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• THE 

PRINCIPLES AND PRACTICE 



OF 



SURVEYING 



• BY 

CHARLES B. BREED and GEORGE L. HOSMER 

IMSTTRUCTORS IN CIVIL ENGINEERING, MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY 



First Edition 




OF THF 

UNIVEPSITY 



NEW YORK 
JOHN WILEY & SONS 



London: CHAPMAN & HALL, Limited 
1906 









S" 



SfNiftU. 



Copyright, 1906 

BY 

CHARLES B. BREED and GEORGE L. HOSMER 



Stanbopc prcM 

r. H. OIL*ON COMPANY 
BOSTON. U. •. A. 



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 
sibjects 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 
nght 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 use- 
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 EUamar, 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 

The authors also desire to express their appreciation of the e: 
celJent work of W. L. Vennard, who made the drawings for illu: 
trations. 

No pains has been spared to eliminate all errors, but th 
authors cannot hope that their efforts in this line have been con 
pletely successful, and they will consider it a favor if their attei 
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. — Gsneral Definitions. — Measurement of Lines. 

An. Pack 

1. Definition 3 

2. PuxposES OF Surveys 3 

3. HosizoNTAL Lines 3 

instruments FOR MEASURING LINES. 

4- The Chain 3 

5 Metric Chain 5 

6. Tta Tape 5 

7. Steel Tapes 5 

8. The Stadia .' 6 

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

ter 6 

lOASUREMBNT OF LINES. 

la Measurement of a Horizontal Line with a Chain 7 

11. Measurement of a Horizontal Line with a Chain on Slop- 

ing Ground 9 

12. Measurement with a Steel Tape 10 

13. Common Sources op Error in Measurement of Lines .... 11 
14- Common Mistakes in Reading and Recording Measurements ii 

15. Avoiding Mistakes 11 

id AccoRAcy Required 12 

17- Amount of Dipferent Errors 12 

18. Pull 12 

19- Temperature 13 

20. Alignment 13 

21. Sag 13 

22. Effect op Wearing on Length of the Chain 14 

33. Accuracy of Measurements 14 

Problems 15 

Chapter II. — Measurement of Direction. 

24. The Surveyor's Compass 16 

25. The Pocket Compass 18 

26. Method of Taking a Magnetic Bearing 18 

vii 



vm CONTENTS 

AiT. Pace 

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

28. Declination of the Needle ip 

29. Variations in Declination 20 

3a IsoGONic Chart 25 

31. Observations for Declination 2$ 

ADJUSTMENTS OF TH£ 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 of 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. ECCENTRiaTY 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 of the Transit 55 

ADJUSTMENTS OF THE TRANSIT. 

70. Adjustment of the Plate Bubbles 56 

71. Adjustment of the Cross-Hairs 57 



CONTENTS IX 

AiT. Page 

73. Adjusticent op the Standards 59 

74- Adjustment op the Telescope Bubble 59 

75. Adjustscent of the Auxiliary Level on the Vernier op the 

Vertical Arc 59 

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

77. Adjustment op the Objective Slide 60 

78. Shop Adjustments 61 

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

the Transit 61 

8a Care op Instruments 61 

81. Common Sources of Error in Transit Work 62 

82. Common Mistakes in Transit Work 62 

THE SOLAR ATTACHMENT. 

83. Description op 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. 

9a Adjustment of the Polar Axis 70 

91. Adjustment op 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 op the Level Tube .... 73 

the level. 

97. The Wye Level 73 

98. The Dumpy Level 74 

99- Comparison op 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 

10$. Special Self-Reading Rods 81 

106. Tape Rod 81 

107- Precise Level Rod 82 

108. Advantages op the Selp-ReXDing Rod 82 

109. Attachments to the Rod for Plumbing 83 

1X0. Effect op Heat and Moisture 83 



X CONTENTS 

USE OF THE LEVEL AND ROD. 

Art. Pagk 

112. To Level the Instrument ... 83 

113. To TAsa: 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 

adjusti£ENts op the level. 

I. adjustments of the wye level. 

120. Adjustment of the Cross-Hair 88 

122. Adjustment of the Level Tube 89 

123. Adjxtotment 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 

13a 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 Azimuth Angles 108 

145. Checking the Fieldwork 109 



CONTENTS xi 

Atr. Page 

146. AccintACY Required in 

147. OlGANIZATION OF TRANSIT PARTY Ill 

148. Note-keeping 112 

149. Survey of a Field for a Deed n4 

15a Deed Description 115 

151. Judicial Functions of the Surveyor 116 

152. Rerunning Old Surveys from a Deed 118 

153. How TO Look up a Recorded Deed 120 

THE UNITED STATES 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. SUMM.4RY 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. 
TRAviRSE 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 Triangulation Points .... 158 

i^>cation op buildings from transit line. 

176. Methods of Locating Buildings 159 

177- Geometric Principles 159 

178- TtEs, Offsets, Swing Offsets, and Range Lines 159 

179- General Suggestions 160 

18a Typical Cases 160 

j8i. Building near Transit Line and Nearly Parallel to it. . . 160 

Plotting 161 

i8a. BuiLDrNG near Transit Line, Making a Slight Angle with 

IT 161 

Plotting 162 



Xll CONTENTS 

An. Page 

183. BxTiLDiNO Located Entirely by Direct Ties 162 

Plotting 163 

184. Building Located at a Considerable Skew to the Transit Line i 63 

Plotting 164 

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

Plotting 164 

186. Buildings Located prom Other Buildings • • • 165 

187. Buildings of Irregular Shape 166 

188. Large City Buildings 166 

189. Location op Buildings by Angles and Distances 167 

190. Location op Buildings and Fences prom Transit Line ... 167 

MISCELLANEOUS SURVEYING PROBLEMS. 

191. Random Line 169 

192. Obstacles on Line 170 

193. Oppsetting 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 Oppset Method 175 

202. By Tangent Oppset Method 176 

203. By Obliqxte Triangle Method 177 

204. To Obtain the Distance Between Two Inaccessible Points 

BY Observation prom Two Accessible Points 178 

205. To Obtain the Inaccessible Distance Between Two Accessible 

Points by Observation on Two Inaccessible Points op 

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 por Meridian on Polaris at Elongation ... 182 

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

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

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

211. Meridian Observ ation on Polaris at any Time with the Transit i 88 

212. Solar Observations 190 

213. Observation por Meridian by Equal Altitudes op the Sun in 

the Forenoon and Afternoon 190 

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

215. Observation por Meridian by Means op the Solar Attachment 195 



CONTENTS xiii 

OBSERVATIONS FOR LATITUDE. 

Arr. Page 

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

217. By the Altitude op the Sun at Noon 196 

Probleics 197 

Chapter VIII. — Leveling. 

218. Definitions 198 

219. Leveling to Establish Bench Marks 198 

222. i>ouble rodded lines 20i 

224. BLENCH Marks and Turning Points 202 

225. Leveling por 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 

2361 "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 

CIT? LAYOUTS. 

245- Streets 219 

246. Location of Streets 220 

247* Size of Blocks and Lots 222 

248. Width of Streets 222 

a49* Street Grades 224 

250^ The Datum Plane 226 

251- Establishing Bench Marks 226 

252. Water and Sewer Systems 227 

staking out city work. 

253- Staking out a New District 227 

254- Monuments 228 

255. Setting Stone Bounds 230 



xiv CONTENTS 

AsT. Pack 

256. CiTRVED 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 

267. 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. Relation 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 

Ar. Page 

302. Mistakes in Sketching Contouks 276 

303. Locating Contouks 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 

31a Intersection of Plane with Surface op Ground 280 

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

312. 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 

MLVING instruments. 

315. Mining Transits 287 

316. Side Telescope 287 

317- Top Telescope 287 

318. Adjustments of Side Telescope 287 

319. Adjustment op 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 Line 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. Page 

343. Surface Surveying in Rugged Mountain Regions 311 

344. Mine BouNDASiES. 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-PlTS 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. 

A«T. Page 

383. Computation of Area 352 

.384. Akea by Double Meridian Distance Method 352 

- 386. Computation of Area of Compass Survey 356 

387. Balancing a Chain and Compass Traverse 358 

39a Double Parallel Distance 359 

391. Error op Closxhue 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 

. 403- 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 in 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 

41a 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. 

KKGINEERING DRAFTING INSTRUMENTS. 

415. Straight Edge 381 

416. Engineer's Scale 382 

417- Protractor 382 

418. Semicircular Protractor 382 

419- Full-Circle Protractor 384 

421. Three-Armed Protractor 384 



xviii CONTENTS 

Aht. Pack 

423. Pantograph 385 

424. Parallel Ruler 386 

425. Beam Compass 386 

426. Contour Pen 386 

427. Proportional Dividers 387 

429-31. lUiLROAD Curves, French Curves, Flexible Curves, and 

Spline 387 

DRAWING papers. 

433. Drawing Paper for Plans 388 

434. Tracing Paper and Tracino 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. 
447 
448 
449 
454 
455 
456 
457 
458- 
459 



Plotting by Protractor and Scale 398 

Checks 398 

Protractor and T-Square 400 

Plotting by Rectangular Coordinates 401 

Checks 405 

Plotting by Tangents 405 

Checks 407 

Plotting by Chords 408 

Use of the Sine 409 

Checks 409 



METHOD OF PLOTTING DETAILS. 

46a Buildings, Fences, Streams, Etc 410 

461. Contours 410 

462. Cross-Sections 411 

463. Profiles 411 

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 

An. Pack 

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. Indexlng Notes 433 

487. Other Records 433 



TABLES. 

I. Logarithms of Numbers 437 

U. Logarithmic Sines and Cosines 455 

in. 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 

Vm. Trigonometric and Miscellaneous Formulas 511' 

K. Circular Curve Formulas 513 

X. Geometric Formulas 513 

XL Linear Measure 514 

Xn. Square Measure 514 

XIIL 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, ADjnsTMEin, aud care of htstrumehts. 






/■ 



d 



CHAPTER I. 

aEBTERAI. DUmnnONS. — MEASUREMENT 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 
curvature 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. 

3- Horizontal Lines. — In surveying, all measurements of 
lengths are horizontal or else are subsequently reduced Jto 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. 

nrSTRUMENTS FOR MSASURmG LINES. 

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

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

I Mile =s8o 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 lOO feet long and is divided into one 
hundred links of one foot each. Each end link 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. 



Fig. 1. 



GuNTER's Half-Chain. 



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^re recorded as chaitis and linksj {or in chaifis 
and decimals) ; while those measured with the engineer's chain 
are recorded ?^feet 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 decimete];s 
long. 

6. THE Tape. — There are three kinds of tape in common 
use,— f&/A, metallic^ and steeL Cloth tapes stretch so easily 
that they are of little use in surveying. The so-called metallic 
tapes are cloth tapes having \^xy 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 loft. length at 
one end of the tape being marked at every foot, and the last 
foot divided into tenths* Some of these tapes- gre 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 100 ft. These tapes are more convenient to 
handle than the heavy ones, but are not suited to very rough 
^ork as they are easily kinked and broken. Thejj can be readily 
"^nded, however, by riveting to the back of the tape a piece of 
^ape of the same width. 

Sinc^the surveyor's measurements are usually in feet and 
jl^als, they are not in convenient form for use by mechanics 
ui 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. 

Decimal op Foot. Inches. 

.01 = t - 

.08 - I- 

.17 = 2 + 

.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" — f"= 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. 

0. Other INSTRUMEITTS. — W^^^^^«^^//jare used in certain 
kinds of work for making short measurements, usually less than 

IS 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 10 chains have been measured the head-chainman will 
be out of pins and calls to the rear-chainman, who brings for- 
ward 10 pins. The pins are then counted by both chainmen. 
Every time 10 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 lo-ft. 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 groxmd 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 
factional 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 bead-chainman 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 looft. points as fol- 
lows : — suppose an angle to occur at 870. i ft. from the point of 
beginning; 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, summiqg 
up the par^ 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. COMMON SOURCES OF ERROR IN MEASUREMENT OF 
UBES. — 

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. COMMON 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 side 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 undefttood. It 
is frequently useful in doubtful cases for the note keeper to use 
difiFerent words in answering, which will remove possible ambi- 
guity. For example, if the chainman calls, "Thirty-six, five," 
the note keeper might answer, "Thirty-six ^d 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 tjie 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. 

18. 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 steel tape is nearly 0.0000063 for i^ F. Hence a change of 
temperature of 1 5® produces nearly 0.01 ft. change in the length 
of the tape. Tapes are usually manufactured to be of standard 
length at 62® F., with a pull of 12 lbs. on them while supported 
throughout their entire length. 

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

^ • 

2C 

where k 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, iif 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 

win be — — = 0.005 ft-» (about ^ inch). The correction 

2 X 100 

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 acurve 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 rig^t triangle, 

{c + a) (^ - a) - k\ 
M wnning e^a and applying it to the first parenthesis only, 

2 r (tf — a) — A' (approximately) ^-**^ 

tf — tf —~ (approximately) ^^00^^^ ^i* 

SunOariy .< — a — - (approximately) 

It is evident that the smaller A 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 — 100 ft, or when >l — 30 ft 
aiMia»3ooft 



14 MEASUREMENT OF LINES [Chap. I. 

at the ends of the tape. With a 12-lb. pull on an ordinary 100- 
ft. steel tape supported at the ends the effect of sag is about 
0.0 1 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 5CXX)- 
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, i.e., 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 
^ 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 o.oi ft. too 



ACCURACY OF MEASUREMENTS IS 

long is SO X o.oi = 0.50 ft. The small (0.01) accumulative 
error is therefore seen to, have far greater dBFect than the larger 
(0.02) compensatmg error. 



u 



PROBLEHa U 

1. A distance is measnred with an en^neer's chain and found to be 7964 ft ^a^j£ 
The chain when compared with a standard is found to be 0.27 ft. too long. What JjS^ 
is the actual length of the line ? 

2. A metallic tape which was originaUy 50 ft. is found to be 50.14 ft. long. 
A bouse 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. 
ixA supported its entire length. Its coefficient of expansion is 0.0000063 for 1° F. 
A fine was measured and found to be 142.67 ft. when the temperature was 8^ 
bdow zero. What is the true length of the line ? 

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

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



CHAPTER 11. 

MBASURBMBUT OF DIRBCTION. 

34* 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 Ufted 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 magnetic 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^, 

z6 



THE SURVEYOR'S COMPASS 



17 



the 0* bemg 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 ot 26' with tin ni m : , 
and is in the southeast quadrant its bearing i^ written "^ :■ \ 
Sometimes the bearing is reckoned in a siiuilar nia: nc *^ "•» 




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 azimuths^ and are commonly 
reckoned from the south point right-handed (clockwise) to 360° ; 
ie., a line nmning due West has an azimuth of 90°, a line due 
North an azimuth of 180®. Sometimes, however, the azimuth 



IS 



Measurement of direction 



[Chap. H. 



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

25. The Pocket Compass. — T\i^ 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- 





Beonng of AB Bearing of AB Bearing of AB 

S60*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 



TAICING 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 
really 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 acqount 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 
tixnc. 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 OP DmECTlOM [Chap, tt 

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 ^ Annual^ 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. 





TABLE 2. 






uINATION 


s OP Needle 


IN Eastern M 


Year. 




Declination 


1700 




10° 


31' 


w. 


1750 




r 


13' 


W. 


i8cx) 




6*' 


28' 


w. 


1850 




9" 


10' 


w. 


1900 




I2<» 


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 Isogonic Chart for 1902, and PrincipaJ 
Facts Relating to the Earth's Magnetism," by L. A. Bauer, Issued in 1902. 




(tnxn 



A.U.S.C"-' 



^iG-arfeS-tW 







D States for the Epoch January, 1902 

at ion Tables and Isogonlc Chart for 1903, and Principal Facts Relating to the Earth's Magnetism/' 
. is<ued in iQoa.) 



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 Unes 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 isogonic 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, passes 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. 1 87. 

ADJUSTMENTS OF THE COMPASS. 

32. The three adjustments which need to be most frequently 
made are (1) 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 



IChap. n 




Fig. 6. 



Bent Compass- 
Needle. 



apaurt, 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 CC, (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'jy the 
error is less than before and equals 
B'ly. When the needle is in the 
line CCj i.e., in the position A^jy^ 
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-WEEDLE. — Level the 
instrument and let the needle down on the pivot. Remove the 
glass cover. By means of a brass wire or a light 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 ^4 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 diflference of the readings, or, the distance C^D is the 

amount by which the needle is bent. 
Carefully remove the needle from the 
pivot and bend it by the amount C^D 
in the direction which will move the 
south end half-way back from C* 
toward C, It is better not to touch 
the needle with the hands more than 
is absolutely necessary as this weakens 
the magnetism. Instrument makers 
usually leave the central part cf the 
*]J — *^ 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-PomT. — 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. II. 



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. COMMON SOURCES OF ERROR IN COMPASS WORK. — 

1. Iron or steel near compass. 

2. Parallax in reading needle. 

39. COMMON 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 ATTRACTIOlf 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 BC, 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 -39° 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 BEARINOS. — 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 = iSo** — sum of bearings. 
AOD = 1 80° — difference of bear- w 

ings. 
AOF = sum of bearings. 




Fig. 10. 



I. 
cases. 



PROBLEMS. 

Compute the angle AOB from the g^ven bearings in each of the following 



(a) 
(b) 



OA, 
OB, 

OAy 

OB, 



N 39^1 E. 
N 76°i E. 
N 35° 15' E. 
S 88° 00' W. 



(c) OA, N i5« 
OB, S 36° 

(d) OAy N40° 
OB, N66° 



E. 
E. 

15' E. 
45' W. 



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

3. (a) In 1859 a 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 1902. 

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

4. The following bearings were observed with a compass : ^^, N 27® J E ; 
BA, S 25**^ W ; iPC, S 88'=' W ; CB, N 87°J E ; CA N 47°^ W ; Z?C, 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. 

MBASURISMENT OF ANGLBS. 

THE TRANSIT. 

42. GEHERAL 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. 12. 
r • F 

E 




Section of Transit. 

Af inner spindle ; B^ outer spindle ; C, upper plate ; Z>, lower plate ; E^ graduated 
circle ; F, vernier ; G, magnetic needle ; Hy upper clamp (turned 90° from its nor- 
mal position so as to show in section, corresponding tangent screw not shown) ; 
/i lower clamp ; K, lower tangent screw ; Z, leveling screws ; My ball-and-socket 
joint ; A^, 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 instrunient, 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 with 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 0° 
each way to 180°; and still others (older types) are numbered 
from qP to 90° in opposite directions, like a compass circle 
Transits are all provided with two opposite verniers. 



THE TRANSIT 



33 




Fig. 13. Engineer's Transit. 







C^ Or 






34 



MEASUREMENT OF ANGLES 



(Chap. HI. 



a 

o 

CO 

H 






o 

O 

H 
U 
td 
CO 

•J 

< 

Q 

s 
2 



43. The normal or rf//r^/ 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 i8o° 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 levd tube and a vertical circle, or arc, 
it is called an Engineer's Transit^ or Sur- 
vcyof's Transit, (Fig. 13.) If it do^s not 
have these attachments it is called a Plain 
Transit. 

45. The Telescope. — The essential 
parts of the telescope are the objective^ the 
crosS'hairSy 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 ^^e 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 Qs!&&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 parallax 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 
2000 ft An instnunent 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 Off 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 \!ti^ 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 X on 
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. BiCoNVEx 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 

.1/ J r .. 1 u Fig- 10- 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- 
\'antages 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 

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 poiiits ; this new angle will be smaller. Then 

the Magnifying Power = . 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. 



6- 



— —9 



—7 



-K) 



S 



5i 

—4 ' 
—3 





.^^ 




c 









^^_ 






K) 




... 


—9 




- 


—6 
—7 


u 

it 




-« a: 


— 


—4 ^ 




■^ 


—5 

2 




^^ 




5- 


— 


^mo 






.. 







K) 


^\ 


—9 


6— 


— e 




—7 




^1 

—4 ^ 


SCALE 
1 


—3 


—2 




J^O 






5- 




• 





Fig. 17. 



Fig. 18. 



Fig. 19. 



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



THE VERNIER 39 

ratdy 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 -fj^ of a space on the scale, i.e., it is -j^^ part 
shorter than a space on the scale, hence o^ =r ^ of a space 
on the scale, ^^/ = ^ of a space, etc. ^ow if the vernier is 
raised until a coincides with ^,- 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 SOI. 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 
mdex 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. 

S3. Verniers used on Transits. — In transits, since angles 
may be measured in either direction, the verniers are usually 
double, i.e., 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 fg 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 o^ 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- 




360 

CIRCLE 
Fig. 20. One-Minut£ Vernier Set at 0^ 



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. 



VERNIER 
A 




TRANSIT VERNIERS 



41 



Fig. 21. — Double vernier reading to i'. Circle divided 
into 30' spaces. 29 divisions of the circle divided into 30 paurts 
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 the circle in the same direction that the telescope 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". 




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

Reading, 59° 15' 50". 



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




VERNIER 

Fig. 26. 

Circle divided into 30/ spaces. 29 divisions of the circle 
cJivided 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 diflference. 

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- 
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 






Fig. 27. 



Eccentricity of 
Circle. 



ECCENTRICITY OF CIRCLES 45 

of opposite verniers will diflfer 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 A, 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'B* 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 0°. 

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

USE OF THE TRANSIT. 

56. SETTUfG UP THE TRAKSIT. — In setting the transit 
over a point, place one leg of the tripod in nearly the right posi- 
tion on the ground, then grasp 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 heady 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 trying to center 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 center. After the instrument is leveled 
the plumb-bob should be examined to see that it has not been 
moved from over tbe point during the process of leveling. 

58. To MEASURE A HORIZOIVTAL 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 clamp, 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 lines 
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 "residing 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 MEASXmEMENT OF ANGLES [Chap. HL 

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 screw. 
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 O^. 

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 focus 
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, sue 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 will be 48" which will be read as 
r. 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 throe 
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 wedking 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, ^ii 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 aod divide the second reading 
by 2 simply as a check on the fiyfrtt reading. 

61. To LAY OFF AN AN«;e BY REPETITION. — There is 
no direct method of laying/^ an angle by repetition as in the 
case of measuring an ^angle, therefore the following indirect 
method is used. J^h 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 bQ 
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 $1 

62. RUHlflNG A STRAIGHT LIKE — 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. RumilHG 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 seccmd 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 releveled 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 Kne 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 
^random line as described in Art. 191, p. 169. 



52 MEASUREMENT OF ANGLES IChap. m. 

64. Prolonging a Straight Line. — If a line is fixed by two 
points A and 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 A, 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 ^ 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 knowri 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 crosshair 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 invened 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 is 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 sighted 
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 great 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 pofait is to be set carefully, as a transit point, 

like a plainb-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 
vith sach colors 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 
(aometimes impossible) 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^ — When 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 signeding 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 verticed 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 correction^ 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, tummg 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 thjf 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 be 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 effect. 

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 
damp, and turn the telescope either up or down. 



s<s 



UEASXJKBMEtn Of AKGLES 



tCBAP. Tit. 



adjustmehts of the trahsit. 

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 eflfect 
of an error is doubled. 

70. Adjustheut 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 




Fig. 29. Adjustment of the Plate Bubbles. 



the bubbles to the center of their respective tubes. Tirni 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 S7 

bubble tube untU 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 
until 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. 

71. 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. HI. 



72. 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. B, 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 point 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 makera 
seldom level the transit carefully. In field adjustments it is desirable, although 
not necessary, to level the instrument. The essential condition is that the verticaJ 
axis shall not alter its position. 



ADJUSTMENTS OF TRANSIT 



59 



73. Ad JUSTlfEIlT OF THE STAIVDARDS. — 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 A is the error of adjust- 
ment. Loosen the screws in the pivot 
cap and raise or lower the adjustable end 
of the horizonted 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- 
han- 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"" method^ or direct tnethod, 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. 67, p. 54.) To adjust the Level 
80 that it is in the Center of the Tube when the Line of Sight 
is Level and the Vernier reads O^ This is adjusted by the <' peg 




Fig. 82. Adjustment 
OF THE Standards. 



6o MEASUREMENT OF ANGLES [Chap. HI. 

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 O^ when the Telescope Bubble is in the 
Center of the Tube. If there is any index error (Art. 67, 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 
Objeotive 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 the 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 mo^ng 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 6i 

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 
sKde 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 
ADJUSTHEHT 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 cameFs 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 instrximent, as it collects dust. Care should be 
taken not to strain the adjusting screws in making adjustments. 



62 MEASUREMENT OF ANGLES [Chap. IH- 

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 WORE. — 

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 Df 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 6i^ 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. 



*p 






UNlVEr-^.jy 



Sa 



I T. 



64 



SOLAR ATTACHMENT 



[Chap. UL 




Fig. 33. Solar Attachment to Transit. 

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



SOLAR ATTACHMENT 65 



THE SOLAR ATTACHMEHT. 



83. DESCRIPTION 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 telescojje 
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 JVES W is the observer's horizon and is the 
boundary between the visible and invisible pirts 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 op the Celestial Hemisphere. 



motion from east to west. The sun's 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 

8$. 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 67 

(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 imtil 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 circle 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 four 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, P^ 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. UI. 

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 declination 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 5^72^/^ 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 
0° 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 0° again. 

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



OBSERVATION FOR MEIODIAN 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. S07. 

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 nooiiy aslBMBf^ther 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 pointing at the sun until the maximum 
altitude is reached. The angle f«ad on the vertical circle is the 
co-latitude (see also Art. 217, p. 195). 

Example. 



\ 4o4n. 



Latitude 40*^N. Longitude 4h 4$^ W. 

Jan. 10, 1900. 
Declination for Greenwich mean noon 21° 59' 04^^ 
DifiFerence for ih H- 22'\2S 



Tnn. 


Dbcximatiok. 


RBrRACTION. 


Sbttiitg. 


7 h. A.M. 


2i*>59'o4" 






8 


58 42 


5' 40" 


2I*»S3'02' 


9 


58 20 


2 51 


21 5529 


ID 


57 57 


2 07 


21 55 so 


II 


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 


5628 


2 07 


21 54 21 


3 


5606 


2 SI 


21 53 15 


4 


55 44 


5 40 


21 50 04 



87. * Comstock's Method of finding the Refraction. — Set 
the vertical cross-hair on one edge (or iimi) 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. 



;C MEASUREMENT OF ANGLES {Chap. Ttt. 

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

, ^ 2000 

equal to — ; 

/in 

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 largie 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 IN USnrG THE SOLAR ATTACHMENT. — 

1. Solar on wrong side of main telescope. 

2. Refraction correction applied wrong way. 

ADJUSTMENTS OF THE SOLAR ATTACHMENT. 

90. Adjustment of polar Axis. — To make the Polar 
Axis Perpsndicular to the Plane of the Line of Sight and the Hori- 
zontal Axis. Level tfie 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 
imtil 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 7 1 

93. 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. 

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

3. A transit is sighting toward B from a point A, In setting up the tranut at 
A it ma 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 AB and 
A*B, (I) when AB - 40 ft., (2) when -r*^- 1000 ft.? 

3. An angle of 90^ is laid off with a '* one minute " transit, and the angle then 
determined by ^x repetitions, the final reading being 179° 58'-!- 360®. The point 
sighted is 185 feet from the transit. Compute the offset to be Isud off in order to 
. correa 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' 2%'\ 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. 

Ded. S 23*» 03' 27" .9 

Dift. for I hour, +11 ''.70 » 

April 16, 1906. 

Decl. N 9*' S3' 34^2 

Diff. for I hour, + S3".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, - s8''.si 



CHAPTER IV. 

MBASXJRBMBNT OF DIFFBRBNCB OF EIJEIVATION. 

93. LEVEL SURFACE. — A level surf ace 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 g^ven 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 Hod 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 Levels and the Hand Level. The Precise Level differs 
in its details from the others but does not really constitute a 
di£Ferent type ; it is essentially a wye level or a dumpy level, 
according to the principle of its construction. The engineer's 
transit, whitjh 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 rings 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 mieans 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 



^ 




76 




.J 

> 

td 



Q 



THE DUMPY LEVEL JJ 

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, chietiy 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'' adjustment^ 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, fails to stand the test by the direct method, but 
whith 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 HAND 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- 
rey for the year 189S-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 
o^ 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 Boston, the New York, 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\ 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 tie 
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 



i 
e 
i 




E 

i 






79 



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. 4 1 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. HEW 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 o.cxDS ft. ; the 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 aooi 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 kvelman 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 (o.oi 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. Unliice 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 advantage 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 tod, 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 Pltimbing. — 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 corner 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 ARB 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 be necessary -under some circumstances, for ex- 
ample, in a strong wind. 

113. TO Take a rod-reading. — Therodman 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. 1x5), 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 rea(Ungs 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 line 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 *f 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-hafa- 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. 

*M// 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 or plus sight and is usually written B. S. 
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 i^ 5. 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 




Fig. 43. 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 



Bdqct 


B. S. 


F.S. 

• 


A. 

T.P. 

T.P. 

T.P. 

B. 


8.160 

7.901 
9.446 
8.005 


2.404 

^^ 

2.107 




33.5" 
14.487 


14-487 



Rbmarks 



Highest point on stone bound, S. W. cor. X and Y Sts. 



N. £. cor. stone step No. 64 M St. 



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 A^^ (see foot-note, p. 339). The effect of 



88 MEASUREMENT 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. 5 5, 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 



S9 



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 Asds of PivotSy 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 1 80** 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 
Method. — (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 



/ 2nd.Pbi 


Kon 


Cro 


P^ 




~~ 






1 TrutPtoi 


»iM 


CfM 


i-h* 


I Iflt Pb6H 


•« 


Crtu, 


t^ 


V 


~_ 




fpOlMT / 



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 clamp 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 

LINE 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 



^ *nnie Pwijhofi of *^'^«* 




Fig. 47. Adjustment of the Wyes. 



ADJUSTMENTS OF THE DUMPY LEVEL 9I 

the Level Tube Perpendicular to the Vertical Aids of the Instrument. 
Bring the two clips down over the telescope and fasten them. 
Level the instrument, bring the bubble precisely to the middle 
erf 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. ABJDSTlfENTS OF THE DUMPY LEVEL, 

126. ADJUSTMEHT 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. 

127. 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. 

Z28. THE DIRECT, OR " PEG/' AD JUSTMENT. — To make the 
Une 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 will 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 wUl be 
found 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 diflFerence 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 B- 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 » 4.062 

Rod-reading on B » 5.129 

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

Rod-reading on B « 5.076 

Rod-reading on A — 4.197 

Diff. in elev. of B and A » 0.949 

Mean of two diff. in elev, =* 1:5-Z ?!?!? « j.ooS true diff. in eleV. 

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 HAND 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. 




Fig. 49. Peg Adjustment for Hand Level. 

when the bubble is in the middle. The level is then taken to 
S, 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. COlfMON 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. COlfMON 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. 

v/ I. 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 zX A\ — Instrument at ^ : — 

B. S. on Aj 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 instmment remaining 
3.900 above B, Was the line of sight inclined upward or downward ? How 
much ? 



ERRORS AND MISTAKES IN LEVELING 95 

3. The target on a Boston rod has been disturbed and it is desired to find out 
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 efifect 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 5.098 is taken on a point 400 ft. away, 
then a F. S. of 3.260 is taken on a point 906 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. 
SXJRVEYmG METHODS. 



PART 11. 

SURVEYING METHODS. 

CHAPTER V. 

ItANB SUflVBTZNO. 

132. SURVEYING FOR AREA. — In surveying a field for the 
puqpose 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 surve)ring 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 CHAIN. — 

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 
^nution 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 Pack) 



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Fig. 60. Notes of 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 TRAIVSIT 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 stfll 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 corners 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 £F 
parallel to AD by making DF =s 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 



i' M " r 

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

angle at A, It is evident that the values of AI/3nd DFznA of 
i4/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 comers so that the telescope can be focused on it. In 
this way all the corners 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 comers 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 sur\'ey, but the 



SURVEY BY A TRAVERSE 



103 



fiilktr , 

J.H.BmoHey£skjie-Clinhn,/V.Y ^n^ 



Beaniy of£F fmmplan ofR/rk ComJ^vtMaidkin) 



Tipe .oasiooshor/: 

207.42C?> ' \ 




Fig. 52. Notes of Survey with Transit and Tape. 



104 



LAND SURVEYING 



[Chap. V. 



Survey d/Land c^S/Vas Cokjmr/r, Ai/icrv/A Af///Sj Afe. 
^Seansf x/u/y6y9o6. 




Fig. 58. Notes of Survey with Transit and Tape. 



SURVEY BY SINGLE SET-UP 10$ 

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. Irregtdar 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 off by 
eye. 

138. SURVEY OF A FIELD BY A SINGLE SET-UP OF 
THE 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 comer. 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 frcJm 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 sides 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 
comers of such land. 

141. Hsth3d of Procedure. — 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 

horizontal ties which intersect at 

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- 

fully recorded, usually by means ^^^ ^ Approximate Ties. 

of a sketch Fig. 54 shows a 

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





lo8 LAND SURVEYING [Chap. V. 

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 
surveyor should mark carefully 
by tack or crow-foot the exact 
Fig. 55. Exact Ties. points from which measure- 

ments (taken to j^ ^^•) ^^ 

made, and record the entire information in the notes as shown 

in Fig. 55. 

143. Measurement 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 interior angle and 180°, 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 0° 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 coUimation 
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 vernier 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 toge.ther. The sum 
of the interior angles of the field should equal («-2) X i8o^ 
where n is the number of sides in the field. If the deflection 
angles are used the sum of all the right 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 
360^ 

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 FIELDWORK III 

rotigli 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 in 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^inj^ (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 j-q^-qq 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 jT^i^ny, and an accuracy of ^i^^^^j^^^ is sometimes reached. 

147. Organization 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 113 

a^ Stone bound. 

»*»«»- Monument. 

A Triangulation Station. 

SMu Stake. 

tk Tack. 

-- Naa. 

•fk. Spike. 

*. Drill-hole. 

cii Crow-foot (a mark like this N or n|^). 

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

c: Center. 

i Center line. 

<* Curb. 

CBt Catch basin. 

MK Manhole. 

Tu 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' 3o". It will 
be seen that this is not consistent with the foregoing. A more 



hb. 



114 LAND SURVEYING [Chap. V. 

proper way of reading this angle would be 6** 47^', 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- S3> P- I04> ar^ go^d 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 The Polytechnic^ 
the student journal of the Rensselaer Polytechnic Institute, Troy, N. Y., Janoaryi 
1894. 



DEED DESCRIPTION IIS 

(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. 

« Banning 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. 



11 6 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 stone 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 1° 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 adjacent property, no bearings or distances being given. 

151. JUDICIAL FUNCTIOlfS 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 
t>oundaries 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 



11 8 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. RERUimiNG 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 information 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 &t 
that time computed. Observations at different places and 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 thercr 
fore one well-defined line which is known to be one of the original 
boundary lines, a bearing taken on this line 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 Isogonic 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, the ^^rantor index and the^ran/e^ 
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 UHITED STATES SYSTEM OF SURVEYIlfG 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 work 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 b«ng accepted, are inspected by a corps of Eataminers 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.) 

jtwraillL SoRffi 



i 

i 

i 



T4M 



T3M 

Rie 



TSN 
RIE 



TIN 

RIE 



T4M 
RZE 



T3M 

nzt 



TtN 
REE 



TIN 
REE 



I 



2nd.pA)«AIU) 



I Iftt. 



T4N 
R3E 



T3N 
R3E 



TEN 
R3E 



TIN 
R3E 



T4R 
R4E 



T3H 
R4L 



TIN 
R4E 



TIN 
R4E 



i 






fWALLELNORlrt 



-i" 






T 



^ 



T 



Fio. 67. 



BASe-LINC 

Diagram Illustratino 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 mUe, 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 numbered, commencing with No. i at the northeast angle of 
the township, and proceed-j 
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 
wUl 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 


io 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


??> 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



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



124 LAND SURVEYING [Chap. 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 ^ 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 stilj 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 wiU 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 comers 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 olliis 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. 

157. '* Principal Meridian. — This line shall conform to a 
true meridian [Chapter VII] and wiU 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 J 

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

"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 meridian 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. 

i6o. " 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, 
b^^nning with those of the southwestern 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 corners 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 true 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 faUs 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 corners 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 foUows : 

" 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 comer 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 correction 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. 



Latilode. 



301035 
35 to 40 
40 to 45 
45 to 50 
50 to 55 
55 to 60 
601065 
65 to 70 



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



mile. 2 miles. 3 miles. 4 miles. 5 miles. 



2 

2 
2 

3 
4 



""'Example. — Latitude, 47**. Range line bears N. 0° 2' 
then parallel meridional section lines will be run as follows: 

From the corner for sections — 



E., 



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 80.00 chains the corner for 
sections 25, 26, 35, and 36. 



I30 LAND SURVEYING [Chap. V. 

" From the last-named corner, a random line wUl 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 corner 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 "random 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 corner for sections 25, 26, 35, and 36, the west and 
north boundaries of sections 25, 24, 13, and 12, will be jcstablished 
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 
comer 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 run 
paraUel to the range line as a true line, the quarter-section cor- 
ner will be placed at 40.00 chains, and a closing corner wUl 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 carefuUy 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, froni 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 estabhshed 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 always be run 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. 



>3>MV 



wa3'53W 



jLss ss n. 




SMb9l 



sesrs^t 



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 133 

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 line, with 
a true bearing, determined as above directed for random lines, 
setting the quarter-section comer at 40.00 chains and the section 
comer at 80.00 chains. 

" Quarter-section corners, 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 reason 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 comers at regular intervals of 80.00 chains, the 
subdivision of the township will be made from west to east, and 



134 LAND SURVEYING [CmAP. 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 corner 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 comersj 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 corners, 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 run, 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 mark. 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 action 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) ^^ 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 run 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 run on the land, is 
the boundary. 



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 chsiins, 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 comer to the next meander 
comer. 

" All courses reported are to be compass courses, taken or 
coupted 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 137 

meander lines should be given by the nearest quarter d^;ree. 
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 comers, 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 corner, 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 corner. 

" 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 comers ; 
and the precise relative position of witness comers to the true 
comers. 

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

"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. 



140 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. 

" lo. 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. 

" 17. 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 presence of iron ores ; and 



LIMITS FOR CLOSING I4I 

the position of such points 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 Unks 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 1 50 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 comers 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 everjrthing done and observed by himself 
and his assistants, pursuant to instructions, in relation to riui- 
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 sxu-veyor 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 theni 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 comer and its accessories, following the proper approved 
forai given in these instructions. 

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

" In all cases where a corner 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 mnning such lines, the 
deputy will note particularly in the 'general description' the 
character of townships on each side of the lines nm. 

"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 tme 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 black 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 wiU 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 comers 

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 gronnd, finnly set, and 
marked and witnessed as described by the surveyor general. 
Thence I run 

N. o*» 01' W., bet. sees. 35 and 36. 
Over level bottom land. 
4 .50 Wire fence, bears E. and W. 

20.00 Enter scattering eottonwood timber, bears E. and W. F. G. Alexan- 
der's house bears N. 28° W. 
29 .30 Leave scattering cottonwoods, bearing E. 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 ground, for witness cor. to \ sec. cor., mariced 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, 1} ft. high. 
W. of cor. 
40.00 Point for } sec. cor. in road. 

Deposit a marked stone, 24 ins. in the gromnd, for i see. 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 J ft. high, W. of cor. 
Thence along £. side of field. 

50 . 50 . N£. cor. of Pat. Curran's field, beais 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 eottonwood and willow undergrowth, bears N. 54° £. 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 N-E., 
R 20 £ S 36 on SE., 
S 35 on SW., and 

S 26 on N W. face ; with i notch on S. and E. faces ; from which 
An ash, 13 ins. diam., bears N. 22^ £., 26 Iks. dist., maiked T 1 5 

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

T 15N R20E S36BT. 
A walnut, 17 ins. diam., bears S. 64° W., 41 Iks. dist., marked 

T15N R20ES35BT. 
A Cottonwood, 13 ins. diam., bears N. 21}^ 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 eottonwood, sycamore, ash, and walnut; under- 
growth, eottonwood and willow. 
Dense undergrowth, 15.00 chs. 



MARKING THE CORNERS I47 

z66. Marldng 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 corners, as follows ; 

1. Standard township comers. 

2. Closing township comers. 

3. Comers common to four townships. 

4- Comers common to two townships only. 

5. Corners referring to one township only. 

6. Standard section comers. 

7. Closing section corners. 

8. Comers 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 comers. 

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 
comers which can only be leamed by experience in this kind of 
surveying. 



148 



LAND SURVEYING 



[Chap. V. 



167. To ESTABLISH A PARALLEL OF LATTTUDE.— A paral- 
lei 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 lines, each six miles long, 



T13Kli.2lE. 







3 



^WD Standard" wkRAU-tL wotcnj^i 




Fig. CO. Secant 



^ 3 s ^ ! ^ 

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 Sscant, and Offsets, in Feet, to the Paeallio. 

Ladtode in left-hand colamn and distance from starting point at top or bottom of the table. 



Utir 


Azimuths and offsets at — 


Defleo- 

tion Angle 

and nut. 


tvdc 


omilM. 


imile. 


zmile. 


zimUes. 


2mUes. 


2^ miles. 


3 miles. 


tan. to 
Bad. 66 ft. 




30 


1.93 H. 


89«»68'.7 
0.87 H. 


0.00 


89<'69'.2 
0.67 s. 


89°B9'J5 
z.zsS. 


89°59'.7 
Z.44S. 


90° (E. or W.) 
X.54S. 


3' 00".2 
0.69 i..s. 


31 


fBPWA 
a.ox N. 


89*»68'.6 
0.9Z H. 


89«68'.9 
0.00 


89° 59' .2 
0.70 S. 


89°59'JJ 
Z.30 S. 


89°69'.7 
Z.50 S. 


90° (E. or W.) 
Z.60 S. 


3' 07" .4 
0.73 ins. 


33 


89«58'.4 
2.09H. 


89«68P.6 
0.94 K. 


89»68'.9 
0.00 


89°69'.2 
0.73 s. 


89°69'.6 
Z.25 s. 


89°59'.7 
Z.56 S. 


90° (E. or W.) 
Z.67 s. 


y 15".0 
0.75 ins. 


33 


8B0 58'.3 
3.17 K. 


89° 58^ 
0.97 H. 


89«68'.8 
0.00 


89° 89'.! 
0.76 S. 


89°69'.4 
z.30 S. 


89°59'.7 
Z.62 8. 


90° (E. or W.) 

1.73 s. 


3' 22".6 
0.78 i.is. 


34 


2^5 H. 


89«»68'.6 
i.oi R. 


89<> 68^.8 
0.00 


89<'fi9M 
0.79 s. 


89°e9'.4 
Z.35 s. 


89°69'.7 
Z.69S. 


90° (E. or W.) 
1.80 S. 


3' 30".4 
o.8z ins. 


35 


89»58'.2 
2.33 N. 


89»68'.6 
Z.05R. 


89o68'.8 
0.00 


89° 69'.! 
0.82 S. 


89°59'.4 
Z.40 S. 


89° 59' .7 
Z.75S. 


90° (E. or W.) 
1.87 s. 


3' 38" .4 
0.84 ios. 


36 


3.43 H. 


89°68'.4 
1.09 R. 


89«»58'.7 
0.00 


89°59'.0 
0.85 s. 


89°59'.4 
Z.46 S. 


89° 59' .7 
Z.82 S. 


90° (E. or W.) 
1.94 s. 


3' 4e".4 
0.87 ins. 


37 


3.51 H. 


89»58'.3 
1. 13 R. 


89«68'.6 
0.00 


89°68'.9 
0.88 S. 


89° 69' .3 
1.5Z S. 


89°59'.7 
Z.89S. 


90° (E. or W.) 
2.01 S 


3' 66".0 
0.90 ins. 


38 


a.6i H. 


89<»68'.3 
1.17 R. 


89<>68':6 
0.00 


89°68'.9 
0.91 S. 


89°69'.3 
Z.56S. 


89°59'.7 

X.95 s. 


90° (E. or W. 
2.08 S.) 


4' 03".6 
0.93 ins. 


39 


89«»B7'.9 
2.70 H. 


89o68'.2 
1.21 R. 


89»58'.6 
0.00 


89°58'.9 
0.94 s. 


89°59'.3 
Z.62S. 


89°69'.7 
2.02 S. 


90° (E. or W.) 
2.16 S. 


4' 12".6 
0.97 ins. 


40 


89«»87'.8 
3.79 H. 


89«68'.l 

1.25 R. 


0.00 


89°58^9 
0.98 S. 


89° 59' .3 
1.68 8. 


89° 69' .7 
2.X08. 


90° (E. or W.) 
2.24 s. 


4' 21".6 
Z.00 ins. 


41 


89»B7'.7 
3.89 H. 


89«68'.0 
1.30 R. 


89«68'.4 
0.00 


89°68'.8 
z.oaS. 


89°59'.2 
Z.74S. 


89°59'.6 
2.Z7 S. 


90° (E. or W.) 
2.33 S. 


4' 31" .2 
Z.04 ins. 


43 


»»B7'.7 
3.00 R. 


89»58'.0 
1.35 H. 


89«68'.4 
0.00 


89°68'.8 
Z.05 s. 


89°69'.2 
Z.80S. 


89°69'.6 
2.25 s. 


90° (E. or W.) 
2.40 S. 


4' 40".8 
Z.08 ins. 


43 


89«»67'.6 
3.IXN. 


89»68'.0 
Z.40N. 


89»68'.4 
0.00 


89°58'.8 
Z.08S. 


89°69'.2 
Z.86S. 


89° 59'.6 

3.33 s. 


90° (E. or W.) 
2.48 S. 


4'50".8 
Z.Z2 ins. 


44 


3.32 R. 


89«»57'.9 
X.45K. 


89»58'.3 
0.00 


89°68'.7 
Z.Z2 8. 


89°69'.2 
Z.93 s. 


89°59'.6 
2.41 S. 


90° (E. or W.) 

2.57 s. 


6' 01".0 
1.16 ins. 


45 


89»67'.4 
3.33 H. 


89»67'.8 
1.50 R. 


890 58'.3 
0.00 


89°68'.7 
Z.16 S. 


89° 59'.! 
3.00 S. 


89°59'.5 
2.49 s. 


90° (E. or W.) 
2.6^8 


5' 11" .8 
1.20 ins. 


46 


89»67'.3 
3-44 R. 


89«67'.7 
1.55 H. 


89«68'.2 
0.00 


89°68'.6 
Z.2Z S. 


89° 59'.! 
2.07 s. 


89° 59'.5 

3.59 s. 


90° (E or W.) 
2.76 S. 


6'22".8 
1.24 i.iS. 


47 


«90 87'.2 

3.57 R. 


89«»67'.6 
1.61 H. 


89«»68'.l 
0.00 


89°58'.6 
Z.25S. 


89° 69'.! 
2.Z4S. 


89°59'.6 
3.67 s. 


90° (E. or W.^ 
2.86 S. 


6* 34".2 
z.28in8. 


48 


89»67M 
3.70 R. 


89»57'.5 
1.66 R. 


89«68'.0 
0.00 


89°58'J5 
Z.30S. 


89°69'.0 
2.22 S. 


89°59'.5 
3.78 S. 


90° (E. or W.) 
2.96 S. 


6' 46".2 
Z.33 ins. 


49 


8B»57'.0 
3^3 R. 


89»67'J5 
1.73 R. 


89«»68'.0 
0.00 


89°68'a5 
X.34 s. 


89°59'.0 
2.30 S. 


89°69'.5 
2.87 s. 


90° (E. or W.) 
3.06 S. 


5' 58".6 
1.38 ins. 


50 


80» 50^.9 
3.96 R. 


89» 67'.4 
Z.78R. 


89° 67'.9 
0.00 


89°68'.4 
Z.39 s. 


89°69'.0 
2.38 s. 


89°59'.6 

3.97 s. 


90° (E. or W.) 

3.17 s. 


6' 11".4 
1.43 ins. 


Uti. 


6 miles. 


SimilM. 


smiles. 


4(mUes. 


4 miles. 


3imUes. 


3mUe8. 


Deflec- 
tion Angle 
and nat. 


tade. 






Azin 


luths and < 


>ff8etsat- 


^ 




tan. to 
Had. 66 ft. 



N9 



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 instnunent is set up south of the township comer 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-mile 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 




jt^^ [0ff9ete--rabte C \ ^i^' 

Fig. 61. Tangent Method for Establishing a Parallel of 

Latitude. 

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 1 51 

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 
bearmg 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 
offset 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 lines in a level open country, where no 
intersections with topographical' features will be required ; but, 
in all cases the secant method will be found most convenient." 



152 



LAND SURVEYING 



[Chap. V. 



TABLE 6. 
Azimuths op the Tangent to the Parallel. 

[The asimuth is the snaller anslc the tanceot nukes with the true mendian and alwajs 
measured from the north ano towards the tangenttal pomts.] 



Lati- 
tade. 


zmile. 


a 


milM. 


3 


miles. 


4 


mUM. 


3 


milM. 


6 miles. 


o 


o 


/ 


ff 


o 


/ 


// 


o 


* 


„ 


o 


/ 


/• 


o 


/ 


// 


o 


/ 


// 


30 
3S 
3a 


89 

88 
89 


69 
60 
60 


300 
28.8 
27.6 


89 
89 
88 


68 
68 
68 


68.9 
67.6 
66.0 


88 
89 
89 


68 

68 
68 


29.9 
26.3 
22.6 


88 
88 
89 


67 
67 
67 


60.0 
66.0 
60.0 


89 
89 
89 


67 
67 
67 


29.9 
23.8 
17.6 


88 
89 
89 


66 
66 
66 


608 
filS 
V6.0 


33 
34 
35 


89 
88 
88 


68 
60 
60 


26.2 
24.9 
2^.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 


449 
39.7 
34.4 


89 
88 
89 


57 
57 
66 


11.2 
04.6 
68.0 


89 
89 
89 


66 
66 
66 


37.4 
296 
21.6 


3« 


89 
89 
89 


60 
60 
60 


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 
67 


28.9 
23.3 
17.6 


89 
89 
89 


66 
66 
66 


61.1 
44.1 
36.0 


89 
89 
89 


66 
66 
66 


13.4 
05.0 
66.3 


30 

40 
4X 


89 
89 
89 


69 
69 
60 


17.9 
16.4 
14.8 


89 
89 
89 


68 

68 
68 


36.8 
32.8 
29.6 


89 
89 
89 


67 
67 
67 


53.7 
40.2 
44.4 


89 
89 
89 


67 
67 
66 


11.6 
05.6 
60.3 


89 
89 
89 


66 
66 
66 


29.6 
21.9 
14.1 


89 
89 
89 


65 

65 
66 


47.6 
38.3 
28.9 


4a 
43 
44 


89 
89 
89 


69 
60 
60 


13.2 
11.6 
08.8 


89 
89 
89 


68 
68 
68 


26.4 
23.1 
19.6 


89 
89 
89 


67 
67 
67 


39.6 
34.6 
29.6 


89 
89 
89 


66 
66 
66 


62.8 
46.2 
39.3 


89 
89 
89 


66 
66 
66 


06.0 
67.7 
40.1 


89 
89 
89 


65 
66 

54 


19.2 
09.2 
58.9 


47 


89 
89 
89 


60 
60 
60 


08.0 
06.2 
04.3 


89 
89 
89 


68 

68 
68 


16.1 
12.4 
08.6 


89 
89 
89 


67 
67 
67 


24.1 
18.6 
12.9 


89 
89 
89 


66 
66 
66 


32.1 
21.8 
17.1 


89 
89 
89 


66 
66 
66 


40.2 
31.0 
21.4 


89 
89 
89 


54 

54 
54 


48.2 
37.2 
26.7 


4« 

49 
50 


89 
89 
89 


60 
60 

68 


02.3 
00.2 
68.1 


89 
89 
89 


68 
68 
67 


04.6 
00.6 
66.2 


89 
89 
89 


67 
67 
66 


06.0 
00.7 
64.3 


89 
89 
89 


66 
66 
66 


09.2 
00.9 
62.6 


89 
89 
89 


66 
66 

64 


11.6 
01.2 
60.5 


89 
88 
89 


54 

54 
63 


13.8 
01. « 
48.6 



Lati- 
tude. 


7 miles. 


Smiles. 


9 


miles. 


zo miles. 


II miles. 


la 


miles. 


o 


o 


/ 


„ 


o 


, 


If 


o 


, 


f» 


o 


, 


„ 


o 


/ 


// 


o 


/ 


t* 


30 
3« 
3a 


89 
89 
89 


66 
66 
66 


29.8 
21.3 
12.6 


89 
89 
89 


66 
65 

66 


69.8 
50.0 
40.0 


89 
89 
89 


66 
65 
66 


29.8 
18.8 
07.6 


89 
89 
89. 


64 
64 
64 


69.7 
47.6 
35.1 


89 
89 
89 


54 
54 
54 


29.7 
16.3 
02.6 


89 
89 
89 


63 
63 
63 


69.7 
45.1 

30.1 


33 

34 
35 


89 
89 
88 


66 
66 
65 


03.6 
54.6 
46.2 


89 
89 
89 


66 
66 

66 


29.9 
19.4 
08.8 


89 
89 
89 


54 
54 
54 


56.1 
44.4 

32.3 


89 
89 
89 


54 
54 
63 


22.3 
09.3 
66.9 


89 
89 

89 


63 
53 
63 


48.6 
34.2 
19.6 


89 
89 
89 


63 
62 
52 


14.8 
60.1 
43.1 


36 


89 
89 
89 


66 
66 
66 


36.6 
25.8 
16.7 


89 

89 
89 


64 

54 
54 


67.8 
46.6 
36.1 


89 
89 
89 


54 

54 
53 


20.0 
07.4 
54.5 


89 
89 
89 


63 
63 
63 


42.3 
28.2 
13.9 


89 

89 
89 


53- 
62 
62 


04.6 
49.1 
33.2 


89 
89 
89 


62 
52 
51 


26.7 
09.9 
62.6 


39 
40 
41 


89 
89 
89 


66 
64 
54 


06.4 
64.7 
43.7 


89 
89 
88 


54 
54 
63 


23.3 
11.1 

68.5 


89 
89 
89 


53 
53 
53 


41.2 
27.6 
13.4 


89 
89 
89 


62 
62 
62 


69.1 
43.8 
28.2 


89 
89 
89 


62 
52 
61 


17.0 
00.2 
43.0 


89 
89 
80 


61 
61 
60 


34.9 
16.6 
67.8 


4a 
43 
44 


89 
89 
89 


64 
54 

64 


82.4 

20.8 
08.7 


89 
88 
88 


53 
63 
63 


45.6 
32.3 
18.5 


89 
89 
89 


62 
62 

62 


68.8 
43.8 
28.4 


89 

88 
89 


62 
61 
61 


12.0 
66.4 
38.2 


89 
89 
89 


61 
51 
60 


25.2 
06.9 
48.0 


89 
88 
89 


60 
60 
49 


38.4 

18.6 
57.8 


47 


89 
89 
88 


63 
63 
63 


66.3 
43.4 
30.0 


89 
89 
89 


53 
62 
52 


04.3 
49.5 
34.3 


89 
89 
89 


62 

61 
61 


12.3 
65.7 
38.6 


89 
89 
89 


61 
61 
60 


20.4 
01.9 
42.9 


89 
89 
89 


60 
50 
49 


28.4 
08.1 
47.2 


89 
89 
89 


49 
49 
48 


36.4 

14.3 

51.4 


4« 

49 
SO 


89 
89 
89 


63 
63 
52 


16.1 
01.7 
46.6 


89 
89 
89 


52 
62 
61 


18.4 
01.9 
44.7 

t 


89 
89 
89 


61 
51 
60 


20.7 
02.1 
42.8 


89 
89 
89 


60 
60 
49 


23.0 
02.4 
40.9 


89 

89 
89 


49 
49 
48 


25.3 
02.6 
39.0 


89 
89 
88 


48 

48 
47 


27.6 
02.8 
37.1 



/ 



TANGENT METHOD 

TABLE & 
Okf-sets, in Chains, from Tangent to Parallel. 



Sf}^' / * BMile, I a miles. 


3 milei. 


4]llilM. 


5 miles. 


6 mile 


_ / CrJka**^- \ CA4M.iHS. 


Chanu, 


Chains. 


Chains. 


ChaU 


31 


0.000 

ooe 
0.00^ 


0.023 

0.024 
0.02& 


0.063 
0.065 
0.067 


0.00 
0.10 
0.10 


0.14 
0.16 
0.16 


0.21 
0.22 
0.23 


33 

34 
3S 


O.OOfT 
0.OO7 

o.oov 


O.026 

O.027 
O.028 


0.060 
0.061 
0.064 


0.10 
0.11 
0.11 


0.16 
0.17 
0.18 


0.2ft 
0.26 
0.26 




O.OOT 
0.008 
0.OO8 


O.029 

O.081 
O.032 


0.066 
0.068 
0.071 


0.12 
0.12 
0.13 


0.18 
0.19 
0.20 


0.26 
0.27 
0.28 


M 0.OO8 

5 o.ooe 


O.033 
O.094 

O.036 


0.074 
0.076 
0.079 


0.13 
0.13 
0.14 


0.20 
0,21 
0.22 


0.20 
0.30 
0.32 


4» 
43 
44 

% 

47 

48 
49 
50 


0.000 

0.OO9 
0.010 


O.036 
O.038 
O.039 


0.082 
0.066 
0.068 


0.14 
0.16 
0.16 


0.23 
0.24 
0.24 


0.38 
0.34 
0.35 


0.010 
0.010 
0.011 


O.OiO 
O.042 
0.044 


0.001 
0.004 
0.097 


0.16 
0.17 
0.17 


0.26 
0.26 
0.27 


0.36 
0.37 
0.39 


0.011 
0.012 
0.012 




O.046 
O.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 


Uti- .miles. 1 
tude. ^J 1 . 

I SIS 


Smilei. 


gmUei. 


lomilei. 


II miles. 


lamile 


0.37 
0.39 
O.40 


Chains. 

0.47 

0.48 

^ 0.61 


Chains, 

0.68 
0.60 
0.63 


Chains. 

0.71 
0.74 
0.76 


Chain 

O.ft 
0.81 
0.9 


3* 
33 

34 


33 
0.3B 


0.42 
0.43 
0.45 


0.53 
0.65 
0.67 


0.66 
0.68 
0.70 


0.79 
0.82 
0.86 


0.91 
0.9( 

i.a 


39 

36 

39 


0.3^ 
0.38 


0.47 
0.48 
O.fiO 


0.69 
0.61 
0.64 


0.73 
0.76 
0.78 


0.89 
0.91 
0.96 


1.01 
1.1 
1.1- 


0.40 

o.« 

0.43 


• 0.62 
O.M 
0.66 


0.66 
0.68 
0.70 


0.81 
0.84 
0.87 


0.99 
1.02 
1.06 


1.1 
1.2' 
1.9 


4' 
43 


0.44 
0.40 
0.48 


0.68 

O.eo 

0.62 


0.73 
0.76 
0.79 


0.90 
0.93 
0.97 


1.09 
1.14 
1.18 


1.3 
1.31 

l.« 


44 

47 

4» 
49 

SO 


0.40 

o.&i 

0.63 

I 0.66 
\ 0.57 


0.64 
0.66 
0.68 

0.71 
0.74 
0.77 


0.8t 
0.84 
0.87 

0.91 
0.93 
0.»7 


1.00 
1.04 
1.07 

1.12 
1.16 
1.20 


1.22 
1.26 
1.31 

1.36 
1.40 
1.46 


1.41 

1.61 
1.6( 

1.6 
1.6 
1.7 






— — 













IS4 



LAND SURVEYING 



[Chap. V. 




Fig. 



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 ^ and J?. AT^'^ BT 
are lines tangent to the meri- 
dians at A and By 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 difiFerence in longitude 
of points A and B, The angle 



between the meridians at A and B is the angle A TB 

In the triangle BOO', 

BCy = BO cos O'BO 

= BO cos BOQ 



In the sector AffB, 
AB 



Ba 



= angle AffB 



In the sector A TBy 
AB 



~— = angle A TB (approximately) 
BT 



But 



^r= 



BO' 



BO' 



sin BTO sin BOQ 
AB 



.-. angle ^7:5=^ sin BOQ 

=angle 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. 
Lbngth of a Degree in Longitude. 





D^reedLoBgi- 




Deg;ree of Longi- 




Degree of Longi- 


LaL 


tnde 


Lat. 


tude. 


Lat. 


tude. 




Statata MUet. 




Statute MUes. 




Statute Miles. 




69.160 


30 


59 944 


60 


34.666 




.150 


31 


«'334 


61 


33.615 




.119 


32 


58.706 


62 


32.553 
31 .481 




.066 


M 


.060 


63 




68.992 


34 


57.396 


64 


30.399 


\ 


".■5? 


% 


56.715 
.016 


§ 


29.308 
28.208 


\ 


.647 


n 


55-300 


67 


27.100 


.491 


38 


54.568 


68 


• 25.983 


9 


.314 


39 


53 819 


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 


«3 


.400 


43 


50.659 


73 


14 


.120 


44 


49.830 


74 


19.122 


II 


66.820 
.499 


% 


48.986 
.126 


?i 


Tt 


17 


.158 


47 


46 .362 


77 


•15.607 


t8 


65.797 


48 


78 


14.425 
13-238 


19 


.416 


49 


45 .459 


79 


20 


65.015 


50 


44.542 


80 


12 .047 


21 


64.594 


51 


43 .611 


81 


9.656 


22 


.154 


52 


42 .667 


82 


23 


63 695 
.210 


53 


41.710 


f3 


8.456 


24 


54 


40.740 


84 


7.253 


% 


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 


1 .211 



CHAPTER VI. 

TRAVSRSB LINES. — LOCATION OF BUILDINOS. — MISCIO.- 
LANEOUS 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 
2ifull station and any fractional distance is called \\v^pliis. 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 statfon with the plus station in addition, e.g., 
the station of a pomt 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 KOT 
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. 129. 

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 

(Left-hand Pagb.) (Right-Hand Pack.) 



r 


/ladmtm. Jfo/f rx.. 


^ 


StOL 


fbi/rt- 


OefiAyk 


^ 


'^'ISSSlg 




IQ 
9 
6 

7 
6 

S 

4, 




/er43l 




NcToes 










1 


3. 












^^ 




+42 










1W^ 


<< 




2 










^ 






/ 








HifUw 


M2r4sn 




0^- 


- 
















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 line 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 triangtilation 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 1 59 

LOCATION OF BUILDINGS FROM TRANSIT LINE. 

176. METHODS OF LOCATING 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 great importance, while 
in topographic 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. 



l6o LOCATION OF BUILDINGS [Chap. VX 

An offset is the distance from a line, usually at right angles. 

A swing offset is the perpendiciJar 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 Suggestions. — By whatever method the 

buildings are located the following suggestions should be carried 
out. 

(i) 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) Ties should intersect at an angle as near 90® as practi- 
cable, and never less than 30®. 

(4) One or more cfieck 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 serve 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 corners 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 ^ 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 Hne 




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 cpmpass 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 AOy Al, 
Bly and B2. 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 11. 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 
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 B2, 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. VI. 

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 figure 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 Fare 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 CS, but it is much longer. 





TRANSrr LINE ^ v "•^^■.^. 



Stal 2 L 

Fig. 65. 

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 in. 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 comer of the building and scal- 
ing from the other station the tie to the same comer. Then the 




9ta.2 

Fig. 66. 

other comer is plotted in the same way or by using the side of the 
buflding 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 m Fig. 6t. The range ties through A are sufficient to 




Fig. 67. 

locate the building, 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, 
BS 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 i4 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 2C or the 
range tie DC produced. 



l64 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 strike at F^ and AF should scale 
the measured distance. 

i8s« Example V. Buildings at a Long Distance from the 
Transit Line. — It is evident that in this case (Fig. 68) the tape 









TRANSIT 

iJJi 2 3 

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 xule the features near the transit line are the important 
ones. This method of "cutting in " the corners 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. Buildings 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 




J ^v^ ^ J ^\ifif{f(((<m(iM 



StD4 



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 KQ and the two range ties 4/ and C/, 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 AF a,nd 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 



i 

I 



i66 



LOCATION OF BUILDINGS 



[Chap. VI. 



poorer intersection at L, 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 



^ ^ 

stoa 




^n, IVwAWi^^^ LINE 



3 

Fig. 70. 




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 and EHy and by the dimensions of the build- 
ing; FA is assumed parallel to GB, If the angle F is a right 
angle the tie EF may be taken instead of the range ties FH and 
EHy but even when /^ 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 VIII. 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 AB^ 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. Location 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. 



I68 



LOCATION OF BUILDINGS 



[Chap. VI. 




Fig 72. 



RANDOM LINE 169 

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 
pointy e.g.. A, 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 
ofiFset 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 A By 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 ^4 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 as i42) 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 5, there is no point on the 
line BA to use as a backsight. But any point such as E can be 

readily set on the line AB by making the oflFset ME = BY^^. 

AY 

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 ^4 to 5 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 ^4. If it is desired to mark 
the line AB by 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 I /I 

The instrument is then set up at B and a right angle ABF laid 
off with the transit. BF\& made any convenient distance which 
will bring the auxiliary line beyond the building. Similarly 
point E is set opposite point Ay and sometimes a second point E' 
opposite -4', points A and A' 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 Fand backsighted on E, the 
sight is checked on E\ the telescope inverted, and points G^ H\ 
and H set on line. Leaving the telescope inverted, another 
backsight is taken on A^ 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. 



TWANSiT B ^^C LINE d 




E^ AUXILIARY F Q LINC 



Fig. 74. 

Then by setting up at Cand sighting ahead on Z>, (DH^ GC), 
and checking on point Z>', (jyiT = 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 AEy 
A'E\ DHf and D'lP 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 A'E' and 
lyir are not absolutely necessary, but they serve as desirable 
checks on the work and in first-class surveying they should not 
he omitted. For obvious reasons the offsets AE and Z>-^ 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 J9, or a foresight of 
some sort (Art. 64, p. 52) should be set on line beyond the house 



172 MISCELLANEOUS SURVEYING PROBLEM [Chap. VI. 

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 onQ 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. 

19s. Bisection Method. — A method which is economical in 
fieldwork but not very accurate is the following. In Fig. 75 
the instrument is set up at Ay backsighted on the transit line, 
and equal angles turned off on each side of the transit line pro- 



MEASURING AROUND OBSTACLES 



^n 



duced. Points B! and C are carefully set on one of these lines 
and at convenient distances from A^ and on the other line points 




Fig. 75. 

ff' and L" are set at the same distances from A, Then point B 
is placed midway between V and ^", 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. 6i, p. 50.) 
In this case the distance AB can be computed from the formula 

"b^b^ 

AR — AB =s ■ - (approximately). (See foot-note, p. 339.) 
2AB , 

196. Measuring Around a Small Obstacle. — In Fig. 76 the 




Fig. 76. 

line AB nms 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^ ^^ A,EB^ ^^ ' (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. yy) 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, back sighted on A, and an angle of 120^ laid 
off ; the line BC is made long enough so that when the instru- 
ment is set up at Cand 60*^ is laid off from it, CZ> 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 off 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. 

ig8. 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. 

199. 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 Hne. The line BC is 
laid off at 90° to £A and is intersected with the transit line and 
the distance ^C 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 transit man. 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 lo 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 pdnt 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 



176 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 swung slowly in 
an arc ai. 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 AC may 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 ^ 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 
ofif at right angles to the transit line and of any convenient length. 



TRANSIT / 


Vff^ ?)w 


LINE 


\ 


A i 


7/ Ilia 


^^^^ 


H. 


1 


U //| 




'^ 


N 


mPQNDJPJ^ 





Y\Q. 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 points 
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 ABy the inaccessible distance. In the tri- 



FiG. 82. 

angle ACBy angle ABC can be computed, which, together with 
the measured angle J5CZ?, 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 179 

This problem occurs when the distance between two triangu- 
lation stations, A and By and the azimuth oi AB are desired and 
when it is inconvenient or impossible to measure the line AB or 
to occupy the points with the transit. 

205. To Obtain the Inaccessible Distance Between Two Ac- 
cessible Points by Observations on Two Inaccessible Points of 
Known Distance Apart. — In this case (Fig. i2) A and B are the 
two accessible points and C and D are the two inaccessible points 
but the distance DC is known '; the distance AB is required. 
With the transit at Ay 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 ABj it is not easy to directly determine AB in terms of CD \ 
it will be well therefore to use an indirect method. Assume 
AB 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 
desu-ed to use this line CD as a base-line for a survey, two pdnts 
A and B could be assumed, and the distance between them and 
the azimuth of AB could be found by this method. 



> 



CHAPTER VII. 

OB0BRVATION8 FOR BffHRIDIAN AND LATITTTDB. 
OBSERVAXIOHS FOR HERIDIAlf. 
206. TO ESTABLISH A TRUE MERIDIAN LDffE BY OBSERVA- 

Tioir oir 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 north pole of 
the celestial sphere^ a. point in 
the sky in the prolongation of 
the earth'^ 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 \\.% 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, 

180 






Lottw ^lonjtf KM 




Fig. 83. 



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 



o 

w 

M 

H 



O 



n 
o 



> 

O 

CO 

n 

> 
9 



w 

H 
W 

s: 

W 

o 

o 
> 

H 

o 



•NOIXVNI WinD HSMOl iv sisiHVTOd 

NSHM SNOIXVTiaXSNOO %IJL dO NOIXISOJ 




Ursa MUior 




Ursa Major 



2 
O 

H 

< 
O 

o 
•-] 

N 

N 
H 

CO 

< 

H 



(IS 

< 
•-) 
O 
Oh 

X 

CO 

O 

H 

< 



H 

CO 

O 

u 

h 
O 

O 



O 



Fig. 84. 



Position op the Constellations when 
Polaris is at UPPER CULMINATION. 

Relathte Position of the Constellations near the 
North Pole. 



1 82 OBSERVATIONS FOR MERIDIAN [Chap. VII. 

condition which is most favorable for an accurate observation. 
At culmination 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 S * Cassiopeia^ the lower 
left-hand star of the W, to f Ursce Majoris^ the middle star of the 
Dipper handle, passes very close to Polaris and also to the pole 
itself. 

207. OBSERVATION FOR MERIDIAN ON POLAI^IS 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 tube 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 apiproaching 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: — 

c- o^ , 'T' T> Sin Polar Distance of Star * 

oin Star s True Beanng = -p, — = — ; — = 

Cos Latitude 

The mean polar distances for the years 1906 to 1920 may be 

* This equation may be derived as follows ; in Fig. 83, let P represent the 
pole, Z the zenith, and E the position of the star at elongation. Then by spherical 
ttgonometry, 

sin PZE sin PE 

sin ZEP^ An ZP' 

^Qt PZE is the angle between the two vertical circles and equals the bearing. 
ZEpmm^o ijecause ZE is tangent to the circle WUEL^ which represents the path 
^ Polaris. PE 18 the polar distance and ZP may be shown to be equsd to 
90** - latitude. 

Hence, sin />Z^-?!!L^. 

cos lat. 



1 84 



OBSERVATIONS FOR MERIDIAN [Chap. VIL 



TABLE 8. 
Mean Polar Distances of Polaris.* 



Yew. 


Meam Polar Distance. 


Year. 


Mean Polar Distance. 




' it 




» „ 


1906 


I II 41.05 


1914 


I 09 12.07 
1 08 53.51 


1907 


I II 22.37 


1915 
1916 


1908 


I II 03.71 


I 08 34.97 


1909 


I 10 45.07 
I 10 20.44 


1917 


I 08^ 16.45 


1910 


1918 


I 07 57-94 


1911 


I 10 07.82 


1919 


I 07 39.45 


1912 


I 09 49.22 


1920 


I 07 20.98 


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 oJ9F 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. 85), and calculating the per- 
pendicular distance £C which must be laid off at the 
north stake B to give a meridian AC, 



Fig. 



* 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 aximuth. If the azimuth is to be 
within about one minute of the true value the polar distance 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 g^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- 
ment. 



OBSERVATIONS AT CULMINATION 185 

208. OBSERVATION FOR MERJDIAH ON POLARIS AT CULMI- 
HATLON. — At the instant when Polaris is above the pole the 
star f Ursa Majoris will be almost exactly underneath Polaris. 
When Polaris is below the pole S 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 caltulated 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 S 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 wfll 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 Culmination and Elonga- 
tion. — The approximate times of culmination and elongation of 
Polaris for the ist and 15 th of each month in the year 1907 may 
be found in Table 9. 

TABLE 9. 
Approximate Timbs of Culmination and Elongation of Polaris 

COMPUTKD for THR 9OTH MERIDIAN WEST OF GREENWICH, 

FOR THE Year 1907. 



Date. 



tqo7 

Jan. I . 

" 15 • 

Feb. I . 

Mar. I . , 

" 15- 
Apr. 1 . 

May I . 

, " 'S • 
Jun. I . 

" 15 • 
Jul. I . 

" 15.- 
Aug. I . 

Sep. I . , 

" 15 • 
Oct. I .. 

" IS -■ 
Nov, I . . 

" 15 • 
Dec. I . . 

" IS • 





Western 


Lower 


Eastern 


Elongation. 


Culmination. 


Elongation. 


k m 


A m 


A m 


A m 


6 44 


12 39 


18 42 


49 


S 49 


II 44 


17 47 


23 50 


4 41 


10 36 


16 39 


22 42 


3 46 


9 41 


15 44 


21 47 


2 51 


8 46 


14 49 


20 52 


I 56 


7 SI 


13 54 


19 57 


49 


6 44 


> 12 47 


18 50 


23 50 


s 40 


11 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 50 


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 S4 


9 57 


14 45 


20 40 


2 47 


8 50 


13 50 


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 

8 47 


15 46 


21 49 


3 56 


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 87 

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 ^ 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 24s e.g., 18^ would mean 
& A.M. The tabular numbers are nearly correct for the Stand- 
ard Meridians, i.e., the 75th, 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. 68). 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- 
tracting it if the place is east of the standard meridian. 

210. MERIDIAII 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-lme. 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 pj9F 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 
carries 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 Aour 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 sun*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 /> — sin Z cos / 



where Z « the azimuth, or true bearing ; / « the hour angle ; Z •- the latitude ; 
D = the declination = 90° — 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 perfonn 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. 

ObBervation on Polaris for azamnth April 15, 1908. Latitude 38^ 58'. Longi. 
tade 92^ 25^. Angle between a mark (approximately N.W.) and Polaris is 
repeated 6 times. Watch i™ 138 fast. The times are 





8h 35m 


40S 




8 37 


20 




8 38 


50 




8 39 


59 




8 41 


30 




8 43 


00 


Mean of 6 readings 


8 39 


26.2 


Watch fast 


I 


13 


True Central time 


8 38 


13 


Longitude of Standard Meridian 


6 



From Nautical Almanac, Right Ascension of ** Mean Sun " at Greenwich Mean 
Noon » ih 32111 579,82 ; Right Ascension of Polaris « ih 25°™ oi*.47 ; Declination 
of Polaris - + 88® 48' 52'' ; Correction from Table III (Nautical Ahnanac) for 
Greenwich Time — I4h 38™ = 2™ 24*. 2 

92° 25' = 6h 09»K 40S 
.', longitude correction = 09m 40s 

Mean of observed times 81* 38m 138 

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 25 01 

Hour Angle Polaris « 8h 38™ 54s 

i = i29*» 43' 30" 
log COS L » 9.8907 1 log sin L = 9.79856 

log tan D « 1. 684 1 3 log cos / « 9.80558 (n) * 

1.57484 9-60414 (n) 

37.570 - .4019 

.402 

37.972 

log sin / » 9.88600 

log denominator -3 1.57946 

log tan Z — 8.30654 

Z« I** 09' 37'' W. of N. 



* The » after the logarithm indicates that the number corresponding is negatire. 



190 OBSERVATIONS FOR MERIDIAN [Chap. VH. 

212. SOLAR OBSERVATIONS. — 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 aijgle 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 will 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 ALTTTUDES 
OF THE SUN IN THE FORENOON AND AFTERNOON. — This obser- 
vation consists in measuring in the forenoon the horizontal angle 
between 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 

m which X = the correction to the mean vernier reading, 
rf= 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. 20th. 30th. 

January +12" +22'' +32" +41" 

February +43 +49 +54 

March +57 +59 +59 +5^ 

April +58 +54 +49 +46 

May +45 +39 +39 +23 

June +21 +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. VBL 

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 maybe 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 if d is minus, 
and subtracted if d is plus, i.e., if the sun \% 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°oo'oo" Reading on Mark, o^oo'oo" 

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 7hi9n»3o8 Time 4^1 2mi^B 



MERIDIAN BY SOLAR OBSERVATIONS 



193 



A elapsed time «« 4b26in22> 
- 66°35'30- 
log sin / 9.96270 
log cos L 9.86902 

9-83172 

log 23o".9 2.36342 

2.53170 

correction 340". 2 •- 



Increase in declination in 4b26°>22>«> 
52'' X 4.44 - 23o"-9 



Mean circle reading — 79*51 '08" 
S40 



S'4o".2 

Azimuth of mark — 280° 14^32" 



S 79«45'28'' E 



214. OBSERVATION FOR MERIDIAll 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 o** 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 h in the formula given below. 



TABLE 11. 

Refraction Correction. 



Altitade. 


Refraction. 


Altitude. 


Refraction. 


.o« 


5' 19" 


200 


2' 39" 


II 


4 SI 


25 


2 04 


12 


4 27 


30 


I 41 


n 


4 07 


35 


I 23 


14 


3 49 


40 


I 09 


15 


3 34 


45 


58 


16 


3 20 


SO 


49 


17 


3 08 


60 


34 


18 


2 57 


70 


21 


19 


2 48 


80 


10 



194 OBSERVATIONS FOR MERIDIAN [Chap. VU. 

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. 216, 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 i' = J (Z + A +/) ; 
then 

cot' i Z = sin {s - 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 found. 



OBSERVATION FOR AZIMUTH 



19s 



Example. 

Obsbrvation on Sun for Azimuth. 

Latitude 42<> 21' N. Longitude 4^ 44m i8» W 

Time, Nov. 28, 1905, A.M. 



Horizontal Circle 


Vertical Circle 


Watch 


Vernier A B 






Maik 238«> 14' 14' 




A.M. 


Rig^t and Lower Limbs 311 48 48.5 


I4<»4i' 


8h 39m 428 


•* «* « " 312 20 20 


IS 00 


8 42 19 


The inst. reversed 






Left and Upper limbs 312 27 26.5 


IS ^1, 


8 4S 34 


- « - " 3" 52 51.5 


16 08 


8 47 34 


Mark 238 14 14 







Mean reading on Mark -• 238^ i4'.o 
" « " Sun — 312 21 .7 

Mark N. of Sun » 74. 

Observed Altitude isf 26^0 



077 



Refraction 
True Altitude 



3.S 



IS**22'.S —A 



X- 

* -84 



42^ 

IS" 
iii« 



2I\0 
2l'.0 



'32.2 
/-X - 42<» Il'.2 

/ - >| — 69** 09'.7 
/ - P - - 26*> 48'.8 



Mean - 15^ 26' 



Mean — 8li 43«»> 4/'* 

5 

Greenwich Time —131^ 43°* 47" 
Sun's apparent declination at 
Greenwich Mean Noon -■ — 21° 14' S4"'4 
Difference for i hour -■ — 26^.8 1 
- 26^'. 81 X 13^73 - -6^08"" I 

Declination -■ — 21® 2i'o2".s 

Polar Distance — iii** 2i'o2''.5 

log sin (x — Z) « 9.82708 
log sin ( J — A) = 9.97062 
log sec X -• 1. 02132 

log sec (j — /*) — o 04940 
2 ) 0.86842 
cot J Z« 0.43421 

J Z - 20<> I2'.4 

Z -40° 24^8 East of South 
Mark N. of Su n 74° 07^.7 

ii4°32'S 
MarkN 65** 27'.s E 



315. 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. 



OBSERVATIONS FOR LATITUDE. 

3X6. ( I ) BT 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 pointai 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. 

317. (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 1 7"* 
from nuan noon.* Furthermore it should be remembered that 
standard time may differ a half hour or so from mean time. 
When the maximum altitude is found the following corrections 
are to be made: first, the refraction correction is to be sub- 
tracted (Table 11, 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 noon 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 passes the meridian i6ni iSs before mean noon, i.e., when it is I2h oom oos 
apparent time it is i ih 43m 42B mean time. 



OBSERVATIONS FOR LATITUDE I97 



Example. 



Observed mazimam altitude of the sun's lower limb on 

Index Correction — + i' 



Jan. 8, 1906. - 


25° 06' 


Observed altitude 


25*>o6'.p 


Index Correction 


" I'i 



25° 0/.0 

«:.o 

2S*> o5'.o 


Declination of sun at 
Greenwich mean noon » 


- 22*>i9'35'' (S) 
+ 1 33 


i6\3 
25** 2i'.3. 

-'22<>l8'.0 




- 2ao 18^)2'' (S) 



Refraction 

Sun's semi-diameter 
Altitude of sun's center 
Declination 

47'' 39'.3 
Ladtnde 42** 20^.7 Diflf. ih - + 19". 58 

+ I9''.s8 X 4*».74 - + \' 33" 



PROBLEMS. 

1. (a) What was the azimuth of Polaris at its greatest western elongation at 
Boston when the polar distance of the star was i ® 1 4' 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- 
pencficular 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 «-> o® oo'. 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 45a P.M. At second pointing on the sun» upper and 
left limbs, vernier A, read 168® 52'; vertical arc, 42** 33'; time, 2b 55m 37s 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 -f 1 8^ 42^ 43'^6 (North). The 
change for i hour was -f 35''-94 (sun going north). The latitude of the place was 
42° 17' 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 ? 

X 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 iih 5oin A.M. The semi-diameter of the sun was 15'" 48^^.7. The 
declination of the sun at Greenwich Mean Noon was North I5°46' i3".3 ( + ). The 
difference for i hour was— 43''.46 (sun going south). What was the latitude of 
the place ? 



CHAPTER VIII. 

USVBLZZfO. 

3x8. DEFIHTnoifS. — 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 law 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, le., 
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 1 99 

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 tiiming 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. VIIL 



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 



20I 



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 Redded 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 instrument are obtained. In 
ordinary bench mark leveling these two values should not dififer 
by more than 0.002 or 0.003 ft. from the previous difference, 
Le., if the two heights of instrument differed by 0.013 at a 
certain set-up they should not dififer 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 difference 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 whfite-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. VIIL 

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 Pomts. — 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 2O3 

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. LEVEUNG 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," 
Le., 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 ai 
pomt of known elevation for the purpose of obtaining the height 
of the instrument. A foresight is a reading taken on a new p3int 
to determine its elevation. For this reason backsights are fre- 
quently called p/us sights ( + 5), and foresights are called minus 
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^i^ 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. VIII. 



sired to the nearest hundredth of a foot, as in the case of a 
raih-oad 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 



205 



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. 




SIA.0" 



PROFILt 
Fig. 80. 



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 



LBn>HAND Pagb. 



Right-Hand Pagb. 



^ Cro^aSact/ans far Gmdinff the 


(Hmfcff ^ 
March »,/90^. ^AJ^m 


Sta. 


f^. 


//./. 


-5 


£t^. 




8J», 
A4 
AS 
A6 
B6 
BS*40 
BS 
B4 

C6 
06 


Jtflz 


/24,02 


/2 
/7 
2.4 
23 
2.8 
2X> 
LB 
1.8 
d.o 
OB 
S.O 
7.2 


/2/.90 
123.7 

asz 

122.6 
I2Z.0 
121.1 

ms 

123.1 
/23./ 
f2IS 
124.1 
119.9 
7/7.7 
116.0 

J 






JOO/ 


\3fua 


iras. 




7 

6 

s 

4, 




















s 














♦r 




/ 


- i 


' L 


r < 


; i 


^ 4 


r 



Fig. 90. Cross-Section Level Notes. 

point / would be called (C, 7) ; the point j, (Z?, 5) ; the point 
r, (5 f 80, 4 4-35) ; 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. 106, 
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 b 



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 mentally. Obviously the only notes 
kept are the columns of stations and elevations.^ 

229. CROSS-SECnomifG 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-pity 
(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 subject see '* Railroad Curves and 
Earthwork,** by Professor C. F. Allen, published by Spon & Chamberlain, 
New York. 



J 



\ 



208 LEVELING [Chap. VHI. 

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 bC. 

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 + 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 bQ 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 

Lbpt-Hano Pagb. 



Cross-SecHdnlvrJbmesfom/toad, 'If^jnn 



Sta. 



n 

10 



Surface 

eitv. 



sas 

98.7 

374 
S7S 



Qraelt 
£ltv. 



96fi3 
96Z5 

96fi0 
3XS> 



Cross6acfKm. BcBe4o'-5k^^/. 



M& 



*€jO 



M^3,0 



^4*8 t4'jO ^^ 



HS 



'f'Z.O 



fS.O iZA 
t4tO ^i.6 



23X) 
*Z.O 



248 

23.0 
^2.0 



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. VHI. 

base of the road, as in Sta. lo, the section is called a Level 
Section, Where the surface of the ground is not paraUd 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 seaion, 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 Fivi^ 
Level Section, If intermediate readings (one or more of them) 
are taken anywhere except over the ends of the base, as in Sta^ 
1 2, 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-Kts. — The ground is 
first staked out in squares or rectangles and the elevation at each 
comer 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 comers 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. 

334* (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 



ESTABLISHING 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 pojnt, 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, while 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 comcides with the 
horizontal cross-hair. 

237. To ESTABLISH A DATUM PLANE BY MEANS OF TIDAL 
OBSERVATIOliS. — 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. VIIL 



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 
llll^ should be connected by a line of levels. This 

P^ 1 1 ^H 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 ^^J structed, and which is sufficient where only a 
I I ^^M short series of observations is to be made. If 
!l f5S made in sections not over 3 feet long, as de- 
I y^\ scribed below, it can easily be packed in a box 
yl I for transportation. Each section consists of 

II two strips of wood about i^ inches square, 

H I 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 

I space is placed a glass tube of about f inch 

I ^lii diameter and held in place by brass hooks. 
W^lljl On one side of the tube is a red strip blown 
' into the glass. When the gauge is set up 

for observations the sections are screwed to 



Fig. 



94. Staff 
Gauge. 






LEVELING ACROSS A RIVER 

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 descf iption of such a gauge 
may be foimd in the Reports of the U. S. Coast and Geodetic 
Survey.* 

239. LEVELING Across a RTVER. — Wl^ile 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 lin^ of bench levels is too large 
a quantity to Aeglect. 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 g^eaf 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.^ with the instrument 

• Report for 1897, pp. 315-320 and pp. 480-489. 
Report for 1853, pp. 94-96. 



214 LEVELING [Chap. VOL 

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.^ 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^cdy but this occurs only when the two sights are taken 
imder the same atmospheric conditions. Therefore it will be 
seen that the two sights must be taken simultaneously. 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 instrumentis interchanged. The 
magnifying powers of the two telescopes and the sensitiveness of 
the two spirit levels should be about equal in order to give the 
best results. It will be noticed that this process is similar to 
that of the peg adjustment (Art. 128, p. 91). 

* Errora 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. 

I. Compute the following set of level notes. 



Sta. 


B.S. 


H.L 


F.S. 


Elev. 


B. M., 
B. M., 
T. P., 
B.M., 
T.P., 
B. M., 


4.702 
11.846 
7.276 
8.760 
0.687 
1.607 


' 


6.727 
9.689 
4.726 

IX.OOO 

8.496 


16^27 



2. Compote 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 






-^.2 




21 






7A 




+42 






S'2 




22 






4.7 


' 


T.P.„ 


4.7«4 




9.976 




23 






1 1.2 




+63 






10.4 




B. M.ji 


0409 




7.482 




24 






11.2 





Y.(''^ 



^ Compute the elevations in the following set of level notes. 



Sta. 


+s. 


H.L 


-s. 


Elev. 


B.M.^ 


6.214 






84.238 


T. P.J L. 


3515 




9.280 




T. P., H. 


2.152. 




7.919 




T. P., L. 


2.971 




8.263 




B.M„H. 


^'33^ 




7.629 




T. P., L. 


4.278 




7.529 




T. P.,H. 


2.646 




S.894 




B. M.„L. 


5.721 




6.072 




T. P., H. 


4.837 




5.187 




B.M.„ 






58x7 





4- Make up a .•^et of cross section notes for road construction which shall be 
consistent with the following data: width of road, 50 ft., slopes x\ to 1 ; grade 
elevation of Sta. o — 107.20; grade, + 1.4. Show complete notes from Sta. o to 
Sta. 3 inclusive as follows : St». o, a level section ; Sta. i, a thzee level section ; 
Sta. 2, a five level section ; Sta. 3, an irregular section. 



CHAPTER IX. 

CITir SURVETINO. 

240. iNSTRUMEins 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 discardal 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 
measurements 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 CArt. 19, p. 13). The 
tape should be compared with the City Standard (Art. 243, p. 
218), at a definite tension, and the temperature noted at the 
time. From this information all of the field measurements can 

216 



INSTRUMENTS USED 2l7 

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 
plumbmg 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 
pomts 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. TkBnsits 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, 

pavingy 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 hundredths 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, 
with 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 -residing 
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 STAIfDARD.* — 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 50-ft. points are so marked 
that they can be readily found and used by any siurveyor 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, No. 4, April, 1897. 



STANDARD OF LENGTH 219 

genenJly 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 imtil it acquires the same 
temperature as the standard before the comparison is made, to 
avoid the necessity of applying a temperature correction. 



CITY LAYOUTS. 

344. In laying 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 topographic map of 
the entire area, for the topographic features of a locality will 
influence its development to a marked degree. 

345. 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 
different 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 s.treets 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 obvious 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 



221 




Fig. 96. 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 portions 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 9CX) 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. IX. 

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 1 5 to 20 ft. wide. If they are made nar- 
rower than 1 5 ft. two teams cannot pass each other unless cer- 
tain parts of 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 
a new 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 
ciu-b 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 off 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 eflfect 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.CX), 
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 ^a^ is 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 gradient^ a word which has some advan- 
tages but is not entirely satisfactory. 



226 CITY SURVEYING [Chap. DC 

falls 0.50 per 100 for 122 ft. to grade 50.40 — thence falls by a 
vertical curve for 100 ft. as follows : 

Sta. Elev. 

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 estabhshment, 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 g^di- 
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 gradients, 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 HEW 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 g^und 
a layout exactly like that on the plan. Sometimes the angles 
and distances necessary for the layout have been computed in 
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 fincj 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 they 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 comer 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 229 

dom placed there. The more practicable method is to define 
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 boimd 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 Jine. 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 
mtermediate 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 
comer 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 corner frequently becomes worn off and 
this practice is therefore not recommended. Some surveyors 



230 



CITY SURVEYING 



[CSAP. 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 comer 
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 corner 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 



awi.^ 



Srrddiing 



B 



E3r 



Line 



-Qshl 



§ Stone Bound 



QNi. 



Fig. 



Setting a Stone Bound. 



SETTING STONE BOUNDS 



231 



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 conaiderable 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. 




Fig. 100. Stone Bound with 
Concrete Foundation. 



232 



CITY SURVEYING 



[Chap. IX. 



In the construction of buildings or fences, monuments are 
frequently disturbed 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 Residential 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 



^33 



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, rim 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 or 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 

0J5=- Radius =/? 

AHB = Length of Arc = L^ 

AB= Long Chord =C 

VA = VB^ 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.T. = Point of Tangency 


Fig. 102. Circular Curve. 


From simple geometric and 

^ I T 

Tan - = — , 

2 R 


I trigonometric relations, 

r = i?tan- 
2 


Exsec l^^ 
2 R 


E = R exsec - 
2 


2 R 


M = R vers- 
2 


Sin^=^ 

2 2R 


C=2iesin - 
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. IX 



258. STAKIHG OUT CXRCULAR 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. 2CX), 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 FEC 
and VCE may be measured from 
which / can be easily computed. 
Then the radius R which is deter- 
mined frohi the plan being known, 
the tangent distance T is obtained 
by the formula, T ^ R tan ^ L 
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 foll6wing 

article. 

259. Deflection angles. — 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. 108. Deflection Angles. 



makes it possible to compute the ItngtA of 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., is an element of the circular curve which is used extensively in railroad en- 



C 



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, n, the angle at the center under each of these arcs 

will be - , and the deflection angle for one chord will be — , 
n 2n 

which m Fig. 103 is the angle YAh, 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 — = — 
2n n 

etc. 

Evidently, after the first deflection VAb is found, the 
other deflections can be obtained by simply adding the incre- 
ment — to the preceding deflection angle, and this is the 
method which should be used. The deflection angle from the 
P. C to t*he 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^ bc^ 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 wiU give the number 
of loo-ft. chords in the length of the carve, i-c* -^ ~ ^ (in loo-ft. stations). 

Therefore L (in feet) — -y- • For a complete discussion of railroad curves see 

** Railroad Curves and Earthwork," by Professor C. F. Allen, published by Spon 
ft Chamberlain, New York. 



236 CITY SURVEYING [Chap. DC 

^ = 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. 37i> P- 340- An approximate value for the chord 
length corresponding to a given arc may be obtained by the 
approximate formula, 

f /» • 

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 0°, and the point b set by measuring Ah and placing b 
on line by means of the transit on which the first deflection 
angle VAh has been laid off. Point c is set by measuring be 
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 \i'ill give some idea of the accuracy of this formula. 
With /^ — 100 and /• = 25, the formula gives c — 25.065, (correct value 

is 25.066). 
With ^ — 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 



237 



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, 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 run out by the use of chords and deflection angles ; Fig. 
104 is an example of a concise form of notes for this work. In 



ofcSrv 



Sfafwft 



Dishum 
(An,) 



Chords 






/hffiark 



7*9666 



f6^72.42 



3008 

SOlOO 
SXOO 



3S^ 

S6^9 
SSSB 



of5tnti'70f\et. 



3OJ0S 
49.87 

49.37 
49S7 



70, 
24J9 
4/.f4 

4iM 

4/./4. 



J4'i9-20 
7-09-40 



BT 



ec. 



^/^ 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 meaning 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 the P.C> These same deflection 



238 



CITY SURVEYING 



[Chap. IX. 



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 rf 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 oP and beyond o® 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 o® the instrument will be pointing along 
the direction of an auxiliary tangent df, for angles VAd and AJf 
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 ^de = — , and lay out the 



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 d, the vernier is set at o** and a backsight taken on A. 
Then an angle equal to the deflection angle VA^ is laid off on 
the arc; this will cause the telescope to point in some such 
direction as d/i. 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 + kdfy 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 CORIVERS. — 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 to 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, 



A 6\ 


V 


< 


V 

/I 

/ / 
// 



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 /8 and the distance FVzr^ measured. In the right triangle 
CPF'in which R and 7" are known, compute angle a and distance 
CV. In the right triangle CA V, CFand y = /3 --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'O 
D^ and A'CC = GCE. 

Angle i4CF= 180^ — /9. 
Angle EOF = ACF — 
A'CC\ from which the 
length of the arc FE 
can be readily com- 
puted, which locates thd 
point E. VG^ A V— 
A'Ci which locates point 
G of the curve GE, and 
any intermediate points 
can be located as ex- 
plained in the previous 
Fig. 106. articles. 

As the radius CE 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'U intersect each other and the curve 
whose center is E and with a given 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 ly Vixova the curve A'B'iy and angle 
a and distances 7* and T' are measured. In the right triangle 
CDV, R and T being known, compute angle CVDBxid distance 
CV. Similarly in the triangle C'VD' compute angle CVU and 
distance C'7. In the oblique triangle CVC, CV, CV and 
angle CVC = 360° - (a + CVD + CViy) being known, com- 
pute CC and the angle CCV and CCV^ In the oblique tri- 



ROUNDING STREET CORNERS 24 1 

angle CCE, CE ^ R + r, CE ^ R' -- r, and CC being known, 
compute the angle C^CEy CCE and CECy which is the sup- 
plement of the central angle of the curve BB- Angle 
DCB « BCY^ VCC — CCE, from which arc DB can be com- 



Fig. 107. 

puted. Similarly angle UCB' = 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 £. 

267. STAKING OUT STREET GRADES. — The field work 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 StJRVEYING 



[Chap. IX. 



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 Fand Fyl/ 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 2S.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 

^^ = 4 px, or y^ = (constant) ;r, (i) 

the;rdimensions being parallel to FX" (vertical) and the 7 dimen- 
sions being along A V. From the equation it is readUy seen that 
the offsets from the tangent vary as the squares of the distances 
along the tangent, or re, : otj = 7 \y}^ The lines VP and NQ are 
"vertical and AQ\& horizontal. Since the curve extends an equal 
distance each side of V, AP = PQ ; and therefore AK = KB. 
NB-^AVff'y VIf=4CC; CQ=^4DD'; etc. (from equation i.) 
Let ^ and ^^ represent the rate of grade oi LV and VM, 
and n the number of 2 5 -ft. stations (in this case 4) on each side 
of the Vertex V^ then 



NB 
KV = (from similar triangles) 



VERTICAL CURVES 243 

but NB = 4 VH (from above) 

therefore KV = 2HV, 

or point If is xnidway between V and K. 

The elevation of Vis determined from the established grade. 
The number of 2S-ft. stations will determine the distance VA 
and 7J5. The elevation of A and of B can be readily computed 
along their respective straight grade lines. 

«., „ Elev. A + Elev. B 
E;iev. K = 



Elev. If = 



2 

Elev. V + Elev. K 



2 
VH = Elev. H- Elev. K 

Elevations of all the other intermediate points along the curve can 
be computed by finding the elevation of the points D'y C, £', 
F', G', and /' and by adding to these elevations the ordinates 
ZKA CCE E, etc. 

VH 

16 

VR 

4 

EE =^ FF^ ^^ 
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 hne 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. EX. 

grade given and a 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 templet for this work is laid 
out by the surveyor. If no templet is used he should put 
in intermediate grade stakes between the center and the curb 
lines. In either case the surveyor must 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 : 

Mean Transverse Slope 



Half the Width of Carriageway 30 
or, Mean Transverse Slope = f " per ft. 

370. Gutters at Same Elevatioii — Fig. 109 represents t^lie 




-* • w - 7 

Fig. 109. Cross-Skction 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 DI/ at any other point on the parabola 
• CC X -TTTzr since in a parabola the offsets from a tangent 



2i 

vary as the square of the distance out along the tangent (Art. 
268, p. 242). But CC = AB\ hence, if ZX is half-way from 

the center to the curb, DI/ = — . 

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 

.„,. Since f-Me».^v.™ .op. Wh«. . i, found .h. 
other offsets can be computed as explained in the previous article. 

At A the offset = - 
4 

at B = x—b 

^ 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. 
/ — distance from the line of the highest gutter to the highest 

point of the pavement. 
b — difference in elevation between the two gutters. 
/ — mean transverse slope, expressed as a ratio of crown to half 

the width of pavement. 
X — 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 jf — * — _ (2) (See (i) in foot-note, p. 339.) 

i,R 

•••'-'+4 

zR 



246 



CITY SURVEYING 



[Chap. DC 



Combiiiing (i) and (2),a>-/> — 2Rb 

iRb 



But 



« + /- »^ 

2Rh 



(« + /) + («-/)- ^ + 

2fl - lV-\- 



From (i), 
But 



2^^ 

IV 

iRb 

W 



2 W^ 






(3) 



- ■- Mean transverse slope < 




From (3), 



.ai?-l'-f 



2J 



fl ■- 1- ** 

2 "^- 



2J 

7r 



■(-^)- 



2 



2 



zWs 



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 DC is readily found and the elevation of such points as E 
or Fy 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 — ^ . 



Similarly, Elevation E = Elevation G + 



Elevation F = Elevation H + 



Elevation / = Elevation K + 



Elevation L = Elevation M + 



3DC 

4 
jDC 

4 
7DC 

16 
7DC 

16 



etc. 



273. IRREGULAR SHAPED BLOCKS. — 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 






T 



60 



eo 



60 



60 



60 



^ 



60 



<^ 



T^ 



g 



^ 



Fig. 112. Arrangement op 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 A9 (Fig. 113), a sight is taken on B, and the front 



_ji 



J 





S 






















•« 


F 


D 


y 


50 


M 


» 


90 


90 


90 


90 


90 


■0 


90 


80 




so 


14 


u 


» 


9 


• 


» 


n 


tl 


n 


28 


wa 


G 
































3' 


2 


3 


4 


i 


C 


7 


e 


» 


n 


II 


«a 




sor 

A 




M 


so 


50 


M 


M 


90 


.»_ 


» 


» 


*• 


-2^ 


t 



"1 



1 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 0.01 ft. It will be well first to measure the line 
AB, 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 must 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 AtoBoi 599.88 ft In this case 
each lot should measure 49.99 ft. 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 is 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 //, 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 comers of all the 
lots established. The check on the lines AB, 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. 



2 so CITY SURVEYING [Chap. IX. 

curb, and at intervals of about 25 ft. The grade stakes are set at 
about the same 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 
i") 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 2$ I 

376. 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 hard 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 grades 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 soft, station or oftener. 
The differences 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 comer 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 comers 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 ofiteets to substantial buildings are often of more perma- 
nent value 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. REVISING STREET LINES. — 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. REVISING 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 building. — 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 buildihg 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 underpinning 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 that 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. IX. 



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 Ay By Cy Z>, Ey 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 FBy and GD and EG 
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 
corner 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 a 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 
lines 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. 

282. City PLAHS and 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 
obtab them. For suggestions regarding the filing and index- 
ing of pkns and records see Arts. 483-7, pp. 431-3. 

RECTANGULAR COORDINATE SYSTEM OF SURVEYING CITIES. 

283. GENERAL DESCRIPTION.— It is customary to disregard 
the effect of curvature of the earth in the survey of a dty 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 



3S6 CITY SURVEYING [Chap. DC 

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 ocx5, lo ood). 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 desigfnated by the distance 
in feet north or south, and east or west, as (locx) 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 equilateral. 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 









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Fig. 116. Triangulation Scheme i 
(Printed by penniasion of Major Joseph W. Shirley, Chief En^ 





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SCRVEY O^ THE CITY OF BALTIMORE. 

'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. MEASUREMEITT OF BASE-LINE. — At least one line in 
the system must be chosen where its length can be very ac- 
curately nieasured; 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. MEASUREMEHT 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. IX. 

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 
vicinity, 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 TRIANGULATION 263 

point given, or assumed, the coordinates of all of the other 
points can be computed as explained in Art. 410, p. 373. 

289. SECOITDARY AND TERTIARY TRIANGULATION.— After 
the principal 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-Jine, 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 corners, are computed. 

291. METHOD OF LOCATING PROPERTY LINES AND BUILD- 
INGS. — Since the coordinates of the property comers 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 SURVETINO. 

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 which are a con- 
siderable distance apart. The details of this triangulation work 
have already been described under the head of ** RectangularCo- 
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 theii 
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 26$ 

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 Ponrrs 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. 1 16 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. 

295. Contour LIN£s. — There are two general systems of 
representing on paper the form of the surface of the ground. 



266 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 




Fig. 116. 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 tines, 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. Sjf the surface of the water 
we^ raised 5 feet the new shore line would then be the 5S-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. 



J 



268 TOPOGRAPHICAL SURVEYING [Chap. X. 

Fig. 1 19 illustrates contour maps of simple solids. 



CLCVAnON 




Fig. 119. 



CONTOUR MAP 

Contour Maps of Sibiple SoLros. 



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 any one 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 5. 

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, in 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 D, 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 from E, 
Le., the contour would be unsymmetrical, EF being longer 
than DE. 

297. RELATION BETWEEN CONTOUR HAP 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.UNLAB 
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 



2/2 TOPOGRAPHICAL SURVEYING [Chap. X. 

construct such a projection from a contour map (Fig. 123), lines 



Hii 




PROJECTION ON.PLANE Aft 

Fig. 123. SmE Elevation Constructed from 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 AB. 



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 
contour 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. SKETCHHIG 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. 




Fio. 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 ix thk 
Map above. 



2/6 



TOPOGRAPHICAL SURVEYING. 



301. SKETCHING COliTOURS FROM KNOWN . ELEVATIOHS. 

— 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- 




TRAVLRSE 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 COITTOURS. — 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. 



2/8 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 140-ft. contour. When he is 
on the 140-ft. contour the elevation of his eye (H.I.) is 145.4 



' 5JM1VERS1TY 



LOCATING CONTOURS BY THE HAND LEVEL 2/9 

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 145-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 i4S-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 w^orking down the hill to locate the i3S-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 01 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 
ao 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 cUrection 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 CDy 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 ol 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, ky /, m, 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 q, s, 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 Ca^ 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 alcyig the line Ca 




Fig. 180. 



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 /, y, 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 
r, 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 A 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 ab^ cdy ef^ gL 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 b, on either edge of the road, the new contour 



284 



TOPOGRAPHICAL SURVEYING 



[Chap. 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 i^ to i, 
then the distance cut from c must be lA X 5 = 7-5 ft.; opposite 
^ it is 10 ft. (belo withe road and similarly the distance out from 
e must be 1 5 ft. Where this new 50 ft. contour meets the old 




Fig. 181. 



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 

MINING SURVETING* 

313. GENERAL REMARKS. — In this chapter the limitations 
and diflBculties 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 difficulties 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 surve)dng 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 rimning 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, 
EUamar, 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 levels, 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. 

StuU. Timber nmning 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 

MIRING INSTRUMENTS. 

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 muiing 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 instilment. This is commonly done by attach- 
ing an auxiliary telescope, usually smaller than the main tele- 
scop)e, 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 measured 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 hetn made; in 



288 



MINING SURVEYING 



[Chap. XL 




Fig. 132. Mining Transit with Side Telescope. 

(From the catalogue of C. L. Berger & Sons, by permlMion.) 



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 telesccpe is sighted at one 
point by means of the damp 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 p)ositions and two 
more azimuths are read. The mean of the two readings of each 
pair ^ves two mean lines of sight which are synmietrically 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 tel^cope. No 
corrections have to be applied for single azimuths readings, if the 
instrument is in adjustment; but the altitude readings 'reqiure a 
correction on account of eccentricity of the telescope* This 
form of instrument will not re\'erse 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. INTERCHANGEABLE 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 
turned end for end in the eccentric supports, leveling up with the 



291 



- « 



r H 



3 Z 



m 




292 MINING SURVEYING [Chap. XL 

striding level each time the sight is taken, the striding level being 
used in both its p)ositions in each of the two positions of the hori- 
zontal axis. 

322. COMBIKED SOLAR ATTACHMENT 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 rejecting 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 should be carried by every mine 
surveyor. 

325. COMPASSES USED IN MINES. — The transit has taken, 
to a great extent, the place of the old miner's dial in which the 
compass was the main 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 
veitical angles is very convenient as it obviates the use of any 
other instrument. A small modern 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 tube pivoted 
at 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. XL 

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 put 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. 



UNDERGROUND 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 

pendiciilar 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 du^ 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 incUnation 
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 vet 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 affect the 
transfer of the meridian and is never taken account of. 



296 



MINING SURVEYING 



[Chap. XI. 



327. PLUMBIirG THE MERIDIAN DOWN A SHAFT. —To 

the mine surveyor the plumb-line is an instrument of precision, 
excelling even the transit, and imder 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 should 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 nm and 
Fig. 134. Triangulating at the the passages leading to the shaft 
Bottom of Shaft. 1. *. u j j ..v 

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 affected by drafts and will also stretch 
less. The plumb-bob should jpot 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 lo a standstill in the shortest possible 
time. The shape of the plumb- bob is of importance in this respect 




r^ 



PLUMBING DOWN A SHAFT 297 

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. 135. Type 
of the vibrations of the plumb-bob decreases in of Plump-Bob 
a fixed ratio with equal increments of time, and j^^^ MERmi>^ 
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-line 
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 thoJength of the plumb-line, the same as iFor 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-Une 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 ej^Jmined 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 fhe 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. TRAlfSFERRIIfG A MERIDIAN nVTO 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 A B C D is run 

TRAVERSE 




<S~ . _ Closinq L ine 

F*" "" UNDERGROUND TRAVERSE 




PLAN 

Fig. 138. 

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 i4' and D' established at the bottom. A.i underground 
traverse A'GF'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'U are missing. The horizontal 



UNDERGROUND TRAVERSES 299 

length and azimuth of each of these Unes 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 difference between the true and assumed azimuths of A'D' is 
a correction to be applied to the azimuths of all of the lines of this 
underground traverse. ^ 

329. UNDERGROUND TRAVERSES. — Surveying in a mine 
is necessarily a process of traversing, for only the working passages 
are avr.ihble 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 
boundaries, if touched, may be accurately established 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 direction 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 sp)eedy 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 surmoimted 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 MIKING SURVEYING [Chap. XL 

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 diffused 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 
telescopp 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'lampy 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 stull, 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. 



Sti. 


Bearing. 


Distance. 


Vert. 
Aoffle. 


Back, 
•il^hton 


May x7, 1906. 
Party : Keene, Chase, Holbrook. 





NSS^K/E 


650.8 


- I-17' 


Sun 


To top of air shaft extending 
to I St level; C, of S, edge 
of air shaft, 4' x 4'. Sta. 2. 





N 2^I0'W 


"74 


-8o«io' 


Sun 


To Sta. loi at ist level, 
line runs 3' from S. side 
and 4.5' from ^. side of 
shaft; shaft 8' X 8'. 


1 01 


N 87*^45' E 


230-8 


+ 0O45' 





To Sta. 102 in ist level. 


101 


N 2*»io'W 


112. 6 


-8o*»io' 





To Sta. 201 in 2nd level. 


102 


N 89°io' E 


750 


+ o-sy 


lOI 


To top of center of raise ex- 
tending to 2nd level, raise 
4' X 4'. Sta. 107. 


102 


N 89*^10' E 


1535 


+ o°53' 


lOI 


To Sta. 103 in ist level. 


103 


S 89^15' E 


105 7 


+ 0-39' 


102 


To Sta. 104 in ist level. 


104 


S 88^12' E 


162. 1 


+ o°48' 


103 


To Sta. 105 at foot of S. 
side of air shaft extending 
to surface. 


105 


S 9*^55' W 


92.3 


+88°25' 


104 


To top of air shaft, Sta. 2. 


105 


S 88^12' E 


15.9 


. level 


Compass 


To Sta. 106 at breast of ist 
level. 


201 


N 2*^10' W 


115.8 


-80*^10' 


lOI 


To Sta. 301 in 3rd level. 


201 


N 85*»52' E 


167.4 


+ o<»5o' 


lOJ 


To Sta. 202 in 2nd level. 


201 


S 85*'46' W 


196.0 


+ 0O47' 


lOI 


To Sta. 205 in 2nd level. 


202 


N 88*^20' E 


1380 


+ o«44' 


201 


To C. of raise extending to 
ist and 3rd levels, 4' x 4' 
Sta. 208. 


202 


N 88*»2o' E 


106.3 


+ 0O44' 


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. 
To Sta. 107. 


208 


S 3^14' E 


113. 7 


+ 77**i9' 


202 


205 


S 86*'io' W 


216.8 


+ o<'48' 


201 


To Sta. 206 in 2nd level. 



302 MINING SURVEYING [Chap. XI. 

Survey of Bear Creek Mine, West Boulder, Montana. (ContU) 



Sta. 


Bearing. 


Distance. 


Vert. 
Angle. 


Back- 
sight on 




206 


S 87^14' W 


118.0 


+ 0%I' 


205 


To top center of winze ex- 
tending to 3rd level, 4' X 
4'. Sta. 209. 


206 


S 87*'i4' W 


152. d 


+ 0^41' 


205 


To Su. 207 at breast of 2nd 
level. 


301 


N 86°2o' E 


304.0 


-f o%6' 


201 


To Sta. 302 at C. of raise ex- 
tending to 2nd and 4tb 
levels, 4' X 4'. 


301 


N 86^20' E 


316.0 


+ o<>46' 


201 


To Sta. 303 in 3ni level. 


301 


S 86°4o' W 


195.0 


+ o°5o' 


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. 3c6 in 3rd level. 


306 


S 89^48' w 


150 


+ o°43' 


305 


To Sta. 307 at C of bottom 
of winze extending to 2nd 
level, A' X 4'. 


307 


S 2<'4i'E 


120.5 


+ 71**"' 


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 


+ o°48' 


301 


To Sta. 402 in 4th level. 


401 


S 88«io' W 


X16.4 


+ 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 £ side. 
Shaft 8' X 8'. 


402 


N 89^56' E 


85.0 


+ o°45' 


401 


To Sta. 403 to C, of raise ex- 
tending to 3rd level, 4' X 4'- 


402 


N 89°56' E 


92.6 


+ o%5' 


401 


To Sta. 404 in 4th level 


403 


S o°o6'E 


116.2 


+ 8i°46' 


402 


To. Sta. 302. 


404 


S 87^20' E 


217.6 


+ 0^43' 


402 


To Sta. 405 at breast of 4th 
level. 




May 17 1906 

Surveyor 



PLAN OF U 
6H0WING L 




LONGIT 
LO 



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 ^re 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 fij-st 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- 



3o6 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. UNDERGROUND 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 j&ve 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 Jn charting, unless some special means of identifying them 
is employed. As these galleries or working passages appertain 
to definite levels or strata, a difiFerent 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 multiplied 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. Groimd 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 MUHNG 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 run. 

Vertical shafts are carefully pltunbed on the inside of the 
frames, and frame by frame, as these are put in. It is best to 
hang the plumb-line 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 
pKJsition 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. UNDERGROUIID SURVEYING PROBLEMS.'— In the prac- 
tice of mine surveying, problems are constantly arising which 
tajc 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 ii^ 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 pcint 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 sur- 
face station at £, the distances de- 
sired are EC and AD, HB and 
H'A are both horizontal, 

then F' = F - ACB. 

AB _ Distance between telescopes 
AC Distance measured 

AD = AC sin F', 
and DC = AC cos V. 
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 of i4C then 
DC = CF -{-FD^ BCcos V + ABsinV. _.^ 

Similariy AD = BC sin V - AB cos V. -^ 7 

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. 

DC -- FC + AB = BC cos F -h i45 
AD ^ BF=^ BC sin F. 



Fig. 138. 



But sin ACB ^ 




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 
undergroimd working in the drift H'J' to a point Kf vertically 
under the boimdary line. The surface traverse BADCS is rim 
out, point S is plumbed down to 5', and the meridian trans- 
ferred into the mine. Then the underground traverse 
S^EF'G'H'J' 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'^ 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. 

3^0. 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 i4 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 \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 *i4 and B can be computed. 

341. HYDRAULIC SURVEYING FOR MIITES. — The miner's 
unit for measuring water is the miner^s inch. By an inch of 
water was originally meant sucR 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 (i^ 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 surve)dng 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 SURVEYING. 

343. SURFACE SURVEYING IN RUGGED MOUNTAIN REGIONS. 

— In accurate work, such as the surveying of mining 
claims for patent,* the ordinary mining transit may be used. 
Measurements are made wth 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 
pull 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 
staflF 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 xiJTTJir ^^ 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 
often prove useful in filling in details of topography. 

344. MINE BOUNDARIES.— APPROPRIATIOIfS 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 — ^ o — 1 

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 over 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 coimtrj^ 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 oflF oa the ground can be exceeded. So when the 
location survey is roughly made certain "fractions " of ground 



314 MINING SURVEYING [Chap XI. 

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 
courses 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 lines must be parallel. In order to 
guard against troublesome litigation, an efiFort 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 groimd 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 agricuhural lands. 

345- SURVEYING FOR PATENT. — The sifrveying 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 surveying 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 comers 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 VBm. — 
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 effect the maximum of economy. The 
surveyor bearing this in mind will he 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- Fig. 142. 
tween the plumb-lines, 
and e its error as ascertained by measuring or other means. 

The controlling error at the connection is £ = 5-j-and is in a 

a 

direction perpendicular to that in which D was measured on the 

chart. This is obvious, for the surveys of the galleries are coit 

sidered acciu-ate, 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 £p £„ fij* etc. found at these localities. These 

are then resolved according to their respective latitudes and 

departures into S„ S„ S3, etc., and W^, W^, W,, etc. The greatest 




3l8 MINING SURVEYING [Chap. XI. 

possible error is then 5i + 5, H- 5, + etc. to the north or south 
and PTj H- W, + PTj + 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 
s quares), is V^i' H- 5,' + S^^ + etc. to the north or south and 
VWi* + W,* + W/ + 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 +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 4- 2.4 % from the breast of tunnel, and what is the 
slope length and azimuth of this drift ? 

2. The strike of a cer^^dn 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 -f 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 
nunc to the bottom of vertical shaft No. 2 as follows: Assumed azimuth o^ 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 411 ft. (horizontal) to a point A from which the center of shaft No. i is sighted 
at azimuth 71** 30'. 

(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" Wy 896.796 ft. from X. The 
following are tiie notes of the survey connecting X and the winze comers A, B, C, 
and Z>: — 

Station. Mean Deflection. Horizontal Distance. Station. 

X o«> 00' 896.796 Y 



Y 


45^ OS' 34" R 


403.086 


I 


I 


74^ 05' 06" L 


587.208 


2 


2 


320 23'43" ^ 


67.000 


3 


3 


54^ 43' 47" R 


44.803 


4 


4 


39^ 51' 57" R 


41.075 


5 


5 


3i«> 10' 10" R 


19-573 


Cor. A 




31^ 10' 10" R 


27.240 


Cor. B 




310 43' 40" R 


21.477 


Cor. C 




24^ 02' 40" R 


25-773 


Cor. D 



Required the location of the shaft comers on the surface. 

6. From k monument M at the mouth of a tunnel a traverse is run in the tunnel, 
azimuth 20® 35', distance 352 ft., vertical angle + i^ to point A; thence azimuth 61^, 
distance 528 ft., vertical angle + 1*40' to point B at the breast of the tunnel. 
From M a surface traverse is run, azimuth 11^ lo', distance 578 ft., vertical angle 
+ 4** 25' to point C; tbence azimuth 11**, distance 407 ft., vertical angle -h 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 ♦he shaft to the breast of the 
tunnel 

7. The course of Tunnel X is N 34® 45' 10" W., the grade 0.42%, and the ele- 
vatbn of the mouth 2570 ft. The course of Tunnel B is N o® 45' oo'' W, the 
grade 0.^3%, 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' i</' W, 
100 ft.; thence N 19** 17' 30" E, 381.60 ft.; thence S io*» 21' 20" E, i 30.60 ft; 
thence N 74*^14' 30" E, 3662.01 ft.; thence N 85** 45' 30" E, 1547.21 ft; thence 
N 73** 48^00" E, 1455.00 ft; thence S i2*» oo' 00" E, 205.40 ft; thence 
S 70® 00^ 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 iodined line? 



PART III. 
COMPUTATIONS. 



3»i 



PART III. 
COMPUTATIONS- 
CHAPTER xn. 

GBNBRAL PRINCIFLBS. — MISCEIJULNEOnS PROBLEMS.— 
BARTHVrORK COBIPUTATIONS. 

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 suflBicient 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 will yield- 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 failure 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 MacmiUan & Co., New York. 

323 



324 COMPUTATIONS [Chap XIL 

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 i-(^j^;^j^-q 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 proi>er 
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 
^hich 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 diflfer- 
ence will be, on an average, about i in the fourth place. There- 
fore, in general, four places v^dll 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 



325 



five-place tables. Similarly it can be seen that five-place tables 
of functions will, in general, give angles to the nearest 10 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 four places are needed rather than to use six 
or seven- place tables and drop oflf tlie last two or three digits. 
The amount 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 b, 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 
logcos44''i2' = 9-8555 
log a = 2.7440 
a = 554.6 



log 773.6 = 2.888s 
log sin 44^12' = 9-8433 
log b = 2.7318 
* = 539-3 



If it is assumed, however, that the angle B is measured by 
repetition and foimd to be 44^12^25'^ the error in the original 
angle then was about 25''. By using the same value for the hypo- 



326 COMPUTATIONS [Chap. XtL 

tenuse (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 diflference 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 hun- 
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 line is therefore multi- 




LOGARITHMIC OR NATURAL FUNCTIONS 327 

plied several times. The same degree of predsion should be 
secured in the measured line BC as is desired in the computed 
lines AC or AB, 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 or 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 AC is desired to the nearest himdredth of a foot the angle 
A might be determined closely by repetition, but this will not 
give the length i4C to 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 BC, then the 
line AC being about 15 times as long as BC will have an error of 
0.075 ^t' ^^ 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. Evidentiy the prac- 
tical way of obtaining an exact value for the inaccessible dis- 
tance ^C is to measure AB to the nearest hundredth, and to 
compute AC from AB and BC, using the angle at i4 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 opportimities 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 eflfort 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- 
pUcations 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 




AB = 207.42 
A^f 32' 
AC =207.42 cos 7^32' 
But AC ^ AB - CD 

= 207.42 — 207.42 vers 7^ 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 himdredth of a foot with much less effort 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 sufficiently exact (Art. 359, p. 330). 

Had AC been given (205.63) and the angle i4, (7® 32') then 

AB ??5^, 

COS ^1^ 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. 

,47x2*5 = ?^^^^^^=^^ 

4 4 

68200 
• 682 X 50 = 

694 X 150 = 69400 -I- 34700 

927 X 62.5 = 92700 X I 
o 

672 X 1002.3 = 672000 •\- 1344 -f 201.6 
547 X .9968 = 547 (i - .0032) = 547 - 5.47 X .32 

~ = -^ (reducing minutes to decimals of a degree) 

^-8.43x8 



i 



330 COMPUTATIONS [Chap, XH. 

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' 18* 
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. ARRANGEMENT OF COMPUTATIONS. — All survey- 
ing computations, should be kept in a sp)ecial 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 copying of data out of the note-book to the final restdts. It 
is not unconmion 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 



SLIDE RULE 33^ 

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 lo inches long, h^^ving 
a groove in one side in which rims a small wooden strip called the 
slide. On one face of the rule are placed two scales, A and Z>, 
Fig. 146, one above and one below the slide which is indicated by 



iiilmiDiMHii i [ I [J[iti[iDD["'il ' iiiD" i| mii: I [ l C i [i[iDD(i 



;l l ]l]l)l] l ]l]l]l] l ]l[llll|llll[llll|llll[ilM| ll ll[ I C I [I [I Ub 



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 custoniary 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 Jower scale, then the unit for the upper scale will be 5 inches. 
Since the length of this upper scale is only half the length of the 
rule there are 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 
nile, 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. XII. 

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 eflFect, 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 Dj 
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 dividend 
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. 



(;r . j.j.j.j . j.j.j . jgH i )niini\iiii|inii:j.i,iM . L ' v ' E ' va'v ' Dv 



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 2), 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 HELD NOTES 333 

obtained with the ordinary lo 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 wo. 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. REDUCING 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 distarces are transcribed into 
the computation book. The errors in the an^es are balanced 
by altering the value of those angles which were taken from short 
sights since the angular errors are most Ukely to occur 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 multiplving 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 diflference 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. S3, 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 oflFsets, 
Ae and A'e = 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 affords 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 = - (Ae + 2S h^ + 4S h,^ + h\) 
3 
where d = common distance between oflFsets, 
Ae and h\ = end oflFsets of the series, 

2 2 Aodd = twice the sum of all the odd oflFsets 

(the 3d, 5th, 7th, etc., from the end) 
4 2 Aeven = twicc the sum of all the even oflFsets 
(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. S3, p. 104. 




1 



By Simpson's OmrTh/tri ffi^f^ 
21 



0y Ihtp^uoklcf/ Ruto <%" 

€jo 
7jO 



00.9 . 



X{A*\IA AO.77 n ,9 



6S'34' 



1019 



I? 

/6.ZK,S6TLn 9 J Bx 

pArg 

^io.ynzrzm 231 ivxzy 
ZIo"wxgv 



Fig. 148. 



336 COMPUTATIONS [Chap. Xn. 

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 -45 is first run, and the distance AB, the angles 
at A and By and the necessary ofTsets 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 diflference 
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 be 
computed. Then in the triangle ABCy 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. x 

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 found by the trigonometric formula: 

Area = J a 6 sin C, 

where C is the angle included between the sides a and ft. 

If all three sides of any of the triangles have been measured 



STRAIGHTENING CROOKED BOUNDARY 



337 



or if the field has been surveyed with the tape alone (Art. 139, 
p. io6)y the area of the triangles can be found by the trigono- 
metric formula: — 



Area = V5 (5 — a ) (:>• — 6 ) ( 5 — c ) 

a + b + c 



where a, 6, and c are the sides and s 






367. AREA OF A QUADRILATERAL BT 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, ABCD 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 5Z>, 
CDy and the angle £>, or from 
ABj ACy and the angle A. 
These two determinations of 
BC should check each other. 
Similarly 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. 




z:^^ 



338 



COMPUTATIONS 



[Chap. XH. 



368. AREA OF CURVED 
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 ABODE 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 I 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). 

FHG X HK _rx 0.0174533* X HK 




KFHG 
VI, p, S06.) 



(See Table 



KFCG ^FCXFK 
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 



= -^t 



8i? 



(approximately). 



Expressed in terms of C and -R, 



Area of Circular Segment = - 



12R 



(approximately). 



* 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 7 and 
H are correct to six significant figures. 

t In Fig. 1 52, OB » Radius of circular curve. 

Cff « Middle Ordinate for chord AB, 
CD is drawn tangent to the curve. 



AREA OF CURVED CORNER LOT 339 

These formulas are fairly accurate when M is very small as 
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. Planimeffer. — One of the commonest ways of checking 
the area of a traverse is to obtain its area by means of an instru- 

DB B Tangent Offset for chord CB. 
OE is drawn perpendicular to CB, 
In the two similar triangles OEB and CBD^ 
VBzCB^ BEi OB 

DBiCB ^ — lOB 

2 

DB^^^ 
zOB 

Offset from Tangent - JChord)» 




a X Radios 
But DB — CJIt and AB ^ 2 X CB (approximately) (i) 

I 2 / AB* 

'' ^^" 2 OB " 80^ (approximately) • 

Middle Ordinate - 3 ^x todiL. (approximately) (2) 

The following will give some idea of the accuracy of this formula : 

When radius — 20 ft. and chord — 10 ft, M — 0.625, (correct valae 180.635). 
When radios » 100 ft and chord » 25 ft., M * 0.781, (correct value is 0.784). 
When radios — 100 ft. and chord — 100 ft., M * 12.500, (correctvalueisij.ji);). 
When radios — 1000 ft. and chord » loo ft., M » 1.250, (correct valoeis 1.251). 

It is evident from the above that this formula will not give accurate results 
when the chord is large in comparison with the radius. 



^' O-THF 




340 COMPUTATIONS [Chap. XIL 

ment called the planimeter^ 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 te 
regulated by setting it at the proper reading <m a scale which is 
marked on the arm, so that a imit 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 £rea. 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 ty the wheel by 
making several tests and by setting the anchor point at diflferent 
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 
shown 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. DEFLECTION 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 



GIVEN yff*200, curve to Right, l^Si^'-JS^zo!; /?C. =i6^7a.4Z 

Width of S treet 70t 

Tsifim. 2^*47*40" s zoo X. 40330 « 36.66 T 
Si* • 890li7B 

3S ^.oioiaa ^ i^eaS 

zo' ''^oooo97o eTie*sz.so 

.9003360 A 200^ ieo.oa Lc 

Def/ectioii Ang/es, 

DefkcHonz. for Soft ^^ x ^^V/^'^^jg^x^f :3W 

Log 7zssin:^^//o496 

D€ftecfkmL for3ao8fi:::iadiL^deH.for30f log i6o,oe •Z^SfeS 

icg.eoie ■^•^^ff ^— ^— — 60 

nay 
4:3086 




311 
4^iO'S/''ci€ff.3aa^ft 



43: 
Tos'43'cf€fi.^/f 

i7^Z24Z = -fioS-AO" 
iS^TZHZ ^i^'i9-ZO 
iS*2ZitZsLll-Z9-iO 
4-iS'3o 
RT iS^SZSi^W^^W^^aiecic^ 



Chords 



SOthArc. 

Sm7^09'4o'^.IZ467 
4^ 



VZ47XZ^ s 



49^68 Ce/f/wCM. 

&7Z7- 
SS.6B Uff drci, 
4iM Right Chd. 



5006 fk Arc 
5in4^lS'30^X>7SIZ 

30048 CM.Chd 
\0WXZX3Ss^ S2S7^ 

35.31 UftCiid. 
24.73 RigtitChd. 



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- 
tioD the desired area found. 



34^ COMPUTATIONS [Chap. Xa 

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 £r, 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. 

COlfPUTATION 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 = i4 X ' "^ ^ '^ ^'^ * 

4 
where A is the area of the cross-section and A„ A„ h^ and h^ sue 
the comer heights. 

For a truncated triangular prism such as abc^ using the same 
notation, 

Volume Truncated Triangular Prism ^ A X -^ * ^• 

In computing a trapezoidal prism, such zsfdhg^ the trape2x>id 
is subdivided into a rectangle /eA^ and a triangle /ate ; or for jhds, 
into two triangles by diagonal lines, as jhs and Ms and their 
volumes may be computed by the above formula. 

Wheix there ere 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 
Carrea aud Eaithwoik by ProfMsor C F. Allen, published by Spon & CJiamhT** 
lain, New York. 



BORROW PITS 



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 found by 





^ 


^ 


rf> 


.> 


i . 




m 


a 


iP- 


a 


«? 




A, 


















<> 


<f 


t> 


s? 


«? u 












9 


fV 

!\ 
1 \ 




s? 


i> 


-u* 


ri> 


f iV 


fi 




4f 


iP' 


k 


h 
1> 


\ \ 








• 


J 


6 


\ 


n 


Cr 


»> 


Af 


*? 


1^^ 

r 


k 



Fig. 164. Plan of Portion of a BoRROW-Prr. 



one computation because it is composed of a series of pristos 
having the same cross-section. In the summation of the heights, 
those at a, m, n, r, and s are taken but once, those at such points 
as Cy gf h, etc. are multiplied by 2, at j the height is multiplied by 
3, and at such points as fc 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. 2c8). 

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±Al X / (End Area Formula) 

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 suffi- 
ciently accurate results for certain classes of work. 

376. Prismoidal Formula. — The correct volume of a pris- 
moid is expressed by the Prtsmoidal Formula: 

Volume of Prismoid = - (i4, -h 4i4m + -4,) 

in which / is the distance between the two bases, A^ and A^; and 
i4m 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 i4,; 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 -; — - 

+6.0 ^ +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 GRADrao.— 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 Jf7, 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 fill must run out somewhere between the 
24-ft. contour and the 25-ft. contour, for if it ran beyond the 25-ft. 



S46 



CX)MPUTATIONS 



[Chap. XIL 



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 





:i:=j.^ 



Scale 



Fig. 166. 

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 siwface, then computing the volume between 
the same plane and the finished surface; the diflference between 
these two volumes will be the amoimt of earthwork, or j) by 
drawing on the plan a line of no cut or fill, a line representing, 



VOLUMES FROM CONTOURS 347 

say, 5 ft. cut or fill, a line representing lo ft. cut or fill and so on. 
Then compute the volume between these successive 5 -ft. 
layers. y 

379* (i ) Referring to Fig. 1 5^ 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 plane. 
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, 
3Jid BUD V may be obtained by planimeter (Art. 370, p. 339) or 
otherwise, and the voliune of the solid AKEL-MNPO may be 
obtained 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 diflference 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 equnl 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. 

EXAKPLE. 

In Fig. 154 the amount of fill on the area ABCDEF'ia computed below. 
Area AELK — 900 sq. ft. 900 X i -> 300 cu. ft (Pyramid) 

" MNPO - 1000 -^--- X I - 950. 

- QRST ^10^ i^2?jLi^ X I - loio. 

- BUVD - 680 '°"^+ ^ X I - 850. 

680 X J » 23a (Pyramid) 

3) 334o« cu. ft. 
9 )' "3 

124. cu. yds. Total Fill. 

380. (2) Referring again to Fig. 156 and applying the sec- 
ond method, the area of ABCDEF\% 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 MBCDPOy QBCDST, and 
BCDV are found. The volume of the soUds 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 ABCDEKy 
MBCDPNf 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 amoimt of £11. 

While in this particular prollem 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 drawTi 
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 b'nes and the contours of the 
proposed surface in dash lines. Through the intersection of 
the new contours with the ori£inal ones is dr^wn 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 




Fig. 157. 



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 



3SO COMPUTATIONS [Chap. XIL 

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 a!n 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, afifording 
results suflSciently accurate for rough estimates. 



PROBLEMS. 

1. A series of perpendicular offsets are taken from a straigfat line to a 
curved boundary line. The offsets are 15 ft. apart and were taken in the foU 
lowing order: 6.8, 7.2, 4.6, 57, 7.1, 6.3, and 6.8. 

(a) Find the area between the straight and curved lines by the Trapexoidal 
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 lo^ 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 7891. ft. It is required to determine the distance BX along 
CD such that AX shall be the straight boundary line desired. Also find the 
length of the line AX. 



ROUGH ESTIMATES 35 1 

3. In Che quadrilateral ACBD the distances and angles which were taken 
m the field are as follows: 

A ^-50.63 A B C-ios** 39' 00" 

B C- 163.78 B A D^Zff 37' 30" 

C />- 93.80 
D A^i(f0.2^ 
Z> iff- 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^ if $&'. 
The comer lot is rounded off by a circular 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 ia straight. 

d 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 earthwork 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. 

X2 -!|f +4.2 J^ 

+ 8.0 +S.6 

,,+60 .-30:2 -iSf +4.0 J5f J4f 
+ 10.0 +4.5 +7.5 +6.0 

sta.11 -^ +6.0 jB 

+4.0 +7.2 

The base of the road is 30 ft. and the slopes are i} to i. 

Find by the End Area Method the quantity of earthwork from Sta. il to la. Re- 
sult in cabic yards. 



i 



CHAPTER XIII. 

ARBA B7 DOUBLE BABRIDIAN DISTANCES.— COORDINATBB. 

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 F, 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 latitude 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 hngitudi difference. 

3S2 



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. 

Latitudes are called North or South and departures East or West, 
depending upon the direction of the course as shown by its letters, 
e.g., if the 1 earing is N 30** E thij course has a North latitude 
and n East departure. North latitudes and East departures are 
considered as positive ( + ), South latitudes and West departures 



3S4 



COMPUTATIONS 



[Chap. XHL 



as negative ( — ). In the figure the courses are assumed to run 
from Fto A, from A to B, 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« 
0es, such as DErs^ 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 + BCtw -h CDst) - {DErs -h EFr + FAq\ 

that is, in this figure the sum of all the areas determined by lines 
nmning 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 
meridi-in. 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, twice 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. oi AB is qA -^ wB = qA + qA -{- 
zB, i.e., the D.M.D. of course FA -h the dep. of FA + the dep. 
of AB. The D.M.D. of BC = tC -^ wB = fv + vC -\- qA + 
zB^ qA +wB ^ zB '\- vC = D.M.D. of AB + 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, 
in 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. XUl. 

(2) The D.M.D. of any other course equals 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 departiu-e, 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 minus. The sum of all the north double areas 
minus the siun of all the south double areas equals twice the area 
of the field, Be careful to divide by 2 after completing the other 
details of the computation. 

386. COMPUTATION FOR AREA OP COMPASS SURVEY BT 
DM.J). 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 B, and that in the following computations the corrected 
bearings are used (Art. 41, p. 30). 

In Fig. 159 the learings, 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 



Area of Wood Lot afJohnSmitii Bm-R27. ^^^^ 


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Fig. 160. Area of Compass and Chain Survey by Double 
Meridian Distance Method. 



358 COMPUTATIONS [Chap. XUI. 

387. Balancing a Chain and Compass Traveise. — Before 
the D.M.D. method can be properly applied 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 ^ ^'^^ | ^ a # \ ^f ^^^ 
course is to the total error *^ j j /, / 1 ^^ ^^ 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 diflFerence 



DOUBLE PARALLEL DISTANCES iSg 

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. 'T 

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 multiplied 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. — Th^re is no particular 
reason for using the trapezoids formed by projecting the courses 
on to the meridian rather than those foi*med 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. 



860 



COMPUTATIONS 



[Chap. xni. 



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Fig. 160. Area of Compass Survey by Double Parallel Distances. 

3QI. 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^i^j^ (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 36 r 

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 error in the 
angles where it probably occurred. If nothing is knowli 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 
coorcfinates, as in case of the compass siurvey. 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 coiurse 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 siurvey (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 



363 



COMPUTATIONS 



[Chap. Xm. 



Area J. H Bradky fsiafe - Bk ^.pS7, 






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Fig. 161. Area of a Transit and Tape Survey by Double 
Meridian Distance Method. 

(The remainder of the computations ia in Fig. i6i A.) 



DOUBLE MERIDIAN DISTANCES 



363 



UaCot.Btar. 

Lo9SJii.Baar 
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Ansa 2311 



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161 A. 
(Theae computations go with Fig. 161.) 

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 
calcidation 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. 

394. 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 
logarithmi need not be taken out to more than five or six places. 
Seven places, of course, are more than are necessary so far as 
precisian 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: — 

( IcUitudc ) 
The correction to be applied ^ ^^^ J ^ ^ * e\ ^^ ^^^ course is 

^ ^h s I 1 • ( IcUitvde } . i latitude ) ^ ,, , 

to the total error tn i , ^ , \ as the \ , ^ ^ \ of thai course 
\ departure \ \ departure \ ^ 

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 probally 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 
BCj GHj and KA were all questioned, i.e., they v^ore measiT'^H 
under such conditions that it is probable ::i 1^ ilu . • mav im- 
or two hundredths error in them. In balanri^- •^. lutitii ! 
departures then, this information is use! \\\ Ti r 101 .1 
be seen that in balancing the survey the In* i a«!'. :t' i (k'T> 



DOUBLE MERIDIAN DISTANCES 



36s 



of these questioned measurements have been changed in such a 
way as to reduce the length of BC, GH, and KA each one him- 
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 line 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. 
transit and tape, the angles heir 7 



The traverse was run with a 
r.i ^ared to the nearest minute 



3*6 COMPUTATIONS [Chap. Xm. 

and the sides to tenths of a foot. Nothing appears in the field 
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 slightly difiFerent 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 F, 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. 



St equal zero; or, to put it in the form of an 

Zi cos A + Zj cos 5 + Z, cos C -h etc. = 
Zj sin i4 H- Z, 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. 

The missing data could be any of the following combina- 
tions : — 

(i) 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 ^67 

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 ^.^^^^CT^?'*'^^*-^ 

bearing of 45 is given. Here ^^^ ^ ^""^"^^^^^^^.^^^^^^^ 

one side and two angles are >^^-^Z[ ' "^-^^*=^c 

missing. The solution is as ^v.^-^"^"^"^ ll/ 

follows. In the triangle EAB > r\ "^-^v^ 

find £5, £54, and 4£5. In " fj^j"*^"" * 
the triangle EDC find EC, 
DCEy and DEC. Then in the triangle EBCy in which EC, 



368 COMPUTATIONS [Chap. XIII. 

EB, and EBC are known, find ECBy 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 surveying 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 siurvey, 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 Une 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. 107, 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 LAUD. — 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 Blnown 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 ofiF a cert; in 
area by a line rimning from F 
which is at a known distance 
fiom A or £. The Une FG' is 
drawn on the plan so as to make 
the area FGDE approximately 
equal to the desired area. The 
line DG is scaled oflf 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 FGG will represent it. In this triangle the base FG 
and its area being known tbe 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 Hne 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, 
v^rhereas in the method explained above the correction triangle 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 ofl 
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 TK' 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 accurately de- 
termined; the diflFerence 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 
ruxkning in a Given Direction from a Given Point in the Traverse. 
— This problem may be readily solved by drawing a line from 
the gi\'fen point in the traverse to the comer which lies nearest 
the other extremity of the cut-oflF 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 




In Fig. 165, ABCDEFG repre- 
sents a plot of a field. It is desired 
to run the line from £ in a given 
direction EH and to compute the 
area HEFGAB cut oflF by this line. 
The latitudeand 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 the angles at B and E. 

It is obvious that the solution 
of such problems as these is greatly facilitated by plotting the 
traverse before attempting the computations. 

CALCULATIONS RELATING TO TRAVERSES WHICH DO NOT CLOSE. 

^ 406. To CALCULATE THE 

Total distance, between 
End Points. — Fig. 166 repre- 
sents the traverse ABCDEF in 
which the distance AF and the 
angle BA F are desired. A B 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. 41c, 
p. 373. AF and 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-oflf lines, 
like the line CF in Fig. 166, is the same problem as was explained 
in Art. 398, p. 367. The angles DCF and EFC have been 
measiured in the field and the traverse CDEF is thus complete 
except that the length of the line CF is imknown. 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. 106. 



3. 



T 



S7^ 



COMPUTATIONS 



[Chap. XHI. 



"" 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) angles have been taken to 5, and from these angles 
the coordinates of point 5, referred to AB and a line perpen- 
dicular to A B as axes, can be computed (Art. 410, p. 373). Co- 
ordinates of 5 referred to the same axes 
should have the same value when 
figured from BC as a hase as when 
calculated from the base CD and so on. 
K, however, when computed by means 
of angles at 2? and £, the point falls 
at 5', and angles E and F give its 
location also at S' 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 DE, the mis- 
take was probably made in the meas- 
urement of the line parallel to 55' 
and the distance SS^ should be ap- 
proximately equal to the amount 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. 




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., 



sin^l _ a 
sin 5 "6 
sin A _ a 
sinC c 



asinB 

sin^ 

asinC 

sin^ 



« c 



COORDINATES 373 

Assuming a to be the base and the angles A, B, and C to 
have been measured the calculations are arranged as follows: 

(i) log a (1400.74) = 3-1463575 

(2) Colog Sin A (Sf42' 16") = 0.0729874 

(3) log Sin B (61^ if ss") = 9-9430639 

(4) log Sin C (60^ 59' 51^) = 9.9418088 

Sum of (i) (2) (3) logft = 3 1624088 
Sum of (i) (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 
measiured parallel to XX' 
are usually called abscissas 
and those parallel to FF' 
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 run 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 imknown point are usually computed 
from the coordinates of some other point to which the unkxiowxi 



Y' 



Fig. 168. 



374 



COMPUTATIONS 



(Chap. XIIL 



point is tied by an angle and distance. The difiference in co jr- 
dinates between the known and unknown points will be obtained 
as follows: — 



Difference in X 
Difference in Y > 



distance x sin azimuth angle, 
distance x cos azimuth angle. 



Sometimes the unknown point is located by angles from two 
other knoim points, in which case the distance between the two 
points whose co-ordinates are known can be computed and then 
the distance from one of the known points to the unknown point. 
The problem is tfeen in the form described in the previous para- 
graph. 




Fig. 169. 



4ZZ. TO DETERMIKE 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) + (ft, 2, 3, c) - (a, I, 4, d) - (rf, 4, 3f ^)- 

Expressed as an equation in terms of the coordinates the 
area is 



AREA BY COORDINATES 375 

From this equation is derived the following rule for obtaining 
the area of a closed field from the coordinates of its comers: — 

(i) Number the comers consecutively around the field. 

(2) Multiply each | ^'bscis^ | ^^ ^j^^ difference between the 

following and the preceding | i^ • I > always subtracting the 

preceding from the following, and take one-half of the sum of 
the products. 

4x2. Fig. 170 is the computation, by coordinates, of an area 



StaJ 



09onng 



Dist 



LMtihMh. 



Nt S- 



Oeparhjom 



i^JtL 



Bakmcaa 



lot: Ikp. 






DoabkArBo 



N57>l 



160 



QS7 



134 



*<xa7 



AS* 



205$ 



2136 



'S36 



tiai 



S37V 



I53Z 



RM 



323 



■>e23 



•aa 



/9Z? 



SL25 



*//J6 



ZIQ3 



S46lt£ 



4.S3 



3,11 



128 



Xll 



tX2d 



10.00 K>XX}^IS34 



IS3.4 



N45'i£ 



1X75 



ajs? — 



<9t6 



f3i3B^3i'fra2e 



eS9 '6.07 



91.2 



4- N26*W SCO 4^ 2.19 ^4^ 'tis 2115 I6B}'J447 



3Z9,2 



tS.33 I&3S t2.74 /2.7$ 



37/7 



l,95jcrts. 

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: 

J»2,3,4 = K'^f:^I-^l^S+^«^2-^sJ'8+^4^8-^8^4+^1^4--^4^l)(3) 

When this formula is to be used the coordinates may be arranged 
in the following simple manner: 



I. 2, 3» 4 =i{ - / -^ /, - / - / -] 



(4) 



37« 



DOUBLE MERIDIAN DISTANCES 



[Chap. XIIL 



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. 



z. The latitude of a line of a traverse is + 106.4a ft.; its departuze is 
— 373.62. What is its bearing ? 

3. From the following notes of a compass survey, compute by the double 
meridian distance method the area in acres. 



Station. 


Bearing. 


Distance 
(Chains). 


A 


N 46** J W 


ao.76 


B 


Nsi-fE 


13 83 


C 


East 


21.3s 


D 


S56*'E 


2-. 60 


E 


S33^JW 


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«>i E 


31-25 


4 


North 


13-50 


s 


(omitted) 


(omitted) 


6 


SS^'iW 


34.25 


7 


West 


32-50 



PROBLEMS 



377 



4. From the notes given in Fig. 52, p. 103, and Fig. 161, p. 362, compute by 
the dottbl& mtfidian distance method the area of the traverse ABCDEK. 



5. In the following traverse there are two mistakes. 
and determine their amounts. 



Find where they occur 



SUtion. 


Observed 
Bearing. 


Deflection 
Angle. 


Distance 
(Feet). 


Calculated 
Bearings. 


Remarks. 


A 
B 
C 
D 

E 


N34^E 

S73"iE 

SioOJW 

N26*>}W 

S52OW 


i64*» 14' R 

620 16' R 

84** 22' R 

142^ 49' R 

103041'L 


240.2 

163-7 
207.6 

273.1 

147-4 


N34*'oo'E 


C£ = i88.i 
BC£«34*»i4' 
DEC=^ 8i«2s' 



6. The following is 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 EF. 



Station. 


Bearing. 


Distance 
(Feet). 


A 


S 89« 14' E 


373.62 


B 


N 130 10' E 


100.27 


C 


N o«i7'W 


91.26 


D 


N27°39'E 


112. 4^ 


E 


N72«i2'W 


346.07 


F 


S s«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 AD is due East; AD is 600 ft., AB is 272.7 ft, and DC 
is 484.6 ft Find the length of a line EF parallel toAB which will cut ofT 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. 



o 
6+40 
9+20 

14+ SS 

17+ 18 
20+64 



Deflection 
Angle. 



6«>i7'L 
18043'L 
12047'R 
45*^24'L 
68«o6'R 



Find the length 
of a straight line 
from o to 20+ 64 
and the angle it 
makes with the 
line from o to 6 
+ 40- 






378 DOUBLE MERIDIAN DISTANCES [Chap.^XHI. 

9. Compute the otea of the following traverse by co5rdinates. 



Station. 


Deflection 
Angle. 


Bearing. 


Distance 
(Feet). 


/ A 


780 10' oo» L 




208.64 


B 


88« 28' oo» L 




436.79 


C 


6f 02' 15* L 




56.48 


D 


33" 39' 15' L 




98.80 


E 


9o« oo' oo» R 




68.69 


F 


22« 51' oo» L 




95 10 


G 


68« 50' 15* L 


N36«>i4'oo»W 


207.41 


H 


890 48' Off L 




103.75 


I 


SS^oo'irL 




96.75 


J 


530 49'oo»R 




420.77 



PART IV. 
PLOTTmO. 



PART IV. 

PLOTTIHG. 
CHAPTER XIV. 

DRAFTXNQ ZNSTRnBiENTS AND MATBRIAZiS. 

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. 

ENOINEERIlfO DRAFTHfG mSTRUMEliTS. 

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. STRAIOHT-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 imeven edge of a drawing board will not produce draw- 
ings of sufl&cient 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 rim 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 untrue. 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, qpe 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 200-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^, ^VtrTj ^^c. 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. 

417. 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. 




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. 



y 



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 suflBciently accurate, and, in fact, paper pro- 
tractors of this size will yield more accurate results than the 
smaller metal ones. 

4^0. Some of the metal protractors are provided with aji 
arm and vernier attachment. These, while giving more precise 
results, require more time for manipulation, and a plain metaJ 
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 off 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 off 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 imtil 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 o® and 180® marks, that the edge 
of the protractor is the arc of a true circle, and that the gradua- 
tions are uniform. 

423. PAHTOGRAPH. — This instrument is composed of several 
flat pieces of metal or wood joined in such a way as to form 
a parallelogram. One of the three points i4, ft and C, (Fig. 
173) is fixed and the other two movable. The remaining bear- 




FiG. 173. 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 t 



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 
Tule 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 
with 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 these 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 PEH. — 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 revcdves inside of the holder, 
and the pen is so shaped that it drags behind taking a position 
in 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 little 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 oflF 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 
radii, 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 conform 
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, which frequently are not circular. 

DRAWING PAPERa 

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 perinanent 



DRAWING PAPERS 38^ 

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 eflFectively. No paper, 
however, after scratching can be expected to take bottle red ink, 
which permeates the fiber with extraordinary ease. 

434. TRACING PAPER AND 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. 

* Erasare of ink lines from a tracing, as well as from any drawing paper, is a 
delicate undertaking. Success will result if the following 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 all of the ink is off,, 
rub the line with a pencil eraser. This will take off the rest of the line except 



390 DRAFTING INSTRUMENTS [Chap. 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 gready 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, AND 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^*', y^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- 



perhaps 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 application of a thin coat- 
ing of collodion on the damaged portion will produce a surface which will take 
the ink. 



PROCESS PAPERS 391 

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 gloth 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 offices 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 
|>arts 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 C* '' ) 

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 



i9^ VHAft&tb iNStftUMENTS [Csap. 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 
heav}' glass, with a hack 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 be examined to see that the trac- 
ing has been put into the frame with its ink lines against the 
glass, that the tlue-print paper is under the entire tracing, and 
that mere are no wrinkles in the tracing. All of the process to 
this stage should be done in subdued light, usually in a room 
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 mnutes 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-PRINTINfe 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 explaned 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 medium 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 oflF 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 himg 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 eflFect 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 siu"e to get rough usage are sometimes made on 
doth. 

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 apparendy 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 yellow 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 have 
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 imcertainty of the 
sxmlight 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 DRAFTING INStfttJMfeNTS [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 be 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 tin place of the bottled colored inks, such as 
P.U3sian blue iiste d of bottled blue ink, or bu^nt 5ien»a instead 
of brown 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 light 
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. 

BOZSTHODS OF PLOTTING. 

444* LATmo 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 tide, 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 TxyiTrxr, but some special maps are 
made on scales as large as ^uVtf. The field maps of the U. S. 
Geological Survey are mostly plotted to a scale of ^-uhjsjs and 
reduced on the lithograph sheets to ^sij^jf or iss^jfis - 

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 past- 
ing practice in this matter. 

METHODS OF PLOTTIirG TRAVERSES. 

446. PLOTTING BY PROTRACTOR AND 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 oflF 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 o® and 180® 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 

U 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 does 
not dose 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 
as many courses. 



400 METHODS OF PLOTTING [Chap. XV. 

Second, take a long straight piece of paper, lay this on the 
drawing, and mark off the length of the first line on the edge of 
the paper; then mark off the length of the second line starting 
from the mark which denotes the end of the first line, 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 line 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. PLOTTmS BY RECTANGULAR COORDINATES In plot- 
ting by this system all points in the traverse are referred to a 
pair of coordinate axes. For convenience these 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 each 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. 175. 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 Ay 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- 
ciuacy 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 vjhich 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 




61232 

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 connection 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 
oflf. 

455. PLOTTING BY TANGEIfTS. — 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 ab on the perpendicular.* Draw Ab 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 ip-ft. or ico>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 AB^ the 
length of the first course. On this line produced lay oflF Bg equal 
to 10 inches and erect a perpendicular, scaling ofif on the perpen- 
dicular the length gd equal to the tangent of the deflection angle 
at B, This determines the direction of BCixova 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 2?e, for example, a 
meridian is drawn through D parallel to Auy De is scaled off 10 
inches, and a perpendicular ef erected. The distance ^is scaled 
and from the table of tangents the angle /D« 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 



408 METHODS OF PLOTTING [Chap. XV. 

off with great care, preferably by the method explained in Art. 
451, 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. 178 represents the traverse 45CI>-E-F 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 tKe distance Aa is scaled off equal to 
10 inches and the arc ab swung from ^ 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 gf 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 a as a center intersecting the arc ab ^\. b\ 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 ^* 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 defle^ction angle at 5, is 
scaled off. Bd is then drawn and BC scaled off on, it. Jn the 
same way the entire traverse is plotted. 

458. Use of the Sine. — It is evident that the chord 

ab^2X loxsm -.- 

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 eksily be done away 
with by laying off the radius with a 20-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. CSiecks. — 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 tiot 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 ^ as a center and with a radius of 10 inches, and 
the chord ^is 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 plottecj. 

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 elevations 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 over 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, S 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 thfe 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 



PLOTTING PROHLES 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 bottom 
of the profile to the grade line at these points. 

466. When the elevations are such that the profile, if con- 
tinued, will run oflf 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 paper or on one of the vertical lines 
where the gradient changes. 

467. Checks. — After plotting the surface and 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. 16S, and 
in Fig. 116, p. 266. 



MMybrk Park CummssfoH 




^^'^^ 



CHAPTER XVI. 

FINISHINO AND FILINO DRAWINGS. 

468. WHAT SHOULD APPEAR ON A DRAWUfG. — 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 LDVES. — 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 FILING DRAWINGS [Chap. XVI. 



which are marked by very short lines bisecting the angles. Fig. 
1 80 illustrates these two methods of mar4cing transit points. 




Fig. 180. Methods of Marking Angle Points on Traverse Lines. 



Triangulation 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 ihdi- . 
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 






Deciduous Trees (Oak). 



^U» "**' *M* 




MU. 


.4J4k 


*»!*. yMft» 


*IU*.. 


nM/m, 




■^•^ "^ 




MfM. 


-^««- ,—. ^_ 


WW. 


M 


«»w*. .. . 




,«U4». 




^lU*. 






^/s 


.\Ub 


«U4k 






*U//i ^,^^ 




#'** 




4I/44 






J*l/^ 


^K*. 


«M««» 










JM/* 


-£:& 



Grass. 




Fresh Marsh. 




Waterlining. 



«» ai» 






Deciduous Trees (Round Leaf) 




K V-, v^:^^A V;V^ ^>: 4<^ 



Cultivated Land. 




Salt Marsh — Sand. 




c '*''^-^,I^**^*** * 



Ledges — Evergreen Trees. 



Fig. 181. Topographic Conventional Signs. 



41 8 FINISHING AND HLING 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 syia- 
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. 

Water-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 pen 
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 espe- 
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. 




ii ■ y* 



1 1 I I I I I I I I 

I I I I M I 14- 



< 



^-^..y 



□ 
o 

BM.XIE32 



Building. (On large scale maps.) 

Barn 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. 0" Tnangularion.) 



Fig. 183. 



420 




421 




422 



FINISHING AND FlUNG DRAWINGS [Chap. XVI. 



and boundaries are usually represented in black ink by the sym- 
bols shown in Fig. 183. 

473. LETTBRniG.* — 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 different 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 mustrations of lettering see any of the fol- 
lowing publications : *< Plain Lettering," by Professor Henry S. Jacoby, published 
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 Grothic. 

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. 1 86 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. 



Crescent ^ecic/t to WbodLcLwn Cemetery. 

Octobers /802^ 
ScCLle 400 f)eet to / in,ch. 

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 
£aaten> Sheet 



Fig. 187. Titles of Plans. 



TITLES 425 



TRACK ELEVATION. 

C- 6c \A/- K R. R- 

Cross- Section of Bridge Showing 

Floor Construction. 
Scale iiru- Ift. 



HORIZONTAL SECTIONS 

THROUGH UPPER THROUGH LOWER THROUGH LOWER 

SLUICE-GATE SLUICE-GATE VALVE WELL 

o • 2 3 4 s <n: 



Pmeliminaiy Phofile 

for a Railroad from 

Redford Junction to North Liberty 

Sta.0 io Sta.498*e8,7 
May 1906 



Fig. 188. Titles of Profiles. 



426 



FINISHING AND FILING DRAWINGS [Chap. XVL 



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:- This re/nforcemenf is 8-0 "/ongf, 
and comes d/recf/y under each track. 
Leave amp/e room for bnd^eseat 

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 Confound confinmed and ex- 
tended by data from map(Lr476)onfile with 
Massacnusetts Harbor and Land Commission. 



Fig. 189. Samples of 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 f " 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 ordinau'y 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. 



S 

Meridians. 



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 sufRciently 
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. 

47g. 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 

SCALE g^ 

V *• *. 1 'T 



SCALE., I INCH - 200 rCET 

Fig. 192. Scales. 

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 affect- 
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 Drawings. — 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 diflFerent 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 



432 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 liiany 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 



tND£XlNG — 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. 



435 



437 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


100 


00000 00043 00087 00180 00178 00217 00260 00303 00346 00389 




0432 0475 0518 0561 0604 0647 0689 0732 0775 0817 




0860 0903 0945 0988 1030 1072 111^ 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 265^^ 2694 273i 2776 2816 2867 2898 




2938 2979 3019 3060 3100 3141 3181 3222 3262 3302 




3342 3383 3423 3463 3503 3543 3583 3623 8663 8703 




3743 3782 3822 3862 3902 8941 3981 4021 4060 4100 


110 


04139 04179 04218 04258 04297 04336 04376 04415 04454 04498 




4532 4571 4610 4650 4689 4727 4766 4805 4844 4888 




• 4922 4961 4999 5038 5077 5116 6154 6192 6231 6269 




5308 6346 5385 5423 5461 6500 6538 6576 6614 6652 




6690 6729 5767 5805 5843 6881 6918 5966 6994 6032 




6070 6108 6146 6183 6221 6268 6296 6333 6871 6408 




6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 




6819 6856 6893 6930 6967 7004 7041 7078 7115 7161 




7188 7225 7262 7298 7&35 7372 7408 7446 7482 7518 




7555 7591 7628 7664"77ro 7737 7778 7809 7846 7882 


120 


07918 07954 07990 08027 08068 08099 0813508171 08207 08248 




8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 




8636 8672 8707 8743 8778 8814 8849 8884 8920 8955 




8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 




9342 9377 9412 9447 9482 9517 9552 9587 9621 9656 




9691 9726 9760 9795 9830 9864 9899 9934 9968 10003 




10037 10072 10106 10140 10175 10209 10243 10278 10312 0846 




0380 0415 0449 JQ^ 0517 0561 0585 0610 0658 0687 
0721 0755 0789 "08^ 0857 0890 0924 0958 0992 1025 






1059 1093 1126 1160 1193 1227 1261 1294 1327 1861 


180 


11394 11428 11461 11494 11528 11561 11604 11628 11661 11694 




1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 




2057 2090 2123 2156 2189 2222 2264 2287 2820 2352 




2385 2418 2450 2483 2516 2548 2581 2613 2646 2678 




2710 2743 2775 280r5840 2872 2905 2937 2969 3001 




8033 3066 3098 3130 3162 8194 8226 3258 8290 8822 




8364 3386 3418 3450 3481 3513 3646 3577 8609 3640 




3672 3704 3736 3767 3799 3830 8862 8893 8925 8956 




8988 4019 4051 4082 4114 4145 4176 4208 4239 4270 




4801 4333 4364 4395 4426 4457 4489 4520 4661 4582 


140 


14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 




4922 4953 4983 6014 6045 5076 6106^ 5137 5168 6198 




6229 5259 5290 6320 6351 6381 6412 6442 6478 6608 




6634 6564 6594 5625 6655 6685 6715 5746 6776 6806 




6836 6866 6897 6927 6957 6987 6017 6047 6077 6107 




6137 6167 6197 6227 6266 6286 6316 6346 6876 6406 




6436 6465 6495 6524 6554 6584 6613 6643 6673 6702 




6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 




7026 7056 7085 7114 7148 7178 7202 7231 7260 7289 




7319 7348 7377 7406 7436 7464 7498 7622 7661 7680 


160 


17609 17688 17667 17696 17726 17764 17782 17811 17840 17869 



438 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


160 


17e09 17638 17667 17696 17725 17764 17782 17811 17840 17869 




7898 7926 7956 7984 8013 8041 8070 8099 8127 8166 




8184 8213 8241 8270 8298 8327 8365 8384 8412 8441 




8469 8498 8626 8654 8683 8611 8639 8697 8696 8724 




8752 8780 8888 8837 8865 8893 8921 8949 8977 9005 




9033 9061 9089 9117 9145 9173 9201 9229 9267 9285 




9312 9340 9368 9396 9424 9461 9479 9607 9685 9562 
9690 9618 9646 9673 9700 9728 9766 97^3 9811 9838 






9866 9893 9921 9948 9976 20003 20030 20068 20086 20112 




20140 20167 20194 20222 20249 0276 0303 0380 0368 0385 


ItW 


20412 20439 20466 20493 20620 20648 20675 20602 206^ 20666 




0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 




0962 0978 1006 1032 1059 1086 1112 1139 1166 1192 




1219 1245 1272 1299 1326 1362 1378 1405 1431 1468 




1484^611 1637 1664 1690 1617 1643 1669 1696 1722 




1748 1775 1801. 1827 1864 1880 1906 1932 1968 1985 




2011 2037 2063 2089 2116 2141 2167 2194 2220 8246 




2272 2298 2324 2850 2376 2401 2427 2463 2479 2505 




2631 2567 2683 2608 2634 2660- 2686 2712 2737 2763 




2789 2814 2840 2866 2891 2917 2^3 2968 2994 3019 


170 


23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 




3300 8326 3360 3376 3401 3426 3452 3477 3602 3628 




8563 8678 3603 3629 3664 3679 3704 3729 3754 3779 




8805 3830 8855 3880 8905 3930 3965 3980 4005 4030 




4055 4080 4105 4130 4155 4180 4204 4229 4264 4279 




4304 4329 4363 4378 4403 4428 4452 4477 4502 4627 




4651 4676 4601 4625 4650 4674 4699 4724 4748 4773 




4797 4822 4846 4871 4896 4920 4944 4969 4993.6018 




5042 6066 6091 6116 6139 6164 6188 6212 6237 6261 




6286 6310 6334 6368 6382 6406 6431 5465 6479 6603 


180 


26627 26661 25675 26600 26624 25648 26672 25696 25720 26744 




6768 6792 6816 6840 6864 6888 5912 6935 5969 6983 




6007 6031 6055 6079 6102 6126 6150 6174 6198 6221 




6246 6269 6293 6316 6340 6364 6387 6411 6435 6458 




6482 6505 6529 6563 6676 6600 6623 6647 6670 6694 




6717 6741 6764 6788 6811 6834 6868 6881 6905 6928 




6961 6975 6998 7021 7045 7068 7091 7114 7138 7161 




7184 7207 7231 7264 7277 7300 7323 7346 7370 7393 




7416 7439 7462 7486 7608 7631 7654 7677 7600 7623 




7646 7669 7692 7716 7738 7761 7784 7807 7830 7852 


190 


27876 27898 27921 27944 27967 27989 28012 28036 28068 28081 




8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 




8330 aS63 8376 8808 8421 8443 8466 8488 8611 8633 




8666 8578 8601 8623 8646 8668 8691 8713 8736 8768 




8780 88aS 8825 8847 8870 8892 8914 8937 8969 8981 




9003 9026 9048 9070 9G92 9115 9137 9169 9181 9203 




9226 9248 9270 9292 9314 9336 9358 9380 9403 9425 




9447 9469 9491 9513 9535 9657 9579 9601 9623 9645 




9667 9688 9710.9732 9754 9776 9798 9820 9842 9863 




9886 9907 9929 9961 9973 9994 30016 30038 30060 30081 


800 


30103 30125 30146 30168 30190 30211 30233 30265 30276 30298 



439 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


200 


80103 30125 30146 3016830100 30211 30233 3025^ 30276 30208 




032e 0341 0363 0384 0406 0428 0449 0471 0492 0514 




0536 0557 0678 0600 0621 0643 0664 0685 0707 0728 
0750 0771 0792 0814 0835 0866 0878 0899 0920 0942 






0963 0984 1006 1027 1048 1069 1091 1112 1133 1154 




1175 1197 1218 1239 1260 1281 1302 1323 1345 1366 




1387 1408 1429 14^0 1471 1492 1513 1634 1655 1676 




.1597 1618 1639 1660 1681 1702 1723 1744 1765 1785 




1806 1827 1848 1869 1890 1911 1931 1962 1973 1994 




2015 2035 2056 2077 2098 2118 2139 2160 2181 2201 


210 


32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 




2428 2449 2469 2490 2510 2631 2652 2672 2693 2613 




2634 2654 2675 2695 2715 2786 2766 2777 2797 2818 




2838 2868 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 8405 3425 




8446 3465 3486 3506 8626 3646 3666 3686 3606 3626 




3646 3666 3686 3706 3726 3746 3766 3786 3806 3826 




3846 3866 3885 3905 3925 8945 3965 8985 4005 4025 




4044 4064 4084 4104 4124 4143 4163 4183 4203 4223 


220 


34242 34262 34282 34301 34321 34341 34361 84380 84400 84420 




4439 4459 4479 4498 4618 4637 4667 4577 4596 4616 




4636 4665 4674 4694 4713 4733 4763 4772 4792 4811 




4830 4850 4869 4889 4908 4928 4947 4967 4986 5005 




5025 6044 6064 608;) 5102 5122 5141 5160 5180 5199 




5218 6238 5267 5276 5295 5315 6334 6368 5372 5892 


g 


5411 5430 6449 6408 5488 6607 5626 5645 6564 5583 




5603 5622 6641 6660 6679 5698 5717 6736 5755 6774 




6793 5813 6832 5861 6870 5889 5908 5927 5946 6965 




5984 6008 6021 6040 6069 6078 6097 6116 6135 6154 


280 


36173 36192 36211 36229 36248 36267 86286 36305 86824 36842 




6361 6380 6399 6418 6436 6455 6474 6493 6611 6630 




6649 6568 6586 6605 6624 6642 6661 6680 6698 6717 


3' 


6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 




6922 6940 6969 6977 6996 7014 7033 7061 7070 7088 




7107 7125 7144 7162 7181 7199.7218 7236 7254 7273 




7291 7310 7328 7346 7305 7383 7401 7420 7438 745^ 




7475 7493 7611 7630 7548 7666 7685 7603 7621 7639 




7658 7676 7694 7712 7731 7749 7767 7785 7808 7822 




7840 7868 7876 7894 7912 7931 7949 7967 7985 8008 


240 


38021 38039 38067 38075 38093 38112 38130 38148 38166 38184 




8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 




8382 8399 8417 8436 8453 8471 8489 8507 8625 8648 




8661 8578 8596 8614 8632 8650 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 9U1 9129 9146 9164 9182 9199 9217 9285 9252 




9270 9287 9305 9322 9340 9358 9375 9393 9410 9428 




9445 9463 9480 9498 9615 9533 9660 9568 9585 9602 




9620 9637 9655 9672 9690 9707 9724 9742 9759 9777 


260 


39794 39811 39829 39846 39863 39881 89898 39915 89938 39950 



440 
TABLE L— LOGARITHMS OF NUMBERS. 



N 


0123456780 


250 


397M 30811 8»829 39846 89863 39881 39898 39916 39933 39960 




9967 998^ 40002 40019 40037 40064 40071 40088 40106 4U123 




40140 40167 0175 0192 0209 0226 0243 0261 0278 029^ 




0312 0329 0346 0364 0381 0398 041^ 0432 0449 0466 




0483 0600 0618 0635 0662 0669 0686 0603 0620 0637 




0664» 0671 0688 0706 0722 0739 0756 0773 0790 0807 




0824 0841 0868 087i 0892 0909 0926 0943 0960 0976 




0993 1010 1027 1044 1061 1078 1095 1111 1128 1146 




1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 




1330 1347 1868 1380 1397 1414 1430 1447 1464 1481 


260 


41497 41614 41631 41547 41664 41681 41697 41614 41631 41647 




1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 




1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 




1996 2012 2029 2046 2062 2078 2095 2111 2127 2144 




2160 2177 2193 2210 2226 2243 2259 2276 2292 2308 




2325 2341 2367 2374 2390 2406 2423 2439 2466 2472 




2488 2604 2521 2537 2653 2570 2586 2602 2619 2635 




2661 2667 2684 2700 2716 2732 2749 2765 2781 2797 




2813 2830 2846 2862 2878 2894 2911 2927 2943 2969 




2975 2991 8008 8024 3040 8066 3072 8088 8104 8120 


270 


43m 48162 43169 43185 43201 43217 43233 43249 43265 43281 




3297 3313 3329 3345 ^361 3377 3393 3400 342i 3441 




8457 3473 3489 3605 3521 3537 3553 3569 3584 3600 




3616 3632 3648 3664 3680 3696 3712 3727 3743 3769 




3776 3791 3807 3823 3838 3854 8870 3886. 3902 8917. 




3933 3949 3965 3981 3996 4012 4028 4044' 4059 4076 




4091 4107 4122 4138 4164 4170 4186 4201 4217 4232 




4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 




4404 4420 4436 4461 4467 4483 4498 4514^529 4645 




4660 4676 4692 4607 4623 4638 4064 4669 4685 4700 


280 


44716 44731 44747 44762 44778 44793 44809 44824 44840 44866 




4871 4886 4902 4917 4932 4948 4963 4979 4994 6010 




6025 6040 6066 6071 6086 6102 5117 6133 6148 6163 




6179 6194 6209 6225 6240 6256 6271 6286 6301 6317 




6332 6347 6362 6378 6393 6408 6423 6439 6464 6469 




6484 6500 6615 5530 6545 5561 6676 6691 5606 6621 




6637 6652 5667 6682 6697 6712 6728 6743 6768 6773 




6788 6803 6818 6834 6849 6864 6879 6894 6909 6924 




6989 6964 6969 6984 6000 6015 6030 6045 6060 6075 




6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 


290 


46240 46265 46270 46285 46300 46315 46330 46345 46369 46374 




6389 6404 6419 6434 6449 6464 6479 6404 6609 6623 




6538 6563 6568 6583 6598 6613 6627 6642 6667 6672 




6687 6702 6716 6731 6746 6761 6776 6790 6806 6820 




6835 6850 6864 6879 6894 6S)09 6923 6938 6963 6967 




6982 6997 7012 7026 7041 7056 7070 7086 7100 7114 




7129 7144 7169 7173 7188 7202 7217 7232 7246 7261 




7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 




7422 7436 7461 7466 7480 7494 7609 7624 7538 7653 




7567 7582 7596 7611 7625 7640 7664 7669 7683 7698 


800 


47712 47727 47741 47756 47770 47784 47799 47818 47828 47842 



V 



441 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456780 


800 


47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 




7867 "7871 7886 7900 7914 7929 7943 7968 7972 7986 




8001 8016 8029 8044 8068 8073 8087 8101 8116 8130 




8144 8169 8173 8187 8202 8216 8230 8244 8269 8273 




8287 8302 8316 8330 8344 8369 8373 8387 8401 8416 




8430 8444 8468 8473 8487 8601 8616 8630 8644 8668 




8672 8686 8601 8615 8629 .8645: 8667 8671 8686 8700 




8714 8728 8742 8766 8770 8785 8799 8813 8827 8841 




8866 8869 8883 8897 8911 8926 8940 8954 8968 8982 




8996 9010 9024 9038 9062 9066 9080 9094 9108 9122 


810 


49136 49160 49164 49178 49192 49206 49220 49234 49248 49262 




9276 9290 9304 9318.9332 9346 9360 9374 9388 9402 
9416 94^9 9443 9467^9471 9486 9499 9613 9627 9541 
9664 9668 9682 9696 9610 9624 9638 9661 9666 9679 








9693 9207 9721 9734 9748 9762 9776 9790 9803 9817 




9831 9845 9869 9872 9886 9900 9914 9927 9941 9955 




9969 9982 9996 60010 60024 60037 6006160065 60079 60092 




60106 60120 60133 0147 0161 0174 0188 0202 0216 0229 




0248 0266 0270 0284 0297 03ll 0325 0338 0362 0366 




0379 0393 0406 0420 0433 0447 0461 0474 0488 0601 


820 


60615 60629 60642 60666 60569 60683 60696 60610 60623 60637 




0661 0664 0678 0691 0705 0718 0732 0746 0769 0772 




0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 




0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 




1065 1068 1081 1095 1108 1121 1135 1148 1162 1175 




118& 1202 1216 1228 1242 1266 1268 1282 1^296 1308 
1322 1336 I34d 1362 1376 1388 1402 14i5 1428 1441 






1465 1468 1481 ]49i 1508 1621 1534 1648 1661 1674 




1687 i601 1614 1627 1640 1664 1667 1680 1693 1706 




1720 1733 1746 1769 1772 1786 1799 1812 1826 1838 


880 


61861 61865 61878 61891 61904 61917 61930 61943 61967 61970 




1983 1996 2009 2022 2036 2048 2061 2075 2088 2101 
2114 2127 2140 2163 2166 2179 2192 2206 2218 2231 






2244 2267 2270 2284 2297 2310 2323 2336 2349 2362 




2375 2388 2401 2414 2427 2440 2463 2466 2479 2492 




2604 2617 2630 2643 2666 2569 2682 2696 2608 2621 




2634 2647 2660 2673 2686 2699 2711 2724 2737 2750 




2763 2776 2789 2802 2815 2827 2840 2863 2866 2879 




2892 2905 2917 2930 2943 2966 2969 2982 2994 3007 




3020 3033 3046 8068 8071 3084 3097 3110 3122 3136 


840 


63148 63161 63173 63186 63199 63212 63224 63237 63250 63263 




3275 3288 3301 3314 3326 3339 a362 3364 3377 3:^90 




3403 3416 3428 3441 3463 3466 3479 3491 3504 3617 




3629 3642 3665 3567 3580 3593 3606 3618 3631 3643 
3k66 3668 3681 3694 3706 3719 3732 3744 3767 3769 






3782 3794 3807 3820 3832 3845 3857 3870 3882 3896 




3908 3920 3933 3946 3968 3970 3983 3995 4008 4020 




4033 4046 4068 4070 4083 4096 4108 4120 4133 4146 




4168 4170 4183 4196 4208 4220 4233 4246 4268 4270 




4283 4295 4307 4320 4832 4345 4367 4370 4382 4394 


Z60 


64407 64419 64432 54444 54456 54469 64481 64494 64606 64618 



/ 



442 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 



850 

1 
9 
S 

4 
6 
6 
7 

8 
9 

1 
9 
S 

4 
6 
6 
7 
8 
9 

870 

1 
9 
8 
4 
5 
6 
7 
8 
9 

880 

1 
9 
8 
4 
6 
6 
7 
8 
9 

890 
1 
9 
8 
4 
6 
6 
7 
8 
9 

400 



2 3 4 5 6 7 8 



64407 64419 64432 64444 64466 64469 64481 64494 64606 64618 

4631 4643 4656 4568 4580 4693 460^ 4617 4630 4642 

4664 4667 4679 4691 4704 4716 4728 4741 4763 4766 

4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 

4900 4913 492i 4937 4949 4962 4974 4986 4998 6011 

6023 6035 6047 6060 6072 5084 6096 5108 6121 6133 

614i 6157 6169 6182 6194 6206 6218 6230 5242 5255 

6267 6279 6291 6303 6315 5328 6340 6352 5364 6376 

6388 6400 6413 542^ 5437 5449 6461 6473 6485 6497 

6509 6622 6534 6546 5568 6570 5682 6594 6606 6618 

66630 66642 56664 65666 65678 66691 65703 6671i 66727 66739 

6751 6763 6775 5787 5799 6811 6823 6835 6847 6859 

6871 6883 5895 5907 5919 6931 6943 6955 6967 6979 

6991 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 6638 6549 6561 6673 

6585 6597 6608 6620 6632 6644 6656 6667 6679 6691 

6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 

66820 66832 66844 66856 66867 66879 66891 56902 56914 56926 

6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 

7054 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 7688 7600 7611 7623 

7634 7646 7657 7669 7680 7692 7703 7715 7726 7738 

7749 7761 7772 7784 7795 7807 7818 7830 7841 7852 

7864 7876 7887 7898 7910 7921 7933 7944 7956 7967 

57978 57990 68001 58013 58024 58035 68047 68058 68070 68081 

8092 8104 8116 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 8612 8524 8635 

8646 8557 8569 8580 8691 8602 8614 8625 8636 8647 

8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 

8771 8782 8794 8805 8816 8827 8838 8850 8861 8872 

8883 8894 8906 8917 8928 8939 8950 8961 8973 8984 

9006 9017 9028 9040 9051 9062 9073 908^ 9096 



59106 69118 59129 59140 59151 59162 59173 69184 69196 69207 
9218 9229 9240 9251 9262 9273 9284 9295 9306 9318 
9329 9340 9351 9362 9373 9384 9395 9406 9417 9428 
9439 9450 9461 9472 9483 9494 9506 9617 9528 9539 
9550 9561 9672 95a3 9594 9605 9616 9627 9638 9649 
9660 9671 9682 9693 9704 9715 9726 9737 9748 9769 
9770 9780 9791 9802 9813 9824 9836 9846 9857 9868 
9879 9890 9901 9912 9923 9934 9945 9956 9966 9977 
9988 9999 60010 60021 60032 60043 60054 60065 60076 60086 

60097 60108 0119 0130 0141 0152 0163 0173 0184 0195 

60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 



443 
TABLE I.— LOGARITHMS OF NUMBERS. 



-K 


0123456789 


400 


60206 60217 60228 60230 60249 60260 60271 60282 60203 60304 




0314 0325 0336 0347 0358 0369 0379 0390 0401 0412 




0423 0433 0444 0456 0466 0477 0487 0498 0509 0520 




0531 0541 0562 0563 0674 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 088i 0895 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 1525 1236 1247 1257 1268 


410 


61278 61280 61300 61310 61321 61331 61342 61352 61363 61374 




1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 




1490 1500 1511 1521 1532 1542 1553 1563 1574 1584 




1596 1606 1616 1627 1637 1048 1668 1669 1679 1690 




1700 1711 1721 1731 1742 1762 1763 1773 1784 1794 




1805 1815 1826 1836 1847 1857 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 2169 2170 2180 2190 2201 2211 




2221 2232 2242 2252 2263 2273 2284 2294 2304 2815 


420 


62325 62335 62346 62366 62366 62377 62387 62397 62408 62418 




2428 2439 2449 2469 2469 2480 2490 2600 2511 2521 




2531 2642 2562 2662 2672 2683 2593 2603 2613 2624 




2634 2644 2665 2665 2675 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 3053 3063 8073 3083 8094 3104 3114 3124 3134 




3144 3155 3165 3175 3185 3196 3206 3216 3226 3230 




3246 3256 3266 3276 3286 3296 3306 3317 3327 3337 


480 


63347 63367 63367 63377 63387 63397 63407 63417 63428 63438 




.3448 3468 3468 3478 3488 3498 3508 3618 3528 3638 




3648 3558 3568 3679 3689 3609 3609 8619 3629 3639 




3649 3669 3669 3679 3689 3699 3709 3719 3729 3739 




3749 3759 3769 3779 3789 3799 3809 3819 3829 3839 




3849 3859 3869 3879 3889 8899 3909 3919 3929 3939 




3949 3959 3969 3979 3988 3998 4008 4018 4028 4038 




4048 4068 4068 4078 4088 4098 4108 4118 4128 4137 




4147 4167 4167 4177 4187 4197 4207 4217 4227 4237 




4246 4256 4266 4276 4286 4296 4306 4316 4326 4335 


440 


64346 64356 64366 64375 64385 64395 64404 64414 64424 64434 




4444 4454 4464 4473 4483 4493 4503 4513 4623 4532 




4642 4652 4662 4672 4582 4591 4601 4611 4621 4631 




4640 4650 4600 4670 4680 4689 4699 4709 4719 4729 




4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 




4836 4846 4866 4865 4876 4885 4895 4904 4914 4924 




4933 4043 4963 4963 4972 4982 4992 5002 5011 5021 




6031 5040 6050 5060 6070 5079 5080 5099 6108 6118 




6128 5137 6147 6167 6167 6176 6186 6196 5205 6215 




5225 5234 5244 5254 5263 5273 5283 6292 6302 5312 


460 


65321 65331 65341 65350 65360 66369 65379 66389 65398 66408 



444 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 
450 


0123456780| 


65321 66331 65341 65350 65360 65360 65379 65389 65398 65408 




5418 5427 5437 5447 5456 5466 5475 5485 549i 5504 




5514 5523 5533^5543 5552 5562 5571 5581 5591 5600 




5610 5619^^9 5639 5648 5658 5667 5677 5686 5696 




5706 5715 5725 5734 5744 5753 5763 5772 5782 5792 




5801 5811 5820 5830 5839 5849 5858 5868 5877 5887 




5896 5906 5916 5925 5935 5944 5954 5963 5973 5982 




5992 6001 6011 6020 6030 6039 6049 6058 6068 6077 




6087 6096 6106 6115 6124 6134 6143 6153 6162 6172 




6181 6191 6200 6210 6219 6229 6238 6247 6257 6266 


iW 


66276 66285 66295 66304 66314 66323 66332 66342 66351 66361 




6370 6380 6389 6398 6408 6417 6427 6436 6445 6455 




6464 6474 6483 6492 6502 6511 6521 6530 6539 6549 




6558 6567 6577 6586 6596 6605 0614 6624 6633 6642 




6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 




6745 6755 6764 6773 6783 6792 6801 6811 6820 6829 




6839 6848 6857 6867 6876 6885 6894 6904 6918 6922 




6932 6941 6950 6960 6969 6978 6987 6997 7006 7015 




7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 




7117 7127 7136 7146 7154 7164 7173 7182 7191 7201 


470 


67210 67219 67228 67237 67247 67266 67265 67274 67284 67293 




7302 7311 7321 7330 7339 7348 7357 7367 7376 7385 




7394 7403 7413 7422 7431 7440 7449 7459 7468 7477 




7496 7496 7604 7614 7623 7632 7541 7650 7660 7669 




7678 7687 7696 7606 7614 7624 7633 7642 7661 7660 




7669 7679 7688 7697 7706 7715 7724 7733 7742 7762 




7761 7770 7779 7788 7797 7806 7815 7825 7834 7843 




7852 7861 7870 7879 7888 7897 7906 7916 7925 7934 




7943 7962 7961 7970 7979 7988 7997 8006 8015 8024 




8034 8043 8062 8061 8070 8079 8088 8097 8106 8115 


480 


68124 68133 68142 68161 68160 68169 68178 68187 68196 68205 




8215 8224 8233 8242 8261 8260 8269 8278 8287 8296 




8305 8314 8323 8332 8341 8350 8369 8368 8377 8386 




8395 8404 8413 8422 8431 8440 8449^ 8458 8467 8470 




8485 8494 8602 8611 8620 8629 8638 8647 8656 8665 




8674 8683 8592 8601 8610 8619 8628 8637 8646 8665 




8664 8073 8681 8690 8699 8708 8717 8726 8735 8744 




8763 8762 8771 8780 8789 8797 8806 8816 8824 8833 




8842 8861 8860 8869 8878 8886 8896 8904 8913 8922 




8931 8940 8949 8968 8966 8975 8984 8993 9002 9011 


490 


69020 69028 69037 69046 69055 69064 69073 69082 69090 69099 




9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 




9197 9205 9214 9223 9232 9241 9249 9258 9267 9276 




9285 9294 9302 9311 9320 9329 93:^ 9346 9356 9364 




9373 9381 9390 9399 9408 9417 9426 9434 9443 9462 




9461 9469 9478 9487 9496 9604 9613 9622 9631 9539 




9648 9667 9666 9674 9683 9692 9601 9609 9618 9627 




9636 9644 9663 9662 9671 9679 9688 9697 9705 9714 




9723 9732 9740 9749 9768 9767 9775 9784 9793 9801 




9810 9819 9827 9836 9845 9864 9862 9671 9880 9888 


600 


00897 09906 69914 69923 69932 69940 69949 69958 69966 69975 



445 



TABLE T.— LOGARITHMS OF NUMBERS. 



N 


0123456780 


500 


69897 69906 69914 69923 69932 69940 69949 69968 69966 69975 




9984 9992 70001 70010 70018 70027 70036 70044 70063 70062 




70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 




0167 0166 0174 0183 0191 0200 0209 0217 0226 0234 




0243 0262 0260 0269 0278 0286 0295 0303 0312 0321 




0329 0338 0346 0365 0364 0372 0381 0380 0398 0406 




0415 0424 0432 0441 0449 0468 0467 0476 0484 0492 




0601 0609 0618 0626 0636 0644 0662 0661 0569 0678 




0586 0695 0603 0612 0621 0629 0638 0646 0665 0663 




0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 


510 


70767 70766 70774 70783 70791 70800 70808 70817 70826 70834 




0842 0861 0869 0868 0876 0885 0893 0902 0910 0919 




0927 0936 0944 0962 0961 0969 0978 0986 0995 1003 




1012 1020 1029 1037 1046 1064 1063 1071 1079 1088 




1096 1105 1113 1122 1130 1139 1147 1165 1164 1172 




1181 1189 1198 1206 1214 1223 1231 1240 1248 1267 




1265 1273 1282 1290 1299 1307 1316 1324 1332 1341 




1349 1367 1366 1374 1383 1391 1399 1408 1416 1425 




1433 1441 1450 1468 1466 1475 1483 1492 I5OO 1608 




1617 1626 1633 1642 1660 1659 1667 1675 1684 1692 


520 


71600 71609 71617 71625 71634 71642 71660 71669 71667 71675 




1684 1692 1700 1709 1717 1726 1734 1742 1760 1769 




1767 1776 1784 1792 1800 1809 1817 1826 1834 1842 




1850 1858 1867 1876 1883 1892 1900 1908 1917 1925 




1933 1941 1950 1968 1966 1975 1983 1991 1999 2008 




2016 2024 2032 2041 2049 2067 2066 2074 2082 2090 




^099 2107 2115 2123 2132 2140 2148 2166 2165 2173 




2181 2189 2198 2206 2214 2222 2230 2239 2247 2256 




2263 2272 2280 2288 2296 2304 2313 2321 2329 2337 




2346 2364 2362 2370 2378 2387 2395 2403 2411 2419 


580 


72428 72436 72444 72452 72460 72469 72477 72485 72493 72601 




2609 2518 2526 2534 2642 2660 2668 2667 2675 2683 




2691 2599 2607 2616 2624 2632 2640 2648 2666 2665 




2673 2681 2689 2697 2706 2713 2722 2730 2738 2746 




2764 2762 2770 2779 2787 2795 2803 2811 2819 2827 




2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 




2916 2925 2933 2941 2949 2957 2966 2973 2981 2989 




2997' 3006 3014 3022 3030 3038 3046 3064 3062 8070 




3078 3086 3094 3102 3111 3119 3127 3135 3143 3161 




3169 3167 3175 3183 3191 3199 3207 3216 3223 3231 


540 


73239 73247 73266 73263 73272 73280 73288 73296 73304 73312 




3320 3328 3336 3344 3362 3360 3368 3376 3384 3392 




3400 3408 3416 3424 3432 3440 3448 3466 3464 3472 




3480 3488 3496 3604 3612 3620 3528 3636 3644 3662 




3560 3568 3576 3684 3592 3600 3608 3616 3624 3632 




3640 3648 3666 3664 3672 3679 3687 3696 3703 3711 




3719 3727 3736 3743 3751 3759 3767 3775 3783 3791 




3799 3807 3815 3823 3830 3838 3846 "3854 3862 3870 




3878 3886 3894 3002 3910 3918 3926 3033 3941 3949 




3957 3966 3973 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 


650 


74036 74044 74062 74060 74068 74076 74084 74092 74009 74107 




4115 4123 4131 4139 4147 4155 4162 4170 4178 4186 




4194 4202 4210 4218 4226 4233 4241 4249 4257 4265 




4273 4280 4288 4296 4304 4312 4320 4327 4336 4343 




4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 




4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 




4507 4516 4523 4531 4539 4547 4654 4562 4670 4578 




4686 4593 4C01 4609 4617 4624 4632 4640 4648 4666 




4663 4671 4679 4087 4605 4702 4710 4718 4726 4733 




4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 


660 


74819 74827 74834 74842 74850 74868 74866 74873 74881 74889 




4896 4904 4912 4920 4927 4935 4943 4950 4968 4966 




4974 4981 4989 4997 6005 6012 6020 6028 6036 6043 




6061 6069 6066 6074 6082 6089 6097 6105 6113 6120 




6128 6136 6143 6161 5159 6166 5174 6182 6189 6197 




5205 6213 52^ 6228 5236 6243 6261 6259 6266 6274 




6282 6289 6297 6305 5312 6320 6328 6336 6343 6361 




6358 6366 6374 6381 53^9 6397 6404 6412 6420 5427 




6435 6442 6460 6458 6465 6473 6481 6488 5496 6504 




6611 6619 6526 6534 6642 6649 6567 6665 6672 6580 


670 


76687 76696 76603 76610 75618 76626 76633 76641 76648 76656 




6664 6671 6679 6686 6694 6702 6709 6717 6724 6732 




6740 6747 6765 5762 6770 6778 6786 6793 6800 6808 




6816 6823 6831 6838 5846 6853 5861 5868 6876 5884 




6891 6899 6906 6914 6921 6929 6937 6944 6962 6959 




6967 6974 6982 5989 6997 6005 6012 6020 6027 6035 




6042 6050 6067 6065 6072 6080 6087 6095 6103 6110 




6118 6125 6133 6140 6148 6156 6163' 6170 6178 6186 




6193 6200 6208 6216 6223 6230 6238 6246 6263 6260 




6268 6276 6283 6290 6298 6305 6313 6320 6328 6386 


680 


76343 76360 76368 76366 76373 76380 76388 76396 76403 76410 




6418 6425 64:^3 6440 6448 6455 64(52 6470 6477 6485 




6492 6500 6507 6515 6522 6530 6537 6645 6562 6569 




6667 6574 6682 6589 6597 6G04 6612 6619^ 6626 6634 




6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 




6716 6723 6730 6738 6746 6753 6760 6768 6775 6782 




6790 6797 6805 6812 6819 6827 6834 6842 6849 6866 




6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 




6938 6946 6953 6960 6967 6975 6982 6989 6997 7004 




7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 


690 


77086 77093 77100 77107 77115 77122 77129 77137 77144 77161 




7159 7166 7173 7181 7188 7105 7203 7210 7217 7225 




7232 7240 7247 7254 7262 7269 7276 7283 7291 7298 




7306 7313 7320 7327 7335 7342 7349 7857 7364 7371 




7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 




7452 7459 7466 7474 7481 7488 7495 7503 7610 7617 




7525 7532 7539 7546 7554 7561 7568 7576 7583 7590 




7597 7605 7612 7619 7027 7634 7641 7648 7656 7663 




7670 7677 7685 7692 7^99 7706 7714 7721 7728 7735 




7743 7750 7767 7764' 7772 7779 7786 7793 7801 7808 


600 


77816 77822 77830 77837 77844 77851 77859 77866 77873 77880 



447 



TABLE I.— LOGARITHMS OF NXJMBERS. 



N 


0123456789 


600 


77816 77822 77830 77837 77844 77851 77859 77866 77873 77880 


1 


7887 7895 7902 7909 7918 7924 7931 7938 7945 7952 


2 


7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 




8032 8039 8046 8053 8061 8068 8075 8082 8089 8097 




8104 8111 8118 8125 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 




8390 8398 8405 8412 8419 8426 8433 8440 8447 8455 




8462 8469 8476 8483 8490 8497 8504 8512 8510 8526 


610 


78533 78540 78547 78654 78561 78569 78576 78583 78690 78697 




8604 8611 8618 8625 8633 8640 8647 8654 8661 8668 




8675 8682 8689 8696 8704 8711 8718 8725 8732 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 8965 8972 8979 8986 8993 9000 9007 9014 9021 




9029 9036 9043 9050 9057 0064 9071 9078 9085 9092 




9099 9106 9113 9120 9127 9134 9141 9148 9155 9162 




0169 9176 9183 9190 9197 9204 9211 9218 9225 9232 


620 


79239 79246 79253 79260 79267 79274 79281 79288 79296 79302 




9309 9316 9323 9330 9337 9344 9351 9358 9365 9372 




9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 




9449 9466 9463 9470 9477 9484 9491 9498 9505 9511 




9518 9525 9532 9539 9546 9553 9560 9567 9574 9581 




9588 9595 9602 9609 9616 9623 9630 9637 9644 9660 




9657 9664 9671 9678 9686 9692 9099 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 79976 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 0164 0161 0168 0175 0182 0188 0196 0202 




0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 




0277 0284 0291 0298 0305 0312 0318 0326 0332 0339 




0346 0353 0369 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 




0660 0567 0564 0570 0577 0584 0591 0598 0604 0611 


640 


80618 80625 80632 80638 80646 80652 80659 80666 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 0876 0882 




0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 




0956 0963 0969 0976 0983 0990 0996. 1003 1010 1017 




1028 1030 1037 1043 1050 1057 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 1258 1265 1271 1278 1285 


660 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81361 



448 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


01234567 89 


660 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 




1358 1365 1371 1378 1385 1391 1398 1405 1411 1418 




1425 1431 1438 1445 1451 1468 14(i5 1471 1478 1485 




1491 1498 1505 1611 1518 1625 1531 1538 1544 1551 




1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 




1624 1631 1637 1644 1651 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 


81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 




2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 




2086 2092 2099 2105 2112 2119 2125 2132 2138 2145 




2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 




2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 




2282 2289 2295 2302 2308 2315 2321 2328 2334 2341 




2347 2364 2360 2367 2373 2380 2387 2393 2400 2406 




2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 




2478 2484 2491 2497 2504 2510 2517 2523 2530 2536 




2543 2549 2556 2562 2569 2575 2582 2588 2595 2601 


670 


82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 




2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 




2737 2743 2750 2756 2763 2769 2776 2782 2789 2795 




2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 




2866 2872 2879 2885 2892 2898 2905 29U 2918 2924 




2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 




2995 8001 3008 3014 3020 3027 3033 3040 3046 3052 




3059 8065 3072 3078 3085 3091 3097 3104 3110 3117 




3123 3120 3136 3142 3149 3155 3161 8168 3174 3181 




3187 3198 3200 3206 3213 8219 3225 8232 3238 3245 


680 


83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 




3315 3321 3327 3334 3340 3347 3353 3359 3366 3372 




8378 3385 3391 3398 3404 3410 3417 3423 3429 3486 




3442 3448 3455 3461 3467 3474 3480 3487 3498 3499 




3506 3512 3518 3525 3531 3537 3544 3550 3556 3563 




3569 3575 3582 3588' 3594 3601 3607 3613 3620 3626 




3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 




3696 3702 3708 3715 3721 3727 3734 3740 3746 3753 




3759 3766 3771 3778 3784 3790 3797 3803 3809 3816 




3822 3828 3835 3841 3847 3863 3860 3866 3872 3879 


690 


83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 




8948 8954 3960 3967 3973 3979 3985 3992 3998 4004 




4011 4017 4023 4029 4036 4042 4048 4055 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 




4823 4330 4336 4342 4348 4354 4361 4367 4373 4379 




4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 




4448 4454 4460 4466 4473 4479 4485 4491 4497 4504 


700 


84510 84516 84522 84528 84535 84541 84547 84563 84559 84566 



449 





TABLE I.— LOGARITHMS OF NUMBERS. 


N 


123456789 


700 


84610 84616 84622 84628 8463^ 84641 84647 84663 84669 84666 




4672 4678 4684 4690 4697 4603 4609 4616 4621 4628 




4634 4640 4646 4662 4668 466^ 4671 4677 4683 4689 




4696 4702 4708 4714 4720 4726 4752 4739 4745 4761 




4767 4763 4770 4776 4782 4788 4794 4800 4807 4813 




4819 4826 4831 4837 4844 48^0 4866 4862 4868 4874 




4880 4887 4893 4899 4906 4911 4917 4924 4930 4936 




4942 4948 4964 4960 4967 4973 4979 4985 4991 4997 




6003 6009 6016 6022 6028 6034 6040 6046 6062 6068 




5065 5071 6077 6083 6089 6096 6101 6107 6114 6120 


710 


86126 86132 86138 86144 86160 86166 86163 86169 86175 86181 




6187 6193 6199 6206 6211 6217 6224 6230 6236 6242 




6248 6264 6260 6266 6272 6278 6285 6291 6297 6303 




6309 6316 6321 6327 6333 6339 6346 6362 6368 6364 




6370 6376 6382 6388 6394 6400 6406 6412 6418 6425 




6431 6437 6443 6449 6465 6461 6467 6473 6479 6486 




6491 6497 6603 6609 6616 6622 6628 6634 6640 6646 




6662 6668 6664 6670 6676 5682 6688 6694 5600 6606 




6612 6618 6625 6631 6637 5643 5649 5665 5661 6667 




5673 6679 6685 6691 5697 6703 6709 6716 6721 6727 


720 


86733 86739 86746 86761 86767 86763 86769 86775 86781 86788 




6794 6800 6806 6812 6818 6824 6830 6836 6842 6848 




6864 6860 6866 5872 6878 6884 686^ 6896 6902 ^908 




6914 6920 6926 5932 5938 5944 5950 6966 5962 5968 




5974 6980 6986 5992 6998 6004 6010 6016 6022 6028 




6034 6040 6046 6062 6068 6064 6070 6076 6082 6088 




6094 6100 6106 6112 6118 6124 6130 6186 6141 6147 




6163 6169 6165 6171 6177 6183 6189 6196 6201 6207 




6213 6219 6226 6231 6237 6243 6249 6265 6261 6267 




6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 


780 


86332 86338 86344 86350 86366 86362 86368 86374 86380 86386 




6392 6398 6404 6410 6416 6421 6427 6433 6439 6446 




6451 6467 6463 6469 6475 6481 6487 6493 6499 6604 




6610 6516 6622 6628 6634 6640 6646 6662 6668 6664 




6670 6676 6681 6687 6693 6699 6606 6611 6617 6623 




6629 6635 6641 6646 6662 6668 6664 6670 6676 6682 




6688 6694 6700 6706 6711 6717 6723 6729 6735 6741 




6747 6763 6769 6764 6770 6776 6782 6788 6794 6800 




6806 6812 6817 6823 6829 6836 6841 6847 6863 6859 




6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 


740 


86923 86929 86935 86941 86947 86963 86968 86964 86970 86976 




6982 6988 6994 6999 7006 7011 7017 7023 7029 7035 




7040 7046 7062 7068 7064 7070 7076 7081 7087 7093 




7099 7105 7111 7116 7122 7128 7134 7140 7146 7161 




7167 7163 7109 7175 7181 7186 7192 7198 7204 7210 




7216 7221 7227 7233 7239 7245 7261 7266 7262 7268 




7274 7280 72d6 7291 7297 7303 7309 7315 7320 7326 




7332 7338 7344 7349 7365 7361 7367 7373 7379 7384 




7390 7396 7402 7408 7413 7410 7425 7431 7437 7442 




7448 7454 7460 7466 7471 7477 7483 7489 7495 7600 


160 


87606 87512 87618 87623 87629 87536 87541 87547 87562 87668 



4SO 

TABLE I.— LOGARITHMS OF NUMBERS. 



N 


123450789 


750 


87606 87512 87618 87623 87629 87636 87641 87647 87662 87568 




7564 7670 767(J 7681 7687 7693 7609 7604 7610 7616 




7622 7628 7633 7639 7645 7661 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 7916 7921 7927 7933 7938 7944 7950 7966 7961 




7967 7973 7978 7964 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 8235 8241 *8247 




8252 8258 8264 8270 8275 8281 8287 8292 8298 8304 




8309 aS15 8321 8326 8332 8338 8343 a349 8355 8360 




8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 




8423 8429 8434 8440 8446 8451 8457 8463 8468 8474 




8480 8485 8491 8497 8502 8508 8513 8519 8525 8530 




8536 8542 8547 8553 8559 8564 8670 8576 8581 8687 




8593 8698 8604 8610 8616 8621 8627 8632 8638 8643 


770 


88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 




8705 8711 8717 8722 8728 8734 8739 8745 8750 8756 




8762 8767 8773 8779 8784 8790 8795 8801 8807 8812 




8818 8824 8829 8835 8840 8846 8852 8857 8863 8868 




8874 8880 8885 8891 8897 8902 8908 8913 8919 8925 




8930 8936 8941 8947 8963 8958 8964 8969 8975 8981 




8986 8992 8997 9003 9009 9014 9020 9025 9031 9037 




9042 9048 9053 9069 9064 9070 9076 9081 9087 9092 




9098 9104 9109 9115 9120 9126 9131 9137 9143 9148 




9154 9159 9165 9170 9176 9182 9187 9193 9198 9204 


780 


89209 89215 89221 89226 89232 89237 89243 89248 89254 89260 




9265 9271 9276 9282 9287 9293 9298 9304 9310 9315 




9321 9326 9332 9:i37 9343 9348 9364 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 9526 9631 9637 




9542 9548 9553 9569 9564 9570 9675 9581 9586 9692 




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 9767 


790 


89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 




9818 9823 9829 9834 9840 9845 9851 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 




0265 0260 0266 0271 0276 0282 0287 0293 0298 0304 


800 


90309 90814 90820 90326 90331 90336 90342 90347 90352 90368 



451 



TABLE 1.— LOGARITHMS OF NUMBERS. 



N 


0^23456780 


800 


90S09 90814 90320 90325 90831 90386 90342 90347 90862 90868 




0303 0869 0374 0380 038i 0390 0896 0401 0407 0412 




0417 0423 0428 0434 0439 0445 04^0 0455 0461 0466 




0472 0477 0482 0488 0498 0499 0504 0609 0515 0520 




0526 0581 0536 0542 0547 0558 0558 0563 0569 0674 




0580 0585 0590 0596 0601 0607 0612 0617 0628 0628 




0684 0639 0644 0650 0655 0660 0666 0671 0677 0682 




0687 0698 0698 0708 0709 0714 0720 0725 0730 0736 




0741 0747 0752 0757 0763 0768 0773 0779 0784 0789 




0795 0800 0806 0811 0816 0822 0827 0882 0888 0848 


810 


90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 




0902 0907 0913 0918 0924 0929 0934 0940 0945 0960 




0956 0961 0966 0972 0977 0982 0988 0993 0998 1004 




1009 1014 1020 1025 1030 1036 1041 1046 1052 1057 




1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 




1116 1121 1126 1132 1137 1142 1148 1153 1168 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 1384 1339 1344 1350 1355 1860 1366 1371 1876 


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 15^ 1529 1535 




1540 1545 1651 1566 1661 1666 1672 1677 1582 1587 




1593 1698 1603 1609 1614 1619 1624 1630 1685 1640 




1646 1661 1656 1661 1666 1672 1677 1682 1687 1693 




1698 1703 1709 1714 1719 1724 1730 1735 1740 1746 




1751 1756 1761 1766 1772 1777 1782 1787 1798 1798 




1803 1808 1814 1819 1824 1829 1834 1840 1845 1850 




1865 1861 1866 1871 1876 1882 1887 1892 1897 1903' 


880 


91908 91913 91918 91924 91929 91934 91939 91944 91950 91965 




1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 




2012 2018 2023 2028 2033 2038 2044 2049 2054 2069 




2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 




2117 2122 2127 2182 2137 2143 2148 2153 2158 2163 




2169 2174 2179 2184 2189 2195 2200 2205 2210 2216 




2221 2226 2231 2236 2241 2247 2252 2257 2262 2267 




2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 




2324 2330 2335 2340 2345 2860 2355 2361 2366 2371 




2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 


840 


^428 92433 92438 92443 92449 92454 92459 92464 92469 92474 




2480 2485 2490 2495 2500 2605 2611 2516 2621 2526 




2531 3586 2542 2547 2562 2557 2562 2567 2572 2678 




2583 2588 2593 2508 2603 2609 2614 2619 2624 2629 




2634 2639 2645 2650 2655 2660 2666 2670 2675 2681 




2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 




2737 2742 2747 2762 2758 2763 2768 2773 2778 2783 




2788 2793 2799 2804 2800 2814 2819 2824 2829 2834 




2840 2845 2850 2855 2860 2865 2870 2875 2881 2886 




2891 2896 '2901 2906 2911 2916 2921 2927 2932 2937 


860 


92942 92047 92962 92957 92962 92967 92973 92978 92983 92988 



452 

TABLE L— LOGARITHMS OF NUMBERS. 



N 


0123456789 


860 


92042 92947 92962 92957 92962 92967 92978 92978 92983 92988 




2998 2998 3008 3008 3013 3018 3024 3029 3034 3089 




3044 3049 3054 3059 3064 3069 3075 3080 308^ 3090 




3095 3100 3105 3110 3116 3120 3125 3131 3136 3141 




3146 8151 3156 3161 3166 3171 3176 3181 3186 3192 




3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 




3247 3252 3258 3263 3268 3273 8278 3283 3288 3293 




3298 3303 3308 3313 3318 8323 8828 3334 3339 3344 




3349 3854 3359 3364 8369 3374 3379 3384 3889 3394 




3899 8404 8409 3414 8420 3425 8430 8485 8440 8445 


860 


98450 98465 93460 98465 93470 98475 93480 93485 93490 93495 




3500 3505 3510 3515 3520 8526 8531 3586 3541 8546 




8551 8566 8661 3666 3571 3576 3581 3686 3591 8596 




3601 8606 8611 8616 3621 3626 3631 3636 3641 8646 




3661 3656 8661 3666 3671 3676 3682 3687 3692 3697 




3702 3707 8712 3717 3722 8727 3732 3737 3742 8747 




8752 3767 8762 3767 8772 8777 3782 3787 3792 8797 




8802 3807 8812 8817 3822--a827 8832 3887 8842 3847 




3862 8867 3862 8867 8872 8877* 8882 3887 3892 3897 




3902 3907 3912 3917 8922 3927 3932 8987 3942 3947 


870 


93962 93957 93962 93967 98972 98977 98982 93967 93992 98997 




4002 4007 4012 4017 4022 4027 4032 4087 4042 4047 




4062 4057 4062 4067 4072 4077 4082 4086 4091 4096 




4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 




4161 4166 4161 4166 4171 4176 4181 4186 4191 4196 




4201 4206 4211 4216 4221 4220 4231 4236 4240 4245 




4260 4255 4260 4265 4270 4275 4280 4285 4290 4295 




4800 4305 4310 4315 4320 4325 4330 4835 4340 4345 




4349 4354 4369 4364 4369 4374 4879 4384 4389 4394 




4399 4404 4409 4414 4419 4424 4429 4438 4438 4443 


880 


94448 94453 94458 94468 94468 94473 94478 94488 94488 94498 




4498 4603 4607 4612 4517 4522 4527 4532 4537 4542 




4647 4562 4667 4662 4567 4671 4676 4581 4586 4591 




4696 4601 4606 4611 4616 4621 4626 4630 4635 4640 




4645 4660 4665 4660 4665 4670 4675 4680 4685 4689 




4694 4699 4704 4709 4714 4719 4724 4729 4784 4788 




4743 4748 4763 4758 4763 4768 4778 4778 4783 4787 




4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 




4841 4846 4861 4866 4861 4866 4871 4876 4880 4885 




4890 4895 4900 4905 4910 4915 4919 4924 4929 4984 


890 


94939 94944 94949 94954 94959 94968 94968 94973 94978 94983 




4988 4993 4998 5002 6007 6012 6017 6022 5027 5032 




6036 6041 6046 6051 6066 5061 5066 6071 5075 5080 




6086 5090 5095 5100 6105 6109 6114 5119 5124 5129 




5134 5139 5143 5148 5168 5158 5168 6168 5178 5177 




6182 5187 5192 5197 5202 6207 5211 5216 5221 5226 




5231 6236 5240 5245 6260 5266 5260 5265 5270 5274 




6279 5284 5289 5294 5299 5303 6308 5313 5318 5328 




5328 5:i32 5887 5342 6347 5362 5357 5361 5866 5871 




5376 5881 5886 5390 6396 5400 5405 5410 5415 5419 


900 


95424 95429 96434 96439 95444 95448 95458 95458 95468 95468 



453 



TABLE 1.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


900 


95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 




5472 5477 5482 5487 5492 5497 5501 5506 5511 5516 




5521 5525 55:^0 5535 5540 554^ 55^0 6554 5559 5564 




5569 5574 5578 5583 5588 5593 5598 5602 5607 5612 




6617 5622 5626 5631 5636 5641 5646 5650 5656 6660 




5665 5670 6674 5679 6684 6689 5694 6698 6703 6708 




5713 5718 6722 6727 6732 6737 5742 6746 6761 6756 




6761 6766 5770 5775 6780 5785 6789 6794 6799 6804 




6809 6813 6818 6823 5828 6832 6837 5842 6847 5852 




6856 6861 5866 5871 6876 5880 5885 6890 6895 5899 


910 


95904 96909 96914 95918 95923 96928 95933 95938 96942 96947 




6952 6957 6961 6966 5971 5976 6980 6986 6990 5995 




6999 6004 6009 6014 6019 6023 6028 6033 6038 6042 




6047 6062 6057 6061 6066 6071 6076 6080 6086 6090 




6095 6099 6104 6109 6114 6118 6123 6128 6133 6137 




6142 6147 6152 6156 6161 6166 6171 6175 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 63lS 6317 6322 6327 




6332 6336 6341 6346 6350 6355 6360 6365 6369 6374 


920 


96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 




6426 6431 6436 6440 6445 6450 6454 6459 6464 6468 




6473 6478 6483 6487 6492 6497 6501 6606 6611 6615 




6520 6625 6530 6534 6539 6544 6548 6563 6568 6662 




6567 6572 6577 6581 6586 6591 6696 6600 6605 6609 




6614 6619 6624 6628 6633 6638 6642 6647 6652 6666 




6661 6666 6670 6675 6680 6685 6689 6694 6699 6703 




6708 6713 6717 6722 6727 6731 6736 6741 6745 6760 




6755 6769 6764 6769 6774 6778 6783 6788 6792 6797 




6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 


980 


96848 96863 96858 96862 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 7026 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 7161 7165 7160 7165 7169 




7174 7179 7183 7188 '7192 7197 7202 7206 7211 7216 




7220 7225 7230 7234 7239 7243 7248 7253 7267 7262 




7267 7271 7276 7280 7285 7290 7294 7299 7304 7808 


940 


97313 97317 97322 97327 97331 97336 97340 97346 97350 97364 




7359 7364 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 7602 7606 7611 7616 7620 7525 7529 7534 7639 




7643 7548 7552 7567 7562 7566 7571 7576 7580 7685 




7589 7694 7598 7603 7607 7612 7617 7621 7626 7680 




7635 7640 7644 7649 7653 7668 7663 7667 7672 7676 




7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 




7727 7731 7736 7740 7745 7749 7764 7759 7763 7768 


960 


97772 97777 97782 97786 97791 97796 97800 97804 97809 97815 



454 

TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


950 


97772 97777 97782 97786 97791 97795 97800 97804 97809 97818 




7818 7823 7827 7832 783« 7841 7846 7860 7865 7869 




7864 7868 7873 7877 7882 7886 7891 7896 7900 7905 




7909 7914 7018 7923 7928 7932 7937 7941 7946 7960 




7955 7959 7964 7968 7973 7978 7982 7987 7991 7996 




8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 




8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 




8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 




8137 8141 8146 8150 8155 8159 8164 8168 8173 8177 




8182 8186 8191 8195 8200 8204 8209 8214 8218 8223 


960 


98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 




8272 8277 8281 8286 8290 8295 8299 8304 8308 8813 




8318 8322 8327 8331 8336 8340 8345 8349 8354 8858 




8363 8367 a372 8376 8381 8385 8390 8394 8399 8403 




8408 8412 8417 8121 8426 8430 8435 8439 8444 8448 




8453 8457 8462 8406 8471 8475 8480 8484 8489 8493 




8498 8502 8507 8511 8516 8520 8525 8529 8534 8538 




8543 8547 8552 8656 8561 8566 8570 8674.8579 8683 




8588 8592 8597 8601 8605 8610 8614 8619 8623 8628 




8632 8637 8641 8646 8650 8655 8659 8664 8668 8673 


970 


98677 98682 98686 98691 98696 98700 98704 98709 98713 98717 




8722 8726 8731 8735 8740 8744 8749 8753 8758 8762 




8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 




8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 




8866 8860 8865 8869 8874 8878 8883 8887 8892 8896 




8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 




8945 8949 8954 8958 8963 8967 8972 8976 8981 8985 




8989 8994 8998 9003 9007 9012 9016 9021 9026 9029 




9034 9038 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 99158 99162 




9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 




9211 9216 9220 0224 9229 9233 9238 9242 9247 9251 




9265 9260 9264 9269 9273 9277 9282 9286 9291 9295 




9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 




9344 9348 9362 9357 9361 9366 9370 9374 9379 9383 




9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 




9432 9436 9441 9445 0449 9464 9468 9463 9467 9471 




9476 9480 9484 9489 9493 9498 9502 9606 9611 9516 




9620 9524 9628 9633 9537 9542 9546 9660 9665 9669 


990 


99664 99568 99572 99577 99581 99586 99690 99694 99699 99603 




9607 0612 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 0839 9843 9848 9852 9866 9861 9866 




9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 




9913 9917 9922 9926 9930 9935 0939 9944 9948 9952 




9967 9961 9965 9970 9974 9978 9983 9987 9991 9996 


1000 


00000 00004 00009 00013 00017 00022 00026 00030 00085 00039 



455 



TABLE II.— LOGARITHMIC SINES AND COSINKS 


/ 


• 


o» 


1 


• 


«♦ 


/ 


Sine 


Oofdne 


Sine 


Ck)eine 


Sine 


Ckwlne 




—00 


10.00000 


8.84186 


9.99998 


8.54888 


9.99974 


60 




6.46878 


00000 


84908 


:'9998 


54648 


99978 


60 




76476 


00000 


85609 


yyjtgo 


64999 


99978 


68 




94085 


00000 


86804 


99993 


65354 


99978 


67 




7.06679 


00000 


86088 


99998 


56706 


99978 


56 




16870 


00000 


87661 


99992 


66054 


60971 


56 




84188 


00000 


28384 


99998 


66400 


99971 


54 




80888 


00000 


88977 


99993 


66748 


99970 


68 




86688 




29681 


99098 


67084 


99970 


58 




41797 


00000 


80856 


9999] 


67481 


99969 


51 




7.46878 


10.00000 


8.30879 


9.99991 


8.67757 


9.99969 


60 




60618 


00000 


81495 


99991 


58089 


99968 


49 




54901 


00000 


88108 


99990 


58419 


99968 


48 




57767 


00000 


88708 


99990. 


68747 


99967 


47 




60985 


00000 


88298 


99990 


69078 


99967 


46 




68988 


00000 


88875 


99990 


59895 


99967 


46 




66784 


00000 


84450 


99989 


59715 


99960 


44 




69417 


9.99999 


85018 


99989 


60088 


99966 


48 


18 


71900. 


99909 


86578 


99960 


60849 


99965 


48 


19 


74848 


OOQOO 


86131 


99989 


60668 


99964 


41 


90 


7.76475 


9.99999 


8.36678 


9.99988 


8.60078 


9.09964 


40. 


31 


78594 


OOOfiO 


87217 


99968 


61288 


99963 


89 


SS 


80615 


99999 


87760 


i 99988 


61680 


99968 


88 


88 


88545 


99999 


88276 


99987 


61894 


99968 


87 


24 


84898 


99999 


88796- 


99967, 


68106 


99968 


36 


25 


86166 


99999 


89.310 


99987 


68497 


99961 


86 


86 


87870 


99999 


39818 


99986 


62796 


99961 


84 


«7 


89509 


99999 


40380 


99986 


68091 


99960 


38 


28 


91068 


99999 


40816 


99986 


63386 


99960 


88 


29 


98618 


99998 


41807 


99986 


68678 


99959 


81 


80 


7.94084 


9.99998 


8.41798 


9.99986 


8.63968 


9.99059 


80 


81 


95606 


QQQOO 
WMPO 


42zn 


99966 


64856 


99958 


89 


88 


96887 


99998 


42746 


AAQUJ 


64543 


99958 


28 


88 


9am 


99998 


43810 


99984 


64827 


99957 


27 


34 


99580 


Aftgno 


48680 


mmf* 


65110 


99966 


86 


as 


8.00779 


QQQQQ 

WvtfO 


44139 


99988 


65391 


99956 


85 


86 


08008 


OQOQR 


44594 


99988 


66670 


99956 


84 


87 


03198 


99997 


45044 


99968 


65947 


99965 


88 


as 


04850 


99997 


45489 


99988 


66828 


99954 


28 


89 


06478 


99997 


45930 


99988 


66497 


90954 


81 


40 


8.06578 


9.99997 


8.46366 


9.99988 


8.66789 


9.99968 


80 


41 


07650 


99997 


46799 


999B1 


67069 


99053 


19 


48 


06606 


99997 


47886 


99961 


67306 


99968 


18 


48 


09718 


99997 


47650 


99981 


67575 


99951 


17 


44 


10717 


VWvQ 


48069 


99960 


67841 


99951 


16 


45 


11698 


99996 


48485 


99980 


68104 


99960 


15 


46 


18647 


99996 


48896 


99979 


68367 


99949 


14 


47 


18581 


99996 


49304 


99979 


e&dam 


99949 


18 


48 


14495 


99996 


49708 


99979 


68886 


99948 


18 


49 


15891 


99996 


50108 


99978 


09144 


90948 


11 


60 


8.16868 


9.99095 


8.50604 


9.99978 


8.00400 


9.99947 


10 


61 


17128 


99995 


50897 


99977 


69654 


99946 





58 


17971 


99995 


51287 


99977 


69907 


99946 


8 


68 


18798 


99995 


51678 


99977 
99976 


70169 


99945 


7 


54 


19610 


99995 


52055 


70409 


99944 


6 


96 


90407 


99994 


58484 


99976 


70668 


99944 


5 


56 


81189 


99994 


58810 


99976 


70905 


99948 


4 


67 


81^ 


99994 


53183 


99975 


71151 


99948 


8 


68 


88713 


99994 


58552 


99(.)74 


71395 


99943 


8 


60 


28136 


99994 


53919 


99974 


71638 


99941 


1 


60 


84186 


99998 


542^ 


99974 


71880 


99940 





t 


OoBlne 


Sine 


Cosine 


Sine 


Ck>sine 


Sine 


t 




890 


88 


« 


87 






456 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



» 




8* 




4« 


«• 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





8.71880 


9.99940 


8.84868 


9.99894 


8.94080 


9.99884 


60 


1 


78180 


99940 


84589 


99698 


94174 


99688 


59 


9 


783S9 


99930 


84718 


99692 


94817 


99682 


68 


8 


78597 


99988 


84887 


99691 


94461 


996S1 


57 


4 


78884 


99988 


85075 


99691 


94608 


99680 


66 


5 


78060 


99987 


86252 


99690 


94746 


99629 


56 


6 


78308 


99986 


86429 


99689 


94887 


09828 


54 


7 


78535 


99936 


85606 


90688 


96029 


96827 


53 


8 


78767 


99985 


85780 


90687 


95170 


99885 


62 


9 


78997 


99934 


85966 


99886 


96810 


99884 


61 


10 


8.74826 


9.99964 


8.86128 


9.99686 


8.96450 


9.90888 


60 


11 


74454 


99938 


8B301 


99884 


96669 


99622 


49 


18 


74680 


99938 


86474 


99883 


957« 


99681 


48 


IS 


74906 


99982 


86646 


99882 


95867 


99620 


47 


14 


76130 


99981 


86816 


99881 


9G006 


99819 


46 


16 


75888 


99980 


86087 


99680 


96143 


99617 


45 


16 


75675 


90929 


87156 


99879 


96280 


99816 


44 


17 


75795 


999629 


8i?)25 


99679 


96417 


99816 


48 


18 


76016 


99928 


87494 


90678 


96553 


99814 


42 


19 


76884 


99927 


67661 


99877 


96680 


99818 


41 


80 


8.76451 


9.99926 


8.87889 


9.99676 


8.96826 


9.99612 


40 


81 


76667 


99926 


87996 


99675 


96960 


99810 


89 


88 


76888 


999S5 


88161 


99674 


97096 


99609 


88 


88 


77097 


99924 


88826 


^99673 


97S29 


99606 


87 


94 


77810 


99928 


88490 


99672 


97868 


99607 


86 


25 


77528 


99928 


68654 


90671 


07496 


99606 


86 


86 


77TS8 


99922 


88817 


99870 


971629 


99604 


84 


87 


Tr943 


99921 


88980 


OQCMQ 
VsODtf 


97768 


99808 


88 


SB 


78163 


99930 


80142 


99668 


97894 


99802 


82 


89 


78860 


99020 


89304 


99867 


96096 


99601 


81 


ao 


8.78668 


9.99919 


8.89464 


9.99666 


'•JUg 


9.99800 


80 


81 


78774 


99918 


89685 


99665 


99798 


29 


38 


7T9979 


99917 


89784 


99864 


98419 


99797 


28 


88 


79188 


99917 


89948 


99663 


98649 


99796 


27 


84 


79386 


99916 


90102 


99602 


98679 


99T06 


28 


85 


79588 


99916 


90260 


99661 


96806 


99793 


25 


86 


79789 


99914 


90417 


99660 


98987 


99791 


24 


87 


79990 


99018 


90574 


99659 


99066 


28 


88 


80180 


99918 


90780 


99658 


99194 


99790 


22 


80 


80388 


99912 


90686 


99657 


998^2 


99788 


81 


40 


8.80585 


9.99911 


8.91040 


9.99866 


8.99450 


9.99787 


80 


41 


80782 


99910 


91195 


99655 


99577 


99786 


19 


48 


80978 


99909 
99909 


91849 


99864 


99704 


99785 


18 


43 


81173 


91502 


99658 


99830 


90783 


17 


44 


81367 


99006 


91655 


99868 


99966 




16 


45 


81560 


99907 


91807 


99851 


9.00062 


99781 


16 


46 


81752 


99906 


91959 


99650 


00207 


99780 


14 


47 


81944 


99905 


92110 


99648 


00332 


99778 


18 


48 


82134 


99904 


92261 


99647 


00456 


99777 


18 


49 


82324 


99904 


92411 


W040 


00681 


90776 


11 


SO 


8.82513 


9.99908 


8.98561 


9.99846 


9.ooro4 


9.99776 


10 


61 


82701 


99902 


92710 


OKXiAA 


008« 


99778 


9 


68 


82888 


99901 


93860 


99843 


00961 


99772 


8 


58 


83076 


99000 


93007 


99648 


01074 


99771 


7 


64 . 


88261 


90890 


98154 


99841 


01196 


99769 


6 


55 


83446 


90808 


93801 


99840 


01318 


90768 


6 


66 


83630 


99898 


98448 


99639 


01410 


99767 


4 


67 


88818 


99897 


93594 


998.38 


01561 


90766 


8 


68 


83906 


99896 


98740 


99887 


01682 


99764 


8 


50 


84177 


99896 


93885 


90836 


01803 


• 99763 


1 


60 


84868 


99604 


94030 




01923 


99761 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 







86« 




85* 




84» 



457 



TABI>E n.— LOGARITHMIC SINES AND COSINES. 



i 


6« 




JO 




8- 


/ 


Sine 


Cofline 


Sine 


Cosine 


Sine 


CoAine 




9.01928 


9.99761 


9.06589 


9.99675 


9.14856 


9.99575 


60 




0^043 


99760 


08092 


99674 


14445 


99574 


69 




02168 


99759 


06705 


99672 


14535 


99572 


68 




02388 


99767 


08897 


99670 


14624 


99570 


67 




0^402 


09756 


•06000 


99669 


14714 


99568 


56 




0-2320 


99755 


09101 


99667 


14808 


99566 


65 




02683 


99768 


00:>02 


90666 


14891 


99566 


54 




02757 


99752 


09804 


99664 


14980 


99568 


58 




02874 


. 99751 


09405 


99663 


15069 


99561 


52 




02992 


99749 


09506 


99661 


15167 


99659 


51 




9.08109 


9.99746 


9.00606 


9.90650 


9.15245 


0.99567 


60 




08^ 


99747 


09707 


99658 


15388 




49 




03842 


99745 


09607 


99656 


15421 


99564 


48 




03458 


99744 


09007 


99665 


15508 


99562 


47 




08574 


99742 


10006 


99668 


16606 


99550 


46 




0:J690 


99741 


10106 


99651 


15688 


99548 


46 




03803 


99740 


10205 


90660 


15770 


99646 


44 




OSOiO 


99788 


10304 


99648 


16867 


99646 


48 




<XI034 


99787 


10402 


99647 


16044 


09648 


42 


19 


04149 


99786 


10501 


99645 


16080 


99641 


41 


30 


9.04262 


9.99784 


0.10699 


9.99648 


9.16116 


9.99539 


40 


21 


0487B 


99788 


10697 


90642 


16208 


99537 


80 


8 


04490 


99781 


10795 


99640 


16289 


99685 


86 


24 


04608 


99780 


10608 


99688 


16874 


99683 


87 


94 


04715 


99728 


10990 


99637 


16460 


99582 


86 


25 


Mses 


99727 


11087 


99685 


16545 


99580 


85 


96 


04940 


99726 


11184 


99638 


16631 


99628 


84 


87 


06053 


09724 


11281 


99632 


16716 


99626 


83 


28 


06164 


99723 


11877 


99680 


16801 


99624 


82 


29 


05275 


99721 


11474 


99629 


16866 


99522 


31 


80 


9.05886 


9.997S0 


9.11570 


9.99627 


0.16970 


9.99520 


30 


81 


05497 


99718 


11G66 


99626 


17055 


991)18 


«29 


83 


05607 


99717 


11761 


99624 


17189 


99617 


28 


88 


05717 


99716 


11857 


99622 


17228 


99516 


27 


84 


06827 


99714 


11962 


99620 


17807 


99518 


26 


85 


06987 


90718 


12047 


99616 


17801 


99511 


25 


86 


06046 


99711 


12142 


99617 


17474 


99509 


24 


37 


06155 


99710 


. 12286 


99615 


17S68 


99507 


38 


88 


06264 


99706 


1288^ 


99618 


17641 


99605 


23 


89 


068?^ 


90707 


12426 


90612 


17724 


99508 


21 


40 


9.06481 


9.99705 


0.1SS19 


(.99610 


9.17907 


9.99601 


20 


41 


06589 


90704 


12612 


99608 


17890 


99499 


10 


49 


06686 


997U2 


12706 


99607 


17978 


99497 


18 


48 


06804 


99701 


12799 


99605 


18053 


99496 


17 


44 


06011 


99699 


12892 


99608 


18187 


99494 


16 


45 


07018 


90696 


12986 


99601 


16220 


99492 


16 


46 


07124 


99096 


18078 


99600 


16302 


99490 


14 


47 


07231 


99695 


18171 


99698 


18888 


99488 


18 


48 


07387 


99603 


18268 


99596 


18465 


90486 


12 


49 


07442 




18365 


99596 


18547 


99184 


11 


00 


9.07548 


9.99090 


9.18447 


9.99698 


9.18628 


9.99482 


10 


51 


07653 


99680 


18689 


99591 


18709 


99480 





02 


07768 


99687 


13680 


99689 


18790 


99478 


6 


58 


07868 


99666 


18722 


99586 


18871 


99476 


7 


54 


07968 


99684 


13818 


99586 


18952 


99474 


6 


66 


06072 


99688 


iaoo4 


99584 


19088 


99472 


. 6 


06 


06176 


99681 


i:}994 


99682 


19118 


99470 


4 


57 


06280 


99680 


14065 


99581 


19193 


99468 


8 


06 


06388 


99678 


14175 


99579 


19278 


99466 


3 


60 


06466 


99877 


14266 


99577 


19358 


99464 


1 


60 


06569 


99675 


14366 


99575 


19488 


99462 





# 


Coelne 


Sine 


Coctlne 


Sine 


Coeine 


Bine 







68« 




82« 




8f 



458 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



1 




BO 


]0» 


11- 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 




9.19488 


9.99468 


9.88007 


9.99886 


9.88060 


9.99195 


60 




10618 


99460 


84089 


99688 


88185 


99198 


59 




19698 


99468 


94110 


99881 


88190 


90190 


68 




19678 


99466 


24181 


99688 


88864 


99187 


67 




19751 


99464 


94858 


99386 


86819 


99165 


66 




19880 


99468 


94824 


99684 


88884 


99188 


66 




19909 


99460 


94895 


09888 


88448 


99180 


64 




19988 


99448 


84466 


99819 


98518 


99177 


68 




80067 


99446 


84586 


99817 


88577 


99175 


58 




80146 


99444 


24607 


99815 


88641 


99178 


51 




9.80888 


9.99448 


9.24677 


9.99818 


9.88706 


9.99170 


60 




80008 


99440 


24748 


99810 


88769 


99167 


49 




80380 


99488 


24818 


99808 


88888 


99165 


48 




90468 


99486 


94888 


99806 


88896 


99168 


47 




90585 


99484 


24958 


99804 


88900 


99160 


46 




80618 


99488 


26088 


99801 


89094 


99157 


45 




80691 


99489 


86098 


99899 


89087 


99155 


44 




80788 


99487 


85168 


99897 


89160 


99168 


48 


18 


80646 


99485 


85887 


99894 


89814 


99150 


48 


19 


80988 


99488 


86807 


99898 


89877 


99147 


41 


90 


9.80999 


9.99481 


9.86876 


9.99890 


9.89840 


9.99145 


40 


21 


81076 


99419 


85445 


99888 


89408 


99148 


89 


88 


91168 


99417 


86514 


99885 


99400 


99140 


88 


88 


81889 


99415 


95588 


99888 


99689 


99187 


87 


84 


81806 


90418 


96668 


99881 


29S91 


99186 


86 


85 


81888 


99411 


95781 


99878 


29654 


99188 


85 


83 


81468 


99409 


85790 


99876 


29716 


99180 


84 


87 


81584 


99407 


85858 


99874 


29779 


99127 


88 


8S 


81610 


99404 


85987 


99871 


29841 


99184 


88 


29 


21685 


99408 


86095 


99869 


29908 


99188 


31 


80 


9.81761 


9.99400 


9.86068 


9.99867 


9.29966 


9.99119 


80 


81 


21886 


99398 


86181 


99864 


80088 


99117 


89 


88 


21918 


99896 


86199 


99868 


80000 


99114 


88 


88 


21987 


99894 


96867 


99860 


80151 


99118 


87 


84 


28068 


99888 


86886 


99857 


80818 


99109 


86 


85 


28187 


99890 


96406 


99865 


80875 


90106 


85 


86 


22811 


99888 


96470 


99868 


80886 


99104 


84 


87 


88386 


99386 


96688 


99860 


80898 


99001 


88 


88 


88861 


99888 


96605 


99848 


- 80450 


99099 


88 


89 


88485 


99881 


96678 


99945 


80621 


99096 


81 


40 


9.88809 


9.99879 


9.86789 


9.99848 


9.80683 


0.9Q098 


80 


41 


88588 


99877 


86806 


99841 


80648 


BOOOT 


19 


48 


88667 


99376 


86878 


99888 


80704 


99088 


18 


48 


88781 


90878 


86940 


90886 


80766 


wm 


17 


44 


88805 


99870 


87007 


99888 


80886 


^IH3 


16 


46 


88878 


OCMMA 
WWOQO 


87078 


90881 


80887 


99080 


15 


46 


88968 


99866 


87140 


99889 


80947 


fl9078 


14 


47 


88085 


99364 


87806 


99886 


81008 


w^m^ 


18 


48 


88098 


99968 




99884 


81068 


9JK572 


18 


49 


88171 


99869 


87889 


99881 


81189 


'Mm 


11 


60 


9.88844 


9.99857 


9.87406 


9.99819 


9.81189 


9.99007 


10 


61 


88817 


99i'« 


87471 


99817 


81860 


99004 


9 


68 


88890 


99358 


87R87 


99814 


81810 


99008 


8 


58 


88468 


99351 


27602 


99818 


81870 


99069 


7 


64 


88585 


99348 


87668 


99909 


81480 


99066 


6 


66 


88607 


99846 


27734 


99807 


81490 


99054 


5 


66 


23679 


99644 


27799 


99804 


81549 


99051 


4 


67 


28758 


993<? 


87864 


99808 


81609 


99048 


8 


68 


88888 


99840 


87930 


99200 


81609 


99046 


8 


60 


88895 


99837 


£7995 


99197 


81788 


99048 


1 


60 


88967 


99885 


.28060 


99195 


81788 


99040 





9 


Ck)slne 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 




80* 




79« 




78» 



459 



TABLE 11.— LOGARITHMIC SINES AND COSINES. 



/ 


12» 


18* 


14* 


1 


Sine 


GoBine 


Sine 


Cosine 


Sine 


Cosine 




9.81788 


9.99040 


9.85809 


9.86878 


9.88868 


9.96690 


60 




8t847 


99038 


85868 


86869 


38418 


98687 


59 




81907 


99065 


86:il8 


98867 


38469 


96684 


58 




81966 


90088 


86373 


96864 


38519 


06681 


57 




880^5 


09000 


85487 


98861 


88570 


98678 


66 




82084 


00087 


85481 


96858 


88680 


98675 


66 




88148 


90084 


85636 


96855 


88670 


98671 


54 




88808 


99028 


85600 


96868 


86781 


96668 


68 




88-^61 


99019 


85644 


96849 


88771 


98666 


58 




88819 


99016 


86098 


96846 


88881 


98668 


51 




9.88878 


9.98018 


9.85763 


9.96848 


9.88871 


9.96659 


60 




88487 


90011 


85806 


96840 


88981 


86666 


49 




88495 


99006 


85860 


9688? 


86971 


96668 


48 




88558 


90005 


85914 


96834 


89081 


98649 


47 




88618 


99008 


86968 


96881 


890n 


98646 


46 




88670 


99000 


86028 


96828 


89181 


98648 


46 




88788 


96097 


86075 


98825 


89170 


96640 


44 




88786 


QAQQ4 


86189 


96888 


89880 


98686 


48 




88844 


96991 


86188 


96819 


89870 


96688 


48 


19 


88908 


ganon 


80886 


06816 


89819 


98680 


41 


90 


9.88960 


9.96966 


9.86889 


9.96818 


8.89809 


9.96687 


40 


21 


88018 


96988 


86818 


96810 


89418 


98688 


88 


as 


88075 


96080 


86896 


96807 


39467 


96680 


38 


SH 


88183 


96978 


86449 


86804 


89517 


96617 


37 


24 


8SI90 


98075 


86608 


96801 


89566 


98614 


86 


85 


88848 


96978 


86665 


96786 


89616 


96610 


36 


86 


88305 


98069 


86606 


88795 


89664 


98607 


84 


87 


88868 


96967 


86660 


96798 


39718 


96604 


88 


88 


88480 


98964 


86718 


98789 


39768 


96601 


88 


89 


88477 


98961 


86766 


96786 


89811 


96597 


31 


80 


9.38684 


9.96058 


9.86819 


9.98783 


9.89860 


9.86594 


80 


81 


88691 


96855 


86871 


98780 


89909 


88591 


89 


88 


88647 


98058 


36984 


96777 


39958 




88 


88 


88704 


96860 


3G976 


98774 


40006 


86584 


87 


84 


88761 


98947 


87088 


98771 


40055 


86581 


86 


8S 


88818 




87081 


06768 


40108 


86678 


85 


86 


88874 


96941 


87133 


98765 


40158 


96574 


94 


87 


83081 


96038 


37185 


66768 


40200 


98671 


88 


88 


83987 


08986 


87287 


86750 


40849 


98668 


88 


89 


84048 


96088 


37889 


96266 


40897 


98565 


81 


40 


9.84100 


9.96080 


9.87841 


0.96768 


9.40846 


9.98561 


90 


41 


84166 


93987 


87888 


96750 


40694 


96558 




48 


84818 


98984 


87445 


98746 


40448 


98666 




43 


84868 


96981 


87497 


96748 


40490 


98661 




44 


81884 


08919 


37549 


98740 


40638 


98548 




45 


84880 


96916 


87600 


98787 


4058G 


98646 




46 


84486 


96918 


87668 


96734 


40G84 


98541 




47 


84491 


96910 


87708 


98731 


40688 


96688 




48 


84547 


98007 


87756 


98726 


40780 


98586 




49 


84608 


98904 


87806 


96786 


40778 


96631 




60 


9.84658 


9.86901 


9.87866 


9.86788 


9.40886 


9.98688 




61 


84718 


98806 


87909 


96719 


40678 


98685 




68 


84769 


96896 


87960 


98715 


40981 


96581 




58 


84884 




38011 


96718 


40968 


96618 




64 


84879 


96890 


38068 


96709 


41016 


98616 




66 


84934 


98887 


88118 


96706 


41068 


9&'>11 




66 


84989 


08884 


88164 


96703 


41111 


98506 




67 


88044 


96881 


88.215 


86700 


41158* 


98505 




68 


86099 


98878 


38266 


96697 


41805 


98501 




60 


85154 


98875 


88317 


98694 


41858 


98408 




.60 


85909 


98872 


38368 


86690 


41300 


98494 




$ 


Oodne 


Sine 


Cosine 


Sine 


CoBlne 


Sine 


/ 




77* 




76* 




76« 



460 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



9 


!«• 


16* 


17* 


/ 


Sine 


CkMlne 


Bine 


Coeine 


Sine 


Cosine 





9.41800 


9.98494 


9.44084 


9.96884 


9.46694 


9.96060 


60 


1 


41847 


96491 


44078 


96281 


46635 


98066 


60 


8 


418M 


98488 


44128 


98877 


46676> 


96068 


66 


8 


41441 


98484 


44166 


96878 


46717 


96048 


67 


4 


41488 


98481 


44810 


98870 


46768 


98044 


66 


5 


41586 


98477 


44258 


96866 


46800 


98040 


66 


6 


416Mi 


98474 


44297 


98862 


46841 


96086 


64 


7 


A\im 


98471 


44341 


96859 


46888 


96088 


63 


a 


41675 


96467 


44385 


96856 


46923 


96029 


68 


9 


41788 


96464 


44428 


98261 


46064 


96085 


61 


10 


9.41788 


9.96460 


9.44478 


9.96848 


9.47006 


0.98081 


60 


n 


41816 


96457 


44516 


98844 


47046 


96017 


49 


18 


41861 


96458 


44559 


96240 


47066 


98018 


48 


18 


41906 


96460 


44608 


98287 


47187 


98009 


47 


14 


41954 


96447 


44646 


98288 


47168 


06006 


46 


15 


AHaOi 


96448 


446K9 


98229 


47209 


98001 


45 


16 


««)47 


98440 


44733 


96226 


47249 


97997 


44 


17 




96436 


44776 


98288 


47290 


97998 


48 


18 


42140 


98488 


44819 


98218 


47880 


97989 


48 


19 


48186 


98429 


44868 


98216 


47371 


97986 


41 


80 


9.4-2388 


9.96426 


9.44906 


9.96211 


9.47411 


9.97088 


40 


81 


42878 


96488 


44948 


98207 


47488 


97978 


80 


88 


42324 


96419 


44908 


96804 


47498 


97974 


88 


88 


42870 


98415 


45086 


98200 


4758.} 


97970 


87 


84 


42416 


98418 


450r7 


98196 


47673 


97966 


86 


26 


42401 


96409 


45120 


98198 


47618 


97968 


86 


26 


42507 


96406 


45168 


96189 


47654 


97958 


84 


27 


48558 


96408 


45206 . 


98186 


47694 


97964 


88 


28 


42&09 


96398 


45249 


96161 


47734 


97960 


82 


28 


48644 


96895 


46298 


96177 


47774 


97946 


81 


80 


9.42690 


0.98891 


9.45834 


9.96174, 


9.47814 


9.97948 


80 


81 


42785 


96388 


45377 


98170 


47854 


97988 


89 


88 


42781 


06884 


45419 


96166 


47894 


97984 


88 


88 


42826 


96881 


45468 


96168 


47934 


97930 


27 


84 


42878 


98377 


45604 


98150 


47974 


97986 


26 


85 


48917 


96878 


45547 


96156 


4^014 


97928 


25 


86 


42968 


96370 


45589 


96151 


48064 


97918 


84 


87 


«i008 


98366 


45638 


98147 


48094 


97914 


83 


88 


48058 


98868 


4.'i674 


98144 


48183 


97910 


88 


80 


46096 


98859 


45716 


96140 


48178 


97906 


81 


40 


0.48148 


9.98856 


9.45756 


9.98186 


9.48818 


9.97908 


80 


41 


48188 


98852 


45601 


96188 


48258 


97898 


19 


48 


43284 


98849 


45848 


96189 • 


48298 


97894 


18 


48 


43278 


96845 


45885 


96185 


48388 


97890 


17 


44 


4&328 


96848 


45987 


96181 


48871 


97886 


16 


45 


48867 


96888 


45909 


96117 


48411 


97889 


15 


46 


48418 


98&34 


46011 


96118 


. 48450 


97878 


14 


47 


48467 


98831 


46058 


96110 


48190 


97874 


18 


48 


43502 


98827 


46095 


98106 


48689 


97870 


18 


49 


48546 


98324 


46186 


96108 


48668 


97866 


11 


60 


9.43591 


9.98880 


9.46178 


9.98096 


9.48807 


9.97881 


10 


61 


48635 


98317 


46220 


98094 


48647 


97867 


9 


58 


43680 


98813 


40'J68 


98090 


48686 


97858 


8 


58 


43?J4 


08309 


46308 


98067 


48726 


97849 


7 


64 


43769 


06306 


4U:i45 


98088 


48764 


97845 


6 


65 


48818 


98808 


46386 


98079 


48808 


97841 


6 


66 


43857 


98299 


46428 


98075 


48842 


97887 


4 


67 


48901 


98295 


46469 


9H071 


48881 


97888 


8 


68 


48946 


96291 


46511 


98067 


48920 


97829 


8 


69 


48990 


98286 


46558 


98063 


48959 


97886 


1 


60 


44084 


96884 


46594 


98060 


48096 


97821 





» 


CoRine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


» 




74» 




78» 




72» 





46i 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



# 


18« 


19* 


20- 


/ 


Sine 


Cofline 


Sloe 


Cosine 


Sine 


Cosine 





9.48996 


9.97821 


9.61264 


9.97587 


9.53405 


9.97299 


60 


1 


49037 


97817 


51801 


97568 


58440 


97894 


60 


8 


49076 


97812 


51338 


97558 


58475 


97289 


68 


8 


49115 


97806 


51874 


97554 


53500 


97285 


67 


4 


49153 


97804 


61411 


97550 


53544 


97260 


66 


6 


49192 


97800 


61447 


97545 


68578 


97276 


66« 


6 


49231 


97T96 


51464 


97541 


58618 


97871 


64 


7 


49269 


97792 


51520 


97586 


58647 


97886 


68 


8 


49306 


97786 


61557 


97532 


63682 


97882 


68 


9 


49347 


97784 


51596 


97B28 


53716 


97857 


51 


10 


9.49385 


9.97779 


9.51629 


9.97B23 


9.58751 


9.97863 


60 


11 


49424 


97776 


51666 


97519 


63786 


97848 


49 


12 


49462 


97771 


51702 


97515 


63819 


97848 


48 


13 


49500 


97767 


61786 


97510 


53854 


97886 


47 


14 


49539 


97768 


61774 


97506 


58666 


97884 


46 


15 


49577 


97759 


51811 


97501 


. 53$&2 


97829 


46 


18 


49615 


97764 


51847 


97497 


63957 


9r284 


44 


17 


49654 


97750 


51868 


97492 


63991 


97220 


48 


18 


49692 


97746 


51919 


97488 


54025 


97215 


48 


19 


49780 


97748 


51965 


97484 


P059 


97210 


41 


90 


9.49766 


9.97788 


9.51991 


9.97479 


9.54098 


0.97806 


40 


31 


49606 


97784 


62027 


97475 


54187 


97801 


89 


22 


49844 




62063* 


97470 


54161 


97196 


88 


23 


49682 


97725 


62099 


97466 


. 54196 


97192 


87 


34 


49920 


^mi 


52185 


97461 


64229 


97187 


36 


25 


49958 


9T717 


58171 


97457 


54268 


97188 


86 


26 


49996 


97718 


52207 


97458 


54297 


97178 


84 


27 


50034 


97706 


52242 


97446 


64381 


97178 


88 


28 


60072 


97704 


62278 


97444 


54365 


97168 


82 


29 


60110 


97700 


52814 


97489 


64899 


97168 


81 


30 


9.50148 


9.97696 


9.62850 


9.97436 


9.54438 


9.97169 


80 


81 


601^ 


9^691 


52385 


97430 


54460 


97154 


89 


82 


50223 


97667 


52421 


97426 


64600 


97149 


86 


33 


50261 


97688 


62456 


97421 


54534 


97146 


87 


84 


50296 


97679 


62492 


97417 


54587 


07140 


86 


85 


50336 


97674 


52627 


97418 


54601 


97186 


26 


86 


50874 


97670 


6S568 


97406 


54635 


97180 


84 


87 


60411 


97666 


52596 


97403 


54668 


97126 


88 


88 


50449 


97662 


52684 


97899 


54702 


97121 


88 


89 


50486 


97667 


52669 


97394 


54735 


97116 


81 


40 


9.80628 


9.97658 


9.52706 


9.97890 


9.54769 


9.9nil 


80 


41 


50661 


97649 


52740 


97365 


54808 


97107 


19 


42 


50598 


97645 


52775 


97381 


54636 


97108 


18 


48 


60635 


97640 


52811 


97876 


54869 


97097 


17 


44 


50G73 


97686 


62846 


97372 


54903 


97098 


16 


45 


60710 


97632 


52881 


97867 


54936 


97087 


16 


46 


50747 


97826 


52916 


97368 


54969 


97068 


14 


47 


50784 


97623 


52951 


971(56 


55008 


97078 


18 


48 


50621 


97619 


52966 


97358 


55036 


97078 


IS 


49 


50666 


97615 


58021 


07349 


56060 


97066 


11 


90 


9.50896 


9.97610 


9.58056 


9.97344 


9.55108 


0.97068 


10 


51 


509:» 


97606 


53092 


97840 


65188 


97059 


9 


62 


50970 


97602 


63126 


97836 


66169 


97054 


8 


58 


61007 


97597 


53161 


97881 


55202 


97049 


7 


54 


61048 


97598 


58196 


97826 


55285 


97044 


6 


56 


51060 


97589 


58231 


97328 


&5.?66 


97039 


6 


66 


51117 


97584 


53266 


97317 


65301 


97085 


4 


57 


51164 


97580 


68301 


97812 


55384 


97080 


8 


58 


51191 


97576 


58336 


97306 


66367 


97025 


8 


50 


51827 


97571 


63370 


97308 


55400 


97090 


1 


60 


51964 


97567 


53405 


97299 


66483 


97016 





» 


Ck>6ine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


# 




71» 




70« 




6»^ 



462 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



/ 


tl» 


»• 


w 1 


/ 


Sine 


Cosloe 


Sine 


Cosine 


Sine 


Cosine 


"T" 


9.65488 


9.97015 


9.57368 


9.96717 


0.59188 


9.96408 


60 




56466 


07010 


67380 


96711 


59218 


96397 


60 




66499 


97006 


67420 


96706 


60247 


96898 


68 




66688 


97001 


57461 


96701 


50277 


968^7 


67 




65564 


96996 


67488 


96686 


68807 


96381 


66 




65607 


90991 


67614 


96691 


50S86 


96876 


65 




65680 


96986 


67545 


96686 


59366 


96370 


54 




66668 


96981 


57676 


96681 


50896 


96865 


53 




66606 


96976 


57607 


96670 


59426 


96360 


62 




66788 


96971 


67688 


96670 


59456 


96854 


51 




9.6OT61 


9.90966 


9.57669 


9.96666 


9.60484 


9.96349 


50 




86'n8 


9C968 


57700 


96660 


60614 


96848 


49 




66ft!6 


969OT 


57781 


96656 


69643 


96338 


48 




56868 


96958 


57T68 


96650 


69578 


96883 


47 




66891 


96047 


67798 


96645 


59608 


968«7 


46 




55088 


96948 


67824 


96640 


69682 


96882 


45 




66066 


96987 


67856 


96684 


69661 


96816 


44 




65968 


96988 


57886 


96629 


59690 


96311 


48 




56081 


96987 


57916 


96624 


60720 


96306 


42 




56068 


96948 


57947 


96619 


69749 


96800 


41 


80 


0.56066 


9.96917 


0.67978 


9.96614 


9.50778 


9.96894 


40 


SI 


66118 


96919 


68006 


96608 


60806 


96289 


80 


29 


56150 


96907 


58089 


96608 


59887 


96284 


86 


88 


66188 


96908 


68070 


96698^ 


68866 


96278 


87 


M 


66»5 


OAfiOA 


68101 


96fiA8 


59696 


96278 


86 


85 


563847 


96898 


68181 


96588 


KAQOlJ 


96267 


85 


86 


66879 


96888 


58168 


96588 


59064 


96462 


34 


87 


66811 


96888 


5819-i 


96577 


59988 


96256 


88 


88 


66848 


96878 


68228 


96578 


60018 


96^51 


88 


89 


66375 


96878 


58258 


06667 


60041 


96245 


31 


80 


9.60408 


9.96868 


9.68884 


9.9G562 


9.60070 


9.96240 


80 


81 


56440 


96863 


58814 


96666 


60099 


96284 


29 


38 


56478 


96858 


58845 


96561 


60128 


96229 


88 


88 


66504 


96868 


68875 


96546 


60157 


968^ 


27 


84 


56586 


96848 


58406 


96541 


60186 


98218 


26 


85 


56568 


96848 


68486 


96636 


60815 


96212 


25 


86 


KKMAQ 

OOaW 


96888 


58467 


96530 


60244 


96207 


M 


87 


56631 


96838 


68497 


96525 


60278 


96201 


28 


88 


56663 


96828 


58627 


96590 


60802 


96196 


28 


89 


56606 


96888 


58557 


96514 


60881 


96190 


81 


40 


0.56727 


9.96818 


9.58588 


9.96609 


9.60369 


0.96186 


80 


41 


56760 


96818 


58618 


96601 


60888 


96179 




4S 


56790 


96806 


58648 


96496 


60417 


96174 




48 


ami 


96803 


66678 


96498 


60446 


96168 




44 


56854 


96796 


58709 


96488 


60474 


96168 




45 


66886 


96798 


68789 


96483 


60503 


96157 




40 


56917 


96788 


68769 


964T7 


60532 


96151 




47 


56049 


96788 




96472 


60561 


96146 




48 


56980 


96778 


68889 


96467 


60580 


96140 




49 


570IS 


96773 


68859 


96461 


60618 


06186 




60 


9.57044 


9.96767 


9.58889 


9.96456 


9.60646 


9.96129 




51 


57075 


967081 


58919 


96451 


60675 


96188 




68 


67107 


96757 


58949 


96446 


60704 


96118 




58 


67188 


96758 


58979 




60788 


96112 




54 


57169 


96747 


50009 


96436 


60761 


96107 




56 


57801 


96742 


59039 


96429 


60789 


96101 




66 


67J88 


96787 


59060 


96424 


60618 


96095 




57 


67264 


9C733 


500B6 


96419 


60646 


96000 




58 


57895 


96727 


69128 


96418 


60675 


960H4 




69 


57826 


96722 


69156 


96406 


60003 


96079 




60 


67858 


96717 


59188 


96403 


60961 


96078 




/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 




68» 




670 




•e» 



463 



TALBE n.— LOGARITHMIC SINES AND COSINES. 






24« 




86» 


W 


t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.80981 


0.06078> 


0.68585 


0.06728 


0.64184 


9.06866 


60 


1 


60880 


06067 


68688 


05728 


64310 


05860 


68 


8 


60088 


96062 


63640 


06716 


64286 


96854 


68 


8 


61016 


06056 


62676 


05710 


64268 


06848 


57 


4 


61046 


96050 


63708 


05704 


64288 


06841 


86 


5 


61078 


96045 


63780 


93698 


64818 


06886 


65 


6 


61101 


06039 


68757 


96608 


64889 


95329 


54 


7 


61129 


06034 


63764 


96686 


64366 


96888 


68 


8 


61158 


00028 


62811 


956S0 


64891 


96817 


68 


9 


61186 


06038 


62888 


95674 


64417 


96810 


51 


10 


9.61214 


0.06017 


0.62885 


9.06666 


9.64448 


9.95804 


60 


11 


61818 


06011 


62603 


05663 


64468 


95208 


40 


IS 


61270. 


06005 


68018 


06657 


64404 


95808 


48 


18 


61806 


06000 


62045 


05661 


64510 


95386 


47 


14 


61888 


06904 


68973 


93646 


64546 


95379 


46 


15 


61854 


05068 


63900 


05630 


64571 


96278 


45 


18 


61883 


05062 


63026 


06638 


64506 


95267 


44 


17 


61411 


OSOTT 


63063 


93627 


64638 


96261 


48 


18 


61488 


05071 


63079 


96631 


64647 


96254 


43 


10 


61466 


95965 


63106 


95616 


64678 


96346 


41 


20 


9.61494 


9.95960 


9.63138 


9.06609 


0.64608 


9.96848 


40 


21 


61582 


06054 


68159 


06603 


64734 


05386 


80 


83 


61550 


05048 


68186 


93507 


64740 


0S389 


88 


88 


61578 


05042 


68318 


05G01 


64775 


96228 


87 


84 


61606 


03037 


63289 


03685 


64800 


96217 


86 


25 


6168t 


05081 


63366 


03570 


64838 


96211 


86 


88 


61668 


05035 


63393 


95578 


64851 


96304 


34 


27 


61689 


05020 


63319 


05567 


64877 


. 05106 


88 


88 


61717 


05014 


63345 


05661 


64003 


96192 


88 


89 


61745 


86006 


63373 


95655 


64087 


96186 


31 


80 


9.61778 


9.96002 


9.63306 


9.96549 


0.64068 


9.06170 


80 


81 


61800 


95897 


63435 


95548 


64078 


05178 


30 


38 


618-28 


95801 


68451 


96587 


60008 


06167 


88 


83 


61856 


068S5 


63478 


06531 


66028 


05160 


87 


84 


61883 


05670 


68504 


05535 


65054 


05164 


86 


vs 


61011 


03878 


63531 


05519 


66079 


95148 


86 


86 


61080 


95868 


68667 


96518 


65104 


93141 


84 


87 


61066 


05863 


63588 


95507 


65180 


93185 


28 


88 


61004 


95836 


68610 


0G600 


65166 


05120 


83 


89 


68081 


95830 


63686 


05401 


66180 


05128 


91 


40 


9.62049 


9.06844 


0.68663 


,0.05486 


9.66205 


9.95116 ' 


90 


41 


62076 


06339 


63C60 


03483 


65880 


95110 




48 


68104 


93838 


63715 


06476 




95108 




48 


62181 


95627 


63741 


06470 


65281 


93007 




44 


68159 


06831 


63787 


95464 


65306 


95000 




45 


62186 


93815 


63704 


96458 


65381 


95061 




48 


62214 


96810 


68880 


95452 


65356 


95078 




47 


62241 


96604 


63846 


95446 


65881 


06071 




48 


62266 


95708 


68S72 


95440 


65406 


95065 




40 


62896 


95799 


63808 


05484 


66481 


95059 




50 


9.62388 


9.95786 


0.63824 


8.06437 


9.65456 


9.06058 




51 


62850 


95780 


63060 


95431 


66481 


05046 




58 


62877 


05775 


68976 


95415 


66606 


03030 




58 


82405 


05760 


64002 


95409 


66531 


95088 




54 


63483 


05768 


04026 


95408 


66565 


95087 




55 


62450 


05757 


64064 


93807 


66580 


95080 




66 


62486 


03751 


64080 


03S01 


66605 


96014 




57 


62518 


95745 


64106 


05884 


66630 


95007 




58 


62641 


96739 


64133 


03378 


66656 


96001 




69 


6B568 


95738 


64166 


05878 


66080 


94906 




80 


62506 


95728 


64164 


06366 


66706 


94088 




r 


Ck>Bine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




86« 




64* 




6S* 



i 



464 



TABLE n.— LOGARITHMIC SINES AND. COSINES. 



t 


27» 


«8» 


29<> 


• 


Sine 


Ckwfne 


Bine 


Cosine 


Sine 


Cosine 





9.65705 


9.94968 


9.67161 


9.94508 


9.68557 


9.94182 


00 


1 


66729 


94982 


67185 


94687 


68580 


94175 


60 


s 


66754 


94975 


67208 


94560 


68603 


94168 


58 


8 


65779 


94969 


67282 


W573 


6862S 


94161 


57 


4 


65804 


94962 


67266 


94567 


68648 


94164 


66 


5 


65828 


94956 


67280 


94560 


68671 


94147 


66 


6 


65858 


94949 


67303 


94668 


68694 


94140 
941^ 


54 


7 


65878 


9494S 


67B27 


94546 


68716 


68 


8 


65902 


94936 


67850 


94640 


68739 


04126 


52 


9 


66927 


M980 


67874 


94583 


68762 


94119 


51 


10 


9.66052 


9.94928 


9.67898 


9.94526 


9.68784 


9.94112 


50 


11 


66978 


94917 


67421 


94.')19 


68807 


94106 


40 


12 


66001 


94911 


67445 


94518 


68829 


94096 


46 


28 


66025 


94904 


67468 


94506 


68652 


94090 


47 


14 


66060 


94898 


67492 


94499 


68876 


94068 


46 


16 


66075 


94891 


67516 


94492 


68897 


94076 


45 


16 


66099 


94885 


67589 


94485 


68920 


94069 


44 


17 


66124 


94878 


67562 


94470 


66942 


94062 


48 


18 


66146 


94871 


67566 


94472 


68966 


94055 


42 


19 


66178 


04866 


67609 


94465 


68987 


94046 


41 


20 


9.66197 


9.94868 


9.67638 


9.94458 


9.69010 


9.94041 


40 


21 


66221 


94862 


67656 


94451 


69082 


94084 


80 


28 


66246 


94846 


67680 


94445 


69055 


94027 


86 


28 


66270 


94889 


67708 


94438 


690T7 


94090 


87 


84 


66295 


94882 


67726 


94431 


69100 


94012 


86 


25 


66819 


94826 


67750 


94424 


60122 


94005 


85 


20 


66348 


94819 




94417 


60144 


98998 


84 


27 


66866 


« 94818 


67796 


94410 


69107 


98991 


83 


28 


66892 


94806 


67B20 


94404 


69169 


98984 


82 


29 


66416 


04799 


67848 


94397 


69212 


98977 


81 


80 


9.66441 


9.94798 


0.67866 


9.94890 


9.69384 


9.93970 


80 


81 


66465 


94786 


67890 


94388 


69256 


98968 


29 


« 82 


66489 


947B0 


67918 


94376 


69279 


98956 


28 


83 


66518 


94778 


67986 


94869 


60801 


98948 


27 


84 


66587 


94767 


67969 


94862 


69323 


98941 


26 


85 


66662 


94760 


67982 


94855 


69846 


93984 


25 


86 


66586 


94753 


68006 


94349 


69868 


98927 


•84 


87 


60610 


94747 


66029 


94349 


69890 


98990 


88 


88 


66684 


94740 


68052 


94335 


69412 


98912 


22 


89 


66668 


94734 


68075 


1M828 


69484 


93906 


21 


40 


9.66682 


9.94727 


9.68096 


9.94321 


9.69456 


9.93898 


80 


41 


66706 


94720 


68121 


94814 


69479 


98691 


19 


42 


66781 


94714 


68144 


94807 


69501 


93684 


18 


48 


66766 


94707 


68167 


94300 


69628 


98876 


17 


44 


66779 


94700 


68190 


94298 


69545 


93869 


16 


45 


66808 


94694 


68218 


94286 


69567- 


98862 


16 


46 


66827 


94687 


68287 


94279 


69589 


98855 


14 


47 


66861 


94680 


68260 


94278 


69611 


93647 


18 


48 


66875 


94674 


68283 


94266 


69638 


98840 


12 


49 


66899 


94667 


68805 


94259 


69655 


98888 


11 


60 


9.60922 


9.94660 


9.68828 


9.94252 


9.69677 


9.93626 


10 


51 


66946 


94654 


68851 


94245 


69699 


93819 


9 


62 


66070 


94647 


66874 


94288 


69721 


98811 


8 


68 


66994 


94640 


68897 


94231 


69748 


93804 


7 


64 


67018 


94634 


66420 


94224 


69765 


98797 


6 


65 


67042 


M627 


68448 


94217 


69787 


93789 


6 


56 


67066 


94690 


68466 


04210 


69809 


98788 


4 


57 


67090 


94614 


68489 


94208 


69831 


98775 


8 


58 


67118 


94607 


68512 


94106 


09858 


98768 


2 


69 


67187 


94600 


66534 


94189 


69675 


98760 


1 


60 


67161 


94698 


68657 


94182 


69697 


98753 





9 


CkMlne 


Sine 


Cosine 


Sine 


CoBine 


Sine 


/ 




W 




6f 




60« 



4<55 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



t 


ao'" 


8l» 


B2^ 1 


* 


3iri« 


Cosrne 


srns 


Cosine 


Sln^ 


CosSne 





S.fiOflOT 


9,Bff75a 


S.TllM 


9.93307 


9.:S431 


%.m^ 


60 


] 


wfm 


93TW 


71305 


e®299 


7^1 


02834 


69 


i 


09^1 


937^ 


712^ 


0.1291 


72461 


92836 


58 


3 


Q900S 


ft3ni 


71247 


oa-^i^ 


?i>482 


0^18 


67 


4 


«99$t| 


B3rJ4 


ivm 


Ki'.^76 


75Sffi! 


9^2glO 


56 


5 


70006 


fl«T(7 


"Am 


Mioy 


^^Si^ 


92803 


65 


6 


T0ft38 


aarod 


71310 


DaJCl 


7554^ 


03795 


54 


7 


70(W0 


9:^09 


71881 


08258 


72662 


92787 


68 


8 


70072 


06G96 


71862 


98246 


72582 


92779 


68 


9 


70098 


93687 


71878 


98288 


72608 


92771 


61 


10 


9.70115 


9.93680 


9.71398 


9.98230 


9.72622 


9.92768 


60 


11 


70187 


03673 


71414 


082:i;8 


72048 


92T55 


49 


12 


70159 


93665 


71486 


98216 


72668 


92747 


48 


18 


70180 


93658 


71456 


98207 


72683 


92789 


47 


14 


70202 


93660 


71477 


08200 


72708 


92781 


46 


15 


70224 


93648 


71498 


93192 


72723 


92728 


46 


16 


70245 


96686 


71519 


93184 


72743 


92715 


44 


17 


70267 


93628 


71589 


98177 


72768 


98707 


48 


18 


70288 


93621 


71660 


93169 


72788 


92699 


48 


19 


70310 


93614 


71681 


98161 


72808 


92691 


41 


90 


9.70882 


9.03606 


9.71602 


9.98164 


9.72828 


9.92688 


40 


81 


70858 


08599 


71628 


98146 


72848 


98675 


89 


23 


70875 


98591 


71648 


93188 


72868 


92667 


88 


28 


70396 


98584 


71664 


98181 


72888 


92659 


87 


24 


70418 


98677 


71686 


98128 


72902 


92651 


86 


26 


70489 


98569 


71706 


93115 


72922 


92643 


85 


26 


70461 


93563 


71726 


98108 


72948 


02636 


84 


97 


70483 


98554 


71747 


98100 


72062 


92827 


S3 


28 


70604 


98547 


71767 


98092 


72982 


92619 


82 


99 


70626 


98589 


71788 


08084 


780(» 


92611 


81 


80 


9.70647 


9.93588 


9.71800 


9.98077 


9.78022 


9.92608 


80 


81 


70508 


98525 


71889 


98069 


78041 


92595 


89 


89 


70690 


08517 


71850 


93061 


73061 


92687 


2& 


88 


70611 


98510 


71870 


98058 


78081 


92579 


87 


84 


70638 


93502 


71891 


98046 


78101 


92571 


86 


86 


70654 


98406 


71911 


98038 


- 78121 


92568 


26 


86 


70675 


93487 


71932 


93080 


78140 


92555 


84 


87 


70697 


98480 


71958 


93023 


78160 


92546 


28 


88 


70718 


98472 


71978 


93014 


78180 


92588 


22 


88 




93465 


71994 


98007 


73200 


98580 


81 


40 


9.70761 


9.03457 


9.79014 


9.92999 


9.78819 


9.92523 


20 


41 


70r82 


93450 


72034 


92991 


78839 


92514 




42 


70808 


98142 


72065 


92988 


78269 


92506 




48 


70824 


93436 


720?6 


92976 


78278 


92498 




44 


70846 


98427 


72096 


92968 


7329R 


92490 




45 


70867 


93420 


72116 


92960 


78318 


92482 




46 


70688 


98412 


72187 


92952 


78837 


98478 




47 


70900 


93406 


72167 


02944 


78857 


92465 




48 


70931 


93397 


72177 


08936 




92457 




49 


70952 


98390 


72198 


98989 


78896 


92449 




60 


9.70978 


9.93882 


9.78218 


9.02921 


9.78416 


9.92441 




61 


70994 


98875 


78288 


92913 


73436 


98438 




62 


71016 


93367 


72259 


93906 


78456 


92425 




68 


71036 


93360 


7*279 


92897 


78474 


92416- 




64 


71058 


93352 


72299 


92889 


78494 


92406 




66 


71079 


93S44 


72320 


92881 


78513 


02400 




66 


71100 


93387 


78340 


92874 


73.')38 


92392 




67 


71121 


98329 


72360 


92866 


78562 


92384 




68 


71143 


98822 


78881 


0^858 


73578 


923T6 




80 


71168 


93314 


72401 


92850 


78591 


92367 




60 


71184 


93307 


72421 


92842 


78611 


92869 




/ 


Oofline 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




6»*» 




ftfto 




fi7«» 



466 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



/ 


ts« 


t4* 


w 1 


# 


Sine 


Oocine 


ffine 


Ooeine 


Sine 


Codne 




0.78611 


0.08880 


0.74766 


9.91857 


9.75869 


0.01886 


60 




78680 


08851 


74775 


91849 


768r7 


01828 


60 




78650 


0S318 


74704 


91840 


76896 


01810 


68 




78660 


08885 


74818 


91888 


75918 


01810 


57 




78680 


08886 


74881 


91888 


76981 


oiibi 


56 




78706 


08818 


74860 


91616 


75049 


01808 


66 




78787 


08810 


74868 


91806 


75967 


01288 


54 




78747 


08808 


74887 


91798 


75986 


01874 


68 




78766 


88806^ 74006 


91789 


76008 


01266 


68 




78785 


08886 


74084 


• 91781 


76021 


01857 


61 




0.78805 


0.08877 


0.74048 


9.91778 


9.76089 


0.01848 


60 




78884 


0«60 


74061 


91768 


76067 


01888 


49 




78848 


08860 


74980 


91766 


76075 


91280 


48 




78868 


08868 


74090 


91746 


76098 


91881 


47 




78888 


08844 


76017 


91788 


761U 


91818 


46 




78001 


08886 


75088 


91789 


76189 


91808 


46 




78081 


98887 


76054 


91780 


76146 


91194 


44 




78040 


0S810 


75078 


91718 


76164 


91185 


48 




78050 


08811 


75091 


91708 


76188 


91176 


48 




78078 


08808 


75110 


91696 


76800 


91 167 


41 




0.78007 


0.08104 


9.7^188 


9.9K66 


9.76818 


9.91158 


40 




74017 


08186 


76147 


91677 


76286 


91149 


89 


29 


74086 


08177 


75166 


91669 


76268 


91141 


88 


$» 


74065 


08160 


76184 


91660 


76871 


91188 


87 


84 


74074 


98161 


76*^ 


91651 


76889 


91128 


86 


85 


74008 


08168 


76881 


91648 


78807 


91114 


85 


86 


74118 


08144 


75889 


916H4 


76384 


91106 


84 


87 


74188 


08186 . 


75868 


91686 


76843 


91096 


38 


88 


74151 


08187 


76876 


91617 


76860 


91087 


88 


89 


74170 


98110 


76894 


91608 


76878 


91078 


31 


80 


0.74180 


0.08111 


9.76818 


9.91699 


9.76895 


9.91069 


30 


81 


74208 


08108 


76831 


91591 


76418 


01060 


89 


88 


74887 


08004 


75850 


91688 


76481 


01061 


86 


88 


74846 


08066 


76868 


91578 


76448 


01048 


87 


84 


W865 


08077 


76886 


91565 


76466 


01083 


86 


85 


74884 


ttoS 


76405 


91566 


76484 


01028 


86 


86 


74806 


76483 


91547 


76601 


01014 


84 


8/ 


74888 


08068 


76441 


91538 


76519 


01006 


88 


88 


74841 


08044 


75450 


91580 


76687 




28 


80 


74860 


08086 


75478 


91521 


76564 


.00087 


81 


40 


9.74879 


0.08087 


0.75496 


9.91518 


9.76572 


0.00078 


80 


41 


74808 


08018 


75614 


91504 


76590 


00060 




48 


74417 


08010 


75588 


91496 


76607 


00060 




48 


74486 


08008 


76561 


91486 


76625 


00961 




44 


74455 


01908 


75?i60 


91477 


76648 


90948 




45 


74474 


01085 


75687 


91469 


76660 


90988 




46 


74498 


01076 . 


76605 


91460 


76677 


90984 




47 


74518 


01068 


76684 


91451 


76695 


90916 




48 


74581 


01060 


76648 


91448 


76718 


90906 




40 


74640 


01061 


76600 


91488 


76780 


90896 




60 


0.74666 


0.0104S 


9.7567R 


9.91425 


9.76747 


9.90887 




61 


74687 


010S4 


75696 


91416 


76785 


90878 




68 


74606 


01025 


75714 


914(17 


76788 


90869 




68 


74685 


01017 


76738 


91398 


76800 


90860 




64 


74644 


01006 


75761 


91389 


76817 


90861 




66 


74668 


01000 


75769 


91881 


7fi836 


90848 




66 


74681 


01801 


75787 


91879 


TtlNiO 


90888 




67 


74700 


01888 


75806 


91.%8 


76870 


90828 




66 


74710 


01874 


76828 


91854 


76887 


90R14 




60 


74787 


01866 


75841 


91346 


78904 


00806 




60 


74756 


01857 


75660 


91836 


76088 


90796 




# 


Ooslne 


Sine 


Cosine 


Sine 


CkMrine 


Sine 


/ 




66* * 




66«» 




»4» • 



467 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



« 


•6« 


•7* 


W 


/ 


Sine 


Cosine 


Bine 


Cofline 


Sine 


Cosine 




0.76928 


9.90796 


9.77946 


9.90886 


9.78084 


9.80658 


60 




76989 


90787 


77968 


90886 


78960 


89648 


59 




76097 


90777 


77980 


90816 


78967 


89638 


58 




76974 


90768 


77997 


90806 


78063 


89624 


57 




76991 


90759 


78018 


90197 


78999 


89614 


56 




77009 


907SO 


78080 


90187 


79016 


89604 


66 




77086 


90741 


78047 


90178 


79081 


89594 


54 




77048 


90781 


78068 


90168 


79047 


89584 


68 




77061 


90788 


78080 


90159 


70068 


89574 


62 




77078 


90718 


78097 


90149 


79079 


89564 


61 




9.7r095 


9.90704 


9.78118 


9.90139 


9.79095 


9.89654 


60 




77118 


90694 


78180 


90130 


79111 


89544 


48 




TTiao 


90685 


78147 


90180 


79188 


89684 


48 




77147 


90676 


78163 


90111 


79144 


89684 


47 




77164 


90667 


78180 


90101 


79160 


89514 


46 




77181 


90657 


78197 


90091 


79176 


89604 


46 




77199 


90648 


78818 


90068 


79198 


89496 


44 




77216 


90639 


78830 


90078 


79806 


89485 


48 




77888 


90630 


78846 


00068 


79884 


89475 


48 




77»0 


90680 


78863 


90058 


79840 


89466 


41 


90 


9.77868 


9.90611 


9.78880 


9.90048 


9.79856 


9.89466 


40 


St 


77885 


90608 


78296 


90084 


79878 


89445 


88 


S2 


77808 


00608 


78818 


90084 


79388 


88485 


88 


S8 


77819 


90588 


78389 


90014 


79B04 


89485 


87 


84 


77886 


90574 


78346 


90005 


79619 


89415 


86 


25 


77858 


90565 


78368 


QQAQK 
OWWO 


79885 


89405 


86 


S8 


77870 


90555 


78379 


89985 


79851 


89396 


84 


27 


77887 


90546 


78895 


89976 


79367 


89386 


88 


28 


77405 


90587 


78418 


89966 


79388 


89875 


88 


89 


77488 


90687 


78428 


80950 


79399 


80364 


81 


80 


9.77439 


9.90618 


9.78445 


9.89947 


9.79416 


9.89364 


80 


81 


77456 


00609 


78481 


89937 


79481 


89344 


89 


88 


77478 


90499 


78478 


89987 


79447 


69884 


88 


88 


77400 


90490 


78494 


80918 


79468 


89884 


27 


81 


77507 


90480 


78510 


89906 


79478 


89814 


86 


85 


77584 


90471 


78587 


80698 


79494 


89304 


86 


86 


77541 


90468 


78648 


89688 


79510 


80894 


24 


87 


77558 


90458 


78560 


89879 


79586 


89284 


88 


38 


77575 


90448 


78576 


88669 


79548 


89274 


88 


88 


77508 


90434 


78598 


89659 


79658 


89864 


81 


40 


9.77609 


9.90484 


9.78609 


9.88649 


9.79678 


9.89864 


80 


41 


77686 


90415 


78686 


69840 


79580 


89844 




48 


77648 


90405 


78648 


89830 


79606 


89838 




48 


77660 


90896 


7S658 


89620 


78681 


89828 




44 


77677 


90386 


78674 


89810 


79636 


89818 




45 


77604 


90377 


78691 


89801 


79653 


80203 




46 


777U 


9086S 


78707 


89791 


79668 


89198 




47 


77788 


90358 


78738 


89781 


79684 


89168 




48 


77744 


90349 


78739 


89771 


79699 


89178 




49 


77761 


90389 


78756 


80761 


79716 


89168 




60 


9.77778 


9.90330 


9.7877a 


9.80758 


0.79781 


9.89158 




51 


77796 


90880 


7RT88 


89748 


79746 


89142 




58 


77818 


90311 


78805 


89788 


79768 


89188 




58 


77S29 


90301 


78821 


89728 


79778 


89188 




54 


77846 


90298 


78837 


89718 


79798 


89112 




55 


77868 


908S3 


78853 


69708 


79809 


89101 




56 




90873 


78869 


80698 


79885 


89091 




57 


77896 


90263 


78886 


89688 


79840 


89061 




58 


77018 


00854 


78908 


89678 


79856 


89071 




60 


7T»:*) 


90844 


7R9I8 


80668 


79878 


89060 




00 


77946 


90835 


78934 


89663 


79687 


89060 




» 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




Br 




62« 




«!• 



468 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



/ 


•»• 


4«» 


41» 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 




0.79887 


0.800SO 


0.80607 


0.88485 


0.81004 


••S22 

W707 


00 




70908 


89040 


60628 


88415 


61700 


SO 




70018 


60080 


80687 


88404 


61728 


87750 


68 




70084 


80090 


80668 


88804 


81788 


87745 


St 




70050 


80000 


80607 




81782 


87784 


60 




70966 


88000 


60682 


86872 


81T67 


87728 


66 




79961 


88860 


80897 


88808 


81781 


87719 


64 




70900 


88078 


80918 


68851 


81700 


9noi 


68 




80018 


88008 


60927 


88840 


81610 


67800 


68 




800W 


88058 


60942 


88880 


81886 


87870 


61 




0.80048 


0.68048 


0.80057 


0.88810 


0.81880 


0.87008 


60 




80068 


88987 


80078 


86808 


81864 


87067 


40 




80074 


88027 


60067 


86896 


81868 


87840 


48 




80060 


88917 


81008 


88887 


81868 


87086 


47 




80106 


88000 


81017 


88870 


81607 


87084 


40 




80120 


86690 


81082 


88860 


81011 


87018 


45 




80180 




81047 


88865 


81086 


87001 


44 




80151 


68875 


81061 


68244 


81040 


87580 


48 




80160 


88605 


81070 


88884 


81066 




42 


10 


80182 


68865 


81001 


88228 


SIOOO 


87608 


41 


80 


0.80197 


0.88844 


0.81100 


0.88818 


0.81088 


0.87857 


40 


21 


80218 


66884 


81121 


68801 


81008 


87540 


89 


tt 


80228 


86824 


81186 


88101 


89012 


87686 


88 


S8 


80244 


88618 


81151 


68180 


89006 


87594 


97 


84 


80850 


6S808 


81160 


88100 


88041 


87518 


80 


86 


80274 


8»708 


81160 


68168 


88056 


87601 


86 


80 


80290 


88788 


81105 


88148 


88060 


87400 


84 


87 


80805 


88772 


81210 


66187 


K084 


87470 


88 


88 


80690 


88761 


61285 


88186 


88008 


87408 


92 


20 


80880 


88751 


81240 


88115 


88118 


87457 


81 


80 


0.90851 


0.88741 


0.81854 


0.66105 


0.88180 


0.87440 


80 


81 


80800 


88780 


81200 


68004 


88141 


87484 


90 


88 


608S) 




81884 


66088 


68166 


87488 


88 


88 


80807 


88700 


61800 


88078 


88100 


87418 


27 


84 


80418 


88600 


81814 


68061 


88184 


87401 


80 


86 


80428 


86668 


81888 


88051 


88196 


87890 


26 


86 


80448 


66078 


81848 


88040 


88818 


87878 


94 


87 


80458 


88668 


81858 


68080 


68880 


87807 


98 


88 


80478 


86657 


61872 


88018 


88840 


87860 


98 


80 


80480 


86647 


81887 


88007 


88806 


8^46 


91 


40 


0.80604 


0.88686 


0.81408 


0.67000 


0.68900 


0.87884 


90 


41 


80510 


68626 


81417 


87065 


88888 


87889 


10 


42 


80684 


88615 


81481 


87076 


88897 


87811 


18 


48 


80550 


88605 


81440 


87004 


68811 


87800 


17 


44 


80565 


88594 


81401 


87058 


68880 


87888 


10 


46 


80580 


88584 


81475 


87048 


88840 


87877 


16 


46 


80506 


8S678 


81400 


87031 


88854 


87800 


14 


47 


80610 


66568 


81506 


87080 


68868 


87866 


It 


48 


80085 


88552 


81510 


87000 


88888 


67848 


19 


40 


80641 


68542 


81584 


87608 


88800 


87288 


11 


60 


0.80656 


0.88531 


0.61540 


0.87687 


0.82410 


0.67281 


10 


61 


80071 


88521 


81663 


87877 


88484 


67200 




63 


80680 


88510 


81578 


87860 


88480 


87196 




68 


80701 


88400 


81508 


83666 


88458 


87187 




64 


80710 


66480 


61607 


87844 


88407 


8717S 




66 


80781 


88478 


81082 


87838 


82481 


67104 




66 


80740 


88468 


81680 


67888 


88405 


87168 




67 


80762 


88457 


81651 


87811 


88600 


87141 




68 


80777 


88147 


81665 


87800 


88680 


87180 




60 


80T02 


88486 


81660 


67780 


68537 


87110 




60 


80607 


88425 


81604 


67778 


88551 


87107 




# 


Cosine 


Sine 


Coeine 


Sine 


Cosine 


Sine 


» 


50* 




49» 




48« 



469 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



1 


42» 


4go 


44«» 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


CoAine 




9.82561 


9.87107 


9.88378 


9.86418 


9.84177 


9.65693 


60 




82565 


87096 


8889;i 


86401 


84190 


85681 


60 




82579 


87095 


88405 


8C8S0 


84208 


85669 


58 




62508 


87078 


88419 


86377 


84216 


&'}657 


57 




82607 


87062 


&3432 


8G866 


64829 


86045 


66 




82621 


87050 


83446 


86854 


84248 


86682 


56 




82685 


87089 


8^69 


86842 


84256 


86680 


64 




82649 


87028 


88478 


8C330 


84269 


85606 


68 




62668 


87016 


83486 


66318 


84282 


85606 


52 




82677 


87005 


88500 


86806 


84295 


85588 


51 




9.82691 


9.86998 


9.88518 


9.86295 


9.84308 


0.85571 


60 




82706 


86962 


88527 


86288 


84881 


85550 


40 




62719 


86970 


68540 


86271 


84884 


85547 


48 






86959 


88654 


86259 


84847 


85584 


47 




82747 


86947 


88507 


86247 


84860 


85622 


46 




82761 


669:^ 


88S81 


86285 


84878 


66610 


46 




82775 




88504 


86828 


84885 


86407 


44 




82788 


86918 


88608 


86211 


84896 


85465 


48 




82809 


86902 


83621 


88200 


84411 


85478 


48 


10 


82816 


86890 


83684 


86188 


84484 


85460 


41 


90 


9.62880 


9.66879 


9.88848 


9.86176 


9.84487 


0.85448 


40 


81 


8'i8l4 


86867 


88661 


86164 


84450 


85486 


80 


29 


82868 


86855 


88674 


66152 


84463 


85428 


88 


iM 


82872 


86844 


a3688 


86140 


84476 


85411 


87 


24 


82886 


86882 


68701 


86128 


84480 


85800 


86 


25 


82899 


R6821 


&3716 


66116 


84502 


65886 


85 


26 


8:!918 


86809 


83728 


86104 


84615 


85874 


84 


27 


82027 


86798 


88741 


86092 


84528 


85861 


88 


28 


82941 


86786 


88755 


86080 


84540 


85340 


82 


29 


82996 


86775 


88768 


860G8 


84558 


66837 


31 


80 


9.82968 


9.86763 


9.88761 


9.86a56 


9.84566 


0.85824 


30 


81 


8-i982 


86769 


83795 


86044 


84579 


86312 


20 


39 


82996 


86740 


88808 


86082 


84592 


85200 


88 


83 


88010 


86728 


88821 


86020 


84605 


85887 


27 


84 


88028 


86717 


88884 


86006 


84618 


86874 


26 


85 


880?}7 


86705 


88848 


85996 


84680 


85262 


25 


86 


88051 


86694 


83861 


85972 


84648 


85250 


84 


87 


8:i065 


86682 


88874 


84656 


86287 


88- 


88 


88078 


80670 


83887 


85060 


84669 


85825 


28 


89 


83092 


86659 


88001 


85948 


84682 


85212 


81 


40 


9.88106 


9.86647 


0.83914 


0.85986 


9.84694 


0.85200 


80 


41 


88120 


86685 


88927 


65924 


84707 


85187 


10 


49 


83188 


86624 


88940 


85912 


84780 


. 85175 


18 


48 


88147 


86612 


83954 


85900 


84788 


66168 


17 


44 


88161 


86600 


88967 


85888 


84745 


85150 


16 


45 


83174 


66589 


83960 


85876 


84758 


85187 


15 


46 


88188 


86577 


a3n93 


85864 


84771 


65125 


14 


47 


88202 


86565 


84006 


85861 


81784 


85112 


18 


48 


88*^15 


86554 


84020 


85839 


W796 


a5100 


18 


49 


882-^ 


86542 


84038 


85827 


84809 


65067 


11 


SO 


9.88242 


9.86530 


9.84046 


0.85815 


9.8IR28 


0.86074 


10 


61 


9l^SSfA 


8651R 


B4059 


a5808 


848.35 


85068 





m 


88-i70 


86507 


84072 


«i791 


84847 


85040 


8 


58 


88•^88 


86495. 


84085 


85779 


84860 


86037 


7 


54 


83297 


86488 


84098 


85766 


64878 


86024 


6 


55 


S9310 


86472 


84112 


86754 


84885 


85012 


6 


56 


88824 


86460 


84125 


85742 


84898 


84000 


4 


57 


83388 


86443 


84188 


85730 


84911 


84986 


8 


58 


88351 


86436 


84161 


86718 


84923 


84974 


8 


50 


88865 


86426 


84164 


85706 


^vm 


84961 


1 


60 


8.'U)78 


86418 


84177 


85698 


84949 


84940 





# 


Ooeine 


Sine 
47« 


Cosine 


Sine 


Cosine 


Sine 


» 




46- 




46« 



470 



TABLE in 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 




0* 




1- 


2 


e 


» 


Tkn 


Ootan 


Tan 


Ck>tan 


Tkn 


Ootan 




— 00 


OD 


8.24102 


11.75808 


8.54806 


11.45692 


60 




6.46878 


18.68027 


24910 


76000 


64669 


45881 


50 




76476 


28524 


26616 


74884 


55027 


44978 


68 




04066 


05016 


26812 


7WJ88 


55882 


44618 


87 




7.06670 


12.93421 


26006 


78004 


60784 


44206 


66 




16270 


68780 


27600 


78331 


66063 


48817 


66 




24188 


76812 


28888 


71668 


66429 


48571 


54 




80682 


60118 


28886 


71014 


66778 


48887 


53 




86682 


68818 


29629 


70871 


67114 


42886 


69 




41707 


68808 


80263 


60787 


67452 


42648 


61 


10 


7.46878 


12.68627 


8.80688 


11.60112 


8.57788 


11.42812 


60 


11 


50612 


40488 


81506 


68495 


58121 


41879 


49 


12 


64201 


46900 


821121 


67888 


58461 


41649 


48 


18 


67787 


42233 


827111 


07889 


68779 


41221 


47 


14 


60066 


80014 


88802 




60106 


40696 


46 


16 


68062 


86018 


83886 


66114 


60428 


40672 


46 


10 


66786 


88215 


84461 


66689 


69740 


40961 


44 


17 


60418 


80582 


35029 


64971 


60068 


89982 


48 


r 18 


71000 


28100 


85500 


64410 


60884 


89616 


42 


10 


74^48 


26752 


86148 


68857 


60696 


89802 


41 


20 


7.76476 


12.28524 


8.86680 


11.68811 


8.61009 


11.88991 


40 


21 


78605 


21405 


87229 


68771 


61819 


88681 


89 


22 


80616 


19386 


87762 


62288 


61626 


88874 


88 


28 


82546 


17454 


88880 


61711 


61931 


88069 


87 


24 


84804 


15606 


88800 


61191 


08284 


87766 


86 


26 


86167 


1S88S 


89323 


00677 


62585 


S7466 


86 


26 


87871 


12129 


89882 


60166 


62834 


87166 


84 


27 


80610 


10400 


40884 


69666 


63181 


86869 


88 


28 


01080 


08011 


40880 


59170 


68486 


86574 


88 


20 


02618 


07887 


41321 


68679 


68718 


86282 


81 


80 


7.04066 


12.05014 


8.41807 


11.58198 


8.64000 


11.85991 


80 


81 


05510 


04490 


42287 


67718 


64298 


85702 


29 


32 


06880 


08111 


42762 


67288 


64686 


85415 


88 


83 


08825 


01776 


48282 


66768 


64870 


• 85180 


27 


84 


00522 


00478 


48696 


50804 


65154 


34846 


26 


86 


8.00781 


11.09219 


44156 


66844 


66486 


84566 


25 


86 


0-2004 


97996 


44611 


66889 


65716 


84286 


84 


87 


03104 


96806 


46061 


64989 


65998 


84007 


28 


88 


04869 


06047 


46607 


64498 


66809 


83781 


22 


80 


06461 


04510 


45048 


64052 


66648 


88457 


21 


40 


8.06581 


11.08410 


8.46885 


11.53615 


8.66816 


11.88184 


20 


41 


07668 


02847 


46817 


63188 


07087 


83918 


19 


42 


06700 


91300 


47246 


62766 


07350 


88644 


18 


48 


00782 


00278 


47600 


62881 


07624 


82370 




44 


10720 


80280 


48060 


61011 


07890 


82110 




46 


11606 


88304 


48505 


61495 


08154 


81846 




46 


12661 


87840 


48917 


51C88 


08417 


81588 




47 


18586 


86416 


49325 


60675 


06678 


81322 




48 


14500 


85500 


49729 


602n 


68088 


81062 




40 


16886 


84605 


60180 


49670 


60196 


80604 




60 


8.iem 


11.88727 


8.50527 


11.49478 


8.69458 


11.80547 




61 


17188 


88807 


60920 


49080 


60706 


80292 




62 


170T6 


80084 


61310 


48690 


69968 


80088 




68 


18804 


81196 


61696 


4R804 


70814 


29786 




64 


10616 


80384 


62070 


47921 


70465 


29586 




66 


20418 


79.%87 


624.'S9 


47541 


70714 


29886 




66 


21105 


78805 


528.% 


47166 


70962 


89088 




67 


21964 


78086 


53206 


46792 


71808 


88792 




68 


28720 


77280 


68578 


46422 


71468 


88647 




60 


28462 


76588 


53945 


46055 


71697 


28808 




60 


24192 


75806 


54806 


45692 


71940 


88060 




# 


Ciotan 


Ton 


Cotan 


Tan 


Ootan 


I^ 


/ 




»•• 




88« 




87» 



471 



TABLE III 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 




!• 


i 


!• 




B» 


/ 


Tftn 


Ootan 


Tan 


Cotan 


Tan 


Cotan 





8.71940 


11.88060 


8.84464 


11.15586 


8.04196 


11.05805 


60 


1 


79181 


87819 


84646 


15854 


94840 


05660 


69 


9 


79490 


97580 


84886 


15174 


94485 


06515 


58 


a 


79659 


27841 


86006 


14994 


94680 


05870 


57 


4 


78896 


27104 


86186 


14815 


94778 


06287 


66 


5 


73183 


86868 


85868 


14637 


94917 


05068 


55 


6 


73386 


26634 


' 65540 


14460 


95060 


04940 


64 


7 


73600 


86400 


85717 


14888 


96209 


04798 


68 


8 


78839 


26168 


85898 


14107 


95844 


04656 


68 


9 


74068 


95987 


86060 


13981 


95486 


04514 


51 


10 


8.74899 


11.95708 


8.66948 


11.18757 


8.95627 


11.04378 


60 


11 


74521 


95479 


86417 


13583 


95767 


04888 


49 


12 


74748 


85859 


86591 


18409 


95908 


04099 


48 


18 


74974 


86086 


86768 


18837 


96047 




47 


14 


75199 


94801 


66935 


13066 


96187 


08818 


46 


16 


75488 


245T7 


87106 


12894 


96385 


08675 


45 


10 


75645 


94365 


87877 


18788 


OttMA 
W0404 


03586 


44 


17 


75867 


94183 


87447 


18558 


96608 


08898 


48 


18 


78087 


83913 


87616 


18884 


96789 


03961 


42 


19 


76806 


98694 


8T785 


18915 


96877 


08188 


41 


80 


8.76685 


11.93475 


8.87068 


11.18047 


8.97018 


11.08967 


40 


SI 


76749 


98858 


88180 


11680 


97160 


08850 


89 


«8 


70968 


88049 


68387 


11718 


97986 


08715 


88 


88 


77178 


288-/7 


88458 


11547 


97421 


08579 


87 


S4 


77887 


89618 


88618 


11388 


07566 


08444 


86 


86 


77600 


99400 


88788 


11917 


97601 


08809 


85 


96 


77811 


99189 


88948 


11069 


97925 


08178 


84 


87 


78089 


91978 


89111 


10680 


97959 


08041 


88 


88 


7S889 


81788 


89874 


10726 


98099 


01906 


88 


89 


78441 


21659 


89487 


10568 


98986 


01775 


81 


80 


8.78849 


11.91851 


6.89598 


11.10409 


8.98858 


11.01649 


80 


81 


78865 


91145 


89760 


10340 


96490 


01510 


99 


88 


79061 


90039 


89080 


10080 


96689 


01878 


96 


83 


79966 


90784 


90080 


09920 


98768 


01947 


97 


84 


79470 


90530 


90840 


00760 


98884 


01116 


96 


85 


79678 


90397 


90399 


09601 


9901^ 


00965 


95 


86 


7987S 


90195 


90557 


09448 


99145 


00655 


84 


87 


80076 


19994 


90715 


09385 


9».'75 


00725 


88 


88 


80977 


19788 


90878 


09188 


99405 


00586 


99 


89 


80476 


19524 


91020 


08971 


99534 


00466 


91 


40 


8.80674 


11.19396 


8.91185 


11.08816 


8.99669 


11.00388 


80 


41 


80879 


19188 


91340 


08660 


9971)1 


00809 




48 


81068 


18939 


91495 


06505 


99919 


00081 




48 


81964 


18786 


91650 


08350 


9.00046 


10.99964 




44 


81450 


18541 


91808 


06197 


00174 


99888 




45 


81658 


18847 


01057 


OS048 


00801 


OQflOO 




46 


81846 


18154 


9-illO 


07890 


00487 


99573 




47 


, 89088 


17969 


98869 


07738 


00568 


99447 




48 


' 88980 


17770 


98414 


07586 


00679 


99881 




49 


89490 


17580 


98565 


07485 


00606 


99195 




60 


8.89610 


11.17890 


8.98716 


11.07884 


9.00930 


10.99070 




61 


89799 


17901 


98866 


07184 


01055 


96945 




69 


88987 


17018 


93016 


06964 


01179 


96891 




68 


83176 


16886 


93165 


06885 


01808 


98697 




64 


83361 


16689 


93313 


00687 


01427 


98578 




66 


83547 


16458 


93468 


06638 


01660 


96450 




66 


88739 


16868 


93609 


06391 


01673 


98387 




67 


88916 


16084 


93756 


06844 


01796 


98804 




68 


84100 


15900 


93003 


06097 


01918 


96089 




69 


84989 


15718 


94049 


05961 


08040 


07060 




60 


UAAttA 


15536 


94195 


05805 


08168 


97888 





# 


Cotan 


Tan 


Cotan 


Tap 


Cotan 


Tan 


f 




86» 




86« 




84« 



472 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


6 


• 


7- 




8<» 


/ 


Tan 


Ootan 


Tan 


Cotan 


Tan 


Cotan 





9.08168 




9.06914 


10.91086 


9.14780 


10.86880 


60 


1 


08888 


97717 


09019 


90981 


1487S 


86138 


69 


8 


08404 


97596 


09188 


90877 


14968 


86087 


68 


8 


085-J6 


97475 


09287 


90778 


15064 


84946 


67 


4 


03645 


97866 


•09830 


90670 


15146 


84856 


56 


6 


08766 


97iM 


09484 


90566 


15886 


64764 


55 


6 


08886 


97116 


09587 


90463 


15837 


84678 


64 


7 


08006 


96096 


09640 


90360 


15417 


64588 


58 


8 


03184 


96676 


09748 


9036Q 


15508 


64498 


58 





03848 


96758 


09845 


90155 


15508 


84408 


61 


10 


9.08861 


10.96689 


9.09947 


10.90058 


9.16688 


10.84818 


60 


11 


08479 


96581 


10049 


89951 


16777 


84838 


49 


18 


08597 


96406 


10150 


69850 


16867 


84138 


48 


18 


08714 


96386 


10358 


89748 


15966 


84044 


47 


14 


08888 


96168 


10863 


69647 


16046 


88954 


46 


16 


03948 


96058 


10454 


89646 


16186 


88866 


45 


16 


04065 


960a5 


10665 


89446 


16884 


88776 


44 


17 


01181 


96619 


10656 


69344 


16818 


83668 


48 


18 


04S97 


98703 


10756 


89344 


16401 


68599 


48 


19 


04418 


95587 


10856 


89144 


16489 


88611 


41 


90 


9.04528 


10.96473 


9.10956 


10.89044 


9.16677 


10.88483 


40 


81 


04648 


95867 


11056 


88944 


16666 


88385 


89 


88 


04758 


96848 


11155 


88846 


16758 


88847 


88 


2» 


04878 


95187 


11354 


88746 


16841 


88160 


87 


84 


04987 


95018 


11868 


88647 


16988 


88078 


86 


85 


05101 


94899 


11458 


88548 


17016 


82984 


36 


86 


06814 


94786 


11651 


' 88449 


17108 


63807 


84 


87 


06898 


94678 


11649 


68361 


17190 


88810 


88 


88 


05441 


94559 


11747 


88358 


17877 


83783 


88 


89 


05568 


94447 


11845 


88155 


17863 


82687 


31 


80 


9.06666 


10.94884 


9.]]948 


10.88057 


9.17460 


10.88560 


30 


31 


05778 


94388 


13040 


87960 


17686 


88464 


89 


88 


05890 


94110 


18188 


87868 


17638 


833718 


88 


88 


06003 


93998 


18335 


877185 


17708 


88898 


87 


84 


06118 


98887 


13388 


87668 


17794 


88206 


26 


85 


06884 


93776 


18438 


87578 


17880 


88180 


85 


86 


06835 


98666 


13585 


87475 


17965 


88085 


84 


87 


06445 


93555 


13631 


87379 


18051 


81949 


88 


88 


06556 


93444 


12717 


87383 


18186 


81864 


28 


89 


06666 


98334 


13813 


87187 


18381 


81779 


81 


40 


9.06r75 


10.93335 


9.18909 


10.87091 


9.18806 


10.81604 


90 


41 


06886 


a3n5 


13004 


86996 


18391 


81609 




48 


06904 


99006 


13009 


80901 


18476 


81586 




48 


07103 


93807 


18194 


86806 


18560 


81440 




44 


07',»11 


93789 


13389 


8G711 


18644 


81856 




45 


07330 


93680 


13884 


8C016 


18788 


81878 




46 


07488 


9357^ 


13478 


86538 


18818 


81188 




47 


07&36 


9-2404 


13573 


86437 


18896 


81104 




48 


07643 


9*2857 


13667 


663S8 


18979 


81081 




49 


07751 


93849 


13761 


86239 


19068 


80967 




BO 


9.07R68 


10.98148 


9.18854 


10.66146 


9.19146 


10.80854 




51 


07964 


93036 


18948 


86058 


19389 


80771 




68 


08071 


91939 


14041 


86950 


19313 


80688 




58 


08177 


9IP28 


14134 


85866 


19396 


80606 




64 


08888 


91717 


14837 


85778 


1047B 






55 


08889 


91011 


14320 


85680 


19561 


80439 




66 


08405 


91606 


14418 


85588 


19648 


80867 




67 


(moo 


91400 


14504 


85496 


19785 


80875 




68 


08705 


91305 


14597 


86403 


19807 


80193 




59 


08810 


91190 


14688 


85313 


19889 


80111 




60 


08914 


91086 


14780 


85380 


19971 


60039 




/ 


Golan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


i 




88* 




82» 




Sf 



473 



TABLE in.— LOG. TANGENTS AND COTANGENTS. 








e» 


10* 


IP 


$ 


Tan 


Ck>tan 


Tan 


Ootan 


Tan 


Cotan 





9.19971 


10.80029 


9.84638 


10.75868 


9.8S866 


10.71185 


60 


1 


80068 


79947 


84706 


75294 


88988 


71067 


69 


8 


90184 


79666 


84779 


75281 


89000 


71000 


58 


8 


80816 


79784 


84858 


75147 


80067 


70988 


57 


4 


20897 


79708 


84928 


75074 


89184 


70806 


56 


6 


80378 


79688 


85000 


75000 


89201 


70799 


55 


6 


20459 


79541 


85078 


74987 


89868 


70782 


54 


7 


20640 


79460 


85146 


74854 


89835 


70665 


58 


8 


206^1 


79379 


85819 


74:«1 


89409 


70506 


68 





80701 


79899 


85298 


74708 


89468 


70538 


51 


10 


9.20782 


10.79218 


9.25366 


10.74685 


9.29536 


10.70465 


60 


11 


20868 


79188 


85187 


74668 


29601 


70899 


49 


18 


80048 


79058 


85510 


74490 


80668 


70838 


48 


18 


81088 


789» 


85588 


74418 


89784 


70266 


47 


14 


81108 


78898 




74845 


89600 


70200 


46 


^16 


81188 


78818 


^85727 


74S73 


89866 


70184 


45 


21261 


78789 


^85799 


74801 


89988 


70068 


44 


17 


81341 


78659 


88871 


74189 


XVtftfO 


70008 


48 


16 


81480 


78580 


85848 


74057 


30064 


69936 


48 


19 


81499 


78501 


86015 


73985 


30180 


69870 


41 


90 


9.81578 


10.78488 


9.86066 


10.73914 


9.30195 


10.69605 


40 


8t 


81657 


78348 


88158 


73848 


80261 


697^ 


89 


83 


81786 


78264 


86289 


78771 


80886 


69674 


36 


28 


81814 


78186 


86801 


73699 


80391 


69609 


87 


84 


81808 


78107 


86878 


78688 


80457 


60548 


36 


85 


81971 


78089 


86448 


73557 


80538 


69478 


35 


86 


82049 


77961 


86514 


78486 


30587 


69418 


34 


87 


38127 


77878 


86586 


78415 


80658 


60348 


83 


88 


82805 


77795 


86655 


73845 


80717 


69288 


88 


89 


82888 


77717 


86726 


73274 


80788 


69818 


31 


80 


9.88361 


10.77689 


0.26797 


10.78808 


9.80846 


10.69154 


80 


81 


88488 


77568 


86867 


78183 


30911 


69069 


89 


88 


88516 


77484 


86967 


78068 


80975 


69025 


28 


88 


88598 


77407 


8700B 


7^998 


31040 


66960 


87 


84 


88670 


77380 


87078 


78928 


81104 


66896 


26 


85 


88747 


77858 


87148 


72858 


81168 


68888 


85 


86 


82884 


77176 


87818 


72788 


81388 


66767 


84 


87 


89901 


77099 


87888 


7^718 


81297 


68708 


88 


88 


82977 


77088 


87857 


72648 


81361 


68639 


28 


89 


81054 


76946 


27427 


72578 


81485 


68575 


81 


40 


9.28180 


10.76870 


9.87496 


10.78504 


9.31489 


10.68511 


80 


41 


88»6 


78794 


87566 


784a4 


81568 


68448 


19 


48 


83«8 


76717 


87685 


78365 


31616 


68384 


18 


48 


88869 


76641 


87704 


72896 


81679 


68881 


17 


44 


83435 


76565 


27778 


72827 


31748 


68257 


16 


45 


88510 


76490 


87848 


72158 


81806 


68194 


15 


46 


88586 


76414 


87911 


78069 


31870 


66180 


14 


47 


88661 


76339 


87980 


78020 


81983 


68067 


18 


48 


88787 


76268 


88049 


71951 


81996 


6H004 


18 


49 


88818 


76188 


88117 


71883 


38059 


67941 


11 


IM) 


9.83887 


10.76113 


9.28186 


10.71814 


9.38183 


10.67R78 


10 


51 


88968 


76088 


28254 


71746 


82185 


67815 


9 


58 


84037 


75963 


8R328 


71677 


82848 


67752 


8 


58 


84118 


75688 


88891 


71609 


8-^11 


6T6R9 


7 


54 


81186 


75814 


88459 


71541 


88873 


67627 


6 


56 


84261 


75789 


88587 


71473 


8843C 


67564 


5 


66 


84335 


75666 


88505 


71405 


82498 


67602 


4 


67 


84410 


75590 


88668 


71838 


82561 


67489 


8 


58 


84484 


75516 


88730 


71870 


82623 


67377 


8 


59 


84558 


75448 


88796 


71803 


82685 


67315 


1 


60 


84688 


75868 


88865 


71185 


82747 


67^58 





/ 


Cotan 


Tan 


Ootan 


Tan 


Cotan 


Tan 






80» 




79» 




!%• 



J 



474 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


W 


18* 


14* 


/ 


Tan 


Cotan 


Tan 


Ootan 


Tan 


Ootan 





0.82747 


10.67958 


0.86336 


10.63664 


0.80677 


10.60828 


60 


1 


82810 


6710O 


86304 


68606 


80781 


60269 


69 


s 


82879 


67128 


86468 


68548 


89785 


60215 


68 


8 


82088 


67067 


86600 


68401 


89838 


60162 


67 


4 


82005 


67006 


86566 


68484 


80099 


60108 


66 


6 


88067 


66048 


86624 


68376 


80045 


60056 


55 


8 


88110 


66881 


86681 


63310 


80000 


60001 


54 


7 


83180 


66880 


86788 


68363 


400B2 


59048 


53 


8 


88^2 


66758 


86705 


68205 


40106 


69894 


62 


• 


83308 


66697 


86852 


68148 


40150 


69811 


61 


10 


0.88885 


10.66686 


0.80000 


10.63001 


9.40212 


10.59788 


50 


11 


83426 


66574 


86066 


68084 


40206 


68784 


49 


18 


88487 


66618 


87098 


62077 


40310 


59681 


48 


18 


83548 


66459 


87060 


62020 


40872 


50628 


47 


14 


88800 


66891 


87187 


62868 


40425 


60575 


46 


15 


88670 


66830 


87108 


62807 


40478 


59522 


46 


16 


88781 


66260 


87950 


62750 


40681 


69469 


44 


17 


88799 


66206 


87306 


69604 


40584 


50416 


48 


18 


88858 


66147 


87968 


62687 


40686 


69364 


49 


10 


83018 


66087 


27410 


62581 


40680 


59811 


^ 


90 


0.88074 


10.66026 


0.87476 


10.62624 


0.40742 


10.59258 


91 


84034 


66066 


87532 


62468 


40795 


60906 


89 


S9 


84006 


66006 


87568 


62419 


40847 


50158 


88 


88 


84155 


66845 


87644 


62856 


40900 


50100 


87 


94 


84215 


65785 


87700 


62300 


40968 


60048 


86 


95 


84276 


66794 


87766 


62244 


41006 


68096 


86 


96 


84886 


65664 


87812 


62188 


41057 


68948 


84 


97 


84886 


6S604 


87866 


62189 


41109 


68891 


88 


98 


84456 


65544 


87024 


62076 


41161 


58889 


89 


99 


84516 


65484 


87080 


62020 


41914 


58786 


81 


80 


0.84576 


10.65424 


0.88036 


10.61965 


9.41266 


10.58784 


80 


81 


84635 


65365 


88001 


61000 


41818 


58682 


29 


39 


84605 


65806 


88147 


61858 


41870 


58680 


28 


88 


84766 


66945 


88209 


61798 


41422 


66578 


97 


84 


84814 


65186 


88257 


61748 


41474 


58526 


96 


85 


84874 


65126 


88813 


61687 


41626 


66474 


95 


80 


84088 


66067 


88368 


61689 


41578 


68422 


94 


87 


84909 


66006 


88423 


61577 


41629 


68871 


98 


88 


85051 


64040 


88470 


61621 


41681 


58819 


99 


99 


86111 


64880 


88534 


61466 


41738 


6ai87 


91 


40 


0.85170 


10.64880 


0.38589 


10.61411 


9.41784 


10.58916 


80 


41 




64771 


88044 


61856 


41836 


56164 


19 


42 


86288 


64712 


88600 


61801 


41887 


68118 


18 


48 


85847 


64658 


88754 


61246 


41989 


58061 


17 


44 


85406 


64605 


88808 


61109 


41990 


66010 


16 


45 


85464 


64536 


88868 


61187 


42041 


57959 


16 


46 


85598 


64477 


88918 


61089 


42098 


57907 


14 


47 


85581 


64410 


88079 


61028 


42144 


57866 


18 


48 


85640 


64860 


30037 


60973 


42195 


57805 




40 


85608 


64809 


30069 


60018 


42246 


57764 




50 


0.85757 


10.64948 


0.80186 


10.60864 


9.42297 


10.57708 




51 


85815 


64185 


89190 


60610 


42348 


67652 




59 


85878 


64127 


89246 


60765 


42399 


57601 




68 


85081 


64060 


80299 


60701 


42450 


57550 




54 


85960 


64011 


39863 


60647 


42501 


57499 




55 


86047 


63958 


39407 


60593 


42559 


57448 




56 


86106 


63805 


89461 


60A30 


4260R 


67397 




57 


86168 


63837 


39515 


60485 


42658 


57347 




58 


86221 


63770 


80M» 


60431 


42704 


57296 




50 


86279 


68721 


80698 


60877 


42756 


67946 




60 




63664 


89677 


60328 


42805 


57195 




/ 


Ootan 


Ton 


Cotan 


Tan 


Ootan 


Tan 


• 




IV 




76» 




76<» 



475 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



* 


16« 


16« 




I7« 


/ 


Tan 


Cotan 


Tftn 


Ck>tan 


Tan 


Cotan 





9.43805 


10.57196 


9.46750 


10.54260 


9.48684 


10.51466 


60 


1 


42866 


67144 


4OT97 


64206 


48579 


61421 


69 


2 


42006 


67094 


46846 


54156 


48624 


61876 


58 


8 


42957 


57048 


45608 


64106 


48669 


61831 


W 


4 


48007 


66093 


46940 


64060 


48714 


61286 


56 


6 


48057 


56948 


46967 


64018 


48760 


61211 


66 


8 


48108 


56898 


46066 


63965 


48804 


61196 


54 


7 


48158 


66842 


46088 


63918 


48849 


61151 


63 


8 


48206 


66792 


46180 


68870 


48894 


61106 


68 


9 


48258 


66742 


46177 


58828 


48989 


61061 


51 


10 


9.48806 


10.56688 


9.46884 


10.68776 


9.48984 


10.51016 


60 


11 


48858 


66642 


46871 


68729 


49029 


60971 


49 


18 


48406 


66598 


46819 


68681 


49078 


50927 


48 


18 


48468 


66648 


46866 


68684 


49118 


50688 


47 


14 


48S06 


56198 


46418 


58587 


49168 


60837 


46 


16 


48558 




46460 


68540 


49207 


60793 


45 


16 


48007 


66393 


46607 


68498 


40858 


60748 


44 


17 


48697 


66348 


46664 


68446 


49296 


60704 


48 


18 


48707 


66298 


46601 


68399 


49841 


50659 


42 


19 


487S6 


56244 


46648 


58852 


49885 


60616 


41 


SO 


9.43806 


10.56104 


9.46694 


10.63306 


9.49480 


10.60670 


40 


SI 


48866 


56145 


46741 


63269 


49474 


60626 


89 


ss 


48905 


66095 


46788 


68218 


49619 


60481 


88 


28 




66046 


46835 


63166 


40668 


60437 


87 


24 


44004 


55996 


46881 


63119 


49607 


60883 


86 


25 


44068 


55047 


46928 


68078 


49668 


60348 


86 


20 


44102 


65696 


46975 


68085 


49696 


60804 


84 


27 


44151 


65649 


47021 


6^70 


49740 


60260 


88 


28 


44201 


66799 


47068 


62932 


497B4 


60216 


82 


29 


44260 


65750 


47114 


69886 


49^8 


60178 


81 


90 


9.44209 


10.55701 


9.47160 


10.62840 


9.49678 


10.50138 


80 


81. 


44848 


56658 


47207 


52798 


49916 


60084 


89 


89 


44397 


65608 


47263 


62747 


49960 


60040 


88 


88 


44446 


66554 


47299 


62701 


60004 


49996 


27 


84 


44495 


65506 


47846 


62654 


60048 


49958 


86 


86 


41544 


56466 


47892 


6-J606 


60008 


49906 


25 


86 


44592 


65108 


47488 


62568 


60186 


49864 


84 


87 


44641 


65359 


47484 


52616 


60180 


49620 


88 


88 


44690 


55810 


47680 


62470 


50228 


49777 


28 


89 


44788 


65262 


47676 


62424 


50207 


49733 


81 


•40 


9.44787 


10.65218 


9.47628 


10.58378 


9.60311 


10.49669 


80 


41 


44686 


66164 


47668 


62338 


60656 


49645 


19 


42 


44884 


65116 


47714 


62286 


60806 


49602 


18 


48 


44988 


56067 


47780 


. 68240 


50448 


40556 


17 


44 


44961 


66019 


47806 


62194 


60486 


49615 


16 


45 


45029 


64971 


47852 


62148 


60629 


49471 


16 


46 


45078 


54922 


47897 


62108 


60578 


49428 


14 


47 


45126 


64874 


47948 


68057 


60616 


49884 


18 


48 


45174 


54886 


47989 


52011 


60660 


49841 


18 


49 


45222 


54778 


48036 


61966 


60708 


49297 


11 


SO 


9.46271 


10.54729 


9.48060 


10.61930 


9.60746 


10.49264 


10 


51 


45319 


64681 


4«15» 


61874 


60789 


49211 


9 


52 


45867 


64683 


48171 


618«9 


60688 


49167 


6 


58 


46416 


64586 


48217 


61788 


80876 


49124 


7 


54 


45468 


54537 


48262 


61788 


60919 


49081 


6 


55 


46511 


64489 


48307 


51698 


60962 


49088 


6 


56 


45559 


64441 


48358 


61647 


61006 


48995 


4 


57 


45606 


54394 


48396 


61608 


61048 


48952 


8 


58 


46654 


54346 


48448 


61557 


61092 


48906 


8 


59 


467G2 


64298 


48489 


61611 


61186 


48865 


1 


60 


45750 


54250 


48584 


51466 


61178 


48822 





/ 


Ootan 


Tan 


Ootan 


Tan 


Cotan 


Tan 


/ 




740 




78* 




78« 



476 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



/ 


18» 


19» 




20» 


/ 


Tad 


Cotan 


Tan 


Cotan 


Tan 


Cotan 




9.61178 


10.48822 


9.58697 


10.46303 


9.66107 


10.48893 


60 




61221 


48779 


68788 


46262 


66146 


43854 


69 




51264 


48736 


63779 


46221 


66186 


48815 


58 




51306 


48694 


68830 


46180 


66324 


4.'fi76 


67 




61349 


48651 


63S61 


46189 


66264 


43736 


66 




61392 


48608 


63902 


46096 


66303 


48697 


56 




61435 


48565 


53943 


46057 


56342 


43658 


64 




61478 


48522 


53984 


46016 


56381 


43619 


53 




61520 


48480 


54035 


45976 


66420 


48580 


62 




51663 


48437 


54066 


45985 


56459 


48541 


61 




9.51606 


10.48394 


9.54106 


10.45894 


9.56496 


10.48602 


60 




61648 


48352 


64147 


45853 


66587 


48468 


49 




61691 


48309 


64187 


46813 


66576 


48424 


48 




61734 


48266 


64238 


45772 


66615 


43886 


47 




61776 


48224 


64269 


45781 


66654 


4MS46 


46 




61819 


48181 


64309 


45691 


66693 


48307 


45 




61861 


48139 


64360 


45650 


66732 


43268 


44 




61908 


48097 


54390 


46610 


66771 


43239 


48 


18 


61946 


48054 


64431 


45569 


66810 


48190 


42 


19 


61968 


48012 


64471 


45629 


66849 


48151 


41 


90 


9.62031 


10.47969 


9.64512 


10.45488 


9.66887 


10.43118 


40 


21 


62073 


47927 


64652 


45448 


66926 


48074 


89 


S2 


62115 


47885 


64598 


46407 


66965 


43085 


88 


23 


62157 


47843 


54633 


46367 


67004 


42996 


^ 


24 


62200 


47800 


64678 


45827 


67042 


43958 


86 


25 


62242 


47768 


64714 


45386 


67081 


42919 


35 


26 


62284 


47716 


64754 


45246 


67120 


43880 


84 


27 


63326 


47674 


64794 


45306 


67158 


42842 


38 


28 


62368 


47632 


64835 


45166 


67197 


42803 


82 


29 


52410 


47590 


64876 


45125 


67235 


42766 


31 


80 


9.52452 


10.47548 


9.54915 


10.45085 


9.67274 


10.42726 


90 


81 


62494 


47606 


54955 


45046 


67312 


42688 


29 


88 


62536 


47464 


64995 


45006 


67^1 


42649 


28 


88 


62578 


47422 


66086 


44965 


67389 


42611 


27 


84 


6S620 


47880 


65075 


44926 


67428 


42572 


26 


85 


62661 


47389 


65116 


44886 


67466 


42584 


85 


86 


52706 


47:»97 


66155 


44846 


57604 


42496 


84 


37 


62746 


47265 


55195 


44806 


67543 


42467 


28 


88 


52787 


47218 


55235 


44765 


57581 


42419 


82 


89 


62829 


47171 


65375 


44725 


67619 


42881 


21 


40 


9.52870 


10.4n80 


9.55815 


10.44685 


9.67668 


10.42342 


20 


41 


52912 


47068 


65836 


44646 


67696 


43304 


19 


42 


52953 


47047 


55896 


44605 


67734 


42266 


18 


43 


62996 


47006 


65434 


44566 


67772 


42326 


17 


44 


53087 


46968 


65474 


446M 


67810 


42190 


16 


45 


58078 


46932 


66514 


44486 


67849 


42151 


16. 


46 


53120 


46880 


65554 


44446 


67887 


42113 


14 


.47 


53161 


46ft39 


65593 


44407 


67925 


42075 


18 


48 


63202 


46798 


65633 


44867 


67968 


42037 


12 


49 


53244 


46756 


65C73 


44327 


58001 ' 


41999 


11 


SO 


9.532a'S 


10.46715 


9.55712 


10.44388 


9.68039 


10.41961 


10 


61 


53827 


46678 


66752 


44248 


68077 


41928 


9 


62 


63868 


46682 


65791 


44309 


68116 


41886 


8 


53 


53409 


46591 


65831 


44169 


68158 


41847 


7 


54 


53460 


46550 


65870 


44180 


68191 


41809 


6 


65 


63492 


46506 


65910 


44090 


68229 


41771 


6 


66 


53533 


46467 


6.5949 


44051 


58367 


41783 


4 


67 


68574 


46426 


65089 


44011 


68804 


41696 


8 


68 


&3G15 


46386 


66028 


48972 


68842 


41658 


8 


69 


63656 


46344 


66067 


48933 


68380 


41630 


1 


60 


63697 


46303 


56107 


43893 


68418 


41583 





/ 


Ootan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


# 




7f 




70* 




«9* 



477 



TABLE in.— LOG. TANGENTS AND COTANGENT& 



/ 


91« 


2SO 


28- 


f 


Tan 


Gotan 


Tan 


Cotan 


Tan 


Cotan 







0.68418 


10.41589 


9.60641 


10.89850 


0.69785 


10.87815 


60 


1 


68465 


41545 


60677 


89898 


69820 


87180 


69 


9 


68498 


41507 


60714 


89286 


69866 


87146 


58 


8 


68681 


41469 


60750 


89250 


68890 


87110 


57 


4 


6f&60 


41481 


60786 


80914 


69996 


87074 


50 


6 


68606 


41894 


60898 


89177 


62961 


87089 


55 


6 


68644 


41856 


60650 


89141 


69996 


87004 


54 


7 


68661 


41819 


60695 


89105 


68031 


86909 


58 


8 


66710 


41981 


60981 


89069 


68066 


86984 


59 





IS«KI 


41948 


60967 


89088 


68101 


86899 


61 


10 


0.68704 


10.41906 


9.61004 


10.88996 


0.63185 


10.86866 


50 


n 


68889 


41168 


61040 


88960 


68170 


86880 


• 49 . 


18 




41181 


61076 


88994 


63906 


86795 


48 


18 


68907 


41098 


61119 


88888 


68240 


86760 


47 


14 


68844 


41056 


61148 


88869 


68976 


86735 


46 


16 


68081 


41019 


61184 


88816 


68310 


86690 


46 


16 


60010 


40981 


61990 


88780 


68345 


86655 


44 


17 


60066 


40944 


61956 


88744 


63879 


86691 


48 


18 


59004 


40906 


61999 


88706 


68414 


86586 


49 


19 


60181 


40860 


61898 


88679 


63449 


86561 


41 


90 


0.60168 


10.40889 


9.61864 


10.88686 


9.68484 


10.86516 


40 


SI 


69905 


40796 


61400 


88600 


63519 


86481 


30 


SS 


69948 


40757 


61486 


88564 


63558 


86447 


88 


28 


68880 


40790 


61479 


88528 


68588 


86412 


87 


94 


60817 


40688 


61506 


88488 


68698 


86377 


86 


86 


60854 


40646 


61544 


88456 


63657 


86318 


85 


96 


60891 


40600 


61579 


88491 


68699 


86806 


84 


97 


50490 


40571 


61616 


88385 


68796 


86974 


88 


S8 


50406 


40634 


61651 


88349 


68761 


86939 


39 


90 


60508 


40497 


61687 


88818 


68796 


86204 


81 


80 


0.59640 


10.40460 


9.61799 


10.88278 


0.63880 


10.86170 


80 


81 


60577 


40498 


61758 


88949 


68866 


86135 


99 


89 


59614 


40886 


61794 


88206 


63899 


86101 


98 


88 


60651 


40849 


61830 


88170 


68934 


86066 


97 


84 


69688 


40819 


61M6 


88186 


63068 


wm 


96 


86 


50785 


40976 


61901 


88090 


64008 


85997 


95 


86 


50708 


40938 


61936 


88064 


64037 


85963 


94 


87 


60790 


40:201 


61979 


88028 


640?2 


85928 


98 


88 


60886 


40165 


69008 


87998 


64106 


35894 


99 


88 


50678 


40198 


69048 


87957 


64140 


85860 


91 


40 


9.69900 


10.40091 


9.69079 


10.87921 


9.64176 


10.85826 


90 


41 




40054 


69114 


87886 


64909 


85791 


19 


49 


50968 


40017 


69150 


87860 


64243 


85767 


18 


48 


60010 


89961 


69186 


87815 


64278 


35729 


17 


44 


60056 


89944 


69991 


87779 


64819 


35688 


16 


46 


60008 


89907 


62956 


87744 


64346 


85654 


15 


46 


60180 


89670 


69299 


87708 


64381 


85619 


14 


47 


60166 


89884 


68397 


87678 


64415 


85585 


18 


48 


60908 


89797 


62869 


87688 


64449 


85551 


19 


40 


60940 


89760 


69398 


87609 


64488 


85517 


11 


60 


9.60976 


10.89794 


9.62438 


10.37567 


9.64517 


10.85488 


10 


61 


60818 


80667 


69468 


87589 


64559 


85148 


9 


69 


00648 


89651 


69504 


87496 


64586 


35414 


8 


68 


60386 


88614 


69SS9 


87461 


64620 


85880 


7 


64 


60429 


89578 


69574 


87496 


64664 


85846 


6 


66 


60459 


89641 


69609 


87391 


64688 


85319 


6 


66 


60496 


89506 


69645 


87855 


64ra} 


85278 


4 


67 




89468 


69680 


87820 


64756 


35944 


8 


66 


60668 


89489 


62715 


37285 


64790 


85910 


9 


60 


60605 


89895 


69750 


37850 


64824 


85176 


1 


60 


60641 


89369 


69785 


87215 


64858 


85149 





/ 


Ootan 


Tan 


€k>tan 


Tan 


Cotan 


Tan 


/ 




68* 




67* 




ee* 



478 



TABLE m 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 


24- 


«6» 


26" 


i 


Tan 


Coton 


Tan 


Cotan 


Tan 


Cotan 





9.64858 


10.35142 


9.66867 


10.33138 


9.68818 


10.81182 


60 


1 


64888 


35108 


66900 


88100 


68850 


81150 


60 


s 


64986 


85074 


66938 


88067 


68882 


81118 


68 


8 


64960 


85040 


66966 


88084 


68914 


81086 


67 


4 


64994 


85006 


66000 


88001 


68046 


81054 


56 


5 


65038 


84972 


6703S 


8S968 


68978 


81022 


65 


6 


65068 


84938 


67065 


82935 


69010 


10990 


54 


7 


65096 


84904 


67098 


82902 


60042 


80958 


53 


8 


66180 


84870 


67131 


82860 


69074 


80986 


62 


9 


65164 


84886 


67168 


82887 


69106 


80894 


61 


10 


9.65397 


10.84803 


9.67196 


10.82804 


9.60188 


10.80662 


60 


11 


66281 


81769 


67229 


32771 


69170 


80630 


49 


12 


65865 


84785 


67268 


82788 


69202 


80796 


48 


18 


6S299 


84701 


67295 


82706 


60284 


30766 


47 


14 


65383 


84667 


67327 


8267B 


60266 


80734 


46 


15 


65866 


84634 


67860 


82640 


69298 


8Or02 


45 


10 


65400 


84600 


67898 


82607 


69829 


80671 


44 


17 


65484 


84566 


67426 


82574 


69361 


30689 


48 


18 


65467 


84538 


67458 


88542 


69303 


806O7 


42 


19 


65501 


84499 


67491 


82509 


69425 


80576 


41 


80 


9.66585 


10.84465 


9.67524 


10.82476 


9.69457 


10.80548 


40 


81 


WXlOO 


84483 


67556 


82444 


69488 


80612 


80 


83 


65608 


84396 


67580 


82411 


60620 


80480 


88 


88 


656S6 


84364 


67628 


82378 


69562 


80448 


87 


84 


65669 


84381 


67654 


82846 


69684 


80416 


86 


85 


65708 


84297 


67687 


82318 


69615 


80885 


85 


86 


657S6 


84264 


67719 


82881 


69647 


80858 


84 


27 


66770 


84280 


67758 


82248 


69679 


80621 


88 


88 


66803 


84197 


67785 


82215 


69710 


80290 


88 


89 


65687 


84163 


67817 


82188 


69742 


80258 


81 


80 


9.66870 


10.84180 


9.67850 


10.82160 


9.69774 


10.80226 


80 


81 


65904 


34096 


67882 


32118 


69805 


80195 


89 


38 


66987 


84063 


67916 


82065 


69687 


80168 


88 


U 


65071 


84029 


67947 


82058 


60868 


80189 


27 


84 


66004 


83996 


67960 


82020 


69900 


80100 


26 


85 


66038 


83962 


68012 


81968 


69982 


80068 


25 


W 


66071 


83929 


68044 


81956 


69968 


80087 


24 


87 


66104 


88896 


68077 


81928 


69995 


80006 


28 


88 


66138 


83862 


68109 


81891 


70026 


29974 


22 


89 


66171 


838^'9 


68142 


31858 


70058 


29942 


21 


40 


0.66804 


10.88796 


9.68174 


10.31826 


9.70089 


10.20911 


80 


41 


66288 


83762 


66206 


81794 


70121 


29879 




48 


66^1 


83729 


68239- 


81761 


70152 


29848 




48 


66804 


88696 


68271 


. 31729 


70184 


29616 




44 


66887 


88668 


68308 


81607 


70215 


29785 




45 


66871 


83629 


68886 


81664 


70247 


29758 




46 


66404 


a3606 


68368 


81632 


70278 


29722 




47 


66487 


Sa563 


68400 


81600 


70809 


29691 




48 


66470 


8:4530 


68438 


81568 


70841 


29659 




49 


66608 


83497 


68465 


81585 


70372 


29628 




50 


9.66687 


10.38463 


9.68497 


10.31503 


9.70404 


10.29596 




51 


66670 


88430 


6^529 


81471 


70485 


29665 




58 


66608 


83397 


6a^61 


81489 


70466 


29584 




58 


66636 


38864 


6KS93 


81407 


70498 


29502 




54 


66600 


88831 


68626 


81374 


70529 


29471 




55 


66708 


a3296 


686.'^ 


81342 


70560 


29440 




56 


667ai 


83265 


68600 


81310 


70598 


29406 




67 


66768 


83232 


68728 


81278 


70628 


29877 




58 


66801 


88199 


68754 


81246 


70654 


89846 




59 


66834 


88166 


68786 


81214 


70685 


29315 




60 


66867 


83183 


68818 


81182 


70717 


29888 




/ 


Ck>taD 


Tao 


Cotan 


Tan 


Cotan 


Tan 


# 




86* 




64« 




68* 



479 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



# 


270 


28« 


29- 1 


/ 


Tan 


Cotan 


Tan 


Cotan 


Ttui 


Cotan 





9.70717 


10.89888 


9.72567 


10.87438 


9.74375 


10.86625 


60 


1 


70748 


89858 


78598 


87408 


74406 


86596 


69 


8 




89381 


78688 


27878 


74485 


85666 


68 


8 


70610 


89190 


72659 


27841 


74465 


25586 


67 


4 


70841 


89159 


72889 


27311 


74494 


86506 


66 


6 


70678 


89187 


78780 


27880 


74624 


86470 


66 


6 


70904 


89096 


7S75C 


27860 


74554 


86446 


64 


7 


70986 


89066 


78780 


27220 


745R8 


95417 


68 


8 


70966 


89084 


78811 


27189 


74618 


85367 


62 





70997 


89008 


72S41 


27159 


74643 


26857 


61 


10 


9.71038 


10.88979 


9.7S87S 


10.2n88 


9.74678 


10.86387 


60 


11 


71069 


S89I1 


72908 


27096 


74708 


85296 


49 


12 


71090 


88910 


78988 


27068 


74738 


25868 


48 


18 


71181 


88879 


78968 


87087 


74768 


85838 


47 


14 


71168 


88847 


78998 


27007 


74791 


26809 


46 


16 


71184 


88816 


78028 


26977 


74881 


25179 


45 


16 


71215 


88785 


78064 


26046 


74851 


85149 


44 


17 


71S46 


88754 


73084 


26916 


74880 


85180 


48 


18 


71877 


88788 


78114 


26886 


74910 


85090 


42 


19 


71808 


88698 


73144 


86866 


74980 


25061 


41 


90 


9.71389 


10.88661 


9.78175 


10.26826 


9.74969 


10.86081 


40 


81 


71870 


86680 


78206 


26795 


74996 


86002 


89 


88 


71401 


88599 


78386 


26765 


76028 


84978 


86 


88 


71481 


88669 


78986 


26736 


76066 


84948 


87 


84 


7146$ 


88588 


78895 


86705 


76087 


84918 


86 


86 


71498 


88607 


73328 


26674 


76117 


84888 


86 


86 


71684 


88476 


78856 


96644 


75146 


84654 


84 


87 


71555 


88446 


78886 


26614 


75176 


24884 


88 


88 


71686 


88414 


78416 


26584 


76806 


94795 


88 


89 


71617 


88888 


78446 


26664 


76885 


84766 


81 


80 


9.71648 


10.28868 


9.78476 


10.36524 


9.75864 


10.94786 


80 


81 


71679 


86881 


78607 


26498 


75894 


24706 


89 


88 


71709 


88891 


78687 


26468 


75888 


24677 


28 


88 


71740 


88360 


78667 


28488 


75368 


24647 


87 


84 


71T71 


88889 


78697 


26408 


75388 


84618 


86 


85 


71808 


88196 


78687 


26373 


75411 


24569 


85 


36 


71833 


88167 


73867 


26348 


75441 


84559 


84 


87 


71863 


88187 


73667 


26318 


75470 


84580 


83 


88 


71884 


88106 


78717 


26288 


75500 


84500 


88 


89 


71985 


88076 


78747 


26358 


75529 


84471 


81 


40 


9.71956 


10.88045 


9.78777 


10.26828 


9.75568 


10.84448 


80 


41 


71966 


86014 


78807 


26198 


75566 


84418 




48 


79017 


87968 


78887 


86163 


75617 


94388 




48 


78048 


87958 


78867 


86188 


75647 


9435:) 




41 


78078 


87928 


78897 


86103 


75676 


84824 




46 


78109 


87891 


78987 


86078 


75705 


84895 




46 


78140 


87860 


78967 


26043 


75786 


84866 




47 


78170 


87880 


73987 


26018 


75764 


94886 




48 


7S901 


87799 


74017 


26983 


75798 


94807 




49 


78881 


87769 


74047 


25968 


75828 


94178 




60 


9.78888 


10.87736 


9.74077 


10.85988 


9.76868 


10.84148 




61 


78898 


87707 


74107 


26698 


76881 


»4119 




68 


78888 


87677 


74137 


26868 


7B910 


' 94090 




68 


78854 


87646 


74166 


85834 


76939 


84061 




64 


78384 


87616 


74196 


86804 


75969 


84081 




66 


78416 


87586 


74886 


25774 


76996 


94008 




56 


78446 


87556 


74856 


25744 


78087 


88978 




67 


78476 


87584 


74286 


25714 


76056 


88944 




68 


79606 


87494 


74316 


85684 


76066 


28914 




59 


78587 


87468 


74845 


25655 


76116 


23685 




60 


78667 


87488 


74875 


25626 


76144 


23666 




/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




8«» 




61« 




80* 



48o 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


«©• 


81* 




18* 


# 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.76144 


10.23856 


9.77877 


10.28128 ' 


9.79579 


10.80481 


60 


1 


76178 


28827 


77906 


82094 


79807 


80888 


69 


2 


7<»02. 


28798 


77986 


88065 


79686 


80866 


68 


8 


76281 


88769 


77963 


82087 


79668 


80887 


67 


4 


76261 


28789 


77998 


88006 


79081 


80809 


66 


6 


76290 


28710 


78080 


81980 


79719 


80S61 


66 


e 


76819 


S9681 


78049 


81961 


79747 


80858 


64 


7 


76848 


28G52 


78077 


81988 


79776 


80884 


68 


8 


76877 


28628 


78106 


81894 


79604 


80196 


68 


9 


76406 


88594 


78186 


81866 


79688 


80168 


51 


10 


9.76485 


10.28566 


9.78168 


10.81887 


9.79660 


10.20140 


60 


11 


764G4 


28536 


78198 


81808 


79688 


80118 


49 


IS 


76498 


28507 


78820 


81780 


79916 


80064 


48 


18 


76522 


28478 


78249 


81751 


79944 


80066 


47 


14 


76651 


88449 


78277 


81728 


79978 


80088 


46 


15 


76580 


88480 


78806 


81694 


80000 


80000 


46 


16 


76609 


88801 


78884 


81666 


80088 


19978 


44 


17 


76639 


83361 


78868 


81687 


80066 


19944 


48 


18 


76668 


88882 


78391 


81609 


80084 


19916 


« 


19 


76697 


88308 


78419 


81681 


80118 


19868 


41 


90 


9.7G7S5 


10.28875 


9.78448 


10.81558 


9.80140 


10.19860 


40 


21 


76754 


88246 


78476 


81524 


80168 


10888 


89 


28 


76788 


88817 


78506 


81495 


80196 


19606 


86 


24 


76812 


88188 


78688 


81467 


80288 


19T77 


87 


24 


76841 


88169 


78568 


81438 


80851 


19749 


86 


23 


76870 


88180 


78690 


81410 


80279 


19781 


86 


26 


76899 


28101 


78618 


81888 


80807 


19608 


84 


27 


76928 


88072 


78647 


81858 


80886 


19666 


88 


28 


76967 


83048 


78675 


81826 


80668 


19687 


88 


29 


76986 


88014 


78704 


81896 


80891 


19609 


81 


80 


9.rroi6 


10.82985 


9.7B738 


10.81868 


9.80410 


10.19581 


80 


81 


77044 


88956 


78760 


81840 


80447 


19668 


89 


SS 


77078 


82927 


T8789 


81811 


80474 


19686 


88 


88 


77101 


82899 


78817 


81188 


80508 


19498 


87 


84 


77180 


82870 


78846 


81156 


60580 


19470 


80 


85 


77169 


82841 


78874 


81186 


80558 


10448 


85 


86 


77188 


22818 


78908 


81098 


60586 


19414 


84 


37 


77217 


827^ 


78930 


81070 


80614 


19986 


88 


88 


77^246 


88754 


78959 


81041 


60648 


19886 


88 


89 


7T274 


827^^ 


78987 


81018 


80609 


19881 


81 


40 


9.77808 


10.22697 


9.79015 


10.80966 


9.80007 


10. 19808 


80 


41 


77882 


22668 


79048 


80057 


80786 


i9sm 


19 


42 


77361 


22639 


79078 


80988 


80768 


19247 


18 


43 


77S90 


22610 


79100 


20900 


80781 


19819 


17 


44 


rr4i3 


ZiSSi 


79128 


80878 


80606 


19198 


16 


45 


77447 


22553 


79156 


80644 


80886 


19164 


16 


46 


77476 


83524 


79185 


80815 


60864 


19186 


14 


47 


77506 


22495 


79218 


80787 


80898 


19106 


18 


48 


77588 


82467 


79241 


80750 


60919 


19061 


18 


49 


77662 


22488 


79269 


80781 


60947 


19068 


11 


60 


9.77691 


10.38409 ^ 


9.79297 


10.80708 


9.80975 


10.19085 


10 


61 


77619 


22881 


79326 


80074 


81008 


18097 


9 


B» 


77648 


82368 


79354 


80646 


81080 


18970 


6 


58 


77677 


22828 


79882 


80618 


81058 


18948 


7 


54 


77706 


22204 


79410 




81066 


18914 


6 


55 


77784 


22266 


79488 


80668 


81118 


18887 


5 


66 


T7763 


82287 


79466 


80684 


61141 


18859 


4 


67 


77791 


82209 


79496 


80606 


81160 


18881 


8 


58 


77820 


82180 


79523 


80477 


81196 


18804 


8 


59 


77849 


82151 


79561 


80449 


61224 


18778 


1 


60 


77877 


82128 


79679 


80421 


81858 


16748 





/ 


Cotan 


Tan 


Ck>tan 


Tan 


Cotan 


Tan 


/ 




«9<» 




68* 




»7» 



48i 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



/ 


SB* 


84«» 




86« 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.81858 


10.18748 


9.82899 


10.17101 


9.84528 


10.164T7 


60 


1 


81.279 


18781 


82986 


17074 


84550 


1R450 


60 


8 


81807 


18698 


88968 


17047 


84576 


15484 


68 


8 


81885 


18665 


82960 


17020 


84608 


16397 


67 


4 


81868 


18688 


88006 


16998 


84680 


15870 


56 


5 


81880 


18610 


83035 


16965 


84657 


15848 


55 


6 


81418 


18588 


88002 


16088 


84684 


15316 


54 


7 


81446 


18555 


88069 


16911 


84711 


15289 


53 


8 


81478 


18587 


88117 


16883 


84788 


16862 


68 


9 


81600 


18500 


88144 


16866 


84764 


15236 


51 


10 


9.81588 


10.18478 


9.88171 


10.16820 


9.84791 


10.15809 


60 


11 


81566 


18444 


83198 


16608 


84818 


15188 


49 


18 


81588 


18417 


88-225 


16775 


84845 


15155 


48 


18 


81611 


18389 


83268 


16748 


8487S 


15188 


47 


14 


81688 


18888 


88880 


16780 


84899 


15101 


46 


16 


81666 


18834 


88307 


16693 


84985 


15075 


45 


16 


81698 


18307 


83384 


16666 


84958 


16048 


44 


17 


8173a 


18279 


88361 


16639 


84979 


15081 


48 


18 


81748 


18268 


83388 


16618 


85006 


14994 


42 


19 


81776 


18824 


88415 


16586 


85033 


14967 


41 


80 


9.81808 


10.18197 


9.88148 


10.16658 


9.86060 


10.14941 


40 


81 


81881 


18169 


83470 


16530 


85086 


14914 


89 


88 


81868 


18143 


83497 


16503 


86118 


14887 


38 


88 


81880 


18114 


88S84 


16476 


86140 


14860 


87 


84 


81918 


18087 


83551 


16449 


85166 


14884 


86 


86 


81911 


18050 


88578 


16428 


86103 


14807 


85 


88 


81968 


18098 


88606 


16395 


85880 


14780 


84 


87 


81990 


18004 


88683 


16368 


86847 


14753 


33 


88 


88088 


\7vn 


88659 


16341 


85278 


14727 


82 


89 


88051 


17949 


83686 


16814 


85300 


14700 


31 


80 


9.88078 


10.17988 


9.83n3 


10.16887 


9.66887 


10.14678 


80 


31 


82106 


17894 


83740 


18860 


86354 


14646 


SO 


88 


88188 


17847 


88768 


16232 


86880 


14620 


28 


88 


82161 


17889 


88795 


16205 


85407 


14503 


27 


84 


88188 


17818 


88828 


16178 


85434 


14566 


86 


85 


8^15 


1T786 


88849 


16151 


85460 


14540 


25 


80 


88848 


17757 


88876 


16124 


85487 


14518 


84 


87 


8WT0 


17780 


88908 


16097 


86514 


14486 


83 


88 


82896 


17708 


83030 


16070 


85640 


14460 


88 


80 


82385 


17675 


88957 


16043 


85507 


14433 


81 


40 


9.82352 


10.17648 


9.83964 


10.16016 


9.86694 


10.14406 


80 


41 


84880 


17620 


84011 


15089 


85620 


14380 


19 


48 


88407 


17593 


84038 


15068 


88647 


14363 


18 


48 


82485 


17565 


84066 


15935 


• 66674 


14826 


17 


44 


82468 


17638 


84092 


16908 


85700 


14300 


16 


45 


82488 


17511 


84119 


15881 


86727 


14278 


16 


46 


82517 


17483 


64146 


15854 


86754 


14846 


14 


47 


825^ 


17456 


84173 


16827 


85780 


14880 


18 


48 


17429 


84800 


16800 


86807 


14193 


12 


49 


88609 


17401 


84227 


15773 


86834 


14166 


11 


60 


9.88686 


iO. 17874 


9.84254 


10.15746 


9.86860 


10.14140 


10 


61 


88658 


17847 


84280 


15780 


85887 


14113 


9 


68 


88681 


17819 


84307 


15693 


8G913 


14087 


8 


68 


88708 


17898 


84334 


isew 


85940 


14060 


7 


54 


ssm 


17866 


84361 


15689 


85967 


14038 


6 


66 


88768 


17888 


84388 


15612 


85998 


14007 


6 


66 


82790 


17210 


84415 


15685 


86020 


13960 


4 


57 


88817 


17188 


84442 


16558 


86046 


13954 


8 


68 


82844 


3^66 


84460 


16531 


86078 


18927 


2 


69 


88871 


17129 


81496 


16504 


86100 


18900 


1 


60 


88899 


17101 


84528 


15477 


86126 


13674 





/ 


Ootan 


Tan 


Cotan 


Tan 


Cotan 


Tftn 


/ 


66* 




56- 




64* 



482 



TABLE in.— LOG 


. TANGENTS AND COTANGENTS. 


i 


se- 


87<' 


88- 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.86126 


10.13874 


9.87711 


10.12389 


9.89281 


10.10719 


60 


1 


86158 


13847 


87738 


12262 


89807 


10698 


59 


8 


86179 


18821 


87764 


12236 


89338 


10667 


66 


8 


86206 


18794 


87790 


12210 


89859 


10641 


67 


4 


86282 


13768 


87817 


12183 


89385 


10616 


66 


6 


86259 


18741 


87848 


12157 


89411 


10589 


65 


6 


86285 


18715 


87860 


12131 


89437 


10568 


54 


7 


86818 


18688 


87896 


12106 


89463 


10687 


53 


8 


86888 


18662 


87928 


12078 


88489 


10611 


62 


9 


86865 


13636 


87948 


12058 


89615 


10485 


61 


10 


9.86888 


10.18609 


9.87974 


10.12026 


9.89641 


10.10459 


60 


11 


86418 


18568 


88000 


12000 


89567 


10483 


49 


12 


86445 


18555 


88027 


11973 


89598 


10407 


48 


18 


86471 


18520 


88068 


11947 


80619 


10381 


47 


14 


86496 


18S02 


88079 


11921 


89645 


10365 


46 


16 


86524 


18476 


88106 


11896 


89671 


10329 


45 


16 


86551 


18449 


68181 


11869 


89697 


10808 


44 


17 


865T7 


18428 


68158 


11848 


89728 


10277 


48 


18 


86608 


13397 


88184 


11816 


89749 


10251 


42 


19 


86680 


18870 


88210 


11790 


89775 


10225 


41 


90 


9.86666 


10.18844 


9.88286 


10.11764 


9.89801 


10.10199 


40 


21 


86688 


18817 


88268 


11738 


89887 


10178 


89 


82 


86709 


18291 


88289 


11711 


89658 


10147 


88 


28 


86786 


18264 


88316 


11686 


89879 


10121 


87 


24 


86702 


18288 


88341 


11650 


89905 


10095 


86 


85 


86789 


18211 


88887 


11683 


89931 


10069 


36 


86 


86815 


18186 


88898 


11607 


89957 


10043 


84 


87 


86842 


18158 


88420 


11580 


89968 


10017 


83 


28 


86868 


18182 


88446 


11554 


90009 


09091 


82 


29 


86694 


18106 


88472 


11528 


90035 


09966 


31 


80 


9.86921 


10.18079 


9.88496 


10.11602 


9.90061 


10.09939 


80 


81 


86947 


13058 


88624 


11476 


90086 


09914 


89 


32 


86874 


18026 


88550 


11460 


90112 


09688 


28 


88 


87000 


18000 


88577 


11428 


90188 


09862 


87 


84 


87027 


12978 


88608 


11897 


80164 


09836 


86 


8A 


87058 


12947 


88629 


11871 


90190 


09810 


85 


86 


87079 


12921 


88665 


11846 


90216 


09784 


24 


87 


87106 


12894 


88681 


11319 


90242 


09758 


88 


88 


87182 


12868 


88707 


11298 


90268 


09782 


28 


89 


87158 


12842 


88788 


11267 


90294 


09706 


81 


40 


9.87185 


10.12815 


9.R8759 


10.11241 


9.90820 


10.09680 


80 


41 


87211 


12789 


887H6 


11214 


90346 


09654 


19 


42 


87288 


12768 


88812 


11188 


90371 


09629 


18 


48 


87264 


12736 


88838 


11162 


90397 


09603 


17 


44 


87290 


12710 


88864 


11136 


90488 


09577 


16 


45 


87317 


12683 


88890 


11110 


90449 


09561 


16 


46 


87348 


12667 


86916 


11084 


90475 


09525 


14 


47 


87369 


12631 


88942 


11058 


90501 


09499 


18 


48 


87896 


12804 


88968 


11082 


90527 


00478 


12 


49 


87422 


12578 


88994 


11006 


90558 


09447 


11 


60 


9.87448 


10.12662 


9.89020 


10.10980 


9.90578 


10.09482 


10 


61 


87475 


12525 


89046 


10954 


90604 


00396 


9 


52 


87501 


12499 


89078 


10927 


90690 


09«70 


8 


58 


87587 


12473 


89099 


10901 


90656 


00344 


7 


54 


87664 


12446 


89125 


10875 


90682 


09818 


6 


55 


87580 


12420 


89151 


10649 


90706 


09292 


6 


66 


87606 


12894 


89177 


10823 


90734 


09266 


4 


57 


87688 


12367 


89203 


10797 


90759 


09241 


8 


56 


87659 


12341 


89229 


10771 


90785 


09215 


2 


59 


87685 


12815 


89-^56 


10745 


90611 


09189 


1 


60 


87711 


12289 


89281 


10719 


90887 


09168 





9 


CoUn 


Tan 


Cotan 


Tan 


Cotan 


Tan 


# 




680 




6«* 




«!• 



483 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



/ 


ao^ 


40» 


41» 


/ 


Tan 


Ck>taii 


Taa 


Cotan 


Tan 


Couui 





9.90887 


10.09168 


9.92881 


10.07619 


9.93916 


10.06084 


60 


1 


90868 


09187 


92407 


07593 


93942 


06056 


69 


8 


00689 


09111 


92488 


07567 


93967 


06088 


58 


8 


90914 


09086 


92458 


07548 


93996 


06007 


67 


4 


90940 


09060 


9:2464 


07516 


94018 


05962 


66 


6 


00966 


09084 


92510 


07490 


94044 


05956 


55 





90902 


09006 


92585 


07465 


94069 


06031 


64 


7 


91018 


08U82 


92661 


07439 


94095 


05906 


58 


8 


91048 


08Itt7 


92587 


07418 


94120 


05680 


52 





91000 


08961 


92G18 


07388 


94146 


05854 


61 


10 


9.91005 


10.08905 


0.93038 


10.07368 


9.94171 


10.058» 


60 


11 


91121 


06879 


92663 


07887 


94107 


05808 


40 


12 


91147 


08863 


92689 


07811 


94222 


05778 


48 


18 


91172 


068J8 


92716 


07286 


94248 


06758 


47 


14 


91198 


08802 


98740 


07260 


94278 


06787 


46 


15 


9J224 


08776 


92766 


07234 


04209 


05701 


46 


16 


91250 


08750 


92r« 


07206 


94824 


06676 


44 


17 


91276 


08724 


9«17 


07163 


94350 


05660 


48 


18 


91801 


08699 


92843 


07157 


94875 


05626 


48 


19 


91827 


08673 


92866 


07188 


94401 


05509 


41 


90 


9.91858 


10.06647 


9.02894 


10.07106 


9.94426 


10.05574 


40 


81 


91879 


06621 


92920 


07060 


94458 


05548 


89 


S8 


91404 


06596 


92945 


07055 


94477 


05528 


88 


28 


91480 


08570 


92971 


07029 


.94608 


05497 


87 


S4 


91466 


08544 


92096 


07004 


01628 


05472 


36 


85 


91482 


08518 


03028 


06078 


94664 


05446 


86 


86 


91507 


08493 


98048 


06952 


94579 


06421 


84 


87 


91588 


08467 


98078 


06927 


94604 


05396 


88 


28 


91569 


08441 


98099 


06001 


94680 


05370 


82 


29 


91585 


08416 


98124 


06876 


94655 


05845 


81 


80 


9.91610 


10.06890 


9.9B160 


10.06850 


9.94681 


10.05319 


80 


81 


91686 


08864 


98176 


.06825 


91706 


05294 


89 


82 


91602 


06338 


98801 


06799 


94738 


06268 


88 


88 


91683 


06312 


93287 


06778 


94757 


06248 


87 


81 


91713 


06:J87 


98252 


00748 


94783 


05217 


86 


85 


91789 


06261 


98278 


06722 


94808 


06192 


86 


86 


91765 


06285 


98808 


06697 


94884 


05166 


84 


87 


91791 


06209 


08829 


06671 


94859 


05141 


88 


88 


91816 


08184 


9:«}54 


06646 


94884 


05116 


88 


89 


91842 


06158 


98380 


06620 


94910 


06090 


81 


40 


9.91868 


10.06182 


9.98406 


10.06694 ' 


9.94985 


10.06065 


80 


41 


91898 


06107 


93481 


06569 


94961 


06089 


19 


42 


91919 


06061 


93457 


06548 


94066 


05014 


18 


48 


91915 


08055 


93462 


06518 


95018 


04968 


17 


44 


91971 


06029 


98606 


06408 


96037 


04968 


16 


45 


91996 


06004 


96588 


06467 


96062 


04988 


16 


46 


92022 


07978 


98559 


06441 


95088 


04912 


14 


47 


92048 


07952 


98584 


06416 


95113 


04887 


18 


48 


92073 


07927 


»^10 


06390 


96139 


04861 


18 


49 


92099 


07901 


93636 


06364 


95164 


04836 


11 


60 


9.92125 


10.07875 


9.08661 


10.06389 


9.95190 


10.04810 


10 


51 


92160 


07850 


93687 


06318 


95215 


04765 


9 


52 


98176 


07H24 


9;J712 


06288 


95240 


04760 


8 


U 


92208 


07796 


93738 


06262 


96866 


04784 


7 


54 


92827 


{mm 


93763 


06237 


95291 


O470O 


6 


55 


98258 


07747 


98789 


08211 


95817 


04688 


5 


56 


92879 


07721 


03814 


06186 


95342 


04658 


4 


57 


9-2804 


07696 


98840 


06160 


95368 


04632 


8 


68 


93880 


07670 


93865 


06135 


95398 


04607 


8 


50 


92866 


07644 


93891 


06109 


95418 


04582 


1 


60 


92881 


07619 


93916 


06064 


96444 


04556 





/ 


Ootan 


Tftn 


Cotan 


Tan 


Cotan 


Tan 


/ 




SO* 




49« 




48* 



i 



484 



TABLE m.— LOG. T.^NGENTS AND COTANGENTS. 



' 


ii- 


48» 




44» 


t 


Tan 


Cotau 


Tan 


Cotan 


Tan 


Cotan 





9.05444 


10.04566 


9.00966 


10.08064 


0.06484 


10.01616 


60 


1 


951(i9 


04531 


96991 


06000 


06600 


01401 


50 


2 


95495 


04606 


97016 


02084 


06684 


01466 


56 


8 


956^0 


04480 


97042 


02068 


96600 


01440 


67 


4 


96545 


04455 


97067 


02088 


96666 


01416 


50 


6 


95571 


04420 


97002 


02006 


96610 


01800 


66 


6 


95596 


04404 


97118 


028R2 


96685 


01866 


64 


7 


96622 


04878 


97148 


02857 


98661 


01880 


68 


8 


96647 


04368 


97168 


02882 


96680 


01814 


59 


9 


95672 


04838 


97198 


02607 


96711 


01280 


51 


10 


9.96696 


10.04802 


9.9?2I9 


10.02781 


9.96787 


10.01268 


60 


11 


95728 


04277 


97244 


02766 


96702 


01288 


40 


12 


95748 


04252 


97269 


03781 


96787 


01218 


46 


18 


95774 


04226 


97295 


02705 


96812 


01188 


47 


14 


95799 


04:201 


97820 


02660 


96888 


01162 


40 


15 


96826 


04175 


97846 


02666 


98868 


01187 


45 


16 


95860 


O4160 


97871 


02620 


98686 


01112 


44 


17 


95875 


04126 


97896 


02604 


96918 


01067 


48 


18 


95901 


04099 


97421 


02670 


96989 


01061 


48 


10 


95926 


04074 


97447 


02558 


OMMJ. 


OlOfW 


41 


20 


9.95952 


10.04048 


9.97472 


10.02526 


9.96969 


10.01011 


40 


21 


96077 


04028 


97497 


02603 


99016 


00086 


80 


22 


96002 


03906 


97528 


02477 


99040 


00060 


86 


28 


96028 


08072 


97548 


02462 


90066 


00086 


87 


24 


9ti058 


08047 


97578 


02427 


00000 


00910 


86 


25 


96078 


08022 


97596 


02402 


00116 


00664 


35 


26 


96104 


03896 


97624 


02876 


00141 


00660 


84 


27 


96129 


08871 


97649 


02851 


00166 


00684 


88 


28 


96155 


06846 


97674 


02826 


00101 


O06O0 


82 


29 


96180 


08820 


97700 


02800 


00217 


00788 


31 


80 


9.96206 


10.08795 


9.97T» 


10.02276 


0.00242 


10.00756 


30 


81 


96281 


08769 


97760 


02260 


00267 


00788 


20 


32 


96266 


08744 


97778 


02224 


00208 


00707 


28 


83 


96281 


06719 


97801 


02100 


00618 


00662 


27 


84 


96807 


08698 


97826 


02174 


00848 


00667 


20 


m 


068ae 


08668 


97851 


02140 


09368 


00682 


25 


86 


96857 


08648 


97877 


02128 


99894 


00606 


24 


37 


96888 


08617 


97902 


02008 


99419 


0C681 


28 


88 


96406 


08692 


97927 


02078 


09444 


00666 


22 


89 


96488 


03567 


97058 


02047 


99460 


00681 


21 


40 


9.96459 


10.08541 


9.97978 


10.02022 


0.00405 


10.00506 


20 


41 


96484 


08616 


9ms 


01907 


00620 


00460 


10 


42 


96510 


08490 


96029 


01071 


00645 


00465 


18 


48 


96585 


08465 


98054 


01046 


0067D 


00480 


17 


44 


96560 


08440 


98079 


01021 


00606 


00404 


10 


45 


96686 


06414 


98104 


01806 


00021 


00870 


15 


46 


96611 


08889 


96180 


01870 


00646 


00854 


14 


47 


06686 


03864 


96155 


01845 


00672 


00828 


18 


48 


96662 


08338 


98180 


01880 


00607 


00808 


12 


49 


96(587 


03813 


08206 


01704 


00722 


00278 


11 


60 


9.06712 


10.08288 


0.06281 


10.01760 


0.00747 


10.00258 


10 


61 


06788 


08262 


06256 


01744 


00778 


00227 





62 


06768 


03287 


06261 


01710 


09708 


00802 


8 


68 


. 06788 


08212 


08807 


01603 


00828 


00177 


7 


M 


06814 


08186 


08382 


01»68 


00648 


00168 





66 


06880 


03161 


oaw 


01648 


00674 


00126 


6 


96 


06864 


08186 


06388 


01017 


00800 


00101 


4 


67 


06890 


08110 


06406 


01592 


00024 


00076 


8 


68 


06015 


08086 


08438 


01567 


00040 


OO061 


8 


69 


06940 


-03060 


08458 


01542 


00075 


00025 


1 


60 


06066 


08084 


08484 


01516 


lOOOOOO 


00000 





/ 


CoUu 


Tan 


Coum 


Tan 


OotflB 


Tan 


# 


47- 




46* 




4fi* 


1 



485 



TABLE rv.— NATURAL SINES AND COSINES. 








** 




Sine ICosin 





.00000 One. 


1 


.(JOO29: One. 


2 


.00058, One. 


8 


.000871 One. 


4 


.001161 One. 


5 


.00145 One. 


6 


.00175 One. 


7 


.00a04 One. 


8 


.00233 One. 


9 


.00262 One. 


xO 


.00291: One. 


11 


.006201.99990 


12 


.00349 '.99909 


18 


.00878 .99999 


14 


.00107*. 99999 


15 


.004361.99999 


le 


.0(M65:. 99999 


17 


.00495 


99990 


18 


.00524 


.99999 


19 


.00558 


99908 


80 


.00682 


.mnnn 


21 


.00611 


.99998 


22 


.00^, .99996 


23 


.006691.99996 


24 


.00698 


.99996 


25 


.00727 


.99997 


26 


.00756 


.99997 


27 


.00785 


.99997 


28 


.006141.99997 


29 


.00844 


.99996 


80 


.00878 


.99996 


81 


.00902 


.99996 


32 


.00931 


99996 


38 


.00060.99995 


84 


.00069 .99995 


35 


.01018 .99995 


86 


.01W7i. 99995 


87 


.010761.99994 


88 


.01105;. 99994 


89 


.01184. 99994 


40 


.On&l' 99908 


41 


.Olios'. 99993 


42 


.01222 .99993 


48 


.012511.99902 


44 


.01280 .99992 


45 


.01309. 99991 


46 


.01888 .90991 


47 


.01367 .99991 


48 


.01306 .99990 


49 


.01425 .99990 


50 


.01454,. 99969 


51 


.01463 1. 99989 


52 


.01518 .99989 


58 


.01542 .99988 


54 


.01571 


.99988 


55 


.01600 


.99967 


56 


.016S9 


.99987 


67 


.01666 '.99966 


58 


.01687 .99966 


69 


.01716 .99986 


60 


.01745. 90966 


/ 


Cosin 1 Sine 




« 


^ 



Sine Codn 



.01746 
.01774 
.01803 
.01832 
.01862 
.01891 
.01930 
.01049 
.01978 
.03007 



.02065 
.02004 
.02128 
.02152 
.02181 
.02211 
.02240 



.02898 



.02414 
.02443 
.02172 
.02501 
.02530 
.02560 
.02589 
.03618 

.026*7 
.0267Q 
.02705 
.02784 
.02768 
.02798 



.02850 
.02879 
.02906 



.02967 
.02906 
.03025 



.03054 
.03068 
.08112 
.03141 
.08170 
.08199 



.08357 



.08316 
.08845 
.08374 
.08408 
.06482 
.08461 
^90 
OosiD 



.99985 
,99984 
,99964 



.99981 
.99980 



.999791 

.99979! 
99978' 
!99977 
.99977 
.99976 
.99976 
.99975 
.99974 
.99974 
.99973 

.99972 
.999^2 
.99971 
.99970 



.99969 



.99967 
.99966 



.99964 
.99963 
.99963 



.99961 
.99960 
.99959 

199958 



.99956 
.99955 
.99954 
.99953 



.99951 
.99050 
.99949 

.99948 
.99947 
.99946 
.99945 
.99944 
.99943 
.99942 
.99941 
.99940 



Sine 



88« 



Slne^ Cosin 

r08490 

.03519 

.03548 

.03577 



08635 



03733 
0:^752 
08781 

08810 



.08926 
.08955 
.03964 
.04013 
.04042 
.(U071 

.04100 
.04129 
.04159 
.04188 
.04217 
.04246 
.04275 
.04304 
.04333 
.04802 

.04891 
.04420 
.04449 
.04478 
.04607 
.04586 
.04565 
.04594 
.04623 
.04653 

.04682 
.04ni 
.04740 
.04769 
.04798 
.04827 
.04856 



.99939 
.99938 
.99937 



.99984 



.04914 
.04948 

.04972 
.05001 
.05030 
.06059 
.05088 
.05117 
.06146 
.05175 
.06206 
.06384 
Ck>sin 



,99939 



.99926 
.99925 
.99924 



.99921 
.99919 
.99918 
.9991 

.99916 
.99915 
.99913 
.99912 
.99911 
.99910 



.99907' 
.99906 
.99905; 

.99904' 
.99903' 
.99901 



.99807 
.99896 
.99894 
.9{)803 
.99892 

.99690 



.99888 



.99882 
.99881 
.99870 
.99678 

.99878 
,'99675 
.99873 
.99872 
,99870 
,99869 
99867 



99804 
99863 



Sine 



Cosin 



.06384 
.05268 



.05321 
.05350 



.05379 
.05406 
.05437 
.05466 
.06495 
.05524 

.06653 



.05582 
.05611 
.05640 
.05669 
.06698 
.05727 
.05756 
.05785 
.06814 

.06<W4 
.05873 
.05902 
.05981 
.05960 
.05969 
.06018 
.0604' 
.06078 
.06105 

.06184 
.06168 
.06192 
.06221 
.06350 
.06279 
.06308 
.06337 
.06366 



.06424 
.00453 
.06482 
.06511 
.06540 
.06569 
.06596 
.06627 
.06656 
.06685 



1.06714 
1.06743 
.06778 
.06802 
.06831 



99861 
.99860 
.99858 
.99857 

99855 
.90854 

99852 
.99851 

90849 
.99817 

.90346 
.99844 
.09842 
.99841 



.99886 
.99884 



.90681 



.99836 
.99834 



Sine 



87« 



.06889 
.06918 
.06947 
.06976 
Cosin 



.99821 
.99819 
.99817 
.99815 
.90813 

.99612 
.99810 
.99808 
.99806 
.99804 
.99803 
.99801 
.99799 
.99797 
.99795 

.99793 
.99^3 
.99790 
.99788 
.997^ 
.99784 
.99782 
.99780 

.99rre 

.99776 

.99774' 
.99772 
.99770 
.99768 
.99766 
.99764 
.99762 
.99760 
.99758 
.99756 



Sine 



86* 



Sine 



Cosin 



.06076 
.07005 
,07084 
.07063 
.07092 
.07121 
.07150 
.07179 
.07208 
.07237 
.07266 

.07205 
.07324 
.07863 
.07882 
.07411 
.07440 
.07469 
.07496 
.07527 
.07556 

.07585 
.07614 
.<'7648 
.07672 
.07701 
.07730 
.07759 
.07788 
.07817 
.07846 



.07875 
.07904 
.07933 
.07962 
.07991 
.08030 
.08049 
.08078 
.08107 
.08186 

.06165 
.06194 



.08810 



.08368 
.08307 
.0&126 

.06455 
.084^ 
.08518 
.08542 
.08571 
.08600 



.08687 
.08716 
Cosin 



99756 
.99754 
.99752 
.99750 
.99748 
.99746 
.99744 
.99742 
.99740 
.99738; 51 
.00736 50 

.09734 49 
.99731 48 
47 
46 
45 
44 
48 
42 
41 
40 

89 
88 
87 
86 
85 
84 
88 
88 
81 
80 



.99727 
.99785 
.99723 
.99721 
.99719 
.99716 
.90714 

.99712 
.99710 
.99706 
.99705 
.99708 
.09701 
.09699 
.99696 
.90694 
.99692 

.99689 
.99687 



27 
26 

.99680 25 
.99678 24 
.99676 23 
.99678 22 
.99671 21 
.99668 20 



.99666 
.99664 



.9966^ 
.99654 



.99649 
.99647 
.99044 

.99642 



.99637 
.99635 



.99630 
.99627 



.99619 



Sine 



85* 



486 



TABLE IV.— NATURAL SINES AND COSINES. 



.08718 
.08745 
.08774 
.08808 



.06860 
.08889 
.08918 
.08M7 
.0897B 
.09006 



.09084 
.09068 
.09098 
.091S1 
.09160 
.00179 
.09308 
.099387 
.09966 

.oa»6 



.09868 



.09411 
.09440 
.09469 
.09496 
.09687 
.09666 
.09686 

.09614 
.0964S 
.09671 
.09700 



.09729 
.09768 
.09787 
.09816 
.09845 
.00874 

.09908 
.09982 
.09961 
.09990 
.10019 
.10048 
.10077 
.10106 
.10186 
.10164 

.10193 
.10331 
.10860 
.10879 
.10908 
.10387 
10866 
.10896 
.10434 
.10468 



Gosin 



Cosin 



.99619 
.99617 
.99614 
.99613 
.99609 
.90607 
.99604 
.99603 
.99599 
.99696 
.99604 

.99591 



,09680 
90678 
00675 
00578 
09570 
90567 

00664 
,00663 
00650 
00566 
00658 
.00651 
.00548 
.00546 
.00548 
.00640 



00587 
,00684 
.00681 



09517 
.09514 
.00611 

.00608 
.00506 
.00503 
.00500 
.09197 
.00404 
.09401 
.99488 
.99485 



00479 
00476 
00473 
.00470 
00467 
00464 
00461 
,00458 
00455 



61x10 



84* 



6 


• 


Sine 


Cosin 
.99458 


.10468 


.10483 


99449 


.10611 


.99446, 


.10540 


.994431 


.10569 


.99440; 


.10507 


.99437 


.10636 


.99434 


.10656 


.994811 


.106&1 


.99438; 


.10713 


.99434 


.10743 


.09431 


.10771 


.00418 


.10800 


.09415 


.10839 


.00413 


.10858 


.09409 


.10887 


.99406 


.10016 


.99402 


.10945 


.99399 


.10978 


.00396 


.11003 


.09303 


.11081 


.00800 


.11060 


.09386 


.11089 


.99888 


.11118 


.99880 


.11147 


.99877 


.11176 


.99374 


.11305 


.99370 


.11384 


.99367 


.11363 


.99364' 


.11391 


.99360 


.11330 


.99357 


.11849 


.99354 


.11878 


.99351 


.11407 


.99347 


.11486 


.99344 


.11465 


.99341 


.11494 


.99337 


.11538 


.99334> 


.11568 


.99831 


.11580 


.90337 


.11609 


.90334 


.11638 


.00830 


.11667 


.09317 


.11696 


.908141 


.11785 


.998101 


.11754 


•99307' 


.11788 


.998031 


.11818 


.998001 


.ll&tf) 


.998971 


.11869 


.99398 


.11898 


.99890 


.11987 


.99386 


.11966 


.90383 


A19e& 


.99379 


.18014 


.99876 


.18048 


.99378 


.18071 


.99869 


.18100 


.99366 


.13139 


.99363 


.18168 


.99358 


1.18187 


.90865 


Cosin 


Sine 


8 


J* 



7 


o 


Sine 


Cosin 


.12187 


.99255 


.12316 


.99351 


.12345 


.99248 


.12274 


.99344 


.12303 .99340 


.12331 .99237 


.12360 .99333 


.13389 .992301 


.12418 .992361 


.12447 


.99323 


.12476 


.99319 


.18504 


.90815 


.18638 


.00311 


.13668 


.09808 


.18591 


.99804 


.18680 


.99300 


.12649 


.99197 


.18678 


.99193 


.13706 


.99189 


.12785 


.99186 


.12764 


.99182 


.18798 


.99178 


.18838 


.99175 


.13851 


.99m 


.18880 


.99167 


.18906 


.99163 


.18987 


.99160 


.12966 


.99156 


.13995 


.99153 


.18034 


.99148 


.18068 


.99144 


.18081 


.99141 


.18110 


.99187 


.18189 


.99133 


.18168 


.99139 


.18197 


.99125 


.13236 


.991S2 


.18354 


.99118 


.13383 


.99114 


.13313 


.99110 


.13341 


.99106 


.18370 


.00103 


.13399 


.99098 


.18437 


.99094 


.18456 


.99091 


.13485 


.99087 


.18514 


.99083 


.18543 


.90079 


.13572 


.99075 


.13600 


.99071 


.13639 


.99067 


.18668 


.99063 


.18687 




.18716 


.99055 


.13744 


.99061 


.18778 


.99047 


.18803 


.99043 


.13831 


.99089 


.13860 


.99035 


.13889 


.99081 


.18917 


.99037 
'Sine 


Cosin 


8i 


J* 



Sine^ 
.13917 
.13946 
.13975 
.14004 
.14033 
.14061 
.14090 
.14119 
.14148 
.14177 
.14306 

.14884 
.14368 
.14398 
.14880 
.14349 
.14878 
.14407 
.14436 
.14464 
.14498 

.14688 
.14661 
.14580 
.14608 
.14687 
.14666 
.14695 
.14^3 
.14758 
.14781 

.14810 
.14888 
.14867 
.14896 
.14935 
.14954 
.14983 
.16011 
.16040 
.16069 

.16097 
.15186 
.15155 
.15184 
.15318 
.16841 
.16870 
.16899 
.16887 
.16866 

.16885 
.15414 
.15448 
.15471 
.15500, 
.155391 
.15557 
.155861 
.156151 
.15648 
Cosin 



Cosin 



.99087 
.99088 
.99019 
.99015 
.99011 
.99006 
.99003 
.98998 
.98994 
.98990 



,96978 
98073 
.08060 
06066 
,08061 
08057 
,06858 
08948 
08944 

,08940 



,08081 
,06987 



96010 
06014 
06910 
96906 



08897 



96884 
:98880 
06876 
,08871 
,08867 



.96854 
.06840 
.96845 
.98841 



.98818 
.96814 



96806 

98800 
98796 
96791 
.98787 
.96788 
.96778 
.96778 
.98769 
Sine 



81* 



Sine^ 
.15648 
.15672 
.15701 
.15730 
.15768 
.15787 
.15816 
.15845 
.15878 
.15908 
.16981 

.16969 
.15068 
.16017 
.16046 
.16074 
.16108 
.16132 
.16160 
.16189 
.16318 

.16846 
.16876 
.16804 
.16888 
.16861 
.16390 
.16419 
.16447 
.16476 
.16606 

.16688 
.16668 
.16691 
.16630 
.16648 
.16677 
.16706 
.16784 
.16768 
.16793 

.16880 
.16849 
.16878 
.16906 
.16086 
.16964 
.16998 
.17081 
.17060 
.17078 

.17107 
.17186 
.17164 
.17108 
.17388 
.17350 
.17279 
.17808 
.17886 
.17866 



Cosin 



.99769 
.96764 
.98700 
.98756 
.96751 
.98746 
.96741 
.96797 
.96788 
.98788 
.98788 

.98718 
.96714 
.08790 
.06^)4 
.06700 



08600 



08676 

06671 
,06667 



08657 
,08668 
06648 
06648 



06610 
.08614 
,08600 



,08604 
06600 



.98500 



.96575 
96570 
06566 



.98561 
.98666 
.18561 
.08646 



96641 
06686 



08681 



Cosin Sine 



06616 
.08511 
08606 
,08801 
,08486 
08491 
06486 
06481 



80* 



487 '"^ 

TABLE IV.— NATURAL SINES AND COSINES. 




1 

t 

8 

4 
6 

7 
8 
9 
10 

11 
IS 
18 
14 
15 
10 
17 
18 
10 
20 

81 
82 
88 
84 
86 
26 
87 
28 
29 
80 

81 
89 
88 
84 
86 
88 
87 
88 
80 
40 

AX 
48 
48 
44 

46 
46 
47 
48 
48 
60 

61 
68 
68 
54 
66 
66 
67 
68 
60 
60 



ig« 



Sine [Cogln 
.178661.98481 
.17898 .96476 
.17488 .86471 
.17461 .96466 
.17479 .98461 
.17606 .96466 
.17687 .98450 
.17666 .96445 
.17604 .98440 
.17883 .98485 
.176611.98480 



II- 



Sine 



.17680 
.17708 
.17787 
.17786 
.1T?94 
.17888 
.178S8 
.17B80 
.17900 
.17987 

.17986 
.17996 
.18088 
.18068 
.18061 
.18100 
.18188 
.18166 
.18195 
.18284 

.18288 
.18881 
.18800 
.18888 
.18867 
.18896 
.18484 
.18468 
.18481 
18609 



.18667 
18605 
18824 
18658 
18681 

.isno 

18788 
18767 
18795 

18884 
188S8 

.18881 
18010 
18988 

.18067 
18996 
19084 
19068 

.19061 



Coein 



.98485 
.98480 
.98414 
.96400 
.96404 



,96804 



96878 
98878 



96857 



98847 
98841 
98886 

98881 



JD6815 
98810 
96804 



96294 



98877 
96278 



96267 
96261 
96256 



.19061 
.19100 
.19188 
.19167 
.19196 
.19824 
.19268 
.19861 
.19809 
.19888 
.19866 

.19806 
.19488 
.19468 
.19481 
.19509 
.19588 
.19566 
.19595 
.19683 
.19658 

.19660 
19709 
19787 
19786 
19794 



CkMln 



96815 
96840 



98218 

98818 
96807 
98801 
96196 
.96190 
96185 
.98179 
.96174 
.96168 
96168 



.19661 
19880 
.19908 
.19987 

.19965 
19994 

90051 
80079 
90106 
80186 
80165 
80193 
80222 



.80879 
.80807 
80886 



Sine 



79- 



.80864 
.80898 
.80481 
.20450 
.80478 
.80607 

.80685 
.80563 

.80508 



.98168 
.96157 
.98168 
.96146 
.98140 
.98185 
.96189 
.96184; 
.96118 
.96118 
.96107. 

.96101 
.98006 
.96090 
.96084 
.96079 
.98078 
.96067 
.98061 
.98056 



12« 



Sine 



.80791 
.80620 

.80648 



98060 



98044 
96039 



96088 
98087 
96081 
96016 
96010 
96004 
97996 
97992, 

97987 
97961 



97975 
97969 
97963 
97958 
97958, 
97946. 
979401 
97934| 

97026 .81966 



20877 
80905 
80988 
.90962 
.80990 
.21019 
.21047 
.21076 

.81104 
.81188 
.81161 
.81189 
.81818 
.81846 
.81875 
.81806 
.81881 
.81860 

.81888 
.81417 
.81445 
.81474 
.81508 
.81580 
.81569 
.81587 
,81616 
,81644 

81678 
81701 
81729 
81758 
81786 
81814 
.81843 
81871 
81899 
81988 



CkMln 



97B15, 
978001 



.81985 
.82018 
.SXMl 



.28096 



80649 
.80677 
.80706 
.80784 
.80763 
.80791 



Oosin 



97922 
97916 
97910 ; 
97906* 
97899! 
9r898| 
97887 
97881' 
97875 

97869) 

97863, 
.978571 
.97851' 
.978451 
.97839 

97833 i .88410 

97887 1.22438 
.97821 j. 88467 



.88155 
.88183 
.88818 

.28840| 



.97808! 
.97797 
.97791 
.97784 
.97778 
.97778 
.97786! 
.97760 
.97754 

.97748' 
.97748 
.97786' 
.97789 
.97788 
.97717 

.9rni 

.97705 
.97606 
.97098 

,97666 
97680 
97078 
97067 
97661 
97065 
97048 
97048 



97086 
97680 

97023 
97017 
97011 
97004 
97598 
97592 
97586 
97579 



.97815 



Sine 



78' 



97573 
97566 

97560 
.97658 

97547 
.97541 
.97584 

97528 
.97581 

97515 
.975061 
.97508 j 

.97496 
.97489' 
.974831 
.97476 
.974701 
.97463' 
.97457 
.97450 
.97444' 
.97487 



ia« 



Cosln I Sine 

I 77^ 



Sine 



.28496 
.88688 



.88580 

.iueo8 

.82687 



.88728 
.88750 
.88778 

.82807 



.22980 
.88948 
.88977- 
.88005 
.88088 
.88008 

.88090 
.88118 
.88146 
.88175 



88816 



.88401 



.88458 
.83486 
.83514 
.83548 

.836n 



.83687 
.88666 



.88718 
.88740 
.88709 
.83797 



88858 



.88910 



.88866 

.88095 
.84023 
.84051 
.84079 
.84108 
.84136 
.84164 
.84192 
Oosin 



Coein 



.97487 
.97480 
.97484 
.97417 
.97411 
.97404 
47896 
.97891 
.97884 
.97878 
.97871 

.97366 
.97868 
.97861 
.97846 
.97888 
.97881 
.97885 
.97818 
.97811 
.97804 

.97898 
.97891 
.97884 
.9785B 



97871 
97864 
97857 
97851 
97244 
97887 

97280 



.97888 
.97217 
.97810 
.97808 
.97196 
.97189 
.97188 
.97176 
.97109 

.97102 
.97155 
.97148 
.97141 
.97134 
.97127 
.97120 
97118 
.9n06 
.97100 

97098 
97086 
97079 
97072 
97065 
97058 
97051 
970441 



97087 
97080 



Sine 



76- 



14* 



Sine 



.84108 
.84880 
.84849 
.84877 
.84805 



.84868 
.84890 
.84418 
.84446 
.84474 

.84608 
.84581 
.84660 
.84587 
.84615 



.84672 
.84700 
.84788 
.84756 

.84784 
.84818 
.84841 
.84869 
.84897 
.84985 
.84954 
.84968 
.85010 
.26088 



.86066 

.26094 
.86188 
.85151 
.85179 
.85807 



.85848 
.85876 
.85404 
.85488 
.85400 
.25488 
.85516 
.85545 
.85678 
.85601 



Ciosin 



.97080 
.97088 
.97015 
.97008 
.97001 
.96094 
.96967 
.90980 
.90978 
.90966 
.96959 



.96968 
.96945 
.96987 
.96980 



90910 
90000 



96008 



.00894 

.90887 

.90860 
.90878 
96866 
96856 
,96851 
96844 
,96887 



96815 

96807 
96800 
96798 
96796 
96778 
96771 
90704 
90750 
90749 
90748 



96784 19 
90787 18 



90719 
90718 
90705 
90097 
96690 



60 
50 
58 
57 
60 
65 
54 
58 
58 
51 
60 

49 
48 
47 
46 
45 
44 
48 
48 
41 
40 

89 

88 

87 

86. 

86 

84 

88 

38 

81 

80 

89 
88 
87 
86 
85 
84 
88 
88 
81 
20 



96675 



96607110 



.86029 
.25657 
.26685 
,25718 
.25741 , 
,25709 
,25798 



.90660 
.96658 
.96645 



26854 



,96680 
,90628 
,96615 
,90006 
,90600 
96598 



Cosin Sine 



488 



r 



TABLE IV.— NATURAL SINES AND COSINES. 



/ 
"0 


1^ 


w 


1 !?• ii 18» 


19* 1 


00 


Sine |C08in 


Sine lOosin 


Rine 


CosinI Sine 
.96680 780908 


Oosin 


: Sine 
.82557 


Oodn 


.25882 .96508 


.27564 .96126 


.29287 


.96106 


.04662 


1 


.26010 .96686 


.27502 .96118 j .29265 


.96622 .80029 


.96097 


.3^84 


.94542 


60 


t 


.26088:. 96578 


.27680[,96110/. 29298 
.27648'. 96102 .29321 


.956131 .80957 


.96068 


.32612 


.94588 


66 


8 


.260661.96570 


.95605 .30985 


.96079 


.32639 


.94523 


87 


4 


.269041.96662 


.27676 1.960941 .29848 


.96596; .81012 


.96070 


1.38667 


.94614 


66 


6 


.260221.96565 


.27704 .96086 .29376 


.95588! .81040 


.96061 


.88694 


.94504 


65 


6 


.26060 .96547 


.27781 .96078 .20404 


.95579, .81068 


.95052 


.38722 


.94495 


54 


7 


.280791.96540 


.27750 -96070 .20432 


.955711 .81095 


.95043 


.32749 


.94485 


58 


8 


.26107'. 96588 


.2',787 .96062 .20460 


.9556211.81123 


.96088 


.82777 


.94476 


52 





.261851.96524 


.27815 .96054 .29487 


.95554 .81151 


.96024 


.38804 


.94466 


51 


10 


.26163 .96617 


.27843 .96046 .29615 


.96645 .81178 


.06016 


.88882 


.94457 


60 


11 


.26191 .966001 


.8787l'.96087V29648 .96686> .81206 


.96006 


.88860 


.94447 


49 


12 


.262191.96502 '.27899. 96029 .29671>. 95528 .812881.94997 


.88887 


.94488 


48 


IS 


.9Q247,.96494! .279871.96021 .29509 .95519 .81261 .94988 


.82914 


.94428 


47 


14 


.26275 .96486 ,.27965 .96013*,. 29626 .96511 .81289'. 94979 


;. 82942 


.94418 


46 


15 


.26806,. 964791 


.27983 .96005 .29654 .96602 .81816 .94970 


.82969 


.94409 


45 


16 


.268811.96471 ' 


.28011' 05097 1 .29682 .96493 .31344 .94961 


.82997 


.94899 


44 


17 


.263591.96468 


.280891. 960891 '.29710 .95485,, .81372 .94952 


1.88084 


.94890 


48 


18 


.268871.96456 


.28067 -.95981' 1.29787 .95476'' .818001.04943 


1.88051 


.94880 


42 


19 


.264151.96448 


.28005 .96972, .29765 .95467 .81427 


.94933 


1.88079 


.94870 


41 


SO 


.26443' 96440 


.28128.96964 .29793J. 05459 .81454 


.94924 


1.88106 


.94861 


40 


81 


.26471 .96488, 


.28160 .96956 


1.29621 .96450' .81482 


.94915 


.88184 


.94851 


89 


SS. 26600!. 9642S 


.28178 .96048 


.29849.95441 .81510 


.94906 


1 .88161 


.048421 88 1 


23 


.266881.96417 


.282061.95040 


.29676.95483 .81537 


.94897 


.88189 


.94882 


87 


24 


.266661.96410 


.28284 .96031 


.29904 '.95421 ',.31505 


.94888 


.83216 


.M828 


86 


25 


.26584 .9<M02 


.28262 .96928 


.29932!. 95415 .81503 


.94878 


1.88244 


.94818 


85 


26 


.26612,. 96304 


.28290 .96915 .29960|. 95407 .31620 


.94869 


.38271 


.94803*84 1 


27 


.26640 


.96886 


.28318. 95907 


.29987 .9539811.81648 


.94860 


1.832981.94293 


88 


28 


.26668 


.96379 


.28346. 95896 


.80015 .95389 .31675 


.94851 


.88886!. 94284 


88 


29 


.26696 


.963711 


.28374;. 96890 


.800481.95380 .81708 


.94842 


.88853 


.94274 


81 


80 


.267^ 


.96868 


.28408 


.95882 


.80071 . 95372, |. 81780 


.94882 


.88881 


.94264 


30 


81 


.26753 


.96855' 


.28420 


.96874 


.80098 .95863 .81758 


.94823 


.88408 


.94254 


29 


88 


.26780 


.968471 


.28457 


.95865 


;. 30126 .95854, .81786 


.94814 


.88436 


.94245 


28 


88 


.26808 


.96340' 


.28485 


.95857 


;. 80154 .05845;. 81813 


.94805 


■88468 


.94285 


27 


84 


26836 


(M.«I9 


28513 1 05849 


1 80188 wuvtr 511R11 


94795 


83490 


94285 


28 


86 


.26864; !96324 


.28541 .95841 


!80209 


.95828 .81868 


!94796 


1 .'88618 


!94215 


25 


86 


.268921.96316! 


.28509 .05832 


' .80237 


.95319 1 .81896 


.04777 


.88545 


.04206 


24 


87 


.20920 


.90308 


.28597 .95821 


1.80965 


.95810 1 .81923 


.94768 


'.83578 


.94196188 1 


88 


.26948 


.96301, 


.28625 .95816 


1.80292 


.95801 , .81951 


.94758 


.88600 


.94186 


22 


89 


.iHMiQ 


.96203 


.28652. 95807 


.80820 


.95893 ' .81979 


.94749 


!. 88687 


.94176 


21 


40 


.27004 


.962851 


.28680 .05799 


.80848 .95284 1 .82006 


.94740 


.88655 


.94167 


20 


41 


.87082 


.96377 


.88708'. 96791 


.80876 .95275 ' .82084 


.94780 


1.88688 


.94157 


19 


42 


.27060 


.962691 


.28:^ .95782 


.80103 .95266,1.32061 


.94721 


.38710 


.94147 


18 


48 


.27088 


.96261, 


.28764 .95774 


1 .80181 .95257 .82080 


.94712 


i.88'i'87;. 94137 


17 


44 


.27116 


.968531 


.287921.05766 


1.804591.95248,1.88110 


.94702 


'.88764 .94127 


16 


45 


.27144 


.96246, 


.28820 .05757 


1.80486 .95240' .88144 


.94693 


.88792>. 04118 


15 


46 


.2n72 


.96288' 


.288471.05740 


'.80514. 95281 .82171 


.94681 


: .83819 .W108 


14 


47 


.27900 


.96230 


.28875 .05740 


.80542 .95282 , .82199 


.94674 


.88846 .94096 


18 


48 


.27228 


.96222, 


.289081. 95732 


.80570 .95213 1.88227 


.9466.^ 


.88874 .94088 


12 


40 


.27256 


.96214 


.28931 .95724 


.80597 


.95201 1.88254 


.94656 


83901 1.94078 


11 


60 


.2?i84 


.96206i 


.28059 .95715 


.30625 


.95195| .88282 


.9464d 


.83029 .94068 


10 


51 


.27812 


.96196 


.28967 .96707 


.80658 


.95186 .82809 


.94637 


.88966!. 94068 


9 


52 


.27340 


.961901 


.29015 .95698 


.30680 


.95177 .82387 


.94627 


.88963 .94049 


8 


53 


.27868 .96182 


.290121.95690 


.807108 


.95168 .82804 


.94618 


1 .34011 i .94089 


7 


54 


.27396 .96174 


.29070 .95681 


.30736 


.95159 .32392 


.94609 


,.840881.94089 


6 


65 


.27424 .961661 


.290981.95678 


.30768 


.95150 .82119 


.94599 


.84065 .94019 


6 


66 


.27452 '.96158 


.29126 .95664 


.80791 


.95142 .82147 


.94590 


.84098 


.94009 


4 


67 


.27480' 96150 


.291 54,. 95656 


.80819 


.95133 .82474 


.94560 


1.84120 


98999 


8 


68 


.27508 .96142 


.29182. 95647 


.80846 


.95124 .32502 


.94571 


.84147 


.'98889 


2 


60 


.27536'. 96134 


.29209 .95639 


.80874 


.95115 .32589 


.94561 


.84122 
j .84202 


.98979 


1 


60 


.27564. 96126 


.292871.05630 


.80902 .95106 .32557 


.94552 


.93969 


J 


/ 




CofiiQ 1 Sine 


Cbsin 1 Sine Cosin 


Sine 


|C<»in 


Sine 


i 


74- 1 


78' 72* 1 IV 1! 70* 1 



489 



TABLE IV.— NATURAL SINES AND COSINES. 



■5 


20^ 1 


21« 1 


230 


23- I 


24« 




Sine 


Cosin 


Sine 


Oogtn 


Sine Cofiin 


Bine 
.89078 


Cosin 


Sine Cosin 


/ 


.84902 


96969 


.36887 


.96868 


.37461 


.92718 


.92060 


.40674 .91855 60 


1 


.34220 


.96959 


.85864 


.98348 


.87488 


.98707 


.89100 .920891 


.40700 .91813 59 


s 


.84257 


.93949 


.86891 


.98337 


.87515 


.98697 


.89127 


.92088 


.407271.91331 68 


8 


.84284 


.98889 


.35918 


.98827 


.87542 


.92686 


.39153 


.92016 


.40753 .91319 57 


4 


.84811 


.93929 


.36945 


.96316 


.87669 


.92675 


.39180 


.92005 


.40780 .913071 56 


6 


.84389 


.93919 


.35973 


.98806 


.87695 


.92664 


.89207 


.91994 


.40606 1. 91895 > 55 


6 


.84866 


.98909 


.86000 .98296 


.87622 


.92653 


.89234 


.91982 


.40833 .91283 54 


7 


.84898 


.98899 


.86027". 93285 


.37649 


.98648 


.39260 


.91971 


.40860 .91272 58 


8 


.84421 


.98889 


.36054 .96274 


.37676 


.98681 


.89287 


.91959 


.40686 .91260 58 


9 


.34448 


.98879 


.38081 .96264 


.37703 


.92620 


.39314 


.91948 


.40913 .91248 51 


10 


.34475 


.96869 


.86106 .93253 


.8T?ao 


.92609 


.89341 


.91986 


.40989 


.91236 50 


11 


.84606 


.93859 


.36185'. 93248 


.377S7 


.92608 


.89867 


.91925 


.40966 


.91884 49 


12 


.84680 


.93849 


.86162 .98882 


.87784 


.98587 


.39694 


.91914 


.40992 


.91212 48 


18 


.84667 


.98839 


.36190 


.93822 


.87811 


.92576 


.39481 


.91902 


.41019 


.91200 47 


14 


.84684 


.93889 


.36217 


.96211 


.87838 


.98585 


.89448 


.91891 


.41M5 


.911881 46 


15 


.84612 


.96819 


.36244 


.98201 


.37865 


.92564 


.89474 


.91879 


.41072 


.91176 45 


16 


.84689 


.98800 


.36271 


.98190 


.37892 


.92543 


.39601 


.91868 


.41096 


.91164144 


17 


.81666 


.93799 


.36296 


.96180 


.37919 


.98582 


.39528 


.91866 


.41185 


.91152 48 


18 


.84694 




.86325 


.98169 


.37946 


.92521 


.39555 


.91845 


.41151 


.91140 42 


19 


.84721 




.3635?. 


.96150 


.37978 


92510 


39581 


91883 


41178 


91188 41 


30 


.84748 




.36879 


.96148 


.87999 


.92499 


.89608 


.91822 


.41204 


.91116 40 


21 


.84775 


.98759 


.36406 


.98187 


.38026 


.92488 


.89685 


.91810 


.41281 


.91101 39 


2S 


.84806 


.98748 


.86434 


.98127 


.88058 


.92477 


.89661 


.91799 


.41257 


.91092 38 


28 


.84830 


.98738 


.36461 


.98116 


.38060 


.92466 


.39688 


.91787 


.41284 


.91080 37 


24 


.84867 


.98728 


.36488 


.98106 


.88107 


.92465 


.89715 


.91775 


.41310 


.91068 86 


25 


.84884 


.96718 


.86615 


.93095 


.88134 


.92444 


.89741 


.91764 


.41387 


.91066 85 


26 


.84912 


.93708 


.86642 


.98084 


.88161 


.92482 


.89768 


.91752 


.41363 


.91044 84 


27 


.84869 


.98608 


SAKAQ 
.OOODV 


.96074 


.38188 


.92421 


.39795 


.91741 


.41390 


.91032 38 


88 


.84966 


.98888 


.36596 


.96068 


.38215 


JB2410 


.39888 


.91729 


.41416 


.91020: 82 


29 


\^ 


.98677 


.86623 '.wwosf 


.38241 


.98399 


.39848 


.91718 


.41443 


.91008 31 


80 


.98667 


.86660,. 96(M2 


.88268 


.92888 


.39875 


.91706 


.41469 


.90996:30 


81 


86048 


96857 


36677 


98061 


88295 


92877 


89908 


91694 


41496 


909641 29 


82 


!35075 


.'98647 


.36704 


!93oao 


!88S22 


!92866 


!89988 


■.91688 


!41S82 


!90g72 28 


83 


.85102 


.96687 


.86781 


.98010 


.88349 


.92865 


.89955 


.91671 


.41549 


.90960,27 


84 


.86180 


.98626 


.36758 


.92999 


.38376 


.92843 


.89988 


.91660 


.41575 


.90948,26 


85 


.85157 


.93616 i .86785 


.92988 


.38403 


.92332 


.40008 


.91648 


.41602 


.90666' 25 


86 


.85184 


.98606 


.36812 


.92978 


.384801.92821 


.40065 


.91636 


.41688 


.90624124 


87 


.85211 


93566 


.36889 


.92967 


.38466 i. 98810 


.40062 


.91625 


.41665 


.90911! 28 


88 


.86239 


.98585 


.86867 


.92950 


.38483;. 92299 


.40068 


.91613 


.41681 


.90899 22 


89 


.35206 


.98575 


.36894 


.92945 


.38510 


.92287 


.40115 


.91601 


.41707 


.90687 21 


40 


.85298 


.93565 


.36921 


.92965 


.38537 


.92276 


.40141 


.91590 


.41734 


.90875 20 


41 


.85820 


.96556 


.86048 '.92924 


.38564 


.92265 


.40168 


.91578 


.41760 


.90863 19 


42 


.86847 


.98544 


.869751.92913 


.38501 


.92254; .40195 


.91566 


.41787 


.90861 18 


43 


.85375 


.98584 


.87002 .92903 


.38617 


.92243 .40221 


.91555 


.41813 


.90839 17 


44 


.85402 


.93524 


.37029 .92892 


.38644 


.92231' .40248 


.91543 


.41840 .90626 


16 


45 


.85429 


.96514 1.370661.92881 


.88671 


.92220 1.40875 


.91631 


.41866 .90814 


15 


46 


.35460 


.93506 


.37068;. 92870 


!. 88696:. 92209 


.40301 


.91519 


.41892 .90802 


14 


1 47 


.85484 


.98493 


.37110 .9285911.88725 .92198 


.40828 


.91506 


.41919 .90790 


18 


48 


.35511 


.96483 


.8n37 1.92849 


.38752 .92186 


.40856 
^40381 


.91496 


.41945 


.90778 


12 


! 49 


.35638. 93472 


.87164!. 92888 


.88778 


.92175 


.91481 


.4197^ 


.90766 


11 


(.60 


.86666 .98462 


.8n91 


.92827 


.38805 


.921&1 


.40406;. 91472 


.41998 


.90758 


10 


61 


.35602'.96462l 


.87218 


.92816 


.38832 


.92152 


.40484 .91461 


.42024 


.90741 


9 


62 


.85619 .984411 


.87245 


.92805 


.38859 


.92141 


.40461 .91449 


.42051 


.90729 


8 


68 


.36647 


.96431! 


.87272 


.92794 


.88886. 921801 


.40488 .91487 


.42077 


.90717 


7 


64 


.86674 


.96420 


.87299 


.92784 


1 .88912 .921191 


.40514 .91425 


.42101!. 90704 


6 


55 


.85701 


.93410 


.37326 


.92778 


1 .88989 .92107 


.40541 .91414 


.42130 '.90692, 5 


66 


.85728 


.96400 


.37853 


.92782 


1.88966 .92096' 


.40567 .91403 


! .42166 .90680! 4 


67 


.35756 


.96389 


.37380 


.92751 


,.889931.92065 


.40594 .91890 .421881.90668 


3 


68 


.85782 


.96379 


.37407 


.92740 


.39020 .92073 ; .40621 .91378/ .42209.90666 


8 


69 


.86810 


J0368 


.37484 


.92729 


.39046 .92062 ' A0M7 .919661 .482851.90648 


1 


60 

/ 


.85887 


.93858 


.37461 
Cosiii 


.92718 
Sine 


.89073 .92050 .40674 .91855 
Cosin Sine j| Cosin Sine 


.42262 .90631 
Cosin Sine 




Cosin 


Sine 


69» 1 


68« 1 


1 e7« !i 6e« 


W 



490 



TABLE IV.— NATURAL SINES AND COSINES. 





25- 1 




Sine 


O'^in 





.48868 


.»kL;l 


1 


.48888 


9«'';i3 


2 


.48815 


»r,^ 


8 


.48841 


.»i.-.n| 


4 


.42867 


»i:.>-8 


6 


.48894 


.»ot:-} 


6 


.48480 


.»i;v,r 


7 


.48446 


.9 CM 


8 


.48478 


911.-.M-8 


9 


.48499 


.fti:.:.-^) 


10 


.48686 


.ftir^iir 


11 


.48668 


»)r^5 


18 


.48578 


giis-^l 


18 


.48604 


9^3^') 


14 


.48681 


9IM-.S 


16 


.48657 


.91MJ1 


16 


48688 


.9(»r^ 


17 


.48700 


.910JI 


18 


.48786 


.9(sr'-i 


19 


.48768 


.fl(X-^'.,J 


80 


.48788 


.9^n<i 


81 


.48816 


.9in7\ 


8S 


.48841 


9^14 


88 


.48867 


91 'ill) 


84 


.48894 


9insi 


85 


.48980 


9ir-l 


80 


.48946 


.»nO'i 


87 


48973 


»i-N^ 


88 


.48999 


.9iri-i 


89 


.48085 


.9-SA 


80 


.48061 


.9te-'3 


81 


.48077 


.9fi'^ii5 


88 


.48104 


.9(^V!J 


88 


.48180 


9r::\ 


84 


.48156 


9^:H 


85 


.48188 


9(M «i 


8fl 


.48809 


.9(K'vJ, 


87 


.43885 


.9nA' 


88 


.43861 


.9(MVi| 


89 


.43887 


.9i'i)<)l 


40 


.48818 


.9('^Ei| 


41 


.48840 


.9(n:>') 


48 


.43366 


.9(i|"^| 


43 


.48898 


.9(i-''5' 


44 


.48418 


9('<^^■i 


45 


.48445 


.9(>'i.*). 


46 


.48471 


.9(-r.r 


47 


.48497 


.9('>^) 


48 


.43588 


.9(»'';3 


49 


.43549 


.9(^ni 


60 


.48576 


.9(*"*ir 


51 


.48008 


.ftj'.K^l 


68 


.48688 


.8!^-.iM 


58 


.48654 


.8!iHiJ 


54 


.48680 


.Slii.^J 


55 


.48706 


.ftNfcl, 


66 


.48788 


.a^i'-^) 


57 


.48769 


.8'.^^. 


66 


.43785 .ai>'5| 


59 


.48811 


.811-:. J 


60 


.48887 


.ft^-V'i 


f 


Ck>8ln 


8J]ji^ 




^ 


!• 



S6« 



Sine^ 
.48837 
.43863 



48916 
.43918 



43994 
.44080 
.44046 
.440?2 
.44096 

.44184 
.44151 
.44177 
.44808 
.44289 
.44855 
.44281 
.44807 
.44383 
.44859 

.44886 
.44411 
.44437 
.44464 
.44490 
.44516 
.44548 
.44668 
.44604 
.44680 

.44646 
.44673 
.44696 
.44724 
.44750 
.44776 
.44808 
.44828 
.44854 
.44880 



.44906 
.44938 
.44958 
.44964 
.45010 
.45038 
.46068 
.45068 
.45114 
.45140 

.46166 
.45192 
.45218 
.45243 
.45269 
.45205 
.46321 
.45347 
.46373 

Cosin 



CoBln 

.89879 
.89667 
.89654 
.89641 



.89616 



.89790 
.89777 
.89764 
.89752 



.89789 
.89736 
.89713 
.89700' 
.89687J 
.89674' 
.896621 
.896491 
.896861 



89610 
80597 1 



895n 
89558 
89545 



89519 
89506 



89480 
89467 
89454 
89441 



,89115 
89402 
89389 
89376 



89311 



89272 
89269 
89245 



.89819 
.89206 
.89193 
.89180 
.89167 
.89153 
.89140 
.89127 
.89114 
.89101 

Sine 



270 



Sine^ 
.45399 
.4M25 
.45451 
.46477 
.45508 
.45529 
.45554 
.45580 
.45606 
.45632 
.45658 

.45684 
.45710 
.45736 
.46762 
.46787 
.46813 



CosIn 

.89101 
.89087 
.89074 
.89061 
.89018 
.89035 
.80021 
.89006 
.88995 



.45891 
.46917 

.46948 
.45968 
.45994 
.46020 
.46046 
.46072 
.46097 
.46123 
.46149 
.46175 

.46801 



.46278 
.46304 
.46330 
.46355 
.46381 
.46407 
.46433 

.46458 
.46464 
.46510 
.46536 
.46561 
.46587 
.46618 



.88966 

.88942 



.88916 



.88876 



.88846 



.88795 



.88768 
.88;55 
.88741 



.88715 
.88701 



.88674 
.88661 
.88647 



.88607 



.88512 
.88499 



es' 



.46664 
.46690 

.46716 
.46748 
.46767 
.46798 
.46819 
.46844 
.46870 
.46896 
.46921 
.46947 



Cosin 



.88472 
.86458 
.88445 
.88431 

.88417 
.88404 



.883771 



28* 



.883491 



.88308 
Sine 



62» 



Sine Cosin 



.46947 
.46978 



.47024 
.47050 
.47076 

.4noi 

.47127 
.47153 
.47178 
.47304 

.47889 
.47855 
.47261 
.47806 
.47838 
.47858 
.47388 
.47409 
.47434 
.47460 

.47486 
.47511 
.47537 
.47662 
.47586 
.47614 
.47689 
.47665 
.47690 
.47n6 

.47741 
.47767 
.47793 
.47818 
.47844 
.47869 
.47895 
.47920 
.47946 
.47971 

.47997 



.88840 



.88199 
.88185 
.88173 



88144 
88180 
88117 
88108 



88075 



.88048 
.88084 
,88080 



,88006 
,87993 
,87979 
,87965 
,87951 



.87037 



,87909 
.87896 



.48048 
.48078 
.48099 
.48184 
.48160 
.48175 
.48801 



.48K7 
.48308 



.48354 

.48379 
.46405 
.48430 
.4»456 
.46481 



Cosin 



.87868 
.87864 
.87840 
.87826 
.87812 
.87796 
.87784 
.87770 
.87756 
.87743 

.87729 
.87n5 
.87701 
.87687 
.87678 
.87869 
.87645 
.87631 
.87617 
.87603 



.87575 
.87561 
.87546 
.87538 
.87518 
.87504 
.87490 
.87476 
.87462 



Sine 



61- 



29^ 



Sine Cosin 



.48481 
.48606 



.48667 



.48608 
.48684 

.48660 
.48684 
.48710 
.48736 

.48761 
.48786 
.48811 
.48887 



.48918 



.48964 



.49014 
.40040 
.49066 
.49000 
.49116 
.49141 
.49166 
.49198 
.49217 
.49848 



.49818 
.49844 



.40894 
.49419 



.49470 
.49486 

.49681 
.49546 
.49571 
.49696 



.49647 
.49673 
.49697 
.49733 
.48748 

.49778 
.49798 
.49624 
.49649 
.49674 
.49609 



.49960 
.40976 
.60000 



.87462 
.87448 
.87484 
.87480 
.87406 
.87891 
.87877 
.87868 
.87349 
.87886 
.87831 

.87806 
.87892 
.87378 
.87364 
.87350 
.87335 
.87381 
.87307 
.87198 
.87178 

.87164 
.87150 
.87186 
.87121 
.87107 
.87093 
.87079 
.87064 
.87060 
.87096 

.87081 
.87007 
.80998 
.80976 
.86964 
.86949 
.86986 
.86921 
.86906 
.86808 

.86878 



.86884 
.86880 
.86806 
.86791 
.86777 
.86768 
.86748 

.80788 
.86719 
.86704 
.86690 
.66675 
.86661 
.86646 
.86682 
.86617 



Cosin Sine 



491 

TABLE IV.— NATURAL SINES AND COSINBSl 



80- 



Cosln 



.60000 
.60025 
.50060 
.500TO 
.50101 
.60126 
.50151 
.50176 
.50301 



.60277 
50602 



.50387 



.50877 
.50108 
.50428 
.50458 
.60478 
.50603 



.50568 
.60578 
.50603 



.60654 
.60679 
.50701 
.50729 
.50754 

.50779 
.60804 



.50654 
.50679 
.50904 



.50954 
.50079 
.51004 

.51029 
.51054 
.51079 
.51104 
.51129 
.51154 
.51179 
.51204 
.51229 
.61254 

.512r9 
.51804 
.51829 
.51354 
.51879 
.51404 
.61429 
.51454 
.61479 
.51504 



Ckwin 



.86588 
.86573 
.86559 
.86514 
.86530 
.86515 
.86601 
.8&186 
.864n 
.86457 

.86442 
.86427 
.86413 



.86384 



.86854 
.86340 



86310 



.86251 



.86207 
.86192 
,88178 
.86163 

.86148 



31« 



.86119 
.86104 
.86069 
.86074 
.86069 
.86015 
.86030 
.86015] 

.86000 
.85965 
,85970 
.86956 
.85941 



.85911 



.85861 



.85821 
.86806 
.85792 
.85777 
.85782 
.85747 



^717 
Sine 



69» 



Sine 


Cosin 


.51504 
.51529 
.51554 
.51579 
.51604 
.51628 
.51653 
.51678 
.51708 
.51728 
.51758 

.61778 
.51803 
.51828 
.51852 
.51877 
.51902 
.51927 
.51952 
.61977 
.60002 

.52026 
.52051 
.52078 
.62101 
.52126 
.52151 
.62175 
.52200 

!62250 

.62275 
.52299 
.52824 
.52349 
.52374 
.52899 
.52423 
.52148 
.52478 
.52196 

.52522 
.52547 
.52572 
.52597 
.52821 
.52648 
.58671 
.52698 
.52730 
.62745 

.627?0 
.62794 
.52819 
.52844 
.52869 
.62898 
.62918 
.62943 
.52967 
.62992 


.85717 
.86702 
.86687 
.86672 
.85657 
.85642 
.85827 
.85612 
.85697 
.86582 
.85687 

.85561 
.85536 
.85621 
.85506 
.85491 
.85476 
.85461 
.85448 
,85431 
.85418 

.85401 
.85885 
.85370 
.85355 
.85310 
.85325 
.85310 
.85294 
.85279 
.85264 

.86219 
.85234 
.85218 
.85208 
.85188 
.85173 
.85157 
.85142 
.86127 
.85112 

.85006 
.85061 
.85066 
.85051 
.85035 
.85020 
.85005 
.81989 
.84974 
.81959 

.84943 
.84928 
.&1913 
.84897 
.84882 
.^866 
.84851 
.84838 
.84820 
.84805 


Cosin 


Sine 



68- 



820 



Sine Cosin 



.530171 
.53041 
53088 
.53091 
.53115 
.53140 
.58164 
.63189 
.53214 



.63263 



.53312 
.53337 
.53361 



.63411 
.53435 
.5&180 
.53484 

.58509 
.63534 
.53558 
.53588 

.53807 



.53658 
.53681 
.63705 
.53730 

.53754 
.63779 
.53801 



.53877 



.53928 
.53951 
.53975 

.54000 
.54024 
.51049 
.54078 
.64097 
.54122 
.64146 
.64171 
.54195 
.54220 

.64244 
.54269 



.54317 
.54»12 
.54368 
.54391 
.54415 
.54440 
.54484 
Ck)sin 



.84805 
.84789 
.84774 
.84750 
.84743 
.84728 
.84712 
.84697 
.84881 
.84888 
.84860 



,84819 
.84804, 
.845881 
.845781 
.84567 
.84542 



.845111 
.844951 

.84480' 
.84484; 
.84448' 
.84433 
.&M17 
,64402 



,81370 
81355 



.&1308 



, 81277 
,81281 
81245 
81280 
84214 
,841981 
84182. 

,84187 
.81151 
,81135 
.84120 
,84104 
.84088 
.84072 
.W057 
.81041 



,84009 
.83994 



.83915 



Sine 



67» 



83* 



Sine 



.64484 
.54488 
.64518 
.64537 
.54581 
.64588 
.54610 
.54835 
.54659 
.54688 
.54708 

.64782 
.54758 

.54781 
.54805 



Oosin 



.83867 



.83819 
.83804 



.88r?2 
.83756 



.54854 
.54876 
.54902 
.64927 
.54951 

.64975 
.54999 
.55024 
.65018 
.55072 
.55097 
.55121 
.55145 
.55169 
.65194 

.55218 
.55242 
.55266 
.55291 
.56315 
.55389 



.55863 
.55888 
.56412 
.56438 

.55460 
.55181 
.56509 
.5&533 
.65557 



.55830 
.55654 
.55678 

.6CT02 
.56728 
.65750 
.65775 
.55799 



.55847 
.55871 
.55895 
.55919 



Cosin 



88740 >'^68112 



.83724 
.837U8 



.88645 
.83829 
.83813 



.83549 



.88517 
.88501 



.88453 
.88437 



.83406 



.83878 
.83358 
,88340 



.83244 



.88212 
,83195 
,83179 
.83168 
.88147 
.88181 1 
,88115! 



84» 



Sine Cosin 



.66019 
.65948 
.65088 
.55992 
.68018 
.58040 
.68084 
.68068 



.88050 
.83084 
.83017 
,68001 



.82963 



^8^ 
Bine 



56<> 



.58186 
.58160 

.68184 



.58258 



.58805 



.58853 
.68877 
.68401 

.68425 
.66449 
.68478 
.58497 
.66521 
.58545 
.58669 
.56598 
.56817 
.56841 

.68685 



.68718 
.68738 
.66760 
.68784 
.68808 



.58868 



.58904 



.58978 
.57000 
.57024 
.67047 
.57071 
.57095 
.57119 

.67143 
.6716: 
.6n91 
.57216 
.57238 
.67262 
.67288 
.67810 
.57834 
.67868 



Cosin 



82904 
,82887 
,88871 



,82790 
.82778 
,82767 
.82741 

.82724 
.82708 



82875 



60 

60 
68 
67 
66 
65 
54 
58 
62 
51 
60 

49 
48 
47 
48 
46 
44 
48) 
42 
41 
82677.40 

,82561189 
,82544 86 
.82528187 
.82511 88 
.82496' 85 
.82478; 84 
.82482 88 
.82448 82 
.82129 81 
.80418 80 

29 



82848 



,82610 



.82847 



.82814 



27 
26 
25 
24 
28 
22 
21 
80 

19 
82214 18 
.82198 17 
.82181 1 18 
15 

.82148 14 
13 
12 
11 
10 

9 
8 
7 
6 
6 
4 
8 
2 
1 




.82115 
.82098 



,82066 



.82082 
.82015 
.81999 
.819Si 
.81965 
.81949 
.81932 
.81915 

Sine i" 



66» 



49a 



TABLE IV.— NATURAL SINES AND COSINES, 



/ 

"o 


86« 1 


86» 


87- 1 


880 1 


89- 


60 


Sine 


Ooain 

.81915 


Sine CoBln 
.58779' 80908 


Sine 
.60182 


CoBln' 


Sine 


CoirinI 

.78801 


Sine 


Cosin 


.57868 


.79664; 


.61566 


.68988 


.77715 


1 


.57881 


.81899 


.688081. 80685, 


.60805 


.798461 


.61589 


.78783 


.68955 


.77696 


69 


8 


.67405 


.81888 


.58828 


.80667 


.60288 


.79689 


.61612 


.78765 


.68977 


.77678 


58 


8 


.57429 


.81865 


.68849 


.80650" 


.60251 


.79811 


.61635 


.78747 


.68000 


.77660 


57 


4 


.57468 .818181 


.58878 


.80883 


.60274 


.79793, 


.61658 


.78729 


.68028 


.77641 


66 


6 


.57477 


.81838 


IU20CM 


.80816 


.60298 .79776 


.61681 


.78ni 


.63045 


.77688 


66 





.57801 


.81815 


.58980!. 80799 


.60321 .79758 


.61704 


.78694 


.68088 


.77806 


54 


7 


.67584 


.81-798 


.58948 .80788 


.60344 .79741 


.61726 


.78676 


.68090 


77566 


68 


8 


.57648 


.81788 


.58967 .80765 


.60367 


.79723 


.61749 


.78668. 


.68118 


.77868 


52 


9 


.67672 


.81766 


.58990 .80748 


.60390 


.79706 


.617r8 


.78640 


.68186 


.77560 


61 


10 


.67596 


.81748, 


.69014 .80^^ 


.60414 


.79688 


.61796 


.78622 


.68168 


.77681 


60< 


11 


.67619 


.817811 


.59037 .80713! 


.60487 


.79671 


.61818 


.78804 


.68180 


.77B18 


49 


12 


.57648 


.81714 


.59061 .80696: 


.60460 


.79663 


.61841 


.78566 


.68808 


.77494 


48 


18 


.57667 


.81698 


.590641.806791 


.60488 


.79685 


.61864 


.78568 


.68886 '.7747« 


47 


14 


.57691 


.81681 


.59108 


.80668 


.60506 


.79618 


.61887 


.78560 


.68848 .77468 


46 


15 


.57715 


.81664 


.59131 


.80644 1 


.60529 


.79600 


.61909 


.78532 


.68871 77489 


46 


16 


.57788 


.81647 


.59154 


.80627 


.60558 


.79583 


.61932 


.78514 


.68893 .77481 


44 


17 


.57768 


.81681 


.59178 


.80610 


.60576 


.79566! 


.61965 


.78496 


.68816 .77402 


48 


18 


.67786 


.61614 


.59201 


.80693 


.60599 


.795471 


.61978 


.78478 


.68388 77884 


48 


19 


.57810 


.81597 


.59885 


.80576; 




.79580 


.62001 


.78460 


.68861 .7r866 


41 


SO 


.57888 


.81580 


.59848 


.80558| 


.60645 


.79512 


.68024 


.78448 


.68888 


.77847 


40 


SI 


.57867 


.81568 


.69272 


.80541 


.60668 


.79494 


! .68046 .78424; 


.68406 


.77889 


89 


i 


.67881 
.57904 


.81546 
.81530 


.59895 
.59318 


.805241 
.805071 


.60691 
.60714 


.79477 
.79459 


.62069 .78406, 
.62098 .78887! 


.68428 


.77810 


88 
87 


.68451 


.77292 


24 


.57988 


.81518 


.59342 


.80489 


.60rd8 


.79441 


.<8115 .78869; 


.68478 .77278 


86 


26 .57958 


.81496 


.598651.80478 


.60761 


.79424 


.62188 .78851 


.68496 .77286 


85 


88 


.5T»76 


.81479. 


.59389!. 80456 


.60784 


.79406 


.62160 .78883 


.68518 


.77888 


84 


27 


.57999 


.81463 


.59412 


.80438 


.60807 


.79388 


.62183 .78315 


.63540 


.77218 


38 


28 


58088 Hii4K> 


59436 


80480 


60630 


79871 


62206 7f^sfi7i 


68668 


77199 


88 


29 


.68047 


.81488: 


.59459 


!80103 


'60R53 


!79353 


.62229 


.78879 


.68686 


.77181 


ffl 


30 


.68070 


.81412! 


.59482 


.80386 


.60876 


.79885 


.62251 


.78861 


.68608 


.77168 


80 


81 


.68094 


.81895! 


.B0SO6 


,80968' 


.60609 


.79818 


.68274 


.78848 


.68680 


.77144 


29 


88 


.58118 .81378 


.59589 


.80651; 


.60928 


.79300 


.68897 


.78885, 


.68653 


.77126 


28 


88 


.58141 .818611 


.69558 


.80834; 


.60045 


.79288 


.62380 


.78806 


.63676 


.77107 


87 1 


84 


.58165 .81344 


.59576 


.806161 


.60068 


.79864 


.62842 


.78188! 


.68608' 77068 


86 


85 


.58189 :.8]887| 


.59599 


.80899: 


.60091 


.79847 


.68865!. 78170 


.68780 .77070 


85 


86 


.582181.818101 


.59682 


.80288; 


.61015 


.79889 


.62888 .78162 


.687421.77061 


24 1 


87 


.58886 .81898 


.69646 


.80264' 


.61088 


.79811 


.68411 .78184 


.687661.77088 


28 


88 


.588601.81276 


.59669 


.80847, 


.61061 


.79193 .684881.781161 


.68787!. T7014 


28 


89 


.58283 .81859 


.59693 .808801 


.61084 


.79176 


.62456 .780081 


.68810 


.76996 


21 


40 


.68807 


.SLSCS 


.69716 


.OUKIX. 


.61107 


.79158 


.62479 


.TBUTV, 


.68882 


.76977 


20 


41 


.58890 


.81885 


.60789 


.801961 


.61180 


.79140 


.62502 


.78061 


.68854 


.76969 


19 


48 


.58354 


.81808 


.59763 


.80178, 


.61153 


.79188 




.78043 


.63877 


.76940 


18 


48 


.58378 


.81191 


.59786 


.80160 


.61176 


.79105 


.62547 


.78085 


.63899 


.76981 


17 


44 


.68401 


.81174 


.59809 


.80143 


.61199 


.79087 


.62570 


.78007 


.63988 


.76008 


16 


46 


.58485 


.'81157 


.59832 .801251 


.61222 


.79069 


.625921.77988 


.68944;. 76884 


15 i 


46 


.58449 


.81140 


.69856!. 80108! 


.61245 


.79051 


.626151.77970 


.63966 .76866 


14 


47 


.58472 


.81128 


.598791.800911 


.61288 


.79083 


.62688 


.77958| 


.68989 .76847 


18 


48 


.58496 


.81106 


.599081.80078 


.61291 


.79016 


.62660 


.77934 


.64011,. 76888 


12 


49 


.58519 


.81089 


.59926 .80056 


.61814 


.78998 


.62683 


.77916; 


.64088 


.76810 


11 


60 


.58518 


.810781 
.81066' 


.50949 


.80038 


.61887 


.78960 


.62706 


.778971 


.64066 


.76791 


10 


61 


.68667 




.50972 


.80021' 


.61860 .7HBQ2 


.68728 


.77879 


.64078 


.76772 


9 


68 


.68690 


.81088 


.59995 


80003 


.618831.78944 


.62751 


.778611 


.64100 


.78754 


8 


63 


.68614 


.81081 


.60019 


.79986 


.614061.78986 


.62774 


.77848 


.64183 


.76736 


7 


64 .58637 


.81004 


.60042 


.79968 


.61489 


.78906, .62796 


.TJW4 


.64146 


.76n7 


6 


65 1.58661 


.80987 


.60065 


.79951 


.61451 


.78891! '62819 


.77806 


.64167 


.76686 


6 


66 1.58684 


.80970 


.60089 


.79934 


.61474 


.78873 i .62842 


.77788 


.64190 


.76679 


4 


57 


.58708 


.80958 


.60112 


.79916 


.61497 


.78855 .02864 


.77769, 


.64212 .76661 


8 


58 


.58781 


.80936 


.60135 


.79899 


.61080 


.78837, .68887 


.77751 


.64284 .76642 


2 


59 


.68765 


.80919 


.60158 


.79881; 


.61543 


.78819 .68909 


.77733 


.64256 .76688 


1 


80 ;. 58779 


.80908 


.60188 


.79864, 


.61566 


.788011 .68938 


.77715 


.64279 .76604 





/ 


Cosin 


Sine 


Cosin 


Sine 


Cosin 


Sine"! 


Cosin 


Bine 


CodniSine 






54- 1 


580 1 


M 


5- 1 


61- 1 


00* 



493 



TABLE IV.— NATURAL SINES AND COSINES. 




494 



TABLE v.— NATURAL TANGENTS AND COTANGENTS. 



/ 


0* 1 


!• 1 


2« , 


8- 1 


/ 
60 


Tang 
.00000 


Cotang 


Tang 
.01746 


Cotang 


Tang 
.0849r 


Cotang 


Tang 


Ootang 


Infinite. 


67 8900 


26.6868 


.06241 


19.0811 




.00029 


8487.75 


.01775 


66.8806 


.03521 


88.8994 


.06270 


18.9765 


60 




.00068 


1718.87 


.01804 


66.4415 


.03550 


88.1664 


.06899 


18.8711 


68 




.00087 


1145.92 


.01833 


64.6618 


.08579 


27.9878 


.06828 


18.7678 


67 




.00116 


869.486 


.01868 


63.7086 


.03609 


27.7117 


.06357 


18.6656 


66 




.00146 


687.549 


.01891 


62.8821 


.03636 


27.4899 


.06387 


18.6645 


55 




.00175 


578.957 


.01920 


52.0807 


.06667 


27.2715 


.05416 


18.4645 


64 




.00204 


491.106 


.01940 


61.8032 


.03696 


87.0666 


.06446 


18.8666 


58 




.00238 


429.718 


.01978 


60.5485 


.08725 


86.8460 


.05474 


18.2677 


68 




.00268 


881.971 


.02007 


49.8157 


.03754 


26.6867 


.06608 


18.17t» 


61 


10 


.00891 


848.774 


.09036 


49.1098 


.03788 


96.4316 


.06688 


18.0760 


60 


11 


.00620 


812.521 


.02066 


46.4181 


.08813 


96.2296 


.05568 


17.9608 


49 


12 


.00849 


286.478 


.02096 


47.7895 


.06848 


26.0307 


.05591 


17.6868 


48 


18 


.00878 


264.441 


.02124 


47.0658 


.08871 


96.8848 


.05620 


17.7084 


47 


14 


.00407 


215.668 


.02153 


46.4489 


.08900 


95.6418 


.05649 


17.7016 


46 


16 


.00486 


229.188 


.02188 


45.8294 


03929 


25.4617 


.06678 


17.6106 


46 


16 


.00165 


214.868 


.02211 


46.2261 


.03958 


25.2644 


.05708 


17.6906 


44 


17 


.00496 


202.219 


.02240 


44.6886 


.08967 


25.0798 


.06787 


17.4314 


48 


18 


.00624 


190.984 


.02269 


44.0661 


.04016 


24.8978 




17.8488 


48 


19 


.00558 


180.988 


02298 


43.5061 


.04046 


94.7185 


.05796 


17.2566 


41 


90 


.00683 


171.885 


.02328 


42.9641 


.04076 


94.6418 


.06894 


17.1608 


40 


SI 


.00611 


168.700 


.02357 


42.4838 


.04101 


24.8676 


.06854 


17.0687 


80 


28 


.00640 


166.260 


.02886 


41.9158 


.04183 


94.1957 


.05868 


16.9990 


88 


S8 


.00669 


149.466 


.09415 


41.4106 


.04168 


84.0268 


.05912 


16.9160 


87 


94 


.00606 


148.287 




40.9174 


.04191 


88.8593 


.05941 


16.8819 


86 


95 


.00727 


187.607 


.02478 


40.4856 


.04220 


23.6946 


.05970 


16.7496 


86 


26 


.00756 


132.219 


.02509 


89.9655 


.04250 


23.5321 


.06999 


16.6681 


84 


27 


.00785 


127.821 


.02531 


89.5059 


.04279 


23.8718 


.06029 


16.6874 


83 


28 


.00815 


122.774 


.02560 


89.0568 


.04306 


98.2187 


.06058 


16.5076 


82 


80 


.00644 


118.540 


.0:^89 


88.6177 


.04887 


93.0677 


.06087 


16.4288 


81 


80 


.00878 


114.689 


.02619 


88.1886 


.04866 


98.9068 


.06116 


16.8490 


80 


81 


.00902 


110.892 


.02648 


87.7686 


.04896 


88.7519 


.06146 


16.2728 


29 


82 


.00981 


107.426 


.02677 


87.8579 


.04424 


22.6080 


.06175 


16.1958 !28l 


88 


.00960 


104.171 


.02706 


86.9560 


.M454 


22.4541 


.06904 


16.1190 


27 


84 


.00989 


101.107 


.02785 


86.5627 


.04483 


22.8061 


.06238 


16.0485 


96 


85 


.01018 


98.2179 


.02784 


86.1776 


.04512 


22.1640 


.06262 


16.9687 


25 


86 


.01W7 


96.4896 


.02798 


85.8006 


.04541 


je2.0217 


.06291 


16.8945 


94 


87 


.01076 


98.9086 


.02829 


85.4812 
86.06« 


.04570 


21.8818 


.06821 


15.6211 


93 


88 


.01105 


00.4688 


.02851 


.04599 


21.7426 


.06850 


16.7488 


22 


89 


.01185 


88.1486 


.02881 


84.7161 


.(M628 


81.6056 


.06879 


16.67B9 


21 


40 


.01164 


85.9886 


.08910 


84.8678 


.04656 


21.4704 


.06406 


16.6048 


90 


41 


.01198 


83.8486 


.02939 


84.0278 


.04687 


21.8869 


.06487 


16.6840 




42 


.01228 


81.8470 


.02968 


88.6985 


.04716 


21.2049 


.06467 


15.4688 




43 


.01251 


79.9434 


.02997 


38.3668 


.04746 


91.0747 


.06496 


15.8948 




44 


.01280 


78.1268 


.06086 


88.0452 


.04774 


90.9460 


.06626 


15.3254 




45 


.01809 


76.8900 


.08066 


82.7308 


.04808 


90.8188 


.06554 


15.95n 




46 


.01888 


74.7292 


.00084 


82.4218 


.04838 


20.6988 


.06584 


15.1893 




47 


.01867 


78.1890 


.03114 


82.1181 


.04862 


80.6601 


.06618 


16.1222 




48 


.01896 


71.6151 


.08148 


31.8206 


.04891 


20.4466 


.06642 


16.0587 




49 


.014S6 


70.1583 


.03173 


81.6284 


.04920 


90.8868 


.066n 


14.9696 




50 


.01465 


68.7501 


.08201 


81.2416 


.04949 


90.2066 


.06700 


14.9944 




61 


.01484 


67.4010 


.08230 


80.9590 


.04978 


90.0678 


.OOVISO 


14.8506 




68 


.01518 


66.1066 


.08269 


80.6888 


.06007 


19.9708 


.06759 


14.7D54 




53 


.01542 


64.8560 


.06286 


80.4116 


.06087 


19.8646 


.06788 


14.7817 




64 


.01571 


63.6567 


.03817 


80.1446 


.05066 


19.7408 


.08817 


14 6665 




55 


.01600 


62.4992 


.03346 


89.8823 


.06095 


19.6278 


.06847 


14.6069 




66 


.01629 


61.3829 


.03376 


29.6246 


.06124 


19.5156 


.06876 


14.5438 




67 


.01658 


60.3066 


.03106 


89.8711 


.06158 


19.4061 


.06906 


14.4828 




58 


.01687 


60.2669 


.08434 


29.1220 


.05182 


19.2969 


.06934 


14.4212 




50 


.01716 


66.2612 


.06463 


28.8771 


.05212 


19.1879 


.06963 


14.3607 




!2 


.01746 


67.2900 


.08492 


28.6868 


.05241 
Ootang 


19.0811 


.06993 
Cotang 


14.8007 


_0 


Cotang 


Tang 


Ootang 


Tang 


Tang 


Tang 


89« 


88<» 


87* 1 


8e« 



495 



TABLE v.— NATURAL TANGENTS AND COTANGENTS. 





4- 


5- 1 


1 6« 1 


I 7- 


/ 
60 


Tang 
.06098 


Cotang 


Tang 
.06749' 


Cotang 


Tang 


Ck)tang 


Tang 

.12278" 


Cotang 


14.8007 


11.4801 


.10610 


9.61436 


8.14485 


1 


.07088 


14.8411 


.08778 


11.8919 


.10540 


9.48781 


.12308 


8.12481 


60 


8 


.07061 


u.iesn 


.08807 


11.3540 


.10569 


9.46141 


.18338 


8.10680 


68 


8 


.07080 


14.1885 I 


.06837 


11.8163 


.10599 


9.43515 


.15367 


8.06600 


67 


4 


.07110 


14.0655 


.08866 


11.8789 


.10628 


9.40904 


.12897 


8.06674 


66 


6 


.07139 


14.0079 


.06895 


11.2417 


.10657 


9.88307 


.12426 


8.04756 


66 


6 


.07168 


13.9507 


.08985 


11.2048 


.10687 


9.35784 


.12456 


8.02848 


64 


7 


.07197 


18.8940 


.08954 


11.1681 


.lono 


9.83155 


.12485 


8.00948 


68 


8 


.07iB7 


13.8378 


.0898^ 


11.1316 


.10746 


9.80599 


.12515 


7.99068 


68 


9 


.07256 


13.7821 


.09013 


11.0954 


.10775 


9.28058 


.12544 


7.97176 


61 


10 


.07886 


18.7267 


.09048 


11.0594 


.10606 


9.25680 


.12574 


7.96808 


60 


11 


.07314 


13.6719 


.09071 


11.0287 


.10884 


9.28016 


.18606 


7.98438 


49 


12 


.07344 


13.6174 


.09101 


10.9688 


. .10603 


9.20516 


.12C33 


7.91588 


48 


18 


.07878 


13.6684 


.09130 


10.9529 1 


.10898 


9.18028 


.12662 


7.89784 


47 


14 


.07402 


13.5096 


.09159 


10.9178 


.10928 


9.16564 


.12698 


7.87896 


46 


16 


.07481 


13.4566 


, .09189 


10.8889 


.10952 


9.13098 


.18728 


7.86064 


46 


16 


.07461 


18.4089 


.09818 


10.8488 


.10081 


9.10646 


.12751 


7.84842 


44 


17 


.07«W 


13.8515 


' .09847 


10.8139 


.11011 


9.06211 


.18781 


7.824S6 


48 


18 


.07519 


18.8996 


.09877 


10.7797 


.11040 


9.057S9 


.12810 


7.60688 


42 


19 


.07548 


13.2480 


.09306 


10.7457 


.11070 


9.08379 


.18840 


7.78886 


41 


90 


.09578 


13.1969 


.09835 


10.7119 


.11099 


9.00988 


.12869 


7.77085 


40 


21 


.07607 


18.1461 


.09865 


10.6788 


.11128 


8.98596 


.18889 


7.76864 


89 


82 


.07686 


18.0058 


.09894 


10.&450 


.11168 


8.9G227 


.18989 


7.78480 


88 


28 


.07666 


18.0458 


.09188 


10.6118 


.11187 


8.98867 


.12958 


7.71716 


S7 


24 


.07605 


18.9962 


.0M53 


10.5789 


.11217 


8.91580 


.12988 


7.69967 


86 


26 


.07724 


12.9469 


.09188 


10.5468 


.11246 


8.89185 


.18017 


7.68806 


85 


26 


.07753 


12.8961 


.09511 


10.5136 


.11876 


8.86862 


.13047 


7.66466 


84 


27 




12.8196 


.09541 


10.4818 


.11805 


8.84561 


.18076 


7.64782 


83 


88 


.07B12 


12.8014 


.09570 


10.4491 


.11335 


8.82252 


.13106 


7.63005 


82 


89 


.07841 


18.7586 


.00600 


10.4172 


.11364 


8.79964 


.13186 


7.61887 


81 


80 


.07870 


18.7062 


.00629 


10.8854 


.11394 


8.77689 


.13165 


7.69675 


80 


31 


.07899 


12.6591 


.09658 


10.8638 


.11483 


8.75425 


.18195 


7.67872 


29 


a2 


.07929 


18.6184 


.09688 


10.3824 


.11468 


8.78172 


.18884 


7.56176 


88 


88 


.07958 


12.5660 


.09717 


10.2913 


.11488 


8.70931 


.13254 


7.64487 


27 


84 


.07967 


18.5199 


.09746 


10.2602 


1 .11511 


8.68701 


.18284 


7.62606 


26 


86 


.06017 


12.4748 


.09776 


10.2294 


' .llMl 


8.66482 


.18313 


7.61138 


86 


86 


.06016 


12.4288 


.09806 


10.1988 


.11570 


8.64875 


.18343 


7.49466 


84 


87 


.08075 


12.8838 


.09834 


10.1683 


.11600 


8.62078 


.18872 


7.47806 


23 


88 


.08104 


13.3390 


.09861 


10.1981 


.11689 


8.59898 


.13402 


7.46154 


82 


88 


.06184 


18.8946 


.09893 


10.1080 


.116*3 


8.57718 


.13432 


7.44509 


21 


40 


.06168 


18.8005 


.09923 


10.0780 


.11688 


8.55655 


.18461 


7.48871 


80 


41 


.06198 


18.2067 


.09962 


10.0488 


.11718 


e.68408 


.18491 


7.41840 


19 


42 


.06881 


18.1688 


.09961 


10.0187 


.11747 


8.51260 


.13581 


7.89616 18 


48 


.06251 


12.1801 


.10011 


9.96931 


' .117T7 


8.49188 


.13550 


7.37999 :i7 


44 


.06280 


i2.orra 


.10040 


9.96007 


.11806 


8.47007 


.13580 


7.86889 .16 


46 


.08809 


12.0846 


.10069 


9.98101 


.11836 


8.44896 


.13609 


7.84786 ;i5 


46 


.06839 


11.9928 


.10099 


9.90811 


.11865 


8.42796 


.18639 


7.38190 


14 


47 


.06868 


11.9504 


.10128 


9.87338 


.11895 


8.40706 


.13669 


7.31600 


18 


48 


.06897 


11.9067 


.10158 


9.81488 


.11924 


6.88625 


.13608 


7.30018 


18 


49 


.08427 


11.8678 


.10187 


9.81641 


.11954 


8.86655 


.18^8 


7.88448 


11 


60 


.06466 


11.8862 


.10216 


9.78817 


.11963 


8.34496 


.18758 


7.86878 


10 


61 


.06486 


11.7858 


.10246 


976009 


.18013 


8.38446 


.18787 


7.86810 


9 


62 


.06514 


11.7448 


.10275 


9.73817 


.18048 


8.3W06 


.18817 


7.83764 


8 


68 


.06544 


11.7015 


.10805 


9.70141 


.12072 


8.88876 


.18846 


7.88804 


7 


64 


.06578 


11.6646 


.10884 


9.67880 


.12101 


8.26355 


.18876 


7.80661 


6 


66 


.06602 


11.6248 


.10863 


9.64935 


.12131 


8.24345 


.18906 


7.19185 


6 


66 


.06632 


11.5858 


.10896 


9.62805 


.18160 


8.22344 


.18936 


7.17604 


4 


67 


.06661 


11.5461 


.10482 


9.69490 


.18190 


6.80858 


.13965 


7.16071 


8 


68 


.06600 


11 5078 


.10452 


9.56791 ' 


.18819 


8.18870 


.13996 


7.14658 


8 


69 


.06780 


11.4685 


.10481 


9.54106 


.18249 


8.16396 


.14084 


7.13048 


1 


60 

/ 


.06749 


11.4801 


.10610 
Cotang 


9.51436 ' 


.18278 


8.14435 


.14054 


7.11587 


_0 


Ciotangi Tang 


Tang { 




Tang 


Cotang 


Tang 


86* 


i a 


4- 1 


1 88* 1 


82* 



496 



TABLE v.— NATURAL TANGENTS AND COTANGENTS. 



/ 


8* 


90 


10» 


!!• 


/ 


Tan^ 


CotanfiT 


Tang 


Cotong 


Tang 


Cotang 


Tang 


Cotang 




.140&4 


7.11587 


.15838 


6.81376 


.1763? 


5.67128 


.19486 


5.14466 


60 




.140&1 


7.10088 


.15868 


6.30189 


.17608 


6.66165 


.19468 


5.18666 


60 




.14118 


7.08646 


.15898 


6.29007 


.17698 


6.66205 


.19406 


6.188l» 


58 




.14148 


7.07059 


.15928 


6.27829 


.177V8 


5.&1248 


.19520 


5.18060 


57 


4; .14173 


7.05679 


.15938 




.17758 


6.68295 


.19560 


5.11879 


56 




.14202 


7.ftll06 


.15988 


6.25486 


.17788 


5.62844 


.19589 


5.10490 


65 




.14282 


7.02687 


.16017 


6.21821 


.17813 


6.61807 


.19610 


5.09704 


54 




.14268 


7.01174 


.16017 


6.23160 


.17848 


5.60458 


.19649 


6.06981 


58 




.14291 


6.99718 


.16077 


6.22003 


.17873 


5.50511 


.19680 


5.08180 


58 




.14321 


6.98268 


.16107 


6.20851 


.17908 


5.58578 


.19710 


5.07360 


61 




.14361 


6.96828 


.16187 


6.19708 


.17988 


5.57688 


.19740 


5.06664 


50 




.14881 


6.96385 


.16167 


6.18660 


.17968 


5.56706 


.19770 


6.06809 


49 




.14410 


6.93952 


.16106 


6.17419 


.17996 


5.55777 


.19601 


5.06087 


48 




.14440 


6.92525 


.16226 


6.16288 


.18088 


5.54851 


.19681 


5.04867 


47 




.14170 


6.91104 


.16256 


6.15151 


.18058 


5.58087 


.19861 


5.08400 


46 




.14199 


6.89688 


.16286 


6.14028 


^.18088 


6.58007 


.19691 


5.08784 


45 




.14529 


6.8827S 


.16816 


6.12899 


.18113 


5.52090 


.19981 


6.01071 


44 




.14559 


6.86874 


.16816 


6.11770 


.18148 


6.61176 


.19968 


6.01810 


43 


18 


.14588 6.85475 1 


.16376 


6.10064 


.18178 


5.50864 


.19968 


6.00461 


48 


19 


.14618 


6.810® 


.18105 


6.09352 


.18808 


5.49356 


.80018 


4.99009 


41 


20 


.14648 


6.82694 


.16486 


6.06444 


.18288 


5.48151 


.80048 


4.96040 


40 


21 


.14678 


6.81812 


.16465 


6.07840 


.18268 


6.47548 


.80078 


4.08188 


89 


22 


J4707 


6.79936 


.16495 


6.06240 


.18298 


6.46648 


.80108 


4.07438 


38 


28 


.14737 


6.78564 


.16525 


6.05148 


.18328 


6.45751 


.80138 


4.O660O 


87 


24 


.14767 


6.77199 


.16566 


6.04051 


.18858 


6.44857 


.80164 


4.06046 


86 


26 


.14796 


6.75838 


.16585 


6.02968 


.18384 


5.43966 


.80194 


4.06801 


85 


26 


.14826 


6.74488 


.16615 


6.01878 


.18414 


6.43077 


.80224 


4.04460 


84 


27 


.14856 


6.78138 


.16615 


6.00797 


.18444 


6.42198 


.80854 


4.08781 


88 


28 


.14886 


6.71780 


.16674 


5.99720 


.18174 


6.41309 


.80885 


4.98984 


88 


29 


.11915 


6.7M60 


.16701 


5.96646 


.18504 


5.40129 


.80615 


4.02240 


81 


80 


.14945 


6.69116 


.16784 


6.97576 


.18584 


5.88668 


.80846 


4.01516 


80 


81 


.14975 


6.67787 


.16764 


6.96510 


.18664 


6.88877 


.80876 


4.00786 


89 


82 


.15005 


6.66168 


.16794 


5.95148 


.18594 


6.87806 


.80406 


4.90066 


88 


88 


.15034 


6.65144 


.16824 


5.94300 


.18684 


5.38966 


.80496 


4.80680 


87 


84 


.15061 


6.63831 


.16854 


5.93885 


.18654 


5.86070 


.80166 


4.88605 


86 


85 


.15094 


6.62528 


.16881 


5.92288 


.18684 


5.86806 


.80497 


4.87888 


85 


86 


.15121 


6.61219 


.16914 


5.91236 


.18714 


5.84846 


.80687 


4.87108 


84 


87 


.15158 


6.59921 


.16944 


5.90191 


.18r45 


6.88487 


.80557 


4.86444 


88 


88 


.15183 


6.58627 


.16974 


5.89151 


.18775 


5.82681 


.80688 


4.85787 


88 


89 


.15218 


6.57339 


.17004 


6.88114 


.18806 


6.31778 


.80618 


4.85018 


81 


40 


.16213 


6.56066 


.17088 


6.87060 


.18886 


6.80988 


.80648 


4.84800 


80 


41 


.15»72 


6.54777 


.17063 


6.86061 


.18865 


6.80060 


.80670 


4.88600 


10 


42 


.15302 


6.53508 


.17098 


5.85084 


.18896 


6.29286 


.80709 


4.82882 


18 


48 


.15882 


6.52384 


.17128 


5.81001 


.18925 


5.28398 


.80739 


4.82175 


17 


44 


.15303 


6.50970 


.17158 


5.82982 


.18955 


5.27553 


.80770 


4.81471 


16 


45 


.15891 


6.49n0 


.17188 


5.81966 


.18966 


5.26716 


.80600 


4.80789 


16 


46 


.15421 


6.48156 


.17218 


5.80958 


.19016 


5.25880 


.80680 


4.80068 


14 


47 


.15451 


6.47206 


.17248 


5.79914 


.19016 


6.25048 


.80661 


4.7D870 


18 


48 


.16481 


6.45961 


.17?78 


5.78938 


.19076 


6.24218 


.80601 


4.78678 


18 


49 


.15511 


6.44720 


.17308 


5.7r936 


.19106 


6.28391 


.80981 


4.77078 


11 


60 


.16540 


6.43484 


.17888 


5.76987 


.19186 


6.29666 


.80968 


4.77880 


10 


51 


.16670 


6.42288 


.17868 


6.75941 


.19166 


5.21744 


.80068 


4.76S06 





62 


.15600 


6.41026 


.17893 


5.74940 


.19197 


5.20925 


.81018 


4.75906 


8 


58 


.15630 


6.89804 


.17428 


5.78960 


.19227 


6.80107 


.81018 


4.7S819 


7 


64 


.15660 


6.38687 


.17153 


6.72974 


.19257 


6.19293 


.81078 


4.74584 


6 


55 


.15689 


6.87374 


.17483 


5.71992 


.19287 


6.18180 


.81104 


4.78851 


5 


66 


.15719 


6.36165 


.17513 


6.71018 


.19317 


6.17871 


.81184 


4.78170 


4 


57 


.15749 


6.84961 


.17548 


6.70097 


.19817 


5.16868