


41649 


48 




67767 


42283 


82711 


67289 


58779 


41991 


47 




60986 


89014 


88302 


6669R 


59106 


40696 


46 




68988 


86018 


83886 


66114 


60428 


40679 


45 




66786 


88215 


84461 


65639 


60749 


40251 


44 




09418 


80582 


85029 


64971 


60068 


89989 


48 




71900 


28100 


86590 


64410 


60884 


89616 


49 




74248 


25758 


86148 


68857 


60696 


89809 


41 


90 


7.76476 


18.23584 


8.86689 


11.68311 


8.61000 


11.88991 


40 


81 


78595 


21405 


87220 


62771 


61819 


88681 


89 


8S 


60616 


19885 


87762 


estm 


61696 


88874 


88 


88 


88546 


17454 


88288 


61711 


61931 


88069 


87 


84 


84394 


15606 


3b809 


61191 


62884 


87766 


86 


85 


86167 


13838 


89328 


60677 


62535 


87466 


85 


86 


»7871 


12129 


89832 


60168 


62834 


87166 


84 


87 


80510 


10490 


40884 


69666 


63131 


86809 


88 


88 


01089 


08911 


40880 


69170 


63426 


86674 


88 


89 


98618 


07887 


41321 


58679 


68718 


86889 


81 


80 


7.94066 


19.05914 


8.41807 


11.68198 


8.64009 


11.86991 


80 


81 


95510 


04490 


42287 


67718 


64998 


85708 


99 


38 


96889 


08111 


42768 


57288 


64586 


85415 


98 


83 


08225 


01775 


48239 


66768 


64870 


85180 


97 


84 


99588 


00478 


48696 


66304 


65154 


84846 


96 


85 


8.00781 


11.99219 


44166 


66844 


65485 


84665 


95 


88 


0^004 


97996 


44611 


65389 


66715 


84285 


94 


87 


08194 


96S06 


45061 


54939 


65098 


84007 


28 


88 


04353 


96647 


46607 


54493 


66869 


83781 


89 


89 


05481 


94510 


46948 


54053 


66548 


88457 


91 


40 


8.06581 


11.93419 


8.46885 


11.63615 


8.66816 


11.88184 


90 


41 


07658 


92847 


46817 


53188 


67087 


88918 


19 


4S 


08700 


91300 


47246 


52756 


67856 


82644 


18 


48 


09728 


90278 


47669 


52331 


67624 


82376 


17 


44 


10720 


80280 


48089 


51911 


67890 


82110 


16 


45 


11696 


88304 


48505 


51495 


68154 


81846 


15 


46 


18651 


87340 


48917 


51C83 


68417 


81583 


14 


47 


13586 


86415 


49335 


60675 


68678 


81322 


18 


48 


14500 


85500 


49729 


60271 


68038 


81062 


19 


49 


16896 


84605 


60180 


49870 


69196 


80804 


11 


50 


e.ioifl^ 


11.88787 


8.80627 


11.49473 


8.69468 


11.30547 


10 


61 


17188 


82867 


50920 


49080 


69708 


80299 


9 


58 


17976 


82024 


61310 


48690 


69969 


80088 


8 


68 


18804 


81196 


61696 


48304 


70214 


99766 


7 


54 


19616 


80384 


62079 


47921 


70465 


29685 


6 


66 


80418 


7».')87 


62459 


47541 


70714 


99286 


5 


66 


21195 


78805 


628:» 


47165 


70968 


99088 


4 


57 


21964 


78036 


58208 


46792 


71208 


98799 


8 


68 


22720 


77280 


53578 


46422 


71458 


2f547 


9 


60 


28462 


76538 


63945 


46055 


71697 


28808 


1 


60 


24192 


76808 


64308 


45692 


71940 


28060 





9 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




89« 




88» 




S7- 



471 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 




8- 




!• 




6» 


/ 


Tan 


Ootan 


Tan 


Cotan 


Tan 


Cotan 





8.71940 


11.88060 


8.84464 


11.15586 


8.94195 


11.06805 


60 


1 


78181 


87819 


84646 


15854 


94840 


05660 


59 


a 


78480 


87580 


84826 


15174 


94485 


06515 


58 


8 


78659 


87841 


86006 


14994 


94680 


05870 


57 


4 


78896 


87104 


85185 


14815 


94778 


05327 


66 


5 


73133 


86668 


85868 


14637 


94917 


05063 


55 


8 


78866 


86634 


85540 


14460 


95060 


04940 


54 


7 


73600 


26400 


85717 


14283 


95808 


04798 


68 


8 


78832 


26168 


85898 


14107 


96844 


04656 


63 


9 


74068 


85987 


86060 


139S1 


95486 


04514 


51 


10 


8.74898 


11.85708 


8.86843 


11.18757 


8.95697 


11.04378 


60 


11 


74521 


85479 


86417 


13588 


95767 


04838 


49 


12 


74748 


85853 


86591 


18409 


95906 


04098 


48 


18 


74974 


26026 


86763 


18237 


96047 


08958 


47 


14 


75199 


84801 


86935 


13066 


96187 


08818 


46 


16 


76428 


24577 


87106 


12894 


96385 


03075 


45 


16 


75645 


84365 


87877 


12728 


96464 


03536 


44 


17 


76867 


84183 


87447 


12568 


96608 


03896 


48 


18 


76067 


83913 


87616 


12384 


96739 


08261 


48 


19 


76806 


28694 


87785 


12215 


96877 


08128 


41 


20 


8.76525 


11.83475 


8.87968 


11.12047 


6.97018 


11.08967 


40 


31 


76748 


83258 


88120 


11880 


97160 


02860 


88 


88 


76968 


83043 


88287 


11718 


97285 


02715 


88 


88 


77178 


82827 


88458 


11547 


97421 


02579 


87 


84 


77887 


22613 


88618 


11383 


97556 


02444 


86 


85 


77600 


38100 


88783 


11217 


97691 


08309 


85 


80 


77811 


88189 


88948 


11053 


97825 


08175 


84 


87 


78088 


21978 


89111 


10689 


97969 


02041 


88 


88 


78238 


81768 


89274 


107^6 


96092 


01906 


82 


29 


78441 


81559 


89437 


10668 


98225 


01775 


81 


SO 


8.78649 


11.81351 


8.89698 


11.10403 


8.96358 


11.01648 


80 


81 


78866 


21145 


89700 


10240 


96490 


01610 


89 


88 


79061 


20939 


89980 


10060 


96628 


01878 


88 


83 


79866 


80784 


90080 


09920 


98753 


01847 


37 


84 


79470 


80530 


90240 


09760 


98884 


01116 


86 


85 


79678 


80327 


90399 


09601 


99015 


00965 


85 


86 


79875 


80125 


90657 


09443 


99145 


00855 


84 


87 


80076 


19924 


90716 


09^85 


99".'75 


00725 


88 


88 


808n 


19728 


90872 


09128 


99405 


00595 


88 


89 


80476 


19584 


910-29 


08971 


99634 


00466 


81 


40 


8.80674 


11.19386 


8.91185 


11.08815 


6.99663 


11.00888 


90 


41 


80878 


19128 


91340 


08660 


99791 


00209 




49 


81068 


18938 


91495 


06505 


99919 


00081 




48 


81864 


18736 


91650 


06350 


9.00040 


10.99954 




44 


81459 


18641 


91803 


08197 


001T4 


99826 




45 


81658 


18347 


91957 


08043 


00301 


99099 




46 


81846 


18154 


92110 


07890 


00427 


99573 




47 


88088 


17962 


92262 


07788 


00553 


99447 




48 


82830 


17770 


92414 


07686 


00679 


99821 




49 


88490 


17580 


92565 


07485 


00805 


99195 




60 


8.88610 


11.17890 


8.92716 


11.07884 


9.00980 


10.90070 




51 


82799 


1T201 


92866 


07134 


010)5 


96945 




58 


82987 


17018 


93016 


06984 


01179 


96881 




58 


83175 


16825 


93165 


06835 


01808 


98697 




54 


83361 


16638 


98313 


06o87 


01427 


98578 




55 


83547 


16453 


93468 


06538 


01660 


98450 




56 


88738 


16268 


93609 


06391 


01678 


98327 




67 


88916 


16084 


93756 


06244 


01796 


96204 




58 


84100 


15900 


93903 


06097 


01918 


98082 




59 


84283 


15718 


94049 


05951 


02040 


07960 




60 


84464 


15536 


94195 


05805 


02162 


97888 




» 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




se* 




85- 




84* 



472 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


« 


e 


7* 




a* 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.02162 


10.97888 


9.08914 


10.91086 


9.14780 


10.66220 


60 


1 


021288 


97717 


09019 


90961 


148ni 


85128 


69 


2 


02404 


97596 


09123 


90877 


14963 


85087 


58 


8 


02.Vi6 


97475 


09227 


90778 


15054 


84946 


67 


4 


09645 


97365 


09830 


90670 


15146 


84855 


56 


6 


02766 


97284 


09484 


90566 


15286 


64764 


55 


6 


02886 


97115 


09537 


90463 


15827 


84678 


64 


7 


08006 


96995 


09640 


90360 


15417 


84583 


58 


8 


08124 


96876 


09742 


90258 


15506 


84493 


58 


9 


08242 


96758 


09845 


90155 


15698 


84402 


61 


10 


9.08361 


10.96689 


9.09W7 


10.90058 


9.16688 


10.84812 


50 


11 


03479 


96521 


J0049 


89951 


15777 


84228 


49 


12 


08597 


96408 


10150 


69850 


16867 


64138 


48 


18 


03714 


96286 


10252 


89748 


15966 


64044 


47 


14 


06882 


96168 


10663 


89647 


16046 


63964 


46 


IS 


08948 


96052 


10454 


89546 


16)35 


63865 


45 


16 


04065 


96983 


10665 


80145 


16284 


83776 


44 


17 


04181 


95819 


10656 


69344 


16812 


83688 


48 


18 


wa97 


95703 


107D6 


89244 


16401 


88599 


42 


19 


04418 


95587 


10656 


69144 


16488 


88611 


41 


20 


9.04528 


10.g54?3 


9.10956 


10.89044 


9.16577 


10.88428 


40 


21 


04048 


95357 


11056 


88944 


16665 


63885 


89 


22 


04758 


95242 


11155 


88846 


16758 


88847 


38 


28 


04878 


95127 


11254 


88746 


16841 


88169 


87 


24 


04987 


95018 


11868 


88647 


16928 


88078 


86 


25 


05101 


94899 


11452 


88548 


17016 


82964 


36 


26 


06214 


947% 


11651 


88449 


17108 


62897 


34 


27 


05328 


94672 


11649 


66361 


17190 


82810 


38 


28 


06441 


94559 


11747 


882C8 


17277 


82723 


88 


29 


05558 


94447 


11845 


68155 


17368 


82687 


31 


80 


9.05666 


10.94384 


9.11943 


10.88057 


9.17460 


10.82560 


30 


81 


05778 


94222 


12040 


879G0 


17536 


82464 


29 


32 


05890 


94110 


12188 


87862 


17622 


82378 


88 


88 


06003 


93998 


12235 


87/65 


17708 


82298 


27 


84 


06118 


93887 


12382 


87668 


17794 


82S06 


86 


85 


06224 


93776 


12428 


87572 


17880 


82120 


26 


86 


06335 


98665 


12525 


67475 


17965 


82085 


24 


87 


06445 


93556 


12621 


87379 


16051 


81949 


28 


88 


06556 


93444 


12717 


87283 


18186 


81864 


22 


89 


06666 


93334 


12818 


87187 


18221 


61779 


21 


40 


9.06775 


10.9322S 


9.12909 


10.87091 


9.18306 


10.81694 


20 


41 


06885 


93115 


13004 


8C096 


18891 


61609 




42 


06994 


93006 


13009 


8(J901 


18475 


61525 




48 


07103 


92897 


13194 


86806 


18560 


81440 




44 


07.211 


92789 


13289 


6G7I1 


18644 


81856 




45 


07820 


92C80 


13884 


8G616 


16728 


81272 




46 


07428 


92572 


13478 


86522 


18812 


81188 




47 


07536 


92464 


13573 


86427 


18806 


81104 




48 


07648 


92857 


13667 


863S8 


18979 


81021 




49 


07751 


92249 


13761 


66239 


19068 


80987 




60 


9.07858 


10.92142 


9.13854 


10.86146 


9.19146 


10.80K54 




61 


07964 


92036 


13948 


86052 


19229 


80771 




52 


08071 


91929 


14041 


85069 


19312 


80688 




58 


08177 


91828 


14134 


85866 


19395 


80606 




64 


08288 


91717 


14227 


85778 


19478 


80522 




56 


08389 


91(311 


14320 


85680 


10561 


60489 




56 


08495 


91506 


14412 


85568 


19643 


80367 




57 


08t>00 


91400 


14504 


65496 


J9725 


80875 




68 


08705 


91295 


14597 


65408 


19807 


80198 




59 


08810 


91190 


14688 


85312 


10889 


80111 




60 


0H914 


91086 


14780 


85220 


19971 


60029 




§ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


# 




88«» 




82* 




81» 



473 



TABLE III.— LOG. 


TANG] 


ENTS AND COTANGENTS. 


$ 




»• 


10" 


II* 


t 


Tan 


Cotan 


Tan 


Ootan 


Tan 


Cotan 





9.19971 


10.80039 


9.94689 


10.75368 


9.38866 


10.71186 


60 


1 


30068 


79947 


. 84706 


75294 


98988 


71067 


59 


9 


80184 


79866 


94T79 


75S91 


99000 


71000 


58 


8 


90916 


79784 


94858 


751^ 


90067 


70883 


57 


A 


30897 


79708 


94936 


75074 


99184 


70666 


56 


6 


80878 


79688 


95000 


75000 


99801 


70799 


56 


6 


20459 


79541 


95078 


74997 


99966 


70788 


54 


7 


80540 


79460 


95146 


74854 


99335 


70665 


58 


8 


80621 


79379 


35919 


74781 


99408 


70598 


63 


9 


80701 


79399 


96299 


74706 


29466 


70539 


51 


10 


9.80788 


10.79918 


9.35865 


10.74635 


9.39585 


10.70465 


SO 


11 


80868 


79188 


95487 


74668 


39601 


70899 


49 


18 


80948 


79a'W 


95510 


74490 


90668 


70889 


48 


18 


81088 


78978 


95588 


74418 


99784 


70286 


47 


14 


31108 


78806 


95655 


74845 


90600 


70200 


46 


15 


81188 


78818 


96737 


74278 


99666 


70134 


45 


18 


91861 


78789 


95709 


74901 


99889 


70068 


44 


17 


81341 


78669 


95871 


74199 


99096 


70009 


48 


18 


31480 


78580 


95948 


74057 


30064 


69936 


49 


19 


81499 


78501 


96015 


78965 


30180 


69670 


41 


80 


9.81578 


10.78499 


9.96066 


10.73014 


9.80106 


10.69606 


40 


81 


21657 


78348 


96158 


73649 


80361 


60789 


38 


88 


81786 


78964 


90230 


78771 


80896 


69674 


36 


88 


81814 


78186 


96801 


7S699 


80891 


69609 


37 


84 


81808 


78107 


96873 


78098 


80457 


69548 


36 


85 


81971 


78039 


96448 


73567 


80539 


69478 


35 


86 


88049 


77951 


96614 


78486 


80567 


69413 


34 


87 


38187 


77878 


8G585 


78416 


30659 


69348 


83 


88 


82903 


77795 


96655 


78845 


30717 


68268 


83 


89 


83883 


77717 


88796 


78374 


80789 


60218 


31 


80 


9.89361 


10.77689 


9.96797 


10.78906 


9.80S46 


10.69154 


80 


81 


38488 


77563 


96867 


78188 


30911 


69069 


99 


88 


98516 


77484 


96967 


78068 


80975 


69035 


28 


88 


98598 


77407 


97008 


7'2992 


81040 


68960 


97 


84 


88670 


77830 


97078 


79939 


81104 


66896 


36 


fti 


98747 


77858 


97148 


73859 


31166 


68839 


95 


86 


9;»34 


77176 


87318 


72789 


81988 


68767 


34 


87 


99901 


77099 


97288 


73719 


81297 


68703 


33 


88 


33977 


77038 


97357 


73648 


31361 


68689 


99 


88 


»M)54 


76946 


97427 


73578 


81495 


68575 


91 


40 


9.38130 


10.76870 


9.97496 


10.73504 


9.31489 


10.68511 


30 


41 


88M6 


76794 


97566 


79484 


31559 


68448 


19 


4'^ 


93«3 


76717 


97635 


73366 


81616 


66384 


18 


48 


88869 


78641 


97704 


72296 


81679 


68321 


17 


44 


93435 


76565 


97778 


73387 


81743 


68257 


16 


45 


98510 


76490 


97849 


73158 


81806 


6S194 


15 


46 


98586 


76414 


97911 


79069 


31870 


68180 


14 


47 


98661 


76839 


97960 


73030 


81938 


68067 


18 


48 


93787 


76368 


98049 


71951 


81906 


68004 


19 


49 


83813 


76188 


98117 


71888 


83059 


67941 


11 


60 


9.33887 


10.78118 


9.3R186 


10.71814 


9.39199 


10.67H78 


10 


51 


93963 


76088 


36S54 


71746 


83186 


67815 


9 


53 


94037 


75963 


88338 


71677 


83348 


6T763 


8 


58 


94113 


75888 


98891 


71609 


8-1811 


67689 


7 


54 


91186 


75814 


38459 


71541 


89878 


67637 


6 


65 


94361 


75789 


38537 


71473 


83430 


67564 


6 


56 


94335 


75665 


38595 


71405 


33498 


67603 


4 


87 


94410 


75690 


98669 


71886 


8SS61 


67489 


8 


58 


84484 


75516 


98730 


71370 


83623 


67377 


9 


69 


94558 


75443 


98796. 


71909 


82685 


67315 


1 


60 


34633 


75368 


98865 


71135 


32747 


67253 





/ 


Cot&n 


Tan 


Cotan 


Tan 


Cotan 


Tan 






80» 




79* 




780 



474 



TABLE in.— LOG. TANGENTS AND COTANGENTS. 



f 


19» 


18« 


140 


/ 


Tan 


Cotan 


Tan 


Otan 


Tan 


Cotan 





9.82747 


10.67258 


9.36336 


10.63664 


9.89677 


10.60828 


60 


1 


82810 


67190 


86394 


63C06 


89781 


60269 


69 


2 


82872 


67128 


86459 


63548 


89785 


60215 


68 


8 


82983 


67067 


86509 


63491 


89838 


60162 


67 


4 


82995 


67005 


86566 


68484 


88693 


60108 


66 


6 


83057 


66048 


86624 


68376 


89945 


60056 


65 


6 


38119 


66661 


86681 


63319 


80999 


60001 


54 


7 


83160 


66830 


86738 


63263 


40062 


69948 


63 


8 


83242 


66758 


86795 


63205 


40106 


69894 


63 


9 


83303 


66697 


86852 


68148 


40159 


69811 


61 


10 


9.83865 


10.66636 


9.36909 


10.63091 


9.40212 


10.59788 


60 


11 


a^26 


66574 


86966 


63034 


40366 


60784 


49 


18 


88487 


66618 


87028 


62077 


40319 


69681 


48 


18 


88548 


66459 


37060 


62920 


40372 


60628 


47 


14 


83609 


66891 


87137 


628C3 


40425 


60575 


46 


16 


88670 


66830 


87193 


62807 


40478 


69523 


45 


i6 


83731 


66269 


8?250 


62750 


40581 


69469 


44 


17 


88792 


66908 


87306 


62G94 


40584 


69416 


43 


18 


83858 


66147 


87363 


62637 


40636 


69364 


42 


19 


83918 


66067 


27419 


62581 


40689 


69811 


41 


90 


9.88974 


10.66026 


9.87476 


10.62524 


9.40742 


10.69268 


40 


81 


84034 


65966 


87532 


624C8 


40795 


69806 


89 


23 


84095 


66006 


87588 


62412 


40647 


69158 


88 


88 


84165 


66845 


87644 


62356 


40900 


69100 


87 


84 


84216 


65765 


87700 


62300 


40062 


69048 


86 


S5 


84276 


66724 


87T66 


62244 


41005 


68996 


86 


96 


84836 


65664 


87812 


62188 


41057 


68943 


84 


97 


84806 


65604 


37868 


62132 


41109 


58891 


88 


98 


84456 


65544 


87924 


62076 


41161 


58839 


88 


99 


84516 


65484 


37960 


62020 


41814 


68786 


81 


80 


9.34576 


10.65424 


9.38036 


10.61965 


9.41266 


10.58784 


80 


81 


84636 


65365 


38091 


61909 


41818 


58683 


29 


33 


84695 


65306 


88147 


61853 


41370 


68630 


28 


88 


84756 


66245 


38202 


61798 


41422 


68578 


27 


84 


84814 


65166 


38257 


61743 


41474 


68526 


86 


85 


84874 


65126 


38313 


61687 


41636 


68474 


26 


86 


84833 


66067 


88368 


61632 


41578 


68432 


84 


87 


84999 


65008 


38423 


61577 


41629 


68371 


88 


88 


85051 


64949 


38479 


61521 


41661 


68319 


28 


80 


86111 


64889 


88534 


61466 


41733 


68367 


21 


40 


9.85170 


10.64830 


9.88589 


10.61411 


9.41784 


10.68316 


90 


41 


85229 


64771 


38644 


61356 


41836 


68164 


19 


42 


85288 


64712 


38G99 


61301 


41887 


68118 


18 


48 


85847 


64663 


88754 


61246 


419:» 


68061 


17 


44 


85406 


64595 


88808 


61192 


41990 


68010 


16 


45 


85464 


64536 


38868 


61137 


42041 


67959 


16 


46 


86523 


64477 


8S918 


61082 


42093 


67907 


14 


47 


35581 


64419 


88972 


61028 


42144 


67856 


18 


48 


85640 


64360 


39037 


60973 


42195 


67805 




49 


85698 


64303 


89062 


60918 


42246 


67754 




60 


9.86757 


10.64248 


9.89136 


10.60864 


9.42297 


10.57708 




51 


85815 


64185 


89190 


60810 


42348 


67652 




63 


85878 


64127 


89246 


60755 


42399 


67601 




68 


85931 


64069 


39299 


60701 


43460 


67550 




54 


35969 


64011 


39863 


60647 


42601 


67499 




55 


36047 


63953 


30407 


60593 


42652 


67448 




66 


36105 


68895 


89161 


60539 


42603 


67397 




57 


36163 


63837 


39515 


60485 


42653 


67347 




68 


86221 


63779 


89569 


60431 


42704 


67296 




SO 


36279 


€87*31 


39^23 


60377 


427S5 


5T24B 




60 


363-16 


ekdc^fl^ 


3t«77 


flO?t2S 


42805 


571flfi 




' 


CoUu 


Tan 


CoUn 


T&n 


(JOULD 


Tan 


* 




37* 




«• 




76- 



47S 



TABLE ra.— LOG. TANGENTS AND COTANGENTS. 



* 


16» 


!«• 




17» 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.49805 


10.57196 


9.4B750 


10.64860 


9.48684 


10.61466 


60 


1 


42856 


67144 


45797 


64208 


48579 


61421 


69 


2 


42000 


67094 


46846 


64156 


48624 


61876 


68 


8 


42967 


67048 


45898 


64106 


48669 


51831 


^7 


4 


48007 


66998 


46040 


64060 


48714 


61286 


66 


6 


48057 


56943 


46987 


64018 


48760 


61241 


56 


e 


48108 


56898 


46086 


63965 


48804 


61196 


54 


7 


48158 


66842 


46088 


68918 


48849 


61151 


63 


8 


48206 


66792 


46180 


68870 


48894 


61106 


68 


9 


48258 


66742 


46177 


68628 


48069 


61061 


61 


10 


9.48808 


10.66608 


9.46884 


10.68770 


9.48964 


10.51016 


60 


11 


48858 


66642 


46271 


68729 


49029 


60971 


49 


12 




66598 


46819 


63681 


49078 


60927 


48 


18 


48468 


66548 


46860 


63634 


49118 


60662 


47 


14 


48506 


56498 
66442 


46418 


68567 


49168 


60887 


46 


16 


4.S558 


46460 


68540 


49207 


60798 


45 


10 


48607 


66898 


46507 


68493 


49858 


60748 


44 


17 


48657 


66843 


46664 


68146 


49296 


50704 


43 


18 


43707 


66298 


46601 


68889 


49341 


60659 


42 


19 


48766 


66244 


46648 


68352 


49885 


60616 


41 


20 


9.48806 


10.56104 


9.46604 


10.68306 


9.49480 


10.60670 


40 


21 


48866 


66145 


46741 


63269 


49474 


60586 


89 


e 


489a5 


66095 


46788 


63212 


49519 


S0481 


88 


28 


48954 


66046 


46885 


63166 


49608 


60437 


87 


24 


44004 


65996 


46881 


63119 


49607 


60893 


80 


25 


44058 


65947 


46928 


68072 


49668 


60848 


86 


M 


44102 


65896 


46975 


6S0S6 


49690 


60804 


34 


27 


44161 


&'^9 


47021 


62979 


49740 


60260 


88 


28 


44201 


66799 


47068 


62082 


49784 


60216 


82 


29 


442Q0 


65750 


47114 


62886 


49628 


60178 


81 


80 


9.44299 


10.66701 


9.47160 


10.62840 


9.49678 


10.60128 


80 


81 


44848 


66668 


47207 


62798 


49910 


60064 


29 


82 


44397 


65608 


47268 


62747 


49960 


60040 


88 


88 


44446 


65554 


47299 


62701 


60004 


49996 


27 


84 


44495 


66506 


47346 


62654 


60048 


49952 


86 


88 


44544 


56466 


47892 


62606 


60098 


49906 


85 


88 


4459-^ 


65408 


47488 


62562 


60130 


49864 


84 


87 


44641 


65359 


47484 


62516 


60180 


40620 


88 


88 


44690 


65310 


47680 


68470 


60228 


49777 


28 


89 


44788 


65262 


47670 


62424 


60267 


49738 


81 


40 


9.44787 


10.66213 


9.47628 


10.68878. 


9.50311 


10.49689 


80 


41 


44886 


65164 


47668 


62332 


60366 


49646 




42 


44884 


66116 


47714 


62286 


60396 


49602 




48 


44933 


55067 


47780 


68240 


60442 


49558 




44 


44961 


6S019 


47806 


62194 


60486 


49515 




46 


46029 


64971 


47a52 


62146 


60629 


49471 




46 


45078 


64922 


47897 


62106 


60678 


49428 




47 


45126 


64874 


47943 


68057 


60616 


49884 




48 


45174 


64826 


47989 


52011 


60660 


49841 




49 


45223 


54778 


48036 


61066 


60708 


49297 




60 


9.45271 


10.64789 


9.48080 


10.51920 


9.60746 


10.49254 




51 


45819 


64681 


4H126 


61874 


60780 


49211 




52 


45367 


64683 


48171 


61829 


60633 


49167 




58 


45416 


64566 


48217 


61783 


60876 


49124 




64 


45463 


64537 


48262 


51786 


60919 


49081 




65 


46511 


64480 


48307 


61698 


60962 


49088 




66 


455.'>9 


54441 


48]i53 


61647 


51005 


48995 




57 


45606 


64^J94 


48:^96 


61608 


61046 


48952 




68 


46654 


64:M6 


4H443 


61567 


61092 


48908 




69 


46702 


64298 


48489 


61611 


51185 


4»«5 




60 


45750 


64250 


48534 


61466 


61178 


48822 





Co tan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


. 




74» 




78« 




72» 



476 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



/ 


18* 


19* 




eo«» 


i 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.61178 


10.48882 


9.53697 


10.46303 


9.66107 


10.48883 


60 


1 


61281 


48779 


58788 


46268 


56146 


43854 


59 


8 


61264 


48736 


53779 


46821 


66186 


48816 


58 


8 


51306 


48694 


53820 


46180 


66284 


48776 


57 


4 


51349 


48651 


58861 


46139 


66864 


43786 


56 


6 


51398 


48608 


53902 


46098 


66308 


48697 


55 


6 


51435 


48565 


53948 


46057 


56342 


43658 


54 


7 


61478 


48528 


53984 


46016 


56381 


48619 


58 


8 


51580 


48480 


54025 


45976 


56480 


48580 


52 


9 


61663 


48437 


54065 


45935 


uOw>V 


48541 


51 


10 


9.61606 


10.48394 


9.54106 


10.45894 


9.56498 


10.43508 


50 


11 


51648 


48858 


54147 


45853 


56587 


43463 


49 


12 


51691 


48309 


54187 


46818 


66576 


48484 


48 


13 


51784 


48266 


54828 


45772 


66616 


48885 


47 


14 


61776 


48224 


54269 


45781 


56654 


4:1846 


46 


15 


51819 


48181 


54309 


45691 


56693 


43807 


45 


16 


51861 


48139 


54350 


45650 


56738 


43868 


44 


17 


51903 


48097 


54390 


45610 


56771 


48829 


48 


18 


51946 


48054 


54431 


45569 


66810 


48190 


48 


19 


51988 


48012 


64471 


45529 


66849 


48151 


41 


90 


9.5S031 


10.47969 


9.54518 


10.45488 


9.56887 


10.48118 


40 


81 


52073 


47927 


54552 


45448 


66986 


43074 


89 


88 


58115 


47885 


54593 


45407 


56965 


43085 


38 


83 


62157 


47843 


54633 


45367 


67004 


42996 


37 


84 


58200 


47800 


54673 


45827 


67048 


42958 


86 


85 


52842 


47768 


54714 


45286 


57081 


48919 


35 


86 


52284 


47716 


54754 


45246 


57180 


42880 


84 


87 


52328 


47674 


54794 


45206 


57168 


48848 


88 


88 


52368 


47638 


54835 


45165 


57197 


42803 


88 


89 


58410 


47590 


54875 


45125 


67235 


48765 


31 


80 


9.52458 


10.47548 


9.54915 


10.45085 


9.57274 


10.48726 


30 


31 


62494 


47506 


54955 


45045 


57318 


48688 


89 


38 


52r>3A 


47464 


54995 


45006 


57851 


48649 


88 


83 


52578 


47422 


55035 


44965 


57889 


48611 


27 


34 


52620 


47880 


55075 


44925 


67428 


48572 


86 


35 


52661 


47339 


55116 


44885 


57466 


42534 


85 


86 


62703 


47,'97 


55155 


44845 


57504 


42496 


84 


37 


62715 


47S55 


55195 


44805 


67543 


42467 


88 


88 


52787 


47:218 


55235 


44765 


57581 


48419 


28 


89 


52889 


47171 


55275 


44'ra5 


57619 


42381 


21 


40 


9.52870 


10.47130 


9.55815 


10.44685 


9.57658 


10.42848 


80 


41 


52918 


47088 


55355 


44645 


57696 


42304 


19 


42 


52953 


47047 


55895 


44605 


67734 


42266 


18 


48 


58995 


47006 


65434 


44566 


57778 


42288 




44 


53037 


46968 


55474 


445-26 


57810 


48190 




45 


53078 


4G922 


65514 


44486 


57849 


48151 




4& 


m-M 


4^880 


55554 


4UiG 


h7W*7 


42113 




47 


531(11 


im'^ 


firsr.03 


4m7 


670-:5 


4ii075 




48 


Mt}02 


4fi:;is 


S-T/^ 


AWil 


670f4 


4m^ 




4» 


63^14 


4(5750 


DGOTS 


44.1ii7 


5^101 


410S9 




GO 


9.r>?^P5 


io.4onjv 


9.,')!i7ia 


t0.4l^ 


9.58039 


30.4ia6l 




51 


53327 


46S73 


55752 


442 J8 


58077 


41923 




W 


5330S 


4fiG3^2 


5&701 


44^09 


6SII5 


41885 




53 


&1509 


46r,oi 


B-VflUl 


44100 


58153 


41847 




54 


53450 


4fir>so 


5.W0 


44130 


68191 


41809 




56 


GSin 


4ft508 


5,-1010 


44000 


58223 


417T1 




66 


^m^ 


4filfj7 


fir/HS 


44051 


.-ie-s; 


41733 




57 


6-^'iT4 


4(hJSfl 


^'>o^^ 


44011 


58804 


41696 




fiS 


53fil5 


mi^ 


50(KS 


490?i 


5834i 


41558 




59 


5.%M 


4fMI44 


5eOG7 


43033 


5>^B0 


41030 




60 


63607 


40303 


56107 


43S)3 


ES418 


4158a 




/ 


Cot&n 


Tan 


Cotan 


Tan 


Cotan 


Ttan 


t 




71* 




70- 




6»' 



477 



TABLE in.— LOG. TANGENTS AND COTANGENTS. 



$ 


21. 


22« 


28- 


1 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 







9.68418 


10.41588 


9.60641 


10.80850 


9.62785 


10.37815 


60 


1 


68166 


41545 


60677 


89383 


62820 


87180 


59 


8 


58498 


41507 


60714 


89288 


62855 


87145 


58 


8 


68581 


41460 


60750 


89250 


62890 


87110 


57 


4 


6R569 


41481 


60788 


89214 


62926 


37074 


56 


5 


68606 


41894 


80828 


89177 


62961 


87039 


55 


8 


68641 


41856 


60859 


89141 


62998 


87004 


54 


7 


58681 


41819 


60695 


39106 


68031 


86909 


58 


8 


56719 


41281 


60981 


89069 


68068 


86984 


58 


9 


58757 


41248 


60067 


89068 


83101 


86899 


51 


10 


9.58794 


10.41908 


9.61004 


10.38996 


9.63135 


10.86866 


50 


11 


58888 


41168 


81040 


88960 


68170 


86880 


49 


13 


58869 


41181 


61076 


88924 


63206 


36795 


48 


18 


5S907 


41098 


81118 


88888 


68240 


86760 


47 


14 


58044 


41056 


61148 


88858 


63275 


86725 


46 


16 


58961 


41019 


81184 


88816 


63310 


86690 


46 


16 


59019 


40981 


81220 


88780 


6a345 


86655 


44 


17 


50066 


40944 


81256 


88744 


6S879 


86621 


48 


18 


50094 


40008 


81292 


38706 


63414 


86586 


48 


19 


59181 


40860 


81326 


86672 


68449 


86551 


41 


90 


9.59168 


10.40688 


9.61864 


10.88688 


9.63484 


10.86516 


40 


81 


59306 


40795 


61400 


88600 


68519 


36481 


39 


28 


59848 


40767 


61486 


88564 


68558 


86447 


88 


£8 


59280 


40720 


61473 


88528 


63588 


86412 


87 


24 


58617 


40688 


61506 


8H498 


68623 


86377 


36 


85 


59854 


40646 


81544 


88458 


68657 


36318 


85 


86 


50891 


40609 


81579 


88421 


68692 


86808 


84 


87 


59429 


40571 


61615 


88385 


68726 


86274 


33 


28 


50466 


40534 


•'61651 


38.349 


68761 


362'J9 


38 


29 


59508 


40497 


81687 


88818 


68796 


86204 


31 


80 


9.50640 


10.40460 


9.61722 


10.88278 


9.63830 


10.36170 


SO 


81 


59577 


40428 


61758 


38242 


68865 


36135 


29 


88 


59614 


40886 


61794 


38206 


68899 


86101 


26 


88 


59651 


40849 


61830 


38170 


68934 


86066 


87 


84 


59688 


40818 


61865 


38135 


63968 


86032 


26 


85 


59725 


40275 


61901 


88099 


64003 


85997 


25 


86 


59768 


40238 


81936 


38064 


64037 


35963 


24 


87 




40201 


61972 


38028 


64072 


85928 


23 


88 


50885 


40165 


62006 


87992 


64106 


35894 


22 


89 


50678 


40128 


62043 


87057 


64140 


35860 


21 


40 


9.59909 


10.40091 


9.82079 


10.37981 


9.64175 


10.35825 


20 


41 


59946 


40054 


83114 


87886 


64209 


85791 


19 


48 


69068 


40017 


62150 


87850 


64243 


85757 


18 


48 


60019 


89961 


82185 


87815 


64278 


85722 


17 


44 


60056 


89944 


68221 


87779 


64312 


85688 


16 


45 


60093 


89907 


62256 


87744 


64346 


85654 


15 


46 


60180 


89870 


62292 


87706 


64381 


35619 


14 


47 


60166 


89634 


62327 


87678 


64415 


35585 


18 


48 


60206 


89797 


62862 


87688 


84449 


36551 


12 


49 


60240 




62398 


37602 


64483 


85517 


11 


60 


9.60876 


10.89784 


9.62438 


10.87567 


9.64517 


10.85488 


10 


51 


60318 


89687 


68468 


87588 


64562 


85448 


9 


58 


60849 


89661 


62504 


87496 


64586 


35414 


8 


68 


60386 


89014 


62539 


87461 


64620 


35880 


7 


54 


60428 


89578 


62574 


87426 


64654 


85346 


6 


65 


60459 


89541 


62609 


87391 


64688 


85312 


6 


66 


60495 


89505 


62645 


37855 


647^ 


85278 


4 


67 


60588 


89468 


62680 


87320 


64756 


35244 


8 


68 


60608 


89438 


62715 


37285 


64790 


35210 


8 


50 


60606 


89895 


68750 


37250 


64824 


35176 


1 


60 


60641 


89859 


62785 


87215 


64858 


85142 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




68« 




87« 




W 



478 



f ABLK ITT 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 


«4« 


«5«» 


28- 


/ 


Tan 


Cotan 


Tan 


Coran 


Tnn 


Cotan 





9.64868 


10.85142 


9.66867 


10.33138 


9.68818 


10.81182 


00 


1 


64892 


85108 


06900 


88)00 


68850 


81150 


60 


2 


64926 


85074 


66933 


88007 


68882 


81118 


68 


8 


64960 


85040 


66966 


83034 


68914 


81066 


67 


4 


64994 


85006 


66999 


83001 


68946 


81064 


66 


6 


65028 


84972 


67032 


82968 


68078 


81022 


65 


6 


65062 


84938 


67066 


82936 


69010 


t0990 


54 


7 


66096 


84904 


67098 


82902 


69042 


80968 


63 


8 


66180 


84870 


67131 


32869 


69074 


80926 


62 


9 


65164 


84886 


67163 


82887 


69106 


80604 


61 


10 


9.65197 


10.84803 


9.67196 


10.82804 


9.60188 


10.30662 


60 


11 


65281 


84769 • 


67229 


82771 


69170 


80630 


49 


12 


65265 


34785 


6?262 


82788 


09202 


80798 


48 


13 


65299 


84701 


67295 


82705 


60234 


30766 


47 


14 


65883 


84667 


67327 


82678 


69266 


80734 


46 


15 


65S66 


84634 


67860 


32640 


69296 


80702 


45 


18 


65400 


84600 


67398 


82607 


69829 


80671 


44 


17 


65484 


34566 


67426 


82574 


69361 


80689 


48 


18 


65467 


84533 


67458 


82542 


69393 


80607 


42 


19 


65501 


84499 


67491 


82509 


694;e5 


80676 


41 


90 


0.65585 


10.84465 


9.67624 


10.82476 


9.69457 


10.80643 


40 


21 


65668 


8448d 


67556 


82444 


69488 


80512 


89 


22 


65602 


84396 


67589 


&2411 


69620 


80480 


88 


28 


66686 


84364 


67622 


82378 


60552 


80448 


87 


24 


65669 


84881 


67654 


82846 


69584 


80416 


86 


25 


65708 


84297 


67687 


82313 


69615 


80885 


85 


26 


65786 


84264 


67719 


82281 


69647 


80658 


84 


27 


65770 


84230 


67752 


32248 


89679 


80821 


88 


28 


66803 


84197 


67785 


82215 


69710 


80290 


88 


29 


65837 


84163 


67817 


82188 


09742 


80258 


81 


80 


9.65870 


10.84180 


9.678IJ0 


10.82150 


9.69774 


10.80220 


80 


81 


65904 


84096 


67882 


32118 


69805 


80196 


29 


32 


66937 


34063 


67915 


82085 


69S87 


80168 


28 


88 


65971 


84020 


67947 


82053 


69868 


80182 


87 


84 


66004 


83996 


67980 


82080 


69900 


80100 


26 


85 


66038 


83962 


68012 


81988 


69932 


80068 


86 


W 


66071 


83929 


68044 


81956 


69963 


80037 


84 


87 


66104 


83896 


68077 


81923 


69995 


80006 


88 


88 


66188 


83S62 


68109 


31691 


70026 


29974 


88 


89 


66171 


838:.>9 


68142 


31858 


70058 


29942 


21 


40 


9.66204 


10.88796 


9.68174 


10.31886 


9.70089 


10.29911 


80 


41 


66288 


83762 


68:208 


31794 


70121 


29870 




42 


66271 


837^ 


68239 


81761 


70152 


29848 




48 


66804 


88696 


68371 


81729 


70184 


29616 




44 


66887 


88668 


68303 


81697 


70216 


29785 




45 


66371 


83629 


68386 


81664 


70247 


29758 




40 


66404 


a36n6 


68308 


81682 


70278 


89722 




47 


66487 


S&563 


68400 


31600 


70809 


29G91 




48 


66470 


aS580 


68432 


81668 


70841 


29669 




49 


66603 


83497 


66465 


81585 


70372 


29628 




60 


9.66637 


10.33463 


9.68497 


10.31.'i03 


9.70404 


10.29596 




51 


66670 


384.30 


68539 


31471 


70485 


29665 




52 


66608 


83.397 


68?)61 


81489 


70466 


29584 




58 


66636 


88864 


es^9Z 


81407 


70498 


89602 




54 


66660 


83881 


68626 


81374 


70529 


204n 




55 


66702 


83298 


68658 


81342 


70560 


29440 




56 


667&5 


88265 


6H690 


81810 


70592 


29406 




57 


66768 


8323> 


68722 


81278 


70623 


99877 




58 


66801 


83199 


68754 


81246 


70654 


89346 




59 


66834 


38166 


68786 


81214 


70685 


29816 




60 


66667 


83133 


68818 


81182 


70717 


29888 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 


• 


86» 




64« 




W 



479 



TABLE ni. 


—LOG. 


TANGENTS AND COTANGENTS. 


# 


87« 


28* 


290 1 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.T0ri7 


10.89288 


9.72667 


10.27438 


9.74875 


10.95625 


60 


1 


70748 


292S8 


72598 


27402 


74405 


25596 


59 


2 




29S21 


78628 


27872 


74435 


25666 


58 


8 


70610 


29190 


72659 


27841 


74465 


85586 


57 


4 


70841 


29159 


72689 


27811 


74494 


26506 


66 


6 


70878 


89127 


7^^720 


87280 


74524 


26476 


55 


8 


70904 


29096 


72750 


87260 


74554 


25446 


54 


7 


70985 


29065 


72780 




74588 


85417 


68 


8 


70966 


29034 


72811 


27189 


74618 


85887 


62 


9 


70997 


29008 


T2S41 


87159 


74643 


26857 


61 


10 


9.71028 


10.88978 


9.78878 


10.2n28 


9.74678 


10.25327 


60 


11 


71059 


:28941 


72902 


27098 


74702 


85296 


49 


1« 


71090 


28910 


72968 


27068 


74788 


26266 


48 


18 


71181 


28879 


72063 


27087 


74762 


25836 


47 


14 


71158 


28847 


72998 


27007 


74791 


85209 


46 


15 


71184 


88816 


78028 


26977 


74821 


95179 


45 


18 


71215 


28785 


78054 


26946 


74851 


25149 


44 


17 


71246 


28754 


78084 


26016 


74880 


26120 


48 


18 


71277 


28728 


73114 


26886 


74910 


25090 


48 


19 


71808 


28692 


T8144 


26856 


74030 


25061 


41 


80 


9.71389 


10.28661 


9.78175 


10.26825 


9.74969 


10.25061 


40 


81 


71870 


28680 


78:205 


26706 


74906 


86002 


89 


88 


71401 


28599 


78285 


26705 


75028 


24072 


88 


£8 


71481 


28569 


73266 


86786 


75058 


84942 


87 


24 


71468 


28588 


78295 


26705 


75087 


84918 


86 


86 


71408 


28607 


78328 


26674 


75117 


24888 


86 


88 


71524 


28476 


78856 


96644 


75146 


84854 


84 


87 


71555 


28445 


78886 


26614 


75176 


24624 


88 


28 


71686 


28414 


78416 


26584 


76206 


84795 


88 


89 


71617 


28888 


78446 


96664 


75285 


84765 


81 


80 


9.71648 


10.28868 


9.78476 


10.26524 


9.76264 


10.84786 


80 


81 


71679 


28821 


73507 


26498 


75294 


84706 


29 


88 


71709 


28291 


78687 


26468 


75828 


84677 


28 


88 


71740 


28260 


78567 


26433 


75368 


24647 


27 


34 


71771 


28229 


78697 


26406 


75388 


24618 


26 


85 


71802 


28196 


78627 


263T3 


75411 


24589 


26 


86 


71833 


28167 


78867 


26343 


75441 


24559 


24 


87 


71863 


28187 


73667 


26818 


75470 


24580 


28 


88 


71894 


28106 


73717 


26283 


75500 


24500 


22 


88 


71085 


28075 


78747 


26258 


75529 


24471 


21 


40 


9.71956 


10.28045 


9.787T7 


10.26228 


9.75558 


10.24442 


80 


41 


71986 


28014 


73807 


26103 


75588 


24412 




48 


72017 


27068 


78887 


26163 


75617 


84383 




48 


72048 


27952 


73867 


26188 


76647 


2435.) 




44 


72078 


27922 


73897 


26108 


75676 


24824 




45 


72109 


27991 


78927 


26073 


75706 


84295 




46 


72140 


27860 


73957 


26048 


75735 


84266 




47 


72170 


27830 


78987 


26018 


76764 


24286 




48 


72201 


27799 


74017 


26883 


75798 


24207 




49 


72231 


27769 


74047 


25958 


76622 


84178 




50 


9.72262 


10.27738 


9.74077 


10.25928 


9.76652 


10.24146 




51 


72298 


27707 


74107 


26806 


75881 


24119 




58 


72823 


27677 


74137 


S6868 


75910 


84090 




58 


72854 


27646 


74166 


2.5884 


76989 


24061 




54 


72384 


27616 


74196 


25804 


75969 


24081 




66 


72415 


27586 


74226 


25774 


75996 


24002 




56 


72445 


27555 


74256 


25744 


78027 


28973 




57 


72476 


27524 


74286 


25714 


76056 


28944 




58 


72606 


27494 


74316 


26684 


76086 


28914 




50 


72587 


27468 


74345 


26655 


76115 


28885 




60 


72667 


27483 


74875 


25625 


76144 


88856 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




6f 




61* 




60» 



48o 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


80« 


81- 




82» 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.76144 


10.23866 


9.77m 


10.22128 ' 


9.79579 


10.80421 


00 


1 


76178 


23827 


77906 


82004 


79607 


20898 


60 


2 


76202 


28798 


77985 


22066 


79685 


80866 


58 


.8 


78281 




77963 


2S087 


79668 


80887 


67 


4 


76261 


28789 


77992 


88006 


79691 


20809 


66 


5 


76290 


28710 


78020 


81980 


79719 


80861 


56 


6 


76819 


23681 


78049 


21961 


79747 


80858 


64 


7 


76348 


23G52 


78077 


81928 


79776 


80294 


58 


8 


76377 


28628 


78106 


81894 


79604 


80196 


59 


9 


76406 


28694 


78185 


21865 


79688 


80168 


61 


10 


9.76435 


10.23666 


9.78168 


10.21887 


9.79660 


10.20140 


50 


11 


764G4 


23536 


78192 


21808 


79688 


20118 


40 


18 


76198 


23507 


78820 


21780 


79916 


80064 


46 


18 


76622 


28478 


78249 


817V1 


79944 


20056 


47 


14 


76551 


23449 


78277 


21728 


79978 


S0OS6 


46 


15 


76580 


23420 


78806 


21694 


80000 


80000 


45 


16 


76609 


28391 


78334 


81666 


80028 


19078 


44 


17 


76639 


28861 


78868 


81687 


80056 


19944 


48 


18 


76668 


28882 


78391 


81609 


80064 


19916 


48 


19 


76697 


23308 


78419 


81681 


80118 


19668 


41 


SO 


9.76725 


10.23276 


9.7B448 


10.81558 


9.80140 


10.19600 


40 


21 


76754 


28246 


78476 


81524 


80168 


19832 


80 


22 


76788 


28217 


78506 


21495 


80195 


19605 


88 


28 


76812 


28188 


78538 


21467 


80288 


19777 


87 


24 


76841 


281.59 


78562 


81438 


60%1 


19749 


86 


25 


76870 


28180 


78590 


81410 


80279 


19781 


85 


26 


76899 


28101 


78618 


81882 


80807 


19698 


84 


27 


76928 


28072 


78647 


81858 


60885 


19666 


88 


28 


76967 


28043 


78675 


81326 


60868 


19687 


88 


29 


76986 


23014 


78704 


81296 


80891 


19609 


31 


80 


9.rroi6 


10.22985 


9.78788 


10.21268 


9.80419 


10.19681 


30 


81 


77044 


22956 


78760 


21240 


60447 


19668 


80 


82 


77073 


22987 


78789 


81211 


80474 


10686 


88 


83 


77101 


22899 


78817 


81188 


80502 


19496 


97 


84 


77180 


22870 


78846 


21165 


60580 


19470 


86 


85 


77159 


22841 


78874 


81126 


60666 


19448 


85 


86 


77188 


22812 


78902 


81098 


60686 


10414 


94 


37 


77217 


22783 


78980 


81070 


60614 


10886 


88 


88 


77246 


22754 


78959 


81041 


60648 


10868 


88 


89 


77274 


22726 


78987 


81018 


60669 


10881 


91 


40 


9.77303 


10.22697 


9.79016 


10.80986 


' 9.60607 


10.10808 


90 


41 


77882 


2-J668 


79043 


20957 


60785 


1087S 




42 


77361 


22639 


79072 


20928 


80768 


10247 




43 


77890 


22610 


79100 


20900 


60781 


10810 




44 


77418 


22582 


79128 


20878 


80806 


10108 




45 


77447 


22553 


79156 


80844 


80886 


10164 




46 


77476 


22524 


79185 


80815 


80864 


10186 




47 


77605 


22495 


79218 


80787 


80808 


10106 




48 


77533 


22467 


79241 


80759 


80919 


10061 




49 


77562 


22488 


79269 


20781 


80947 


10068 




50 


9.77591 


10.32409 


9.79297 


10.80708 


9.800ni 


10.10085 




51 


77619 


22381 


79826 


20674 


61008 


16097 




52 


7764S 


22352 


79354 


20646 


61080 


18970 




58 


77677 


22328 


79888 


80618 


61056 


18948 




54 


77706 


22294 


79410 


80590 


81086 


18914 




66 


77784 


22266 


79488 


20668 


81118 


16867 




66 


77763 


22237 


79466 


20584 


61141 


18659 




57 


77791 


22209 


79496 


80506 


81169 


18881 




58 


77820 


22180 


79528 


20477 


61196 


16804 




59 


77849 


22151 


79551 


80449 


81224 


18778 




60 


77877 


22128 


79679 


80421 


61252 


16748 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




6»» 




58* 




67« 



48i 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


SS* 


S4' 




•S- 


/ 


Tan 


Ootan 


Tun 


Cotan 


Tan 


Cotan 




9.81858 


10.18748 


9.88809 


10.17101 


9.84588 


10.15477 


60 




81-279 


18781 


82986 


17074 


81560 


15460 


60 


81807 


18608 


88958 


17047 


84678 


15484 


58 




81886 


18665 


88980 


17080 


84606 


16397 


57 


81868 


18688 


88008 


16008 


84680 


15870 


66 


81890 


18610 


88085 


16965 


84657 


16348 


66 


81418 


18588 


83068 


16888 


84684 


16816 


64 




81446 


18565 


88089 


16911 


84711 


15889 


68 


81478 


18587 


88117 


16888 


84788 


15868 


58 




81600 


18500 


88144 


16856 


84764 


15886 


61 


10 


9.81688 


10.18478 


9.83171 


10.16889 


9.84701 


10.16809 


60 


11 

13 


81566 


18444 


83198 


16808 


84818 


16188 


49 


81588 


18417 


83835 


16775 


84845 


16165 


48 


18 
14 
16 
16 


81611 


18880 


88258 


16748 


84873 


16188 


47 


81638 


18868 


83280 


16780 


84899 


16101 


46 


81666 


18881 


88307 


16698 


84925 


15075 


45 


81098 


18807 


88334 


16666 


84958 


15048 


44 


17 


817^ 


18879 


83861 


16639 


84970 


16081 


48 


18 


81748 


18858 


838S8 


16618 


86006 


14994 


48 


19 


81776 


18884 


88415 


16586 


85068 


14967 


41 


SO 


9.81806 


10.18197 


9.88148 


10.16668 


9.86060 


10.14941 


40 


81 
S8 


81681 


18169 


83470 


16580 


85088 


14914 


89 


81868 


18148 


88497 


16508 


86118 


14887 


88 


81886 


18114 


88584 


16476 


86140 


14860 


87 


M 


81918 


18087 


88551 


16449 


85166 


14884 


86 


S6 


81941 


18060 


88578 


16488 


86198 


14807 


86 


26 


81968 


18088 


83605 


16895 


85880 


14780 


84 


87 


81906 


18004 


88688 


16868 


86847 


14758 


88 


88 


88088 


1T977 


83650 


16841 


85878 


14787 


88 


SO 


82051 


17949 


88686 


16814 


86300 


14700 


81 


80 


9.88078 


10.17988 


9.88718 


10.16887 


9.86387 


10.14678 


80 


81 
88 


88106 


17894 


88740 


16860 


85364 


14646 


80 


88188 


17897 


83768 


16288 


85880 


14620 


88 


88 
84 


88161 


17830 


88795 


16805 


85407 


14508 


87 


88188 


17818 


88888 


16178 


85484 


14666 


86 


86 


88815 


17786 


83849 


16151 


85460 


14540 


85 


80 


«2848 


17757 


88876 


16184 


85487 


14618 


84 


87 


88-^ro 


17780 


83008 


16007 


86514 


14486 


88 


88 


82898 


17708 


83930 


16070 


85540 


14460 


88 


89 


88385 


17676 


83957 


16048 


85567 


14488 


81 


40 


9.83858 


10.17048 


9.83984 


10.16016 


9.85594 


10.14406 


80 


41 


8a2880 


17680 


84011 


15969 


86680 


14880 




48 


88407 


17598 


84038 


15968 


86647 


14368 




48 


88435 


17566 


84066 


15935 


85674 


14886 




44 


82468 


17588 


84002 


15908 


86700 


14300 




45 


88489 


17511 


84119 


16881 


85T27 


14878 




46 


88517 


17483 


84146 


15864 


86754 


14846 




47 


88544 


17456 


84178 


16887 


85780 


14890 




48 


88671 


17489 


84800 


15800 


85807 


14198 




49 


88509 


17401 


84837 


15778 


85834 


14166 




60 


9.88086 


/O. 17874 


9.84854 


10.15746 


9.85860 


10.14140 




51 


8S658 


17847 


84880 


15780 


86887 


14118 




68 


88681 


17819 


84807 


15698 


86913 


14087 




68 


88706 


17898 


84884 


15666 


85940 


14060 




64 


82786 


17866 


84361 


15689 


86967 


14033 




66 


88768 


17888 


84888 


15618 


85993 


14007 




66 


88790 


mio 


84415 


16685 


86080 


18980 




57 


88817 


17188 


84448 


15558 


86046 


18964 




68 


82844 


37:56 


84460 


16631 


86078 


18987 




60 
60 


88871 


17189 


81406 


16504 


86100 


13900 




88899 


17101 


84583 


16477 


86186 


13874 





/ 


Ck>tan 


Tan 


Cotan 


Tan 


Ck>tan 


Tan 


/ 




M* 




66» 




64» 



482 



TABLE m.— LOG 


. TANGENTS AND CO' 


FANGEN 


TS. 


/ 


86«» 


' $7» 


88« 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.86186 


10.18874 


9.87711 


10.18289 


9.89881 


10.10719 


60 


1 


86168 


18847 


87788 


18868 


89807 


10693 


60 


a 


86179 


18881 


87764 


18236 


89888 


10667 


68 


8 


86806 


13794 


87790 


18810 


89859 


10641 


57 


4 


86888 


13768 


87817 


18188 


89885 


10616 


66 





86859 


18741 


.87848 


18157 


89411 


10580 


55 


6 


86886 


18716 


•87869 


12131 


89487 


10568 


54 


7 


86818 


18688 


87896 


18106 


89463 


10587 


63 


8 


66388 


18668 


87928 


18078 


89489 


10511 


68 


9 


86865 


13635 


87948 


18068 


89516 


10485 


61 


10 


9.86888 


10.13608 


9.87974 


10.18026 


9.89541 


10.10459 


60 


11 


86418 


18568 


88000 


12000 


89567 


10483 


49 


18 


86446 


18666 


88087 


11978 


89598 


10407 


48 


18 


86471 


18629 


88068 


11947 


89619 


10881 


47 


14 


86496 


13608 


88079 


11981 


89646 


10866 


46 


16 


86584 


18476 


88105 


11895 


89671 


10889 


46 


16 


86561 


13449 


88181 


11869 


89697 


10806 


44 


17 


86677 


18488 


88158 


11848 




10877 


48 


18 


86608 


13397 


88184 


11816 


89749 


10251 


48 


19 


86680 


18370 


88810 


11790 


89776 


10886 


41 


90 


9.86666 


10.18844 


9.88886 


10.11764 


9.89601 


10.10199 


40 


81 


86688 


18817 


88868 


11738 


88887 


10178 


89 


23 


86709 


18891 


88289 


11711 


89658 


10147 


88 


88 


86786 


18864 


88316 


11685 


89879 


10181 


87 


84 


86768 


13888 


88341 


11659 


89906 


10095 


86 


86 


86789 


18811 


88867 


11633 


89931 


10069 


85 


88 


86816 


18185 


88393 


11607 


89957 


10043 


34 


87 


86849 


18168 


88420 


11560 


89983 


10017 


38 


88 


86868 


18188 


88446 


11554 


90009 


09991 


88 


89 


86894 


18106 


88478 


11528 


90085 


09966 


31 


80 


9.86981 


10.18079 


9.88498 


10.11508 


9.90061 


10.09989 


80 


81 


86947 


18058 


88524 


11476 


90086 


09914 


89 


38 


86974 


• 13086 


88550 


11450 


90118 


09888 


88 


88 


87000 


18000 


88577 


11488 


90188 


09868 


97 


84 


87027 


18978 


88608 


11397 


90164 


00686 


86 


85 


87068 


18947 


88629 


11871 


90190 


09610 


85 


86 


87079 


18981 


88655 


11845 


90216 


09:84 


84 


87 


87106 


12894 


88681 


11819 


90248 


09758 


88 


88 


87188 


18868 


88707 


11293 


90268 


09782 


88 


89 


87168 


18848 


88783 


11867 


90894 


09706 


81 


40 


9.87186 


10.18815 


9.88759 


10.11841 


9.90380 


10.09680 


80 


41 


87811 


18789 


88786 


11814 


90346 


09654 




48 


87888 


18768 


88818 


11188 


90S71 


09689 




48 


87264 


18786 


88838 


11168 


90897 


09608 




44 


87290 


18710 


88864 


11186 


90488 


09577 




45 


87817 


18688 


88800 


11110 


90449 


09561 




46 


87343 


12657 


88916 


11084 


90475 


09585 




47 


87369 


12631 


88942 


11058 


90501 


09499 




48 


87396 


18604 


88968 


11088 


90587 


09478 




49 


87488 


18578 


88994 


11006 


90558 


09447 




50 


9.87448 


10.18858 


9.89020 


10.10980 


9.90578 


10.09498 




61 


87475 


18525 


89046 


10964 


90604 


08396 




68 


87501 


18499 


89078 


10927 


• 90630 


09370 




68 


87537 


18478 


89099 


10901 


90656 


09844 




64 


87664 


18446 


89185 


10875 


90688 


09818 




55 


87580 


12420 


89151 


10849 


90708 


09298 




66 


87606 


18894 


89177 


10823 


90734 


09266 




67 


87638 


12367 


89203 


10797 


90769 


09841 




68 


8T659 


12841 


89229 


10771 


90785 


09815 




59 


87685 


12315 


89256 


10746 


90611 


00189 




60 


87711 


12289 


89281 


10719 


90837 


09168 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




58- 




««• 




61« 



483 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



• / 


sr 


40- 


41' 


/ 


Tau 


Cotan 


Tan 


Ctotan 


Tan 


Cotan 





9.90687 


10.09163 


9.92381 


10.07619 


9.93916 


10.06084 


60 


] 


90668 


09187 


92407 


07593 


93942 


06058 


69 


2 


00889 


09111 


92438 


07567 


93907 


06038 


68 


8 


90914 


09086 


92458 


07642 


98906 


06007 


67 


4 


00940 


09060 


92484 


07516 


94018 


05982 


56 


6 


90966 


00034 


92510 


07490 


94044 


06956 


65 


6 


90992 


09008 


92535 


07465 


94069 


06931 


64 


7 


91018 


08982 


92661 


0:439 


94095 


06906 


68 


8 


91048 


08S)S7 


02587 


07418 


94120 


05880 


62 


9 


91000 


08081 


92612 


07888 


94146 


06854 


61 


10 


9.91095 


10.08905 


9.92688 


10.07368 


9.94171 


10.05829 


60 


11 


91121 


08879 


92663 


07887 


94197 


05803 


49 


12 


91147 


08853 


92689 


07811 


94222 


06778 


48 


18 


91172 


088-^8 


92715 


07286 


91248 


06762 


47 


14 


91198 


08802 


92740 


07260 


94278 


06787 


46 


16 


91224 


08776 


92766 


07234 


94299 


05701 


46 


16 


91250 


08750 


92792 


07208 


94824 


06676 


44 


17 


91278 


08724 


92817 


on83 


94a'30 


05660 


48 


18 


91801 


08699 


92813 


07167 


94376 


05626 


42 


19 


91827 


08673 


92868 


07182 


94401 


06599 


41 


20 


9.91338 


10.08647 


9.92894 


lo.onoo 


9.94426 


10.06574 


40 


21 


91379 


08621 


92920 


07080 


94452 


05548 


89 


22 


91404 


08596 


02946 


07066 


94477 


05528 


86 


128 


91480 


08570 


92971 


07029 


94603 


05497 


87 


24 


91466 


oa>i4 


92006 


07004 


94628 


06472 


86 


25 


9148^ 


08518 


93022 


06978 


94564 


06446 


86 


26 


91507 


08493 


93048 


06052 


94579 


. 05421 


84 


27 


91538 


08467 


98078 


06927 


94604 


05896 


88 


28 


91559 


08441 




06001 


94680 


06370 


82 


29 


91585 


08416 


93124 


06876 


04656 


05345 


81 


80 


9.91610 


10.08890 


9.0615O 


10.06850 


9.94681 


10.05819 


80 


81 


91036 


08364 


98176 


06825 


94706 


a^294 


29 


82 


91662 


0»i88 


93201 


06799 


94782 


05268 


28 


88 


9168S 


08312 


93227 


06778 


04757 


06248 


27 


34 


91713 


08J87 


93252 


06748 


94788 


05217 


26 


85 


917B9 


08261 


93278 


06722 


94806 


06192 


25 


86 


91765 


08285 


98308 


06697 


94834 


05166 


24 


87 


91791 


08209 


06329 


06671 


94859 


06141 


28 


88 


91816 


08184 


93354 


06646 


94884 


06116 


22 


89 


91842 


08158 


93380 


06620 


94910 


06090 


21 


40 


9.01868 


10.08132 


9.93406 


10.06594 


9.94985 


10.06066 


20 


41 


91893 


08107 


93481 


06569 


94961 


05089 


19 


4;! 


91919 


0S081 


93457 


06548 


94986 


05014 


18 


4.3 


91945 


08055 


93482 


06518 


95012 


04968 


17 


44 


91971 


08029 


93508 


06492 


96037 


04968 


16 


45 


91996 


08004 


06583 


06467 


06062 


04988 


16 


46 


92022 


07978 


93559 


06441 


95088 


04912 


14 


47 


92048 


07962 


06584 


06416 


95118 


04887 


18 


48 


93073 


07927 


fr^eio 


06390 


96139 


04861 


12 


49 


92099 


07901 


98686 


063&4 


95164 


04886 


11 


50 


9.92125 


10.07875 


9.08661 


10.06389 


9.06190 


10.04810 


10 


51 


92160 


07850 


93687 


06318 


95216 


04786 





52 


92176 


07H24 


93712 


06288 


95240 


04760 


8 


f>3 


92»2 


07798 


93738 


06262 


95266 


04784 


7 


54 


92227 


07V/8 


93763 


06237 


95v>91 


04709 


6 


55 


92258 


07747 


93789 


06211 


95817 


04688 


6 


56 


92279 


07721 


93814 


06186 


95842 


04658 


4 


57 


9-.'304 


07696 


93840 


06160 


96368 


04C32 


8 


58 


92380 


07670 


93865 


06185 


95898 


04607 


2 


59 


92366 


07644 


98891 


06109 


95418 


04682 


\ 


60 


92881 


07619 


93916 


06064 


96444 


04656 





» 


Ck>taii 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




M* 




490 




48» 



TABLE ni.— LOG. TANGENTS AND COTANGENTS. 



' 


42* 


48* 




44» 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.95444 


10.04666 


9.90966 


10.08084 


9.96484 


10.01616 


60 


1 


9&4(i0 


0453] 


96991 


06009 


96600 


01491 


69 


8 


95495 


04606 


97016 


08984 


96684 


01466 


68 


8 


95620 


04480 


97048 


08968 


96660 


01440 


67 


4 


96545 


04465 


97067 


02988 


96686 


01416 


60 


6 


96571 


04429 


97098 


08908 


96610 


01890 


66 


6 


96596 


04404 


9ni6 


09888 


96686 


01865 


64 


7 


96628 


04878 


97143 


02867 


98661 


01839 


68 


8 


96647 


04868 


97166 


02838 


96686 


01814 


68 


9 


96678 


04888 


97196 


08607 


98711 


01889 


61 


10 


9.96698 


10.04308 


9.97819 


10.0S781 


9.96787 


10.01868 


60 


11 


96728 


04277 


Vm4 


02766 


96768 


01886 


49 


18 


96748 


04288 


97209 


02781 


96787 


01818 


48 


18 


95774 


04226 


97296 


08705 


96818 


01188 


47 


14 


96799 


04^1 


97880 


08680 


96888 


01168 


46 


16 


95826 


04176 


97845 


02656 


96868 


01187 


46 


16 


95660 


04160 


97871 


08089 


98886 


01118 


44 


17 


95876 


04126 


97896 


08604 


96918 


01067 


48 


18 


95901 


04099 


97421 


08679 


96989 


01061 


48 


19 


96926 


04074 


97447 


08668 


98964 


01066 


41 


90 


9.96068 


10;04048 


9.97478 


10.08626 


9.98969 


10.01011 


40 


21 


96977 


04088 


97497 
97588 


08603 


99016 


00966 


89 


28 


96002 


08996 


08477 


99040 


00960 


88 


23 


96028 


08978 


97546 


02468 


99066 


00965 


87 


24 


9tK)5S 


08947 


97678 


08427 


99090 


00910 


88 


25 


96078 


03922 


97696 


02403 


99116 


006B4 


36 


86 


96104 


08896 


97624 


08876 


99141 


00669 


84 


27 


96129 


08871 


97649 


02851 


99166 


00684 


88 


88 


96156 


08846 


97674 


08326 


99191 


00609 


88 


89 


96180 


08880 


97700 


02800 


99817 


00788 


31 


80 


9.96206 


10.08796 


9.977S5 


10.08276 


9.99848 


10.00766 


ao 


81 


962JJ1 


08769 


97760 


08260 


99867 




89 


88 


96866 


08744 


97778 


09884 


99898 


OOiW 


88 


88 


96281 


06719 


97801 


08199 


90816 


00662 


87 


34 


96307 


08698 


978M 


02174 


99848 


00667 


86 


8R 


96338 


08668 


97851 


08149 


99368 


00688 


86 


86 


96857 


08648 


97877 


02128 




00606 


84 


37 


96888 


08617 


97908 


08006 


99419 


0C681 


88 


88 


96406 


08692 


97927 


02073 




O0666 


28 


89 


96438 


08567 


97B68 


02047 


99460 


00681 


81 


40 


9.964!)9 


10.08641 


9.97978 


10.08028 


9.99496 


10.00606 


80 


41 


96484 


08516 


96008 


01997 


99620 


00480 


19 


48 


96610 


03490 


98029 


01971 


99645 


00466 


18 


48 


96586 


08466 


96054 


01946 


9067U 


00480 


17 


44 


96560 


08440 


98079 


01981 


99606 


00404 


16 


45 


9C688 


08414 


96104 


01896 


99081 


00879 


16 


46 


96611 


08389 


96180 


01870 


mow 


00864 


14 


47 


96686 


03364 


96166 


01845 


99678 


O0BS6 


18 


48 


96668 


03338 


96180 


01890 


99697 


00808 


18 


49 


96667 


03318 


98806 


01794 


99788 


00878 


11 


60 


9.90718 


10.03288 


9.96231 


10.01769 


9.99747 


10.00258 


10 


61 


96738 


08268 


96256 


01744 


99773 


00827 


9 


68 


96763 


03237 


96261 


01719 


99798 


00008 


8 


68 


96788 


03212 


98807 


01693 


99688 


00177 


7 


64 


96814 


08186 


983S8 


01668 


99848 


00168 


6 


66 


96889 


08101 


96367 


01648 


99674 


00180 


6 


66 


96864 


06186 


96868 


01617 


99889 


00101 


4 


67 


96690 


08110 


96406 


01698 


99994 


00076 


8 


68 


96915 


08066 


96488 


01667 


99949 


00061 


8 


69 


96940 


03060 


96466 


01548 


99976 


00085 


1 


60 


96966 


00084 


98484 


01516 


10X10000 


00000 





/ 


CotAD 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




4T 




46» 




♦«• 





485 



TABLE IV.— NATURAL SINES AND COSINES. 



/ 


0» 


V 1 


2- 1 


8- 1 


4- 1 




Sine ;CosIn 
TOOOOO One. 


Sine 
.01745 


Cosin 

.99985 


Sine 
:03490 


Cosln 

.9i««i9 


Sine 
.06284 


Cosin 
799863 


Sine 


Cosin 




.06976 


.99756 60 


1 


.000291 One. 


.01774 


.99984 


.03519 


.99ff« 


.05263 


.99861 


.07006 


.99754169 


2 


.00068 One. 


.01803 


.99984 


.03548 


.99937 


.05292 


.99660 


.07084 


.99752168 


8 


.00087 One. 


.01833 


.99963 


.08577 


.99936 


.05321 


.99858 


.07063 


.99750! 57 


4 


.001161 One. , .018621 


.99988 


.08606 


.90935 


.05350 


.99657 


.07092 


.99748166 


5 


.00145 One. 


.01891 


.99962 


.06635 


.99934 


.05379 


.99855 


.07121 


.99746! 65 


6 


.00175, One. 


.01920 


.99962 


.06664 


.99933 


.05406 


.99854 


.07150 


.997441 54 


7 


.00204 One. 


.01949 


.99981 


.08693 


.99932 


.05487 


.99652 


.07179 


.997421 53 


8 


.00283 One. 


.01978 


.99960 


.03723 


.99931 


.06466 


.99661 


.07208 


.99740 52 





.00262 One. 


.02007 


.99980 


.08752 


.99930 


.05495 


.99649 


.07237 


.99V38I 51 


10 


.00891 One. 


.02086 


.99979 


.08781 


.99929 


.06624 


.99647 


.07266 


.00786. 60 


11 


.00620 .99909 


.02065 


.99979 


.06610 


.90927 




.98846 


.07296 


.99784' 49 


12 


.00849 .99909 


.02004 


.99978 


.03839 


.99020l 


.05582 


99644 


.07824 


.99731,48 


18 


.006781.99999 


.02123 


.999'i7 


.08866 


.99925, 


.06611 


!99642 


.07868 


.99729 47 


14 


.00407 .99999 


.02152 


.99977 


,08897 


.999241 


.05640 


.99641 


.07382 


.99727 46 


15 


.0(M36,. 99909' 


.02181 


.99976 


.08926 


.99923 


.06609 


.99639 


.07411 


.99725 45 


10 


.00465 .999991 


.02211 


.99976 


.03955 


.99922 


.06696 


.99688 


.07440 


.99733 44 


17 


.0M96 .90999' 


.02240 


.99975 


.08084 


.99921 


.05727 


.99686, 


.07469 


.99721148 


18 


.00604 .99999, 


.02269 


.99974 


.04013 


.999191 


.05756 


.998841 


.07498 


.99719 42 


19 


.00658 .999981 


.02296 


.99974 


.04042 


.99918 


.05786 


.99833; 


.07527 


.99716 


41 


20 


.00682 .99996; 


.02827 


.99973 


.04071 


.99917 


.06814 


.998811 


.07556 


.99714 


40 


21 


.00611 .99998* 


.08366 


.99972 


.04100 


.99916' 


.05H44 


.99629 


.07585 


.09712 


89 


22 


.00610 .990981 


.02385 


.99972 


.04129 


.99915, .05873 


.99627 


.07614 


.09710 


88 


28 


.00669. 999981 


.02414 .99971 


, .04159 


.09913 .06902 


.99826 


.<7643 


.99706 


87 


24 


.00698 .09998' 


.02443 


.99970 1.04186 


.99912, .06931 


.99824 


.07672 


.99705 


36 


26 


.00727' 99997 


.02472 


.9996911.04217 


.99911! .06960 


.996:^ 


.07701 


.99708 


85 


26 


.00756 .99997 


.02501 


.99969 


.04246 


.99910 .06989 


.99821 


.07780 


.09701 


84 


27 


.00785 .99997 


.02530 


.99968 


.04275 


.9990911.06018 


.99819 


.07759 


00699 


88 


28 


.008141.99997 


.02560 


.99967 


1.04304 


.99907; 1.06047 


.99817 


.07788 


.99696 


82 


29 


.00844 .99996 


.02589 


.99966 


.04883 


.99906,,. 06076 


.99815 


.07817 


.99694 


81 


ao 


.00673 .99996 


.02618 


.99966 


.04862 


.99905 !|. 06105 


.99813 


.07846 


.99602 


80 


81 


.00902 .99996 


.02647 


.99966 


.04891 


.099041,. 06184 


.99612 


.07875 


.09689 


29 


82 


.00981 .99996 


.02678 


.99964 


.04420 


.99903 .06163 


.90810 


.07904 


.99687 


28 


83 


.00960 .99995 


.02706 


.9996b 


.04449 


.999011!. 06192 


.99608 


.07938 


.99685 


27 


84 


.00969*. 99995 


.08784 


.99968 


.04478 


.99900 ' .06221 


.99806 


.07962 


.99683126 


86 


.01018 .99995 


.02763 


.99962 


.04507 


.99698; .06250 


.99804 


.07991 


.99680 25 


86 


.010471.99995 


.02792 


.99961 


.04586 


.99697 


.06279 


.99803 


.08Q!» 


.99678! 24 


87 


.01076 .99991 


.02821 


.99960 


.04565 


.99896 


.06908 


.99801 


.08049 


.99676! 23 


88 


.01105 .99994 


.02850 


.99959 


.04504 


.99894 


.06387 


.99799 


.08078 


.99673' 22 


80 


.01184 .99994 


.02879 


.99959 


.04823 


.9989311.06366 


.99797 


.08107 


.99671121 


40 


.01164 .99998 


.02906 


.99958 


.04658 


.99692, '06396 
.99890 '1.06424 


.99795 


.06186 


.99668 


20 


41 


.01196'. 99993 


.02988 


.99957 


.04682 


.99793 


.06165 


.99666 


19 


42 


.01^ .99998 


.02967 


.99956 .04711 


. 99889 1I.OO453 


.99792 


.08194 


.99664 


18 


48 


.01251 .99992 


.02996 


.999551 .04740 


.99888; .06482 


.99790 


.08223 


.99661 


17 


44 


.01280 .999921 


.08025 


.999541 .04769,. 99886 1.06511 


.99788 


.08252 


.99659 


16 


45 


.01809 .99991 > 


.03054 


.999531 .04796 


.99885 .06&40 


.99786 


.06281 


.99657 


15 


46 


.01388 .99991 1 


.09063 


.99952 ,'.04827 


.998831 .06569 


.99^ 


.08310 


.99654 


14 


47 


.01367. 99991 


.08112 


.999521!. 04866 


.09883 .06596 


.99782 


.08339 


.99652! 13 1 


48 


.01396 .99990 


.08141 


.99951 1 .04885 


.99881 1.06627 


.99780 


.06368 


.99649 


12 


49 


.01425 .999901 


.08170 


.99060, .04914 


.99879 1.06656 


.99778 


.08397 


.99647 


11 


50 


.014M|.99989 


.08199 


.999491,. 04948 


.99678 i .06686 


.99776 


.08426 


.99644 


10 


61 


.014881.99960 


.08828 


.09048 1.04972 


.09876 


.06714 


.99774' 


.06466 


.99642 


9 


02 


.016181.999691 


.09257 


.99947 1 1.05001 


.99675 


'. 06743 i.OOTTS 


.08484 '.99689 


8 


58 


.01542'. 99968 


.03286 


.99946 .05080 


.99673 


.067731.99770 


.06518!. 99687 


7 


54 


.015711.99966 


.08316 


.99945 ,.05050 


.99872 


.06802 .99768 


.Oa'>42,. 99635 


6 


66 


.01600 .90967 


.08845 


.99944. .06068 


.99670 


.06831 .99766 


.08571 


.99632 


6 


66 


.01629 .99967 


.06374 


.99943,1.06117 


.99669;!. 06860!. 99764 


.06600 


.99630 


4 


67 


.01668 .99966 


.06403 


.99042 , .05146 


.99867 1.068801.99762 
.99866". 06918'. 99760 


.08629 


.99627 


8 


68 


.01667 .99966 


.03482 


.99041 


1.06175 


.06658 


.99625 


2 


60 


.017161.99966 


.08461 


.99940 


.06206 


.09664! 1.06947 1.99758 


.08687 


.99622 


1 


60 

/ 


.01746 .99985 
Cosln fSlne 


.08490 
CkwiD 


.99989 
Sine 1 


.05284 
Cosln 


.99863 .06976 .99756 


.06716 


.99619 


_0 
/ 


Sine 


Coain|Sine | 


Cosin 


Sine 


88* 


81 


!• 1 


87- 1 


1 86- i 


>9- 



486 



TABLE IV.— NATURAL SINES AND COSINES. 



"o 


6- 


6o , 


70 , 


8» 


9* 


/ 
60 


Sine 
.08716 


Oodn 
.99619 


Sine 


Cosin 
.99452 


Sine 
.12187 


Cosin 
.9ft255| 


Sine 
:i3917 


Cosin 

.99027 


Sine 
, .15643 


Cosin 


.10453 


.96769 


1 


.08746 


.99617 


.10482 


.99449 


.12816 


.99251' 


.13946 


.99023 


1.15672 


.98764 


60 


2 


.08774 


.99614 


.10511 


.99446 


.122451. 99248, 


.13975 


.99019 


.15701 


.96760 


58 


8 


.08803 


.99612 


.10540 


.99443 


.12274 .99844 


.140W 


.99015 


1.15780 


.96755 


67 


4 


.06831 


.99609 


.10569 


.99440 


.12308 .99240' 


.14033 


.99011 


.15758 


.96751 


66 


6 


.08860 


.99607 


.10597 


.994371 


.12331 .998371 


.14061 


.99006 


1 .15787 


.98746. 56 1 


6 


.06889 


.99804 


.10626 


.99134' 


.12360 .992331 


.14090 


.99008 


.15816 


.98741 


54 


7 


.08918 


.99602 


.10655 


.99131; 


.123891.99230, 


.14119 


.98998 


1.15845 .96737 


53 


8 


.08947 


.99599 


.10684 


.994281 


.12418 


.992361 


.14148 


.98994 


.15878 


.98782 


58 


9 


.08976 


.99596 


.10713 


.99421' 


.12447 


.99823' 


.14177 


.96990 


.15002 


.98728 


61 


10 


.09006 


.99594 


.10742 


.99421 


.12476 


.992191 


.14205 


.96966 


.15981 




60 


11 


.09084 


.99691 


.10771 


.99418 


.12504 


.99215 


.14234 


.96962 


1.16959 


.98718 


49 


12 


.09063 


.99588 


.10800 


.99415 { 


.12533 


.99211 


.14263 


.98978 


.15968 


.96714 


48 


18 


.09092 


.99586 


.10829 


.99412 


.12562 


.99208: 


.14292 


.98973 


.16017 


.98709 


47 


14 


.09121 


.99583 


.10858 


.99409' 


.12591 


.99204 


.14320 


.96969 


'.16046 


.96704 


46 


15 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200' 


.14349 


.98965 


.16074 


.98700 


46 


16 


.09179 


.99578 


.10916 


.99408 


.12649 


.99197 


.14878 


.98961 


.16103 


.98605 


44 


17 


.09206 


.99573 


.10945 


.99399 


.12678 


.99193 


.14407 


.98957 


,.16132 


.98690 


48 


18 


.09237 


.995?2 


.10973 


.99396 


.12706 


.99189 


.14436 


.98953 


1.16160 


QfiUfiA 


48 


19 


.09266 


.99570 


.11008 


.99393 


.12735 


.99186 


.14464 


.98948 


I .16189 


.96681 


41 


20 


.09295 


.99667 


.11081 


.99390 


.12764 


.99183 


.14493 


.96944 


.16218 


.98676 


40 


21 


.09324 


.99564 


.11060 


.99386 


.12798 


.99178 


.14582 


.98940 


1.16946 


.96871 


80 


22 


.09353 


.99562 


.11089 


.99383 


.18822 


.99175 


.14651 


.98936 


,.16875 


.98667 


88 


28 


.09382 


.99559 


.11118 


.993801 


.12851 


.99171 1 


.14580 


.98931 


M6804 


.98662 


87 


24 


.09411 


.99556 


.11147 


.993771 


.12880 


.991671 


.14608 


.98927 


.16838 


.96657 


86 


25 


.09440 


.99553 


.11176 


99374 


18<¥V^ 


99163 


14687 


98983 


1 16361 


96652 


85 


28 


.09469 


.99551 


.11205 


.99370 


.12937 


!99160 


!l4666 


.'98919 


:!l6S90 


!96648 


84 


27 


.09498 


.99548 


.11231 


.99367 


.12966 


.99156' 


.14695 


.98914 


1 .16419 


.96643 


83 


28 


.09527 


.99545 


.11263 


.99364 


.12995 


.99158 


.14';^ 


.98910 


.16447 


.96688 


38 


29 


.09556 


.99M2 


.11291 


.99360 


.18024 


.99148 


.14758 


.98906 


.16476 


.98638 


81 


SO 


.09585 


.99540 


.11820 


.99337 


.13053 


.99144 


.14781 


.98902 


1.16505 


.98689 


80 


81 


.09614 


.99637 


.11849 


.99354 


.18081 


.99141 


.14810 


.98897 


.16688 


.98684 


89 


32 


.09642 


.99534 


.11878 


.993311 


.13110 


.99137 


.14838 


.98893 


1.16662 


.98619 


88 


88 


.09671 


.99531 


.11407 


.99347 


.13139 


.99133 


.14867 


.98889 


.16691 


.98614 


87 


84 


.09700 


.99528 


.11436 


.99344 


.13168 


.99129 


.14896 


.96884 


.16680 


.96609 


86 


85 


.09729 


.99526 


.11465 


.99341' 


.13197 


.99125 


.14925 


.98880 


.16648 


.98604 


85 


86 


.09758 


.99523 


.11494 


.99337, 


.18226 


.99128 


.14954 


.98876 


.16677 


.96600 


84 


87 


.09787 


.99520 


.11523 


.99334 1 


.13854 


.99118 


.14982 


.96871 


.16706 


.96595 


83 


88 


.09816 


99517 


.11552 


.993311 


.13883 


.991141 


.15011 


.98867 


.10784 


.96690 


88 


80 


.09845 


.99514 


.11580 


.993271 


.13312 


.991101 


.15040 


.98863 


.16768 


.98585 


21 


40 


.09874 


.99511 


.11609 


.99384 


.13311 


.99106, 


.15069 


.98868 


.16798 


.96680 


80 


41 


.00908 


.99508 


.11638 


.99380' 


.18370 


.99108 


.15097 


.98854 


.16820 


.98576 


19 


42 


.09932 


.99506 


.11667 


.993171 


.13399 


.990981 


.15126 


.98849 


.16849 


.96570 


18 


48 


.09961 


.99503 


.11696 


.99314' 


.13427 


.99094' 


.15155 


.98845 


.16878 


.96565 


17 


44 


.09990 


.99500 


.11725 


.99310 1 


.13456 


.99091, 


.15184 


.98841 


.16906 


.96661 


16 


46 


.10019 


.99497 


.11754 


•99307 


.13485 


.99087' 


.15218 


.98836 


.16935 


.98556 


16 


46 


.10048 


.99494 


.11783 


.993031 


.13514 


.99083 


.15241 


.98832 


.16964 


.96551 


14 


47 


.10077 


.99491 


.11812 


.99800 


.13543 


.99079 


.15270 


.98827 


.16992 


.96546 


18 


48 


.10106 


.99488 


.11840 


.992971 


.13572 


.99075 


.15299 


.98883 


.17021 


.98541 


18 


40 


.10135 


.99485 


.11869 


.992931 


.18600 


.99071 


.15327 


.96818 


.17060 


.98536 


11 


60 


.10164 


.99482 


.11898 


.99290' 


.13629 


.99067 


.15866 


.96814 


.17078 


.98531 


10 


61 


.10192 


.99479 


.11927 


.998861 


.18858 


.99063 


.15886 


.96809 


.17107 


.98686 


9 


62 


.10221 


.99476 


.11956 


.99883] 


.18687 


.99a59 


.15414 


.98805 


.17186 


.96621 


8 


68 


.10250 


.99473 


.11985 


.99279 


.13716 


.99055 


.15448,. 98800 


.17164 


.96616 


7 


64 


.10879 


.99470 


.18014 


.99276 


.13744 


.99051 


.15471 .98796 


.17193 


.98611 


6 


65 


.10808 


.99467 


.12043 


.99878! 


.13773 


.90017 


.15500 .98791 


.17282 


.98606 


5 


66 


.10337 


.99464 


.12071 


.992691 


.13802 


.99043 


.15529!. 98787 


.17250 


.96601 


4 


67 


10366 


.99461 


.12100 


.99885 


.18831 


.99039 


.155571.98782 


.17879 


.98496 


8 


68 


.10895 


.99158 


.12189 


.998621 


.13860 


.99035 


.15586 .98778 


.17806 .96491 


8 


69 


.10424 


.99455 


.12158 


.99258' 


.13889 


.99a31 


.15615 .98r?8 


.17386 


.96488 


1 


!2 
/ 


.10453 


.99152 
Sine 


.12187 
Cosin 


.99255 
Sine 


.13917 
Cosin 


.99027 
Sine 


.15643 .98769 
Cosin Sine 


.17865 


.98481 


_0 

t 


Cosin 


CoBln 


Sine 


84- 1 


88- 1 


82- 1 


81* 


80* 1 



487 



^' TABLE IV.— NATURAL SINES AND COSINES. ^ 



Sine ,Co8in 
[7865 .96481 
[7888 .96476 
[742S .96471 
17461 .96466 
17479 .96461 
17506 .96455 
17587 .96460 
17565 .96445 
17594 .96440 
[7623 .96485 
[76511.96480 



17660 
17T06 
17787 
17766 
17794 
I78S3 
17852 
17880 
17909 
17987 



16424 
18452 
18481 
18509 



18861 
18010 
18988 
18967 
18995 
19024 
19052 
19061 



Cosin 



98425 
96420 
96414 
96409 
98404 
96899 
98894 



98357 



96825 



II' 



96250 
96245 



,98218 I 



.20861 



.20635 
.20563 



Cofllnl 
.98163 
.98157 
.96152 
.96146 
.96140 
.98135 
.96129 
.96124 
.^118 
.96112 
.98107 

.98101 
.96096, 
.98090 
.960641 
.960791 
98073, 
96067' 
.980611 
96066| 
.96060 

96OI4I 
96089 
98033 
96027, 
960211 
.98016 
98010, 
960O4I 
97996, 
97992 

979671 
97961 1 
97975I 
97969 
97963 I 
97958 
97952 I 
97946 I 
97940 I 
97931 I 



120 



Sine 



.20791 



.20648 
.20877 
.20905 



.20962 
.20990 
.21019 
.21017 
.21076 

.21104 
.21188 
.21161 
.21189 
.21218 
.2t»16 
.21275 
.21808 
.21381 
.21860 

.21886 
.21417 
.21445 
.21474 
.81502 
.21580 
.21550 
.21587 
.21616 
.21644 

.21672 
.21701 
.21729 
.21758 
.21786 
.21814 
.21843 
.21871 
.21899 
.21928 

97928 '.21956 
97922,1.21965 
97916 ' .22013 
97910 .22041 
97905 '.22070 



978991 
97893; 
978871 
978811 
97875 



22096 
.22126 
.22155 
.221831 
.22212, 



OosinI 



Sine 



97869' .28240 
9786311.22268 
97857 1.22297 1 
97851:1.22325 
978451 .22353 
97839 '.22382 
97833 ,. 22110 : 
97827 1.22138 
97821 .22467; 
,97815 .22495, 



.97815, 
.97809 
.97808 
.97797 
.97791 
.977841 
.97778 
.97772 
.97766 
.97760, 
.97754' 

.97748' 
.97742 
.977861 
.97789 
.977831 
.977171 
.97ni| 
.97705 
.97696 
.97692 

.97688* 
.97680 

97673; 

976671 

97661: 
.97655, 
.97648' 

97612 1 
.97638 

97630 1 

97823| 
97617 
976111 
976011 
97596 
.97592 
97585 
.97579 
97573 
.97566; 

97S60 
97553! 
97547, 
975411 
97584' 
97528 
975211 
97515 1 
,975061 
,97602 1 

,97496 

.97489 

.97483 

.97476 

.97470 

.97468 

.97467 J 

.97450 

.97444 

.97437 



18« 



Sine 



.82496 



.82722 
.22760 
.22778 



.82807 



.22977 
.83005 



.88090 
.88118 
.83146 
.28175 



,88316 
,83845 

.88878 
83401 
,83429 
.83458 
.83486 
.23514 
,28542 
.23571 



.88687 
.88656 



.83712 
.23740 



.23797 



Cosin I Sine Cosin | Sine 
78* il 77» 



.28995 
.24023 
.24051 
.24079 
.24106 
.24136 
.84164 
.81102 
Cosin 



Cosin 



.97487 
.97480 
.97484 
.97417 
.97411 
.97404 
4)7896 
.97891 
.97364 
.97878 
.97871 

.97866 
.97868 
.97851 
.97846 
.97886 
.97881 
.97885 
.97818 
.97811 
.97804 

.97896 
.97891 
.97884 
.97878 
.97871 
.978U 
.97857 
.97851 
.97844 
.97887 

.97880 
.97!!^ 
.97217 
.97810 
.97208 
.97190 
.97189 
.971821 
.97176 
.97169 

.97162, 
.97156 
.97148 
.971411 
.97184, 
.97127 1 
.971201 
.971181 
.971081 
.97100 

.97093' 

.97066 

.97079 

.970781 

.97065 

.97058' 

.97061, 

.97044' 

.97087 1 

■97030 I 

Sine ! 



14* 



Sine Cosin 



.24198 



.24849 
.24877 
.84805 



.84390 
.84418 
.84446 
.84474 

.84608 
.84581 
.84560 
.84587 
.84616 
.84644 
.84672 
.84700 
.84786 
.84756 

.84784 



.84841 



.84897 



.84954 



.85010 
.85086 

.86066 
.85091 
.25122 
.25151 
.25179 
.26807 



.25380 



.85848 
.85876 
.8&104 



.85460 



.85616 
.85645 
.85573 
.85601 



.97030 
.97023 
.97016 
.97006 
.97001 
.96091 
.96967 
.96960 
.96978 
.96966 
.96960 



.96946 
.96987 
.96030 



.96016 



.96008 
.96894 



.96880 
.96878 
.96866 
,96858 
.96851 
,96644 
.96887 
,96829 



,96815 

,96807 
96800 
96793 
96786 
96778 
96771 
96764 
96756 
96749 
96742 

96734 
96787 
96719 
96712 
96705 
96697 
966901 18 
96662 18 
96675 11 
96667 10 



.85657 



.96660 
.96658 
.96645 

.85713 

.85741 

.85769 

.85796 .966151 
96606) 
966001 

25682 .96593 



966301 



.25864 
.25682 
Cosin 



Sine 



76- 



76» 



488 



TABLE IV.— NATURAL SINES AND COSINES. 



/ 

"o 


16" 


ie« 


17* 1 


18« 


1 19« 


/ 
60 


Sine ICosin 


Sine iCosin 


Sine 


Cosin 


Sine 


Cosin 
.95106 


1 Sine 
.82567 


Cosin 


.258821.96598, 


.275641.96128 


.29237 


.95630 :80902 


.04558 


1 


.25910 .965851, .27592 .96118' .29265 


.95622. .80929 


.95097 


.32584 


.94542 


60 


?. 


.25988|.96578|l .27820,. 96110|; .29293 


.96618; .80967 


.96088 


.82612 


.94583 


58 


8 


.25966.96570' .27&18'. 96102,1 .29821 


.96606! .80985 


.95079 


.a»89 


.94523 57 


4 


.260941.966621 


.27676 1.960941;. 29348 


.96696, .81012 


.96070 


.82667 


.94514 56 


5 


.26022;. 96555 


. 27704,. 960861 1.29376 


.95588; .81040 


.95061 


.82694 


.04504 


65 


6 


.26060 .96&17I 


.27781 .96078 1.29404 


.96579! 1.81068 


.960^2 


.38722 


.04495 


54 


7 


.28079 


.96540 


.27769|-96070 .2W32 


.955n .81096 


.96048 


.82749 


.04485 


58 


8 


.26107 


.96632 


.27787 .96062 1.29460 
.278161.96054 1.29487 


.95662 ; .81128 


.96088 


.82777 


.04476 


52 


9 


.26185 


.96624 


.96664 1 .81161 


.96024 


1.82804 


.04466 


51 


10 


.26168 


.96617 


.27848 .96046 1,. 29515 


.96646 , .81178 


.06016 


.82882 


.04457 


60 


11 


.26191 


.96609 


.27871 


.96087 .29648 


.96686 ' .81206 


.96006 


'.82860 


.04447 


40 


12 


.26219 .966021 


.27899 


.96029 .29671 


.9562811.31233 


.94997 


1.82887 


.04488 


48 


13 


.262471.96494, 


.27927 


.96021 1 .29599 


.95519, .81261 


.94968 


.82914 


.04488 


47 


14 


.26275 .96486' 


.27955 


.96013 .29626 


.95511 II. 81289. 94979 


.82942 


.04418 


46 


16 


.268081.96479! 


.27988; .960051 1 .29654; .96502 1 .31316, .94970 


1 .82060 


.04409 


45 


16 


.26331 


.96471 .280111.959971,. 29682I.9&493 .81344 .94961 


.82097 


.94899 


44 


17 


.26359 


.96463 


.28039!. 96989 .297101 .95486 ; .31372 


.94962 


.88024 


.94890 


48 


18 


.26887 


.96466) 


.28067 .95981 .29737 '.96476, .81899 


.94943 


.88051 


.94880 


48 


19 


.26415 


.96448 


.28096.95972 .297651.95467 .81427 


.94983 


! .88070 .94870, 41 


20 


.26443 


.96440 


.281281.96964 


.29793 .95459 .81464 


.949»4 


' .88106 .04861 


40 


21 


.26471 


.96488! 


.28160 1.96966 


1.29821 .96450!;. 81482 


.94915 


.88184 .04351 


80 


22 .26600 


.96425 


.28178. 96948 


.29849 .95441, .81510 


.94906 


.881611.04842! 88 


28 1.26528 


.964171 


.282061.95940 


1.29876 .95483'!. 81537 


.94897 


1 .88189. 94832 87 


24 


.26566 


.96410, 


.28234. 96031 


1.29904'. 95424 11.31565 


.94888 


.382161.94822 88 


25 


.26684 


.96402 


.28262 .9»928 


, .29932 .95415 .81503 


.94878 


1.38244,. 04818 85 


26 


.26612 


.96394, 


.282901.96915 


1.29960 .95407, .81620 


.94869 


.83271 


.94803 


84 


27 


.26640 


.96386, 


.28818 .96907 


.29987 .95898l|. 81648 


.94860 


.83208 


.04293 


38 


28 


.26668 


.963791 


.28346 .96808 


.800151.95389, .81675 


.94861 


.88826 


.94284 


82 


29 


.26696 


.96371 


.28374 .95890 


.800431.95380 .81708 


.94842 


.88858 


.04274 


81 


80 


.26721 


.96863 


.28402 .96682 


.80071 .96872 1 .81780 


.94882 


.88881 


.04264 


80 


81 


.26752 


.96865 


.28429 .96874 


.80098 .95863'i. 81758 


.04828 


.88408 


.04254 


20 


82 


.26780 


.963471 


.28457 .95865 


.80126 .95354,1.81786 


.94814 


.88486 


.04245,28 


38 


.26808 


.96340, 


.28185 .95857 


.801541.95345 !. 81813 


.94806 


;. 83468 


.04285 27 


84 


.26836 


.96382' 


.28513 .95*49 


.80182 


.953371 1.81841 


.947% 


.88490 .04225126 


85 


.26864 


.96S34 


.28541 .95841 


.80209 


.95828 I .81808 


.94786 


1.88518 .04215 25 


86 


.26892 


.96316 


.28569 .95882 


.80237 


.95319 ' .81896 


.94777 


.88645 .04206124 


87 


.26920 


.90308 


.28597 .95824 


.30265 


.95310 1 .81923 


.04788 


1.88573 .04106,28 


38 


.26948 


.96301 


.28625 '.95816 


!. 30292 


.95301 .31951 


.04758 


1. 88600 1.04180 122 


39 


.26976 


.96293 


.28652 .aVW 


1. 30320;. 95293 1.81979 


.94749 


.88627 '.04176 21 


40 


.27004 


.96285 


.28680 .95799 


.80848 .95284 1.82006 


.04740 


.88655 .94167:20 


41 


.27T)82 


.9627?' 


.28708 '.95791 


.80876 .95275 '1.82034 


.04780 


'.88682 .04157119 


42 


.27060 


.96269, 


.28736 .95;^ 


.30403 .95266 i .82061 


.04721 


1.88710 1.04147 18 


43 


.27088 


.96261 


.28764 .95774 


.30431 . 95257 1{. 32089 


.04n2 


'.88787,. 04137 
1 .8»r64 .04127 


17 


44 


.27116 


.962531 


.28792 .95766 


.30459 .95248 .82116 


.04702 


16 


45 


.27144 


.96246 


.28820 


.95757 


.80486 .95240 .82144 


.04693 


1.33702 '.941 18 


15 


46 


.2n72 


.962381 


.28847 


.96749 


.30614 


.952311 


.82171 


.046*4 


1.38819 .04108 


14 


47 


.27200 


.96280 


.28875 


.95740 


1.30542 


.95222 


.82199 


.04674 


.88846 .04098 


18 


48 


.27228 


.962221 


.28903' 96732 


1.80570 


.962131 


.82227 


.04665 


.88874 .94088 


12 


49 


.27256 


.96214 


.28931 .95724 


.80597 


.952041 


.82254 


.04656 


1 83901 .94078 


11 


50 


.27284 


.962061 


.28959 .95716 


.30625 


.96195, 


.82282 


.0464d 


.88929 .04068 


10 


51 


.27812 


.96198 


.289871.96707 


'.80658 


.961861 


.82809 


.04637 


.889561.94068 





52 


.27340 


.96190 


.29015'. 96698 .30680 


.95177 


.82387 


.04627 


' .38983 .04049 


8 


58 


.27368 .96182 


.29042,. 95690 


.30708 


.95168' 


.82364 .94618 


.84011 1.94089 


7 


54 


.27396'. 96174 


.290^/0 .0)681 


.30736 


.95159 


.82392 


.04609 


.84038 .94029 


6 


65 


.274241.96166 


.29098 .95673 


.30763:. 96150 1.32419 


.04699 


1 .84065 .04010 


5 


56 


.27452 .96158 


.29126 .95664 .307911.95142 ' .82447 


.94690 


.34093!. 04009 


4 


57 


.27480 1.96150 


.29154. 95656 


.80819 .95133 ; .82474 


.94680 


, .84120 .98999 


8 


58 


.27508. 96142 


.29182 .95647 


.80846 .96124 .82502 


.04571 


' .84147 .98969 


8 


69 


.27536 .96134 


.29209 .95639 


.30874 .95115 1 .82529 


.94561 


.84175 .08979 


1 


60 


.27564 {.96126 


.292:^ .95680 


.30902 95106 | .32557 


.94.V>2 


,.84202 .98969 


_0 


/ 


Cosin, Sine 


Cosin Sine 


Cosin I Sine Ck>8in 


Sine 


Cosin {Sine 


9 


740 


78- 


72» 71« 1 


70* 



489 



>>-. 



TABLE IV.— NATURAL SINES AND COSINES. 



20» 



Sine Cosiii 



.84208 
.84229 
.84257 



84811 
84839 



,84888 
.84421 
.84448 
.84475 

.84608 
.84680 



84667 
.84584 
.84612 
.84689 
.84666 
.84694 
.84721 
.84748 

.84775 
.84808 
.84880 
.84867 
.84884 
.84912 



.84966 
.84996 
.86021 



.85048 
.86075 
.85102 
.85180 
.85157 
.85184 
.85211 



.85847 
.86875 
.85402 
.85429 
85456 



.85484 
.86511 



.86565 



.85608 
.85619 
.85047 
.85674 
.85701 
.85728 
.86756 
85782 
.85810 



98969 
98949 
98989 



.98919 



98879 



.98849 



.98819 
.98809 
.98799 
.98789 
.98779 
.98769 

.98759 
.96748 
.98788 
.98728 
.98718 
.98708 



.986Tr 
.98667 

.98657 
.98M7 
.996371 



.98616 
.93606 
93596 



.98575 
.93565 

.98555 
.93544 
.93581 



.98514 
.98508 
.98493 



.93472 
.93462 

.98462 
.98441 
.98431 
.93420 
.98410 
.98400 



.98879 



Cosin I Sine 
69^ 



^Ine Coain __ 
.40674 .91355 60 
.40700 .9iai3 59 
.40727 1. 91831 58 
.40753,. 91319, 57 
.40780 .91307 156 
.406061.91296 56 
.40833 '.91283 54 
.40860 .91272 58 
.40686,. 91 260 52 
.409181.91248 51 
.40989 




.91236 50 

.91224)49 
.91212 48 
.91200 47 
.91188 46 
.91176, 45 
.91164 1 44 
.91152 48 
.91140 42 
.91128 41 
.911161 40 

.91104 39 
.91092 38 
.91080 87 
.91068 36 
.91056,36 
.91044 34 
.91082 38 
.91020, 82 
.91006' 81 
.9099^130 



.36948 .92924 
.86975 .929131 
.87002 '.92002 



.87029 


.92892 


.87066 


.92881 


.37088 


.92870 


.37110 


.92859 


.371371.928491 


.37164 


.92888 


.8n91 


.92827 


.87218 


.98816 


.87245 


.98805 


.87272 


.92794 


.87899 


.92784 


.87826 


.92778 


.87863 


.92782 


.87380 


.92751 


.87407 


.98740 


.87434 


.92729 


.37461 


.92718 


Cosin 


Sine 


61 


!• 



490 



TABLE rV.— NATURAL SINES AND COSINES. 





25* 1 


26<' 1 


27» 1 


28* 1 


29* 


/ 
60 


Sine 


Coslii 
.90631 


Sine 
.48887 


Cosin 
789879 


Sine 
.45399 


Codn 
.89101 


Sine 

.46947 


Codn 


Sine ICosIn 


.42262 


.88295 


.48481 


.87462 


1 


.42288 


90618 


.48863 


.89867 


.45425 


.89067 


.46973 


.8^81 


.48506 


.87448 


59 


2 


.42815 


90606 


43889 


.80664 


.46451 


.89074 


.46999 


.88267 


.48582 


.87484 


58 


8 


.42841 


.90594 


48916 


.89841 


.46477 


.89061 


.47024 


.88264 


.48567 


.87420 


57 


4 


.42387 


90582 


.43942 


.89888 


.45503 


.89048 


.47050 


.88240 


.48588 


.87406 


66 


6 


.42394 


.90569 


.48968 


.89816 


.45529 


.89035 


.47076 


.88226 


.48606 


.87891 


65 


6 


.42420 


.90557 


43994 


.89803 


.45554 


.89021 


.47101 


.88213 


.48684 


.87877 


64 


7 


.42446 


.90545 


.44020 


.89790 


.45580 


.89008 


.47127 


.88199 


.48659 


.87863 


58 


8 


.42478 


90532 


.44046 


.89777 


.45600 


.88995 


.47153 


.88185 


.48684 .87849 


62 


9 


.42499 


.90520 


.44072 


.89764 


.46632 


.88981 


.47178 


.88172 


.48710 


.87886 


61 


10 


.42626 


.90607 


.44098 


.89752 


.45658 


.88968 


.47904 


.88168 


.48786 


.87821 


60 


11 


.42652 


90495 


.44124 


.89789 


.45684 


.88965 


.47220 


.88144 


.48761 


.87806 


49 


12 


.42678 


90483 


,44151 


.89726 


.45710 


.88942 


.47255 


.88180 


.48786 


.87802 


48 


18 


.42604 


90470 


.44177 .89713 


.45786 


.88928 


.47281 


.88117 


.48811 


.87278 


47 


14 


.42681 


90458 


.44203 .89700 


.45762 


.88915 


.47306 


.88108 


.48887 


.87264 


46 


16 


.42667 


.90446 


.44229 


.89687 


.45787 


.88902 


.47332 


.88089 


.48862 


.87250 


45 


16 


42683 


.90483 


.44255 


.89674 


.46813 


.88888 


.47858 


.88075 


48888 


.87235 


44 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


.47883 


.88062 


.48918 


.87281 


48 


18 


.42786 


9040S 


.44307 


.89649 


.45865 


.88862 


.47409 


.88048 


.48988 


.87207 


42 


19 


.42762 


.90896 


.44333 


.89686 


.45891 


.88848 


.47434 


.88034 


.48964 


.87198 


41 


20 


.42788 


.90883 


.44859 


.89623 


.45917 


.88835 


.47460 


.88020 


.48060 


.87178 


40 


21 


.42816 


.90371 


.44885 


.89610 


.45942 


.88822 


.47488 


.88006 


.49014 


.87164 


89 


22 


.42841 


90858 


.44411 


.89597 


.45968 


.88808 


.47511 


.87998 


.40040 


.87150 


88 


28 


.42867 


90346 


.44437 


.89584 


.45994 


.88795 


.47537 


.87979 


.49066 


.8n86 


87 


24 


.42894 


90834 


.44464 


.89571 


.46020 


.88782 


.47562 


.87965 


.49090 


.87121 


96 


25 


.42920 


90321 


.44490 


.89558 


.46046 


.88768 


.47588 


.87961 


.49116 


.87107 


86 


26 


.42946 


.90809 


.44516 


.89545 


.46072 


.88755 


.47614 


.87937 


.49141 


.87098 


84 


27 


42972 


90296 


.44%42 


.89532 


.46097 


.88741 


.47039 


.87923 


.49166 


.87X)7» 


88 


28 


.42999 


.90284 


.44568 


.89519 


.46123 


.88728 


.47665 


.87909 


.49192 


.87064 


82 


20 


.48025 


.90271 


.44694 


.8950G 


.46149 


.88715 


.47090 


.87896 


.49217 


.87060 


81 


80 


.48061 


.90259 


.44620 


.89493 


.46175 


.88701 


.47716 


.87882 


.49242 


.87086 


30 


81 


.48077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


.40268 


.87081 


29 


82 


.48104 


.90233! 


.44672 


.89467 


.46226 


.88674' 


.47767 


.87854 


.40298 


.87007 


28 


88 


.48180 


90221; 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


.49318 


.80098 


27 


84 


.48156 


90208! 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


.40844 


.86078 


26 


86 


.48182 


90196, 


.44750 


.89428 


.46304 


.88634 


.47844 


.87812 


.49860 


.86064 


25 


86 


.48209 


.90183 


.44776 


.89415 


.46330 


.88620 


.47869 


.87798 


.49894 


.86948 


24 


87 


.48285 


.90171 


.44802 


.89402 


.46355 


.88607 


.47895 


.87784 


.49419 


.86086 


28 


88 


.43261 


: 90158 


.44828 


.89389 


.46381 


.88593 


.47920 i.srr/o 


.49446 


.86981 


22 


89 


.43287 


.90146 


.44854 


.89376 


.40407 '.885801 


.47W6 


.87756 


.49470 


.86906' 21 1 


40 


.48818 


.90133 


.41880 


.89363 


.46483 


.88566 


.47971 


.87743 


.48496 


.86802 


20 


41 


.48840 


.90120 


.44906 


.80350 


.46458 


.88553 


.47997 


.87TO9 


.40621 


.88878 


10 


42 


.48866 


.90108, 


.44932 


.89337 


.40484 


.88589 


.48022 '.87715 


.49546 


.86868 


18 


48 


.43892 


.900951 


.44958 


.89324 


.46510 


.88526 


.48048 .87701 


.49571 


.86840 


17 


44 


.48418 


90082 


.44984 


.89311 


I .46536 .885121 


.480731.87687 


.49596 


.86884 


16 


45 


.43445 


.900'.^ 


.45010 


.89298 


1 .46561 1 .88499 


.48099 .87678 


.49622 


.86820 


16 


46 


.48471 


.90057 


.45036 


.89285 


.465871.88485 


.48124 .87659 


.49647 


.86806 


14 


47 


.43497 


.90045 


.45062 


.89272 


1.406131.88472 


.48150 .87645 


.49678 


.86791 


18 


48 


.43528 


.90032 


.45088 


.89259 


.46689;. 88458 


.48175 .87681 


.49697 


.86777 


12 


40 


.43549 


.90019 


.45114 


.89245 


1.46664 


.88445 


.482011.87017 


.49788 


.86762 


11 


60 


.48575 


.90007 


.45140 


.80232 


'.46600 


.88481 


.48226 .87608 


.48748 


.86748 


10 


51 


.43602 


.89994 


.46166 


.89219 


.46716 


.88417 


.48252 .87589 


.49778 


.86788 


9 


6S 


.48628 




.45192 


.892061 .46742 


.8^404 


.48277 .87575 


.49798 


.86n9 


8 


68 


.48654 


!89968 


.45218 


.89193 


.46767 


.SK^IK)' 


.48308 .87561 


.40624 


.86704 


7 


64 


.43680 


.89956, 


.46243 


.89180 


,.46793 


.88:^77 


.48828 


.87546 


.49849 


.86600 


6 


66 


.48706 


.899431 


.45269 


.89167 


1.46819 


.88363 


.48354 


.87582" 


.49674 


.88676 


5 


56 


.48783 


.89930 


.45295 


.891.53 .46844 


.88*19 ; .48379 


.87518,:. 49809 


.86661 


4 


57 


.43759 


.89918 


.45321 


.89140, .46870 


.rnVVS .48405 


.87504 


.49924 


.86646 


8 


68 


.43785 


.89905 


.4&^7 


.89127 .46896 


,88:^22 .48430 


.87490 


.49950 


.86682 


2 


69 


.438111.89892 


.45373 


.89114 


, .46921 


.saws , .48456 


.87476 


.49976 


.86617 


1 


60 

9 


.438371.89879 


.45399 
Cosin 


.89101 
"Sine 


' .46947 
(Josin 


.88205 
Sine 


.484811.87462 
Cosin 1 Sine 


.50000 
Cosin 


.86608 
Sine 


_0 

/ 


Cosin 1 Sine 


64- 


e3» 


i 62- "1 


ei' 


1 0O<» 



491 





TABLE IV.— 


NATURAL SINES AND COSINSSl 




"o 


80* 1 


t 81«» 1 


V 82- 


88« 


84- 


i 
00 


Sine 

.50000 


CoBin 
.86608 


Sine 
:51'504 


Cosin 


Sine I Cosin 

.52992 .84805 


SinelCosin 


Sine 


Cosin 


.85717 


.54464 '.83867 


.56019 


.82904 


1 


.50025 


.86588 


.61529 


.85702 


.63017 .84789 


.64488 .88851 


.55948 


.82887; 6© 


2 


.50050 


.86578 


.61554 


.85687 


.68041 .84774 


.64518 .88835 


.55968 


.828n|68 


8 


.50076 


.86659 


.51579 


.85672 


.68066 .84759 


.64587 .83819 


.55992 


.a»65!67 


4 


.50101 


.86544 


.51604 


.85657 


.530911.84748 


.64661 .83804 


.56016 


.82889 66 


5 


.50126 


.86530 


.61628 


.85642 


.531151.84728 


.54586 .88788 


.56040 


.82822 


65 


6 


.50151 


.86515 


.61653 


.85627 


.68140 


.84712, 


.64610 .8877% 


.56064 


.82806 


64 


7 


.50178 


.86501 


.6167« 


.85612 


.68164 


.84697 


.54685 .83756 


.50088 


.82790 


68 


8 


.50201 


.8ft486 


.61703 


.86597 


.58189 


.846811 


.54659 .88740 i .56112 


.82778 


62 





.50227 


.86471 


.61728 


.a'5582 


.53214 


.84666 1. 54688. 83724; .56186 


.82757 


61 


10 


.60252 


.86467 


.61763 


.85667 


.68838 


.84660 


.647061.83708 .56160 


.82741 


60 


11 


.60277 


.86442 


.61778 


.85561 


.68263 


.84686* 


.54782!. 88092 1.66184 


.8S724 


49 


12 


.50802 


.86427 


.51803 '.855361 


.63288 


.84619 


.54756 .83676 1.50208 


.82706 


48 


18 


.50827 


.86413 


.518281.85521 1 


.53312 


.84604 


.64781 .83660 


.66282 


.82692 


47 


14 


.50852 


.86398 


.51853 


.85506 


.63337 


.84588 


.64805 .88645 


.66256 


.82875 


46 


15 


.50377 


.86384 


.61877 


.85491 


.53361 


.84573 


.64829 .83629 


.66280 


.82659 


45' 


16 


.50108 


.86:369 


, .61902 


.8&476 


.58386 


.84557 


.64854 .88613 


.66805 


.82648:44 


17 


.50428 


.86354 


.51927 


.85461 


.6.3411 .84542 


.54878 •.88597 


.66329 


.82626 48) 


18 


.50458 


.86810 


.51952 


.85446 


.68435 


.84526, 


.64902 .88581 


.56858 


.82610 42 


19 


.50478 


.86325 


.61977 


.85431 


.63460 


.84511 


.54927 .88565 


.66877 


.82503 41 


SO 


.50608 


.86310 


.62002 


.85416 


.5»m 


.844951 


.64961 


.88649 


.66401 


.82677,40 


21 


.60528 


.86295 


.52026 


.85401 


.68509 


.84480 


.64975 


.88688' 


.66425 


.82561189 


22 


.50553 


.86281 


.5aaji 


.85885 


.63584 


.84464 ! .64999 


.83517, 


.66449 


.82544 86 


28 


.50578 


.86266 


'52076 


.85370 


.5^58 


.84448 


.550241.88501 


.66473 


.82528; 87 


»t 


.50603 


.86251 


1 .52101 


.85355 


.53383 


.84433 


.55048 .88485 


.66497 


.82511 86 


25 


.50628 


.86237 


.52126 


.85840 


.53607 


.SU17 


.65072 .88469 


.66521 


.82495, 85 


26 


.50654 


.86222 


.62151 


.85325 


.53632 .84402 


.55097 .88453 


.66645 


.82478 84 


27 


.50679 


.86207 


.52175 


.83310 


.53666 .84886 


.65121 .88487 


.56569 


.82462 88 


28 


.50704 


.86192 


.52200 


.85294' 


.53681 


.84370 


.55145 .834211 


.66593 


.82446 82 


29 


.50729 


.86178 


.52225 


.85279 


.63705 


.ai;J55 


.55169 .8ai05' 


.66617 


.82429 81 


SO 


.60754 


.86168 


.62250 


.85264 


.63730 


.84889 


.65194 .88389 j 


.66641 


.88418, 80 


81 


.50779 


.86148 


.52275 


.85249 


.5875-4 


.84324 


.65218 .888781 


.66665 


.82896 


28 


82 


.506&1 


.86183 


.52299 


.85234 


.63779 


.84308 


.55242. 88856' 


.66689 


.82380 


28 


88 


.50829 


.86119 


.52324 


.85218 


.63804 


.84292 


.55286,. 88340 


.66713 


.82368 


27 


Si 


.50854 


.86104 


.52349 


.852031 


.53828 


.84277 


.55201 .83324 


.66736 


.82847 


26 


85 


.50879 


.86089 


.52374 


.85188' 


.53853 


.84261 


.558151.83308, 


.66760 


.82380 


25 


86 


.50904 


.86074 


.52399 


.851731 


.538771.84245 


.55389 .83292 


.66784 


.82814' 24 


87 


.50929 


.86059 


.52423 


.85157 


.5.3902,. 84230 


.5536:3 .83276 


.568081.82297 28 


88 


.50954 


.86045 


.52448 


.85142 


.53926 


.84214 


.55388 .88260, 


.66832 


.82281 22 


39 


.50979 


.80030 


.5247:} 


.85127 


.53951 


.ail98 


.554121.83244 


.66856 


.82264 


21 


40 


.61004 


.86015, 


.52498 


.85112 


.58975 


.84182, 


.55436 83228, |. 56880 


.82248 


20 


41 


.51029 


.860001 


.52522 


.86096 1 


.54000 


.84167 


.65460 1.88212' .66904 


.82281 


19 


42 


.510M 


.859851 


.52547 


.85081 1 


.M024 


.84151 


.554*4 '.83193 .66928 


.82214 


18 


48 


.510791.85970 


.525?2 


.85006: 


.54049 


.84185 


.55509,. 83179 j .66962 


.82198) 17 


44 


..')1104.. 85956, 


.52397 


.85031 


.54073 


.84120 


.5f>333'. 88163 1 1.56976 


.821811 16 


45 


.51129 


.85941 


.52621 


.85035 


.54097 


.84104 


.655571.88147 i .57000 


.82165 15 


46 


.51154 


.85926 


.52646 


.86020 


.54122 


.W088 


.555811.88181 1.67024 


.82148 14 


47 


.51179 


.85911 


.52671 


.8500) 


.54146 


.84072 


.65605 .88115 1 .57047 


.82182, 13 


48 


.51204 


.85896 


.52696 


.84989 


.54171 


.84057! 


.65680 


.83098 .67071 


.82115 12 


49 


.51229 .a'>881 • 


.52720 


.84974 


. 54195 ;.&4041' 


.65654 


.83082 .67095 


.820981 11 


60 


.61254 


.85866 


.52745 


.84959 1 


.54220 


.84025 


.65678 


.83066 .57119 


.62082^ 10 


61 


.61279 


.86861 1 


.62770 


.849481 


.64244 


.8i009 


.56702 


.88050 .67143 


.82066 


9 


62 


.51304 


.85836 


.52794 


.84928 


.64269 


.8.3994 


. 55726 


.88084 ; .67167 


.82048 


8 


63 


.51829 


.85821 


.62819 


.&1913 


.54293 


.88978 


.66750 


.83017; 


.6n91 


.82032 


7 


64 


.51854 


.85806 


.52844 


.848{>7 


.54317 


.83962 


.56775 


.830011 


.67215 


.82015 


6 


65 


.61879 


.85792 


.52869 


.WH82 


.54.S12 


.&3946 


.66799 


.82985 


.67238 


.81999 6 


66 


.61404 


.85777 


.52893 


.^4^^ 


.54.366 


.83930 


.55823 


.82969' 


.57262 


.81962 4 


67 


.51429 


.86762 


.62918 


.ai851 


.54391 


.83916, 


.65847 


.82953 


.67286 


.81965 8 


68 


.51464 


.85747 


.62948 


.ai836 


.544 15. 88899! 


.55871 


.829361 


.67310 


.81949 2 


69 


.61479 


.85792 


.62967 


.84820 


.54440 .83883! 


.5^895 


.82920!;. 67834 


.81932 1 


60 


.51504 .85717 


.62992 


.84805 


.544W .838071 


.55919 


.82904 .67358 


.81915; I 


/ 


Cosln Sine 


Cosin 


Sine , 


Cosin rsine" 


Cosin 


Sine l| Cosin 


Sine 


^ , 


69« 


58- 1 


67« 


56' II SS' 


_ 



493 



TABLE IV.— NATURAL SINES AND COSINES. 



"o 


860 I 


86* 


ar* 1 


88* 1 


89- 


/ 
60 


Sine 


Ooain 

.81915 


Sine Cosin 

.587791.80902 


Sine 
.60188 


Cosin 

.79864 


Sine 
.61566 


Cotdnl 

:'re80i 


Sine 


Cosin 


.57S58 


.62982 


.77715 


1 .67881 


.81899 


.58808 


.806851 


.60805 


.79646 


.61589 


.78783 


.62965 


.77696 


59 


2 


.57405 


.81882 


.58886 


.80867' 


.60228 


.79829 


.61618 


.78766 


.62977 


.77678 


58 


8 


.57429 


.81865 


.68S49 


.80650 


.60251 


.79811 


.61685 


.78747 


.63000 


.77660 


57 


4 


.57458 


.81848 


.58878 


.808881 


.60874 


.79798 


.61658 


.787^ 


.63022 


.77641 


66 


6 


.57477 


.81832 


.58896 


.808161 


.60296 


.79';76 


.61681 


.78711. 


.68045 


.77828 


55 


6 


.57501 


.81815 


.58920 


.807991 


.60821 


.79758 


.61704 


.78694 


.68068 


.77606 


54 


7 


.57524 


.81798 


.58948 


.807881 


.60344 


.79741 


.61726 


.786761 


.68090 


77586 


58 


8 


.67548 


.81788 


.58967 


.80765 


.60367 


.79728 


.61749 


.78658 


.68118 


.77568 


62 





.67572 


.81766 


.58990 


.80748 


.60890 


.79706 


.61772 


.78640 


.68185 


.77860 


51 


10 


.57596 


.81748 


.59014 


.80780 


.60414 


.79688 


.61796 


.78622 


.68168 


.77681 


60< 


11 


.57619 


.81781 


.59037 


.80718 


.60487 


.79671 


.61818 


.78604 


.68180 


.77518 


49 


IS 


.57648 


.81714 


.59061 


.80696 


.60460 


.79658 


.61841 


.78686 


.68806 


?74fM 


48 


18 


.57667 


.81698 


.69064 


.80679 


.60488 


.79685 


.01864 


.78668 


.68225 .T747« 


47 


14 


.57891 


.81681 


.59108 


.80668 


.60506 


.79618 


.61887 


.78660 


.68248 .77458 


46 


15 


.57n5 


.81664 


.50181 


.80644 


.60529 


.79600 


.61909 


.78582 


.68871,. T7489 


46 


16 


.57788 


.81647 


.59154 


.80687 


.60658 


.79588 


.61982 


.78514 


.68293 !.774S1 


44 


17 


.67762 


.81681 


.59178 


.80610 


.60676 


.79565 


.61955 .7B496 


.68816,. 77402 


48 


18 


.57788 


.81614 


.59801 


.80503' 


.60599 


.79547 


.61»re 


.78478 


.68888 1.77884 


42 


19 


.57810 


.81597 


.59885 


.805761 


.60622 


.79580 


.62001 


.78460 


.68361 .77866 


41 


20 


.57888 


.81680 


.69248 


.80558 


.60645 


.79512 


.62024 


.78442 


.68888 .77847 


40 


21 


.57867 


.81668 


.69872 


.80541 


.60668 


.79494 


.62046 


.78424 


.68406 .77889 


89 


22 


57881 


81546 


69295 


80524 


60091 


79477 


62069 


TRJnK 


68428 77810 


88 


28 


.57904 


.'81580 


!59818 


!80507 


!60714 


.'79450 


.62092 .78887 


.68451 .77282 


87 


24 


.57988 


.81518 


.59842 


.80489 


.60788 


.79441 


.62115 .78869 


.68473 .77278 


86 


25 


.57952 


.81496 


.59365 


.80472 


.60761 


.79424 


.62188 


.78851, 


.68496 .77866 


85 


26 


.57976 


.81479 


.59389 


.80455 


.60784 


.79406 


.62160 


.78838, 


.68618. T7886 


84 


27 


.57999 


.81468 


.59418 


.80488 


.60607 


.79888 .62188 


.783151 


.68540 .77218 


88 


28 


.58028 


.81445 




.80480 


.60830 


.79371 


.62806 


.78297 


.68668 


.77199 


82 


20 


.58047 


.81488 


.59459. 804031 


60653 


.79358 


.62829 


.78879 


.68585 


.77181 


81 


SO 


.58070 


.81412 


.59482,. 80386; 


.60676 


.79885 


.62251 


.78861 


.68606 


.77168 


80 


81 


.58094 


.81395 


.59506 


.80368 


.60899 


.79318 


.62274 


.78248 


.68680 


.77144 


29 


82 


.58118 


.81878 


.59529 


.80351, 


.60928 


.79300 


.62297 


.78225 


.68858 


.7n26 


28 


88 .58141 


.81361 


.59552 


.803341 


.60M5 


.79888 


.62820 


.78206 


.68676 


.77107 


27 


84 


58165 


81844 


59570 


80316 


60968 


79264 


.68842 


78188 


68698 


.77068 


26 


85 


!58189 


.81887 


.59599 


!80899; 


.60991 


! 79247 


62865 


i-ffilTO 


!68720 


.77070 


25 


86 


.58212 


.81810 


.59622 


.80882: 


.61015 


.79229 


.68888 


.78152 


.68742 


.77051 


24 


87 


.56286 


.81298 


.59646 


.808641 


.61088 


.79211 


.62411 


.78184 


.68765 


.75X188 


28 


88 


.58260 


.81276 


.59669 


.802471 


.61061 


.79198 


.62488 


.78116 


.68787 


.77014 


28 


89 


.58888 


.81250 


.59693 


.808801 


.61084 


.79176 


.62456 


.78098 


.63810 


.76996 


21 


40 


.58807 


.81242 


.59716 


.80812 


.61107 


.79158 


.62479 


.V80'<»| 


.68882 


.78977 


80 


41 


.58380 


.81225 


.59789 


.80195 


.61180 


.79140 


.62602 


.TWWll 


.68854 


.76960 


19 


42 


.68354]. 812081 


.59763 


.80178; 


.61153 


.79128 


.62524 


.78048 


.68877 


.76940 


18 


48 


.58878 .81191' 


.59786 .80160: 


.61176 


.79105 


.62547 


.78025 


.68899 


.76921 


17 


44 


.58401 .81174 


.59809 


.80148' 


.61199 


.79087 


.62570 


.78007 


.63982 


.76908 


16 


45 


.58425 '.'81157 


.69632 


.801251 


.61228 


.79069 


.62598 


.77988 


.639441.76884 


15 


46 


.58449,. 81140 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970 


.68966 


.76866 


14 


47 


.58472 


.81128 


.59879 


.80091 


.612681.79088 


.62688 


.77952 


.68989 


.76W7 


13 


48 


.58496 


.81106, 


.59902 


.80078 


.61291 


.79016 


.68660 


.77984 


.64011 


.76888 


12 


49 


.58519 


.81089 


.59986 


.80066 


.61814 


.78998 


.68688 


.77916 


.64083 


.76810 


11 


60 


.58548 


.81078 


.59949 


.80038 


.61887 


.78980 


.62706 


.77897 


.64056 


.76791 


10 


61 


.58667 


.81065 


.59972 


.80021 


.61360 


.78962 


.68788 


.77879 


.64078 


.76772 


9 


52 


.58590 


.81038, 


.59995 


.80003 


.61883 .789441 


.68751 


.77861 


.64100 


.7B7H 


8 


58 


.58614 


.81081 


.60019 


.79986 


.61406 


.78986 


.62774 


.77843 


.64128 


.78736 


7 


54 


.58687 


.81004, 


.60048!. 79968 


.61429 


.78008 


.62796 


.77884; 


.64145 


.76717 


6 


66 


.58661 


.80987 


.60065 


.79951! 


.61461 


.78891 


.68819 


.77806- 


.64167 .76696 


5 


56 


.58684 


.80970 ; .60089 


.79934 


.61474 


.78878 


.68842 


.77788 


.641901.76679 


4 


67 


.58708 


.80953 


.60118 


.79916 


.61497 


.78865 


.62864 


.77769, 


.64212 .76661 


8 


68 


.58781 


.809381 


.60135 


.79899 


.61680 


.78887 1.68887 


.77751 


.64284 .78642 


8 


59,. 58786 


.80919 


.60158 


.79881 


.61548 


.78819 


.62909 


.Tr?83, 


.64256 .76688 


1 


60 .58779 


.80908 


.60188 


.79864 


.61566 


.78801 


.62938 


.77715 


.64279 .76604 


J9 


/ 


Cofiin 


Sine 


Ck>ain 


Sine 


Cosin 


Sine 


Cosin 


"StaVi 


Cosin, Sine 






54- 1 


6J 


J» 


52- 1 


61- 1 


W i 



493 



TABLE IV.— NATURAL SINES AND COSINES. 





40* 




Sine 


Codn 





Mm 


.76604 


1 


.64801 


.70586 


8 


.6438S 


.76567 


8 


.648M 


.70548 


4 


.64368 


.76530 


6 


.64390 


.76511 


6 


.6441S 


.76493 


7 


.64485 


.75473 


8 


.64457 


.76455 


9 


.61479 


.76436 


10 


.64501 


.78417 


11 


.64584 


.76896 


12 


.64546 


.78880 


18 


.64568 


.76361 


U 


.64590 


76343 


15 


.64618 


.76388 


16 


.64685 


.78804 


17 


.64057 


.78886 


18 


.64679 


.76867 


19 


.64701 


.76848 


SO 


.64788 


.70889 


81 


.64746 


.70810 


88 


.64788 


.78198 


88 


.64790 


.76178 


84 


.64818 


.78154 


85 


.64884 


.76185 


86 


.64866 


.76116 


87 


.64878 


.780971 


88 


.64901 


.78078 


89 


.64983 .760691 


80 


.64945 


.76041 


31 


.64967 


.76083 


88 


.64969 


.70003 


88 


.6S011 


.75984 


84 


.65088 


.75065 


85 


.65065 


.759461 


86 


.66077 


.75087 


87 


.65100 


.75906 


88 


.65183 


.75889 


89 


.65144 


.75870 


40 


.65166 


.75851 


41 


.66188 


.75888' 


48 


.65310 


.75813 


48 


.65883 


.75794 


44 


.651^ 


.75775 


46 


.658r6;. 75756! 


46 


.66806 


.757381 


47 


.65880 


.76719 


48 


.66848 


.75700 


49 


.66864 


.75680 


50 


.65886 


.75661 


51 


.06408 


.75648 


68 


.65480 


.76083 


58 


.65458 


.75604 


54 


.66474 


.75585 


55 


.65496 .75566 


66 


.66518 .75547 


57 


.66540 


.75528 


m 




.75509 


69 


tMKHA 

.00004 


.75490 


60 


.66606 


.75471 


/ 


Oosin 


Sine 




41 


^ 



41» 



Sine 



.66606 



.66050 
.05678 
.05004 
.05710 
.06738 
.65759 
.65781 
,65808 



.65847 



.65913 



.66956 
.66078 
.60000 



,00044 
66000 



.00068 



.00109 
,60181 
,00153 
60175 
86197 
60818 



00384 
00300 
00887 
00340 
00371 



06414 
66436 
66158 
.66480 

66501 



6651o 
66566 



66610 



Cosin 
.7W71 
.75152 
.75133 
.75414 
.75395 
.75375 
.75356 
.75387 
75318 
.75299 
.75380 

.75861 
.75841 



66653 
.66675 
.66697 

.66718 
.66740 
.60708 
.00783 
.06806 



75803 
.75184 
75106 
.75146 
.75186 
,75107 
7B088 

75069 
.75050 
75090 
75011 
.74932 
•4978 
.749531 
74984 
749151 
74800 

74878 
748571 
74888 
748181 
74799 
74780 
74760 
74741 i 
74722 
747031 

746831 
74664 
74644' 
74625 
74006 
74586 
74567 
745481 
745281 
74509 1 

74489 
74470 
,744511 
.74431' 
.74412 



.66870 



.66018 
Cbsin 



.74373 
.74358 
.74884 
.74314 
Sine 



48» 



42« 



Sine 



,66918 



.66956 



.67021 
.67048 
.67064 
.67086 
.67107 
.67189 

.07151 
.67178 
.67194 
.67315 
.07887 
.07858 
.07880 
.07301 
.07823 
.07844 

.67866 
.67887 
.67409 
.67490 
.67458 
.67473 
.67495 
.67516 
.67538 
.67559 

.67580 
.67608 



Cosin 



.74314 
.74295 
.74276 
.74256 



.67615 
,67866 



,67709 
,67730 
67752 
,67773 

,67795 
.67816 
,67837 
.67859 
.67880 
.67901 



67944 
,67965 
67987 

.08006 



.74217 
.74198 
.74178 
.74159 
.74139 
.74180 

.74100 
.74080 
.74061 
.74041 
.74082 
.74008 



.73963 
.73044 
.73084 

.78904 



.73865 
.78816 



.78806 
.78787 
.73767 
.73747 
78788 

73706 
.78688 

78669 

78649 
.78629 

78610 
.73590 

73670 
.73551 

78531 

78511 
73101 
73172 
73462 
73432 
73413 
73398 



73373 
78853 
73333 

,78814 
78894 
,73274 
.73254 



Cosinj 



.73815 
.73195 
.73175 
.73155 
,78135 
Sine 



47* 



48* 



Sine 



.68864 



.68306 



.68370 
.08891 
.08418 

.08484 
.08455 
.08476 
.68497 
.68518 



.68561 



.68645 



.68709 
.68730 
.68751 
.68773 
.68793 
.68814 



.68857 
.68878 



.68920 
.68941 



,69001 



,69016 

,69067 
,69088 
,69109 
,69130 
,69151 
,69172 



,69214 



.69377 



,69340 
.09361 



.69103 
.69424 
.69445 
.69166 
Cosin 



Cosin 



.73135 
.73116 
.73090 
.78070 
.73050 
.73030 
.73016 
.72996 
.72976 
.72957 
.78987 

.72917 
.72897 
.72877 
.78857 
.72837 
.72817 
.72797 
.72Trr 
.72757 
.72787 

.78717 
.72697 
.786771 
. 78667 ■ 
.78637, 
.786171 
.72597 
.78577, 
.72557 
.78537 

.78517 
.78497 

.784rr' 

.724571 
.78437 
.7TM17 
.72397 
.72377 
.72357 
.78387 

.78817 



72277 
72257 



78210 
72196 
78176, 
72156 
78136 

72116 
72095 
72075 
73056 
72035 
72015 
71995 
,71974 
.71964 
.71934 



440 



Sine 



46» 



Sine 



.69487 
.69508 
.69529 
.60549 
.69570 
.69591 
.69618 
.69688 
.69654 
.09075 

.09090 
.09717 
.09787 
.09758 
.09779 
.09800 
09821 



C06in{ _ 
.71934,00 
.71914 69 
.71894 58 
.71873 57 
.71853 50 
.71838 55 
.71813 
.71792 
.71772 
.71752 
.71788 

.71711 
.71091 
.71071 
.71050 
.71080 
.71610 
.71590 



.09904 
.09925 
.00946 



.69642 .71569 
.69862 .71549 
.71589 

.71508 
.71488 
.71468 
.71447 
.71427 
.71407 
.71886 
.71366 
.71345 
.71885 



.69987 
,70008 



.70049 
.70070 
.70001 

.70118 
70132 
.70153 
.70174 
.70195 
.70215 
.70236 
.70257 
.70277 
.70298 

.70319 
.70339 
.70360 
.70381 
.70101 
.70422 
.70143 
.70468 
.7W^ 
.70505 

.70525 
.70546 
.70567 
.70687 
.70608 
.70828 
.70649 
.70670 
.70690 
.70711 



.71805 
.71284 
.71264 
.71243 



.71203 
.711821 28 
.71162 28 
71141 81 
71181 80 

.71100 19 
71080 18 
.71059 17 
.71089 
.71019 
.70998 
.70978 
.7095: 
.70937 
.70916 



70896 
,70875 
,70855 
,70634 
,70813 
,70793 



.70772 
.70752 
.70731 
.70711 



Cosin I Sine 
46^* 



494 



TABLE v.— NATURAL TANGENTS AND COTANGENTS. 





O* 1 


1- 1 


1 2« 1 


8<» 


/ 

60 


Tang 
.00000 


Cotangr 


Tang 
.01746 


Ck>tang 


Tang 


Cotang 


Tang 
.05241 


Cotang 


Infinite. 


67.2900 


.03492 


28.6868 


19.0811 


1 




3487.75 


.01776 


66.8506 


.08521 


28.8994 


.06270 


18.9755 60 


2 


.00058 


1718.87 


.01804 


66.4416 


.08560 


28.1664 


.06299 


18.8711 68 


8 


.00087 


1145.92 


.01888 


64.5618 


.08579 


27.9872 


.05828 


18.7678 67 


4 


.00116 


859.486 


.01862 


68.7066 


.08609 


27.7117 


.05857 


18.6656 66 


5 


.00146 


687.649 


.01891 


62.8821 


.06688 


27.4899 


.06387 


18.5646 I55 


6 


.00175 


672.957 


.01920 


62.0807 


.08667 


27.2715 


.05416 


18.4645 ,64 


7 


.00204 


491.106 


.01949 


61.8032 


.03696 


27.0566 


.06445 


18.8655 


68 


8 




429.718 


.01978 


60.5485 


.03725 


26.8450 


.05474 


18.2677 


52 


9 


.00262 


381.971 


.03007 


49.8167 


.03754 


26.6367 


.05508 


18.1708 


61 


10 


.00291 


848.774 


.02036 


49.1039 


.08788 


26.4316 


.06588 


18.0760 


60 


11 


.00820 


812.521 


.02066 


48.4121 


.08812 


26.2296 


.05662 


17.9602 


49 


12 


.00349 


286.478 


.02095 


47.7396 


.03842 


26.0307 


.05591 


17.8888 


48 


13 


.00878 


264.441 


.02124 


47.0853 


.03871 


25.8348 


.05620 


17.7934 


47 


14 


.00407 


245.552 


.02158 


46.4489 


.03900 


25.6418 


.OoG49 


17.7016 


46 


15 


.00436 


829.182 


.02182 


46.8294 


.03929 


25.4517 


.05678 


17.6106 


45 


16 


.00465 


214.838 


.022U 


45.2261 


.03958 


25.2644 


.05708 


17.5805 


44 


17 


.00495 


2U2.219 


.02240 


44.6386 


.03987 


25.0798 


.05787 


17.4814 


48 


18 


.00321 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.05766 


17.8482 


42 


19 


.0a">.53 


180.932 


.02298 


43.5081 


.04046 


24.n85 


.05796 


17.2558 


41 


20 


.00682 


171.886 


.02328 


42.9641 


.04075 


24.6418 


.06824 


17.1008 


40 


21 


.00611 


168.700 


.02857 


42.4835 


.04104 


24.8675 


.05854 


17.0887 


89 


22 


.00640 


156.258 


.02386 


41.9158 


.04138 


24.1957 


.05888 


16.9990 '881 


28 


.00669 


149.466 


.02416 


41.4106 


.04163 


24.0268 


.05912 


16.9150 


87 


24 


.00098 


148.287 


.02144 


40.9174 


.04191 


23.8598 


.05941 


16.8310 


86 


25 


.00727 


187.607 


.02478 


40.4358 


.04220 


23.6946 


.06970 


16.7496 


85 


26 


.00756 


182.219 


.02508 


89.9655 


.04250 


23.6321 


.06999 


16.6681 


84 


27 


.00785 


127.821 


.02531 


89.5059 


.04279 


28.8n8 


.06029 


16.5874 


88 


28 


.00816 


122.774 


.02560 


89.0568 


.04308 


28.2187 


.06058 


16.5075 


82 


29 


.00844 


118.540 


.02589 


88.6177 


.04387 


28.0577 


.00087 


16.4288 


31 


80 


.00878 


114.589 


.02619 


88.1885 


.04366 


22.9038 


.06116 


16.8490 


80 


81 


.00902 


110.892 


.02648 


87.7686 


.04896 


22.7519 


.06145 


16.2722 ^29 


82 


.00931 


107.426 


.02677 


87.8579 


.04424 


22 6020 


.06175 


16.1952 {S8 


88 


.00960 


104.171 


.02706 


86.9560 


.04454 


22.4541 


.06204 


16.1190 27 


84 


.00989 


101.107 


.02735 


86.5627 


.04483 


22.3081 


.06238 


16.0485 


28 


85 


.01018 


98.2179 


.02764 


86.1776 


.04512 


22.1640 


.06262 


15.9687 


85 


86 


.01047 


95.4895 


.02798 


85.8006 


.04541 


22.0217 


.06291 


15.8946 


24 


87 


.01076 


92.9085 


.02822 


85.4818 


.04570 


21.8813 


.06821 


15.8211 


23 


88 


.01105 


90.4688 


.02851 


85.0695 


.04599 


21.7426 


.06350 


15.7488 


22 


89 


.01185 


88.1436 


.02881 


84.7151 


.04628 


21.6056 


.06379 


15.6782 


21 


40 


.01164 


85.9898 


.02910 


84.8678 


.04658 


21.4704 


.06406 


15.6048 20 


41 


.01198 


83.8435 


.02989 


84.0273 


.04687 


21.8869 


.06487 


15.6840 19 


42 


.01222 


81.8470 


.02968 


88.6935 


.04716 


21.2049 


.06467 


15.4688 18 


43 


.01251 


79.9434 


.02997 


83.3662 


.04746 


21.0747 


.0^96 


16.3948 17 


44 


.01280 


78.1268 


.03026 


88.0452 


.04774 


20.9460 


.06525 


15.8254 16 


45 


.01309 


76.3900 


.03055 


32.7303 


.04803 


.20.8188 


.06554 


15.2671 15 


46 


.01388 


74.7292 


.08084 


32.4218 


.04838 


20.6982 


.06584 


15.1893 14 


47 


.01867 


78.1390 


.08114 


32.1181 


.04862 


20.6691 


.06618 


15.1222 


13 


48 


.01396 


71.6151 


.03148 


31.8206 


.04891 


20.4465 


.06642 


15.0557 


12 


49 


.01426 


70.1583 


.03172 


81.5284 


.04920 


20.3253 


.06671 


14.9898 


11 


50 


.01465 


68.7501 


.08301 


81.2416 


.04949 


20.2066 


.06700 


14.9244 


10 


61 


.01484 


67.4019 


.03280 


80.9609 


.04978 


20.0872 


.06780 


14.8696 





62 


.01518 


66.1055 


.03259 


80.6838 


.0o<T07 


19.9708 


.06759 


14.7954 


8 


53 


.01542 


64.8580 


.03288 


80.4116 


.05037 


19.8546 


.06788 


14.7317 


7 


54 


.01571 


63.6567 


.03317 


80.1446 


.06066 


19.7403 


.06817 


146685 


6 


65 


.01600 


62.4992 


.03346 


29.8828 


.06095 


19.62'/8 


.06847 


14.6059 


6 


66 


.01629 


61.3829 


.08376 


29.6246 


.06124 


19.5156 


.06876 


14.5438 


4 


57 


.01638 


60.3a'58 


.03406 


29.3711 


.05153 


19.4051 


.06905 


14.4823 


8 


58 


.01687 


59.2659 


.08434 


29.1220 


.06182 


19.2969 


.96934 


14.4212 


2 


59 


.01716 


68.2612 


.03463 


28.8771 


.05212 


19.1879 


.06968 


14.3607 


1 


60 

/ 


.01746 


57.2900 


.03492 
Cotang 


28.6:363 


.06241 
Cotang 


19.0811 


.06993 
Ctotang 


14 8007 


_0 


Cotang 


Tang 


Tang 1 


Tang 


Tang 


\ S9- 


8«» 1 


87- 1 


1 86« 1 



495 



TABLE V.~NATURAL TANGENTS AND COTANGENTS. 



/ 
"0 


4* 


6'> 1 


1 e» 1 


1 7« 


/ 

60 


Tang 


Cotang ' 


Tang 
.06749 


Cotang 


Tang 1 Cotang 1 


Tang 


Cotang 


.06998 


14.3007 1 


11.4801 


.10610 


9.51488 


.12278 


8.14485 


1 


.07028 


14.8411 


.08778 


11.8919 


.10540 


9.48781 


.12308 


8.18481 


59 


2 


.07051 


14.1881 


.08807 


11.8540 


.10569 


9.46141 


.12338 


8.1068G 


58 


8 


.07080 


14.1835 


.06837 


11.8168 


.10599 


9.48515 


.12367 


8.06600 


57 


4 


.07110 


14.0655 


.08866 


11.2789 


.10688 


9.40904 


.12397 


8.08674 


66 


5 


.07189 


14.0079 


.08895 


11.2417 


.10057 


9.38807 


.12426 


8.04756 


55 


6 


.07168 


13.9607 


.06925 


11.2048 


.10087 


9.85784 


.12456 


8.02848 


54 


7 


.07197 


13.8940 


.08954 


11.1681 


.10716 


9.83155 


.12485 


8.00948 


68 


8 


.07227 


18.8378 


.08988 


11.1316 


.10746 


9.80599 


.12515 


7.99068 


52 


9 


.07256 


13.7821 


.09013 


11.0954 


.10775 


9.28058 


.12544 


7.97176 


51 


10 


.07286 


18.7887 


.09042 


11.0594 


.10606 


9.1^580 


.12574 


7.96808 


50 


11 


.07814 


13.6719 


.09071 


11.0237 


.10834 


9.28016 


.18808 


7.96488 


49 


IS 


.07844 


13.6174 


.00101 


10.9682 


.106G3 


9.20516 


.12638 


7.91588 


48 


18 


.07878 


13.5634 


.09130 


10.9529 1 


.10693 


9.18028 


.12668 


7.89784 


47 


14 


.07402 


18.5098 


> .09159 


10.9178 


.10922 


9.15554 


.18098 


7.87895 


46 


15 


.07431 


18.4566 


.09189 


10.8829 


.10952 


9.18098 


.12722 


7.86064 


46 


16 


.07461 


18.4089 


1 .09218 


10.8483 


.10961 


9.10&46 


.12751 


7.84242 


44 


17 


.07490 


13.8515 


1 .09847 


10.8189 


.11011 


9.08211 


.12781 


7.88428 


48 


18 


.07519 


18.2996 


.00277 


lo.vvyy 


.11040 


9.05789 


.12810 


7.80688 


42 


10 


.07548 


18.2180 


.09306 


10.7457 


.11070 


9.03379 


.12840 


7.78825 


41 


80 


.07578 


18.1969 


.09385 


10.7119 


.11099 


9.00988 


.12869 


7.77085 


40 


81 


.07fi07 


18.1461 


.09865 


10.6788 


.11128 


8.98598 


.12809 


7.75854 


89 


28 


.07686 


18.0058 


.09394 


10.0450 


1 .11158 


8.90227 


.12929 


7.73480 


88 


88 


.07665 


18.0458 


.09423 


10.6118 


.11187 


8.93867 


.12958 


7.71715 


87 


24 


.07695 


18.9968 


.09453 


10.5789 


.11217 


8.91520 


.12988 


7.69967 


86 


25 


.07724 


12.9469 


.(ms2 


10.5462 


.11246 


8.89185 


.18017 


7.68206 


85 


26 


.07753 


12.8981 


.09511 


10.5136 


.11270 


8.86862 


.13047 


7.66466 


84 


27 


.07782 


12.8196 


.09541 


10.4813 


.11806 


8.84551 


.13076 


7.64738 


38 


28 


.07812 


12.8014 


.09570 


10.4491 


.11885 


8.82252 


.13100 


7.68005 


38 


29 


.07841 


12.7536 


.00600 


10.4172 


.113ft4 


8.79964 


.13186 


7.61887 


81 


ao 


.07870 


12.7062 


.09689 


10.8854 


.11894 


8.77689 


.18165 


7.69575 


80 


81 


.07899 


12.6591 


.09658 


10.8538 


.11423 


8.76425 


.18195 


7.57872 


29 


82 


.07989 


12.6121 


.00688 


10.8224 


.11452 


8.7^172 


.18224 


7.56176 


88 


83 


.07958 


12.5660 


.09717 


10.2913 


.11188 


8.70931 


.13254 


7.54487 


87 


84 


.07987 


12.5199 


.09746 


10.2602 


.11511 


8.68701 


.13284 


7.52806 


86 


85 


.08017 


12.4742 


.09776 


10.2894 


.11541 


8.66482 


.18313 


7.51188 


86 


86 


.06046 


12.4288 


.09806 


10.1988 


.11570 


8.64276 


.18343 


7.49465 '24 


87 


.08075 


12.8838 


.09834 


10.1683 


.11600 


8.62078 


.18372 


7.47B06 83 


88 


.08104 


i2.a^ 


.09864 


10.1381 


.11629 


8.59893 


.13402 


7.46154 182 


39 


.08184 


12.2916 


.09893 


10.1080 


.iie>^ 


8.57718 


.13432 


7.44609 '21 


40 


.08163 


12.2505 


.09923 


10.0780 


.11688 


8.56555 


.18461 


7.42871 laO 


41 


.08192 


12.2067 


.09952 


10.0488 


.11718 


&.6W08 


.18491 


7.41840 '19 


42 


.08221 


12.1632 


.09981 


10.0187 


.11747 


8.51250 


.13521 


7.89616 


18 


48 


.08251 


12.1201 


.10011 


9.98931 


.11777 


8.49128 


.13550 


7.87999 


17 


44 


.06280 


12.077^ 


.10040 


9.96007 


.11806 


8.47007 


.13580 


7.86389 


16 


45 


.06809 


12.0346 


.10069 


9.98101 


.11836 


8.44896 


.13609 


7.84786 


15 


46 


.06339 


11.9983 


.10099 


9.90211 


.11865 


8.42795 


.13689 


7.83190 


14 


47 


.06868 


11.9504 


.10128 


9.87838 


.11895 


8.40705 


.18669 


7.31600 


18 


48 


.08397 


11.9087 


.10158 


9.84488 


.11924 


8.88685 


.13698 


7.80018 'l8| 


49 


.06427 


11.8673 


.10187 


9.81641 


.11954 


8.86555 


1 .18728 


7.88448 


11 


50 


.06456 


11.8262 


.10816 


9.78817 


.11963 


8.84496 


.18758 


7.26873 


10 


51 


.06485 


11.7868 


.10846 


9.76009 


.12018 


8.82446 


.18787 


7.25310 




68 


.06514 


11.7448 


.10275 


9.73217 


.12042 


8.3O406 


.18817 


7.23754 




68 


.06644 


11.7045 


.10305 


9.70441 


.12072 


8.28376 


.13846 


7.22804 




64 


.08578 


11.6645 


.10334 


9.67680 


.12101 


8.26355 


.13876 


7.20661 




66 


.08602 


11.6248 


.10363 


9.64935 


.12131 


8.24345 


.18906 


7.19125 




66 


.06638 


11.5853 


.10393 


9.62206 


.12160 


8.22344 


.18035 


7.17594 




57 


.08661 


11.5461 


.10428 


9.59490 


.12190 


8.20352 


.13965 


7.16071 




68 


.08690 


11.5078 


.10458 


9.56791 


.12819 


8.18870 


.18996 


7.14653 




69 


.06780 


11.4685 


.10481 


9.54106 


.12249 


8.16398 


.14084 


7.13042 




60 

/ 


.08749 


11.4301 


; .10510 
Cotang 


9.51486 


.12278 
Cotang 


8.14436 


.14054 


7.11587 


/ 


Ck>tangi Tang 


Tang 


Tang 


Cotang 


Tang 


860 


1 840 1 


>s° 1 


8S- 



49* 



TABLE V.—NATURAL TANGENTS AND COTANGENTS. 



/ 


8» 


90 


10- 


!!• 


/ 


Taofl^ Cotang 


Tang 


Cotang 


Tifcng 


Cotang 


Tang 


Cotang 


"o 


.14054 7.11587 


.15838 


6.31875 


.17688 


5.07128 


.19488 


6.14466 


60 


1 


.140^ 7.10088 


.15868 


6.80189 


.17668 


6.66165 


.19488 


6.18668 


60 


2 


.14118 


7.08M6 


.15898 


6.29007 


.17698 


6.66205 


.19496 


6.18688 


68 


8 


.14148 


7.07069 


.15928 


6.27829 


.17728 


6.M248 


.19529 


6.18060 


67 


4 


.14178 


7.05579 


.15958 


6.26665 


.17758 


6.68296 


.19KW 


6.11879 


56 


6 


.14202 


7.04106 


.15988 


6.25486 


.17788 


6.62844 


.19589 


6.10400 


65 


8 


.14232 


7.08687 


.16017 


6.24821 


.17818 


6.61897 


.19619 


6.00704 


54 


7 


.14262 


7.01174 


.16047 


6.23160 


.1W48 


6.60452 


.19649 


6.06021 


58 


8 


.14291 


6.99718 


.16077 


6.22008 


.17878 


6.59511 


.19680 


6.06180 


62 


9 


.14821 


6.96268 


.16107 


6.20651 


.17903 


6.58578 


.19710 


6.07360 


51 


10 


.14851 


6.96828 


.16187 


6.19708 


.17988 


6.67686 


.19740 


6.06684 


60 


11 


.14881 


6.95886 


.16187 


6.18550 


.17968 


6.66706 


.19770 


6.06800 


40 


12 


.14410 


6.93852 


.16196 


6.17419 


.17998 


6.65777 


.19801 


6.06087 


48 


18 


.14440 


6.92525 


,WSA 


6.16288 


.18028 


6.64851 


.19681 


6.(Vt867 


47 


14 


.14470 


6.91104 


.16256 


6.15151 


.18058 


6.68927 


.10681 


6.08480 


46 


15 


.14499 


6.89688 


.16286 


6.14028 


.18068 




.19801 


6.02784 


45 


16 


.14529 


6.88278 


.16316 


6.12899 


.18118 


6.62090 


.10021 


6.oion 


44 


17 


.14559 


6.86874 


.16846 


6.11779 


.18148 


6.61178 


.19068 


6.01810 


48 


18 


.U'iSS 


6.8547B 


.16376 


6.10864 


.18178 


6.60264 


.10068 


6.00451 


42 


19 


.14618 


6.&4082 


.16405 


6.09562 


.18206 


5.49856 


.20018 


4.0060ff 


41 


20 


.14648 


6.88694 


.16485 


6.06444 


.18288 


6.48451 


.80048 


4.06040 


40 


21 


.14678 


6.81312 


.16465 


6.07840 


.18288 


6.47548 


.80078 


4.06188 


80 


2d 


.14707 


6.79936 


.16405 


6.06240 


.18298 


6.46&48 


.80108 


4.97488 


88 


28 


.14737 


6.7^564 


.16525 


6.06148 


.18828 


6.45761 


.80188 


4.06800 


87 


24 


.14767 


6.77199 


.16556 


6.04061 


.18858 


6.44857 


.80164 


4.06045 


86 


26 


.14796 


6.75838 


.16686 


6.02962 


.18884 


6.48966 


.20104 


4.06801 


85 


26 


.14826 


6.74483 


.16615 


6.01878 


.18414 


6.48ar7 


.80224 


4.04480 


84 


27 


.14856 


6.73138 


.16(M6 


6.00797 


.18444 


6.42192 


.20254 


4.08m 


88 


28 


.14886 


6.71789 


.16674 


6.99720 


.1W74 


6.41809 


.80886 


4.08064 


82 


29 


.14915 


6.70450 


.16704 


5.98646 


.18604 


6.40420 


.80816 


4.02240 


81 


80 


.14945 


6.69116 


.16784 


6.97576 


.18584 


6.89662 


.80846 


4.01616 


80 


81 


.14975 


6.67787 


.16764 


6.96510 


.18664 


6.88877 


.80878 


4.00765 


89 


32 


.16005 


6.66468 


.16794 


6.95448 


.18504 


6.87806 


.80406 


4.00066 


88 


83 


.16034 


6.65144 


.16824 


6.94890 


.18624 


6.86986 


.80436 


4.80880 


87 


84 


.15064 


6.63831 


.16854 


6.98385 


.18654 


6.86070 


.80486 


4.88805 


86 


86 


.15094 


6.62528 


.16884 


6.92288 


.18684 


6.86206 


.80407 


4.878ffl 


85 


86 


.15124 


6.61219 


.18914 


6.91236 


.18714 


6.84845 


.80687 


4.87188 


84 


87 


.15158 


6.69921 


.16944 


6.90191 


.18746 


6.88487 


.80667