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^ 




WORKS OF CHARLES B. BREED 
GEORGE L. HOSMER 



PUBLISHED BT 



JOHN WILEY & SONS 



THE PRINCIPLES AND PRACTICE OP SURVBYINO 
Volume L Elementary Surveying. 

Part I. Use, Adjustment, and Care of Instruments. 
Part II. Surveying, Meridian Observations, Leveling; 

City, Mine, and Topographic Surveying. 
Part III. Computations. 

Part IV. Plotting, Finishing, and Filing Plans. 
Tablrs. Logarithmic, Trigonometric, Stadia, and others. 
Third Edition. Rtvistd. 8vo, 540 pages, 200 cuts. Cloth, 
ftj.oo. 

Volume II. Higher Surveying. 

Part I. Triangulation, Astronomical Observations; Pre- 
cise, Trigonometric and Barometric Leveling. 

Part II. Stadia, Plane Table, Photographic Surveying, 
Relation of Geology to Topography. 

Part III. Hydrographic Surveying, Stream Gauging. 

Part IV. Constructing and Finishing Maps. 

Tables. Geodetic Position, Polyconic Projections. Baro- 
metric, Stadia, and others. 
First Edition. 8vo, 430 pages, 155 cuts and 7 full page 

plates. Cloth, fa.50. 



THE 

PRINCIPLES AND PRACTICE 

OF 

SURVEYING 



Volume I. Elementary Surveying 



by 
CHARLES B. BREED and GEORGE L. HOSMER 

Associate Professor Assistant Professor 

of Civil Engineering of Civii Engineering 

MASSACHUSETTS INSTITUTE OP TECHNOLOGY 



Third %w*j5r& • ; 
Total Issue Fourteen Thousand 



NEW YORK 
JOHN WILEY & SONS 



London: CHAPMAN & HALL, Limited 
1912 



TA <^ 
if// 

AMron. Dtpt. 




Copyright, 1906, 1908 

BY 
CHARLES B. BREED and GEORGE L. HOSMER 



PRINTED AND BOUND AT THE PLIMPTON PRESS, 
NORWOOD, MASS., U.S.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 
subjects being left for a later volume. 

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

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

iii 

263636 



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



PREFACE V 

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

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

C. B. B. 

G. L. H. 
Boston, Mass., September, 1906. 



PREFACE TO THE THIRD EDITION 

In this third edition the portion of Chapter V which deals with 
the Public Lands System and Chapter XI on Mine and Mountain 
Surveying have been thoroughly revised. The authors have been 
fortunate in securing the services of Professor William C. Hoad, of 
the University of Kansas, in revising the articles on Public Lands 
Surveys. The chapter on Mine and Mountain Surveying has 
been rewritten by Mr. Blarney Stevens. 

In order to meet the requirements of the schools in which the 
subject of stadia is included in the elementary course, the authors 
have deemed it advisable to add Appendix A, containing the 
fundamental principles of this method of surveying. While all 
of the important principles have been included in this appendix, 
the complete treatment of the subject, especially on the practical 
side, will be found in Volume II, Chapter IV. 

Appendix B, which treats of the theory and use of the Plani- 
meter, and Appendix C, in which is explained in detail the process 
of mounting drawing paper, appear for the first time in this edition. 

C. B. B. 

G. L. H. 

Boston, Mass., July, 1908. 



CONTENTS. 

PART I. 

USE, ADJUSTMENT, AND CARE OF INSTRUMENTS. 

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

Art. Pack 

i. Definition 3 

2. Purposes of Surveys 3 

3. Horizontal Lines 3 

instruments for measuring lines. 

4. The Chain 3 

5 Metric Chain 5 

6. The Tape 5 

7. Steel Tapes 5 

8. The Stadia 6 

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

ter 6 

MEASUREMENT OF LINES. 

ia Measurement of a Horizontal Line with a Chain 7 

xi. Measurement of a Horizontal Line with a Chain on Slop- 
ing Ground 9 

12. Measurement with a Steel Tape 10 

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

14. Common Mistakes in Reading and Recording Measurements ii 

15. Avoiding Mistakes n 

16. Accuracy Required 12 

17. Amount of Different Errors 12 

18. Pull 12 

19. Temperature 13 

20. Alignment 13 

21. Sag 13 

22. Effect of Wearing on Length of the Chain 14 

23. 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 



V1U CONTENTS 

Ait. Paoi 

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

28. Declination of the Needle 19 

29. Variations in Declination 20 

3a Isogonic Chart 25 

31. Observations for Declination 25 

ADJUSTMENTS OF THE 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 Remagnetlze 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. Eccentricity 44 

USE OF THE TRANSIT. 

56. Setting up the Transit 45 

58. To Measure a Horizontal Angle 47 

59. To Measure an Angle by Repetition 48 

61. To Lay Off an Angle by Repetition 50 

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

63. Neither Point Visible from the Other 51 

64. Prolonging a Straight Line 52 

65. Methods of Showing Sights 52 

66. Signals 53 

67. To Measure a Vertical Angle 54 

68. Precautions in the Use 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. Adjustment of the Standards 59 

74. Adjustment of the Telescope Bubble 59 

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

Vertical Arc 59 

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

77. Adjustment of the Objective Slide 60 

78. Shop Adjustments 61 

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

the Transit 61 

8a Care of Instruments 61 

81. Common Sources of Error in Transit Work 62 

82. Common Mistakes in Transit Work * 62 

THE SOLAR ATTACHMENT. 

83. Description of Solar Attachment 65 

84. The Celestial Sphere 65 

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

86. Computation of Declination Settings . 68 

87. Comstock's Method of 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 of the Cross-Hairs 70 

92. Adjustment of Telescope Bubble 71 

Problems 71 

Chapter IV. — Measurement of Differences of Elevation. 

93. Level Surface 72 

94. The Spirit Level 72 

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

the level. 

97. The Wye Level 73 

98. The Dumpy Level 74 

99. Comparison of Wye and Dumpy Levels 77 

100. The Locke Hand Level 77 

leveling rods. 

102. Boston Rod 78 

103. New York Rod 80 

104. Philadelphia Rod 81 

105. Special Self-Reading Rods 81 

106. Tape Rod 81 

107. Precise Level Rod 82 

108. Advantages of the Self-Reading Rod 82 

109. Attachments to the Rod for Plumbing 83 

no. Effect of Heat and Moisture 83 



X CONTENTS 

USE OF THE LEVEL AND ROD. 

Art. Pac» 

112. To Level thz Instrument ... 83 

113. To Take a Rod-Reading 84 

114. Waving the Rod 84 

115. Signals * 85 

116. Differential Leveling 85 

117. The Proper Length of Sight 87 

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

Leveling 87 

119. Precautions in Level Work 88 

ADJUSTMENTS OF THE LEVEL. 

I. ADJUSTMENTS OF THE WYE LEVEL. 

120. Adjustment of the Cross-Hair 88 

122. Adjustment of the Level Tube 89 

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

125. Adjustment of the Wyes 90 

II. ADJUSTMENTS OF THE DUMPY LEVEL. 

126. Adjustment of the Cross-Hairs 91 

127. Adjustment of the Bubble Tube 91 

128. The Direct or "Peg" Adjustment 91 

129. Adjustment of the Locke Hand Level 93 

130. Common Sources of Error in Leveling 94 

131. Common Mistakes in Leveling 94 

Problems 94 



PART II. 
SURVEYING METHODS. 

Chapter V. — Land Surveying. 

132. Surveying for Area 99 

133. Surveying for Area with Compass and Chain ....... 99 

SURVEY OF FIELD WITH TRANSIT AND TAPE. 

134. Survey of a Field by a Traverse 101 

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 

Art. Pass 

146. Accuracy Required m 

147. Organization of Transit Party m 

148. Note-keeping 112 

149. Survey of a Field for a Deed 114 

150. 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 

154a. Process of Subdivision 123 

1546. Methods of Designating Lines and Areas 127 

154c. Field Methods 128 

155. To Establish a Parallel of Latitude 128 

155a. The Secant Method 129 

1556. The Tangent Method 131 

156. Initial Points 134 

157. Base-Line 134 

158. Principal Meridian 136 

159. Standard Parallels 136 

160. Guide Meridians 136 

161. Township Exteriors 138 

162. Subdivision of Townships 139 

163. Fractional Sections 142 

164- Summary of Objects and Data Intersected by the Line . . . 145 

165. Marking Corners 146 

165a. Witnessing Corners 148 

165ft. Witness Corners 149 

165c. Meander Corners 149 

166. Meandering 151 

167. Access to Records and Maps . 153 

168. Relocating Lost Corners 153a 

169. Subdivision of Sections 153d 

170. Convergence of the Meridians 154 

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

traverse lines. 

171. Traverses which do not Form Closed Figures 156 

172. Method of Checking Traverses which do not Form Closed 

Figures, — Checking by Astronomical Method 156 

173. Checking by Cut-Off Lines , 157 

174. Checking by Angles to a Distant Object 158 

1 75. Checking by Connecting with Tri angulation Points .... 158 

location op buildings from transit line. 

176. Methods of Locating Buildings 159 

177. Geometric Principles 159 



Xll CONTENTS 

Art. Page 

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

179. General Suggestions 160 

180. Typical Cases 160 

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

Plotting 161 

182. Building near Transit Line, Making a Slight Angle with it . 161 

Plotting 162 

183. Building Located Entirely by Direct Ties 162 

Plotting 163 

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

Plotting 164 

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

Plotting 164 

186. Buildings Located from Other Buildings 165 

187. Buildings of Irregular Shape 166 

188. Large City Buildings 166 

189. Location of Buildings by Angles and Distances 167 

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

MISCELLANEOUS SURVEYING PROBLEMS. 

191. Random Line 169 

192. Obstacles on Line 170 

193. Offsetting Transit Line 170 

194. Short Transit Sights 172 

195. Bisection Method 172 

196. Measuring Around a Small Obstacle 173 

197. Equilateral Triangle Method 174 

198. Inaccessible Distances 174 

199. By Right Triangle Method 174 

200. Intersecting Transit Lines 175 

201. By Swing Offset Method 175 

202. By Tangent Offset Method 176 

203. By Oblique Triangle Method 177 

204. To Obtain the Distance Between Two Inaccessible Points by 

Observation from Two Accessible Points 178 

205. To Obtain the Inaccessible Distance Between Two Accessible 

Points by Observation on Two Inaccessible Points of 

Known Distance Apart 179 

Chapter VII. — Observation for Meridian and Latitude. 

OBSERVATIONS FOR MERIDIAN. 

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

with the Transit 180 

207. Observation for Meridian on Polaris at Elongation 182 

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

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

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

211. Meridian Observation on Polaris at any Time with the Transit 188 



CONTENTS X1I1 

An. Pace 

212. Solar Observations 190 

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

the Forenoon and Afternoon 190 

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

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

OBSERVATIONS FOR LATITUDE. 

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

217. By the Altitude of the Sun at Noon 196 

Problems 197 

Chapter VIII. — Leveling. 

218. Definitions 198 

219. Leveling to Establish. Bench Marks 198 

222. Double Rodded Lines 201 

224. Bench Marks and Turning Points 202 

225. Leveling for Profile 203 

227. Cross-Sectioning 206 

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

229. Cross-Sectioning for Earthwork 207 

230. Road Cross-Sections 207 

231. Setting Slope Stakes 208 

232. Earthwork Notes for Road Cross-Sections 209 

233. Cross-Sections for Borrow-Pits 210 

234. Cross-Sections for Trench Excavation 210 

235. Leveling to Establish a Grade Line 210 

236. "Shootokj m" a Grade Ltne 211 

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

238. The Staff Gauge 212 

239. Leveling Across a River 213 

Problems 215 

Chapter IX. — City Surveying. 

240. Instruments Used 216 

241. Tapes and Tape Measurements 216 

242. Transits and Levels 217 

243. City Standard . 218 

city layouts. 

245. Streets 219 

246. Location of Streets 220 

247. Size of Blocks and Lots 222 

248. Width of Streets 222 

249. Street Grades 224 

250. The Datum Plane 226 

«5i. Establishing Bench Marks 226 

25a. Water and Sewer Systems 227 



XIV CONTENTS 

STAKING OUT CITY WORK. 

Art. Paob 

253. Staking out a New District 227 

254. Monuments 228 

255. Setting Stone Bounds ..." 230 

256. Curved Layouts 232 

257. Elements of a Circular Curve 233 

258. Staking out Circular Curves 234 

259. Deflection Angles 234 

260. Keeping the Notes 237 

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

262. First Method 238 

263. Second Method 239 

264. Curved Street Corners 239 

265. One Street Line Straight, the Other Curved ....!.. 239 

266. Both Street Lines Curved ....*. 240 

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 24Q 

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. Revisdsg 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 Descrdttion 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 Ld>je 265 

295. Contour Lines 265 

296. Characteristics of Contours 268 

297. Relation Between Contour Map and Profile ........ 270 



CONTENTS XV 

Aw. Paob 

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 

302. Mistakes in Sketching Contours 276 

303. Locating Contours 277 

304. Locating Contours by Cross-Sections 278 

305. Locating Contours by Profiles 278 

306. Locating Points on Contours 278 

307. Locating Contours by Means of the Hand Level 278 

308. Location of Streams and Shore Lines 280 

309. Contour Problems 280 

310. Intersection of Plane with Surface of Ground 280 

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

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

Chapter XI. — Mine and Mountain Surveying. 

313. General Remarks 285 

314. Definitions of Mining Terms 285 

315. Mining Transits 287 

316. Eccentric Telescopes 289 

317. Correction for Eccentricity 289 

318. Relative Merits of Attachments 289 

319. Adjustments of MrNTNG Transits 290 

320. Interchangeable Side and Top Telescope 293 

321. Combined Solar Attachment and Top Telescope 293 

322. Use of the Ordinary Transit in Mining Surveying 293 

323. Compasses used in Mines 294 

324. Magnetic Surveying 295 

UNDERGROUND SURVEYING. 

325. Transferring a Meridian into a Mine by use of the Transit 296 

326. Plumbing the Meridian down a Shaft 298 

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

Shafts 300 

328. Underground Traverses 301 

329. Establishing Station Points 302 

330. Underground Measurements 303 

331. Notes of a Mine Traverse 303 

332. Plotting a Mine Traverse 303 

333. Underground Leveling 304 

334. Mine Maps and Charts 304 

335. Laying out Mining Work 305 

336. Underground Surveying Problems 306 

336a. To Find an Ore Shoot by Driving a Level 306 

337. Vertical Angle Correction for Eccentricity of the Top Tele- 

scope 306 

337a. Vertical Angle Correction for Eccentric Bearing Telescope 307 



XVI CONTENTS 

Act. Pao* 

33&. To Establish a Boundary Line of the Claim Underground . . 307 

339. To Lay Out a Connection d* a Mine 308 

340. Plotting Geological Data 308 

340a. Location op Ore Shoot Displaced by Fault 309 

341. Hydraulic Surveying for Mines 311 

342. Testing for Ore by Electric Currents 312 

SURFACE SURVEYING. 

343. Surface Surveying n* Rugged Mount aw Regions 312 

344. Mine Boundaries. Appropriations Under United States Laws 313 

345. Surveying for Patent 315 

346. The Surveying of Boreholes 316 

347. Staring out the Probable Apex of a Vein 317 

348. Economic Princd?les 317 

Problems 319 



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-Thhud Rule 334 

365. Straightening Crooked Boundary Lines 336 

366. Area by Triangles 33 6 

367. Area of a Quadrilateral by Triangles 337 

368. Area of a Curved Corner Lot 338 

369. Rough Checks on Areas 339 

370. Plandceter 339 

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

372. Computations of Observations 34* 

computation of volume. 

373. Borrow-Pits 34 a 

374. Volume of Prismoid 344 

375. End Area Formula 344 

376. Prismoidal Formula 344 

378. Estimates for Grading 345 

382. Rough Estimates 349 

Problems 35° 



CONTENTS XVH 

Chapter XIII. — Area by Double Meridian Distances — 

Coordinates. 

Art. Paox 

383. Computation op Area 352 

384. Area by Double Meridian Distance Method 352 

386. Computation of Area of Compass Survey 356 

387. Balancing a Chain and Compass Traverse 358 

390. Double Parallel Distance 359 

391. Error of Closure 360 

39a. Computation of Area of a Transit and Tape Survey 360 

395. Balancing a Transit and Tape Survey 364 

306. Fractional Areas 365 

397. Supplying Missing Data 366 

401. Detecting Mistakes 368 

40a. The Subdivision of Land 368 

403. To Cut Off from a Traverse a Given Area by a Straight Ljne 

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 Fton> the Area Cut Off from a Traverse by a Line Running 

in a Given Direction from a Given Point in the Traverse . 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 Distant 

Object 372 

409. Calculation of Triangulation 372 

410. Coordinates 373 

411. To Determine the Area of a Field by Rectangular Coordi- 

nates 374 

Problems . . 376 



PART IV. 

PLOTTING. 

Chapter XIV. — Drafting Instruments and Material. 

engineering drafting instruments. 

415. Straight Edge 381 

416. Engineer's Scale 382 

417. Protractor 38a 

418. Semicircular Protractor 382 

4x9. Full-Circle Protractor 3*4 

491. Three- Armed Protractor 3&4 



XV111 CONTENTS 

Art. Page 

423. Pantograph 385 

424. Parallel Ruler 386 

425. Beam Compass 386 

426. Contour Pen 386 

427. Proportional Dividers . 387 

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

Spline 387 

DRAWING PAPERS. 

433. Drawing Paper for Plans . 388 

434. Tracing Paper and Tracing Cloth 389 

436. Cross-Section and Profile Papers 390 

438. Process Papers, — Blue-Prints . 391 

441. Vandyke Solar Paper 393 

442. Electrical Printing Frames 395 

443. Inks and Water-Colors 395 

Chapter XV. — Methods of Plotting. 

444. Laying Out a Plan 397 

445. Scale . . ." 397 

METHODS OF PLOTTING TRAVERSES. 

446. Plotting by Protractor and Scale 398 

447. Checks 398 

448. Protractor and T-Square 400 

449. Plotting by Rectangular Coordinates 401 

454. Checks 405 

455. Plotting by Tangents . . 405 

456. Checks 407 

457. Plotting by Chords . . 408 

458. Use of the Sine 409 

459. Checks 409 

METHOD OF PLOTTING DETAILS. 

460. Buildings, Fences, Streams, Etc 410 

461. Contours 410 

462. Cross-Sections 411 

463. Profiles 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 

Aw. Page 

473. Lettering 422 

474- Titles 423 

475. Notes 426 

476. Border Lines 427 

477. Meridians 427 

478. Scales 428 

479. Shrinkage op Drawing Papers 428 

480. Maps op Large Extent 429 

481. Inking in a Profile 430 

482. Cleaning Drawings 430 

483. Filing Drawings 431 

484. Indexing Drawings 432 

485. Filing Note-books 432 

486. Indexing Notes 433 

487. Other Records 433 



TABLES. 



I. Logarithms op Numbers 436 

II. Logarithmic Sines and Cosines 455 

III. Logarithmic Tangents and Cotangents 470 

IV. Natural Sines and Cosines 485 

V. Natural Tangents and Cotangents 494 

VI. Lengths op Circular Arcs: Radius — i 506 

VII. Mean Repractions in Declination 507 

VIII. Trigonometric and Miscellaneous Formulas 511 

IX. Circular Curve Formulas 513 

X. Geometric Formulas 513 

XI. Linear Measure 514 

XIL Square Measure 514 

XIII. Linear Measure — Metric System 514 

XIV. Square Measure — Metric System 514 

XV. Constants 515 

XVI. Stadia Reductions 516 

Greek Alphabet 519 



Appendix A — Principles op the Stadia 523 

Appendix B — The Planimeter 533 

Appendix C — Mounting Paper por Drawings 5 4 , 



THE PRINCIPLES AND PRACTICE OF 
SURVEYING. 



PART I. 
USE, ADJUSTMENT, AND CARE OF INSTRUMENTS. 



CHAPTER I. 

GENERAL DEFINITIONS. — MEASUREMENT OF LINES. 

i. 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 to hori- 
zontal distances. As a matter of convenience, measurements 
are sometimes taken on slopes, but the horizontal projection is 
afterward computed. The distance between two points as 
shown on a map then is always this horizontal projection. 

INSTRUMENTS FOR MEASURING LINES. 

4. THE CHAIN. — There are two kinds of chain in common use, 
the Surveyors (or Gunter*s) C/iain t and the Engineer's Chain 
(Fig. 1). 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. 

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

1 Mile =80 Chains. 

1 Acre =10 Square Chains. 

Evidently each link is T 6 \ of a foot (or 7.92 inches) long. 
The inch, however, is never used in surveying fieldwork. 

3 



MEASUREMENT OF LINES 



[Chap. I. 



•*: V; 3Tie<eftginjeer 2 s "chain is ioo 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 io-ft. and 90ft. 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 admeasurement falls in 
order to read the distance correctly. Distances measured with 
the surveyor's chain are recorded as cJiains and /inks, (or in cftains 
and decimals) ; while those measured with the engineer's chain 
are recorded as 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 decimeters 
long. 

6. THE TAPE. — There are three kinds of tape in common 
use, — cloth, 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 very fine brass wires woven into 
them to prevent stretching. They are usually graduated into 
feet, tenths, and half -tenths and are made in lengths of 25 ft., 
50 ft., and 100 ft. When precise results are required a steel 
tape should be used. While a steel tape varies a slight 
amount in length with the temperature and with the pull, it is 
possible to determine the amount of these variations and hence 
to arrive at accurate results. 

7. Steel Tapes. — Steel tapes may be obtained in lengths 
up to 500 ft., but the most common in use are the 50ft. and 
100-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 10-ft. length at 
one end of the tape being marked at every foot, and the last 
foot divided into tenths. Some of these tapes are marked every 
foot throughout their entire length. The light tapes are divided 
throughout their entire length into feet, tenths, and hundredths, 
each line being etched on the steel. The numbering is contin- 
uous from o ft. to 100 ft. These tapes are more convenient to 
handle than the heavy ones, but are not suited to very rough 
work as they are easily kinked and broken. They can be readily 
mended, however, by riveting to the back of the tape a piece of 
tape of the same width. 

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



6 MEASUREMENT OF LINES [Chap, i 

TABLE 1. 
Decimals of Foot in Inches. 

Decimal op Foot. Inches. 
.01 - |- 

.08 - I- 

.17 " * + 

.25 - 3 (exact) 

.50 — 6 (exact) 

.75 - 9 (exact) 

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

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

8. THE STADIA. — Where it is desired to measure distances 
with great rapidity but not with very great accuracy the stadia 
method is coming to be very generally used. The distance is 
obtained by simply sighting with a transit instrument at a grad- 
uated rod held at the other end of the line and noting the space 
on the rod included between two special cross-hairs set in the 
instrument at a known distance apart. From this observed in- 
terval on the rod the distance from the transit to the rod can be 
easily calculated. (See Appendix A, p. 517.) 

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

i 5 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 7 

MEASUREMENT OF LINES. 

IO. 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 




HeadChainman. 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- 
cbainman 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 10-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 M 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 ioo ft. In 
doing this he should take his natural step and avoid any attempt 
to take steps just 3 ft. long. 

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

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



IO MEASUREMENT OF LINES [Chap. I. 

on the ground so that the head-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 ioo-ft. points. 
Mistakes will often be avoided by setting the ioo-ft. points as fol- 
lows : — suppose an angle to occur at 870. 1 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 + 70.1 and the stake at station 9 placed at the ioo-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 100 ft. apart, summing 
up the partial measurements when the end of the line is reached. This guards 
against the mistake of omitting a whole tape-length. Another advantage is that 
it is easier to read the distance to a fixed point than to set a point accurately at 
the end of the tape ; this is especially true in measurements where plumbing is 
necessary. This method takes less time than the usual method, but it is not 
applicable when it is necessary to mark the ioo-ft. points on the line. 



ERRORS AND MISTAKES IN MEASURING II 

13. COMMON SOURCES OF ERROR IN MEASUREMENT OF 
LINES. — 

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. — 

i. 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 understood. It 
is frequently useful in doubtful cases for the note keeper to use 
different words in answering, which will remove possible ambi- 
guity. For example, if the chainman calls, "Thirty-six, five," 
the note keeper might answer, "Thirty-six and a half." If the 



12 MEASUREMENT OF LINES [Chap. L 

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

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

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

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

16. ACCURACY REQUIRED. — If, in a survey, it is allowable 
to make an error of one foot in every five hundred feet the chain 
is sufficiently accurate for the work. To reach an accuracy of 1 
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 1 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 1 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 1 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 13 

19. Temperature. — The average coefficient of expansion for 
a steel tape is nearly 0.0000063 for i° F. Hence a change of 
temperature of 15 produces nearly 0.0 1 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. 

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

# * 

c — a = — 

2C 

where h is the distance of the end of the tape from the line, c is 

the length of the tape, and a is the distance along the straight 

line. For example, if one end of. a 100-ft. tape is held 1 ft. to 

one side of the line the error produced in the length of the line 

i 2 
will be — = 0.005 ft., (about ^ inch). The correction 

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

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

* In the right triangle, 

<* - a* = h\ 
(c + a)(c-a)= h\ 
assuming c «= a and applying it to the first parenthesis only, 

2 c {c — a) — A 2 (approximately) 
c — a =— - (approximately) 

Similarly e — a —-— (approximately) 

It is evident that the smaller h 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 T fo ft. when h «■ 14 ft. and a « 100 ft., or when h — 30 ft. 
and a « 300 ft. 




14 MEASUREMENT OF LINES [Chap. I. 

at the ends of the tape. With a 12-lb. pull on an ordinary ioo- 
ft. steel tape supported at the ends the effect of sag is from 
005 ft. to .01 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 
4- or a — error. There will then be 50 tape-lengths in the 5000- 
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 
in marking the tape-lengths on the ground. Since the error due 
to erroneous length of tape increases directly as the number of 
measurements, and since these errors are not compensating, the 
total error in the line due to the fact that the tape is 0.01 ft. too 



ACCURACY OF MEASUREMENTS IS 

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

PROBLEMS 

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

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

3. A steel tape is known to be 100.000 ft. long at 62 F. with a pull of 12 lbs. 
and supported its entire length. Its coefficient of expansion is 0.0000063 f° r l ° F. 
A line was measured and found to be 142.67 ft. when the temperature was 8° 
below zero. What is the true length of the line ? 

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

5. In measuring a line with a 100-ft. tape the forward end is held 3 ft. to the 
side of the line. What is the error in one tape-length ? 



CHAPTER II. 

MEASUREMENT OF DIRECTION. 

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

The 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°, 



THE SURVEYOR'S COMPASS \J 

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




Fig. 3. 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 
Teckoned from the south point right-handed (clockwise) to 360 ; 
i.e., a line running due West has an azimuth of 90 , a line due 
North an azimuth of 180 . Sometimes, however, the azimuth 



18 



MEASUREMENT OF DIRECTION 



[Chap. II. 



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

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

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





Bearlng of AB Bearing of AB Bearing of AB 

S60°E S60°W N45 # W 

Diagram Illustrating Reading of Bearings. 

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

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



TAKING A MAGNETIC BEARING 1 9 

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

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

In reading the compass-needle, the surveyor should take 
care to read the farther end of the needle, always looking along 
the needle, not across it. By looking at the needle sidewise it is 
possible to make it appear to coincide with a graduation which is 
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 account of its magnetic influence 
a permanent magnet, such as a compass-needle, when freely 
suspended will take a definite direction depending upon the 
direction of the lines of magnetic force at any given place and 
time. If the needle is perfectly balanced before it is magnetized 
it will, after being magnetized, dip toward the pole. In the 
northern hemisphere the end of the needle toward the north 
pole points downward, the inclination to the horizon being slight 
in low latitudes and great near the polar region. In order to 
counteract this dipping a small weight, usually a fine brass wire, 
is placed on the higher end of the needle at such a point that 
the needle assumes a horizontal position. 

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



20 MEASUREMENT OF DIRECTION [Chap. IL 

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

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






-INATION 


s of Needle 


in Eastern M 


Year. 




Declin 


ATION 


1700 




10° 


3i' 


w. 


1750 




7° 


13' 


W. 


1800 




6° 


28' 


W. 


1850 




9° 


10' 


w. 


1900 




12° 


00' 


W. 



In the United States all declinations are now increasing (except 
those in the region just west of the agonic line) at an average 
rate of about 3 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 
44 U. S. Magnetic Declination Tables and Isogonic Chart for 1902, and Principal 
Facts Relating to the Earth's Magnetism," by L. A. Bauer, issued in 1902. 




Fig. 5. Isogonic Chart of the United 5 

(From the U. S. Coast an 

The lines of equal magnetic declination, or isogonic lines, are given for every degree, and are baa 
marked zero, the magnetic needle points true north and south. 

The north end of the compass needle » moving to the westward for places east of the line of no chj 
equal annual change (dash lines). Or, east of the line of no change the isogonic lines are moving westw? 




States for the Epoch January, 1910. 

id Geodetic Survey.) 

ped on all known observations to July i, 191 1. They apply to January 1, 1910. Along the line 

jajige, and to the eastward for places west of that line, at an annual rate indicated by the lines of 
■xd, and west of that line they are moving eastward. 



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. Isogonic Chart. — If lines are drawn on a map so 
as to join all places where the declination of the needle is the 
same at a given time, the result will be what is called an isogonic 
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. 187. 

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 



[Chap. II 




Fig. 6. Bent Compass- 
Needle. 



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

direction the compass is turned, it 
follows that the needle is bent but 
the pivot-point is in the center of the 
circle. (See Fig. 6.) The bent needle 
is represented by the line A OB 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, Le., 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 AD, 
perpendicular to CO, (where C is 
the true center and C is the 
position of the pivot-point), then 
the error is a maximum. If B is 
a point 180 from A then the dif- 
ference of the two readings is BD. 
When the needle is at A'D* the 
error is less than before and equals 
B r iy. When the needle is in the 
line CC y i.e., in the position A"D", 
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-NEEDLE. — 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. 



A M 

pivot-point out 
Center. 



of 



ADJUSTMENTS OF THE COMPASS 27 

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

"^^^v^ Carefully remove the needle from the 

>. pivot and bend it by the amount CD 
\ in the direction which will move the 
\ south end half-way back from C 
3 1 toward C. It is better not to touch 

\ \ / the needle with the hands more than 

y 1 / is absolutely necessary as this weakens 

\ I / the magnetism. Instrument makers 

V. \ ^/ usually leave the central part cf the 

^ needle quite soft so that it can be 

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

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

36. TO CENTER THE PlVOT-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 bax 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 mistaess.— 

1. Reading wrong end of needle. 

2. Not letting needle down on pivot. 

3. Reading the wrong side of the 10th degree, 

viz., reading 6l° instead of 59 . 



DETECTING LOCAL ATTRACTION 29 

40. DETECTING LOCAL ATTRACTION OF THE NEEDLE. — As 

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

In Fig. 10 suppose that the compass is at A and that the 




Fig. 10. 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 9 hence S 49 W 
is the correct bearing of line BA, and there is i°{ error in all 
bearings taken at A. If the bearings of BC and CB had not 
agreed it would have been necessary to take the bearing and 
reverse bearing of a new line CD. This process is continued 
until a line is found whose bearing and reverse bearing differ by 
exactly 180 . 



30 



MEASUREMENT OF DIRECTION 



[Chap. IL 



41. Calculating Angles from bearings. — In calcu- 
lating the angle between two lines it is necessary only to 
remember that the bearing is in all cases reckoned from the 
meridian, either N or S, toward the E and W points. In Fig. 1 1, 



A OB = difference of bearings. 
AOC = 180 — sum of bearings. 
AOD = 180 — difference of bear- w 

ings. 
A OF = sum of bearings. 




Fig. 11. 



PROBLEMS. 

1. Compute the angle A OB from the given bearings in each of the following 

case*. 

(a) OA, N 39 °i E. (c) OA, N 15° E. 
OB, N 76 } E. OB, S 36 E. 

(b) OA, N 35° 15' K (d) OA, N 40 15' E. 
OB, S 88° 00' W. OB, N 66° 45' w - 

2. The bearing of one side of a field in the shape of a regular hexagon is 
S io°i 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 
io° 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 
o° 13' E. In 1902 the declination was i° 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° 21' E in 1800, i° 03' W in 1850, 4 00' W in 1900. If a line had a bearing of 
S 65°i W in 1900, what was its bearing in 1700, 1750, 1800, and 1850 ? 

a The following bearings were observed with a compass : AB, N 27°$ E ; 
BA S 25°* W; 2?C, S 88° W ; CB, N 87 } E; CD, N 4 7°i W ; DC, S 4 7°i E. 
Find the true bearing of AB. Where is the local attraction ? Which way 11 the 
needle deflected at each point, and how much ? 



CHAPTER III. 

MEASUREMENT OF ANGLES. 

THE TRANSIT. 

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




Fig. 12. Section of Transit. 

A, inner spindle ; B, outer spindle ; C f upper plate ; D y lower plate ; £ y graduated 
circle ; F, vernier ; G, magnetic needle ; H y upper clamp (turned 90 from its nor- 
mal position so as to show in section, corresponding tangent screw not shown) ; 
/, lower clamp ; K y lower tangent screw ; Z, leveling screws ; M, ball-and-socket 
joint ; iV, shifting head ; 0, 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 

3* 



32 MEASUREMENT OF ANGLES [Chap. Ill 

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

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

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

At the center of the ball-and-socket joint is a ring to which the 
plumb-line may be attached. The plumb-bob used with the tran- 
sit is generally heavier than that used in taking tape measure- 
ments. Modern transits are so made that the entire head of the 
instrument can be shifted laterally 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 o° to 360 in a 
right-hand direction and, by a second row of figures, from o° 
each way to 1 8o° ; and still others (older types) are numbered 
from o° to 90 in opposite directions, like a compass circle. 
Transits are all provided with two opposite verniers. 



THE TRANSIT 



33 




Fig. 13. Engineer's Transit. 



34 



MEASUREMENT OF ANGLES 



[Chap. m. 



El 



O 
u 

w 

J 

H 






Pu 

O 

2 
O 

H 
U 
Id 

3 

H 

3 

Z 

o 



g 



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

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

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

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



THE TRANSIT TELESCOPE 35 

of this tube is parallel to the line of sight. The eyepiece is 
simply a microscope for viewing the image and the cross-hairs. 
When the plane of the image coincides with the plane of the 
cross-hairs, both can be viewed at the same instant by means of 
the eyepiece. The adjustment of the eyepiece and the objective, 
to enable the cross-hairs and the image to be clearly seen at 
the same time, is called 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 10 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 instrument can be quickly focused on a distant 
object if the objective is first moved in as far as it will go and 
then moved out slowly until the image is distinct. The objec- 
tive should not be moved too rapidly as it may pass the correct 
position before the eye can detect the distinct image. If an in- 
strument is badly out of focus it may be pointing directly at an 
object and yet the image may not be visible. 

46. The Objective. — The objective might consist of a simple 
bi-convex lens, like that shown in Fig. 15, which is formed by 
the intersection of two spheres. The line 0(7 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 the principal focus. If light falls on 
the lens from any direction there is one of the rays such as 



36 



MEASUREMENT OF ANGLES 



[Chap. in. 



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 

yA 




Fig. 15. Bi-Convex Lens. 

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

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

47. Cross-Hairs. — The 

cross-hairs consist of two 

very fine spider threads 

stretched across a metallic 

ring at right angles to each 

*v ir * jl Fig. 16. Cross-Hair Ring. 

other and fastened by means 

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

four capstan-headed screws which permit of its being moved 




MAGNIFYING POWER 37 

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

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

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

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

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

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. 



—9 



—7 



3 



-K> 



—4 
—3 

-1 



9 oc 

UJ 

-si 



6- 



—9 



i — 






5- 



h- 3 

2 

I 





K> 




—9 


6— 


—8 


— 


—7 




fi 


— 


B 

UJ 

-4> 


SCALE 

1 1 


—3 


—2 




— 1 






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 

rately 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. 1 7, is constructed 
by taking a length equal to 9 divisions on the scale and dividing 
this length into 10 equal parts. One space on the vernier is 
then equal to -^ of a space on the scale, i.e., it is -^ part 
shorter than a space on the scale, hence ab = -^ of a space 
on the scale, cd = ^ of a space, etc. Now if the vernier is 
raised until a coincides with b> 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 -fa of a space and the reading 
will be 501. If the vernier is moved -fa space higher then 
line 2 coincides with the next higher line on the scale and 
the reading is 502, as shown in Fig. 18. Similarly Fig. 19 
shows reading 526. Thus it is seen that the number of the line 
on the vernier which coincides with a line on the scale is the 
number of tenths of the smallest division of the scale that the 
index point (zero) lies above the next lower division on the scale. 
Furthermore it will be seen from its construction that it is im- 
possible to have more than one coincidence at a time on a single 
vernier. The type of vernier just described is used on leveling 
rods. 

53. Verniers used on Transits. — In transits, since angles 
may be measured in either direction, the verniers are usually 
double, 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 £# 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. m. 



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 1' of arc. 
The second line on the vernier falls 2' short of the i° mark, 
the third line 3' short of the i° 30' mark, etc. If the vernier 
should be moved one minute to the left the first line would coin- 



30 




A 


* 


\0 y IO 


* 


ife 


1 ' 1 


360 


VM 
10 


Fig. 


20. 


CIRCLE 
One-Minute Vernier Set 


AT 


o°. 



39 




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



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




TRANSIT VERNIERS 



41 



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

Reading, outer row of figures, 9 16'. 
Reading, inner 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 




VERNIER 
B 



m 



30 
330 

CIRCLE 

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, 3 2 8° 42' 30". 



42 



MEASUREMENT OF ANGLES 



[Chap. UL 



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

Reading, 73 48' 40". 




CIRCLE 

FiG.,28. 

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



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




CIRCLE 

Fig. 24. 



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



TRANSIT VERNIERS 



43 



angles are to be measured in the opposite direction the index at 
the left end should be set at o°. To avoid this inconvenience of 
resetting, some surveyors set the middle, line (10' 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 



VERNIER 
5 




into 10' 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 5 




VERNIER 

Fig. 26. 

Circle divided into 3c/ spaces. 29 divisions of the circle 
divided into 30 parts to make one division of the vernier. 

Reading, 2° 23'. 



44 



MEASUREMENT OF ANGLES* 



[Chap. IIL 



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, (1) because the center of the vernier plate does 
not coincide with the center of the graduated circle, (2) because 
the vernier zeros have not been set exactly 180 apart. The 
first cause produces a variable difference while the second 
produces a constant difference. 

It will be noticed that the effect of these errors is similar 
to that described in Art. 34, p. 25, on Adjustments of the Com- 
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 t the readings 




Fig. 27. 



Eccentricity op 
Circle. 



ECCENTRICITY OF CIRCLES 45 

of opposite verniers will differ by the maximum amount. Suppose 
that the graduations are numbered from o° right-handed to 360 . 
When the vernier is at an intermediate position, as at A 9 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 f . Since AB and 
A'ff are parallel, BB f and AA f 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; (1) 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. SETTING UP THE TRANSIT. — In setting the transit 
over a point, place one leg of the tripod in nearly the right posi- 
tion on the ground, then 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 dn 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. III. 

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

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

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




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

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



SETTING UP THE TRANSIT 47 

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

58. TO MEASURE A HORIZONTAL ANGLE. — After setting 
the instrument up over the point, first set the zero of one of the 
verniers opposite the zero of the circle. This is done by turning 
the two plates until the two zeros are nearly opposite, clamping 
the plates firmly together with the upper 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 "reading glass," 
is generally used for setting and reading the vernier. Never 
touch the clamp after a setting has been made by means of the 



48 MEASUREMENT OF ANGLES [Chap. IIL 

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 0°. 

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

59. TO MEASURE AN ANGLE BY REPETITION. — The 
eyepiece magnifies the image so much tint 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 caabe read, consequently 
the full value of the instrument is not utilized by single readings 
of an angle. To obtain the value of an angle more accurately 
proceed as follows. After the first angle has been measured 
leave the two plates clamped together, loosen the lower clamp 
and turn back to the first point. Set on the first point, using 
the lower clamp and its tangent screw. Then loosen the upper 
clamp and set on the second point, using the upper clamp 
and its tangent screw, thus adding another angle, equal to the 
first one, to the reading on the circle. Repeat this operation, 
say, six times. The total angle divided by six will give a more 
precise result than the first reading. Suppose that the angle is 
actually 18 12' 08'' ; if a "one-minute" instrument is being used 
it is impossible to read the 08" on the vernier, so the reading 
will be 1 8° 12'. Each repetition will add 08" (nearly) and after 
the 6th repetition, the amount will be 48" which will be read as 
i'. After the 6th pointing the total angle will then be read 109 
13' which divided by 6 gives 18 12' 10", a result in this case 
correct to the nearest 10". To eliminate errors in the adjust- 
ment of the transit the above process should be repeated with 
the instrument reversed and the mean of the two values used. 
(See Art. 79, p. 61.) It is customary to take only the 1st 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, £ 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 6o° 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. in. 

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

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

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

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



LAYING OFF AN ANGLE BY REPETITION 5 1 

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

63. RUNNING A STRAIGHT LINE — Neither Point Visible 
from the Other. — If neither point can be seen from the other 
then it is necessary to find some point, by trial, from which the 
terminal points can be seen. The transit is set up at some point 
estimated to lie on the line, a " backsight " is taken on one of 
the points and the instrument clamped. The telescope is then 
reversed on its horizontal axis. If the vertical cross-hair strikes 
the second point the instrument is in line ; if not, then the error 
in the position of the instrument must be estimated (or meas- 
ured) and a second approximation made. In this way, by suc- 
cessive trials, the true poiht 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 line of sight. After one point has been found by this 
method other points may be set as described in the previous 
article. 

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



52 MEASUREMENT OF ANGLES [Chap. Ill 

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 B a backsight 
is taken on A and the instrument clamped. The telescope is 
inverted and a point set ahead in line. The process is repeated, 
the backsight being taken with the telescope in the inverted 
position. The mean of the two results is a point on the line AB 
produced. The transit is then moved to the new point, a back- 
sight is taken on B, and another point set ahead as before. 

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

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

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

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

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

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

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

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

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

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

like a plumb-bob). This may be fastened so that its axis coincides with the 
string and so that it can be raised and lowered on the string. It should be painted 
with such 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 
(sometimes 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 signaling is to be done flags are attached to the 
lining-poles. Special signals may be devised for different kinds 
of work and conditions. 

67. TO MEASURE A VERTICAL ANGLE. — In measuring a 
vertical angle with a transit, first point the vertical cross-hair 
approximately at the object, then set the horizontal cross-hair 
exactly on the point by means of the clamp and tangent screw 
controlling the vertical motion. Next read the vertical arc or 
circle. Then, without disturbing the rest of the transit, unclamp 
the vertical arc, and bring the telescope to the horizontal position 
by means of the level attached to the telescope, and the clamp 
and tangent screw of the vertical arc. When the telescope 
bubble is in the center read the vertical arc again. This gives 
the index 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, turning it through 180 
azimuth, and remeasuring the angle. The mean of the two 
results is free from such errors. If the transit is provided with 
only a portion of a circle the vernier will be off the arc when 



PRECAUTIONS IN USE OF TRANSIT 55 

the telescope is inverted, consequently with a transit of this type 
the elimination cannot 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 mojion 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 1 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 
clamp, and turn the telescope either up or down. 



56 



MEASUREMENT OF ANGLES 



[Chap. III. 



ADJUSTMENTS OF THE TRANSIT. 

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

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

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



•*«A 




Fig. 29. Adjustment of the Plate Bubbles. 



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



ADJUSTMENTS OF TRANSIT $7 

bubble tube until the bubble moves half-way back to the center 
as nearly as this can be estimated. Each bubble must be ad- 
justed independently. The adjustment should be tested again 
by releveling and reversing as before, and the process continued 
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. Adjustment 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. 30. 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. 



$8 MEASUREMENT OF ANGLES [Chap. in. 

72. 2nd. To make the Line of Sight Perpendicular to the 
Horizontal Axis.* (See Fig. 31.) Set the transit over a point 




Fig. 31. Adjustment of the Cross-Hairs (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 C is 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. 
It D falls to the right of Cthen the cross-hair ring should be 
moved to the left; but if the transit has an erecting eyepiece 
the cross-hair will appear to move to the right when viewed 
through the telescope. If the transit has an inverting eyepiece 
the cross-hair appears to move in the same direction in which 
the cross-hair is actually moved. 

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



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



ADJUSTMENTS OF TRANSIT 



59 



73. ADJUSTMENT OF THE STANDARDS. — 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 horizontal axis by means of the 
capstan-headed screw under the end of 
the axis. Repeat the test until the high 
and the low points are both on the cross- 
hair in either the direct or reversed posi- 
tions of the transit. The adjusting screw 
should be brought into position by a right- 
hand turn, otherwise the block on which 
the horizontal axis rests may stick and 
not follow the screw. The cap screws 
should then be tightened just enough to 
avoid looseness of the bearing. 

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

75. Adjustment of the Auxiliary Level on the Vernier of the 
Vertical Arc— (See Art. 67, p. 54.) To adjust the Level 
so that it is in the Center of the Tube when the Line of Sight 
is Level and the Vernier reads 0°. This is adjusted by the " peg 




Fig. 82. Adjustment 
of the Standards. 



60 MEASUREMENT OF ANGLES [Chap. III. 

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

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

76. Adjustment of the Vernier of the Vertical Circle. — To 
make the Vernier read 0° when the Telescope Bubble is in the 
Center of the Tube. If there is any index error (Art. 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 
Objective Slide move Parallel to the Line of Sight. If the tube 
holding the objective is adjustable it must be placed so that the 
direction of the line of sight will not be disturbed when the 
telescope is focused. The adjustment may be made as follows. 
Adjust 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 moving the adjustment 
screws of the objective slide so as to apparently increase the 
error making, by estimation, one-quarter the correction required. 



HOW TO ELIMINATE ERRORS 6 1 

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

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

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

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

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

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

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



62 MEASUREMENT OF ANGLES [Chap. Ill 

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. 

8l. COMMON SOURCES OF ERROR IN TRANSIT WORK. — 

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

2. Changes due to temperature and wind. 

3. Uneven settling of tripod. 

4. Poor focusing (parallax). 

5. Inaccurate setting over point. 

6. Irregular refraction of atmosphere. 



82. COMMON MISTAKES IN TRANSIT WORK. — 

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

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

3. Reading the circle wrong, e.g., reading 59 for 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. 



«4. 



•9QLAR ATTACHMENT 



[Chap. HI. 




Fig. 33. Solar Attachment to Transit. 



(The authors are indebted to'C. L. Berger & Son 
for the photograph from w)u?h this out was nude.) 



SOLAR ATTACHMENT 65 



THE SOLAR ATTACHMENT. 

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

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

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

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

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



66 



MEASUREMENT OF ANGLES 



[Chap. III. 



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




Fig. 34. Diagram of the Celestial Hemisphere. 



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

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



OBSERVATION FOR MERIDIAN WITH SOLAR 6j 

(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 as the main telescope by sighting both on some dis- 
tant object, and then making the angle between the two telescopes 
equal to the sun's declination. Some instruments are provided 
with a declination arc upon which the declination angle can be 
laid off directly. Others have a small spirit level attached to 
the small telescope, in which case the vertical circle of the tran- 
sit is used for laying off the declination angle. Incline the main 
telescope until the reading of the vertical circle equals the de- 
clination, and clamp ; then level the solar telescope by means of 
the attached level. The angle between the polar axis and the 
solar telescope is then 90 plus or minus the reading of the 
vertical circle. 

(2) By means of the vertical 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 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. HI. 

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

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

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

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



OBSERVATION FOR MERIDIAN WITH SOLAR 69 

rected for refraction of the atmosphere. The effect of refrac- 
tion is to make the sun appear higher up in the sky than it 
actually is.. In the northern hemisphere, when the declination 
is North the correction must be added, when South, subtracted. 
This correction may be taken from Table VII, p. 507, the decli- 
nation being given at the top and the number of hours from 
noon at the left 

The co-latitude which must be set off on the vertical circle 
may be obtained from a map or may be determined by an ob- 
servation which is made as follows. Set off the sun's declination 
for noon, as for any other observation, the two telescopes being 
in the same vertical plane, and point the small . telescope at the 
sun. By varying the angle of elevation of the main telescope, 
keep the solar telescope pointing at the sun until the maximum 
altitude is reached. The angle read on the vertical circle is the 
co-latitude (see also Art. 217, p. 196). 

Example. 

Latitude 40 N. Longitude 4b 45m W. 

Jan. 10, 1900. 
Declination for Greenwich mean noon S 21 59' 04" 
Difference for 1 h + 22" .25 

Time. Declination. Refraction. Setting. 

7 h. A.M. 21 59' 04* S 

8 58 42 4' 59" 21° 53' 43" 

9 58 20 2 36 21 ss 44 

10 57 57 I 57 21 56 00 

11 57 35 1 48 21 55 47 

12 M - 57 13 (1 39) (21 55 34) 

1 P.M 56 51 1 48 21 55 03 

2 56 28 1 57 21 54 31 

3 56 06 2 36 21 53 30 

4 55 44 4 59 21 50 45 

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



* Sec Bulletin of the University of Wisconsin, Science Series, Vol. I, No. 3. 



JO MEASUREMENT OF ANGLES [Chap. III. 

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

equal to - f h being expressed in degrees. 

hn 

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

89. MISTAKES IN USING 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 Perpendicular to the Plane of the Line of Sight and the Hori- 
zontal Axis. Level the transit and the main telescope. Bring the 
bubble of the solar telescope to the center of its tube while it is 
parallel to a pair of opposite adjusting screws which are at the 
foot of the polar axis. Reverse the solar telescope 180 about 
the polar axis. If the bubble moves from the center position, 
bring it half-way back by means of the adjusting screws just 
mentioned and the other half by means of the tangent screw 
controlling the vertical motion of the solar. This should be 
done over each pair of opposite adjusting screws and repeated 
until the bubble remains central in all positions. 

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



ADJUSTMENTS OF SOLAR ATTACHMENT ^\ 

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



PROBLEMS. , 

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

2. A transit is sighting toward B from a point A. In setting up the transit at 
A it was carelessly set 0.01 ft. directly to one side of A y as at A'. What would 
be the resulting error, i.e., the difference in direction (in seconds) between AS and 
A'B, (1) when AB « 40 ft., (2) when ^# = 1000 ft.? 

3. An angle of 90 is laid off with a '« one minute " transit, and the angle then 
determined by six repetitions, the final reading being 179 s8' + 36o°. The point 
sighted is 185 feet from the transit. Compute the offset to be Jaid off in order to 
correct the first angle. Express the result in feet and also in inches. 

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

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

January 1, 1906. 

Decl. S 23 03' 2 7". 9 

Dift. for 1 hour, +11 "70 
April 16, 1906. 

Decl. N 9 53' 3 4 ".2 

Diff . for 1 hour, + 53"-44 
July 2, 1906. 

Decl. N 23 05' 49". 5 

Diff. for 1 hour, — io // .39 

Sept. 25, 1906. 

Decl. S o° 35' 49".4 
Diff. for 1 hour, - 58". 51 



CHAPTER IV. 

MEASUREMENT OF DIFFERENCE OF ELEVATION. 

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

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

72 



THE WYE LEVEL 73 

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

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

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

THE LEVEL. 

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

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



74 



MEASUREMENT OF ELEVATION 



[Chap. IV. 




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

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



j* 



S 

8 



H 
w 










76V 




W 
> 

w 



Oh 

Q 

w 
a 

H 



g 



THE DUMPY LEVEL 77 

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

99. Comparison of Wye and Dumpy Levels. — The wye level 
has long been a favorite in this country, chiefly on account of 
the ease with which it can be adjusted, which depends upon the 
fact that when the telescope is reversed in the wye supports the 
line through the centers of the pivots is exactly coincident with its 
first position. While this feature of the wye level is of practical 
advantage in adjusting the instrument it is based on the assump- 
tion that both pivots are circular and of exactly the same diame- 
ter, which may or may not be true. For, even supposing the 
pivots to be perfect when new, they soon wear, and perhaps 
unevenly, and consequently the method of adjusting by reversal 
will then fail and the "peg" 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 
which is capable of excellent work when adjusted by the latter 
method. 

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




Fig. 38. 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- 
vey for the year 1898-99, p. 351, and the year 1900, p. 525. 



78 MEASUREMENT OF ELEVATION [Chap. IV. 

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



LEVELING RODS. 

101. 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 the 
ground, as shown in Fig. 39. For readings over 5.8 ft. the rod 
is turned end for end so that the target is at the top and can be 
moved from 5.8 to 11.4 ft., the limit of the rod. The terms 







9 



CO 




6 
4 
I 
\ 
8 



* © 





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. 41 is sometimes used, in which the white strip in 
the center may be bisected by the horizontal cross-hair. 
Bisection is more precise under all conditions than setting on a 





Fig. 40. Boston Rod Target. 



Fig. 41. Bisection Target. 



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

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

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



LEVELING RODS 8 1 

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

104. PHILADELPHIA ROD. — This rod has the graduations 
plainly painted on its face so that it can be used as a sdlf-reading 
rod (Fig. 39). It has a target reading to thousandths, like that 
of the New York rod. In some cases the target has no vernier 
but is graduated directly to 0.005 ft- 1 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 0.00 1 ft. by means of a scale fastened on the back of- the rod. 
When the rod is fully extended, the graduations on the front 
face are continuous and the readings can be made directly by 
the levelman if desired. 

105. SPECIAL SELF-READING RODS. — There are a large 
number of self-reading rods of special design. One of the com- 
monest types shown in Fig. 39, is similar to the Philadelphia 
rod except that it has no target and is not graduated closer than 
tenths. The figures on the face of the rod are made of definite 
height (0.06 or 0.08 ft.) and of definite thickness (0.01 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. 1 ft. wide, on the outside of which for 
its entire length is painted a scale graduated to feet, tenths, and 
half-tenths, with the details of the numbers so designed that 
readings to the nearest 0.0 1 ft. can readily be made. Unlike the 
other rods mentioned the scale reads down on the face of the 
rod instead of up. It is provided with a clamp so that the metal 
band, or tape, can be set at any desired reading and held firmly 
in that position. The use of this type of rod is limited to cer- 



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



82 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



tain kinds of work, its 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 rod, but the errors of estimating 
fractional parts are compensating, so that in the long run the 
results are found to be very accurate. Precise leveling carried 
on by the U. S. Coast and Geodetic Survey and by European 
surveys has demonstrated the superiority of such rods. The 



SELF-READING RODS 83 

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

109. Attachments to the Rod for Plumbing. — In accurate 
work it will be convenient to use some device for holding the 
rod plumb. Spirit levels attached to brass "angles" which may 
be secured to a 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 AND ROD. 

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

While the levelman is sighting the target, the rodman should 
stand beside the rod so that he can hold it as nearly vertical as 
possible in the direction of the line of sight. The levelman can 
tell by means of the vertical cross-hair whether it is plumb in 
the direction at right angles to the 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 " waving the rod" To do this the rodman stands 
directly behind the rod and inclines it toward the instrument so 
that the target will drop below the line of sight. He then slowly 
draws it back, causing the target to rise. It will be highest 
when the rod is plumb. If at any point the target appears 
above the cross-hair it should be lowered. If, while the rod is 
being waved, the target does not reach the cross-hair the target 
must be raised and the process repeated until as the rod is 
waved there appears to be just one place where the target coin- 
cides with the horizontal line of sight. Whenever close results 
are desired it will be well to take several readings on each point 
and use the mean. 

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

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

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

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

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

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

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

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



86 



MEASUREMENT OF ELEVATION 



[Chap. IV. 



This is called a backsight ox plus sight and is usually written £. 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 a. foresight 
or minus sight and is written F. S. or — S. The difference be- 
tween the two readings gives the difference in elevation between 
this new point and the first point. . This second point is called a 
turning point and is written T. P. The level is next set up in 
a new position and a backsight taken on the turning point. A 




Diagram Illustrating Differential Leveling. 



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



DIFFERENTIAL LEVELING 



87 



Point 


+ s. 


— S. 


Remarks 


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

B. 


8.160 
7.901 
9.446 
8.005 


3.404 
3.070 
6.906 
2.107 


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




33.512 
14.487 


14.487 





Diff. 19.025 B above A. 

117. The Proper Length of Sight. — The proper length of 
sight will depend upon the distance at which the rod appears 
distinct and steady to the levelman, upon the variations in read- 
ings taken on the same point, and upon the degree of precision 
required. Under ordinary conditions the length of sight should 
not exceed about 300 ft. where elevations to the nearest 0.01 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 jPZ 1 (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 



8 9 



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

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

telescope 180 in the wyes, so that 
the level tube is above the tele- 
scope. The intersection of the 
cross-hairs should still be on the 
point. If not, move the horizontal 
cross-hair half-way back t6 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 
:he direction in which the adjustment should be made. Move 



/ 2«d.F* 




Cra i-htk \ 






._■_. 


— ■ 


1 TmtPw 


Lm 


Cm 


i-haV 1 


I l«t font 


M 


£3 


I 1 "* / 








* POINT J 



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



go 



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 OF 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 maks the Axis of 



rjruc Position of ****** ^S^ 



Tjruc Poet Hon of 




Fig. 47. Adjustment of the Wyes. 



ADJUSTMENTS OF THE DUMPY LEVEL 9 1 

the Level Tube Perpendicular to the Vertical Axis of the Instrument. 

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



II. ADJUSTMENTS OF THE DUMPY LEVEL, 

126. ADJUSTMENT OF THE CROSS-HAIR. — If the horizon- 
tal cross-hair is not truly horizontal when the instrument 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. 

128. THE DIRECT, OR " PEG," ADJUSTMENT. — To make the 
Line of Sight Parallel to the Axis of the Bubble. (See Fig. 48.) 
Select two points A atid 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 will be 
found convenient to set a pencil-point on the roc! at the center of 



92 MEASUREMENT OF ELEVATION [Chap. IV. 

the small field of view. Turn the telescope toward B and take 
a rod-reading on it in the usual way, being certain that the bub- 
ble is in the middle of the tube. The difference between these 
two rod-readings is the difference of elevation of the two points 
plus or minus the error of adjustment. The level is next taken to 
B and the above operation is repeated. The result is the differ- 
ence in elevation minus or plus the same error of adjustment. 
The mean of the two results is the true difference in elevation of 
points A and 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 

Diff. in elev. of A and B -■ 1 .067 
Instrument at B. 

Rod-reading on B «= 5.076 

Rod-reading on A — 4.127 

Diff. in elev. of B and A — 0.949 

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

2 

Instrument is now 5.076 above B. 

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

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



THE PEG ADJUSTMENT 



93 



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




Fig. 48. Peg Adjustment. 



129. Adjustment of the Locke 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 
B, held so that its center is at the height of this mark, and 
sighted toward the first point. A third point C is marked in 



94 MEASUREMENT OF ELEVATION [Chap. IV. 

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

130. COMMON SOURCES OF ERROR IN LEVELING. — 

1. Improper focusing (parallax). 

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

3. Rod not held plumb. 

4. Foresights and corresponding backsights on turning 

points not equally distant from the instrument. 

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

131. COMMON MISTAKES.— 

1. Foresight and Backsight not taken on exactly the same 

point. 

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

used. 

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

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

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

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

vice versa. 

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

PROBLEMS. 

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

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

Instrument at A : — Instrument at B : — 

+ S. on A, 4.139 + S. on B y 3.900 

— S. on B, 4589 — S. on A, 3.250 

Find the rod-reading on A to give a level line of sight, the instrument remaining 
3.900 above/?. 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 effect this will have in finding the difference in eleva- 
tion between two points. 

5. (a). A level is set up and a + S. of 5.098 is taken on a point 400 ft. away, 
then a — S. of 3.260 is taken on a point 900 ft. away. What is the curvature and 
refraction correction ? What is the difference in elevation of the two points? 

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



PART II. 
SURVEYING METHODS. 



PART II. 

SURVEYING METHODS. 

CHAPTER V. 

LAND SURVEYING. 

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

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

closure expected is about 1 part in 500, expressed as 

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

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

99 



IOO 



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 
corner 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 cbmputing 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 



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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 corner ; 
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 corners 
of the property, and in such cases assumed lines running 
parallel or approximately parallel to the property lines can be 



SURVEY BY A TRAVERSE - 1 01 

surveyed as described in Art. 134, and the 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 TRAHSIT 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. 

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

(2). When the property lines are so occupied by buildings 
or fences that the transit cannot be set up at the corners, but 
the distances can still be measured along the property lines, then 
the angles at the corners 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 case (2) the parallel lines are established in 
the following manner. Set the transit up at some point E 



102 LAND SURVEYING [Chap. V. 

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



y>K 




Fig. 51. Transit Lines Parallel to the Sides of Field. 

angle at A It is evident that the values of AH and ZVand of 
^4/and BG are of no permanent use and are therefore not recorded 
in the notes. When practicable it is advisable to choose the 
transit point, K for example, on one of the property lines or its 
prolongation. Fig. 52 is a set of notes illustrating either 
case (1) or (2). 

136. In case (3) the transit can be set up at an arbitrary 
point marked by a stake and chosen far enough from one 
of the corners so that the telescope can be focused on it. In 
this way all the 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 corners of the property 
before the instrument is moved to the next point. Fig. 53 is a 
set of notes illustrating this case. Time can be saved in the 
computations and a good check on the work may be obtained if 
the property lines are also measured when possible. These are 
not only useful as checks on the accuracy of the survey, but the 



SURVEY BY A TRAVERSE 



I03 



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H G_ 

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



ro4 



LAND SURVEYING 



[Chap. V. 



Survey of Land cff//as Cofema/r, fia/?cn>// Af/Y/s, Afe. 



rfeJ/s 

7ape o.o? foo shark 

ff&r//rgs, Jfoynetic* Dec/. /# \ iy: 




Fig. 53. Notes of Survey with Transit and Tape. 



SURVEY BY SINGLE SET-UP 105 

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

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

137. Irregular Curved Boundaries. — When a tract of land is 
bounded by an irregular curved line such as a brook it is custom- 
ary to run the traverse line near it, sometimes crossing it several 
times, and to take perpendicular offsets to the brook. If it is a 
winding brook with no distinct turns in it, offsets at regular in- 
tervals are measured from the transit line as in the portion near 
point A of Fig. 53. Near point B in this figure the brook has 
practically a direct course between its turns, in which case the 
proper measurements to make are the offsets to those points 
where the course of the brook changes and the distances 
along the transit line between these offset lines. Since they 
are usually short the right-angle offset lines are laid 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 corner. In this way the field is divided into several oblique 
triangles in each of which two sides and the included angle have 
been measured and from these the area and third side (property 
line) can be computed. As a check on the measured angles 
their sum should be 360 ; there is no check on the property 
lines unless they are measured directly. 

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



106 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 150 . 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 luv\ but only approximate 
side measurements can be obtained from such a line. 

140. SELECTING THE CORNERS. — 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 comfes directly on the line it is blazed on both 
sides where the line strikes it. • A small pile of stones, sometimes 
with a stake in the center of the pile, is often used to mark the 
corners of such land. 

141. METHOD OF PROCEDURE. — In deciding where the 
traverse shall be run the surveyor should keep in mind both con- 
venience 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 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 the 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 (1) 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 \_ J^?.6"Wrch 

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 F ^ M AppROXIMATE TlES 
of a sketch. Fig. 54 shows a 

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




I08 LAND SURVEYING [Chap. V. 



Sfo* 



simply to aid in finding the 
stake. 
\ J*^\\ It is often desired to take the 

&vt&^* \£^ t^ es so ^ at the exact point can 

*<\^ be replaced! In such cases the 

'^x^ l 56^ ^ "txctf 1 ^. surveyor should mark carefully 

"■*• by tack or crow-foot the exact 

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

ments (taken to jfo ft.) are 
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 o° and the 
telescope inverted, then revolving the telescope about its hori- 
zontal axis to the direct position and turning the upper limb tc 
the right or left until the next point is sighted. The deflection 
angle as recorded in the notes is marked R or L 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 witt 
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 



IO9 



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 AB, 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. 
( 1 ) Invert the telescope and backsight on A, the vernier remain- 
ing at the reading it had at A ; then clamp the lower plate, turn 
the telescope to its direct 
position, and sight on C 
The angle on the vernier is 
the azimuth of BC referred 
to the same meridian as the 
azimuth of AB, The disad- 
vantage of this method is 
that the error of collimation 
enters the azimuth angle each 
time. (2) Add 180 to the 
azimuth of AB, set' this off 
on the vernier, and sight on 
A. The telescope may then 
be turned directly to C (with- 
out inverting) and the azi- 
muth of BC can be read directly on the vernier. The disad- 
vantages of this method as compared with the former are that 
the error of eccentricity of the circle enters, that time is con- 
sumed in setting the 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 <o be of unusual precision a 




Fig. 56. Azimuth Angles. 



I IO LAND SURVEYING [Chap. V. 

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

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

After the angles have been measured, the accuracy of the 
transit work may be tested by adding them together. The sum 
of the interior angles of the field should equal (n-2) X 180 , 
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 



CHFXKING FIELDWORK III 

rough check on the distances in the field and a close check after 
the survey has been calculated. It is often advisable to run a 
line across the traverse, especially when there are many sides 
to the field, thus dividing the field into two parts, as in 
Fig- 5 2 > P- io 3- ^ 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 -g^ff* but would sel- 
dom be less than y^ta^ ^ rt " 1 $ 2 ' P" ^' 

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 ^^-^ 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 2"5"$"jnF» and an accuracy of i^Tff is sometimes reached. 

147. ORGANIZATION OF TRANSIT PARTY. — Transit surveys 
can be readily carried on by a party of three men. The note 



112 LAND SURVEYING [Chap. V. 

keeper 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-chain- 
man 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 ijote-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 1 13 

sa Stone bound. 

won. Monument. 

•a Triangulation Station. 

shu Stake. 

* Tack. 

««. Nail. 

ftffc. Spike. 

<*h. Drill-hole. 

ci. Crow-foot (a mark like this NJ or >k). 

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

c Center. 

4 Center line. 

cb Curb. 

ca* Catch basin. 

mh Manhole. 

in Telegraph pole. 

Fence. 

_ Fence, showing on which side the posts are. 

Kb. Base-board of fence. 

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

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



1 14 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. 104, are good examples of field notes. 

149. SURVEY OF A FIELD FOR A DEED. — In this case 
the lengths and bearings of all the boundaries are desired. The 
traverse lines should therefore follow the property lines, if 
possible. The bearings desired are not the observed magnetic 
bearings, but are those calculated by means of the transit angles 
as explained in Art. 145, p. 109, 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. 

(1) 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., January. 
1894. 



DEED DESCRIPTION 115 

(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 f ollowing is an example 
of a deed description of the property shown in the form of notes 
in Fig. 53, p. 104. 

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

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



Il6 LAND SURVEYING [Chap. V. 

N 86° 45' W a distance of two hundred and five and eight-tenths 
(205.8) feet to the middle of another stone wall at land of said 
Barnes; thence turning and running by latter 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 i° 51' W a distance of one hundred 
and sixty-nine and four-tenths (169.4) feet to the point of begin- 
ning ; all the bearings being magnetic and the parcel containing 
a calculated area of 79,305 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 FUNCTIONS OF THE SURVEYOR. — In rerun- 
ning old property lines which have been obliterated, the surveyor 
is called upon to set aside temporarily his strict adherence to the 
mathematical side of surveying and must endeavor to find if pos- 
sible where the lines originally ran. He should therefore be fa- 
miliar with the relative importance of various evidence regarding 
the location of the property lines, as determined by court deci- 
sions. It is distinctly his duty to find the position of the original 
boundaries of the property and not attempt to correct the original 
survey even though he may be sure that an error exists in it. 
Very often it is true that, owing to the cheapness of land, the 
original survey was roughly made with little thought of the effect 
it would have when the land became valuable'. 

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



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



DISPUTED BOUNDARIES 117 

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

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

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

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

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



Il8 LAND SURVEYING [Chap. V. 

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

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

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

A more complete statement of the principles mentioned above 
particularly with reference to the U. S. Public Land Surveys 
will be found in an address on " The Judicial Functions of Sur- 
veyors," by Chief-Justice Cooley of the Michigan Supreme Court, 
read before the Michigan Association of Engineers and Survey- 
ors, and published in the proceedings of the society for 1882, 
pp. .112-122. 

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



RERUNNING OLD SURVEYS 1 1 9 

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 at 
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 there- 
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. 



120 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 grantor index and the grantee 
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 U1HTED STATES SYSTEM OF SURVEYING THE PUBLIC LANDS. 

154. The System. — The United States System of Sur- 
veying the Public Lands, which was inaugurated in 1784, and 
modified since by various acts of Congress, requires that the 
public lands "shall be divided by north and south lines run 
according to the true meridian, and by others crossing them at 
right angles so as to form townships six miles square," and that 
the corners 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- 
ment that the lines shall conform to true meridians and that 
townships shall be six miles square, is mathematically impos- 
sible. In order to conform as nearly as practicable to the spirit 
of the law, and also to make its application both uniform and 
effective, an elaborate system of subdivision has been worked 
out. This system will be described in this chapter, first in its 
general and afterward in its more detailed features; this will 



122 LAND SURVEYING [Chap. V. 

then be followed by a discussion of the ways in which the work 
of present-time county and other local surveyors is related to 
the Public Lands System. 

The work of the Public Lands Surveys is and has been carried 
on under the direction of the Commissioner of the General Land 
Office. Usually the area comprised in each State or Territory 
has been denominated a District, and has been placed in direct 
charge of a Surveyor General. The functions that the Sur- 
veyor General exercises in his District may be likened to those 
of a division engineer on construction work; he examines the 
Deputy Surveyors, approves their contracts, and inspects their 
fieldwork. The maps, field notes and other records are kept at 
his office until all the subdivision work in his district is com- 
pleted, when they are turned over to the State to which they 
pertain, and the office of the Surveyor General is then dis- 
continued. (See Art. 167, p. 153.) 

The actual surveying operations are performed by Deputy 
Surveyors, who run the lines in the manner specified in the 
Manual * or as directed in detail by the Surveyor General. 
This work is usually done under contract, at stipulated prices 
per mile for lines of various degrees pf importance or difficulty. 
These prices have varied from time to time, with the demand 
for and supply of deputies, the relative degree of accuracy with 
which the work' was required to be done, and with other con- 
ditions. Those at present prescribed by lawf are shown in 
Table 3. 

It will be observed from the schedule of prices given that 
higher rates are paid for standard lines, which» constitute the 
general framework or control for the subdivision work, and that 
the township lines in turn are rated higher than the section lines. 
It is the obvious intention, and has been the general practice, to 
secure a somewhat higher degree of accuracy for these more 
important lines by awarding them to the more experienced and 
skilful deputies, while inexperienced or less skilful surveyors 
were employed on subdivision work. It follows from this that 

* Manual of Surveying Instructions for the Survey of the Public Lands 
issued by the Commissioner of the General Land Office, Washington, D. C. 
t Act of Congress approved, March 3, 1905. 



PROCESS OF SUBDIVISION 



123 



in the relocation of lost comers more weight may properly be 
given to the more important lines. 



TABLE 3. 

Showing Prescribed Rates of Payment Per Linear Mile por 
Surveying Public Lands. 



Minimum rates: to be used under ordinarily 
favorable conditions. 

Intermediate rates: to be applicable to lands 
"heavily timbered, mountainous, or covered 
with dense undergrowth, but not exception- 
ally difficult to survey." 

Maximum rates: to be allowed only in cases of 
exceptional difficulties in the surveys. 

Special maximum rates: to be allowed in cases of 
exceptional difficulties in the surveys, in cer- 
tain remote districts, at the discretion of the 
Secretary of the Interior. 



Standard 




and 


Township 


Meander 


Lines. 


Lines. 




*9 


*7 


*i3 


$.1 


$18 


*>S 


**S 


**3 



Section 
Lines. 



*5 

*7 

$12 

$20 



In the following named States and Territories the surveying of 
Public Lands is still in progress, the work being under the super- 
vision of Surveyors General.* 



Alaska. 


Louisiana. 


Oregon. 


Arizona. 


Minnesota.! 


South Dakota. 


California. 


Montana. 


Utah. 


Colorado. 


Nevada. 


Washington. 


Florida, f 


New Mexico. 


Wyoming. 


Idaho. 


North Dakota.f 





154a. Process of Subdivision. — It will be convenient to con- 
sider the process of subdivision as separated ihto several distinct 
operations, to be carried out in sequence. It must be understood, 
however, that one operation, for instance, the division of the area 
into 24-mile tracts, is rarely or never completed over the entire 

* From Report of the Commissioner of the General Land Office for the Year 
ended June 30, 1906. 

t Practically completed. 



124 



LAND SURVEYING 



[Chap. V. 



area to be covered before the next operation in order is begun; 
a single surveying camp may be carrying on two or three different 
operations before removing from the neighborhood, for example, 
running township exteriors and immediately afterward subdivid- 
ing the townships into sections. 

Briefly stated, the subdivision work is carried on as follows: 

First. The establishment of 

(a) An Initial Point by astronomical observations. 

(b) A Principal Meridian conforming to a true meridian of 
longitude through the Initial Point, and extending both north and 
south therefrom, and 

(c) A Base-Line conforming to a true parallel of latitude 



i 



2nd. Standard 



1st. Standard 



Parallel 



1 



Parallel 



North 



124 Miles less corvV 

Is 



•North 



ui 



u 



$ 



IA MiltS 



s] 

c 

1st. Standard 



Initial Point 
I Base 



I^IK 



_Cl05inq 
" Townsnip .g 
Corners ."g 



Line 



\ 



o 



Parallel 



South 



L 



Standard 
Township 12 
Corners b 




Fig. 57. Showing Division into 24-MiLE Blocks. 



through the Initial Point, and extending both east and west there- 
from. This initial operation is indicated in Fig. 57. 

Evidently the principal meridian will be marked out on the 
ground as a straight line, while the base-line will follow the curve 
of a due east and west line, being at every point at right angles to 



PROCESS OF SUBDIVISION 1 25 

the meridian through that point. The field methods prescribed 
for running out the principal meridian and the base-line on the 
ground are described in detail in Arts. i55a-b, pp. 129-31. 

Second. The division of the area to be surveyed into tracts 
approximately 24 miles square (Fig. 57) by the establishment of 

(a) Standard Parallels conforming to true parallels of latitude 
through the 24-mile points previously established on the principal 
meridian, and extending both east and west therefrom, and 

(b) Guide Meridians conforming to true meridians of longi- 
tude through the 24-mile points previously established on the 
base-line and standard parallels, and extending north therefrom 
to an intersection with the next standard parallel or to the base- 
line. 

Since the guide meridians converge, these 24-mile tracts will 
be 24 miles wide on their southern and somewhat less than this 
on their northern boundaries. Theoretically, both the east and 
the west boundaries should be just 24 miles in length, but, owing 
to discrepancies of field measurements, this is rarely or never the 
case. 

Third. The division of each 24-mile tract into Townships, 
each approximately 6 miles square, by the establishment of 

(a) Meridional lines, usually called Range. Lines, conforming 
to true meridians through the standard township corners previ- 
ously established at intervals of 6 miles on the base-line and stand- 
ard parallels, and extending north therefrom to an intersection 
with the next standard parallel, or to the base-line, and 

(b) Latitudinal lines, sometimes called Township Lines, 
joining the township corners previously established at intervals 
of 6 miles on the principal meridian, guide meridians, and range 
lines. The division resulting from the first three operations is 
indicated in Fig. 57a. 

It will be apparent that, neglecting the effect of discrepancies 
and irregularities in measurement, both the east and the west 
boundaries of all townships will be just 6 miles in length, but the 
north and south boundaries will vary in length from a maximum 
at the standard parallel or base-line forming the southern limit 
of the 24-mile tract 'to a minimum at that forming its northern 
limit. 



126 



LAND SURVEYING 



fCHAP. V. 



Fourth. The subdivision of each township into Sections, each 
approximately i mile square and containing about 640 acres, by 
the establishment of Section Lines, both meridional and latitudinal, 
parallel to and at intervals of 1 mile from the eastern and southern 
boundaries of the township. (See Fig. 60, p. 147.) 



1 



# 



1st Sfa 



I 2ndJ Stlandfardj PnfraUJel Nlorttj | 



2. 
nda 



rd 



Base, 



T2S 
R3W 



* 



Par 



allel 



f 



1 



1 



No rth 



itial 
Li 



T3N 
R3E 



Pomp- J> 

ne 



2 V^ 

fl Ctesing" 



VD 



^, Stand 

5^ 



{ES&Sf* **"** 



|Tw nhipO men 



BE 



,1 ,1 J . 



etc. 



I 



Tow nship 



Tow nshif 



Tbw nshif 



Tow nship I Ni 



Tow nship 



Tow nship 




1st 



alnd 



T3S 
R7E 



ett. 



Stalnda rd Parallel 



So 



uth 



BMito 



M I 1 I 1 1 1 I 1 1 1 I 

Fig. 57a. Showing Subdivision of 24-MiLE Blocks into Townships. 



Assuming all fieldwork to be done with mathematical exact- 
ness, this subdivision would result in sections exactly 80 chains 
(1 mile) on each of the four sides,* except the most westerly range 
of 6 sections in each township, which would be less than 80 chains 
in width by an amount varying with the distance from the southern 
boundary of the 24-mile tract. The extent to which this condi- 
tion is realized in practice is indicated in Art. 162, p. 139, wherein 
the usual field methods of subdividing a township are described 
in detail. 



* These theoretical Sections would not be exactly square, as may be readilj 
perceived, but would be rhomboids. 



METHODS OF DESIGNATING LINES AND AREAS 12? 



154b. Methods of Designating Lines and Areas. — The 

various principal meridians and base-lines of the Public Lands 
Surveys are designated by definite names or by number, as, for 
example, "The Fifth Principal Meridian and Base-Line, ,, or 
"The Cimarron Meridian." 

The standard parallels are numbered in order both north and 
south from the base-line, and are so designated. The guide 
meridians are numbered in a similar manner east and west from 
the principal meridian. Fig. 57 illustrates the method. 

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. 

The tiers of townships are numbered in order, to both the 
north and the south, beginning with number 1 at the base-line; 
and the ranges of townships are numbered to both the east and 
the west, beginning with number 1 at the principal meridian. 
A township is designated, therefore, by its serial number north 
or south of the base-line followed by its number east or west of 
the principal meridian, as 
"Township 7 south, Range 
19 east, of the Sixth Prin- 
cipal Meridian." This is 
usually shortened to "T. 7 
S., R. 19 E., 6th P. M." 

The sections of a town- 
ship are numbered commen- 
cing with No. 1 at the 
northeast angle of the town- 
ship, and proceeding west to 
No. 6, and then proceeding 
east to No. 12, and so on, 
alternately, to No. 36, in 
the southeast angle as illus- 
trated by Fig. 57b. In all 
cases of surveys of fractional 

townships the sections will bear the same numbers they would 
have if the township were complete. 

The regular subdivisions of a Section are indicated by stating 



6 


5 


4 


3 


2 


1 


7. 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



Fig. 57b. Diagram of a Town- 
ship Illustrating Method op 
Numbering the Sections. 



128 LAND SURVEYING [Chap. V. 

briefly the aliquot part of the section intended together with its 
location in the section, as "the N. £ of the S.W. i of Sec. 27, 
T. 12 N., R. 5 W." 

154c. Field Methods. — The work of subdivision of the 
Public Lands has already been largely completed, and the sur- 
veyor of to-day is usually concerned only with the retracing of 
old lines, the relocation of lost corners, or with the subdivision 
work that comes with increase in population. For all these, 
however, a thoroughgoing knowledge of at least the common 
field processes and methods that have been used in the original 
surveys is essential. Certain details of field practice have varied 
somewhat from time to time, but the leading features have 
remained fairly constant for all those areas that have been sur- 
veyed since the system became well established. 

In the following pages is given a somewhat detailed descrip- 
tion of the methods commonly employed in carrying out the 
operations briefly indicated in Art. 154a. Inasmuch, however, as 
certain of the east and west lines are required to be established 
as true parallels of latitude, the two commonly accepted methods 
of accomplishing this will first be described. 

155. TO ESTABLISH A PARALLEL OF LATITUDE. — A parallel 
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 using 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. 



THE SECANT METHOD 



129 



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 
Lands Surveys. 

155a. The Secant Method.* — (Fig. 58.) " This method con- 
sists of running a connected series of straight lines, each six miles 
long, 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, measur- 
ing north or south, as the case may be, to attain other required 



T13KR.21E. 

\ -t* " JP^NT LINE 

- < ™W> STANDARD PARAUXL #>?&% \ 

* § 3 S t i « s * * 




1 \\tfWL\ 




[Offsets -ToWe 43 

Fig. 58. Secant Method for Establishing a Parallel of Latitude. 

points on the latitude curve." The o and 6 mile points of a 
parallel will be north of the secant, and the 2, 3, and 4 mile points 
will be south of the secant. 

The instrument is set up south of the township corner where 
the survey is to begin, the distance from the corner being found 
in Table 4 in the column headed "o miles." For example, in 
latitude 40 the transit would be set 2.79 feet south of the corner. 
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. 58.) This angle should be repeated 
several 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. 

* The quotations are from the "Manual of Surveying Instructions for the 
Survey of the Public Lands of the United States, " prepared by the Commissioner 
of the General Land Office in 1902. 



TABLE 4. 

Azimuths of the Secant, and Offsets, in Feet, to the Parallel. 

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



Lati- 


Azimuths and offsets at — 


Deflec- 
tion Angle 


tude 


miles. 


indie. 


x mile. 


1 J miles. 


2 miles. 


2 Y miles. 


3 miles. 


tan. to 
Bad. 66 ft. 


o 

30 


89° 56' .5 
1.03 H. 


89°58'.7 
0.87 N. 


0.00 


89° 59* .2 
0.67 s. 


1.15S. 


144S. 


90° (E. or W.) 
1.54 s. 


3' 00" .2 
0.69 ins. 


31 


89°58'.4 
2.ox N. 


89°58'.6 
0.91 11. 


89°58'.9 
0.00 


89°59'.2 
0.70 S. 


89° 6V J6 

x.20 S. 


89° 59' .7 
X.50S. 


90°(E.or W.) 
x.60 S. 


3' 07" .4 
0.72 ins. 


3a 


89° 58' .4 
2.09N. 


89° 68 .6 
0.94 H. 


89° 58' J9 
0.00 


89° sy.2 
0.73 s. 


89° 69> .5 
X.25S. 


89° 69". 7 
X.56S. 


90° (E. or W.) 
X.67S. 


& 15" .0 
0.75 ins. 


33 


89° 58' .3 
2.17 N. 


89° 58 .5 
0.97 N. 


0.00 


89°59'.l 
0.76 S. 


89° 50\4 
X.30 S. 


89°59'.7 
x.62 S. 


90° (E. or W.) 

1.73 s. 


3' 22".6 
0.78 Ins. 


34 


89° 58\2 
a. 25 N. 


89°58'.5 
I.OI 11. 


89°58'.8 
0.00 


89°59'.l 
0.79 s. 


89°59'4 
x.35 s. 


89° 69*. 1 
x.69 S. 


90° (E. or W.) 
X.80S. 


3' 30".4 
0.8 x ins. 


35 


89° 58\2 
2.33 H- 


89°58'.5 
1.05 N. 


89°58'.8 
0.00 


89° SPA 
0.82 S. 


89°59'.4 
x.40 S. 


89°59'.7 

1.75 s. 


90° (E. or W.) 
1.87 s. 


3' «".4 
0.84 ins. 


36 


89°58'.l 
2.42 N. 


89°58'.4 
1.09 11. 


89°58'.7 
0.00 


89°59'.0 
0.85 S. 


89° 6V A 
X.46S. 


89° 69' .7 
x.82 S. 


90° (E. or W.) 
X.94S. 


V 46".4 
0.87 ins. 


37 


89°58'.0 
2.51 11. 


89° 58'.3 
1. 13 N. 


89°68'.6 
0.00 


89°58'.9 
0.88 S. 


89°69'.3 
1.51 s. 


89° 6V.1 
X.89S. 


90° (E. or W.) 
2.ox S 1 


3' 65" .0 
0.90 ins. 


)8 


89°V.O 
2.61 N. 


89°58'.3 
1.X7N. 


89° 58' .6 
0.00 


89° 58'.9 
0.91 S. 


89°59\3 
X.56S. 


WW.! 
1.95 s. 


90° (E. or W. 
2.08 S.) 


4' 03" .6 
0.93 ins. 


39 


89° 57' .9 
2.70 N. 


89° 58' .2 
1.21 N. 


89° 58' .6 
0.00 


89° 58' .9 
0.94 s. 


89° 59' J3 
1.62 S. 


89°59\7 
2.02 S. 


90° (E. or W.) 
2. 16 S. 


4' 12".6 
0.97 in*. 


40 


89° 57' .8 
2.79 N. 


89°58'.l 
x.25 N. 


89°58'.5 
0.00 


89°58'.9 
0.08 S. 


89°69 > .3 
X.68S. 


89° 59' .7 
2.xoS. 


90° (E. or W.) 
2.24 s. 


4'21".6 
x.ooins. 


41 


89° 57' .7 
2.89 N. 


89°68'.0 
1.30 N. 


89°58'.4 
0.00 


89° 58' .8 
x.02 S. 


89° 6V.2 
1.74 S. 


89° 5^.6 
2.X7 S. 


90° (E. or W.) 
232 S. 


4' 31" .2 
x.04 ins. 


4» 


89° 57' .7 
3.00 N. 


89*58\0 
x.35 N. 


89° 58' .4 
0.00 


89°68'.8 
x.05 s. 


89° 59\2 
X.80S. 


89°69 f .B 
2.25 s. 


90° (E. or W.) 
2.40 s. 


4' 40".8 
1.08 ins. 


43 


89° 57' .6 
3- 11 N. 


89°58'.0 
1.40 N. 


89°58'.4 
0.00 


89° 58' .8 
X.08S. 


89° 6W.1 
X.86S. 


89°59\6 
2.33 s. 


90° (E. or W.) 
2.48 S. 


4' 50" .8 
x.x2 ins. 


44 


89° 57' .5 
322 N. 


89° 57'.9 
1.45 H. 


89° 58' .3 
0.00 


89°58'.7 
x.12 S. 


89° W.I 
x.93 s. 


89° 59> .6 
2.41 S. 


90° (E. or W.) 
a. 57 S. 


5' 01".0 
1. 16 ins. 


45 


89° 57'.4 
3.33 H. 


89° 57'.8 
X.50N. 


89° 68* A 
0.00 


89°58'.7 
1.16S. 


89°59M 
2.00 S. 


89° 69** 
2.49 s. 


90° (E. or W.) 
2.6/ S 


5' ll'\8 
x.20 ins. 


46 


89° 57'.3 
3-44 N- 


89° 57'.7 
1.55 H- 


89°58'.2 
0.00 


89° 5V.6 
X.2X S. 


ww.i 
2.07 s. 


89° 69'.6 
a.59S. 


90° (E or W.) 
2.76.S. 


6* 22".8 
x.24 ins. 


47 


89° 57' .2 
3-57 H. 


89° 57'.6 
1.61 N. 


89°58'.l 
0.00 


89°58'.6 
x.25 s. 


89°59U 
2.X4S. 


89°59'.5 
2.67 s. 


90° (E. or W.) 
2.86 S. 


6* 34".2 
x.28 ins. 


48 


89° 57M 
3.70 H. 


89° 57'.5 
X.66N. 


89°58'.0 
0.00 


89° 58' .5 
X.30S. 


89°59'.0 
2.22 S. 


89° 6P& 
2.78 S. 


90° (E. or W.) 
2.96 S. 


1.33 ins. 


49 


89° 57'.0 
3.82 N. 


89°57\5 
1.72 N. 


89°58'.0 
0.00 


89° 58'.5 
x.34 s. 


89°59'.0 
2.30 S. 


89°59'.5 
2.87 s. 


90° (E. or W.) 
3.06 S. 


6' 68".6 
1.381ns.. 


50 


89° W.9 
3.06 N. 


89° 57' .4 
X.78N. 


89° 57\9 
0.00 


89°58'.4 
1-39 s. 


89°59M> 
2.38 S. 


89°59'Ji 
a. 97 S. 


90° (E. or W.) 

3.17 s. 


V 11".4 
1.43 ins. 




6 miles. 


similes. 


1 
5 miles. | 


4} miles. < 4 miles. 


3f miles. 


3 miles. 


Deflec- 
tion Angle 


Lati- 







■' i 


1 


1 and nat. 


tude. 


Azim 


uths and 


ffsetsat — 






1 tan. to 
|Kad. 66 ft. 



130 



THE TANGENT METHOD 131 

■» 

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

" With ordinary field instruments, usually reading to single 
minutes only, fractional parts of the 'least count' are generally 
estimated by the eye. Greater accuracy may be attained by 
making use of a linear measure to lay off deflection angles." 
In the right-hand column of Table 4 are given linear dimensions 



31 
3*5 H. 




PARALLEL NOim*,UCrm^ .^J?- 

JL It*,!.. \ V 8 f\ ft *. «\ 



*d£' * 8 I \ ^r** 53 * 

f~* [Offsets -TOte 6] 1 Jg/ 

Fig. 58a. Tangent Method for Establishing a Parallel- of Latitude. 

suitable for use in laying off the deflection angles corresponding 
to the various latitudes noted. In using this extremely valuable 
method of laying off small angles, a point is first carefully marked, 
by double centering, at a distance of one chain (66 feet) from the 
instrument. A scale divided to decimals of an inch is then used 
to measure off toward the north the appropriate distance taken 
from the Table, and the vertical cross- wire of the transit is moved 
through the angle subtended. The reading of the vernier will 
check the measurement and guard against large errors. The 
direction thus determined is then prolonged in the usual manner. 
155b. "The Tangent Method. — This method consists in 
laying off from a true meridian, established by observations on 
Polaris at elongation, an angle of 90 , producing the direc- 
tion thus determined, a distance of 6 miles, in a straight line, and 



132 



LAND SURVEYING 



[Chap. V. 



TABLE 5. 
Azimuths of the Tangent to the Parallel. 

[The azimuth is the smaller angle the tangent makes with the true meridian and always 
measured from the north ana towards the tangential points.] 



Lati- 
tude. 


1 mile. 


a 


miles. 








1 


n 





/ 


„ 


33 

3« 
3a 


89 
89 
39 


59 
59 
59 


30.0 
28.8 
27.5 


89 
89 
89 


58 
58 
58 


59.9 
57.5 
56.0 


33 
34 
35 


89 
89 
89 


59 
59 
59 


26.2 
24.9 
23.6 


89 
89 
89 


58 
58 
58 


62.5 
49.9 
47.2 


36 

u 


89 

89 
89 


59 
59 
59 


22.2 
20.8 
19.4 


89 
89 
89 


58 
58 
58 


44.4 

41.6 
38.8 


39 
40 
4* 


89 
89 
89 


59 
59 
59 


17.9 
16.4 
14.8 


89 
89 
89 


58 
58 
58 


35.8 
32.8 
29.6 


4« 
43 
44 


89 
89 
89 


59 
59 
59 


13.2 
11.6 
09.8 


89 
89 
89 


68 
68 
58 


26.4 
23.1 
19.6 


45 
46 
47 


89 
89 
89 


59 
59 
59 


08.0 
06.2 
04.3 


89 
89 
89 


58 
58 
58 


16.1 
12.4 
08.6 


48 

49 
50 


89 
89 
89 


59 
59 
58 


02.3 
00.2 
58.1 


89 
89 
89 


58 
58 
57 


04.6 
00.5 
56.2 



Lati- 
tude. 



30 
3* 
3a 

33 
34 
35 

36 

% 

39 
40 
41 

4« 
43 



9 

47 

48 
49 
5Q 



7 miles. 



89 56 29.8 

89 56 21.3 

89 56 12.5 

89 56 03.6 

89 65 54.5 

89 56 45.2 

89 55 35.6 

89 55 25.8 

89 55 15.7 



89 



55 05.4 
54 54.7 
54 43.7 



89 54 32.4 

89 54 20.8 

89 54 08.7 

89 53 56.3 

89 53 43.4 

89 53 30.0 

89 53 16.1 

89 53 01.7 

89 52 46.6 



8 miles. 





1 


„ 


89 
89 
89 


55 
55 
55 


59.8 
50.0 
40.0 


89 
89 
89 


55 
55 
55 


29.9 
19.4 
08.8 


89 

89 
89 


54 
54 
54 


67.8 
46.6 
35.1 


89 
89 

89 


54 
54 
53 


23.3 
11.1 
58.5 


89 
89 
89 


53 
53 
53 


45.6 
32.3 

18.5 


89 
89 
89 


53 
52 
52 


04.3 
49.5 
34.3 


89 
89 
89 


52 
62 
51 


18.4 
01.9 
44.7 



3 miles. 



89 58 
89 58 
89 58 



29.9 
26 3 
22.5 



89 



58 18.7 

58 14.8 

58 10.8 

58 06 8 

58 02.5 

57 58.2 

57 53.7 

57 49.2 

57 44.4 

57 39.6 

57 34.6 

57 29.5 



57 
57 
57 

57 
57 
56 



24 1 

18.6 
12.9 

06.9 
00.7 
54.3 



4 


miles. 


3 


miles. 


6 miles. 





/ 


/• 





, 


„ 





/ 


„ 


89 
89 
89 


57 
57 
57 


59.9 
55.0 
50.0 


89 
89 
89 


57 
67 
57 


29.9 
23.8 
17.5 


89 
89 
89 


56 
56 
56 


59 8 
52.5 
45.0 


89 
89 
89 


67 
57 
57 


44.9 
39.7 
34.4 


89 
89 
89 


57 
57 
56 


11.2 
04.6 
58.0 


89 
89 
89 


56 
56 
66 


37.4 
29.6 
21.6 


89 
89 
89 


57 
67 
67 


28.9 
23.3 
17.5 


89 
89 
89 


56 
56 
56 


51.1 
44.1 
36.9 


89 
89 
89 


56 
56 
55 


13.4 
05.0 
56.3 


89 
89 
89 


57 
57 
56 


11.6 
05.5 
59.3 


89 
89 
89 


56 
56 
56 


29.6 
21.9 
14.1 


89 
89 
89 


55 
55 
65 


47.5 
38.3 
28.9 


89 
89 
89 


56 
56 
56 


52 8 
46.2 
39.3 


89 
89 
89 


56 
55 
55 


06 
67.7 
49.1 


89 
89 
89 


55 
55 

54 


19.2 
09.2 
58.9 


89 
89 
89 


56 
56 
56 


32.1 
24.8 
17.1 


89 
89 
89 


55 
55 

55 


40.2 
31.0 
21.4 


89 
89 
89 


54 
54 
54 


48.2 
37.2 
25.7 


89 
89 
89 


56 
56 
56 


09.2 
00.9 
52.6 


89 
89 
89 


55 
55 
54 


11.5 
01.2 
50.5 


89 
89 
89 


54 
54 
63 


13.8 
01.4 
48.6 



9 miles. 


xo miles. 


ix miles. 
/ // 


1a 


miles. 


1 


11 





/ 


11 





/ 


/» 


89 55 
89 55 
89 55 


29.8 
18.8 
07.6 


89 
89 
89 


54 
54 
54 


59.7 
47.6 
35.1 


89 
89 
89 


54 
54 
54 


29.7 
16.3 
02.6 


89 
89 
89 


53 
53 
53 


59.7 
45.1 
30.1 


89 54 
89 54 
89 54 


56.1 
44.4 
32.3 


89 
89 
89 


54 
54 
53 


22.3 
09.3 
55.9 


89 
89 
89 


53 
53 
53 


48.5 
34.2 
19.5 


89 
89 
89 


53 
52 
52 


14.8 
59.1 
43.1 


89 54 
89 54 
89 53 


20.0 
07.4 
54.5 


89 
89 
89 


53 
53 
53 


42.3 
28.2 
13.9 


89 
89 
89 


53 
52 
52 


04.5 
49.1 
33.2 


89 
89 

89 


52 
52 
51 


26.7 
09.9 
52.6 


89 53 
89 53 
89 53 


41.2 
27.5 
13.4 


89 
89 
89 


52 
52 
52 


59.1 
43.8 
28.2 


89 
89 
89 


52 
52 
61 


17.0 
00.2 
43.0 


89 
89 
89 


51 
51 
50 


34.9 
16.6 
67.1 


89 52 
89 52 
89 52 


58.8 
43.8 

28.4 


89 
89 
89 


52 
51 
51 


12.0 
55.4 
38.2 


89 
89 
89 


51 
51 
50 


26.2 
06.9 
48.0 


89 
89 
89 


50 
50 
49 


38.4 
18.6 
67.8 


89 52 
89 51 
89 51 


12.3 
55.7 
38.6 


89 
89 
89 


51 
51 
50 


20.4 
01.9 
42.9 


89 
89 
89 


50 
50 
49 


28.4 
08.1 
47.2 


89 
89 
89 


49 
49 
48 


36.4 
14.3 
61.4 


89 51 
89 51 
89 50 


20.7 
02.1 

42.8 


89 
89 
89 


50 
50 
49 


23.0 
02.4 
40.9 


89 
89 
89 


49 
49 
48 


25.3 
02.6 
39.0 


89 
89 
89 


48 
48 

47 


27.6 
02.8 
37.1 



TANGENT METHOD 

TABLE & 
Offsets, in Chains, from Tangent to Parallel. 



133 



Lati- 
tude. 


1 mile. 


a miles. 


3 miles. 


4 miles. 


5 miles. 


6 miles. 


o 


Chain*. 


Chains. 


Chains. 


Chains. 


Chains, 


Chains. 


30 
3* 
33 


0.006 
006 
0.006 


0.023 
0.024 
0.025 


0.053 
0.055 
0.057 


0.09 
0.10 
0.10 


0.14 
0.15 
0.16 


0.21 
0.22 
0.23 


33 
34 
35 


0.007 
0.007 
0.007 


0.026 
0.027 
0.028 


0.059 
0.061 
0.064 


0.10 
0.11 
0.11 


0.16 
0.17 
0.18 


0.24 
0.25 
0.25 


36 

2 


0.007 
0.008 
0.008 


0.029 
0.031 
0.032 


0.066 
0.068 
0.071 


0.12 
0.12 
0.13 


0.18 
0.19 
0.20 


0.26 
0.27 
0.28 


39 
40 
41 


0.008 
0.008 
0.009 


0.033 
0.034 
0.035 


0.074 
0.076 
0.079 


0.13 
0.13 
0.14 


0.20 
0.21 
0.22 


0.29 
0.30 
0.32 


4« 
43 
44 


0.009 
0.009 
0.010 


0.036 
0.038 
0.039 


0.082 
0.085 
0.088 


0.14 
0.15 
0.16 


0.23 
0.24 
0.24 


0.33 
0.34 
0.35 


45 

46 
47 


0.010 
0.010 
0.011 


0.040 
0.042 
0.044 


0.091 
0.094 
0.097 


0.16 
0.17 
0.17 


0.25 
0.26 
0.27 


0.36 
0.37 
0.39 


48 
49 
50 


0.011 
0.012 
0.012 


0.045 
0.046 
0.048 


0.101 
0.104 
0.108 


0.18 
0.19 
0.19 


0.28 
0.29 
0.30 


0.40 
0.42 
0.43 



Lati- 
tude. 


7 miles. 


8 miles. 


9 miles. 


10 miles. 


xx miles. 


xa miles. 





Chains. 


Chains. 


Chains. 


Chains. 


Chains. 


Chains. 


30 
3> 
3a 


0.29 
0.30 
0.31 


0.37 
0.39 
0.40 


0.47 
0.49 
0.51 


0.58 
0.60 
0.63 


0.71 
74 
0.76 


0.84 
0.88 
0.91 


33 
34 
35 


0.32 
0.33 
0.35 


0.42 
0.43 
0.45 


0.53 
0.55 
0.67 


0.65 
0.68 
0.70 


0.79 
0.82 
0.86 


0.95 
0.98 
1.02 


3« 


0.36 
0.37 
0.38 


0.47 
0.48 
0.50 


0.59 
0.61 
0.64 


0.73 
0.76 
0.78 


0.89 
0.91 
0.95 


1.06 
1.10 
1.14 


39 

40 

. 4i 


0.40 
0.41 
0.43 


0.52 
0.54 
0.56 


0.66 
0.68 
0.70 


0.81 
0.84 
0.87 


0.99 
1.02 
1.06 


1.18 
1.22 
1.26 


4^ 
43 
44 


0.44 
0.46 
0.48 


0.58 
0.60 
0.62 


0.73 
0.75 
0.79 


0.90 
0.93 
0.97 


1.09 
1.14 
1.18 


1.31 
1.35 
1.40 


3 

47 


0.49 
0.51 
0.53 


0.64 
0.66 
0.68 


0.81 
0.84 
0.87 


1.00 

1.04 
1.07 


1.22 
1.26 
1.31 


1.45 
1.50 
1.56 


48 
49 
50 


0.55 
0.57 
0.59 


0.71 
0.74 

0.77 


0.91 
0.93 
0.97 


1.12 
1.16 
1.20 


1.36 
1.40 
1.45 


1.61 
1.67 
1.73 



134 LAND SURVEYING [Chap. V. 

measuring north therefrom, at half-mile intervals, distances of 
correct length, taken from Table 6 (interpolated if necessary), 
for the given latitude, to attain other points on the latitude 
curve passing through the tangential or initial points. 

" The azimuth or bearing of the tangent at successive mile 
points will be taken from Table 5 to the nearest whole minute 
only, and will be inserted in the field notes, no interpolation 
being required, except when test sights are taken. The true 
bearing between two points on a standard parallel will be derived 
from Table 5 by taking it in the column headed with one-half the 
distance between said points. The offsets at intervals of one 
mile are inserted in Table 6; to obtain the length of offsets at the 
half-mile points, take one-fourth of the 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." 

" 156. 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." 

157. Base-Line. — From the initial point the base-line is 
extended both east and west on a true parallel of latitude, one of 
the methods described in the foregoing paragraphs being used. 
Great care is taken to secure instrumental accuracy. Two 
back and two fore sights are taken at each setting of the instru- 
ment, the horizontal limb being revolved 180 in azimuth 
between the observations, in one method, taking the mean of 
observations. Another method, called double back and fore 



INITIAL POINTS AND BASE-LINE 135 

sights, is still more exact, and therefore 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. (See Art. 64, p. 52.) 

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

"In order to detect errors and insure accuracy in meas- 
urement, two sets of chainmen will be employed; one to note 
distances to intermediate points and to locate topograph- 
ical 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 measurements; but if the discrepancy exceed 8 links on 
even ground, or 25 links on mountainous surface, the true 
distance will be found by careful re-chaining by one party or 
both. 

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

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

The proper township, section, and quarter-section corners 
are established at the appropriate intervals, and meander and 
witness corners (Arts. i6sa-c, pp. 148-9.) are set wherever the 
line crosses such streams, lakes, bayous, or other objects as may 
make their use necessary. Stones or posts used to mark the 
positions of the township or section corners are marked on 
their north face with the letters SC, for "standard corner," for 



136 LAND SURVEYING [Chap. V. 

the purpose of easily distinguishing them from the "closing 
corners," to be set later. 

158. Principal Meridian. — The principal meridian is 
extended as a true meridian of longitude both north and south 
from the initial point. The methods used for the determina- 
tion of directions, and the precautions observed to secure 
accuracy of measurement, are the same as those described in 
the preceding article, under the subject of " Base-Line." 

Also, as in the case of the base-line, all township, section, 
quarter-section, and other necessary corners are established in 
the proper places as the survey proceeds. 

159. Standard Parallels. — Standard parallels, which are also 
sometimes referred to as correction lines, are extended both east 
and west from every fourth township corner previously estab- 
lished on the principal meridian. Sometimes, however, the dis- 
tance between them is more or less than 24 miles, depending 
upon the requirements of the particular survey in question. 
For example, in Kansas the correction lines occur at regular 
intervals of 30 instead of 24 miles. In all cases deviations from 
the regular order are made only under the written special 
instructions of the Surveyor General. The Manual provides 
further that "where gross irregularities (in previous surveys) 
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." 

Standard parallels are established as true parallels of latitude, 
and are run in the same manner and with the same precautions 
for accuracy as in the survey of the base-line. 

Appropriate corners are established at the proper intervals, 
and the township and section corners are marked SC on their 
north face, the same as those on the base-line. 

160. Guide Meridians. — Guide meridians are extended north 
from the base-line, or standard parallels, at intervals of 24 miles 
east and west from the principal meridian. They are run as 
true meridians of longitude, and are extended to an intersection 
with the next correction line north. At the point of intersection 



STANDARD PARALLFXS AND GUIDE MERIDIANS 1 37 

of the guide meridian with the correction line a closing corner is 
established, and the stone or post is marked on its south face with 
the letters CC, to distinguish it from the standard corners already 
in place. Also, the distance of the closing corner from the 
nearest standard corner is measured and recorded in the field 
notes. This correction offset will vary with the latitude and 
with the distance of the corner from the principal meridian. 
At a distance of 15 or 20 ranges from the principal meridian it 
may be so great that the closing corner will be nearer to the 
adjacent quarter-section corner than to the standard township 
corner. Furthermore, it is obvious that the closing corners will 
be west of the corresponding standard corners on the east 
side of the principal meridian, and east of them on the west 
side. 

The mile and half-mile distances on the guide meridians are 
made full 80 and 40 chains in length until the last half-mile is 
reached, into which all excess or deficiency due to discrepancies 
of measurement is thrown. 

The general method of running the guide meridians is the same 
as that used in running the principal meridian, and all the pro- 
visions for securing accuracy of alignment and measurement, and 
for establishing corners, prescribed for the latter apply to the 
former also. 

Provision is made for running guide meridians from north to 
south where existing local conditions require this departure from 
the usual practice. In such a case the closing corner is first 
established on the correction line by calculating the proper cor- 
rection distance and laying it off from the standard corner; and 
then the guide meridian is run due south from this point. This 
method may be used in case the standard corner from which the 
guide meridian would ordinarily originate is inaccessible, or for 
other adequate reasons. 

The Manual also provides that " where guide meridians have 
been improperly placed at intervals greatly exceeding the author- 
ized 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 



138 land surveying [chap. v. 

Guide Meridian. ' These additional guide meridians will be 
surveyed in all respects like regular guide meridians." 

161. Township Exteriors. — The usual method of subdividing 
a 24-mile tract into townships is as follows (see Fig. 57a). 

Beginning at the standard corner at the southeast comer of the 
southwest township in the tract, the surveyor runs north on a 
true meridian of longitude a distance of 6 miles, setting all neces- 
sary corners by the way. From the township corner thus estab- 
lished he runs due west on a random line (Art. 191, p. 169) to 
intersect the guide meridian (or the principal meridian, in case 
he is working in Range 1 East), setting temporary section and 
quarter-section corners as he goes. When he intersects the 
meridian, he notes the "falling"* of his random line, and, in 
case this is within the limit prescribed, he then calculates the 
course of the true line joining the two township corners and runs 
back on it, setting permanent corners opposite the temporary ones 
previously set on the random line. In this way all the deficiency 
due to the convergence of the meridional boundaries of the town- 
ship, together with whatever excess or deficiency may arise from 
inaccuracies in measurement, are thrown into the most westerly 
half-mile of the latitudinal boundary. 

The range line is now continued as a true meridian for another 
6 miles, permanent corners being set as before. Then another 
random line is thrown across to the western boundary of the 
range of townships, and is corrected back to the true line, in the 
same manner as that just described. This process is continued 
until the most northerly township in the 24-mile tract is reached, 
when the range line is merely continued as a true meridian to 
an intersection with the correction line, at which point a closing 
township corner is established. The half-mile intervals on the 
range line are made full 40 chains for the entire 24 miles, except 
the most northerly half-mile, into which all excess or deficiency 
due to irregularities of measurement is thrown. 

The two other range lines of the 24-mile block are run in a 
similar manner, the latitudinal township lines being extended to 



* That is, the distance of the point at which the random line intersects the 
meridian from the objective corner. 



SUBDIVISION OF TOWNSHIPS 1 39 

the westward at the proper intervals and made to connect with 
the township corners previously established. From the township 
corners on the last range line, however, random lines are run also 
to the eastward to meet the guide meridian, and are then cor- 
rected back to the westward on a true line between the township 
corners. This is done in such a way that the excess or deficiency 
of this line also is thrown into the most westerly half-mile. 

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

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

The Manual provides for a maximum allowable limit for 
closing the random line upon the township corner, as follows: 
"If in running a random township exterior, such random exceeds 
or falls short of its proper length by more than 3 chains, allow- 
ing for convergency, or falls more than 3 chains to the right or 
left of the objective point (or shows a proportionate error for 
lines of greater or less length than 6 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 the cause of misclosure." A lateral displace- 
ment of 3 chains in a distance of 6 miles is equivalent to an 
angular deviation of 21 minutes. 

162. Subdivision of Townships. — In the subdivision of a 
township into sections the following routing is followed in the 
field. The surveyor sets up his instrument at the southeast 



140 LAND SURVEYING [Chap. V 

corner of the township, observes the meridian, and retraces the 
range line northward for a distance of one mile, and the township 
line westward for the same distance. This is for the purpose of 
comparing his own meridian and needle observations and the 
length of his chain with those of the previous surveyor who laid 
off the township exteriors.* Then from the southwest corner of 
Section 36 he runs north on a line parallel with the east boundary 
of the township, setting a quarter-section corner at 40 chains 
and a section corner at 80 chains. Then from the section corner 
just set he runs east on a random line, parallel to the south 
boundary of the section, setting a temporary quarter-section 
corner at 40 chains. When he intersects the range line he notes 
the falling of his random and also the distance it overruns or falls 
short of the length of the south boundary of the section. If the 
falling is not more than 50 links (33 feet, representing an angular 
deviation of 21 minutes), and if the distance overruns or falls 
short of the length of the southern boundary of Section 36 by not 
more than the same amount, a return course which will join the 
two section corners is calculated; this new line is then run toward 
the west, the permanent quarter-section corner being set at its 
middle point. 

From the section corner just regained the survey is now con- 
tinued north between Sections 25 and 26, the direction being 
changed slightly to the east or west according to whether the 
latitudinal section line just completed exceeded or fell short of 
the desired length. At 40 and 80 chains on this line the quarter- 
section and section corners, respectively, are set, and from the 
section corner a random is run across to the range line, and a 
return course is calculated and run as before. This process is 
continued until five of the six sections in the series are inclosed. 
Then, if the north boundary of the township is not a correction 
line, from the section corner last established a random is run 
north to the township boundary, and from the data thus secured 
a true line is calculated and run from the section corner on the 
township line back to the initial corner. If the north boundary 
of the township is a correction line, however, the point at which 



* See specimen field notes, p. 144. 



SUBDIVISION OF TOWNSHIPS 



141 



the random intersects this boundary is established as a closing 
corner and its distance from the nearest standard corner is meas- 
ured and recorded. In either case the permanent quarter- 
sfection corner is established at 40 chains north of the initial 
corner, the excess or deficiency being thrown into the most 
northerly half-mile. 

In a similar manner the succeeding ranges of sections are 
enclosed, randoms being run across eastward to the section cor- 
ners previously established and true lines corrected back. From 
the fifth series of section corners thus established, however, 
random lines are projected to the westward also, and are closed 
on the corresponding section corners in the range line forming 
the western boundary of the township. In correcting these lines 
back, however, the permanent quarter-section corners are estab- 
lished at points 40 chains from the initial corners of the randoms, 
thereby throwing all fractional measurements into the most 
westerly half-miles. Reference to Figs. 59 and 60 will help 
toward an understanding of this method of subdivision. 

Table 6a, taken from the Manual, gives (to the nearest whole 
minute) the angular convergency of meridians from one to five 
miles apart. The meridional section lines, therefore, by reason 
of being (theoretically) parallel to the range line on the east 
boundary of the township, will depart from true meridians by 
the amounts indicated in the table. 

TABLE 6a. 
Corrections for Convergency within a Township. 



Latitude. 



30 to 35 
35 to 40 
40 to 45 

45 *> 50 

5° to 55 
55 to 60 

60 to 65 

65 to 70 



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



i mile. 


a miles. 


3 miles. 


4 miles. 


5 miles. 


/ 


' 


' 


/ 


/ 




I 


2 


' 2 


3 




I 


2 


3 


3 




2 


2 


3 


4 




2 


3 


4 


i 




2 


3 


5 




3 


4 


5 


7 


2 
2 


3 
4 


1 


8 


8 
10 



142 LAND SURVEYING [Chap. V. 

From a consideration of the foregoing it will be apparent 
(i) That interior meridional section lines are 80 chains in 
length, except those next to the north boundary of the town- 
ship; and that the south half of these is 40 chains. 

(2) That interior latitudinal section lines are within 50 links 
of the length of the line forming the southern boundary of the 
range of sections, except those section lines next to the west 
boundary of the township; and that the east half of these is 40 
chains. 

(3) That interior section lines, whether meridional or latitu- 
dinal, are ordinarily straight for one mile only. 

(4) That except in those section lines next to the north and 
west boundaries of the township, the quarter-section corners 
are placed equidistant from the two section corners on either 
side. 

(5) That meridional section lines are intended to be parallel 
to the range line forming the eastern boundary of .the town - 
ship; and similarly, that latitudinal section lines are intended to 
be parallel to the township line forming its southern boundary. 

(6) That the cumulative deficiency in latitudinal lines due 
to the convergence of the meridians is thrown into the most 
westerly half-mile of the township. 

(7) That no quarter-section closing corners are established on 
correction lines for the use of the sections south of these lines. 

163. Fractional Sections. — In sections made fractional by 
rivers, lakes, or other bodies of water, lots are formed bordering 
on the body of water, and numbered consecutively through the 
section. The boundaries of these lots usually follow the quarter 
lines of the section, and contain, as nearly as may be, forty 
acres each. Fig. 59 indicates the method. Also, the quarter 
quarter sections along the north and west boundaries of a 
township, into which the discrepancies of measurements or 
the deficiencies due to the convergence of the range lines are to 
be carried when the sections are subdivided, are usually num- 
bered and sold as lots. (See Art. 169, p. 153d.) These lot lines 
are not actually run in the field, but, like the quarter-section 
lines, are merely indicated on the plates, and the areas by which 
the lots are sold are computed in the office. 



FRACTIONAL SECTIONS 



H3 



Field notes taken in connection with the survey of public 
lands are required to be returned to the General Land Office 
in the narrative form. A sample page of notes, somewhat con- 
densed from those given in the Manual, is herewith presented 



NB9 J5ff 




5£* 



sjraft 



5 S3" S3 I 



Fig. 59. Showing Part of a Typical Township; also Showing Method 
op Forming Lots in Fractional Sections. 



for its value not only as an illustration of a section line survey 
but also as an instructive example of this very valuable method 
of note keeping. Fig. 59, also from the Manual, shows the 
nine sections in the southeastern part of a typical township, and 
illustrates the specimen notes given. 



144 



LAND SURVEYING 



[Chap. V. 



Chains. 



4.50 
20.00 

29.30 
30.00 
39.50 



40.00 
40.50 



50.50 
51.50 

57.50 

72.50 

80.00 



Specimen of Field Notes. 
Subdivision of T. 15 N., R. 20 E. 

From the Tp. cor. already described,* I run 

North, on the 5th Guide Meridian and £. bdy. of sec. 36: and, at 
40.01 chs., intersect the { sec. cor.; and, at 79.98 chs., fall 1 Ik. W. 
of the cor. of sees. 25, 30, 31, and 36; therefore, the line bears north. 
From the Tp. cor. I run N. 89° 57' W., on the S. bdy. of sec. 36; at 
39.99 chs., fall 0* Ik. N. of the £ sec. cor.; and at 80.01 chs. fall 
1 Ik. S. of the cor. of sees. 1, 2, 35, and 36, on S. bdy. of the Tp.; 
consequently, the S. bdy. of the sec. 36 bears N. 89° 57' W. 
Therefore, the bearings are as stated by the surveyor general, and my 
chaining practically agrees with the field notes of the original survey. 
I commence at the cor. of sees. 1, 2, 35, and 36, on the S. bdy. of the 
Tp., which is a sandstone, 6x8x5 ins. above ground, firmly set, 
and marked and witnessed as described by the surveyor general. 
Thence I run 

N. 0° Or W., bet. sees. 35 and 36. 
Over level bottom land. 
Wire fence, bears E. and W. 

Enter scattering cottonwood timber, bears E. and W. F. G. Alexan- 
der's house bears N. 28° W. 
Leave scattering cottonwoods, bearing E. and W. ; enter road, bears N. 
SE. cor. of F. (f. Alexander's field; thence along west side of road. 
To crossroads, bears E. 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 i sec. cor., marked W C J 835 
on W. and 36 on E. 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. 
Point for ± sec. cor. in road. 

Deposit a marked stone, 24 ins. in the ground, for \ sec. cor. 
The 8E. cor. of Pat. Curran's field bears W., 5 Iks. dist. 
Set a limestone, 15 X 8 X 6 ins. 10 ins. in the ground, for witness cor. 
to } sec. cor., marked W C J 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, 1$ ft. high, W. of cor. 
Thence along E. side of field. 

NE. cor. of Pat. Curran's field, bears W. 4 Iks. dist. 
Leave road; which turns to N. 70° W., leads to ferry on Yellowstone 

River; thence to Lake City. 
Enter dense cottonwood and willow undergrowth, bears N. 54° E. and 

S. 54° W. 
Leave undergrowth, enter scattering timber, bears N. 60° E. and S. 

60° W. 
Set a locust post, 3 ft. long, 4 ins. sq., 24 ins. in the ground, for cor. of 
sees. 25, 26, 35 and 36, marked 
T 15 N S 25 on NE., 
R 20 E S 36 on SE., 
8 35 on SW., and 

S 26 on NW. face; with 1 notch on S. and E. faces; from which 
An ash, 13 ins. diam., bears N. 22° E., 26 Iks. dist., marked T 15 

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

T 15 N R 20 E S 36 B T. 
A walnut, 17 ins. diam., bears S. 64° W., 41 Iks. dist., marked 

T 15 N R 20 E S 35 B T. 
A cottonwood, 13 ins. diam., bears N. 21 \° W., 36 Iks. dist., 
marked T 15 N R 20 E S 26 B T. 



* Description omitted. A description of the determination of a true meridian 
by both solar and Polaris observations is also omitted. 



DATA TO BE NOTED 145 

164. " Summary of objects and data intersected by the line 
or in its vicinity, to be noted. — 1. The precise course and length 
of every line run, noting all necessary offsets therefrom, with 
the reason for making them, and method employed. 

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

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

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

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

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

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

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

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

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



I46 LAND SURVEYING [Chap. V. 

•"11. 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 
extent, 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, with their directions, whence and 
whither. 

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

" 17. Precipices, caves, 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 the 
position of such points will be perfectly identified in the field 
notes. " 

165. Marking Corners. — Corners are marked on the- 
ground by various kinds of monuments, depending upon the 
character and importance of the corner to be perpetuated, 
the soil, the materials available, and upon other local and special 
conditions. In places where stone is plentiful monuments of 
this material are usually set. In timbered districts where suitable 
stones are difficult to obtain, posts are driven to mark the points. 
In prairie regions where neither stones nor timber are available 



MARKING CORNERS 



147 



a mound of earth may be raised over the corner, a small marked 
stone, a charred stake, a quart of charcoal, or some other per- 
manent and distinguishable mark being deposited beneath it. 
Occasionally in the timber the corner falls on a spot occupied by 
a tree, in which case the tree itself may stand as the monument. 




.2 



16 



30 



SI 



Townfchip 



20 



29 



32 



Lfc* 



21 



28 



33 



fcwp 



22 



27 



34- 



(a) 



23 



26 



35 




Fie 60. Showing the Subdivision op a Township into Sections and the 
Method Commonly used por Marking Township and Section Corners. 



In case stones or posts are set they are marked with notches 
as shown in Figs. 60 and 60a, in order to indicate their respective 
positions in the township. Section corners on range lines, 
including under this term principal and guide meridians, are 
marked with notches on their north and south faces, the number 



148 



LAND SURVEYING 



[Chap. V. 




Fig. 



60a. Sketch of Stone Monu- 
ment, Showing Notches. 



of notches being equal to the number of miles to the next adjacent 
township corner north or south. In a similar manner the section 
corners on the township lines, including base-lines and standard 
parallels, are notched on their east and west faces. Township 
corners, being located on both range and township lines, are 
marked with six notches on each of the four sides. In addition 
to being notched as just indicated, corners on correction lines are 
marked SC on their northern or CC on their southern faces, 

depending upon whether they 
are standard or closing corners. 
Section corners in the interior of 
a township are given notches on 
their east and south faces corre- 
sponding to the number of miles 
to the east and south boundaries 
of the township. Thus, the 
corner common to sections 20, 
21, 28, and 29 would have two 
notches on the south and four on the east face, as sketched 
in Fig. 60a. Quarter-section corners are marked with the 
fraction "£", those on meridional lines on their west and those 
on latitudinal lines on their north faces. 

165a. Witnessing Corners. — Wherever possible the monu- 
ment set at a corner is witnessed by several nearby objects, which 
may be easily found by anyone looking for the corner itself, which 
are not readily moved or obliterated, and which are compara- 
tively permanent. In timbered country the stone or post is 
usually witnessed by "bearing trees " located near the corner. 
The process of establishing a witness tree is to take its bearing 
and distance from the corner, then to blaze off the bark from a 
short section of the trunk on the side facing the corner and to 
cut into the wood with scribing tools certain letters and numerals 
indicative of the section in which the tree is located. For 
example, the tree northeast from the corner shown in Fig. 60a 
might be marked 

7 s 

15 E 

31 

T, 



T 
R 
S 
B 



WITNESS CORNERS 1 49 

the letters and figures being abbreviations of "Township 7 south, 
Range 15 east, Section 21, Bearing Tree." Usually one tree is 
marked in each of the sections to which the corner refers, pro- 
vided suitable trees can be found within a reasonable distance of 
the corner. 

In prairie regions small rectangular pits are dug near to the 
corner, the earth taken from them being used to form a mound. 
These pits are placed either on the section lines leading from the 
corner or at angles of 45 degrees with these lines, depending on 
the kind of corner witnessed; and the mound may be either 
alongside the monument or, in case the monument is merely a 
deposit beneath the surface of the ground, may be placed immedi- 
ately over it. Fig. 60b, adapted from illustrations given in the 
Manual, indicates the manner of using this method of witnessing 
corners of the several classes. Marks of this kind are of much 
greater value than might at first be supposed, for, although the 
sharp outlines are quickly worn away, the grass sod soon covers 
the mound and grows down into the pits and preserves them from 
entire obliteration. In many places on the plains four slight 
depressions in the prairie sod with a little mound between have 
perpetuated the location of the section corner for a generation or 
more, until the country has been settled up and the fence lines 
strung. Under other prevailing conditions corners have been 
witnessed by mounds of stone, by prominent boulders, and by 
various other suitable objects. 

165b. Witness Corners. — In case a regular corner falls in a 
creek, pond, or in any other place where it is impracticable to 
set or maintain a monument, witness corners are set on all the 
lines leading to this corner. These ^re marked with the letters 
WC in addition to the markings that would be appropriate to the 
corner of which they are witnesses. Witness corners are, in 
turn, referenced by bearing trees, pits and mounds, and other 
objects, the same as true corners. 

165c. Meander Corners. — Where a surveyed line intersects 
the bank of a stream whose width is more than three chains, or 
of a lake, bayou or other body of water having considerable 
extent, a meander corner is established. The distance from the 
nearest section or quarter-section corner is measured and recorded 



ISO 



LAND SURVEYING 



[Chap. V. 



€3 



.© 



« 



Standard Township, r 1 Mf> 



Corners. 



©■ 




S 



«■! pi Standard Section 
Corners. 






& 




^&£&& "^if-^^6 



Closing Township 
Corners. 



JO 

t3 



V.^ Closing*.**. %(Jj 

"£3 



* 






'€3 



© 



common to 
4 Townships 




Standard 

Quarter Section 

Corners 



jjYjj " wS®^ 




© 



<3 

Comers -J* V**/ '" 

"*[_ *J Quarter ^ ^ 



Section Cor. on a 
Meridional line 



rYrspactive 
Of Standard T> Cor, wffli Closing T> Cor, inthe 

PLANS OF CORNERS 

■ ■ Corner Stone or posh 

M« Mound. 

M'" Mound with deposit-. 

P«Ph\ 



Fig. 60b. Showing Scheme for Designating Corners by Means of 

Mounds and Pits. 



MEANDERING 151 

in the notes, and the stone or post set as a monument is marked 
MC on the side facing the water, and the point is referenced by 
bearing trees or by mounds of earth in much the same manner 
as a quarter-section corner. If practicable, the line is then carried 
across the stream or other body of water by triangulation to 
another meander corner set in line on the further bank, and the 
survey is continued, 

166. MEANDERING. — After the regular subdivision work 
has been done traverses are run, usually by the needle, joining 
the successive meander corners along the banks of the streams or 
lake. A traverse of this kind, or a meander line, as it is called, 
originates at a meander corner and follows as closely as may be 
practicable the various sinuosities of the bank until the next 
meander corner is reached. Here the traverse is checked by 
calculating the position of the new meander corner and comparing 
this with its known position on the surveyed line, and the mean- 
dering is then continued. Fig. 61 illustrates the relation of 
meander corners and lines to the regular lines of the survey. 
These meander lines are used in plotting the stream on the map 
and in calculating the areas of the sections or quarter-sections 
made "fractional" by the presence of the body of water. 

The following quotation from the Manual indicates the 
location of meander lines, their functions in the survey, and their 
authority as boundaries. 

"Lands bounded by waters are to be meandered at mean 
high-water mark. This term has been defined in a State decision 
(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 



is* 



LAND SURVEYING 



[Chap. V. 



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 affect 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 




Fig. 61. 



Showing the Relation of Meander Corners and Meander 
Lines to the Section Lines. 



windings of the high-water line, the U. S. Supreme Court has 
given the principles governing the use and purpose of meandering 
shores, in its decision in a noted case (R. R. Co. v. Schurmeier, 
7 Wallace, 286-7) as follows: 

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



ACCESS TO RECORDS AND MAPS 1 53 

terms will be universally used to distinguish the two banks of 
a river or stream. 

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

167. Access to Records and Maps of the Public 

LANDS SURVEYS* — It is evident that a present day surveyor, 
practicing in a region originally owned as Public Land and sur- 
veyed as such, should have frequent access to the field notes of the 
original survey of his district. The Circular previously referred 
to states that "The original evidences of the public land surveys 
in the following States have been transferred to the State authori- 
ties, to whom application should be made for such copies of the 
original plats and field notes as may be desired, viz. : 

Alabama: Secretary of State, Montgomery. 

Arkansas: Commissioner of State Lands, Little Rock. 

Illinois: Auditor of State, Springfield. 

Indiana: Auditor of State, Indianapolis. 

Iowa: Secretary of State, Des Moines. 

Kansas: Auditor of State and Register of State Lands, Topeka. 

Michigan: Commissioner of State Land Office, Lansing. 

Mississippi: Commissioner of State Lands, Jackson. 

Missouri: Secretary of State, Jefferson City. 

Nebraska: Commissioner of Public Lands and Buildings, Lincoln. 

Ohio: Auditor of State, Columbus. 

Wisconsin: Commissioners of Public Lands, Madison. 

" In other public land States the original field notes and plats 
are retained in the offices of the United States Surveyors General." 

In many if not all these States named either the original records 
or copies of the same have been distributed among the various 



IS 3a LAND SURVEYING [Chap. V. 

Counties of the State, and are kept for reference and inspection 
in the office of the County Register of Deeds, County Surveyor, 
or other official. 

Township maps of much of the area covered by the Public 
Lands Surveys may be obtained from the General Land Office at 
Washington, at nominal prices. 

168. Relocating lost Corners. — it has been the com- 
mon experience that many of the monuments and marks origi- 
nally established on the lines of the Public Lands Surveys become 
lost or obliterated by the time the country has been settled for a 
generation or two. Witness and line trees * are cut down when 
the land is cleared, the pits and mounds marking the corners on 
the prairie are quickly destroyed when the sod is broken up, 
posts rot away, and no one takes the trouble to see that new and 
more durable marks are set to perpetuate the location of the 
points. Largely owing to the fact that in the areas covered by 
the Public Lands Surveys the public roads are usually located 
along the section lines, even substantial stone monuments often- 
times are carelessly knocked out of place and eventually are 
thrown into the ditch or the fence corner. Particularly during 
the pioneer period, when the country is just being settled and 
land is cheap, the greatest indifference prevails relative to the 
preservation of the original monuments, resulting in serious 
disputes and costly litigation later when the land becomes more 
valuable. A considerable part of the work of the present day 
County Surveyor is concerned with the relocation of corners that 
have been carelessly destroyed. 

An act of Congress approved February n, 1805, specifically 
provides that corners actually located in the field shall be estab- 
lished as the proper corners of the sections or quarter-sections 
which they were intended to designate, irrespective of whether 
they were properly located in the first place or not. A further 



* "Line trees" are those directly on a line of the survey. They arc blazed on 
opposite sides, the blazes facing backward and forward along the line. Trees 
near the line are scored with two blazes "quartering" toward the line; the further 
the trees are from the line the nearer together the two blazes are placed, and vice 
versa. These blazed trees are of great service in marking the approximate position 
of the line through the timber. 



RELOCATING LOST CORNERS 153b 

provision is that "the boundary lines actually run and marked" 
(in the field) "shall be established as the proper boundary lines 
of the sections, or subdivisions, for which they were intended, 
and the length of such lines as returned by . . . the surveyors 
aforesaid shall be held and considered as the true length thereof." 
These are the principles upon which is based the present practice 
in the relocation of the corners of the original survey.* 

The General Land Office distinguishes between an obliterated 
and a lost corner, as follows : 

" An obliterated corner is one where no visible evidence remains 
of the work of the original surveyor in establishing it. Its loca- 
tion may, however, have been preserved beyond all question by 
acts of landowners, and by the memory of those who knew and 
recollect the true situs of the original monument. In such cases 
it is not a lost corner. 

"A lost corner is one whose position cannot be determined 
beyond reasonable doubt, either from original marks or reliable 
external evidence." 

In the case of a corner that is merely obliterated the method of 
procedure, obviously, is to establish a new monument in the same 
location as the old one, this location being determined by the 
evidence presented, which should be adequate for the purpose. 
Instances of this kind occur when old witness trees, or their 
stumps, or the depressions left in the forest floor by their decay, 
may be identified; or when the point is marked by the intersection 
of hedge or stone or other permanent fences which admittedly 
were constructed on line when the monument was still in place; 
or when the "true situs of the monument" is testified to by other 
competent witnesses. (See Art. 151, p. 116.) 

In the case of lost corners the true location must ordinarily 
be determined from data obtained by actually rerunning the old 
lines, as nearly as may be. But here, in accordance with the 
principle first stated above, the aim should be to relocate the 

* The General Land Office publishes a Circular on the Restoration of Lost or 
Obliterated Corners and Subdivision of Sections, which states these principles 
and suggests methods of procedure in conformity therewith. Many of the 
methods referred to in the following paragraphs are condensed from this 
Circular. 



1 5 3C LAND SURVEYING [Chap. V. 

corner at the exact point at which it was originally established, 
irrespective of whether it was properly located in the first place 
or not. As a help toward this end the following suggestions are 
offered, taken mainly from the Circular of the General Land 
Office to which reference was made in the earlier part of this 
Article. 

A lost corner on a principal or guide meridian or range line 
will be located by proportional measurements from the nearest 
original corners in place north and south of the lost corner. It 
will be located on the straight line joining these original corners, 
irrespective of whether the measurements to corners east or west 
of the lost corner correspond with the original field notes or not. 

A lost standard corner on a base-line or standard parallel or 
other correction line will be located by proportional measurements 
to the nearest original corners east and west of the lost corner, and 
will be located on the true line joining them, irrespective of 
whether the required distances to corners north or south would 
tend to pull it off this line or not. In like manner, a lost corner 
on an interior latitudinal township line will be recovered by 
proportional measurements to corners in place east and west of it. 

In other words, a lost corner is to be relocated by proportional 
measurements from corners which were established at the same 
time and with the same degree of care as the lost corner. It 
sometimes happens that errors are discovered which throw a 
doubt upon the accuracy of the field notes of the line upon which 
the corner in question was located, in which case the lateral 
measurements might prevail and the corner be located in accord- 
ance with them; but ordinarily the rule as stated is the one to be 
observed. 

A lost section corner in the interior of a township is to be 
located by proportional measurements from the corners nearest 
to it in all four directions. It is sometimes found, however, that 
the meridional section lines have been run with greater care than 
the latitudinal lines, owing to the operation of certain routine 
methods in use on some surveys. In case this is found to be 
true, the measurements on the meridional section lines may be 
given a certain precedence over those on the latitudinal lines. 

A lost closing corner on a correction line should be located at 



SUBDIVISION OF SECTIONS 153d 

the intersection of the correction line with the meridional line 
closing upon it, even though the distance called for would place 
the corner north or south of the correction line* It is even held 
true of a closing corner actually in place, if it happens to be a 
little off the correction line, that such corner is to be construed 
as establishing merely the direction and not the termination of 
the meridional line upon which it is located. This is one of the 
very few cases in which the statutory provision quoted in the 
early part of this Article is not rigidly adhered to. 

A lost quarter-section corner is always to be established at the 
middle point of the section line upon which it was originally 
located, except those next to the west and north boundaries of 
the township. On correction lines the quarter-section corners 
referring to the sections south of the line* are to be located mid- 
way between the adjacent closing section corners, except that 
in the north boundary of Section 6, which is to be placed forty 
chains west from the east boundary of the section. 

169. Subdivision of Sections. — When the Public Lands were 
parceled out to settlers the quarter-section was usually the unit 
area granted as a "homestead." To locate the lines of a "quar- 
ter," however, obviously required the establishment of the 
quarter-section corner at the center of the section. Also, the 
subsequent division of the original "quarters" into "eighties," 
"forties," or other minor subdivisions has necessitated the loca- 
tion of numerous corners in addition to those originally estab- 
lished by the Government. Much of the routing work of the 
present day surveyor is in connection with subdivisions of this 
kind. In the following paragraphs a number of typical examples 
will be given, illustrating common practice in the location of 
subdivisional corners. It will be noted that the methods given 
are based upon those employed in the original surveys of the 
sections. 

The interior quarter-section corner of a section is always to be 
located at the intersection of straight lines joining the quarter 
corners on opposite sides of the section. This method holds 
wherever the section may be located within the township; that is, 

* It will be remembered that these corners were not established on the 
original survey. 



I53e 



LAND SURVEYING 



[Chap. V. 



it applies to those in the north and west tiers as well as to the other 
sections of the township. For example, the center of Section 19, 
in Fig. 61a, would be properly located at the intersection of 
ab and ef. This rule would still hold even though, through 
some error in the original field work, the corner/, for instance, had 
been set too far east by one chain-length. If the corner is still 
in the place at which it was actually set by the Deputy Surveyor, 
this location is established for all time as the proper corner for 
the northern quarters of the section, and the line from e will be 
run to it. 





A f*k 


B J 




^^^ 




D 




1 


^ 


!\ 1 


n 


c ? 


1 ^ 






1 


P 


^. *- 


u 


1 L. 


• 1 






H 


I 


|g 1 




If d 


E 



Fig. 61a. Illustrating Methods of Locating Interior Quarter- 
Section Corners. 



In case one or more of the exterior quarter-section corners are 
not accessible, lines are run through the interior of the section 
from whatever quarter corners are in place, having as nearly as 
possible the ^ame directions as they would have were all the 
exterior corners actually in place, and the interior quarter-section 
corner is located at their intersection. For example, in Fig. 61a 
the corner at the center of Section 21 would be located by the 
intersection of a line run north from d in a direction which is a 
mean between that of ED and that of FC, with a line run 
east from c having a direction which is a mean between those of 
CD and FE. In case only one of the interior lines can be 
run, the corner may be established by proportionate measurements 
along that line. For example, the center of Section 20 would be 
located by measuring out from c on the line cb a distance equal 
to half the mean length of CB and FG. A modification of 
this method is required in the west and north tiers of sections 



SUBDIVISION OF SECTIONS 



I53f 



in a township. In the case of sections lying in the west tier the 
meridional line run from the quarter corner on the north or south 
boundary of the section is run parallel to the east line of the sec- 
tion; and similarly, in the case of sections lying in the north tier 
the latitudinal interior line initiated at the quarter corner on the 
east or west boundary is run parallel to the south line of the sec- 
tion. The reasons for this method of procedure in these special- 
cases are easily apparent from a consideration of the methods 
previously described for the establishment of the original quarter- 
section corners in these sections. 

For subdivisions smaller than quarter-sections the same general 
methods are employed. For example, to subdivide the north- 
east quarter of Section 10 (Fig. 6ib) into quarter-quarters or 




Fig. 61b. Illustrating the Subdivision of a Quarter of an Interior 

Section. 

Note. — Figures on the outside lines of the illustration are those of 
the original survey. 

"forties," straight lines are run connecting the middle points of 
the opposite sides of the quarter. In case one or more of these 
starting corners are inaccessible, the quarter will be subdivided 
by the application of methods similar to those just outlined for 
the location of the quarter-section corner at the center of a section. 
In the subdivision of quarter-sections adjacent to the west or 



I53g 



LAND SURVEYING 



[Chap. V. 



north boundaries of the township the excesses or deficiences 
originally thrown into these quarters are not divided up between 
the different subdivisions, but are carried forward into the western 
or northern tiers of forty-acre lots. Fig. 6ic illustrates this point 
by showing an ideal subdivision of the northwest quarter of 
Section 6, in a township not immediately south of a correction 



3&76 Township Line 4000 



tt.64 



I 



5 j 
iaae 



6 



-I 



7 

19.08 



20.00 

3 I 



.2000 



JOOO. 



i 3 



40^00 



Fig. 61c. Showing the Subdivision op the North-west Quarter of 

Section 6. 

Note. — Figures on the outside lines of the illustration are those of 
the original survey. 



line. In order to make them apparent in the Figure, the original 
excesses and deficiences are shown on an exaggerated scale. 
The tract marked 4 might be properly described as "the north- 
west quarter of the northwest quarter of Section 6," or simply as 
"Lot 4, of Section 6," of the appropriate township and range. 



CONVERGENCE OF MERIDIANS 



154 




170. CONVERGENCE OF THE 
MERIDIANS. — The angular con- 
vergence of the meridians, given 
in Table 6a, 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 A and B. A T and BT 
are lines tangent to the meri- 
dians at A and B, meeting the 
earth's axis, prolonged, at T. It 
will be seen that the angle 
BTO equals the angle BOQ, 
which is the latitude of points 
A and B. The angle AO B 
is the difference in longitude 
of points A and B. The angle 



Fig. 62. 

between the meridians at A and B is the angle A TB. 
In the sector AO'B, 



— = angle A (7 B 



In the sector A TB, 
AB 



But 



BT 
BT= 



= angle A TB (approximately) 
BO' BO' 



sin BT<y sin BOQ 
AB 



,\ angle ATB = ^—- sin BOQ 
BO 

=angle AO B sin BOQ, 
i.e., the angular convergence equals the difference in longitude 
times the sine of the latitude. 

The linear convergence of two meridians equals the dis- 
tance run (N. or S.) times the sine of the angular conver- 
gence. 



iSS 



LAND SURVEYING 



[Chap. V 



Example. — To find the angular convergence between two 
meridians 6 miles apart in latitude 37 . The length of i° of 
longitude in latitude 37 is 55.30 miles (Table 7). 



5530 



x sin 37 x 6o = 3'.9. 



TABLE 7. 
Length of a Degree of Longitude. 





Degree of Longi- 




Degree of Longi- 




Degree of Longi- 


Lat. 


tude 


Lat. 


tude. 


Lat. 


tude. 




Statute Miles. 




Statute Miles. 




Statute Miles. 





69.160 


30 


59-944 


60 


34-666 


I 


.150 


31 


«' 33 * 


61 


33 615 


2 


.119 


3 2 


58.706 


62 


32-553 
31 .481 


3 


.066 


33 


.060 


63 


4 


68.992 


34 


57.396 


64 


3°- 399 


5 


68.898 


35 


56.715 


S 


29.308 


6 


.783 


36 


.016 


28.208 


7 


.647 


37 


55- 3°° 


67 


27 .100 


8 


.491 


38 


54568 
53-8i9 


68 


25 983 


9 


•3M 


39 


69 


24.857 


10 


68.116 


40 


53.053 


70 


. 23.723 


11 


67 .898 


41 


52 .271 


7' 


22.582 


12 


.659 


42 


5i -473 


72 


21 435 


'3 


.400 


43 


50.659 


73 


20.282 


H 


. 120 


44 


49.830 


74 


19.122 


'5 


66.820 


45 


48 .986 


75 


17.95 6 
16.784 


16 


-499 


46 


.126 


76 


17 


.158 


47 


47.251 


77 


15.607 


18 


65797 


48 


46 .362 


78 


14.425 
13 238 


19 


.416 


49 


45 -459 


79 


20 


65.015 


5° 


44542 


80 


12.047 


21 


64.594 


5 1 


43.611 


81 


lo i s l 
9.656 


22 


•'54 


52 


42 .667 


82 


23 


63 695 


53 


41.710 


P 


8.456 


24 


.216 


54 


40.740 


84 


7253 


M 


62.718 


55 


39-758 


85 


6.048 


.201 


56 


38.763 
37756 


86 


4841 


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. 

4 

TRAVERSE LINES. — LOCATION OF BUILDINGS. — MISCEL- 
LANEOUS SURVEYING 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 1," the next " Station 2" etc. Every 100-ft. length is 
a. full station and any fractional distance is called the plus. The 
distance from Station o to any point, measured along the traverse 
line, is the station of that point and is recorded always by the 
number of the last station with the plus station in addition, e.g., 
the station of a point at 872.4 ft. from Station o is 8 + 72.4. 

At the angle points it is customary to measure the deflection 
angles rather than the interior angles because the former are 
usually the smaller. These should be checked in the field by 
"doubling "the angles. (See Arts. 143-5. PP- 108-10.) 

The notes are kept so as to read up the page. The left-hand 
page is for the traverse notes and the right-hand page for the 
sketch, the stations in the sketch being opposite the same station 
in the notes. Fig. 63 is a set of notes illustrating this type of 
traverse. Frequently no notes are kept in tabular form, all of 
the data being recorded on the sketch. 

172. METHODS OF CHECKING TRAVERSES WHICH DO NOT 
FORM CLOSED FIGURES. — Checking by Astronomical Methods. — 
The angles of any traverse can be checked by determining the 
azimuth of the first and last lines by astronomical methods. 
(See Chapter VII.) But since the meridians converge it is neces- 

156 



157 



CHECKING TRAVERSES 



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 Pagc.) 



falminQry Survey FcrX** XX%M*m/h#,M& 


• ftedmatt. Jlotf- rx.. 


N 


Sta. 


fbint 


Deff.A*ylt 


3manng 


Calculated 
etoartng 


* • 


. 


10 
9 


•+94* 


4?i7Jt 


Nift 


NO m 06£ 


( 


) 




6 

+ 46S 










m t *2&fftL 








W/ttmrSti 


—• >""8Tirt 




7 


• *7te 


MT43L 


AT&fW 


M30MW 


i { 


> 




6 

s 

4. 


**£/ 


j?/7/r 


whw 


M/faw 




> 




3 












<&> 




+42 










v£6 


*T 




2 










^^ 






1 
O 







nh'Mw 


#2f4srt 






- 


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) 



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. 



[Chap. VI. TRAVERSE LINES I5& 

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 triangulation points which can be easily identified. (See 
Art. 283, p. 255.) The latitude and longitude of these triangula- 
tion points and the distances between them can be obtained from 
the proper authorities. Sometimes the distances between them 
are not known but they can be computed. Then by connecting 
the traverse lines with these triangulation points by angles and 
distances a closed traverse is obtained, which serves as a good 
check. 

Many surveyors fail to appreciate the value of this method of 
checking and do not realize how many such points are available. 
The information concerning such triangulation points can be 
obtained from The U. S. Coast and Geodetic Survey, The U. S. 
Geological Survey, State surveys, and frequently from City or 
Town surveys. 



1 59 GENERAL PRINCIPLES 

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 : — 

(1) 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 : — 

(1) By two points on the line. 
t (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. 



[Chap. VI. LOCATION OF BUILDINGS 160 

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

A swing offset is the perpendicular distance to a line and is 
found by trial. The zero end of the tape is held at the point to 
be located and the tape is swung in a short arc about the point 
as a center, the tape being pulled taut and kept horizontal. 
The tape is read from the transit in various positions, and the 
shortest reading obtainable is the perpendicular distance desired. 

A range line is a line produced to intersect the transit line 
or some other line. 

179. GENERAL SUGGESTIONS. — By whatever method the 
buildings are located the following suggestions should be carried 
out. 

(1) 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 check 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 



l6l TYPICAL CASES 

« 

obvious that they are not. The tie from C to station 2 is a 
check against an error in the other measurements. 

Plotting. — This building would be plotted thus: — scale 
the distance AX perpendicular (estimated) to the transit line 




and draw a line with triangles parallel to the transit line ; then 
scale Al from station 1 to this parallel line. Point A is then 
located. Point B is located in the same way, AB being used as 
the tie from A. Then by means of triangles and scale the 
building is completed and the distance C2 scaled and compared 
with the notes. Another way to plot point A would be to set 
on the compass the distance 1A and swing an arc about 1 as a 
center ; then, keeping the scale perpendicular to the transit line, 
find where the distance XA will cut this arc, thus locating point 
A, Point B can be similarly located after A has been plotted. 
For the same degree of accuracy distances can be measured 
more rapidly with a scale than they can be laid off with a com- 
pass, therefore the former method is usually more practicable. 

This building might have been located by four ties AO t Al, 
Bl, 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 II. Building Near Transit Line and Making 
a Slight Angle with it. — Fig. 65 illustrates two ways of locating 
a building in such a position that the intersection of the transit 
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 B$ the fieldwork necessary is slightly greater. If B2 is 



[Chap. VI. LOCATION OF BUILDINGS 1 62 

taken Bl might be measured as a check tie although A I 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 F are measured. The range lines CE and DF are 
also measured and the check tie C3. C2 could have been taken 
as a check tie ; it would have given a better intersection at C 
than the tie C3, but it is much longer. 





TRANSIT LINE 

Fig. 66. 

Plotting. — The left-hand building is plotted as described 
in Example I. In plotting the right-hand building the plus sta- 
tions on the transit line are first scaled. Then with the com- 
pass set at the distance EC an arc is swung from E 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 CS 
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 HI. 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. 



1 63 TYPICAL CASES 

Plotting. — The plotting of this building is done by swing- 
ing the tie from one station to a corner of the building and scal- 
ing from the other station the tie to the same corner. Then the 




Sta.2 

Fig. 06. 

other corner is plotted in the same way or by using the side of the 
building as one of the ties in case it gives a better intersection. 

184. Example IV. Building Located at a Considerable Skew 
to the Transit Line. — A building which is at a- considerable skew 
to the transit line can best be located by range ties as illus- 
trated in Fig. 67. The range ties through A are sufficient to 




twamsit w ^ ^ \ LINE 

Sta2 

Fig. 67. 

locate the building, provided AE and AF are not too short in 
comparison with the sides of the building. If these ranges are 
long enough, then B3 is a check tie ; but if the ranges are short, 
B3 must be depended upon to determine the position of point 
B and in this event one of the range ties becomes a check. But 
if A is within two or three feet of the transit line it will be well 
to omit one of the ranges and take the additional tie 2C or the 
range tie DC produced. 



[Chap. VL LOCATION OF BUILDINGS 1 64 

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. 

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



A 



// X 

T KAW3IT /' X LINE 

Stal 2 3 4 5 

Fig. 68. 



#- 



is not long enough to allow the use of swing offsets. Range 
ties may be used provided the building is not so far away that 
the eye cannot judge the range line with reasonable accuracy. 
Sometimes the only methods available are long ties or angles or 
a combination of the two. In any specific case there may be 
some objections to any of these methods, and the surveyor will 
have to decide according to circumstances which method he will 
use. For example, where there are obstacles to the measure- 
ment of ties, the corners of the building may have to be located 
entirely by angles from two points on the transit line. Loca- 
tion by angles is objectionable because it is difficult to plot an 
angle quickly and at the same time accurately. It often hap- 
pens, however, that when a building is at a considerable distance 
from the transit line its accurate position is not required, since 
as a rule the features near the transit line are the important 
ones. This method of "cutting in " the 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 



1 65 TYPICAL CASES 

with a protractor and the proper intersections determine the 
corners of the buildings. If the building is measured the side 
between the corners 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 M 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 



Sto4 5 

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 barn is located by the distance BK, the 
range tie KC and the tie BC, and checked by the tie BE. An- 
other location of the barn is the distance AK or BK, the range 
tie KC, and the two range ties AJ and CJ. By this latter method 
the directions of both sides of the barn are checked. Still an- 
other location of the point C would be to substitute in the place 
of the range tie CK a swing offset from C to the house. The 
shed is located by the range ties ^4^and FG and by the tie AG. 
The check tie HD in general checks the location of both the 
barn 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 



[Chap. VL 



LOCATION OF BUILDINGS 



166 



poorer intersection at Z. If convenient a tie from L to 4 or 
the range GF continued to the transit line may be measured as 
a check. 

187. Example VII. Buildings of Irregular Shape. — Occa- 
sionally a building of irregular shape has to be located. For 
example, the shop in Fig. 70 is located on the front by ties and 




swing offsets like Example I ; then the direction of AB is deter- 
mined by the range tie BC, The back corner E is determined 
by the ranges FH and EH, 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 tange ties FH and 
EH, but even when F is a right angle it will be well if time will 
permit to take these range distances as they give valuable checks 
on the other measurements which the single tie EF does not 
furnish. ED is scaled along HE produced and the rest of the 
building plotted by its dimensions and checked by AD. 

The ties shown on Fig. 70 to locate the factory chimney 
will locate its sides even if they do not form a regular polygon. 
If such a structure is situated at a considerable distance from 
the transit line probably the best way to locate it is by angles 
and distances to the corners, by the measurements of the sides, 
together with a few such ranges as NJ or KJ. 

188. Example Vm. Large City Buildings. — Fig. 71 illus- 
trates the location of several buildings in a city block where 
the transit line runs around the block. The fronts of the build- 
ings are located from the transit line and the rear corners are 
tied together. The range ties are shown by dotted lines and 
other ties by dashes. The angles measured are marked by 



\6j 



TYPICAL CASES 



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. 



[Chap. VI. 



LOCATION OF BUILDINGS 



168 




Fig. 72. 



RANDOM LINE 1 69 

MISCELLANEOUS SURVEYING PROBLEMS. 

191. RANDOM LINE. — Not infrequently in attempting to 
run a straight line between two points A and B (Fig. 73) it is 
impossible to see one point from the other or to see both points 
A and B from an intermediate set-up on a straight line between 
them. When this condition exists it is necessary to start at one 
point, e.g., 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 
offset YB is measured and also the distance to point Y along AC. 
Unless the random line is very close, say, within about two feet of 
the line AB> the point Y where a perpendicular to AC 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 A Y of the 
right triangle A YB. The distance YB is then measured, and 
AB and angle YAB are easily calculated. 

Angle DAY has been measured from some previous course 



170 MISCELLANEOUS SURVEYING PROBLEMS [Chap VL 

such as AD and the addition of the angle YAB together with 
the known distance- AB makes the traverse complete to the 
point B without any further fieldwork. If the transit is now 
moved to B with a view to carrying on the survey it will be 
found that, since A cannot be seen from B 9 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 offset 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 A to B 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 A. 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 171 

The instrument is then set up at B and a right angle ABF laid 
off with the transit. BF is made any convenient distance which 
will bring the auxiliary line beyond the building. Similarly 
point E is set opposite point A, and sometimes a second point E' 
opposite^', points A and 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 f unless AE is quite long. 

The instrument is then set up at F and backsighted on E, the 
sight is checked on E\ the telescope inverted, and points G, If, 
and If set on line. Leaving the telescope inverted, another 
backsight is taken on E, and the process repeated as described 
in Art. 64, p. 52. Then the transit is moved to point G, and a 
right angle turned off, and point C set on the right angle line, 
the distance GC being made equal to BF. 




Fig. 74. 

Then by setting up at Cand sighting ahead on D 9 (DIf= CC), 
and checking on point D' y (VIP=sGC), the transit line is again 
run forward in its original location. The distance FG is care- 
fully measured which gives the distance BC, and thus it appears 
why it is so necessary that the lines BF and GC shall be laid off 
at rigth angles by means of the transit. The other offsets AE, 
A'E', DH, and D'H' are not in any way connected with the 
measurement along the line ; they simply define the direction of 
the line so that if convenient it is often only necessary to show 
these distances as swing offsets for the transitman to sight on. 
From what has been said it will be seen that offsets A'E f and 
D'H 9 are not absolutely necessary, but they serve as desirable 
checks on the work and in first-class surveying they should not 
be omitted. For obvious reasons the offsets AE and DH should 
be taken as far back from the obstacle as is practicable. 

Should the house be in a hollow so that it is possible to see 
over it with the instrument at A, the point D, or a foresight of 
some sort (Art. 64, p. 52) should be set on line beyond the house 



172 MISCELLANEOUS SURVEYING PROBLEMS [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 one behind the transit, can carry on the work under these 
conditions more accurately and conveniently. This method in- 
sures an accurate prolongation of the line, but the distance 
through the building must be measured by an offset method, 
unless it can be done by plumbing from the edge of the flat roof. 

194. Short Transit Sights. — Sometimes the offset BF 
(Fig. 74) does not need to be more than 2 or 3 feet. The 
shorter this offset line can be made, and still clear the build- 
ing, the better. But to lay off the short line BF will require a 
method somewhat different from any that has been hereto- 
fore explained. As the ordinary transit instrument cannot be 
focused on a point much less than about 5 ft. distant it is im- 
possible to set point F directly. The method employed is to 
set a temporary point, say 10 ft. distant, on which the transit 
can be focused, and on a line perpendicular to the original 
transit line. From the transit point to this auxiliary point a 
piece of string may be stretched and the point F set at the re- 
quired distance from B and directiy under the string. 

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



MEASURING AROUND OBSTACLES 



173 



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




Fig. 76. 

B" and C" are set at the same distances from A. Then point B 
is placed midway between B' and B ', and similarly point C is 
set midway between C and C". The line BC is the prolonga- 
tion of the transit line. Of course the distance B'C should be 
made as long as practicable. The inaccuracy in this method 
lies entirely in laying off the two angles. (See Art. 61, p. 50.) 
In this case the distance AB can be computed from the formula 

~BB n 
AB' — AB = — — r (approximately). (See foot-note, p. r 3.) 
2AB 

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




line AB runs through a tree, and points A, D, and B have been set 
on line. DE is made equal to some convenient short distance and 
laid off at right angles to the transit line by eye. Then AE and 
EB are measured. The distance 



AB = AE- 



DE* 



+ EB- 



DE> 



(See foot-note, p. 13.) 



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 PROBLEMS [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. 77) is set 
on the transit line as near the building as practicable but so that a 
line BC at 6o° with the transit line can be run out. The instru- 
ment is set up at B, backsighted on A, and an angle of 120 laid 
off ; the line BC is made long enough so that when the instru- 
ment is set up at Cand 6o° is laid off from it, CD 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 
I20°the line DE 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. 

198. 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 line. The line BC is 
laid off at 90 to BA and is intersected with the transit line and 
the distance BC measured. AC is calculated from AB and cos 
A and checked by BC and sin A. Also the angle ACB can 
be measured which will check the transit work. 



INACCESSIBLE DISTANCES 



175 




Fig. 78. 

200. Intersecting Transit Lines. — In many kinds of 
surveying work it is necessary to put in points at the intersection 
of two transit lines. It would be an easy matter to set the 
point if two transits could be used, one on each line, and the 
sight simultaneously given by each transitman. As it is seldom 
practicable to use more than one transit in a surveying party the 
following method is resorted to. 

An estimate is made by eye where the lines will cross each 
other and temporary points not more than 10 ft. apart are set on 
one of the transit lines by means of the instrument, enough points 
being marked to make sure that the second line will cross some- 
where among this set of temporary points. A string is then 
used to connect two of these temporary points and the transit 
is set up on the other transit line and the point where the 
second line cuts the string is the intersection point. Some- 
times when the lines cross each other at nearly 90 the intersec- 
tion point can be estimated so closely that only two temporary 
points need be placed on the first line. In other cases, where 
the two transit lines cross at a very small angle, it is impossible 
to tell by eye within several feet where the lines will intersect 
and a number of points must be used because in practice the 
stretching line is seldom applicable for distances much over 
15 ft. For short distances the plumb-line can be used as a 
stretching line. 

201. Inaccessible Distance by Swing Offset Method. — If the 
distance across a pond or river is not great the following method 



176 MISCELLANEOUS SURVEYING PROBLEMS [Chap. VI. 

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 100-ft. swing offset 
from point A. 

A pencil is held vertically at the 100-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 ab. 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 A if more convenient. 

202. Inaccessible Distance by Tangent Offset Method. — In the 
method described above the distance across the pond may be so 
great that 100 ft. will be too short a base to use, or point A may be 
situated on ground sloping upward towards B so that a swing offset 



INACCESSIBLE DISTANCES 



177 



cannot be made. In such cases the line AB (Fig. 80) can be laid 
off at right angles to the transit line and of any convenient length. 



TRANSIT , 




LINE 


1 


A i 




.x^C 


<*>. 








*H 


1 


mPQNDjljjL^ 





Fig. 80. 

Then the angle ACB is measured and the line AC computed. 
By another set-up of the instrument the angle B can be meas- 
ured as a check, and if the line BC does not cut across the pond 
its length can also be measured as a further check. 

203. Inaccessible Distance by Oblique Triangle Method. — 
Often the shores of a stream are covered with trees so that none 




of the above methods are applicable. It may be convenient to 
measure a line AB (Fig. 81) in but one direction along the shore. 
In this case the point C is first carefully set on the opposite side, 
the line AB measured along the shore, and the angles at A and 



178 MISCELLANEOUS SURVEYING PROBLEMS [Chap. VI. 

at Care measured. The distance AC can then be computed. 
It will be well also to set up at B and measure the angle B 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 sure 
measured, and similarly at C the angles ACB and BCD are meas- 
ured. AB can then be calculated as follows : — in the triangle 
CBD compute CB; in triangle ACD compute AC; and in the 
triangle ACB calculate AB, the inaccessible distance. In the tri- 



Fig. 82. 

angle ACB, angle ABC can be computed, which, together with 
the measured angle BCD, will give the difference in direction 
between AB and CD. It is not at all necessary that DC should 
have been measured as one straight line in the traverse ; the trav- 
erse might have run as indicated by the dotted lines, but in such 
an event the distance CD and the necessary angles could have 
been easily figured so that it could be reduced to the above prob- 
lem. 



INACCESSIBLE DISTANCES 179 

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

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. 82) 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 A, the angles CAD and DAB are measured ; 
at B the angle CBD and ABC are measured. The length of 
the line CD is known. While it is simple to obtain CD in terms 
of AB, 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 
CD, and the actual length of CD being known the distance AB 
can be computed. 

A problem of this sort would occur under the following 
circumstances. If the distance CD between two church spires 
were accurately known (from a triangulation system) and it is 
desired to use this line CD as a base-line for a survey, two points 
A and B could be assumed, and the distance between them and 
the azimuth of AB could be found by this method. 



CHAPTER VII. 

OBSERVATIONS FOR MERIDIAN AND LATITUDE.* 
OBSERVATIONS FOR MERIDIAN. 

206. TO ESTABLISH A TRUE MERIDIAN LINE BY OBSERVA- 
TION ON POLARIS WITH THE TRANSIT. — On account of the 
earth's daily rotation on its axis all heavenly bodies appear "to , 
revolve once a day around the earth. Stars in the south appear to 
^^ revolve in large circles parallel 

to the daily path of the sun. 
As we look farther north the 
apparent size of the circles 
grows smaller. The center of 
these circles is the north pole of 
the celestial sphere^ a point in 
the sky in the prolongation of 
the earth's axis. The pole-star 
(Polaris) revolves about the 
pole in a small circle whose 
radius is less than a degree and 
a quarter (Fig. 83). This an- 
gular distance from the pole to 
a star is called its polar distance. 
When the star is directly 
above the pole its bearing is 
the same as that of the pole 
itself and the star is said to be 
at upper culmination. At this 
instant it is in the true meri- 
dian. About twelve hours later 
it will be below the pole at 
lower culmination and will be 
again in the true meridian. About half-way between these 
two positions the star reaches its greatest east or west bearing, 

* See also Chapter II, Volume II. 
180 



nBStvm CnndotiOTi 



Costirn Etongcf w* 




Fig. 83. 



MERIDIAN OBSERVATIONS ON POLARIS 



181 



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 



o 

O 

*i 

H 

X 
W 

o 
o 
z 

c/> 
H 
W 

r 
r 
> 

s 

*/: 

x 
w 

O 

r 



> 

H 

w 
c/5 

H 
W 

W 

r 
o 

o 
> 

H 
O 

as 



NOIXVNI WIAD H3AA0T xv sisihvioj 

N3HAI SNOLLVIiaXSNOO 3HX JO NOLLISOJ 




Ursa Major ^ 



o 

►•* 

H 

< 

o 

W 

w 

H 

CO 



< 

a. 

55 
M 

X 

o 
B 
< 

H 

z 

o 

U 

tii 
X 
H 

h 

o 

O 



o 

Oh 



Fig. 84 



Position of the Constellations when 
Polaris is at UPPER CULMINATION. 

Relative Position of the Constellations near the 
North Pole. 



1 82 OBSERVATIONS FOR MERIDIAN [Chap. VIL 

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 decMnation 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 8 * Cassiopeia, the lower 
left-hand star of the W, to £ Ursa Majoris, the middle star of the 
Dipper handle, passes very close to Polaris and also to the pole 
itself. 

207. OBSERVATION FOR MERIDIAN ON POLARIS AT 
ELONGATION. — When the Dipper is on the right and Cassiopeia 
on the left, Polaris is near its western elongation ; when the dipper is 
on the left Polaris is near eastern elongation. When the constella- 
tions are approaching one of these positions the transit should be 
set over a stake and leveled, and the telescope focused upon the 
star.f Unless the observation occurs at about sunrise or sunset 
it will be necessary to use an artificial light to make the cross-hairs 
visible. If the transit is not provided with a special reflector for 
throwing light down the tube a good substitute may be made by 
cutting a small hole in a piece of tracing cloth or oiled paper and 
then fastening it over the end of the telescope tube by a rubber 
band. If a lantern is then held in front and a little to one 
side of the telescope the cross-hairs can be plainly seen. The 
star should be bisected by the vertical wire and followed by 
means of the tangent screw in its horizontal motion until it no 

* The Greek Alphabet will be found on p. 519. 

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 
1 object. 

I 



OBSERVATIONS AT ELONGATION 1 83 

longer changes its bearing but moves vertically. (It will be seen 
from Fig. 83 that when the star is approaching eastern elonga- 
tion it is moving eastward and upward ; when approaching western 
elongation it is moving westward and downward.) As soon as 
this position is reached the telescope should be lowered and a 
point set in line with the vertical cross-hair at a distance of sev- 
eral hundred feet from the transit. Everything should be ar- 
ranged beforehand so that this can be done quickly. Immediately 
after setting this point the instrument should be reversed and 
again pointed on the star. A second point is then set at one 
side of the first. The mean of these two points is free from the 
errors of adjustment of the transit. If the instrument is in ad- 
justment, of course, the first and second points coincide. On 
account of the great difference in altitude between the star and 
the mark the elimination of instrumental errors is of unusual 
importance (Art. 79, p. 61). For 10 minutes of time on either 
side of elongation the bearing of the star does not change more 
than 5 seconds of arc and therefore there is sufficient time to 
make these two pointings accurately. 

After the direction of the star at elongation has been found, 
the meridian may be established by laying off an angle equal to 
the azimuth, or true bearing of the star. Since this angle to be 
laid off is the horizontal angle between the star and the pole, it 
is not equal to the polar distance but may be found from the 
equation : — 

o- o^ , T* t> • Sin Polar Distance of Star * 

Sin Stars True Bearing = — - — : — . 

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 
trigonometry, 

sin PZE sin P E 

sin ZEP** sin ZP ' 

But PZE is the angle between the two vertical circles and equals the bearing. 
ZEP=*$o because ZE is tangent to the circle WUEL, which represents the path 
of Polaris. PE is the polar distance and ZP may be shown to be equal to 
90 — latitude. 

Hence, sin PZE = s _ in -^l . 

cos lat. 



1 84 



OBSERVATIONS FOR MERIDIAN [Chap. VII. 



TABLE 8. 

Mean Polar Distances of Polaris.* 



Year. 


Mean Polar Distance. 


Year. 


Mean Polar Distance. 




' n 




» 


1906 


I II 41.05 


1914 


I 09 12.07 
1 08 53.51 


1907 


I II 22.37 


- J 9i5 
1916 


1908 


I 11 03.71 


I 08 34.97 


1909 


i 10 45-°7 


1917 


I 08 16.45 


1910 


1 10 26.44 


1918 


1 07 57-94 


191 1 


1 10 07.82 


1919 


1 07 3945 
1 07 20.98 


1912 


1 09 49.22 


1920 


1913 


1 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 
POLAR& angle thus computed must be laid off to the right if 
the elongation is west, to the left if the elongation is 
east. A convenient and accurate way of laying off 
the angle is by measuring the distance between the 
two stakes A and B (Fig. 85), and calculating the per- 
pendicular distance BC which must be laid off at the 
north stake B to give a meridian AC. 



* The above table was derived from data furnished by the Sup- 
erintendent of the United States Coast and Geodetic Survey. The 
Mean Polar Distance is the polar distance the star would have if 
unaffected by small periodic variations. 

In taking the polar distance from the table for the purpose of 
looking up its sine the student should keep in mind the degree of 
precision desired in the computed azimuth. If the azimuth is to be 
within about one minute of the true value the polar 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 given 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 1 85 

208. OBSERVATION FOR MERIDIAN ON POLARIS AT CULMI- 
NATION. — 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 & Cassiopeia will be almost di- 
rectly below Polaris (Fig. 84). In order to know the instant 
when Polaris is exactly on the meridian it is necessary first to 
observe the instant when one of these two stars is vertically 
below Polaris. From this the time when Polaris will be on the 
meridian can be calculated by adding a certain interval of time, 
and the meridian line can thus be directly established. This 
interval of time was, for £ Ursa Mqforis, about 2 m 36 8 in the year 
1900, and it increases about 2 1 9 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 8 Cassiopeia. The interval 
for this star was 3 m 24 s for 1900, with an annual increase of 
about 20 s . 

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 1 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. VDL 



It will be seen that in this method the actual error of the 
watch has no effect on the result since it is used only for meas- 
uring the interval of a few minutes. The error in the meridian 
obtained by this method will seldom exceed one minute of angle. 

209. To Find the Standard Time of Culmination and Elonga- 
tion. — The approximate times of culmination and elongation of 
Polaris for the 1st and 1 5th of each month in the year 19Q7 may 
be found in Table 9. 

TABLE 9. 

Approximate Times of Culmination and Elongation of Polaris 

Computed for the 90TH Meridian West of Greenwich, 

for the Year 1907. 



Date. 



I Upper 

Culmination. 



'907 
Jan. 1 . 

" 15 • 
Feb. 1 . 

44 i5 • 
Mar. 1 . 

" '5- 
Apr. 1 . 

44 15 ■ 
May 1 . 

" 15 • 
Jun. 1 . 

44 J S • 
Jul. 1 . 

« 15 . 
Aug. 1 . 

44 15 • 
Sep. 1 . 

44 IS • 
Oct. 1 . 

" i5 - 
Nov. 1 . 

44 15 • 
Dec. 1 . 

44 IS • 



44 
49 
4i 
46 

5i 
56 
o 40 

23 5° 

22 47 

21 52 

20 45 

19 5 1 

18 48 

17 53 
16 47 

15 52 

14 45 

13 5° 

12 47 

11 53 
10 46 

2 5I 

8 47 
7 52 



Western 
Elongation. 



h m 

12 39 

II 44 

IO 36 

9 4i 

8 46 

7 5i 
6 



44 
40 
46 
5i 
44 

50 

o 47 

23 48 
22 42 

21 47 
20 40 

19 45 
18 42 

17 48 
16 41 

IS 46 

14 42 
13 47 



Lower 
Culmination. 



k m 

18 42 

17 47 
16 39 

15 44 
14 49 

13 54 

12 47 

11 52 

10 49 

9 54 
8 47 

7 53 
6 50 

55 
49 
54 



47 
S 2 
49 



23 5 1 

22 44 

21 49 

20 45 

19 50 



Eastern 
Elongation. 



k m 

o 49 

23 5° 

22 42 

21 47 

20 52 

19 57 

18 50 

17 55 

16 52 

15 57 

14 5° 

13 56 

12 53 

11 58 

10 52 

9 57 

8 50 

7 55 
6 



52 
58 
5i 
56 

52 
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 i minute per year on the aver-, 
age, so that this table will give approximate results for other 
years. Astronomical time begins at noon of the civil day of the 
same date and is reckoned from o h to 24 h , e.g., i8 h would mean 
6 h 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. 

2IO. MERIDIAN OBSERVATIONS ON POLARIS WITH THE 
COMPASS. — In determining a meridian with the compass the 
observations are made as described for the transit except that 
the following modifications will be necessary. Suspend a long 
plumb-line a few feet away from the point where the instru- 
ment is to be set. Since the rear sight is the only part of the 
compass to be used in the observation it may be unscrewed 
from the compass and fastened to a piece of board. This board 
should be placed on a table. The compass sight may then be 
shifted to the right or left to bring it in line with the star and 
the plumb-line. The plumb-line should be illuminated by means 
of a lantern. The direction of the star may be marked by set- 
ting stakes in line. If the observation is made at elongation the 



1 88 OBSERVATIONS FOR MERIDIAN [Chap. VII. 

meridian should be laid out as described in Art. 207. In finding 
the declination of the needle the compass is set up over one of 
the meridian stakes and sighted at the other, when the declina- 
tion can be read off directly. In order to obtain as nearly as 
possible the mean value of the declination this should be done 
at about 10 A.M. or 5 to 6 P.M. because at these times the 
needle is in its mean position for the day. 

211. Meridian Observation on Polaris at any Time 

WITH THE TRANSIT, — In order to make this observation, it is necessary to 
know the local time very closely. As in most cases the time which the surveyor 
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 hour angle of the star at the time of the observation. Take from the Nautical 
Almanac : (1) 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 24k 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 D — sin L cos / 

where Z — the azimuth, or true bearing ; / — the hour angle ; L — 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 24I1 it is east of the 
meridian (see Example below). 

In the " Manual of Surveying Instruction " issued by the General Land Office 
a set of tables is given which will enable the surveyor to perform all of the above 
work by simple inspection and without the aid of the Nautical Almanac. 



• Sm Hayford't Geodetic Attronomy, p. an, Art. 193. 



AZIMUTH OF POLARIS AT ANY HOUR 



189 



Example. 

Observation on Polaris for azimuth April 15, 1908. Latitude 38 58'. Longi. 
tude 92 ° 25'. Angle between a mark (approximately N.W.) and Polaris is 
repeated 6 times. Watch in* 13s fast. The times are 




Mean of 6 readings 
Watch fast 
True Central time 
Longitude of Standard Meridian 
Greenwich time 



From Nautical Almanac, Right Ascension of * Mean Sun " at Greenwich Mean 
Noon -= ih 32m 57 9 .82 ; Right Ascension of Polaris = ih 25m oi s .47 ; Declination- 
of Polaris « -f- 88° 48' 52"; Correction from Table III (Nautical Almanac) for 
Greenwich Time «= 14b 38™ — 2 m 243.2 



92° 


25' a 6h 09 m 40 s 


.\ longitude correction ■- 09m 40 s 


Mean of observed times 


8h 38m 13s 


Longitude correction 


9 40 


Local time 


8 28 33 


Right Ascension M Mean Sun 


1 32 58 


Correction (Table III) 


2 24 


Sidereal time 


10 °3 55 


Right Ascension Polaris 


1 25 01 


Hour Angle Polaris 


= 8h 38^543 




'- 129 43' 30" 


log cos L «= 9.89071 


log sin L = 9.79856 


log tan D — 1.684 13 


log cos / ^ 9.80558 (n) * 


1.57484 


9.60414 (n) 


37.57o 


- .4019 


.402 




37.972 




log sin / «= 


9.88600 


log denominator — 


1.57946 


log tan Z — 


8.30654 


Z« 


i° 09' 37" W. of N. 



• The n after the logarithm indicates that the number corresponding is negativ* 



I go 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 angle solar observations will answer. In 
making these observations with the ordinary transit it is neces- 
sary to have some means of cutting down the sun's light so 
that it will not be too bright for the eye while making point- 
ings. This is usually effected by placing a dark glass over the 
eyepiece. A dark glass in front of the objective 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 ALTITUDES 
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 

x= f . 

cos L sin / 



MERIDIAN BY SOLAR OBSERVATIONS 191 

in which X = the correction to the mean vernier reading, 
d = the hourly change in declination of the sun taken from 
Table 10 and multiplied by half the number of hours between 
the two observations, L = the latitude, and / = half the elapsed 
time converted into degrees, minutes, and seconds. Since the 
hourly change for any given day is nearly the same year after 
year an almanac is not necessary but the table given below is 
sufficient. 

TABLE 10. 

Hourly Change in the Sun's Declination. 

1st. 10th. 20th. 30th. 

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

February +43 +49 +54 

March +57 +59 +59 +58 

April +58 +54 +49 +46 

May +45 +39 +39 +23 

June +21 +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 -f-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 o°. 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. VH. 

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 is going south the mean 
vernier reading is east of the south point, and vice versa. 
When the circle reading of the south point is known the true 
bearing of the mark becomes known and the bearings of other 
points may be fpund (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 ?h 1901308 Time 4b 12 "1158 



MERIDIAN BY SOLAR OBSERVATIONS 



193 



I elapsed time = 4^26^220 
- 66 35' 3 o" 
log sin / 9.96270 
log cos L 9.86902 

"9-83I72 

log 230 // .9 2.36342 

2.53170 



Increase in declination in 4^26 m 228= 
52" X 4-44 - 23o' , -9 



Mean circle reading «= 79°si / o8 // 
5 40 



S 79°45 , 28" E 



Azimuth of mark «= 28o°i4 / 32*' 

214. OBSERVATION FOR MERIDIAN BY A SINGLE ALTI- 
TUDE OF THE SUN. — The azimuth of a line may be obtained 
by measuring a single altitude of the sun with the transit and 
computing the azimuth by spherical trigonometry. The instru- 
ment is set at 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 1 1 . This corrected 
mean is called h in the formula given below. 

TABLE 11. 
Refraction Correction. 



Altitude. 


Refraction. 


Altitude. 


Refraction. 


IO° 


5' '9" 


20° 


*' 39" 


II 


4 5 1 


2 S 


2 04 


12 


4 27 


30 


I 41 


13 


4 07 


35 


I 23 


14 


3 49 


40 


I 09 


15 


3 34 


45 


O 58 


16 


3 20 


50 


O 49 


17 


3 08 


60 


34 


18 


2 57 


7o 


21 


19 


2 48 


80 


• 10 



194 OBSERVATIONS FOR MERIDIAN [Chap. TO. 

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 ; //, the altitude ; /, the distance from the north 
pole to the sun (or 90°-declination) ; and s = J (L + h +/) ; 
then 

cot 2 y z = sin (j - Z) sin (j - A) 
2 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 



195 



Example. 

Observation on Sun for Azimuth. 

Latitude 42 21' N. Longitude 4k 44m i8» W 

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



Horizontal Circle 


Vertical Circle 


Vernier A B 






Mark 238 14' 14' 






Right and Lower Limbs 311 48 48.5 




14 41' 


•4 *« « ** ^12 20 20 




15 00 


The inst. reversed 






Left and Upper Limbs 312 27 26.5 




i5 55 


u u u 11 3I2 ^ 2 ^ ^ 




16 08 


Mark 238 14 14 


Mean 




Mean reading on Mark *» 238 i4'.o 


= 15° 26' 


u " " Sun — 312 21 .7 







Watch 





A.M. 


8h 


39 m 


42a 


8 


42 


19 


8 


45 


34 


8 


47 


34 



Mean 



, 8h 4311 
7 



47* 



Mark N. of Sun — 74. 

Observed Altitude 15 26'.o 



07-7 



Refraction 
True Altitude 

Z- 

h - 

.r — i 
j-Z - 



3.5 



I5°22 , .S -X 



4** 

15° 

iii c 



2I'.0 
22'.5 

■5'.7 



84 29'.6 

42 08' .6 

69°07'.i 

-26 4 6'.i 



ih 43 m 47> 



Greenwich Mean Time * * 
Sun's apparent declination at 
Greenwich Mean Noon — — 21 14' 54 // .4 
Difference for 1 hour =» — 26" .81 

— 26" .81 X ih-73 - - o' 46" 4 

-Declination = — 21 15' 4o".8 

Polar Distance «= 1 1 1° 15' 4o".8 

log sin (s — Z) = 9.8267 1 
log sin (j -'/«) = 9.97049 
log sec j « x. 01 791 

log sec (s -/) = 004923 
2 ) 0.86434 
log cot \ Z = 0.43217 
JZ = 20°i 7 '. 3 

Z = 4 o°34'° 
Mark N. of Sun = 74 07'7 



Mark 



ii4°42'. 3 
N.6s°i7'.7E. 



215. Observation for Meridian by Means of the 
SOLAR ATTACHMENT. — This observation has been described 
in detail in Art. 85, p. 66. 



See Art. 86, p. 68. 



196 OBSERVATIONS FOR LATITUDE [Chap. VH. 



OBSERVATIONS FOR LATITUDE. 

216. ( I ) BY THE ALTITUDE OF POLARIS AT UPPER OR LOWER 
CULMINATION. — When Polaris is approaching either culmination 
(see Art. 206, p. 180, and Fig. 83) set up the transit and point the 
horizontal hair on the star. Keep the cross-hair pointed on the 
star until the culmination is reached. Read the vertical arc and 
determine the index correction. The altitude is to be corrected 
for refraction by Table 1 1, p. 193. This gives the true altitude. 
If Polaris is at upper culmination subtract from the true altitude 
the polar distance of the star at the date of the observation (Table 
8, p. 184). If the star is at lower culmination the polar distance 
is to be added. The result is the latitude of the place of obser- 
vation. 

217. (2). BY THE ALTITUDE OF THE SUN AT NOON. — The ob- 
servation consists in finding the greatest altitude of the sun's 
lower limb. This will occur when the sun is on the meridian 
(very nearly). Begin the observation a little before apparent 
noon, remembering that this differs sometimes more than i6 m 
from mean 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 1st, the 
sun passes the meridian 16m igs before mean noon, i.e., when it is 12b com oos 
apparent time it is I ih 43m 42s mean time. 



OBSERVATIONS FOR LATITUDE 1 97 



Example. 

Observed maximum altitude of the sun's lower limb on 



Jan. 8, 1906. — 


25 06' 


Index Correction — + 1' 


Observed altitude 


25° o6'.o 






Index Correction 


I'.O 








25° 07'.o 


Declination of sun at 


Refraction 


2\o 


Greenwich 


app. noon - — 22°io/33* (S) 




25° os'.o 
i6 7 .3 




+ 1 33 


Sun's semi-diameter 


- 22° i8'oo" (S) 


Altitude of sun's center 
Declination — 

Latitude 


25 2l'. 3 
22° i8'.o 

47° 39'-3 
42° 20 / 7 




Longitude — 4b 44m i8» W. 
- 4 h -74 
Diff. ih - + 19/'. 59 
+ i9"-59 X 4 h -74 - + 1' 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 1 ° 1 4' 1 2" ? The latitude of Boston 
is 42° 21' N. 

(b) In making an observation for meridian two stakes were set 329 feet apart, 
marking the direction of the star at elongation. Compute the length of the per- 
pendicular offset to be laid off at one end of the line to obtain the true meridian. 

2. What is the approximate Eastern Standard Time of the eastern elongation 
of Polaris on August 10th at a place in longitude 7 2° 56' West ? 

3. Observation on May 15, 1906, for determining the azimuth of a line from 
an altitude of the sun. Reading of vernier A of the horizontal circle while point- 
ing on the azimuth mark = 0° 00'. At first pointing on sun, lower and right limbs, 
vernier A, horizontal circle read 168° 59' ; vertical arc read 43° 36'; the Eastern 
Standard Time was 2h 52m 45s P.M. At second pointing on the sun, upper and 
left limbs, vernier A, read 168 52'; vertical arc, 42 33'; time, 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 + 1 8° 42' 43".6 (North). The 
change for 1 hour was + 35^.94 (sun going north).. The latitude of the place was 
42 17' N. ; The longitude was 7i°.o5' 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 1° 12'. What was the 
latitude of the place ? 

5. Observation for latitude. The observed maximum altitude of the sun's 
lower limb on August 10th, 1906, was 66° 29'. The Eastern Standard Time was 
approximately nh 50m 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 / 13"3 ( + ). The 
difference for 1 hour was— 43^.46 (sun going south). What was the latitude of 
the place ? 



CHAPTER VIII. 

LEVELING. 

218. DEFINITIONS. — 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 
niar£s (B.Ms.), the relative heights of which are accurately known. 
These heights are usually referred to some definite zero plane, 
such, for instance, as mean sea-level or mean low water, and the 
height of a point above this plane is called its elevation. This 
plane is called the datum. (See Art. 237, p. 211, and Art. 250, 
p. 226.) Strictly speaking it is not a plane but a level surface, 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. I.) 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 

108 



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 turning point. The elevation of this 
bench mark could be obtained by simply taking a foresight upon 
it and not using it as a turning point, but by making the bench 
mark also a turning point it becomes a part of the line of levels 
and if the levels check, the elevation of the bench mark is also 
checked. Each bench mark established should be carefully re- 
corded by a description or a sketch, or both. The elevations 
of the remaining turning points are as accurate as the elevations 
of the bench marks themselves, so that any of the turning 
points might be used as a bench mark. Consequently it is 
advisable to describe those turning points which can be readily 
identified so that they may be used when it is not convenient or 
possible to use one of the established bench marks. 

In leveling up or down slopes the levelman should be able to 
judge quickly where to set his instrument in order to have it the 
desired height above the turning point. In going downhill the 
rod-reading of the backsight should be as small as possible in order 
to overcome the height with the minimum number of set-ups of 
the level But while the levelman may waste much time by 
having large backsights necessitating additional set-ups, it is 
also possible for him to waste quite as much time in attempting 
to place his instrument so as to get very small backsights. The 
proper way to handle the instrument is as follows. Set up 
roughly (without pressing the tripod legs into the ground), turn 
the telescope toward the rod and then level it, approximately, 
in that direction. By sighting along the outside of the telescope, 
the approximate place where the line of sight will strike the rod 
can be noted and the distance the instrument should be moved 
up or down the slope can readily be estimated. Then move to 
the new position, level up carefully, and proceed to take the back- 
sight. This general procedure should be followed whether level- 
ing up or down a slope. 

220. In this work it is very important to eliminate as far as 
possible errors of adjustment in the instrument. If at every set- 
up of the level the foresight and its corresponding backsight are 



200 



LEVELING 



[Chap. VIII. 



taken at points which are equally distant from the instrument such 
errors will be eliminated. If the level is not in perfect adjustment 
the resulting error in any reading is proportional to the distance. 
At equal distances from the instrument the errors are equal, and, 
since it is the difference of the rod-readings that gives the dif- 
ference in elevation, the error is eliminated from the final result 
by this method. By making the length of foresights and back- 
sights equal on turning points it is possible to eliminate not only 
the error due to non-adjustment of the bubble but also any error 
due to non-adjustment of the objective tube, since this will occupy 
the same position in the telescope in each sight. The distance 
to the backsight is determined by the place where the instrument 
is set up, and the rodman, as he passes from one turning point to 
the next, can by pacing make the foresight distance approximately 
equal to that of the backsight. The line of levels should be "closed" 
by continuing the leveling until the original bench mark, or some 
other bench mark whose elevation is well established, is reached. 
221. The notes for this work may consist of five columns, as 
shown in Fig. 86. The height of instrument is obtained by add- 
ing the backsight to the elevation of the point on which it is taken. 
The elevation of any point is found by subtracting the foresight 
for that point from the height of the instrument. Notice that the 





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Fig. 86. Bench Mark Level Notes. 



DOUBLE RODDED LINES 



20 1 



calculations maybe checked by adding the foresights and the back- 
sights. The difference of these sums should be the same as the 
difference in elevation between the first and last points. 

222. Double Rodded Lines. — A good check on the line of 
levels may be secured by running a double line of turning points. 
Instead of taking a foresight on a single turning point, foresights 
may be taken on two different points near together, from the 
same set-up of the instrument. When the level is set up again 
a backsight is taken on each turning point and two independent 
values of the new height of instrument are obtained. In 
ordinary bench mark leveling these two values should not differ 
by more than 0.002 or 0.003 ft- fr° m the previous difference, 
i.e., if the two heights of instrument differed by 0.013 at a 
certain set-up they should not differ by more than 0.016 nor 
less than 0.010 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 where no established bench marks are available for checking. 

223. A set of notes illustrating double turning points is 
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202 LEVELING [Chap. VILL 

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 Harks and Turning Points. — Both the bench 
marks and the turning points should be such that their elevations 
will not change during the time they are needed. The only 
difference between the two is that turning points may be of use 
for only a few minutes while bench marks may be needed for 
many years. Bench marks should be very carefully and accu- 
rately described, and their heights should be checked before 
being accepted as correct. They are frequently taken on such 
points as these: — stone bounds, tops of boulders, spikes in 
trees, and on sills, stone steps, or underpinning of buildings. 
Curb stones or tops of hydrants are also used but are not so 
permanent. As it is often impossible in a new country to find 
existing points where bench marks can be established, it is usual 
in such cases to set stone monuments or iron rods and to care- 
fully determine their elevation. The U. S. Geological Survey, 
for example, sets an iron pipe with a cap on the top of it ; or in 
some cases a plate with a horizontal line across it in the masonry 
wall of a building. Some of the bench marks of the U. S. 
Coast and Geodetic Survey and of the Missouri River Com- 
mission consist of stones buried 3 or 4 ft. under ground. The 
exact bench is the top of a spherical headed bolt set in the top 
of the stone. This is reached by lowering the rod through an 
iron pipe which extends to the surface of the ground. 

Bench marks should be established at frequent intervals for 
convenience in dependent work. Some surveyors consider it 
advisable to have two bench marks in the same locality to serve 
as checks on each other. In choosing a bench or a turning 
point it is best to select a point which is slightly raised so that 
the rod will always rest on exactly the same point. A rounded 
surface is better than a sharp point, especially when it is on a 
rock, as the rod may chip off a small piece and alter the eleva- 
tion. If a turning point is taken on a flat surface it is difficult 
to get the rod at exactly the same height each time. Bench 



BENCH MARKS AND TURNING POINTS 203 

marks are, however, sometimes established on flat level surfaces 
such as the coping stone of a masonry structure, because per- 
manence is of more importance than great precision. Bench 
marks are not only described in the notes, but are themselves fre- 
quently marked by red chalk, by chisel marks, or drill-holes. 

225. LEVELING FOR PROFILE. — Profile leveling is for the 
purpose of determining the changes in elevation of the surface of 
the ground along some definite line. The line is first " stationed/' 
i.e., marked at every hundred feet or such other interval as is 
desired. The level is set up and a backsight taken on a bench 
mark to determine the height of the instrument. Foresights 
are then read on as many station points on the line as can be 
conveniently taken from the position of the instrument. Inter- 
mediate sights are taken at any points where marked changes 
of slope occur, and the plus stations of these intermediate points 
are recorded with the rod-readings. It will be noticed that here 
the terms foresight and backsight do not refer to the forward 
and backward directions. A backsight is a reading taken on a 
point of known elevation for the purpose of obtaining the height 
of the instrument. A foresight is a reading taken on a new point 
to determine its elevation. For this reason backsights are fre- 
quently called plus 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 is reached. 

A line of levels should be checked by connecting with some 
reliable bench mark if possible. If there are any bench marks 
along the line of levels they should be used as turning points if 
convenient, or at least check readings should be taken on them 
in order to detect mistakes. In such a case it is evident that 
the reading taken on the bench mark is really a foresight since 
its elevation is being found anew from the height of instru- 
ment. Readings on bench marks and turning points should be 
taken to thousandths or to hundredths of a foot, depending upon 
the accuracy desired. If the elevations of the profile are de- 



204 



LEVELING 



[Chap. VIII. 



sired to the nearest hundredth of a foot, as in the case of a 
railroad 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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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. 




5TA.0 



PROFILE. 
Fig. 89. 



Fig. 89 represents a rough plan and profile of the line of 
levels shown by the notes in Fig. 88. Angle points in the 
transit line are shown in the plan, but they do not appear in the 
profile of the line. It will be noticed that the T. Ps. and B. Ms. 
are not on the transit line in plan, and that they consequently 
do not appear on the profile. It is not customary to introduce 



206 



LEVELING 



[Chap. VIII. 



any sketches into the profile notes except those used in describ- 
ing bench marks or turning points. 

227. CROSS-SECTIONING. — If it is desired to know the shape 
of the surface of a piece of ground, the area may be divided 
into squares and the elevation taken at each corner of these 
squares and at as many intermediate points as seem necessary 
to determine the changes of slope. These surface elevations 
are obtained to tenths of a foot. The squares which may be 
anywhere from 10 ft. to 100 ft. on a side are laid out with the 
transit and tape, stakes being driven at the corners. It is well 
to choose some long line of the traverse as the primary line 
from which the cross-section system is to be laid out. The 
points are usually designated by a system of rectangular coor- 
dinates, one set of parallel lines being marked by letters and the 
other by numbers, as shown in Fig. 90. For example, the 





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point / would be called (C, 7) ; the point s, (D, 5) ; the point 
r, (B f 80, 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 is 



CROSS-SECTION LEVELS 20? 

set up at a convenient point and the rod held on a bench mark. 
The tape, or band, on the rod is then moved up or down as di- 
rected by the levelman until he reads the feet, tenths, and hun- 
dredths which are the same as those of the elevation of the bench 
mark, e.g., if the elevation of the B. M. is 195.62, the tape will be 
moved until it reads 5.62. If the rod is then held on a point 
1.6 1 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-Sectioning for Earthwork. — Whenever it 
is desired to ascertain the quantity of earthwork in an excava- 
tion or an embankment, it is necessary to take levels to determine 
the vertical dimensions, and to obtain the horizontal dimensions 
by means of the transit and tape. The three general cases where 
the quantity of earthwork is to be estimated by the engineer 
are : (1) an excavation or embankment having a known base and 
side slopes as in the construction of a railroad or a highway, (2) 
an irregular excavation from a bank of earth called a borrow-pit, 
(3) a trench excavation such as is used for sewer construction. 

230. (1) 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. 



208 LEVELING [Chap. VTIL 

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£ 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 (£ 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 bC, Fig. 92). If the measured 
distance does not agree with the calculated distance a second 
trial must be made by holding the rod at another point and 
repeating the operation. The difference between the measured 
and calculated distances is an aid in judging where the rod 
should be held at the second trial. After a little practice it 
will be possible to set the slope stake at the second or third trial. 
232. Earthwork Notes for Road Cross-Sections. — 
The notes for this work will contain the cut or fill at the 
center, the cut or fill at either side, and the corresponding 
distances out. A cut is usually written in the notes as a plus 
( + ) height and a fill as a minus (— ) height; but the stakes 

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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 



2IO LEVELING [Chap. VIII. 

base of the road, as in Sta. 10, the section is called a Level 
Section. Where the surface of the ground is not parallel to 
the base and where three cuts or fills only are recorded, as at 
Sta. 1 1, the section is called a Three Level Section. If, besides 
the three readings which are taken for a three level section, two 
more intermediate readings are taken one directly over each end 
of the base, as at Sta. 1 1 + 50, the section is called a Five 
Level Section. If intermediate readings (one or more of them) 
are taken anywhere except over the ends of the base, as in Sta. 
12, the section is called an Irregular Section. For methods of 
computing the amount of earthwork see Chapter XII. 

It will be noticed that in the column of the notes headed 
" Cross-Sections " the distances out appear above and the corre- 
sponding cuts below the lines. Besides this set of notes there is 
a simple set of level notes similar to Fig. 86, p. 200, from which 
the height of instrument is determined. This is conveniently kept 
in another part of the note-book, often at the back of the book. 

233. (2) Cross-Sections for Borrow-Pits. — The ground is 
first staked out in squares or rectangles and the elevation at each 
corner and at every change in slope is determined as explained 
in Art. 227, p. 206. Then the work of excavating is carried on, 
and when it is desired to determine the amount that has been 
excavated, the same system of cross-sections is again run out 
and the new elevations at the corners and at the necessary 
intermediate points are determined. 

The notes are kept as shown in Fig. 90, p. 206. For methods 
of computing the earthwork in borrow-pits see Art. 373, p. 342. 

234. (3) Cross-Sections for Trench Excavation. — The sur- 
face elevations are determined by making a profile of the 
line. The grade of the bottom of the trench is obtained either 
from the plan or by direct leveling. The width of the trench is 
measured wherever it changes and the stations of these places 
noted. For methods of computing the quantity of earthwork 
see Chapter XII. 

235. LEVELING TO ESTABLISH A GRADE LINE. — The level 
may be used for setting points at desired elevations as, for 
example, in establishing the grade line of a sewer. To set any 
point at a given elevation, set up the level and take a backsight 



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 point, and the dis- 
tance from the top of the first stake to the axis of the telescope 
is measured with the tape or rod.* Then the rod, which is set at 
this reading, is carried to the last point on the straight grade line, 
and, 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 coincides with the 
horizontal cross-hair. 

237. TO ESTABLISH A DATUM PLANE BY MEANS OF TIDAL 
OBSERVATIONS. — Whenever it is necessary to establish a datum 
from tidal observations it may be determined as follows. Set up 



* Where the grade is flat some surveyors prefer to set the instrument just 
behind the point instead of to one side of it. 



212 



LEVELING 



[Chap. vm. 



a vertical staff, graduated to feet and tenths, in such a manner 
that the high and low water can be read. Read the positions 
of high and low water for each day for as long a period as prac- 
ticable. The mean value obtained from an equal number of 
high and low water observations will give the approximate value 
of mean sea-level. If the observations extend over just one lunar 
month the result will be fairly good, whereas in less than one 
month a satisfactory result cannot be obtained ; to determine this 
accurately will require observations extending over several years. 
The proper location of the gauge is an important factor in 
obtaining the true mean sea-level. The place chosen for setting 
up the gauge should be near the open sea, so that local conditions 
will not influence the tide. It should be somewhat sheltered 
against bad weather. The water should be deep so that at the 
lowest tide the water will stand at some height on the gauge. 

At the beginning of the series the zero of 
the staff and some permanent bench marks 
should be connected by a line of levels. This 
should be tested occasionally to see if the staff 
is moved. After the reading of the rod for 
mean sea-level is found the elevation of the 
bench mark can be computed. 

238. The Staff Gauge. — This is a form 
of gauge (Fig. 94) which can be easily con- 

/| ^_ struct ed, and which is sufficient where only a 
short series of observations is to be made. If 
a made in sections not over 3 feet long, as de- 

j ^j scribed below, it can easily be packed in a box 
for transportation. Each section consists of 
two strips of wood about i£ inches square, 
and 3 feet long, fastened together at the 
ends by strips of brass, leaving a space 
between them of about 1 inch. In this 
space is placed a glass tube of about $ inch 
V <flff diameter and held in place by brass hooks. 
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 21 3 

a long vertical piece of joist. The ends of the tube are nearly 
closed by corks, in which small glass tubes of approximately 1 
mm. (inside) diameter have been inserted. When the water rises 
in the main tube, the red strip appears to be much wider than it 
really is on account of the refraction of light by the water. Above 
the water surface the strip appears its true width. By observing 
the position of the wide strip the height of the water surface can be 
read within a hundredth of a foot. The heights are read on a scale 
of feet painted on the wooden strips. If the size of the small 
glass tube is properly chosen, the fluctuations of the water sur- 
face outside will not disturb the water in the tube, so that the 
reading is a fair average of the water surface. A gauge of this 
sort may be read by means of a. transit telescope or field glass 
at a distance of several hundred feet. 

When a long series of observations is to be made a self-regis- 
tering tide gauge should be used. A description of such a gauge 
may be found in the Reports of the U. S. Coast and Geodetic 
Survey.* (See Volume II, Arts. 258-9, p. 288-91.) 

239. LEVELING ACROSS A RTVER. — While the effect of cur- 
vature and refraction (Art. 118, p. 87) is usually negligible in 
leveling operations, it may in certain special cases become of 
great importance to eliminate this error. For example, it is 
sometimes necessary to carry a line of levels across a river of 
considerable width, say, half a mile. In this distance the correc- 
tion for curvature and refraction amounts to about 0.143 ft. un- 
der normal conditions, which in a line of bench levels is too large 
a quantity to neglect. If the correction as derived from formu- 
las could be depended upon under all circumstances it would be 
sufficient to compute and apply it to the rod-reading. But the 
amount of the refraction correction is so variable that the actual 
value often differs considerably from the computed value. 

If it is desired to obtain the difference in elevation between 
two distant points with great accuracy it will be necessary to use 
a method which will eliminate the effects of curvature and re- 
fraction no matter what their actual amount may be. In Fig. 
95 suppose a backsight were taken on T. T. x with the instrument 

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



214 LEVELING [Chap. VIIL 

at A and then a foresight taken on T. P. 8 The elevation of T. P., 
as computed from T. P. x will be too low by the amount ab 9 
since the foresight on T. P. a is too great by this amount. If 
the difference in elevation is determined by the instrument at 
B the backsight on T. F. Y is too large by the amount cd. Hence 
the H. I. of the instrument at B is too great, and consequently 



Fig. 96. Leveling Across a River. 

the elevation of T. P. 8 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 
under 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 instruments 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).. 

* Errors due to non-adjustment are of unusual importance because the sight is 
much longer than that used in adjusting the instrument. 



LEVELING PROBLEMS 



215 



PROBLEMS. 

1. Compute the following set of level notes. 



Sta. 


+ s. 


H. I. 


- s. 


Elev. 


B. M., 


4.702 






16.427 


B. M. 2 


11.846- 




6.727 




T. P., 


7.276 




9.689 




B. M. 8 


8.760 




4.726 




T. P., 


0.687 




11.000 




B. M.« 


1.607 




8.496 





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



Sta. 


B. S. 


II. I. 


F. S. 


Elev. 


B.M.„ 


6.427 




4273 


62.473 


20 






6.2 




21 






74 




+42 






5-2 




22 






4-7 




T. P. w 


4724 




9.976 




23 






11. 2 




+63 






10.4 




B.M. 22 


0.409 




7.482 




24 




IT.2 





y, Compute the elevations in the following set of level notes. 



Sta. 


+ s. 


H.I. 


-s. 


Elev. 


B.M. M 


6.214 






84.238 


T. P.j L. 


3-5*5 




9.280 




T. P., H. 


2.152 




7.919 




T. P. 2 L. 


2.971 




8.263 




B.M.„H. 


2.338 




7.629 




T. P., L. 


4.278 




7.529 




T. P., H. 


2.646 




5.894 




B.M.j.L, 


5.721 




6.072 




T. P. 4 H. 


4.837 




5.187 




B.M. W 






5.817 





4. Make up a set of cross-section notes for road construction which shall be 
consistent with the following data: width of road, 50 ft., slopes 1} to 1; grade 
elevation of Sta. o = 107.20; grade, -f 1.4. Show complete notes from Sta. o to 
Sta. 3 inclusive as follows: Sta. o, a level section; Sta. I, a three level section; 
Sta. 2, a five level section ; Sta. 3, an irregular section. 



CHAPTER IX. 

CITY SURVEYING. 

240. INSTRUMENTS USED. — Owing to the comparatively 
high value of land in cities and to the fact that a large propor- 
tion of city surveying is the establishing of lines and grades for 
construction work, the chain and compass are discarded entirely 
and the steel tape and transit are used. 

241. Tapes and Tape Measurements. — The tape most com- 
monly employed is the light 100-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 (Art. 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 217 

be reduced to agree with the City Standard and very accurate 
results may be obtained. 

Where the ground is not level and there is call for frequent 
plumbing it is impossible to obtain accurate results unless the 
plumbing is carefully done by experienced tapemen. For very 
accurate work it may be desirable to entirely eliminate the 
plumbing. This is sometimes done by measuring directly on 
the surface (on the slope) from point to point, and by means of 
the level instrument and rod the relative elevations of these 
points arc obtained and the horizontal projection of the slope 
distances computed. Instead of measuring the difference in 
elevation between the two ends of the line, the angle of inclina- 
tion of the slope line is often measured on the vertical arc of a 
transit which is set up over one of the end points. 

The government Bureau of Standards at Washington will, 
for a nominal charge, standardize tapes ; and city and private 
engineers frequently avail themselves of this opportunity. This 
Bureau will give the exact length of the tape at a given temper- 
ature or the temperature at which the tape is of standard length, 
whichever is desired by the engineer. It is well to have the 
tape also tested at a few intermediate points, e.g., the 25 ft., 50 
ft., and 75 ft. marks. One tape which has been standardized 
should be kept in reserve, with which tapes in service can be 
compared both when new and after being mended. 

Besides the ordinary steel tape, steel or metallic tapes 
reading to tenths of a foot are used in taking measurements for 
making approximate estimates of construction and for measuring 
earthwork, paving, and the like. 

242. Transits and Levels. — The transits usually employed 
in city work read to 30" or to 20" ; and for most city work no 
finer graduation is necessary. With these instruments the re- 
quired precision in reading angles on triangulation work 
can be obtained by repeating the angles as explained in Art. 
59, p. 48. In such work, however, it will be of advantage to 
have an instrument reading to 10". It is well also to have 
one or more transits equipped with stadia hairs for use on 
rough surveys. 

Much of the city work, such as the staking out of new streets, 



2l8 CITY SURVEYING [Chap. DC 

paving, sewers, or curbs, requires the establishment of both lines 
and grades. Since this class of work dqes 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 -reading 
rods. (See Chapter IV.) For bench leveling it is customary, in 
large cities at least, to use a precise level, an instrument which is 
similar in principle to the ordinary level but which has a more 
delicate bubble and a telescope of higher power, and is therefore 
capable of yielding more accurate results. 

243. CITY STANDARD.* — It is customary in all large cities to 
have a standard of length, usually 100 ft. long, established in 
some convenient place, often near the office of the City Engineer. 
It sometimes consists of two brass plugs set in a stone pavement, 
or it may be a long steel rod supported on rollers on the side of 
a wall or building in such a way that the rod can expand or con- 
tract freely. The end points and the 50-ft. point are so marked 
that they can be readily found and used by any surveyor who 
desires to test his tape. • 

A city standard is often established by carefully transferring 
the length of some other standard, by means of different tapes 
and under different weather conditions ; or it can be established 
by means of a tape which has been standardized, by the U. S. 
Bureau of Standards (Art. 241, p. 216). The City Standard is 



* See a paper entitled "The 100 -foot Standard of Length of the Boston 
Water Works at Chestnut Hill Reservoir," by Charles W. Sherman, published in 
the Jour. Assoc. Eng. Soc., Vol. XVIII, No. 4, April, 1897, 



STANDARD OF LENGTH 219 

generally placed where it will not be exposed to the direct rays 
of the sun, and with this end in view it is sometimes covered with 
a wooden box. 

When a tape is tested it should be stretched out at full length 
beside the standard and left there until it acquires- the same 
temperature as the standard before the comparison is made, to 
avoid the necessity of applying a temperature correction. 



CITY LAYOUTS. 

244. In 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. 

245. STREETS. — In planning the arrangement of the streets 
for a city such features as a water front, a river or lake, the loca- 
tion of an existing railroad, or the probable location of some 
projected railroad line will determine to a large degree where 
the business section of the city will be located. This section 
should then be so divided as to yield the greatest convenience 
for business purposes. Other sections will be reserved for 
residential districts, and their design will be of a different 
character. Easy access should be provided from the business 
to the residential districts and to outlying towns or adjacent 
cities. 

The streets must be of the proper width to accommodate 
the traffic they are to carry, and their alignment and grades 
must be carefully studied with the topographic map as a guide. 
Adequate drainage of the streets is, of course, one of the most 



220 CITY SURVEYING [Chap. DC 

important features, for which ample provision must be made 
in establishing the alignments and grades. 

In the business section the traffic will move in certain 
directions, e.g., to and from important points such as a river, 
railroad station, or freight yard, and this traffic must be pro- 
vided for by wide streets with easy grades. In the residential 
portions, narrower streets and steeper grades are permissible 
when made necessary by the topography of the district. 

246. Location of Streets. — In establishing the location of 
city streets in hilly districts it is probable that to obtain the 
essential requisites of easy grades and good drainage the topog- 
raphy will govern the street layout. Whereas in a practically 
level country, with no steep grades in any direction, the street 
layout can be such that the most direct communication between 
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 streets which is difficult to drain. Fig. 97 
shows a layout which will obviate this difficulty to some extent, 
the diagonal streets being located in the valleys to take the sur- 
face drainage of surrounding property. It is 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 900 feet. In New York the typical blocks are 200 X 900 
ft., and 200 X 400 ft. ; in Boston they vary in width from 125 
to 252 ft. and in length from 200 to 700 ft., depending upon the 
locality. 

The frontage of lots is frequently 25 ft. in business and 
congested residential districts and 50 feet or more in sub- 
urban districts, but these dimensions are by no means 
universal. 

248. Width of Streets. — The widest streets should in gen- 
eral be the ones which have the greatest traffic. Important 
business streets should be from 100 to 150 ft. in width, while 
streets of secondary importance in business districts may be 
from 60 to 80 ft. wide. In residential districts the main streets 



223 




V 






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i a 



iTim ti: 
mm ti-, 

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miTm.-rrn 









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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 15 to 20 ft. wide. If they are made nar- 
rower than 15 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 
curb lines to 3 or 4 per cent, if the grade of either street is 
greater than this amount. 



STREET GRADES 225 

On account of drainage it is well to build a street with a 
slight grade rather than level. A grade of 6 inches in ioo 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 effect on abutting property, and 
the general appearance of the street. 

At points where there is a decided change in grade it is 
customary to introduce a parabolic vertical curve. (Art. 268, 
p. 242.) 

For the purpose of establishing the grades, profiles are 
made of each street. Levels taken for the purpose of making 
a profile should include elevations at the center of the street and 
along both side lines, and it is often desirable to have a cross- 
section plan of the entire area of the vicinity where the street is 
to be located. A description of the street grade is written up 
for acceptance by the proper municipal authorities. When this 
description has been formally accepted by an order of the City 
Government the grade is said to have been " established.' ' 
Such an order may refer to the profile by title or recorded 
number, instead of a description of the grade. The profile of 
each street should contain one or more cross-sections on which 
is indicated to what part of the cross-section the profile refers, 
i.e., whether the profile grade is the grade of the center of the 
street, the curb, or the sidewalk at the property line. 

The following is an example of a description of an estab- 
lished street grade : — 

"Beginning at Station 146 (Maple St.) at the junction of 
the center lines of Maple St. and Ocean Ave., at grade* 52.00, 
the grade line falls 0.50 per 100 for 726 ft. to grade 48.37 — 
thence rises 0.82 per 100 for 322 ft. to grade 51.01 — thence 



* The word grade 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. IX 

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

Sta. Elev. 

i57+6o 50.40 

157 + 85 49.90 

i5 8 + 10 49-3° 

158 + 35 48-55 

158+60 47.70 • 

thence falls 3.60 per 100 for 239 ft. to Station 160 4- 99 
(Maple St.), grade 39.10." 

250. THE DATUM PLANE. — One of the first tasks of the 
surveyor in laying out a town site is to establish a datum plane to 
which all elevations may be referred. It is customary to choose 
a datum that bears an intimate relation to the topography of the 
locality. For example, if the town is located on the seashore a 
series of tidal observations may be taken to determine the mean 
sea-level or mean low water either of which is often used as a 
datum* (Art. 237, p. 211). The mean level of lakes is used as 
a datum for many inland cities. Frequently the elevation of 
some point not far from the town site has been established by 
the U. S. Geological Survey, the U. S. Coast and Geodetic Sur- 
vey, or by the line of levels of a railroad ; and by careful level- 
ing the elevation of some permanent point in the town site can 
be established which will serve as the starting point for all the 
elevations in the town. Where nothing of this sort is available, 
the elevation of some point is found by barometer so that the 
recorded elevation may approximate the actual height above sea- 
level. 

251. ESTABLISHING BENCH MARKS. — When the datum has 
been determined, bench marks are established by the method 
explained in Art. 219, p. 198. The establishment, at the start, 
of a reliable system of bench marks is of utmost importance, in 
order that the elevations of all parts of the city shall refer to the 
same datum. In laying out construction work it is absolutely 
necessary that bench marks which can be relied upon shall be 
available and sufficiently numerous to be of use in any section 
of the city without requiring several set-ups of the level to con- 
nect a bench mark with the level work that is to be done. 



WATER AND SEWER SYSTEMS 227 

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 A5D SEWER SYSTEMS. — The water and sewer 
systems of any community are of vital importance and provis- 
ion for them must be made in the layout of every town site. 
The location of the water supply and the storage and distributing 
reservoirs is a matter of such magnitude that it cannot be dis- 
cussed in this short treatise.* The conditions essential to an 
economical water or sewer system will sometimes radically affect 
the alignment and grades of many of the streets. The gradi- 
ents of water pipes are of little importance since the water is 
working under pressure, and the pipes can be laid uphill and 
downhill so long as there is sufficient " head " to force the water 
through the pipes. 

In a sewer system the problem is far different ; every sewer 
must have proper 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 NEW DISTRICT. — In staking out 
a new district the information at hand is usually a plan of the 
proposed layout of the streets which has been studied out in 
the office from a map of the district. If this layout has been 
approved by the municipal authorities the street lines as they 
appear on the plan are the "established lines." 

It is the surveyor's duty to stake out these lines on the 
ground, connecting them properly with the street lines of the 

* See Public Water Supplies by Tourneaure and Russell, published by John 
Wiley & Sons, New York. 

t See Sewerage, by Professor A. P. Folwell, published by John Wiley & 
Sons, New York. 



228 CITY SURVEYING [Chap. DC 

older portion of the city, and in short, to produce on the ground 
a layout exactly like that on the plan. Sometimes the angles 
and distances necessary for the layout have been computed 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 find that the exact dimen- 
sions given on the plan do not correspond with his fieldwork 
owing probably to the fact that his tape differs in length from 
that used by the surveyor who made the original plan. In such 
a case he must distribute the discrepancies (unless 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 corner bounds at first on account of the likelihood 
of their being disturbed. In such cases it is the duty of the 
surveyor to properly reference the points by cross transit lines 
or otherwise before construction work begins ; for it is important 
that the layout, as recorded in the city order, shall be accurately 
and definitely defined so that when the streets are brought to 
the proper grade and the monuments are finally set they will 
mark the exact position of the original layout. 

254. MONUMENTS. — It is important and at the same time 
customary to define street lines by setting stone bounds, often 
called monuments y 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 set 



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 anjj end of the 
curves. It is not necessary that all four corners of a street 
intersection shall be marked, as a bound on one corner will define 
the side lines of the two streets and, the width of the streets 
being known, the other sides can easily be determined. Nor 
is it necessary to place a bound at one of the corners of every 
street intersection, provided a street is straight for several blocks, 
although it is good practice to do so. On account of the liability 
of bounds which are placed on the side lines of the street being 
disturbed by building operations, some surveyors prefer to place 
them on an offset line, say 2 ft. from the street line. All mon- 
uments should be placed with extreme care as regards both 
their accuracy of position and their stability. If any bounds 
are set with more care than others, they should be the ones 
which occur at angle points in the street lines rather than the 
intermediate bounds which are set along a straight line. 

Monuments are usually roughly squared stone posts about 4 
to 8 inches square and 3 to 4 feet long, the length depending 
upon the severity of the climate, e.g., in New England a monu- 
ment less than 4 ft. long is likely to be disturbed by frost action. 
They are carefully squared on top and a drill-hole in this end 
marks the exact point. This drill-hole may be made before the 
stone is set in place, or after it has been placed so that its 
center is about in position the exact point may be defined by 
drilling a hole in the top of the bound. Frequently the hole is 
filled with lead and a copper nail set in the lead is used to mark 
the exact point. For nice definition of the point, a copper bolt 
is inserted and two lines scratched across it ; the intersection 
marks the exact point. When the stone bound is placed at the 
intersection of the side lines of the streets it is sometimes 
located entirely in the sidewalk in such a way that its inside 
corner is exactly on the intersection of the street lines. In such 
a case the three other corners of the bound are usually chipped 
off so that there may be no mistake as to which corner defines 
the line, but the line corner frequently becomes worn off and 
this practice is therefore not recommended. Some surveyors 



230 



CITY SURVEYING 



[Chap. IX. 



use, in the place of stone bounds, a t piece of iron pipe or iron 
plug with a punch-hole in the top of it, driven into the ground 
or embedded in cement concrete. Long heavy stakes are 
employed to temporarily define intermediate points or points of 
secondary importance. 

255. Setting Stone Bounds. — When the street lines are 
laid out the corners are marked by tacks in the top of ordinary 
wooden stakes. The monuments which are to take the place of 
the stakes should be set before the frost has entered the ground 
or before any other disturbance of the stakes has taken place. 
When the bound is ready to be set the first thing to do is to 
drive four temporary stakes around the corner stake about two 
feet from it and in such a way that a line stretched from two 
opposite stakes will pass over the tack in the head of the corner 
stake (Fig. 99). Then tacks are carefully set in the tops of 

these temporary stakes in 
such positions that a stretch- 
ing line running from the 
tack on one stake to the 
tack on the opposite stake 
will pass exactly over the 
tack in the corner stake. 

Then the 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 



a*B- 



Strttching 



f 



<0 



fcdb 

•* en 



E3r 



Line 



-BSHl 



Stone Bound 



WK. 

Fig. 99. 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 considerable care, 
the material being properly rammed as the filling proceeds and 
the bound kept in such a position that the drill-hole in the top 
of it, if there is one, shall be exactly under the intersection of 
the strings. It is sometimes desirable to put in a foundation 
of concrete and to fill with concrete around the monument to 
within a foot of the surface, as shown in Fig. 100, where a 
very substantial bound is re- 
quired, or where the ground is 
so soft as to furnish an insecure 
foundation. If the top of the 
bound is plain and the hole is 
to be drilled after the bound 
is in place, care should be taken 
to place the monument so that 
this hole will come practically 
in the center of the top in order 
that it may present a workman- 
like appearance. After the 
bound is set exactly in place the 
temporary stakes are removed. 

Some surveyors prefer to use only two opposite stakes and 
one stretching line, the position of the monument being deter- 
mined by a measurement along the stretching line from one or 
both of the temporary stakes. Still another method of tempo- 
rarily tying in the stone bound, and one which many surveyors 
use, is to set two stakes such as A and B in Fig. 99, and 
either measure the distance from them to the bound or set 
them at some even distance from the bound. This process 
of using temporary stakes and the stretching line is employed 
also in setting other types of bounds such as gas pipes or 
iron rods. 




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. 101.) 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 



233 



the radii of the curves are noted on the plan, but the street 
widths are often the only dimensions given. If the radii are 
not given the surveyor must determine from the plan either 
these radii or some other distances; such as the tangent lengths, 
so that he can go into the field, and, beginning with some known 
street line, run out the new street lines in such a way that when 
the data he determines are plotted the lines will coincide with 
those on the plan prepared by the landscape architect. As a 
rule these curved lines can be made up of a combination 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 

OB =■ * Radius =R 

AHB — Length of Arc =L C 

AB=* Long Chord =C 

VA = VB = Tangent Distance = T 

VII = 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 trigonometric relations, 




~ I T 

Tan = — , 
2 R' 

Exsec 7 = E 



2 

Vers I = 

2 

Sin L = 



R 
~R 
2R 



T = tftan- 
2 

E = R exsec - 
2 

M = R vers- 



es 2 R sin 



L c = 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. STAKING OUT CIRCULAR CURVES. — In Fig. 102 the 
two lines BC and EA are produced in the field and a point is set 
at their intersection V, as described in Art. 200, p. 175. The 

instrument is then set up at V 
and the central angle / carefully 
measured, or if point V is inacces- 
sible other angles such as VEC 
and VCE may be measured from 
which / can be easily computed. 
Then the radius R which is deter- 
mined from the plan being known, 
the tangent distance T is obtained 
by the formula, T = R tan \ I. 
Points P. T. and P.C. are then set 
and the curve is usually laid out 

by the method of deflection angles as explained in the following 

article. 

259. DEFLECTION ANGLES. — A deflection angle is usually 
referred to as an angle between a tangent and a chord, e.g., in 
Fig. 103 angles VAb, VAc, etc., are deflection angles. Since 




Fig. 102. Circular Curve. 




Fig. 108. Deflection Angles. 



makes it possible to compute the length 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 extenoivelyin railroad en- 



DEFLECTION ANGLES 235 

the angle between a tangent and a chord is measured by half the 
included arc these deflection angles must be equal to half the 
angle at the center subtended by the same chord or arc. 

If the total length of the curve is divided into an even num. 
ber of parts, «, the angle at the center under each of these arcs 

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

which in Fig. 103 is the angle VAb. Angle bAc = angle VAb, 

both being measured by one-half of equal arcs. It follows then 

that the deflection angle to point 

/ / 

c — 2 x — = - 

2n n 

2ft 2ft 
I 2/ 

e — 4 x — — — 
2» 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 

2n 

* 

method which should be used. The deflection angle from the 

P. C. to the P. T. should be equal to - , and this check should 

always be applied to the computations before they are used in 
laying out the curve. 

The chords A6, 6c, cd f etc. are equal since their arcs are 

equal. With the radius and the central angle [ - for one chord) 

given, the chord length can readily be found from the formula, 

gineering. The central angle divided by the degree of curve will give the number 
of 100-ft. chords in the length of the curve, i-e.,yj — L (in 100-ft. stations). 

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

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



236 CITY SURVEYING [Chap. DC 

c = 2 sin Since the angle at the center is usually small 

and the radius large the angle will have to be carried out in 
some instances much closer than to the nearest minute in order 
that the length of the chord may be obtained to hundredths of 
a foot (Art. 371, p. 341). An approximate value for the chord 
length corresponding to a given arc may be obtained by the 
approximate formula, 

c* / 8 • 

in which l e is the length of the arc, c is the chord length, and 
R the radius. 

The fieldbooks in use by most surveyors contain tables of 
chords and corresponding arcs for curves of different radii, 
which assist greatly in shortening these computations. 

When the deflection angles have been computed and checked 
and the chord length found, the instrument is set up at A, 
(Fig. 103) a foresight taken on the vertex with the vernier 
reading o°, and the point b set by measuring Ab and placing b 
on line by means of the transit on which the first deflection 
angle VAb 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 will give some idea of the accuracy of this formula. 

With R =» 100 and c = 25, the formula gives l c — 25.065, (correct value 

is 25.066). 
With R = 100 and c = 50, the formula gives le ■■ 50.521, (correct value 

k 50-536). 
With R ■= 1000 and c = 100, the formula gives l 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 



vfcttrre 



Station 



Distent* 



Chords 



left J Can her \ ftighr 



Dtflecfo* 



Aiyks 



ffanark 



To Right 
#•200 
T*9646 

T'JfyM 



J8+S2J0 
&Z2A2 

I7+2ZAZ 
/6'72.42 



3008 
S3JOO 
SXOO 
SDjOO 



Wctfr 

3X3/ 

J&S9 
SBS9 



ofSfrtef- 70 fiief. 



jaas" 
49.87 

49.87 
49JB7 



70 

24.79 
4Z./4 

4/.J4 

4f.t4 



2sU7*-40 m 

Z/ -29-/0 

/4-&-20 

7-09*U> 



RT. 



M. 



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 rim. 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. DC 



angles are used in running out the side lines for the chords which 
have been computed for the side lines run between points which 
are radially opposite the corresponding points on the center line. 
The computation of these notes will be found in Art. 371, p. 341. 

261. When the Entire Curve Cannot be Laid Out from One 
End. — It is often impossible to see from the P.C. to the P.T m 
of a curve on account of intervening obstructions. In such a 
case the curve is run from the P.C. as far as is practicable and a 
point is carefully set on the curve ; then the transit is brought 
forward and set up at the point thus fixed, and the curve extended 
beyond. There are two different methods employed in this case. 

262. First Method. — Assume the circular curve in Fig. 
105 to be laid out from A to d as described above. Point d is 




Fig. 105. Intermediate Set-up on Curve. 

carefully set and the instrument then taken to that point and 
set up. The vernier is turned back to o° and beyond 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 Adf 
are equal. It is well to note whether the instrument appears 
to point in the direction of the tangent. Then reverse the tele- 
scope, set off on the vernier the angle gde = — , and lay out the 

2ft 



INTERMEDIATE SET-UPS. 239 

curve from d to B 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 VAe is laid off on 
the arc ; this will cause the telescope to point in some such 
direction as dh. The line of sight is reversed and point e set on 
A/ 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 + /id/, being measured by half of equal arcs. 

This second method is sometimes to be preferred since the 
original deflection angles figured can be used throughout the 
curve. The first method calls for the calculation of a. few more 
angles ; but this is so simple a process that there is probably 
little choice between the two methods. 

264. CURVED STREET CORNERS. — It is the practice in many 
cities to curve the corners of the streets by introducing a cir- 
cular curve of short radius. Where both street lines are straight 
the problem is handled as explained in Art. 258, p. 234. 

O v 265. * One Street 

Line Straight, the Other 
Curved. — In Fig. 1 06 
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 
Fig. 106. GE on the ground. In 

* The authors are indebted to I. T. Farnham, City Engineer of Newton, 
Mass., for the solution of the problems given in Arts. 265-6. 



A <K 


V 


< 


V 

A 

1/ 



240 



CITY SURVEYING 



[Chap. DC 



the field any tangent line, such as FV 9 is run off from some 
known point on the curve and intersected with A V, and the 
angle and the distance FVare measured. In the right triangle 
CFVia which R and Tare 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'C 

and A'CC = £C'£\ 
Angle 4 OF = 180 — 0. 
Angle £CF = 4CF — 
i4'CC, from which the 
length of the arc FE 
can be readily com- 
puted, which locates the 
point E. VG=* A V— 
A'C, 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 C'E 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 D' V from the curve A'B'U and angle 
a and distances ^Tand T' are measured. In the right triangle 
CDV, R and T being known, compute angle CVD and distance 
CV. Similarly in the triangle CVD' compute angle CVU and 
distance CV. In the oblique triangle CVC, CV, CV and 
angle CVC = 360 —( a + CVD + CVD') being known, com- 
pute CC and the angle CCV and CCV. In the oblique tri- 




ROUNDING STREET CORNERS 



24I 



angle CC'E, CE = R + r, C'E = R' — r, and CC being known, 
compute the angle C'CE, CCE and C'EC, which is the sup- 
plement of the central angle of the curve B'B. Angle 
DCB = DCV+ VCC — CCE, from which arc DB can be com- 

C 



w v 




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 K 

267. STAKING OUT STREET GRADES. — The fieldwork neces- 
sary in setting grade stakes is explained in Arts. 235-6, p. 210. 
When new streets are constructed the excavation or embankment 
is first brought to sub-grade, i.e., to the grade of the bottom of the 
road covering or pavement. The grade stakes set for this work are 
usually the center and the two side slope stakes, properly marked 
with the cut or fill, as described in Arts. 230-2, pp. 207-10. 

As the work progresses the center stake is dug out or covered 
up and when the construction has progressed nearly to the sub- 



242 



CITY SURVEYING 



[Chap. 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 VM 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 
O 




^vJN 



Fig. 108. Vertical Parabolic Curve. 
curve at every 25-ft. station ; in such a case, then, Fig. 108 rep- 
resents a vertical curve 200 ft. long on which the elevation of 
nine points must be determined. The equation of the parabola is 

y 2 = 4 px, or y 2 = (constant) x, (1) 

the ;r dimensions being parallel to VK (vertical) and the y dimen- 
sions being along A V. From the equation it is readily seen that 
the offsets from the tangent vary as the squares of the distances 
along the tangent, or x x : x 2 = y x 2 :y 2 2 . The lines VP and NQ are 
vertical and AQ is horizontal. Since the curve extends an equal 
distance each side of V, AP = PQ\ and therefore AK = KB. 
NB = 4VH\ VH=4CC; CC = aDD'\ etc. (from equation 1.) 
Let g and g x represent the rate of grade of LV and VM % 
and n the number of 25-ft. stations (in this case 4) on each side 
of the vertex V, then 

AT TO 

KV = (from similar triangles) 

2 



VERTICAL CURVES 243 

but NB = 4 VH (from above) 

therefore KV = 2HV, 

or point H is midway between V and K. 

The elevation of Fis determined from the established grade. 
The number of 25-ft. stations will determine the distance VA 
and VB. The elevation of A and of B can be readily computed 
along their respective straight grade lines. 

„ _ Elev. A + Elev. B 
Elev. K = 



Elev. H = 



2 
Elev. F + 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 Z?', C", E\ 

F'> G', and /' and by adding to these elevations the ordinates 

UD, CCB E, etc. 

VH 

16 

VH 
CC = G'G = — 
4 
gVH 



EE = F'F = 



16 



269. CROSS-SECTION OF STREET. — On account of the ne- 
cessity for draining the surface of a road the center is raised or 
" crowned" above the grade of the gutters by an amount depend- 
ing on various conditions. The shape of the road surface is 
sometimes two planes, running straight from the gutter to a 
summit or ridge in the center of the street, this ridge being 
rounded off by rolling ; but more frequently it is a curved sur- 
face in the form of a parabola or a circle. The ordinary width 
and crown of streets are such that the parabola and the circle 
are practically coincident. 

When a street is to be paved the curbstones are first set to 
proper line and grade, then stakes are set for the finished grade 
of the roadway. The center grade stake is frequently the only 



244 CITY SURVEYING [Chap. DC 

grade given and 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 : 

,, t, 01 Crown I 

Mean Transverse Slope = 



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

270. Gutters at Same Elevation. — Fig. 109 represents the 




Fig. 109. Cross-Section of Pavement ; Gutters at same Elevation 

cross-section of a pavement and sidewalks. The crown AB is 
computed from the mean transverse slope and the width of the 
pavement. 

The ordinate Dlf at any other point on the parabola 

I 2 
-CC X -jTjjr since in a parabola the offsets from a tangent 

1* 



,2i 

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

the center to the curb, DD f = . 

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. no the maximum ordinate x is at a dis- 



< w > 

Fig. 110. Cross-Section of Pavement: One Gutter Higher 
than the other. 

tance a from the lower gutter. The first step is to find this 
distance a and then x is readily found from the mean transverse 

x 

slope since - = Mean transverse slope. When x is found the 

other offsets can be computed as explained in the previous article. 

At A the offset = - 
4 
at 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. 
s ■■ 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?— (1) (See (1) in foot-note, p. 339.) 

2 R 

and x — b = — (2) (See (1) in foot-note, p. 339.) 

2R 



246 



CITY SURVEYING 



[Chap. IX. 



Combining (1) and (2), a 8 — /* — zRb 
{a + I) (a - t) - 2Rb 

<* + / 
a + l-W 



But 



.*-/- 



(a + + (* - D 



From (x), 
But 



From (3), 



zRb 
W 

W 



2a- W + 



2Rb 

W 



2 W 



C3) 



2*-!L 

x 

— — Mean transverse slope — s 
a 

x — as 

as s 

is 

ab 

W 

a - _ + _ 
2 ^ 



2J 



\ 2 ^> / 2 



2 



2*tt 



272. If, instead of assuming the mean transverse slope of 
the pavement, the elevation of the center of the pavement D 
(Fig. in) 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 F, which are midway between D and the gutters, are com- 
puted from the method explained in Art. 270, eE and fF both 

being equal to _ 1 . 
4 



Similarly, Elevation E = Elevation G + 



Elevation F = Elevation H 4- 



Elevation J = Elevation K + 



Elevation L = Elevation M + 



jPC 

4 
IDC 

4 
7PC 

16 
7PC 

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 



8 



60T 



60 


60 


60 


60 


60 

6^ 


GO 



TY^ 



^ 



% 



<&>> 



<&» 



Fig. 112. Arrangement of Lots in a Wedge-Shaped Block. 



each lot as much street front as is possible consistent with mak- 
ing the side lines of the lots at right angles to the street lines. 
If the side lines do not run at right angles to the street there 
will be portions of the lot which are not available for the custom- 
ary rectangular style of building construction and which are 
therefore not so desirable for business purposes. This is not of 



248 



CITY SURVEYING 



[Chap. DC 



so much importance in residential districts where the rectangu- 
lar system is often purposely avoided to some extent, to obtain 
a layout which has an attractive appearance, as illustrated by 
Fig. 101, 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 corner lots. 

274. STAKING OUT CITY LOTS. — In staking out the lots of 
a rectangular block, the corners of which have been established, 
the most direct method is as follows. The transit is set up on 
the S. B. at A, (Fig. 1 13), a sight is taken on B 9 and the front 



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1L 





s 






















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so' 


J> 


50 


M 


JO 


SO 


SO 


90 


SO 


SO 


so 


so 


so 






3«3 














N 


V 


n 





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£ 






























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2 


3 


4 


9 


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7 


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sor 

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SO 


50 


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Fig. 113. Rectangular City Block. 



corner stakes of lots 1, 2, 3, 4, etc., are set, with a tack, exactly 
on line, in the top of each stake. All such work should be done 
to the nearest 0.0 1 ft. It will be well first to measure the line 
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 A to B of 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 H, points 
F and G are set. Then by setting up at F and sighting on t) 
the front corners of lots 13, 14, 1 S, etc., are determined. Another 
set-up of the transit at G with the line of sight on E will allow 
the "back bone" to be run out and the back corners of all the 
lots established. The check on the lines A B, 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 corner and 
locate the back corner stakes by turning 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 edges tones in some localities. 



250 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 1 foot) above or below the grade of the 
curb. Sometimes the grade stakes are not driven so that their 
tops bear any relation to the finished grade, but a horizontal 
chalkmark is made on the side of the stake marking the proper 
grade. A stake can be marked much more quickly than the 
top can be driven to the exact grade. 

When new curbstones are being set in an old street, stakes 
cannot as a rule be used. The sidewalks are too hard to permit 
the driving of stakes, and even if they could be driven those 
projecting above the surface of the sidewalk would be a source 
of danger to pedestrians. In such cases it is customary to use 
heavy spikes about 6" long. These are driven into the side- 
walk on the offset line and the elevation of their tops determined 
by leveling. The difference between the elevation of each spike 
and the grade of the curb opposite it is calculated. A list of the 
stations and the distances the spikes are above or below the curb 
is given to the foreman in charge of the work. These distances 
should always be transposed into feet and inches (to the nearest 
J") before being given to the foreman, as it is seldom that the 
men employed to lay the curbstones have any conception of the 
meaning of tenths and hundredths of a foot. (See Art. 7, 

P. 5.) 

Where there are trees growing *in line with the curbs, a nail 
can sometimes be set in the side of a tree on the line of the curb 
as well as at its grade. Points like these, of course, should be set 
in preference to offset stakes or spikes wherever possible, as 
there is little liability of the workmen misinterpreting such 
marks. They can fasten their string directly to the nail and set 
the curb to agree with it. 

Before the curbstones are ordered the surveyor usually meas- 
ures the distances between trees and locates driveways, and 
then makes out a list of the lengths of straight, of curved, and of 
chamfered stones (opposite driveways) to be used on the job. 
This list is used in ordering the stones, and when they are de- 
livered they should be found to fit the conditions without the 
necessity of cutting any of them. 



STAKING OUT CURBS AND SEWERS 251 

• 276. STAKING OUT SEWERS. — The lines and grades of 
sewers are sometimes run out in the same way as those described 
for curbstones. The stakes or spikes (in 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 frohi the 
offset 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 50-ft. 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 corner lot the corner bound 
has been disturbed or that it has been removed entirely ; and 
before the line can be properly staked out it may be necessary 
to begin at some reliable S. B. farther down the street or even 
on some other nearby street line. 

When the line has finally been rerun it is customary to take 
and record swing offsets from the corners of the underpining of 
several of the buildings located along the street and near to the 
line. By this record of offsets, then, this street line can very 
easily and quickly be run out at any future time, and any dis- 
turbance of the S. Bs. at the corners can readily be detected. 
Several offsets 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-sectioa 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 building is 
recorded on the plan itself. In a brick or stone building the 
lines to be defined are the outside neat lines of the building, and 
the elevation desired is usually the top of the first floor. In the 
case of a wooden building the line usually given is the outside 
line of the brick or stone 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 underpinning 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. 



J - 



— h 



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. 1 14, at A, B> G, D> £, and F. A stake 

should be set at G also 
so that the entire work 
can be checked by meas- 
uring the diagonals AG 
and FB, and GD and EC 
These checks should 
always be applied where 
possible. Then the posts 
for the batter-boards are 
driven into the ground 3 
or 4 ft. outside the line of 
the cellar so that they 
will not be disturbed 
when the walls are being 
constructed. On these 
posts, which are usually 
of 2" x 4" scantling, 1" 
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. 114), 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 




•f c 

Fig. 114. 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 // andy. 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 PLANS 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 
obtain 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 city on 
account of its limited extent, and to use a system of rectangular 
coordinates based upon plane surveying. In a coordinate system 
two arbitrary lines are chosen for coordinate axes, one usually 
coinciding with some meridian and the other at right angles to it. 
All points in the city are located by distances from these two 
axes, these distances being known as X's and Y's, or sometimes 



256 CITY SURVEYING [Chap. IX. 

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 (10 ooo, 10 ooo). By either of these arrangements 
negative values for coordinates are avoided. The coordinates are 
usually considered positive toward the north and the east, in 
accordance with the custom of analytic geometry, as is the case 
in ordinary land surveying. The convergence of the meridians 
is neglected and all points having the same X coordinate there- 
fore lie on a straight line parallel to the initial meridian and 
are not all on the same true meridian line. 

In the survey of the city of Baltimore (Fig. 115) the origin 
of coordinates was taken through the Washington Monument 
in the central part of the city, and the map divided into squares 
1000 feet on a side. Each square mile is shown on a separate 
page of the atlas of the city and these squares are designated 
by their number north or south, and east or west of the origin, 
as 1S2W, 3N4E, etc. Any point is designated by the distance 
in feet north or south, and east or west, as (1000 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 



• : • • 




Fig. 115. Triangulation Scheme foi 
(Printed by permission of Major Joseph W. Shirley, Chief Engim 




r > Survey of the city of Baltimore. 
.-*r 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. (See Volume II, Chapter I.) 

The system of triangles should cover the entire area but 
should not contain more lines than are necessary to establish 
a sufficient number of points to control the subsequent work 
of the survey. 

285. MEASUREMENT OF BASE-LINE.— At least one line in 
the system must be chosen where its length can be very ac- 
curately measured; this is called the base-line. The lengths of 
all the other lines are to be computed from this line by means 
of the measured angles, hence it will be seen how important 
it is that this line should be measured with great accuracy, 
and that it should also form well shaped triangles with the con- 
necting triangulation stations. 

It should be chosen if possible in some level spot where there 
are no serious obstacles to the measurement. It is sometimes 
an advantage to have the ends of the base-line slightly elevated 
above the general level. The base should be measured with a 
steel tape the exact length of which is known. The tension 
should be kept constant by means of a spring balance, and the 
temperature carefully taken. If the work is done on a cloudy or 
rainy day the thermometer readings will represent the tempera- 
ture of the tape much more nearly than when taken in sunshine. 
The points should be lined in with a transit and the tape held 
horizontal, or, if the measurements are taken directly from stake 
to stake, the slope should be determined, by means of a leveling 
instrument. There should be at least two independent measure- 
ments of the line. 

286. MEASUREMENT OF ANGLES. — If possible all of the 
angles of each triangle should be measured by repetition. An 
"inverting" instrument reading to 20" or to 10" is to be pre- 
ferred for this work. The angles are repeated at least six times 
with the telescope direct and the same number of times with 
the telescope inverted. Several of these sets of readings are 
made beginning each time with a different initial setting on the 
circle. For example, if the first setting was at o° and four sets are 
to be taken the second would begin with a setting of 90 , and 
so on. In each case both verniers should be read and the mean 



262 CITY SURVEYING [Chap. DC. 

of the two taken. Sometimes the direction of the measurement 
is changed during the set, the first six repetitions being taken 
from left to right, and the second six from right to left. In this 
work it is important that the instrument should be carefully 
centered over the point, and that the signals are also carefully 
centered. It is al§o 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 180 . 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 o° 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 Ys 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. SECONDARY 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 corners, and important monuments. 

It frequently happens that, owing to the large number of 
angle measurements and the consequent accumulated error, the 
lengths of the sides of the small triangles become much less 
accurate than they would be if measured directly ; and since 
many of these lines naturally lie in places where the distance 
can easily be measured, this measurement should be made 
as a check, in which case this line becomes a secondary 
base-line. It is a good plan to introduce these measurements 
frequently, where it can be conveniently done without great 
expense, in order to prevent the errors of the survey from 
accumulating unnecessarily. 

290. TRAVERSES. — After all of the triangulation is com- 
pleted the system is extended by running traverses with the 
transit and tape, from one known point to another. The tri- 
angulation points are regarded as fixed and the errors of closure 
of the traverses are assumed to be entirely in the traverse sur- 
veys, the traverses being made to fit in exactly between the 
triangulation points. 

All street lines, or parallel offset lines, are connected with 
the coordinate system so that the azimuth of every street line in 
the city may be known, and 'the coordinates of all important 
points, such as street corners and lot corners, are computed. 

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

TOPOGRAPHICAL SURVEYING. 

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 then 
positions are plotted on the map. This may be done conven- 
iently by the method of rectangular coordinates described in 
Art. 283, p. 255. The extension of the system to smaller systems 
of triangles, called secondary and tertiary, may be made if neces- 
sary. After the triangulation system has been extended far 

264 



TRIANGULATION 265 

enough to furnish a sufficient number of points for controlling 
the accuracy of the map, traverses may be run wherever con- 
venient or necessary for locating topographic details. In all 
cases the traverses should be connected with the triangulation 
points at frequent intervals in order that the relative positions 
of all points may be kept as nearly correct as possible. Where 
a high degree of accuracy is necessary these traverses should 
be run with a transit and tape ; if, however, errors of a foot or 
two would not be appreciable on the map it will be sufficiently 
accurate to use the stadia method of measuring the distances 
and thus save time. (See Volume II, Chapters I, IV, and V.) 

2<H. LOCATION OF POINTS FROM THE TRANSIT LINE.— 

Where a tape is used for measuring the distances, such objects 
as fences, walls, and buildings may be located as described in 
Chapter VI, but it will not be necessary to make the measure- 
ments with as great precision. Fig. 116 is a sample page of 
notes of a topographical survey where the transit and tape were 
used. On city plans, which are frequently drawn to a scale of 
40 feet to an inch, a fraction of a foot can easily be shown. On 
a topographic map the scale is often such that an error of a frac- 
tion of a foot becomes insignificant in the side measurements 
from the transit line, where such errors cannot accumulate. In 
some cases it may be sufficient to obtain the distances by pacing, 
and the angles or directions by means of a pocket compass. Lo- 
cations may frequently be checked by noting where range lines 
intersect the transit line. In making a series of measurements 
it is well to take each measurement with a little more precision 
than is actually needed for plotting, in order to be sure that the 
accumulated error does 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 LINES. — 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 lines, lines joining points of equal elevation, are used. 
In the latter system elevations may be read directly from the 
map, and for this reason it is much more used by surveyors. 




Fig. 117. Hachure Lines. 



Fig. 118. Contour Lines. 



A contour line is the intersection of a level surface with the 
surface of the ground. A clearer conception of a contour line 
may be obtained from the following. Imagine a valley, or de- 
pression in the surface of the ground, partly filled with water. 
The shore line of this body of water will then be a contour line, 
since it is the intersection of a level surface with the surface of 
the ground. If the water stands at an elevation of 50 feet the 
shore line is the 50-ft. contour. If the surface of the water 
were raised 5 feet the new shore line would then be the 55-ft. 
contour. Contour lines if extended far enough will therefore 
be closed curves, and all of the points on any one contour will 
have the same elevation above the datum. It is customary to 
take contours a whole number of feet above the datum, spacing 
them in regard to height, so as to make the contour intervals 
equal, e.g., a contour may be taken at every 5 feet or every 10 
feet of elevation. Since the contours are equidistant in a verti- 
cal direction their distance apart in a horizontal direction shows 
the steepness of the slope. 



268 TOPOGRAPHICAL SURVEYING [Chap. X. 

Fig. 1 19 illustrates contour maps of simple solids. 



ELtVATWN 




Fig. 119. 



CONTOUR MAP 

Contour Maps of Simple Solids. 



296. Characteristics of Contours. — The chief characteristics 
of contours are illustrated in Fig. 1 20, and may be summed up 
as follows. 

1. All points on anyone contour have the same elevation, as 
at A. 

2. Every contour closes on itself, either within or beyond 
the limits of the map. In the latter case the contour line will 
not end within the limits of the map but will run to the edge of 
the map, as at B. 

3. A contour which closes within the limits of the map in- 
dicates either a summit or a depression. In depressions there 
will usually be found a pond or a lake ; but where there is no 
water the contours are usually marked in some way to indicate 
a depression, as at C 

4. Contours can never cross each other except where there 
is an overhanging cliff, 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. 




Fig. 120. Illustrating Characteristics of Contours. 



270 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 




Fig, 



Contour Crossing 
Street. 



Fig. 121 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 5. 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, F, would be farther up the road from E, 
i.e., the contour would be un symmetrical, EF being longer 
than DE. 

297. RELATION BETWEEN CONTOUR MAP AND PROFILE. — 
If a line is drawn across a contour map the profile of the surface 
along that line may be constructed, since the points where the 
contours are cut by the line are points of known elevation and 
the horizontal distances between these points can be scaled or 
projected from the map. The profile shown in Fig. 122 is con- 
structed by first drawing, as a basis for the profile, equidistant 
lines, corresponding to the contour interval, and parallel to AB. 
From the points where AB cuts the contours lines are projected 



PROFILE FROM A MAP 



271 



to the corresponding line on the profile. Conversely, if the pro- 
files of a sufficient number of lines on the map are given it is pos- 
sible to plot these lines on the map, mark the elevations, and 
from these points to sketch the contours as described in Art. 
301, p. 276. 




PROFILE ON LINE. A B 
Fig. 122. Profile Constructed from a Contour Map. 



298. RELATION BETWEEN CONTOUR MAP AND SIDE ELE- 
VATION OR PROJECTION. — A photograph of a landscape 
represents approximately a side elevation of the country. To 



272 TOPOGRAPHICAL SURVEYING [Chap. X 

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




projection on, plane ab 
Fig. 128. Side 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 
indicated by the shaded section. The drainage area for the por- 
tion of the stream above the dam is the area included within the 
heavy dotted line, which follows the line of the divide. 

300. Sketching Contours from Streams and sum- 
mits. — The present topography of some parts of the country 
is due almost entirely to erosion by streams. Consequently the 
position and fall of the streams give more information regarding 
the position of the contours than any other topographic features. 
If a definite position of the contours is desired it will be necessary 
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. 125. (See Volume II, Chapter VII.) 

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 con- 
tours in any particular case will depend upon the geological 
formation. Fig. 126 represents the same country as Fig. 125 
but with the contours sketched on it, following out the general 
suggestions which have just been mentioned. 



274 



TOPOGRAPHICAL SURVEYING 



[Chap. X. 




Fig. 124. Illustrating Flooded Area and Drainage Area. 



CONTOUR SKETCHING 



275 




Fig. 125. Map Showing the Location and Elevation of Streams 
and Summits. 




Fig. 126. Contours Sketched from the Data given in the 

Map above. 



276 



TOPOGRAPHICAL SURVEYING. 



301. sketching contouhs FROM known ELEVATIONS. 

— A portion of the country can be cross-sectioned as described 
in Art. 227, p. 206, or profiles can be run on any desired lines 
as explained in Art. 225, p. 203. From these known elevations 
contours can be sketched by interpolation. This is usually done 
by estimation and the principle involved is the same whether the 
elevations were obtained by cross-sectioning or by profiles. 

Fig. 127 illustrates how contours can be sketched from cross- 




Fig. 127. 



T RAVER*. LINE 

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 2JJ 




Fig. 128. Contours INCORRECTLY Sketched. 

303. LOCATING CONTOURS. — 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 brief explanation of the principles of the Stadia will be found in 
Appendix A, p. 517 of this volume. A complete discussion of the Stadia and the 
Plane Table will be found in Volume II, Chapter IV. 



278 TOPOGRAPHICAL SURVEYING. [Chap. X 

304. Locating Contours by Cross-Sections. — A very com- 
mon as well as expensive method of locating contours is that of 
taking cross-sections. Elevations on the surface of the ground 
are usually taken to tenths of a foot. From these elevations 
the contours may be sketched by interpolating between these 
known elevations as explained in Art. 301. The accuracy may 
be increased by taking a larger number of intermediate points. 
The size of the squares used should depend upon the roughness 
of the surface. 

305. Locating Contours by Profiles. — In some cases where 
the ground is fairly smooth it is sufficient to take a few profiles 
on known lines, not necessarily at right angles to each other. 
These lines are stationed and elevations are taken at every full 
station and at the points of marked change in slope. From 
these data the contours are sketched on the map by interpola- 
tion as described in Art. 301. 

306. Locating Points on the Contours — Where the contour 
interval is small, say one or two feet, and the topography is to 
be determined with considerable accuracy, it is advisable to find, 
in the field, points actually on the contours and thus avoid the 
errors of interpolation. The rodman moves up or down the 
slope until the rod-reading indicates that the foot of the rod is 
on a contour. The position of the rod may then be located by 
an angle and a distance from some known line, the distance 
being taken with a tape. 

307. Locating Contours by the Hand Level.— -A more rapid 
but less accurate way of putting in contours is by means of the 
hand level. The work is done by making profiles of lines whose 
positions on the map are known. A point on some contour is 
found in the following manner. 

The first step to take is to measure to the nearest tenth of 
a foot the distance from the ground to the eye of the leveler, 
which may be, say, 5.4 ft. If the B. M. is at elevation 143.43 
and it is desired to locate a point on the 140-ft. contour, the 
rodman holds the rod (or a tape) on the B. M. while the leveler 
attempts to place himself on the 140-ft. contour. When he is 
on the 140ft. contour the elevation of his eye (H.I.) is 145.4 



LOCATING CONTOURS BY THE HAND LEVEL 279 

and the rod-reading at the B. M. must be 145.4 — r 43»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 145-ft. contour. By trial then the point is 
found where the rod reads 0.4 ft.* Then the leveler walks up 
the hill and, standing on the point just found, places the rodman 
on the next higher contour by the same process. 

In working down the hill to locate the 135-ft. contour, if the 
leveler is standing on the 140-ft. contour, the rod will be on the 
135-ft. contour when it reads 10.4 ft. Or, when the 140-ft. 
contour has been found by the leveler the rodman comes forward 
and holds the rod on this spot and the leveler backs down the 
hill until he reads 0.4 ft. on the rod ; he is then standing on the 
135-ft. contour. Some surveyors prefer to cut a stick just 5 ft. 
long and hold the hand level on the top of it in taking sights. 

The points thus found at regular contour elevations are then 
plotted on the corresponding lines and the contours sketched by 
joining points of equal elevation. Where the lines which are 
profiled are far apart or where the country is very rough it is 
frequently necessary to obtain the correct position of the con- 
tours, to locate extra points on them between these profiled lines. 
The extra points are located by right-angle offsets from the 
lines. Most 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 
so that his feet will be on the proper contour. 



28o TOPOGRAPHICAL SURVEYING [Chap. X. 

308. LOCATION OF STREAMS AND SHORE LINES. — Streams 
or shore lines of ponds may be very rapidly located by stadia 
measurements. If the shore lines are to be located by tape 
measurements, however, a convenient way is to run a transit, 
line aproximately parallel to the general direction of the shore 
line, and to take perpendicular offsets at regular intervals and 
at all points where there is a marked change in the direction of 
the shore line, as was done in the notes in Fig. 53, p. 104. 

309. CONTOUR PROBLEMS. — There are many surveying 
problems involving earthwork which can be worked out approxi- 
mately by use of a contour map. As a rule the smaller the 
contour interval, the more accurate will be the result of such 
work. Contour studies occur in a variety of problems, so 
numerous that it would be useless to attempt to cover the sub- 
ject fully. Three typical problems, however, are illustrated and 
explained ; and these contain the essential principles applicable 
to practically all contour studies. 

310. Example i. — (Fig. 129). Given a contour map, the 
surface being represented by contours shown by full lines, a 
plane (extended indefinitely) is passed through the straight lines 
AB and CD, which are level and parallel, AB being at elevation 
12.5 and CD being at elevation 40. It is required to find where 
this plane intersects the surface, and to shade the portion which 
is above the plane. 

Since the proposed surface is a plane, contours on it will be 
parallel to AB and CD, The elevations of AB and CD being 
known, other contours, such as ef and gh> can be interpolated 
between AB and CD. Their interval is made 5 ft. the same as 
the contour interval for the original surface. Evidently the 
point where any of these parallel lines crosses an original con- 
tour of the same elevation, as/, k, /, 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 t 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 



28l 



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 referen.ee 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 
includes 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 Cc, the points a, b, and 
c being points on the upper side of the road between which it 
may be assumed that the slope is uniform (Art. 301, p. 276). 
Along these lines interpolate points which will lie on the uniform 
slope from C to the road and also on the regular 5 ft. intervals 
which correspond to the contours. For example along the line Ca 




Fig. 130. 



from the summit which is at elevation 89 to the road at a which 
is at elevation 55, there is a drop of 34 ft., or a little less than 
7 contour intervals. Points e, f, g, h, etc. are therefore plotted so 
as to divide Ca as follows : ea, ef, etc., are each -fa of AC, and 
the upper division is ^ of AC Similarly points i, j, k, etc., 
are plotted along the line Cb, but the point b, being at elevation 
56, is plotted so that the distance ib is four-fifths of the other 
distances ij,jk, 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 m, n, 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 of 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 oe 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 1 inch= 150 




Fig. 131. 



feet. Side slopes of road to be \\ horizontal to 1 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, cd, ef, gh. These will be 125 ft. apart, as a 4 % 
grade falls 5 ft. in a distance of 125 ft. If the road is assumed 
to be level on top, then these lines will cross the road at right 
angles to its general direction as shown in the figure. From 
points a and b, on either edge of the road, the new contout 



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 1 J to 1, 
then the distance out from c must be \\ x 5 = 7-5 ft.; opposite 
e it is 10 ft. above the road and similarly the distance out from 
e must be 15 ft. Where this new 50 ft. contour meets the old 




Fig. 131. 



50 ft. contour at o, is a point at the top of the slope. Similarly 
all of the new contour lines, which are represented on the 
figure by dash lines are plotted and their intersections with the 
corresponding contours of the original surface give points of " no 
cut" or "no fill," or top of slope (in excavated portions) and 
foot of slope (in embankment portions). These lines are shown 
in the figure by heavy dotted lines. Where this heavy dotted 
line crosses the road it marks a "no cut" and "no fill" line, 
i.e., the road bed cuts the surface of the ground. 



CHAPTER XL 

MINE AND MOUNTAIN SURVEYING* 

313. General Remarks. — In this chapter the limitations 
and difficulties met with in surveying a mine will be pointed out 
and some of the instruments f 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 surveying are the locating of the ownership boundaries 
underground and the laying out of passageways so as to connect 
with one another, thereby facilitating the working of the mine. 
Such passages are usually highly inclined and therefore it is 
frequently necessary to use entirely different methods from those 
employed in ordinary surveying. 

3 14. DEFINITIONS OF MINING TERMS. —The following terms 
are in common use in mine surveying. 

Adit. An approximately horizontal underground passageway 
running from the surface into the mine workings and used 
only for drainage and ventilation. 

Apex. The portion of the surface of the undisturbed rock forma- 
tion which is included between the walls or sides of the 
mineral deposit. 

Bed. A stratum in the earth's crust which has been formed or 
deposited in an approximately horizontal layer. 

* This chapter was written by Blarney Stevens, M. Sc., Mining Engineer, 
Seattle, Wash. 

285 



286 MINING SURVEYING. [Chap. XI. 

Compartment. One of the smaller passageways of a large shaft 

divided by timber partitions. Fig. 134 is the plan of a three 

compartment shaft. 
Connection. A passageway which is driven from one accessible 

part of the mine to another. 
Cross-cut. A horizontal passageway across or approximately at 

right angles to the strike. 
Dip. The inclination of any rock plane to the horizon. 
Drift. A horizontal passageway along, or approximately parallel 

to, the strike. 
Fault. A fracture in the earth's crust along which slipping or 

shearing has occurred. • 

Floor. The bottom of the passageway or of a seam or bed. 
Heading. Any preliminary passageway driven to explore the 

mine or to facilitate the future operations. 
Heave. The distance between the two parts of the same vein 

which is divided by a fault, measured along the strike of the 

fault. 
Levels. Horizontal passageways run at different levels along the 

deposit or adjacent to it 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. That portion of the vein which is exposed on the sur- 
face of the ground. 
Pitch. The direction (in azimuth and altitude) of the longest 

axis of an ore body. 
Prop. A piece of timber which prevents any rock in the roof from 

falling. 
Raise. A passage driven steeply upward from any portion of 

the mine. 
Roof. The top of a passageway or of a seam or bed. 
Room. A place other than a passageway from which material has 

been extracted. The term usually refers to bed deposits. 
Seam. A bed of mineral or a small vein. 
Shoot. A gate used for drawing off broken ore from a stope or 

raise. Also an ore body within a vein. 



MINING TRANSITS 287 

Slopes. Rooms formed by the excavation of ore above or below. 

a level, sometimes filled with broken ore or rock. 
Strike. The direction (bearing) of a horizontal line in the plane 

of a deposit. The strike is always at right angles to the dip. 
Stull. A piece of timber wedged in crosswise between the side 

walls of a passageway. 
Throw. The vertical distance between the planes representing 

two parts of the same vein which is divided by a fault. The 

term is used only in regard to nearly horizontal deposits. 
Tunnd. A horizontal working passageway from the surface to 

the mine. 
Vein. (Also lode, ledge, lead, etc.) A mineral body of the 

flattened shape. These terms are also applied to mineral 

bearing faults. 
Wall. The rock on each side of the mineral body. The upper 
• wall is called the "hanging wall," and the lower the "foot 

wall," 
Winze. A subsidiary shaft not starting from the surface. 

MINING 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 modern mining, all the 
accurate angle measurements are taken with a transit, the details 
being filled in with a miner's dial or other light compass instru- 
ment. Several forms of transit are designed for mining and moun- 
tain work. The essentials are lightness and adaptability for 
measuring accurate azimuths of nearly vertical or of very short 
sights. A high power telescope is more necessary than for ordi- 
nary surveying, as the small variation of the line of sight means a 
large error in azimuth .when sighting with high altitudes. 

With an ordinary transit one cannot take a downward sight 
more steeply inclined than 55 or 6o° to the horizon. For sight- 
ing larger angles of depression various devices have been used by 
which telescopic sight may be taken over the edge of the horizontal 
circle of the instrument. This is commonly done by attaching 



288 



MINING SURVEYING 



[Chap. XL 




Fig. 132. Mining Transit with Side Telescope. 

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



ECCENTRIC TELESCOPE 289 

an auxiliary telescope to the side or to the top of the ordinary 
engineers' transit so that the instrument will afford all the advan- 
tages of the ordinary transit and also make it possible to sight 
down a vertical shaft. 

316. Eccentric Telescopes. — Fig. 132 shows a mining 
transit in which the auxiliary telescope is attached to an end exten- 
sion of the horizontal axis. Such an instrument is known as a 
side telescope transit. The distance and direction between the 
centers of the main and auxiliary telescopes is called the eccen- 
tricity. In this case it is measured along the horizontal axis. 
Fig. 133 shows an eccentric bearing mining transit which has an 
extra pair of supports for the horizontal axis. The eccentricity in 
this case is also measured horizontally but at right angles to the 
horizontal axis. In the type of transit known as the top telescope 
transit the auxiliary telescope is mounted on the top of the main 
telescope. The eccentricity in this case is at right angles to the 
horizontal axis and to the main telescope. 

317. Correction for Eccentricity. — The eccentricity of a 
mining transit has to be allowed for when either of the auxiliary 
attachments is made use of. The correction may be made by 
regarding the line between the centers of the eccentric and main 
telescopes as one of the lines of the traverse. Instead of making this 
correction as a link of the traverse it may sometimes be more 
convenient to sight the auxiliary telescope to an auxiliary point 
which bears the same relation to the station required to be observed 
as the center of the auxiliary telescope does to the center of the 
main telescope. It is also possible to make a correction to apply 
to the bearing of the auxiliary telescope which renders it the 
same as the bearing from the main telescope would be if it were 
possible to make the observation with it. In some forms of 
instrument the correction may be eliminated by using the instru- 
ment in both the direct and reversed positions. 

318. Relative Merits of Attachments. — In comparing the 
relative merits of the various forms of attachment it must be 
remembered that the object to be accomplished is to transfer the 
meridian accurately from one station to another, these stations 
being close together in plan, and distant in elevation. 

The side telescope has the merit of being easy to work. Since 



290 MINING SURVEYING [Chap. XL 

this telescope is easily detached the transit need not be encumbered 
with it when the main telescope can be used, which is the case 
in the majority of the observations. When this attachment is 
used the effect of eccentricity in the measured azimuths is easily 
allowed for. Accuracy may be obtained by reversing the instru- 
ment and by leveling the horizontal axis each time by means of 
the striding level. 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. 

The eccentric bearing telescope, though not so convenient to 
use as either of the other forms of mining transit, has advantages 
for very accurate work, since the telescope is placed symmetrically 
with respect to the horizontal axis and its supports. It also has 
the advantage of greater rigidity and of being adjusted directly, 
like an ordinary transit, instead of indirectly. The correction for 
eccentricity is simple, and the telescope may be reversed in its 
horizontal bearings and used in connection with a striding level. 
It has, however, the disadvantage of being more bulky, which is 
objectionable in ordinary work, and also the disadvantage that 
in a dirty mine grit is liable to accumulate in the horizontal 
bearings while the telescope is being changed from one set of 
bearings to the other. 

With a top telescope the instrument cannot be reversed. With 
such a transit the correction for eccentricity is only applied to alti- 
tude readings. For rapidly conveying an azimuth where extreme 
accuracy is not paramount this type of attachment is useful. 

319. ADJUSTMENTS OF MINING TRANSITS.— It is assumed 
that all ordinary adjustments of the transit have been made. In 
addition the adjustment of the object slide (Art. 77, p. 60) is of 
unusual importance, because often in mining work the azimuth 
has to be transferred over very short sights. Exceptional care 
must be taken to get the horizontal axis of the telescope truly 
horizontal and the line of sight exactly perpendicular to it. 

The side telescope is generally adjusted 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 assumed that the instrument 



291 



3 
S 



q5 

3 2 



8tf 

| I 
5 2 

P 3 

r h 

».» 

IS 

12 W 

8 ?o 

% 5 

^ o 

H 

w 

r 

M 
c/j 

O 

o 
w 




292 MINING SURVEYING [Chap. XL 

maker has made the optical axis parallel to the axis of the tube, 
but the accuracy of this should be tested. 

Secondly, the line of sight is made parallel to that of the main 
telescope. It is first brought into a vertical plane parallel to the 
vertical plane of the main telescope by means of the adjusting 
screws on the vertical trivet plate of the side telescope, the sight of 
each telescope being taken to the same very distant object. If it 
is not convenient to sight on a distant object, a piece of paper 
with two vertical marks connected by a horizontal line may be 
used, the distance between the marks being equal to the distance 
between the telescopes. This piece of paper should be set at right 
angles to the line of sight and not too near the instrument. The 
vertical cross-hair of the main telescope is sighted at one point by 
means of the clamp and tangent screw of the plates and then the 
vertical cross-hair of the side telescope is sighted at the other point 
by means of the trivet plate adjustment on the side telescope. 

The side telescope and main telescope are then brought into 
the same plane at right angles to the vertical plane as follows: — 
the horizontal cross-hair of the main telescope is sighted at some 
point, preferably a distant one; then the horizontal cross-hair of 
the side telescope is sighted at the same point by means of the 
tangent screws on the side telescope. 

As the adjustment of the side telescope is not direct, but is made 
by comparison with the central telescope, the instrument should 
be reversed when accurate observations are required. This also 
eliminates the correction for azimuth due to eccentricity. 

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 
inverted the striding level is again used in both positions and two 
more azimuths are read. The mean of the two readings of each 
pair gives two mean lines of sight which are symmetrically related 
to the vertical plane passing through the two station points and 
the correct azimuth reading is therefore the mean of these two 
azimuths. There is a small correction to be applied to the mean 
of the altitude readings but this is not usually of any importance. 

The top telescope is adjusted in much the same manner as 
the side telescope. 



ORDINARY TRANSIT USED IN MINES 293 

In the eccentric bearing telescope no extra adjustment is neces- 
sary other than that of the bearings of the horizontal axis when 
the telescope is used in the eccentric position. This is done with 
the striding level, which is also used in taking accurate sights, the 
telescope being reversed by lifting out of the horizontal bearings 
and turning the horizontal axis end for end. As no attempt is 
made to adjust the two pairs of horizontal bearings to be abso- 
lutely parallel, the foresight and backsight in ordinary work 
should both be taken with the horizontal axis in the same pair of 
bearings. 

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. Combined 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 obser- 
vations with the main telescope, and the surveyor is advised not 
to get any such complex attachment for mining work. 

322. 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 accom- 
plish the same result as an eccentric telescope instrument 
(Art. 316). The instrument, firmly screwed on to the tripod, is 
inclined over the shaft at an angle just sufficient for the Kne 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 respec- 



294 MINING SURVEYING [Chap. XL 

tively 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 reversing 
the striding level each time and taking the mean position of the 
four points so set. 

An attachment which is very necessary in performing some of 
the work required of mining transits is the diagonal or prismatic 
eyepiece. This makes it possible to take any sight whatever 
above the horizon, and being a convenient instrument to use and 
not requiring any adjustment, it should be carried by every mine 
surveyor. 

323. 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 mining 
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 
horizon, and a sighting clinometer attachment for measuring 
vertical 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. 

The circumferenter is a dial with an additional revolving plate 
which makes it possible to measure angles independently of the 
compass needle. Other improvements have so modified the 
instrument that it differs little from a light mining transit with an 
auxiliary top telescope instead of the main telescope. 



MAGNETIC SURVEYING 295 

A mounted compass is more accurate than one simply held in 
the hand, but any hand compass may always be mounted when 
the conditions permit. Perhaps the best form of compass for 
details and reconnoissance work is the Brunton mining dial. 
The observer looks down on this instrument and the line of sight 
is reflected upward toward him by a hinged mirror so that the 
object and the compass box are seen simultaneously. A clinom- 
eter is attached, which is suitable either for measuring dips or 
sighting inclinations. 

324. MAGNETIC SURVEYING. — 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. More accurate methods have 
been devised by Swedish engineers who make charts of magnetic 
intensities. 

The latter method depends in principle on the measurement of 
the horizontal and vertical components of magnetic intensity. 
The horizontal intensity is found by noting the angle which a 
horizontal compass needle is deflected by the introduction of a 
magnet placed in the swinging plane of the compass needle and 
a fixed distance from it measured at right angles to the position of 
rest of the needle before the magnet is introduced. This is com- 
pared with the deflection produced by the magnet alone when 
there is no ore body affecting the needle. The vertical intensity is 
found by the deflection of a needle swinging in a vertical magnetic 
east and west plane. The deflection is against a gravity righting 
moment, the needle being previously weighted so as to balance 
the vertical component of the earth's magnetism. 

More detail may be obtained by rendering both the compass 
and dipping needle astatic, i.e., by placing them in a framework 
about which permanent magnets are fixed so as to just neutralize 
the ordinary magnetism of the earth. The lines of force due 
simply to the magnetic ore body may then be mapped together 
with the intensities of these forces as determined by the principles 
previously cited. We may generally take it for granted that the 
magnetic axis of the ore body is parallel to the earth's magnetic 
lines of force. As we probably also have some general notion of 
the strike and dip of the ore shoots it will be possible in many 



296 MINING SURVEYING [Chap. XL 

cases to determine approximately the position of the ore body even 
though it is covered to a considerable depth. 

The above methods may be used in a limited way underground. 
Sometimes an artificial ore body made of a large permanent mag- 
net is used for determining the length and direction of a required 
connection where traversing is not entirely to be relied upon. 



UNDERGROUND SURVEYING. 

325. 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 accu- 
rately 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 riot 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 of meridian due to the line of sight not 
being perpendicular to the horizontal axis are eliminated if the 
readings are made with the telescope in the same position at 
both sights, whereas errors of meridian due to the horizontal axis 
not being perpendicular to the vertical axis are eliminated 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. Obvi- 
ously both are eliminated when the direct and reversed position 
are used at both the top and the bottom of the shaft and the mean 
readings taken in each case. 

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 bottom 



UNDERGROUND SURVEYING 297 

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. 

The great importance of having the horizontal axis truly 
horizontal should not be overlooked and elimination of the 
errors by reversing and taking mean readings should not be 
relied upon without the aid of the sensitive striding level which 
also eliminates any error due to the vertical axis not being truly 
vertical. 

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 in the bottom of the mine. Errors due to a slight 
inclination of the horizontal axis are not important when this 
method is used for a vertical shaft, and for that reason it is 
also useful in cases where a sensitive striding level is not to be 
had. The effect of a slight inclination of the horizontal axis is 
simply to shift the line slightly to one side but parallel to the 
true position. 

When no extra telescope or eccentric bearings are to be had, 
an ordinary transit with a prismatic eyepiece attached may be 
used to drop the meridian down a vertical or highly inclined 
shaft, provided it is not so wet as to prevent sighting upward 
from below. To accomplish this a thin wire is stretched hori- 
zontally across the top of the shaft at a known azimuth; the 
wire should be prolonged one or both ways in order to give 
a good base-line. Two points may be fixed at the top of the 
shaft if preferred. The transit is then set up on the bottom 
and it is brought by trial into the same vertical plane as the 
wire. The striding level is used in both positions and the transit 
is used in both the direct and reversed positions to eliminate 
errors. 

It is to be noted that in mining and fountain work slight errors 
of altitude measurements occur in sighting up steep inclines 
owing to the refraction of the atmosphere, but this is so slight 
as not to affect the transfer of the meridian at all. 



298 



MINING SURVEYING 



[Chap. XL 



326. Plumbing the Meridian down a Shaft. — To 

the mine surveyor the plumb-line is an instrument of precision, 
excelling even the transit, and under most conditions, the work 
of transferring the meridian down a mine can be accomplished 
more accurately by means of the plumb-line than by any other 
method accessible to the surveyor. 

The method usually followed is to suspend two bobs from the 
staging above the mine so that a horizontal line in their plane can 
be sighted both from above and from below. The transit is 
set up both above and below on this line and thus an azimuth 
connection is established between the surface and the workings. 
Sometimes a much longer base-line than can be directly sighted 
can be obtained by plumbing down at the corners 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 a minimum error to the 
meridian. All kinds of drafts in 
the shaft should be avoided dur- 
ing the alignment at the bottom. 
No cages or skips should be run 
and drafts from the passages 
leading to the shaft 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 not weigh less than five pounds 
and should be heavier for a deep shaft. A good working weight 
is one-third of the load at which the wire will break. 

The plumb-bob is hung in a bucket or a barrel of viscous 
liquid so as to bring it to a standstill in the shortest possible 
time. The shape of the plumb- bob is of importance in this respect 




Fig. 184. Triangulating at the 
Bottom of Shaft. 



PLUMBING DOWN A SHAFT 



299 




Fig. 186. Type 
of Plumb-Bob 
Used in Plumb- 
ing Meridian 
Down a Shaft. 



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 
of the vibrations of the plumb-bob decreases in 
a fixed ratio with equal increments of time, and 
the viscosity of the fluid should be such as to 
make each oscillation, say, about one-quarter 
of the preceding. The ratio of decrease during equal incre- 
ments of time is nearly 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 roughly as the square root of 
the length of the plumb-line, the same as for a pendulum swing- 
ing in air. 

Sometimes it seems impossible to stop all oscillation of the 
pendulum. This may be on account of the wabbling of the 
vertical currents of air in the shaft. Perfect rest however is not 
necessary. The telescope may be directed at a mean position 
by placing a graduated scale immediately behind the wire and 
taking readings on it through the telescope when the wire reaches 
the limits of its oscillation. 

If the shaft is wet the vessel should be covered with a sloping 
lid having a hole in it of an inch or so in diameter so that the 
wire can swing freely. In order to obtain as long a base-line as 
possible the wire should be hung as near to the casing of the shaft 



3oo 



MINING SURVEYING 



[Chap. XI 



as is consistent with the precaution that it shall be perfectly 
plumb. It should be carefully examined along all its length to 
make sure that there are no obstacles to interfere with it. In 
some cases it may be sufficient 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, 
as this gives the best test of the accuracy of the plumbing opera- 
tion. If four lines, one in each corner 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 meridian 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. 317). 

327. Transferring a Meridian into a Mine when 
THERE ARE TWO SHAFTS. — The above methods presuppose 
that the mine has so far been opened only by one shaft. If 
there is a second shaft or an adit, it is, of course, only necessary 




UNDERGROUND TRAVERSE 



PLAN 

Fig. 136. 



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 ABCD is run 



UNDERGROUND TRAVERSES 301 

out on the surface to connect the two shafts at A and D. The 
points A and D are plumbed down the shafts and the correspond- 
ing points A' and D' established at the bottom. An underground 
traverse A'G'F'E'D' is then run out. In the surface traverse 
the length and azimuth of AD and in the underground traverse 
the length and azimuth of A'D' are missing. The horizontal 
length and azimuth of each of these lines can be determined from 
their respective traverses as explained in Art. 397, p. 366. The sur- 
face traverse is referred to the true meridian, and, since nothing is 
yet known in regard to the direction of the meridian in the mine, 
the underground traverse is referred fo 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. 

328. UNDERGROUND TRAVERSES. — Surveying in a mine is 
necessarily a process of traversing, for only the working passages 
are available for lines of survey. The line of traverse is not always 
in the center of the passage but is often varied from it in order 
that the longest possible sight may be taken. In the tortuous 
passages of a mine it is frequently necessary to take very short 
sights on the main traverse and since the azimuth is transferred to 
distant connections through these short lines great care should be 
exercised; and instrumental errors should be eliminated by revers- 
ing the telescope and using the mean of the two results. The 
positions of the walls of the passages are noted as the work pro- 
ceeds and are sketched in approximately on the plot. After the 
main traverses have been run, the surface boundaries, if touched, 
may be accurately established underground 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. 



302 MINING SURVEYING (Chap. XL 

A speedy and convenient manner of running an underground 
traverse is to use three tripods having leveling heads and centering 
plates like those of the transit. The transit fits on to any of 
these heads and while it is attached to one of them the other two 
are surmounted by lamp targets in which the sighting center has 
exactly the same position as the sighting center of the transit 
would have if set on the same tripod. These tripods are placed 
vertically over or under the stations and the transit is attached to 
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 
telescope tube in front of the hairs, preferably by a tube reflector 
(Fig. 133) in front of the object glass. 

Sometimes a brass lamp with a small central flame, called a 
plummet-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. 

329. Establishing Station Points. — The station point is 
established either on the floor or on 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. The 
station point, however, should be so placed that it will be pos- 
sible to set the instrument either over or under it. In a vein 
mine a timber in the roof, especially a stull, is often more per- 
manent 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, and the roof liable to 
cave, 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 



NOTES OF MINE SURVEY 



302a 



Survey of Bear Creek Mine, West Bowlder, Montana. 



True 
Bearing. 



Distance 



Vert. 
Angle. 



Back, 
sight on 



May 17, 1906. 
Party : Keene, Chase, Holbrook. 



N88°io'E 



N 2°io'W 



N89°io'E 

N8 9 °io' E 
S89°i5'E 
S 88 i 2 ' E 

S 9 55' W 
S S8°i$' E 

N 2°io'W 

N Ss°S2' E 
S 85 46' W 
N 88°20 / E 

N 88°20 / E 
S Sg°os' E 
S 3 i 4 'E 
S 86°io' W 



650.8 



117. 4 



N87°45 / E 230.8 
N 2°io'W 112. 6 



75o 

'53. 5 
105.7 
162. 1 

92.3 
15.9 

115. 8 
167.4 
196.0 
138.0 

106.3 
176.9 

"3-7 
216.8 



- i°i7' 
-8o°io' 

+ o°45' 

- 8o°io' 

+ 0053' 



Sun 
Sun 

o 

o 

101 



+ o° 53 ' 


IOI 


+ o° 39 ' 


102 


+ o°48' 


103 


+88°25 / 


104 


level 


Compass 


-So !© 7 


IOI 


+ o° 5 o' 


IOI 


+ o°47' 


IOI 


+ o°44 / 


201 


+ o°44' 


201 


+ o° 4 2' 


202 


+ 77°i9' 


202 


+ o° 4 8' 


201 



To top of air shaft extending 
to 1 st level ; C. of S. edge 
of air shaft, 4' x 4'. Sta. 2. 

To Sta. 101 at 1st level. 
Line runs 3' from S. side 
and 4.5' from W. side of 
shaft ; shaft 8' X 8'. 

To Sta. 102 in 1st level. 

To Sta. 201 in 2nd level. 

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

To Sta. 103 in 1st level. 

To Sta. 104 in 1st level. 

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

To top of air shaft, Sta.' 2. 

To Sta. 106 at breast of 1st 
level. 

To Sta. 301 in 3rd level. 

To Sta. 202 in 2nd level. 

To Sta. 205 in 2nd level. 

To C. of raise extending to 
1st and 3rd levels, 4' x 4' 
Sta. 208. 

To Sta. 203 in 2nd level. 

To Sta. 204 at breast of 2nd 

level. 
To Sta. 107. 

To Sta. 206 in 2nd level. 



302b MINING SURVEYING [Chap. XI 

Survey of Bear Creek Mine, West Bowlder, Montana. (Cont'd.) 



Sta. 


True 
Bearing. 


Distance. 


Vert. 

Angle. 


Back, 
sight on 




206 


S87 I4' W 


118.0 


+ o° 4 l' 


205 


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


206 


S 87°I4' W 


152.0 


+ o° 4 i' 


205 


To Sta. 207 at breast of 2nd 
level. 


301 


N 86°2o' E 

* 


304.0 


+ 0*46' 


20I 


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


3Pi 


N 86°2o' E 


316.0 


+ o° 4 6' 


20I 


To Sta. 303 in 3rd level. 


301 


S 86V' W 


I9S-0 


+ o° 5 o' 


20I 


To Sta. 305 in 3rd level. 


301 


N 2°io' W 


116. 8 


-So !©' 


20I 


To Sta. 401 at 4th level. 


302 


S 5°35'E 


116. 5 


+ 78°29' 


301 ' 


To Sta. 208. 


303 


S 89°o7 / E 


289.0 


+ o°39' 


3°« 


To Sta. 304 at breast of 3rd 
level. 


305 


S 88°52' W 


186.2 


+ o° 4 6 / 


301 


To Sta. 306 in 3rd level. 


306 


S 89° 4 8' W 


150 


+ o°43' 


305 


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


307 


S 2° 4 i / E 


120.5 


+ 71°"' 


306 


To Sta. 209. 


3<>7 


S 89°45 / W 


10.9 


level 


Compass 


To Sta. 308 at breast of 3rd 
level. 


401 


N Ss°4&' E 


219.7 


+ o° 4 8' 


301 


To Sta. 402 in 4th level. 


401 


S 88°io' W 


116. 4 


+ 0*52' 


301 


To Sta. 406 at breast of 4th 
level. 


401 


N 2 °io / W 


49-7 


-8o°io' 


301 


To bottom of shaft, 3' from 
S. side and 4' from E side. 
Shaft 8' x 8'. 


402 


N 89°56 / E 


85.0 


+ o°45' 


401 


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


402 


N 8 9 °s6 / E 


92.6 


+ o°4S' 


401 


To Sta. 404 in 4th level. 


403 


S o°o6'E 


116. 2 


+ 8i 4 6' 


402 


To. Sta. 302. 


404 


S 87°2o / E 


217.6 


+ o°43' 


402 


To Sta. 405 at breast of 4 ih 
level. 



PLOTTING A MINE TRAVERSE 303 

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 traverse 
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. 

330. Underground Measurements. — 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 convenient. Great care should be 
taken not to let cars run over the tape as this will invariably 
snap it. Some mine waters are also very severe on tapes and it 
is often impossible to keep them dry. 

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 as shown on pp. 302a-b. 

The different station points of this survey are numbered for 
identification only, their numbers bearing no relation to the 
distances between them. For convenience the stations on the 
first level are numbered 101, 102, etc.; on the second level 201, 
202, and so on. In larger and more complex mines the system 
of numbering and lettering stations is carried out still farther. 

332. Plotting a Mine Traverse. — This survey can be plotted 
by computing three sets of coordinates which give all the data 
needed for showing the mine in plan, longitudinal section, and 
transverse section. If the three coordinate planes are the meri- 
dian plane, the east and west plane, and the horizontal plane, then 
the vertical and horizontal distances from each slope measure- 
ment are first obtained by multiplying the measured distance by 
the sine and the cosine respectively of their vertical angles. The 
vertical distance is the difference in elevation between the two 
points. From the horizontal projection and the azimuth, or 
bearing, the latitude and departure of the course can be computed 
as usual (Art. 384, p. 352). A plot of these notes will be seen in 

Fig- 137- 

It is assumed in plotting these notes that all the transit lines 



304 MINING SURVEYING * [Chap. XL 

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 
simplicity. 

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 advan- 
tage 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. Ordinarily this 
rod is five or six feet long, but may be shorter for use in small 
coal seams. 

334. MINE MAPS AND CHARTS. — The galleries of a mine 
are often so nearly over one another that confusion is liable to 
arise in charting, unless some special means of identifying them 
is employed. As these galleries or working passages appertain 
to definite levels or strata, a different color may be assigned to 
each level or stratum and adhered to throughout. The lines of 
survey are in a colored ink and the passages or workings are of a 
faint 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 posi- 
tion 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. 



MINE MAPS 305 

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 
with other particulars, such as thickness of waste or value of ore. 
(See Fig. 140a.) 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 con- 
forming particularly to any one plane, the above system does not 
apply and some other method must be devised by the surveyor. 

The best way of estimating amounts not mined is to sketch 
their probable extent on such a chart from the data available and 
to make use of the area and thickness method as suggested in the 
preceding paragraph. Ground explored by boreholes but not 
opened by headings may be best shown by plotting in plan the 
positions of both of the walls, where struck in the boreholes, 
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 MIKING 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 plumbed 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. 



306 



MINING SURVEYING 



[Chap. XL 



Hang the line an even fraction of an inch each way from the true 
position of the corners 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. UNDERGROUND SURVEYING PROBLEMS. — In the prac- 
tice of mine surveying, problems are constantly arising which 
tax the ability and ingenuity of the surveyor, although the actual 
solution of most of them is quite simple. A few of the common 
problems met with in such work are given below. 

336a. To Find an Ore Shoot by Driving a Level. — The pitch 
being given by its altitude and azimuth, this serves as a course 
from any point on the ore shoot whose coordinates are known. 
The difference in elevation between this point and that of the 
level to be driven is divided by the sine of the altitude (or vertical 
angle) of the ore shoot, which gives the slope distance along the ore 
shoot. The horizontal coordinates of the point where the level 
will intersect the ore shoot may then be calculated. 

337* Vertical Angle Correction 
for Eccentricity of the Top Tele- 
scope. — As has been stated in Art. 
317, all vertical angles taken by 
means of the top telescope must be 
corrected for the eccentricity of this 
attachment. In Fig. 138 the vertical 
angle has been taken to a point C in 
the bottom of a shaft. The distance 
AC was measured, A being the hori- 
zontal axis of the main telescope. 
Since the transit is set up over a sur- 
face station at £, the distances de- 
sired are DC and AD. HB and 
H'A are both horizontal, 




Fig. 188. 



then V 9 = V - ACB 



UNDERGROUND SURVEYING PROBLEMS 



307 



where V is the angle as measured by the top telescope, v is the 
angular correction for eccentricity, and V is the corrected angle. 

AB __ Distance between telescopes 
AC Distance measured 



Since sin v = 



we may easily construct a table giving the value of v for any 
measured distance. 

The vertical and horizontal components of AC are then 

AD - ACsin V - ACsm (F- v) 
and DC = AC cos V - AC cos (F- v) 

Had the measured distance been BC instead of A C we should 
have 

DC - CF + FD = BCcos V+ ABsin V 
similarly AD - 5C sin 7- AB cos F 

the last term in each equation representing a component of the 
extra link in the traverse introduced by 
the eccentricity as explained in Art. 317, 
p. 289. 

The horizontal angle correction for -_i 
the side telescope is derived by similar 
reasoning. 

* 337a. 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 V + AB 
AD - BF = BC sin V. 




V7777 



Fig. 189. 



336. 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 under- 
ground working in the drift H'J' to a point K' vertically under 
the boundary line. The surface traverse BADCS is run out, 



308 MINING SURVEYING [Chap. XL 

point S is plumbed down to S', and the meridian transferred 
into the mine. Then the underground traverse S'E'F'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 AW and its bearing can be calculated as described 
in Art. 398, p. 367. In the horizontal triangle AWK', AH' 
and all the angles being known, the line WK! can readily be 
computed. If the drift H'J' is not level the distance from W 
along the drift to the boundary plane will be equal to the horizontal 
distance H'K' divided by the cosine of the vertical angle. 

339. 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 A 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 with 
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 
A and B together with the length of the horizontal projection of 
AB will give the vertical angle; from these data the direction 
and distance between the points A and B can be computed. 

340. Plotting Geological Data. — A complete survey should 
show the boundaries of the various formations, the planes of 
bedding and the foliation, the fault planes with their displace- 



PLOTTING GEOLOGICAL DATA 309 

ments, and all veins and dikes encountered. These data could of 
course be shown to some extent by three coordinate planes as in 
solid geometry, but it is usually better to plot the geological data 
taken in each level. Any inclined rock surface is then repre- 
sented by a series of contour lines corresponding to the levels 
from which the information is actually obtained. Strikes, dips, 
and intersections may then be determined by use of a protractor, 
a scale and a table of cotangents, or a cotangent scale. (See 
Fig. 140a.) 

In a metal mine a plan of each level when filled in with all the 
geological data will have as much detail as can conveniently be 
shown. It is usual in such cases to make a geological plan of each 
level separately on thin tracing paper so that any two or possibly 
three consecutive ones may be superimposed. The particular 
position, strike, and dip of any shoot or surface may then be as 
easily found as though they were all plotted on the same piece of 
paper. 

340a. Location of Ore Shoot Displaced by Fault. — A fault 
is a displacement along a plane or approximately plane surface. 
Such a displacement may of course be divided into two direct 
components and a rotation. 

Three distinct measurements have to be made to determine 
these three variables. A fault wholly in a homogeneous forma- 
tion supplies none of these measurements. A fault in a foliated 
formation can in general have only its rotation determined. A 
fault cutting a homogeneous vein may supply one other measure- 
ment but such a vein must have shoots or some other equally 
distinctive feature in order that all three measurements may be 
made. 

In a general way only one of these measurements appears on 
the plan of one individual level but two of them may appear by 
comparison on the plans of two levels by the tracing paper method 
previously mentioned and all three may be shown by a similar 
comparison of three levels. The components of displacement 
need not be always determined in order to plan for future excava- 
tions. It will in general be sufficient to use a scale and a straight- 
edge to determine the position of an ore shoot on the next level 
to be opened up. Fig. 140a shows how these components may be 



3°9a 



MINING SURVEYING 



[Chap. XL 



PLAN OF A VEIN. 

Cut by a NORMAL FAULT having no rotary displacement-. 
To illustrate the interpretation of Geological Data . 



100- 
110- 
120 

8 uo 
"-mo- 

1 '»■ 

> IW- 

«*.»I80- 



5210 
220- 
230- 
240- 
250- 
2€0- 
270- 
280- 
280- 
300- 




.JtL. l \ \/ B Mean Course \ of Vein 



LEVEL 



fe^ 



Natural 
y 30 



Scale:- 3d feet to one inch 

T ■ T 1 T 



T » » T 



80' 



"73 s 70*"^ •3' CO* 33 1 

Cotangent Scale, for Measurement of pips. 



Fic 140a. 









309b 




ftc. 



J 4oa. 



310 MINING SURVEYING [Chap. XL 

determined and made use of, however, in cases of necessity. 
The left portion of the figure shows a plan of a vein the two parts 
of which have been separated by a fault. The directions of the 
horizontal lines marked "mean course of vein" and of the horizon- 
tal lines on the fault planes have been determined at each level in 
the mine. The line AB represents the horizontal distance between 
the two portions of the vein, measured along the fault plane, and 
its length may be scaled directly from the plan. The dips of the 
vein and of the fault plane are indicated by arrows. The dis- 
tances CD, EF, and GH, when scaled off with the cotangent 
scale, will give the angle of dip of the vein, fault, and ore shoot 
respectively, since these distances are taken between points which 
differ just ioo feet in elevation. The vertical distance between 
the two portions of the vein may be measured by scaling JK with 
the vertical scale shown at the left of the figure. The "feet" on 
this scale are not shown their true length but are reduced in the 
ratio of i : cot (dip of vein), i.e., if a right triangle were constructed 
having i foot as its vertical side and one of these scale units as its 
horizontal side the hypotenuse would have the same slope as the 
vein. This scale cannot be used, of course, in any other direc- 
tion than perpendicular to the strike of the vein. 

On the right of the figure are shown a view looking perpendicular 
to the plane of the vein and one perpendicular to the fault plane. 
The former shows the method of recording the quantities of ore 
removed and the results of assays. The latter shows the actual 
displacement, the point L having been at M before the faulting took 
place. 

Where only one level is opened up the dips of faults and veins 
must be determined as accurately as the short distance between 
roof and floor affords. This determination can only be approxi- 
mate as local irregularities of strike and dip are very numerous. 
Where there are several consecutive beds which have been explored 
either by headings or boreholes it is best to represent each bed by 
contours on a separate plan; these can be plotted on tracing paper 
if it is necessary to examine the relations of consecutive beds to 
one another or to the surface. Fault planes cutting folded beds 
will show as curved lines but can be readily plotted or examined 
as to strike, dip, and displacement from the plan of only one 



HYDRAULIC SURVEYING 3" 

curved or folded bed, by the aid of a section on the plane of the 
fault which can be plotted from the data on this plan. 

341. HYDRAULIC SURVEYING FOR MINES. — The miner's 
unit for measuring water is the miner 1 s inch. By an inch of 
water was originally meant such continuous flow as will go 
through a one inch square hole, the head of water behind it 
being usually six to nine inches. This very loose definition has 
been done away with but the name still applies, being defined 
more exactly as ninety cubic feet of water per hour (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 
is multiplied by the area of the corresponding portion of the 



312 MINING SURVEYING [Chap. XI. 

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 roughly twice 
the percentage of the grade of the channel. For details regard- 
ing stream measurements, see Chapter IX of Volume II. 

The surveying and staking out of mining ditches, flumes, and 
pipe lines follow the general practice for this work in other fields 
of engineering. 

342. Testing for Ore by Electric Currents. — Methods of 
testing the earth for ores by means of electrical currents and 
waves are being experimented upon, and the working out, record- 
ing, and plotting of the results are likely to become a part of the 
mine surveyor's work. 

SURFACE 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 with a steel wire tape, 300 to 500 feet 
long and marked every 10 feet (or 20 feet) so as to be used with 
a short auxiliary steel ribbon tape which is divided to hun- 
dredths of a foot. The measurements are taken from the center 
of the instrument to the object at which it is pointed, care being 
taken not to overstretch the tape nor to kink it. The most 
accurate work is done by stretching the tape with a tension 
handle (a spring balance) which can be attached by a clamp to 
any part of the tape. Where it is feasible, just enough tension 
is given so that the stretch of the tape compensates for the short- 
age due to sag. In many cases assistants will have to hold the 
middle point or the points at one-third and two-thirds the length 
of the tape up to the line of sight, giving at the same time enough 
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. 



SURFACE SURVEYING 313 

station point, and then to sight an equal height on a graduated 
staff held on the back and forward stations, recording the azimuth, 
vertical angle, and distance. Another method is to sight and 
measure to targets set at a fixed height above the stations, record- 
ing the vertical angle only at alternate stations. If the vertical 
angles are read at every station there will 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 employed, each sighting 
to the other's telescope and measuring the distances between 
them. Each of these methods has its advantages and disadvan- 
tages, 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 
vertical angles and in some cases twice as many distance read- 
ings 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 iTTtftftf or smaller, and therefore such 
instruments as the hand compass, clinometer, and aneroid baro- 
meter can be used. With such instruments one man can do the 
entire work. The plane table cannot be used to advantage in 
mountain or mine surveying, but photographic surveying may 
often prove useful in filling in details of topography. 

344. MINE BOUNDARIES.— APPROPRIATIONS UNDER UNITED 
STATES LAWS.* — In most countries mineral rights are defined 

* For further information with regard to this subject see the Manual of 
Instructions for the Survey of the Mineral Land of the United States, issued 
by the Commissioner of the General Land Office, Washington, D. C. 



314 MINING SURVEYING [Chap. XI. 

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 

NW NE 

J- hL £ Z 

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 need not of necessity be 
parallel and are not usually run out on the ground. 

A monument with explanations is placed at each of the seven 
points marked. If in a timbered country, the lines run should 
be blazed, and squared trees may be used as monuments. At 
D (Fig. 141) a location notice is posted, defining the boundaries 
of the claim and containing such explanation as would identify 
the claims in case of dispute. The miner usually makes the loca- 
tion survey himself, using approximate courses and distances. 
There is legally no objection to this work being done roughly, 
but when a patent survey comes to be made, neither the dimen- 
sions specified in the location notice nor the limits of the claim 
as marked off on the ground can be exceeded. So when the 
location survey is roughly made certain "fractions " of ground 



PATENT SURVEYING 31$ 

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; this may save 
much time and considerably simplify the work, especially in 
thickly timbered regions (Art. 329, p. 30a). 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 conditions 
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 effort is sometimes made to 
surround a valuable claim with others, thus forming a "group." 
The more valuable claim is then protected as regards all "extra- 
lateral rights." 

Flat deposits, such as coal and placer, are subject only to 
vertical bounding planes, and, provided the boundaries are 
approximately north and south lines, marked plainly on the 
ground, and the legal area is 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 or state laws can regulate the size, provided the 20 acre 
limit per claim is not exceeded. The coal lands law is made sub- 
ject to the general system of public land surveys for agricultural 
lands. (See Chapter V.) 

345. SURVEYING for Patent. — The surveying of claims 
for patent from the United States Government can only be 
obtained by those who have received appointment of United 
States Deputy Mineral Surveyor and they must have an order 
from the Surveyor General of the state or territory in which the 
claims are located before making any such survey. 

In 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 corners and other boundary marks are computed and 



316 MINING SURVEYING [Chap. XL 

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 corners 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 corners 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 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 dishonest work. 
Patented claims may overlap, and in fact do, in all mining dis- 
tricts, 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 sur- 
vey 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- 
mation and the position of any particular body or stratum struck 



ECONOMIC PRINCIPLES 3 17 

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 VEIN. — 
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 in azimuth. 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, a small plane table 
may be improvised so that all points in the plane of the deposit 
can be conveniently sighted, the plane table having been brought 
into the plane of the vein. 

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 be neither too careless nor too 
exact. He will not, for example, close down the mine in order 
to carry a meridian into it when no important connections are 
needed, and there are no boundary disputes. On the other 
hand, in some cases accuracy of a survey is of such prime impor- 



3i8 



MINING SURVEYING 



[Chap. XL 



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 results of known 
accuracy than to go over 
the lines several times 
with varying conditions, 
but this is not always 
good economy, especially 
in such work as plumb- 
ing a shaft which neces- 
sitates a temporary stop- 
page of all hoisting opera- 
tions. Often the con- 
trolling error will be due 
to the plumbing of the 
meridian. In Fig. 142, D 
is the horizontal distance 
in a straight line from 
the shaft to the connec- 
tion, d is the distance be- 
tween the plumb-lines, 
and e its error as ascertained by measuring or other means. 

The controlling error at the connection is E = e -j and is in a 

direction perpendicular to that in which D was measured on the 
chart. This is obvious, for the surveys of the galleries are con- 
sidered accurate, the error being one of relative rotation around 
the shaft as an axis. Where there are many angular errors of 
the same degree of magnitude, such as occur when a number of 
short sights are included in the traverse, the distances may be 
measured from these short lines to the connection and their 
respective errors E v JB„ £ 8 , etc. found at these localities. These 
are then resolved according to their respective latitudes and 
departures into S„ S 8 , S 8 , etc., and W v W v W v etc. The greatest 




Shaft 



ECONOMIC PRINCIPLES 319 

possible error is then S t + S 2 + S 9 + etc. to the north or south 
and W x + W 2 + W 9 + etc. to the east or west, these summa- 
tions being made without regard to any sign. 

Likewise the mean probable error is y/Sf + 5 a 2 + SJ + etc. 
to the north or south and Vw* + Wf + Wf + etc. to the east 
or west. Errors due to the measurement of distances are not 
likely to be great, but if necessary these may be divided into 
latitudes and departures directly and compounded 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 under- 
stand their instructions; for many mining men 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 -f 2 10'; thence azimuth 62 42', 
slope distance 278.5 ft., vertical angle -f 2 18' to breast. From the same monu- 
ment a line is run on the surface, azimuth 98 33', slope distance 318.5 ft., vertical 
angle — 3 22'; thence azimuth 38 02', slope distance 647 ft., vertical angle -f 14 13' 
to the center of a vertical shaft. How deep must the shaft be to meet a connect- 
ing drift run on a grade of -f 2.4 % from the breast of tunnel, and what is the 
slope length and azimuth of this drift? 

2. The strike of a certain vein at point of outcrop is N 43 E and the dip is 
71 5c/, 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 + 3.8% until it intersects the vein. 



320 MINING SURVEYING [Chap. XL 

(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 strike 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. 1 a horizontal traverse was run in the 
mine 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 strike of 6o° (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 11 6° 20', dis- 
tance 41 1 ft. (horizontal) to a point A from which the center of shaft No. 1 is sighted 
at azimuth 7i°3o'. 

(a) How much deeper will shaft No. 2 have to be sunk to reach the vein ? 

(b) What is the true strike of the vein? 

5. A vertical winze has been sunk below the level of a tunnel. It is desired 
to sink a vertical shaft from the surface to connect with the winze. The monu- 
ment X is established at the mouth of the tunnel and the monument Y is near the 
site of the proposed shaft. Y bears S 88° 58' 56" W, 896.796 ft. from X. The 
following are the notes of the survey connecting X and the winze corners A, B, C, 
and £>: — 

Station. Mean Deflection. Horizontal Distance. Station. 

X o° 00' 896.796 Y 

403.080 1 

587.208 2 

2 32 23' 43" L 67.000 3 

3 54° 43' 47" R 44.803 4 

4 39° 5i' 57" R 41.075 5 



45° 05' 34" 


R 


74 05' 06" 


L 


32 23' 43" 


L 


54° 43' 47" 


R 


39° 5i' 57" 


R 


31° 10' 10" 


R 


31° ic/ 10" 


R 


3i° 43' 4o" 


R 


24 02' 40" 


R 



5 31° 10' 10" R 19-573 Cor. A 

27.240 Cor. B 

21.477 Cor. C 

25- 773 Cor. D 

Required the location of the shaft corners on the surface. 

6. From a monument M at the mouth of a tunnel a traverse is run in the tunnel, 
azimuth 20 35', distance 352 ft., vertical angle + i° to point A; thence azimuth 61 °, 
distance 528 ft, vertical angle + o° 40' to point B at the breast of the tunnel. 
From M a surface traverse is run, azimuth 25 io*, distance 578 ft., vertical angle 
+ 4 25' to point C; tbence azimuth n°, distance 407 ft., vertical angle 4- i4°ao* 
to point D, which is the center of a vertical shaft 120 ft. deep. Find the length 
and grade of a connecting incline from the bottom of the shaft to the breast of the 
tunnel. 

7. Assuming the transit to be in perfect adjustment what is the error in hori- 
zontal angle in sighting down a 500-ft. shaft, 5 ft. in breadth, when the telescope 
cannot be sighted closer than 3 seconds along the inclined line ? 



PART III. 
COMPUTATIONS. 



3" 



PART III. 

COMPUTATIONS. 

. CHAPTER XH. 

GENERAL PRINCIPLES. — MISCELLANEOUS PROBLEMS.— 
EARTHWORK COMPUTATIONS. 

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- f 
ing Methods it has been pointed out that in all surveys there 
should be a proper relation between the precision of measure- 
ment of the angles and distances. To secure final results to any 
given degree of precision, the measurements in the field must be 
taken with sufficient precision to yield such results. In com- 
puting from a given set of field notes the surveyor should first 
determine how many places of figures he should use in the com- 
putations, the aim being to obtain all the accuracy which the 
field measurements 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 Macmillan & Co., New York. 

3 2 3 



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 im^Tnnr as large the result would have been 47. 

Again, if a series of rod-readings are taken on different points 
to thousandths of a foot and three of the readings are 4.876, 
5.106, and 4.000 it is evident that each of these readings contains 
•four significant figures; if each of them is multiplied by 1.246 the 
respective results are 6.075, 6.362, and 4.984. But had the 
results been measured to the nearest tenth of a foot and found to 
be 4.9, 5.1, and 4.0 these values when multiplied by 1.246 should 
appear as 6.1, 6.4, and 5.0. This illustration indicates the proper 
use of significant figures. Since the rod-readings 4.9, 5.1, and 4.0 
are reliable only to about 1.5 to 2 per cent, the multiple 1.246 
should be used in this computation as 1.25. Similarly in the use 
of such a constant as ^=3.1415927 it is a waste of time to use any 
more significant figures in the constant than exist in numbers with 
which the constant is to be combined in the computation. 

351. In deciding how many places of decimals to use in the 
trigonometric functions the student should examine the tabular 
differences and determine what percentage error is introduced 
by any error in an angle. For example, suppose an angle of a 
triangle to have been measured in the field to the nearest minute. 
There may be an error of 30 seconds in this angle, and it will be 
seen from the table of natural sines that the tabular difference 
for one minute in the fourth decimal place varies from 3 for a 
small angle to less than 1 for a large angle, and that the vari- 
ation is about the same for cosines, and for tangents and cotan- 
gents of angles under 45 . Then for half a minute the differ- 
ence will be, on an average, about 1 in the fourth place. There- 
fore, in general, four places will be sufficient when the angles 
have been measured to the nearest minute only. But if there 
are several steps in the computations it may be advisable to use 



SIGNIFICANT FIGURES 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 off the 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 0.0001; 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 fig. 143. 

being the fourth significant figure 

should be retained, and that the resulting length of a or b will not 
be reliable to more than four significant figures. 

log 773.6 = 2.8885 lo g 773-6 = 2.8885 

log cos44°i2 / = 9»8555 lo 8 sin 44°i2 / = 9.8433 

log a = 2.7440 ' log b = 2.7318 

<* = 554-6 b = 539.3 

If it is assumed, however, that the angle B is measured by 
repetition and found to be 44°i2 / 25 /r the error in the original 
angle then was about 25". By using the same value for the hypo- 




326 COMPUTATIONS [Chap. XIL 

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 difference between the values of a 
and b obtained by use of the angle 44 12' and similar results by 
use of 44 12' 25* is due entirely to the 25" and not to the fact that 
four-place tables were used in the former case and six-place 
tables in the latter, for in both cases the result has been obtained 
to four significant figures only. 

It is also evident that when the angle B was measured to the 
nearest minute it was inconsistent to measure the hypotenuse 
closer than to the nearest tenth of a foot. But if angle B was 
measured to the nearest 10 seconds the line AB should have been 
measured to the nearest hundredth. It should not, however, 
be assumed that in all cases where angles are only measured to 
the nearest minute the sides should be recorded to tenths of a 
foot. It is the percentage error in the measurement of the sides 
which must be the same as the percentage error in the angles. 
If the sides are very short, they should be measured to hun- 
dredths of a foot to be consistent with angles to the nearest min- 
ute. In general, when the angles are read to nearest minute 
only, the sides should be measured to four significant figures; 
with angle to nearest 10 seconds they should be measured to 
five significant figures; and with angles measured to 1 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 precision 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 5 6' = 8.8373 
log AC = 2.2540 
AC = 179.5 

If AC is desired to the nearest hundredth of a foot the angle 
A might be determined closely by repetition, but this will not 
give the length AC 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 y then the 
line AC being about 15 times as long as BC will have an error of 
0.075 ft- no matter how exact the angle at A may be measured. 
In other words, if A C is desired correct to five significant figures 
BC should contain five significant figures. Evidently the prac- 
tical way of obtaining an exact value for the inaccessible dis- 
tance AC is to measure AB to the nearest hundredth, and to 
compute AC from AB and BC, using the angle at A as a check 
on the measured distances. In both of the above examples it 
is assumed that the 90 is exact. 

354. LOGARITHMIC OR NATURAL FUNCTIONS. — The ques- 
tion as to whether logarithmic or natural functions shall be 
used will depend upon the computation in hand. Many sur- 
veyors have become so accustomed to using naturals that they 
will often use them when logarithms would require less work 
and offer fewer opportunities for mistakes. Each method has 
its proper place, and the computer must decide which will be the 
better in any given case. The use of logarithms saves consid- 
erable time spent in actual computation because the process is 



328 



COMPUTATIONS 



[Chap. XII. 



simpler, but, on the other hand, looking up the logarithms con- 
sumes time. The result is in many cases, however, a saving of 
time over that required to do the arithmetical work of multi- 
plying or dividing. While the multiplication of two numbers 
of three or four digits each can possibly be done directly more 
quickly than by logarithms, still it takes more mental effort and 
there is more opportunity for making mistakes ; but in case sev- 
eral such multiplications are to be made logarithms are almost 
always preferable. Furthermore when there are several multi- 
plications of the same number logarithms will save time since the 
logarithm of this common number has to be taken from the table 
but once. Frequently, however, the computation is so simple 
that the use of logarithms would be almost absurd, e.g., the multi- 
plication of any number by a simple number like 20, 25, 150, or 
500. If a function of an angle is to be multiplied or divided by 

any such number the natural 
function should of course be 
used. 

355. SHORTCUTS.— The so- 
lution of a right triangle, when 
one of the angles is small, involv- 
ing the use of the cosine of this 
small angle, can often be more 
easily obtained by the use of the 
versed sine or external secant of 
the angle. In Fig. 145 




-^i 



AB = 207.42 

4 = 7° 3/ 

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 '79i by slide rule.) 
= 207.42 — 1.79 
- 205.63 



(1) 



SHORT CUTS 329 

Obviously, when the angle is quite small, the result of the multi- 
plication indicated in ( 2) can be taken from the table to the near- 
est hundredth of a foot with much less effort than is required for 
the computation called for in (1). 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 A, (7 32') then 



AB = 



_ 205.63 



t 



cos 7 32 
But AB = AE + EB 

= 205.63 + 205.63 exsec 7 32' 
= 205.63 + 205.63 X 0.00871 
(205.63 X 0.00871 = 1.79, by slide rule.) 
= 205.63 + 1.79 
= 207.42 

356. There are many "short cuts " in arithmetical work 
which are of great value to the computer, and the student should 
endeavor to learn the most common and simple ones. The fol- 
lowing are a few illustrations. 

247 X 100 24700 

247x25 = ^— — =^f- 

4 4 

68200 
682 X 50 = 

694 X 150 = 69400 + 34700 

927 X 62.5 = 92700 X ^ 
o 

672 X 1002.3 = 672000 + 1344 + 201.6 
547 X .9968 = 547 (1 - -0032) = 547 - 5-47 X .32 

A = jA (reducing minutes to decimals of a degree) 
60 o 

-^=8.43x8 
12.5 ,o 



330 COMPUTATIONS [Chap. XEL 

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 special computation 
book. At the head of the page should appear the title of the 
work, the number and page of the field note-book from which 
the data are copied, the names of the computer and checker, 
and the date. The work should be arranged neady 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 results. It 
is not uncommon to find two men computing the same area 
where only one of them looks up the logarithms. In case a mis- 
take is made in looking up the logarithms the results may check 
but both are wrong. The computer should also check his work 
roughly by estimating approximately what the result should be. 

359. SLIDE RULE. — A valuable aid in checking calculations is 
an instrument known as the slide rule, which enables the computer 



SLIDE RULE 33 1 

to multiply and divide numbers by logarithms by a purely mechan- 
ical process. It is really the equivalent of a table of logarithms. 
It consists of a wooden rule, usually about 10 inches long, having 
a groove in one side in which rims a small wooden strip called the 
slide. On one face of the rule are placed two scales, A and D % 
Fig. 146, one above and one below the slide which is indicated by 



:liiiiliin[i i iiliii i [ 1 [1 ( i [i(i[ i [ i [iin l ini[ii ii l i ni[ 1 [ 1 [1 [i[ i [i[ift 



;li]i) i ]i]i)i]i) i 3 i ] i [ii i iliiii[miliiii[iiiilii ii [ I [ 1 [ 1 [1 [ l [t 



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 1 is o, so this is taken as the 
left end of the scale and the number 1 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 jYdV °f *he IO inches, or 3-oi 
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 1 and 2 is 
subdivided by plotting log 1.1, log 1.2, etc. The subdivision is 
continued until the spaces are as small as will admit of rapid 
and accurate reading of the scale. 

It is customary to make the spacing on the upper scale just 
half that on the lower, i.e., if 10 inches is chosen as the unit for 
the lower scale, then the unit for the upper scale will be 5 inches. 
Since the length of this upper 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 
rule, every line on the slide is exactly opposite its corresponding 
line on the rule. A runner is usually attached to the rule for 
convenience in setting and reading the scales. This runner is 
a small metal slide which fits over the face of the rule in such 



332 COMPUTATIONS [Chap. X1L 

a way that it can be slid along the rule and set at any reading of 
the scale. It is usually provided with a fine line running cross- 
wise of the rule which is used in marking the exact setting. 

Multiplication or division of numbers is performed by add- 
ing or subtracting the scale distances corresponding to these 
numbers. The scale distance is the logarithm of the number. 
Adding two scale distances is, in effect, adding two logarithms, 
and the resulting scale distance is the logarithm of the number 
marked opposite on the scale. For example, if the left end of 
scale C, Fig. 147, is set opposite the number 2 of the scale D 9 
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. 



i 



,. .i M. , r .... i . M.r ' i^r ■■ i r . s r^» r " ^ c^'^r ^, !, ■ i f l 1 i ■ d ^'L ' i ' u**^' 1 ^ c ■' ' r ' ^ r ^ ^^^c ^s^^ 

j i jijijijijijij i jH i iiiii i i i inii|iflit i.i ii4 i v i C i vl i v i Dv i L i v i & l ^ 



Fig. 147. 

The scales A and B may be used in a like manner. It is evident 
that, by setting the runner on the result of one operation and 
then moving the slide so that one of its ends coincides with the 
runner setting, continued multiplication and division can be 
performed without the necessity of reading intermediate results. 
Scale D may be used in connection with scale A for obtain- 
ing squares or extracting square roots. Since the spaces on scale 
A are one-half those on scale D the number 4 on scale A is oppo- 
site number 2 on scale D y 9 is opposite 3, and so on, every num- 
ber on scale A being the square of the corresponding number on 
scale D. Other scales, generally log sines and log tangents, are 
placed on the reverse side of the slide, so that trigonometric cal- 
culations can also be performed with this instrument. Results 



REDUCING FIELD NOTES 333 

obtained with the ordinary 10 inch slide rule are usually correct 
to 3 significant figures, so that this slide rule is the equivalent 
of three-place logarithm tables. 

360. Thacher Slide Rule. — The Thacher slide rule consists 
of a cylinder about four inches in diameter and eighteen inches 
long working 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 distances are transcribed into 
the computation book. The errors in the angles are balanced 
by altering the value of those angles which were taken from short 
sights since the angular errors are most likely 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 multiplying them by the versed sine of the vertical 
angle and subtracting the result from the corrected slope dis- 
tance; the correction for error in the tape being made before this 
is done. Sometimes instead of a vertical angle the slope dis- 
tance and the difference 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 h + ^) 



where d = common distance between offsets, 
h % and h\ = end offsets of the series of trapezoids, 
and 2 h = 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 = - (h t + 2S h^ + 4S Ac™ + h\) 

where d = common distance between offsets, 
h t and h\ = end offsets of the series, 

2 S A odd = twice the sum of all the odd offsets 

(the 3d, 5th, 7th, etc., from the end) 
4 2 Aeven" four times the sum of all the even offsets (the 2d, 
4th, 6th, etc., from the end). 



For this rule to apply there must be an even number of 
trapezoids; if there is an odd number, an even number of them 
may be computed by this rule and the extra trapezoid must be 
computed separately. Or, if there is a triangle or trapezoid at 
the end of this series, which has a base greater or less than d, 
it must also be computed separately. 



TRAPEZOIDAL RULE 



335 



Fig. 148 shows the computation of a series by both methods 
and also the computation of several trapezoids and triangles at 
the ends of the series. The data are taken from the field notes 
in Fig. 53, p. 104. 




By Simpson's On*Jfr/nf/RsM 
-72 



is. 

16.9 

P 

/a. 

38.S 

JoJT'x& a /o/ 



By Thtptzoictaf ftuM 



'g'JS&njst 




tozs- 



.'IIj+*0.77*.9 
paUA- 



8S-34 * 

I019 

tit 

Z3C 
103 
ZSzT sfBypA 

Fig. 148. 



FfSf 



/AZ*<562.« 3J 3z 
pArg 

*/0.7xZ7Z* 2BI WXZU 

IJo'wxBv 



336 COMPUTATIONS [Chap. XII. 

365. STRAIGHTENING CROOKED BOUNDARY LINES. — In 

Fig. 149, AEFGH represents a curved boundary between two 



Fig. 149. Straightening a Crooked Boundary. 

tracts of land, and it is desired to run a line from A so as to make 
the boundary a straight line and to leave each tract of the same 
area as before. 

The trial line AB is first run, and the distance AB, the angles 
at A and B, and the necessary offsets to the curved boundary are 
measured in the field. Then the areas of the property between 
this trial line and the curved line are computed as explained in 
the previous articles. The sum of the fractional areas on one side 
of the trial line and the sum of the areas on the other side of it 
should be equal. If not made so by the trial line, the difference 
between these sums is the area of a correction triangle ABC which 
must be taken from one tract and added to the other because the 
trial line has taken this difference from one of the tracts and it 
should therefore be restored. The area and the base AB being 
known the altitude dC can be computed. Then in the triangle 
ABC, the lines BC and AC and the angle at A are calculated ; and 
the line AC is staked out, its calculated length being checked by 
measuring the line AC in the field and the angle at A being 
checked by the measured distance BC. 

366. AREA BY TRIANGLES. — If the field has been sur- 
veyed by setting the transit in the middle of the field and taking 
angles between the corners (Art. 138, p. 105), the areas of the 
triangles may be found by the trigonometric formula: 

Area = \ a b sin C, 

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

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. 106), the area of the triangles can be found by the trigono- 
metric formula: — 



Area = Vs (5 — a ) (s — b ) ( s — c ) 

a + b + c 



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



367. AREA OF A QUADRILATERAL BY TRIANGLES.— 

Most city lots have four sides, and while the Double Meridian 
Distance Method (Art. 384, p. 352) is often* employed in com- 
puting their areas, it is not at all uncommon in computing such 
quadrilateral lots to divide them into triangles, checking the field- 
work and computations, and computing the areas by triangles. 

In Fig. 150, A BCD represents an ordinary city lot in which 
all the sides and angles have 
been measured. It is evident 
that the diagonal BC can be 
computed either from BD, 
CD, and the angle D, or from 
AB, AC , and the angle A. 
These two determinations of 
BC should check each other. 
Similarly two independent de- 
terminations of 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. 




Tr^D 



Fig. 160. 




338 COMPUTATIONS [Chap. XIL 

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 J is known and 
the radius KF of the curve is given or can be computed from data 
such as CH or CF obtained in the field (Art. 257, p. 233). 

KFHG = FHG * HK = 7 ° X °- OI 74533* X&Kf ^ ^^ 

VI, p. 506.) 

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 



C 2 1 

M = -— - * (approximately). 
oK 



the middle ordinate and C is the chord length, 

Expressed in terms of C and R, 

C 3 

Area of Circular Segment = - (approximately). 

12R 

* The length of the arc of curve whose radius is 1 and whose central angle 
is i° is 0.0174533, which will give results to six significant figures, provided /and 
K are correct to six significant figures. 

t In Fig. 152, OB — Radius of circular curve. 

CH « 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 anc^ 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. Planimeter. — One of the commonest ways of checking 
the area of a traverse is to obtain its area by means of an instru- 

DB = Tangent Offset for chord CB. 
OE is drawn perpendicular to CB. 
In the two similar triangles OEB and CBD y 
DB : CB-BE.OB 

DB.CB ~ — :0B 

2 

DB- CB1 



2 OB 




Fig. 162. 



Offset from Tangent - (C1 y r ^ (1) 
* ax Radios l ' 

But DB = CH, and AB « 2 X CB (approximately) 

2 AB* 

.-. CH- 20B - - g-^ (approximately) 

Middle Ordinate - 8 ^ RadL. ( a PP roximatel y) ( 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 value is 0.635). 
When radius — 100 ft. and chord — 25 ft., M = 0.781, (correct value is 0.784). 
When radius — 100 ft. and chord — 100 ft., M « 12.500, (correct value 1313.3^7). 
When radius — loco ft. and chord — 100 ft., M = 1.250, (correct value is 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. 



340 COMPUTATIONS [Chap. XIL 

ment called the planimeter. It is a small instrument consisting 
of an arm, carrying a tracing point, which is fastened to the 
frame of the instrument; the arm can be adjusted to any desired 
length. The frame touches the paper at only three points; the 
anchor point, the tracing point, and the circumference of a small 
wheel which is free to revolve. On the rim of this wheel is a 
scale and beside it is a vernier which is used in reading the scale. 
The length of the arm can be regulated by setting it at the 
proper reading on a scale which is marked on the arm, so that 
a unit on the wheel scale will represent any desired unit area 
such as a square inch or a square centimeter. (See Appendix B 
on the Planimeter.) 

In using the instrument the anchor point is set at some con- 
venient position on the drawing outside of the area to be measured 
and then the tracing point is run around the perimeter of the area 
to be determined. The reading on the wheel is recorded when 
the tracer is at the starting point. The tracer, in passing around 
the perimeter, should be kept as closely as possible on the bound- 
ary line and should return exactly to the starting point. Then 
the scale is again read, and the difference between the two read- 
ings is the area which has been traced out, expressed in some unit 
depending on the length of the arm. The result can be easily 
transposed into the unit of the scale of the map. 

Usually the settings for the scale on the arm are furnished by 
the maker for various units of area. It is safer to test this setting 
by running the instrument around a known area, such as 4 square 
inches and determining the interval passed over by the wheel by 
making several tests and by setting the anchor point at different 
positions. This interval divided by 4 will be the value of one 
square inch of plan area and this is equivalent to a certain number 
of square feet of surface, depending upon the scale of the map. 
It is important that the sides of the trial square should be laid off 
so that they agree with the present scale of the map which, owing 
to swelling or shrinking of the paper, is frequently not quite the 
same as when it was first drawn (Art. 479, p. 428).* 

* When areas are desired from U. S. Geological Survey maps on which are 
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 :-f?*20O, Curve to Right, I*Sf-3S-20"fiC. =/6+7Z42 
Width of S treet 70f 

T*fi tan. 2S*47'40* - ZOO X.46330 * 36.66 T 
St* :8SOI/7S 
3f' s .0los8l/ 



20 **0000970 



/?C./6+72.42 
I+8O.O8 

er. /e+sz.so 



.9003960 x zoo » /eo.oe L c 
Deflection A/rg/es. 




Def/ecHanJL for^oft^^ x ^V7^*-^x<&ag 

Log /28S.M =£1/0496 

Deflection*. for3ao8#=32£Lxde#./brJ&f tog I6O.O8 *Z *&*6S 

*.60/6 X 

log. 60/6 •& 779308 

0.634339 
*:3oS6 

to 43': 

'*''*(& T09 9 43 m d*HS>ft 

3 L fc 16+7242 

.4-'/8'3/"deff.3o.09it. ' '"**% :S£"!*" 

18+71.42, =!+-&- 20 

/8+224Z*2l-29-/0 
4-l8'3o 
RT. J8^SZJd-25 9 -4T-40 t, Check^ 



Chords 



SOfh. Art. 

Sin 7*09'4o"*. 12467 
4QQ 



*IZ47XZ*35 = 



49868 Center Chd. 
8.7Z7- 
S8.S3 Left- Ch<t. 
41.94 Right Chct. 



30.08 fh Arc 

Sin4*/8'3d'*.07S/2 
400 

30.048 Gen. Chef. 

Ko7S/XlX35z S.2S7+ 

35.31 UftChd. 

24.73 Right ChdL. 



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 propor- 
tion the desired area found. 



342 COMPUTATIONS [Chap. XIL 

applied to city surveying (Art. 259, p. 234) will be found the for- 
mulas which have been used in the computations in Fig. 153. 
The length of the curve L e is found by taking from Table VT, 
("Lengths of Circular Arcs: Radius = 1 "), the length of an axe 
for 51 , for 35', and for 20" successively and adding them, which 
gives the arc of a curve whose radius is 1 and whose central angle 
is 51 35' 20". This is then multiplied by the radius (200) which 
gives the value of Lc, which is added to the station of the P.C. to 
obtain the station of the P.T. 

372. COMPUTATION OF OBSERVATIONS. — The computations 
relating to observations for meridian and latitude will be found 
in Chapter VII. 

COMPUTATION OF VOLUME. 

373. BORROW-PITS.* — Fig. 154 is a plan of a portion 
of a borrow-pit, at the corners of which the depth of excavation 
is marked in feet and tenths. Each of the regular sections of 
earthwork is a truncated rectangular prism whose volume is 
equal to the average of the four corner heights multiplied by the 
area of the cross-section, or expressed as a formula, 

Volume Truncated Rectangular Prism = A X — - 

4 
where A is the area of the cross-section and h v h v h t9 and h A are 
the corner heights. 

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

notation, 

h 4- h 4- h 
Volume Truncated Triangular Prism = A X — ? -' 

In computing a trapezoidal prism, such zsfdhg, the trapezoid 
is subdivided into a rectangle /eA# and a triangle fde\ or for jhds, 
into two triangles by diagonal lines, as jits and hds and their 
volumes may be computed by the above formula. 

When there are several prisms with the same cross-section, as 
shown in Fig. 154, these rectangular prisms can be computed as 
one solid by assembling them as follows: — multiply each corner 

* For a complete discussion of the computation of Borrow- Pits see Railroad 
Curves and Earthwork by Professor C. F. Allen, published by Spon & Chamber- 
lain, New York. 



BORROW 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 





3? 


& 


yfi . 


> 


( 


i 














a 




*3 


* 


rf 


$ 


c 


A^> 
















T\ 




*> 


V* 


*J 


* 




<? Y* 








• 






3 


f \ 




sP 


# 


4 


* 




* \ V 










k 




h 


\ e a 




4 


& 


$ 


A> 




\ \ 

\ \ 


«? 












J 


s 


\ 




# 


<* 


* 


* 




*° 


A? 


n 












r 


\ 



Fig. 154. Plan of Portion of a Borrow-Pit. 



one computation because it is composed of a series of prisms 
having the same cross-section. In the summation of the heights, 
those at a, m, n, r, and s are taken but once, those at such points 
as c, g, h, etc. are multiplied by 2, at ; the height is multiplied by 
3, and at such points as k it is multiplied 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. XH. 

divided into rectangles, their corner heights taken, and from these 
the volume can be computed. 

374. VOLUME OF PRISMOID. — The data . obtained from 
field notes are usually in the form of cross-sections which are 
taken at right angles to some general line of the construction, 
thereby dividing the earthwork into prismoidal solids with their 
bases parallel and their sides either plane or warped surfaces. 
The bases of the solids are the cross-sections which are obtained 
by taking sections of trench excavation or of road construction 
(Figs. 91 and 92, p. 208). 

375. End Area Formula. — The simplest method of com- 
puting the volume of a prismoidal solid is to average the areas of 
the two bases and multiply by the distance between them, which, 
expressed as a formula, is 

V =±= £i_±A* x / (End Area Formula) 

2 

in which A l 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 Prismoidal Formula: 

I 
Volume of Prismoid = - (A x + 4^4 m + A J 

in which /is the distance between the two bases, A x and A % ; and 
A m 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 x and A t \ it should not be taken as 
the mean of A x and A M . 

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 

*t * . 2 9«° 2I «S 

Notes of section: , , + 4.0 

+60 * +1.0 

Area M 4 X (21.5 4- 29) ( 20 X (1 + 6 ) 
2 2 

= 2 X 50.5 + 10 X 7 = 171. 

It is also the custom with some surveyors to plot each section 
carefully to scale and to obtain its area by use of the planimeter 
(Art. 370, p. 339). This is probably the most practical method 
when the sections are very irregular since the field work does 
not warrant the use of very accurate methods. 

There are several other methods employed in computing 
earthwork but the above are by far the most common. 

Several sets of Earthwork Tables and Diagrams have been 
published which reduce the work of computation very materi- 
ally. 

378. ESTIMATES FOR GRADING.— Estimates for grad- 
ing may be conveniently made by means of a topographic map. 
On this map will appear the contours of the original surface. 
The contours representing the finished surface are also sketched 
upon the map, and the smaller the interval between the contours 
the more accurate will be the result. In Fig. 156 the full lines 
represent the contours of the original surface which is to be al- 
tered so that when the necessary cutting and filling has been done 
the new surface will have the appearance indicated by the dash 
contours. At contour 20 and at contour 25 no grading is to be 
done. On the plan, first sketch the lines ABCDEF and 
AGHIJB which are lines of "no cut " and "no fill," i.e., lines 
which enclose areas that are either to be excavated or filled. The 
amount of excavation and embankment must be computed sepa- 
rately. In sketching such lines the lines AB, ED> and HI, as 
will be seen, follow the intersection of the original contours with 
the new ones, since at these points there is no cut or fill. There 
are no direct data on the plan which define where the earthwork 
ends at C but the assumption is here made that the fill will run 
out to meet the original surface at about the next contour at C. 
In this example the fill must run out somewhere between the 
24-ft. contour and the 25-ft. contour, for if it ran beyond the 25-ft. 



346 



COMPUTATIONS 



[Chap. XII 



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; (1) by computing directly 
the amount of cut or fill between successive contours, (2) by 




25 — r-'-e ,w -< 




Scale 



Fig. 156. 

assuming a horizontal plane below the lowest part of the earth- 
work and computing the volume of the earth between this plane 
and the original surface, then computing the volume between 
the same plane and the finished surface; the difference between 
these two volumes will be the amount of earthwork, or (3) by 
drawing on the plan a line of no cut or fill, a line representing, 



VOLUMES FROM CONTOURS 347 

say, s ft. cut or fill, a line representing 10 ft. cut or fill and so on. 
Then compute the volume between these successive 5-ft. 
layers. 

379. (1 ) Referring to Fig. 1 56 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 elevatidh between 21 and 22, or will be 1 ft. 

The areas of the horizontal planes AKEL, MNPO, QRST, 
and BUDV may be obtained by planimeter (Art. 370, p. 339) or 
otherwise, and the volume of the solid AKEL-MNPO may be 
obtained by the End Area Method (Art. 375, p. 344), its altitude 
being 1 ft. If it is desired to obtain the volume by the use of 
the Prismoidal Formula the volume of the solid AKEL-QRST 
may be found by using AKEL as one base, QRST as the other, 
and MNOP as the middle area, the altitude, or length, of the 
solid being the difference between 21 and 23, or 2 ft. The solid 
AKEL-F may be considered to be a pyramid with a base AKEL 
and an altitude equal to the vertical distance between the con- 
tour 21 and the point F which is in this case on contour 20, or 
a vertical distance here of 1 ft. 

Example. 

In Fig. 156 the amount of fill on the area ABCDEF is computed below. 

Area AELK — 900 sq. ft. 900 x}= 300 cu. ft. (Pyramid) 

.~., n ,t 9°o + 1000 
" MNPO - 1000 -£- X 1 - 950. 

1000 4- 1020 
" QRST - 1020 X I - 1010. 

- BVVD - 680 I020+ ^ X 1 - 8 5 o. 

2 

680 X i - 227 (Pyramid) 

3 ) 3337 cu. ft. 
9 ) 1 1 1 2 

124. cu. yds. Total Fill. 

380. (2) Referring again to Fig. 156 and applying the sec- 
ond method, the area of ABCDEF is found (by planimeter); this 
is the area of a plane at, say, elevation 20, since none of the fill 



348 COMPUTATIONS [Chap. XII. 

extends below contour 20. Then the area of A BCD EL is found, 
which is the area of the plane cutting the original ground at ele- 
vation 21. Similarly the areas of MBCDPO, QBCDST, and 
BCDV are found. The volume of the solids between these 
planes may be computed by the End Area Method or by use of 
the Prismoidal Formula, in which case every other contour 
plane is used as a middle area as explained in the preceding 
paragraph. The volume of solid whose base is BCDV is a pyr : 
amid whose altitude is the vertical distance between the 24-ft. 
contour and point C, which in this case is 1 ft. 

By the same general method the areas of ABCDEK, 
MBCDPN, etc., which refer to the new surface of the ground, 
may be obtained, and the volume of the solids between succes- 
sive contour planes computed. The difference between this 
quantity and the quantity between a plane at elevation 20 and 
the original surface will give the amount of fill. 

While in this particular problem the first method is the 
shorter, still there are cases where the second method will be 
somewhat simpler. It is particularly useful when the actual 
amount of cut or fill is not desired but when it is required to know 
if the proposed alterations will require more or less earth than 
can be easily obtained on the premises and, if so, about how 
much the excess will be. In this case the portions of cut and 
fill will not have to be computed separately. A line is drawn 
around the limits of the entire area where the grading is to be 
done, the volume between an assumed plane and the original 
surface is found, and then the volume between the same plane 
and the proposed surface. The difference between the two 
values will give the amount of excess of earthwork. 

381. (3) Fig. 157 illustrates a third method of computing 
earthwork from the data given on a topographic map. The 
original contours are shown in full lines and the contours of the 
proposed surface in dash lines. Through the intersection of 
the new contours with the original ones is drawn the line of "no 
cut " (zero line), the line where 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 



34S 




Fig. 167. 



15 ft., etc., above the final surface. The solids included De- 
tween these 5 ft. irregular surfaces are layers of earth each 5 ft. 
thick, and their volumes may be computed by either the End 
Area Method or by the Prismoidal Formula as explained in the 
preceding methods. The areas of these horizontal projections 
are obtained from the map and the vertical dimensions of the 
solids are the contour intervals. 

382. ROUGH ESTIMATES. — Rough estimates of the quantity 
of earthwork are often required for preliminary estimates of the 
cost of construction or for monthly estimates of the amount of 
work done. For preliminary estimates of road construction, very 



3 JO COMPUTATIONS [Chap. 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 an opportunity to exercise his judgment to an unusual degree. 
Rough estimates do not, as a rule, call for a large amount of field- 
work. It is important that as few measurements as possible 
should be taken and that these should also be at the proper 
places to give complete data and to allow simple computations. 
Too often engineers, as soon as they arrive on the work and 
before making a study of their problems, begin to take measure- 
ments, consequently they return to the office after hours of 
hard work with a mass of figures from which it will take several 
more hours to compute the quantities. Whereas, a few mo- 
ments' thought given to the choosing of the proper measure- 
ments to be taken in the field would give data which could be 
computed in a few moments by use of the slide rule, affording 
results sufficiently accurate for rough estimates. 



PROBLEMS. 

1. A series of perpendicular offsets are taken from a straight line to a 
curved boundary line. The offsets are 15 ft. apart and were taken in the fol- 
lowing order: 6.8, 7.2, 4.6, 5.7, 7.1, 6.3, and 6.8. 

(a) Find the area between the straight and curved lines by the Trapezoidal 
Rule. 

(b) Find the same area by Simpson's One-Third Rule. 

2. It is desired to substitute for a curved boundary line a straight line which 
shall part off the same areas as the curved line. A trial straight line AB has 
been run; its bearing is S io° 15' W, its length is 418.5 ft., and point B is on 
a boundary line CD which has a bearing S 8o° 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 351 

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

A £-50.63 A B C-105 39/ oo" 

B C- 163.78 B A Z?-89° 37' 30" 

CZ?- 93.80 
Z?i4 « 160.24 

z> £-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 17' 30". 
The corner lot is rounded off by a circular curve of 40 -ft. radius. 

(a) Find the length of this curve to the nearest T fo 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 a si is straight. 

6. At station 6 a rectangular trench was measured and found to be 3 ft 
wide and 4 ft. deep. At station 6+70 it was found to be 3.2 ft. wide and 8.6 ft. 
deep. 

(a) Find by use of the Prismoidal Formula the quantity of 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. 

12 -J£? +4.2 -IM 

+ 8.0 +5.6 

11+ 60 _J2£ J±o j±o J4* 

+ 10.0 +4.5 +7.5 +6.0 

Sta.ii -1L? +6.0 J&5 

+ 40 +7. 2 

The base of the road is 30 ft. and the slopes are ij to 1. 

Find by the End Area Method the quantity of earthwork from Sta. 1 1 to 12. Re- 
sult in cubic yards. 



CHAPTER XIII. 

AREA BY DOUBLE MERIDIAN DISTANCES.— COORDINATES. 

383. COMPUTATION OF AREA. — The computation of the 
area of any -piece of property which has been surveyed as a 
traverse will in general consist of (1) 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 longitude difference. 

352 



DOUBLE MERIDIAN DISTANCES 



353 




Fig. 168. 



and parallel to the coordinate axes. It is evident from the 
figure that 

Latitude = Distance X cos Bearing, 
and Departure = Distance X sin Bearing. 

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 bearing is N 30 E this course has a North latitude 
and an East departure. North latitudes and East departures are 
considered as positive ( + ), South latitudes and West departures 



354 



COMPUTATIONS 



[Chap. XHL 



as negative ( — ). In the figure the courses are assumed to run 
from F to 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- 
gles, such as DErs, EFr, etc., formed by (1) the courses, (2) 
their projections on the meridian, and (3) the perpendiculars 



DOUBLE MERIDIAN DISTANCES 35 r 

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 + CDst) - (DErs + EFr + FAq), 

that is, in this figure the sum of all the areas determined by lines 
running northward minus the sum of all the areas determined by 
lines running southward. These are known as north areas and 
south areas.* In computing the areas of these trapezoids it is 
convenient as well as customary to find the double areas and 
divide the final result by 2 instead of dividing by 2 in figuring 
each trapezoid. The area of any trapezoid equals the average 
distance of the extremities of the course from the meridian, 
multiplied by the length of the projection of the course on the 
meridian. This average distance of the ends of the line from 
the meridian is known as the meridian distance of the course, 
i.e., the meridian distance of the middle point of the course. In 
computing the double areas, 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 inconsecutive order around 
the traverse, whether they were so taken in the field or not. The 
D.M.D. of the course FA is qA which is the departure of the 
course FA. The D.M.D. of AB is qA + wB = qA + qA + 
zB, i.e., the D.M.D. of course FA + the dep. of FA + the dep. 
of AB. The D.M.D. of BC = tC + wB = tv + 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: — 

(1) 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. 



356 COMPUTATIONS [Chap. XIIL 

(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 departure, but with opposite sign. 

The double areas of all the trapezoids may now be found 
by simply multiplying the D.M.D. of each course by the latitude 
of the same course, North latitudes being regarded as plus and 
South latitudes as minus. The sum of all the north double areas 
minus the sum of all the south double areas equals twice the area 
of the field. Be careful to divide by 2 after completing the other 
details of the computation. 

386. COMPUTATION FOR AREA OF COMPASS SURVEY BY 
D.M.D. METHOD. — The details of the above are illustrated 
in Fig. 159, which is the computation of the area of the traverse 
given in the compass notes in Fig. 50, p. 100. It will be seen 
from a study of the notes that there was local attraction of i° at 
station B, and that in the following computations the corrected 
bearings are used (Art. 41, p. 30). 

In Fig. 159 the bearings, distances, latitudes, departures, and 
D.M.D.'s, which are recorded on a line with station F are those 
corresponding to the course FA; those recorded on a line with 
station A refer to the course AB; etc. After the bearings and 
distances are entered in the table the places which are to be 
blank in the remaining columns are cancelled as shown; this is 
a check against putting the results of the computations in the 
wrong spaces. In computing the latitudes and departures the 
log distance is first entered; the log sin bearing is written below 
this and the log cos bearing is recorded above. To obtain the 
log latitude add the upper two logarithms; to obtain the log 
departure add the lower two logarithms. When the latitude and 
departure of a course have been obtained see if the results appear 
to be consistent with the given bearing and distance; when the 
bearing of a course, for example, is less than 45 its latitude is 
greater than its departure and vice versa. 



DOUBLE MERIDIAN DISTANCES 



357 



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Fig. 169. Area of Compass and Chain Survey by Double 
Meridian Distance Method. 



358 COMPUTATIONS [Chap. XIII. 

387. Balancing a Chain and Compass Traverse. — 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 exacdy 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 lengths of the lines 
but also their directions. 

388. The correction to be applied to the J , \ °f an y 

course is to the total error in \ , I as the length of the course 

is to the perimeter of the field. 

This rule is based upon purely mathematical considerations 
and should be applied only when nothing is known as to where 
the errors probably occurred. Usually the surveyor knows where 
the error is probably greatest and consequently in balancing the 
survey he will place the largest corrections where, in his judg- 
ment, they belong. In measuring with the chain, the recorded 
distances tend always to be too long, because the sag, poor align- 
ment, and poor plumbing, all serve to shorten the chain; conse- 
quently the probability is that the recorded measurements are 
too long, therefore in balancing it is more logical to subtract 
from the latitudes and departures in the columns whose sums 
are greater rather than to add anything to the latitudes and de- 
partures in the smaller columns. The corrections should of 
course be applied in such a way as to decrease the difference 



DOUBLE PARALLEL DISTANCES 359 

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 colufhns headed balanced latitudes and balanced de- 
partures. 

389. From the balanced departures we then compute the 
D.M.D. of each course as shown in the next column. Observe 
that the last D.M.D. (point jF), 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 area columns, those having N latitudes 
being placed in the column of north ( + ) double areas and those 
having S latitudes in the column of south ( — ) double areas. The 
sums of these columns differ by 2245.8. One-half of this, or, 
1 1 23. is the area of the field in square chains, which equals 
1 1 2. 3 acres. 

By proceeding around the field in the reverse- direction the 
letters of all of the bearings would be changed, in which case the 
column of south double areas would be the larger. 

390. Double Parallel Distance. — There is no particular 
reason for using the trapezoids formed by projecting the courses 
on to the meridian rather than those formed by projecting them 
on to the other axis. In the latter case the Double Parallel Dis- 
tance (D.P.D.) should be computed, and the result multiplied 
by the departure for each course. 

In the D.M.D. method the computations have been checked 
at every step with the exception of the multiplication of the 
D.M.D. *s by the latitudes. A check on this part of the work 
can be obtained by figuring the area by use of the D.P.D.'s. 
This furnishes an example of a very desirable method of check- 
ing, as a different set of figures is used in computing the double 
areas, and the opportunity for repeating the same error is thus 
avoided. Fig. 160 shows the computation by the D.P.D. method 
of the area of the same survey as is calculated by the D.M.D. 
method in Fig. 159. 



36o 



COMPUTATIONS 



[Chap. XIII. 



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

391. Error of Closure. — An indication of the accuracy of 
the survey is found in the error of closure. If a complete traverse 
of the field has been made the final point, as computed, should 
coincide with the first. The amount by which they fail to coin- 
cide is the total error of the survey and may be found by the 
formula 

E = VP + <p 
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^W ( see Art. 132, p. 99). 

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 1 

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 known as to 
where it probably occurred the corrections should be made in 
the angles adjacent to the short lines, as any error in sighting or 
setting up the transit causes a greater angular error in a short 
line than in a long one. 

The transit survey is referred to a system of rectangular 
coordinates, as in case of the compass survey. If the direction 
of the true meridian is known (either from a special observation 
or by connection with some other survey referred to the meridian), 
it is advisable to use this meridian as one of the coordinate axes. 
If the direction of the true meridian is not known the magnetic 
meridian may be used. This of course is convenient in some 
respects because the bearings taken in the field already refer 
to this meridian. If not even the magnetic meridian is known it 
will then be advisable to choose some line of the survey (pre- 
ferably a long one) as the axis, for using one of the traverse lines 
as an axis -saves computing the latitude and departure of one 
course. ' 

Whatever line is chosen as an axis, the bearings used for 
computing the latitudes and departures are to be obtained from 
the measured angles (after correction), and not from the observed 
bearings. For instance, if some line is selected and its mag- 
netic bearing used, then the bearings of all of the other lines 
should be computed from this one by means of the (corrected) 
transit angles. In this way the bearings are relatively as accurate 
as the transit angles, even though the whole survey may be 
referred to an erroneous meridian due to the error of the magnetic 



362 



COMPUTATIONS 



[Chap. XIII 



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

(The remainder of the computations is in Fig. x6i A.) 



DOUBLE MERIDIAN DISTANCES 



363 







Latitudes and Departures 






H 


J 


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1.70568 


1.712*37 


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1.836*5 


1.17818 


2.316830 


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Log Dep. 


1.WH 
1.75130 


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*0.1* 


8M3 


162.72. 






Double Areas 








H 


J 


K 


A 


B 


Log DM0 


1.80650 


2.35161 


2.76<|281 


3.0I6837 


3.01*3** 


Log Urt- 


I.1M80 


0.72181 


2.5I26I8 


I.183**6 


2.31*138 


Log Area 


3.7I830 


3.073*3 


5.281911 


5.000283 


5.608*82 


Area 


J228 


\m 


11 1381 


I00066 


H05951 




C 


D 


£ 


F 


G 


LoqDMD 
Log Lafc 
Log Area 


2.815*1 


2.813207 


2.72**56 


2.601168 


2211*1* 


0.16835 


1.770*8* 


1.7*0115 


1.683857 


2.1012*6 


3.3638* 


*.58361l 


*.*€537l 


*.213825 


*.320660 


Area 


23M 


383*3 


ZWY\ 


11671 


20125 



161 A. 

(These 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 
calculation of a new bearing of the first line of the traverse which 
must agree with the magnetic bearing assumed for it, provided 
the deflection angles have been adjusted so that their algebraic 
sum is 360 . The observed magnetic bearings of the different 
courses will serve as a check against large mistakes in this cal- 
culation. 

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 J « 
a minute, seven-place logarithmic tables may profitably be em- ' 



364 m COMPUTATIONS [Chap. XIIL 

ployed, as much interpolation is avoided by their use, but the 
logarithms need not be taken out to more than five or six places. 
Seven places, of course, are more than are necessary so far as 
precision is concerned (Art. 351, p. 324). 

The computation of the latitudes and departures may be 
conveniently arranged as shown in Fig. 161 which is the com- 
putation of the survey in Fig. 52, p. 103. 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: — 

The correction to be applied totheX . J of any course is 

j *i * * 1 • \ latitude } Al ( latitude ) r Ml J 

to the total error tn < , A . > as the \ , L M \ of that course 
[ departure ) { departure ) J 

11 %:r of * of the i £££« 1 {without regard » **■ 

As in the case of a compass survey, the surveyor's knowledge 
of the circumstances should always take precedence over the rule, 
and it is probably more nearly correct to shorten the latitudes or 
departures in the larger columns than to lengthen them in the 
smaller columns. This is because distances are usually recorded 
longer than they actually are; the only cases where the distance 
is probably too short is when an excessive pull has been given 
to the tape or a mistake made in measurement. It will be 
observed in the original notes (Fig. 52, p. 103) that the distances 
BC, GH, and KA were all questioned, i.e., they were measured 
under such conditions that it is probable that there may be one 
or two hundredths error in them. In balancing the latitudes and 
departures then, this information is used. In Fig. 161 it will 
be seen that in balancing the survey the latitudes and departures 



DOUBLE MERIDIAN DISTANCES 



36S 



of these questioned measurements have been changed in such a 
way as to reduce the length of BC, GH, and KA each one hun- 
dredth of a foot. 

In balancing the angles, in which there was an error of 15 
seconds, it will be noticed that the correction for this error, 
being small, was put into one angle, that at C, one of whose 
sides is the shortest 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 



Area ofLancfofS/bsCakma*. &36p6Z. %&t <&&£ * 



Sta 


taring 


Diet. 


Latirudt I Departure Balanced 


QHD. 


Double Ar*a 


r; n ■- £■ ■ 4m rza ■ *?« ■ 'Ifb 




— 


A 


N 


2BL2 


2dtt 









»*99.l 




570.4 


I706O7 






B 


N87-30W 


I7S2 


78 


— 




17910 


♦ 7.8 


-17*1 


3913 


30 SI 






C 


SI-2IW 


1646 




M6 




3.9 


-164.6 


-3.9 


2083 




34286 







N87-09W 


SBJ 


5.0 






996 


♦ 50 


-»7 


104.7 


524- 






C 


SO-5IW 


1693 




169.5 




i* 


-1694 


-25 


2.5 




424 




F 


N85-34E 


2862 


22.1 





285.3 




1-22.1 


♦ 2852 


285.2 


6303 







1198.2 334.1 333.9 28&3 2850 
lrr»r in Lah 04, Error if» 0*p.a5 
Dosing Linear Error •/tM i « .3* 




>fOo*un* 



180486 34710 
34710 
2 )143776 
72888 sait. 



»«4;jt &ZOM-9.7JJ t6£«***l308j 



IS5-4I 
51 -0» 



»44*-0.1*lS.7« 177.1 *qk 
fm-kg>6.» 

fm» 177.2 m«M.|7ZI 
72688 Traverse 



ie4«*74«IZ*«f I&4 1.737 *»ec« 

44-15 »4 * JZS » I J3jf 184 U*\ * « 7 C5 
64*.o3*o.2kD 6.4 itnk 

l3A+l792*l&0*2OJ8*ej 
€f-ij«2.7 

>.2©5.7 



2&2018 
Jf« 205.8 , 



Total Area of Fannf 



Areas. 
J27a^J2J 3 |744 .22&^£LL«2274 
♦ .* *3.l * » I -I5.ne.2 .■ - 84 
-»7»ttJ !1 lf -9.7*83 --63 



Fig. 162. 



Computation of Transit and Tape Survey, including 
Fractional Area. 



survey shown in Fig. 53, p. 104. The traverse was run with a 
transit and tape, the angles being measured to the nearest minute 



366 COMPUTATIONS [Chap. XUI. 

and the sides to tenths of a foot. Nothing appears in the field 
notes to indicate that any of the lines were difficult 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 survey, the rule given in Art. 395 is 
applied. In balancing the angles, in which there was an error 
of 1 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 different 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, 



Z\ cos A + Z 2 cos B + Z z cos C + etc. = 
Z x sin A + Z 2 sin B + Z, sin C + etc. = 



where Z v Z„ Z 8 , 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 : — 

(1) The bearing and length of a line. 

(2) The length of a line and the bearing of another line. 

(3) The length of two lines. 

(4) The bearings of two lines. 



SUPPLYING MISSING DATA 367 

398. Case (1) 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 of 
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 D 

shown by full lines. The ^^T^"******^ 

bearing of A B is given. Here t ^^T ___ ^^"^^^ 

one side and two angles are N^^-ZI ^^c 

missing. The solution is as ^v^/^"***^ Vx// 

follows. In the triangle EAB > T\ ^ ^-^/ 

find EB, EBA, and AEB. In A FlG H ^~* B 
the triangle EDC find EC, 
DCE, and DEC. Then in the triangle EBC y in which EC, 



368 COMPUTATIONS [Chap. XHI. 

EB y and EBC are known, find ECB, CEB, and BC. All the 
angles and sides are then known. Other special cases may be 
solved in a similar manner. 

401- DETECTING MISTAKES. — Mistakes in fieldwork may 
often be detected by means of the calculations. One of the 
easiest mistakes to make in 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 survey, of course, 
will prevent these bearings from agreeing exactly. If two 
mistakes have been made it may be difficult and sometimes im- 
possible to determine where they occurred. When an error of 
this sort is indicated by the computation the line should be re- 
measured. It is bad practice to change an observed measure- 
ment because it is found by calculation to disagree with other 
measured distances. 

It may, and frequently does, happen that there is more than 
one line in the traverse which has about the same bearing. In 
such a case it is impossible to tell in which of these lines the mis- 
take occurred. But if a cut-off line is measured as was sug- 
gested in Art. 145, p. 109, and one portion of the survey bal- 
ances, the other part will contain the mistake. By proceeding 
in this way the number of lines in which the mistake could occur 
is reduced so that its location can be determined and checked 
by field measurement. 

402. THE SUBDIVISION OF LAND. — There are a great 
many different problems which may arise in the subdivision of 
land and which may be solved simply by the application of the 
principles of trigonometry. A few of these problems are so 
common and so frequently involved in the working out of more 
complicated cases that their solution will be given. 



SUBDIVISION OF LAND 



369 



403. To Cut Off from a Traverse a Given Area by a Straight 
Line starting from a Known Point on the Traverse. — In Fig. 
164, ABCDE represents the traverse which has been plotted and 
whose area has been computed. 
It is desired to cut off a certain 
area by a line running from -F 
which is at a known distance 
from A or E. The line FG' is 
drawn on the plan so as to make 
the area FGDE approximately 
equal to the desired area. The 
line DG f is scaled off 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 f 
and its area being known the altitude hG and the distances GG' 
and FG can be readily computed. In the traverse FGDE, 
which is the required area, the length of the missing side FG 
and its bearing can be supplied. 

Instead of using the trial line FG the line FD might have 
been first assumed and the correction triangle would then be 
FDG. This method has the advantage of containing one less 
side in the first trial area, but the correction triangle is large, 
whereas in the method explained above the correction 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 bo cut off a given area by a 
line running at a given angle (BJK) with A B. On the plot of the 




370 



COMPUTATIONS 



[Chap. XIIL 



traverse draw the line J'K' in the given direction cutting off 
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! by the method suggested in Art. 400, i.e., by dividing 
J'BCK' into two oblique triangles. The difference between 

this area and the required area 
is then found, which is a cor- 
rection trapezoid to be added 
to or subtracted from J'BCK'. 
In this case it will be assumed 
that it is to be added to J'BCK'. 
In this correction trapezoid 
the area and one base J'K' are 
known; also the base angles, J' 
and K'. From these data an 
approximate value for the alti- 
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 difference between this and the required correction 
will be small and the dimensions of the second correction trape- 
zoid J"K n K'J' can probably be readily computed from its area 
and the length of J'K" which are known. By successive trials, 
probably not more than two, the correct line JK can be found. 
If lines AB and CD are approximately parallel the trapezoid is 
nearly a parallelogram and its correct altitude can then be 
quickly determined. 

405. To Find the Area Cut Off from a Traverse by a Line 
running in a Given Direction from a Given Point in the Traverse. 
— rThis problem may be readily solved by drawing a line from 
the given point in the traverse to the corner which lies nearest 
the other extremity of the cut-off line. The area of the traverse 
thus formed is then computed, and this area corrected by means 
of a correction triangle. 




TRAVERSES WHICH DO NOT CLOSE 



371 




Fig. 166. 



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 off by this line. 
The latitude and departure of points 
B and E being known the bearing 
and length of BE and the area of 
ABEFG can be computed. Then 
the area and the remaining sides of 
the triangle BEH can be obtained 
from BE and 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 BAF are desired. AB can 
be assumed as one of a pair of 
rectangular coordinate axes and 
the coordinates of point F (AH 
and HF) computed by the 
method explained in Art. 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. 
The calculation of cut-off 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 
measured in the field and the traverse CDEF is thus complete 
except that the length of the line CF is unknown. The length 
of CF and the angle it makes with AB can be readily computed 
since the coordinates of C and F can be found. 




407. 



Fig. 166. 
CUT-OFF 



LINES. 



372 



COMPUTATIONS 



[Chap. XIII. 



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 AB 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 base as when 
calculated from the base CD and so on. 
If, however, when computed by means 
of angles at D and E, the point falls 
at 5', and angles E and F give its 
location also at 5' there is evidence 
of a mistake in the traverse at D. If 
the two locations of 5 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 f 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. 




Fig. 167. 



409. CALCULATION OF TRIANGULATION. — In a triangu- 
lation system the base-line is the only line whose length is 
known at the start. The sides of any triangle are found from 
the law of sines, i.e., 



sin A a 

sin 2* b 

sin A __ a 

sin C c 



a sin 5 

sin A 

a sin C 

sin A 



= b 



= c 



COORDINATES 373 

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

(i) log a (1400.74) = 3-I463S75 

(2) Colog Sin A (57° 42' 16") = 0.0729874 

(3) log Sin B (6i° 17' 53") = 9.9430639 

(4) log Sin C (6o° 59' 51") = 9.9418088 

Sum of (1) (2) (3) log b = 3.1624088 
Sum of (1) (2) (4) log c = 3.1611537 

410. COORDINATES. — In many cities the coordinate sys- 
tem of surveying is used (see Chapter IX). In this system the 
position of each corner of the different lots is fixed by rectangu- 
lar coordinates measured from two lines at right angles to each 
other. 

Often the origin of coordinates O (Fig. 168) is so chosen that 
the whole city is in the first 
quadrant YO X. Distances 
measured parallel to XX' 
are usually called abscissas 
and those parallel to YY' 
ordinates. 

The advantage of this x' 

system of surveying lies in 
the fact that since all surveys 
refer to the same reference 
lines, they are therefore tied 
to each other; and also in the 
fact that a lot can be relo- 
cated from the coordinates of 
its corners even if all of the corner bounds have been destroyed. 

Generally the coordinate lines rim N and S, and E and W, 
but when city streets have been laid out at right angles to each 
other and not on N and S, and E and W lines, it may be more 
convenient to have the system of coordinates parallel to the street 
lines. 

The coordinates of any unknown point are usually computed 
from the coordinates of some other point to which the unknown 



Y' 



Fig. 168. 



374 



COMPUTATIONS 



[Chap. XHL 



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

Difference in X = distance X sin azimuth angle. 
Difference in Y = distance X cos azimuth angle. 

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



X,-- 




--v 



— 1— x 3-r 



# 



Fig. 169. 



411. TO DETERMINE THE AREA OF A FIELD BY REC- 
TANGULAR COORDINATES. — The area of the field i, 2, 3, 4 
(Fig. 169) is equal to the trapezoids 

(a, 1, 2, b) + (6, 2, 3, c) - (a, 1, 4, d) - (d, 4, 3, c). 

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



AREA BY COORDINATES 375 

. -C, 4 -X>^-(*-*> ^ « 

-i {*(*•—*«> +■*(*•-*») +^(^-^+^(^1-^)} ( 2 ) 

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

(i) Number the corners consecutively around the field. 

(2) Multiply each j al *cissa j by the diff erence between the 

following and the preceding j °L c j ss ^ j > always subtracting the 

following from the preceding (or always subtracting the preceding 
from the following), and take half the sum of the products. 
412. Fig. 170 is the computation, by coordinates, of an area 



Ste 


Bearing 


Out 


Latitv<* 


ac/Jarfvrr 


AriMotf 


X 


Y 




DoubM Area 


N* 


5- 


£ + 


te- 


Lot 


Pep 


f 


— 





JYS7*W 


160 


as? 








rn 


*G87 


-/,J4 


zass 


2136 


~S36 




/mi 




/ 


S37*W 


&32 





G24 




$23 


-£22 


^t 


!%& 


2Z23 


* //. 36 


^/6tJ 






^ 


54§'i£ 


4S3 





3JI 


3,28 


— 


-3M 


+316 


to.oo 


iO.OO 


*&34 


tS34 






J 


N43*$E 


a?s 


3.3? 


— 


$46 




*33§ 


*$47 


J328 


6S9 


-6.8? 




Sf.F 


* 


NZ6**f 


s.oo 


449 


— 


— 


BB 


'449 


2i$ 


tt;;s 


!6tf7 


»d7 




3ZS.2 










tSSJ 


tf-tf 


mm 


it.* 












37/7 


S306 

17 i -* 



?3.SS*ctt. 

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 (1) may be developed into the following form: 

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



*\yy \j>, / -a / \7. t / a / 



(4) 



376 



DOUBLE MERIDIAN DISTANCES 



[Chap. XHL 



From equation (3) it will be seen that the area is equal to the 
sum of the products of the ordinates joined by full lines in (4) 
minus the sum of the products of the ordinates joined by broken 
lines. This formula involves the multiplications of larger num- 
bers than in (2), but does not require any intermediate sub- 
tractions. 



PROBLEMS. 



1. The latitude of a line of a traverse is + 106.42 ft.; its departure is 
— 273.62. What is its bearing ? 

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



Station. 


Bearing. 


Distance 
(Chains). 


A 


N 46 J W 


20.76 


B 


N 5 i°iE 


13.80 


C 


East 


2i-3S 


D 


Ss6°E 


27.60 


E 


S 3 3°1W 


18.80 


F 


N74°JW 


30.98 



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). 


1 


S40°W 


I7-SO 


2 


N45°W 


22.25 


3 


N 36 } E 


31-25 


4 


North 


13.50 


5 


(omitted) 


(omitted) 


6 


S8 JW 


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 double meridian distance method the area of the traverse ABCDEK. 

5. In the following traverse there are two mistakes. Find where they occur 
and determine their amounts. 



Station. 


Observed 
Bearing. 


Deflection 
Angle. 


Distance 
(Feet). 


Calculated 
Bearings. 


Remarks. 


A 
B 
C 
D 
E 


N 34 °E 
S73°1E 
S io°l W 
N 26 J W 
SS2°W 


164 14' R 

62 i& R 

84 22' R 

142 49' R 

103 41' L 


240.2 

163.7 
207.6 

273-1 
147-4 


N34°oo'E 


C£ = 188.1 
BCE = 34° 14' 
DEC= 8i c 25' 



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 13 10' E . 


100.27 


C 


N o°i7'W 


91.26 


D 


N2 7 °39'E 


112.48 


E 


N72°i2'W 


346.07 


F 


S s° 07' 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 to AB which will cut off an area 
ABFE equal to half an acre. Also find the length of the lines AE, and BF. (b) 
What is the area of EFCD ? 

8. Given the notes of a traverse, which does not close, as follows: — 



Station. 


Deflection 
Angle. 









Find the length 


6+40 


6°i7'L 


of a straight line 


9+20 


i8°43'L 


from to 20+ 64 


14+ 55 


I2°47'R 


and the angle it 


17+ 18 


45°*4'L 


makes with the 


20 +64 


68°o6'R 


line from to 6 
+ 40. 



378 DOUBLE MERIDIAN DISTANCES [Chap. XIIL 

9. Compute the area of the following traverse by coordinates. 



Station. 


Deflection 
Angle. 


Bearing. 


Distance 
(Feet). 


A 


78 10' 00* L 




208.64 


B 


88° 28' 00" L 




436.79 


C 


67 02' 15* L 




56.48 


D 


33° 39' 15* L 




98.80 


E 


90 00' 00* R 




68.62 


F 


22° 51' OO* L 




95.10 


G 


68° 50' 15' L 


N36°i4'oo"W 


207.41 


H 


89 48' 00* L ' 




103-75 


I 


55° <*>' 15" L 




96.75 


J 


53° 49' 00* R 




420.77 



PART IV. 
PLOTnUG. 



PART IV. 

PLOTTING. 
CHAPTER XIV. 

DRAFTING INSTRUMENTS AND MATERIALS. 

It is assumed in this section that the student is familiar with 
the ordinary drawing instruments such as the T-square, triangles, 
dividers, compasses, and scales, as well as with their use. 

ENGINEERING DRAFTING INSTRUMENTS. 

414. There are several drafting instruments which are used 
by engineers and surveyors but which are not so generally em- 
ployed in other kinds of drafting work. The most common of 
these are briefly described in the following articles. 

415. STRAIGHT-EDGE. — Engineering drawings are made 
with greater accuracy than much of the drafting work of other 
professions. In fact many engineering drawings are limited in 
precision only by the eyesight of the draftsman. It is evident, 
then, that to use a T-square which is run up and down the more 
or less uneven edge of a drawing board will not produce draw- 
ings of sufficient accuracy. For this reason in many classes of 
engineering work the edge of the drawing board is not relied 
upon. Furthermore, in most plots of surveying work the lines 
are not parallel or perpendicular to each other except by chance, 
but run at any angle which the notes require; and there is there- 
fore not so much call for the use of a T-square as there is in archi- 
tectural, machine, or structural drawings. All drawings are 
usually laid out starting from some straight line drawn on the 
paper by means of a straight-edge, which is simply a flat piece 
of steel or wood like the blade of a T-square. Steel straight- 
edges are more accurate and are more commonly used by engi- 
neering draftsmen than the wooden ones, the edges of which are 
likely to nick or warp and become untrue. They can be ob- 

381 



382 DRAFTING INSTRUMENTS [Chap. XIV. 

tained of almost any length and of any desired weight, the com- 
mon length being about 3 feet. 

416. ENGINEER'S SCALE. — Practically all engineering 
plans are made on a scale of 10, 20, 30, etc. feet to an inch. In 
the engineer's scale, therefore, the inch is divided into 10, 20, 30, 
etc. parts, instead of into eighths and sixteenths as in the archi- 
tect's scale. Engineer's scales are made 3, 6, 12, 18, and, 24 
inches long. One form is the flat wooden rule with both edges 
beveled and a scale marked on each bevel. Some flat rules are 
beveled on both faces and on both edges of each face, thereby 
giving four scales on one rule. Still another very common form 
is the triangular scale, made of wood or metal, and having six 
different scales, one on each edge of the three faces. In such 
rules the scales are usually 20, 30, 40, 50, 60, and 80 ft. or 10, 20, 
30, 40, 50, and 60 ft. to an inch. Scales are, however, often 
made having the inch divided into 100 parts, but in plotting a 
map which is on a scale of 100 ft. to an inch the work is probably 
more easily done by using a scale of 10, 20, or 50 divisions to an 
inch and estimating the fractional part of a division than by try- 
ing to plot with a 100-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 
10-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 1 to 500, 1 to 2500, etc., mean- 
ing that one unit on the map is 7 J 7 , s*W> etc - °* 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 3«3 

grees, half-degrees, and sometimes into quarter-degrees. Fig. 
171 represents a semicircular protractor divided into degrees. 




C 

Fig. 171. Semicircular Protractor. 

In plotting an angle with this protractor the bottom line of the 
instrument is made to coincide with the line from which the angle 
is to be laid off, 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 o° and 180 of the protractor on its 
lower edge some instruments are made as shown in Fig. 172. 



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

420. Some of the metal protractors are provided with an 
arm and vernier attachment. These, while giving more precise 
results, require more time for manipulation, and a plain metal 
protractor with a diameter of, say, 8 inches will give sufficiently 
close results for all ordinary work. As a matter of fact a pro- 
tractor with a vernier reading to minutes can be set much closer 
than the line can be drawn, and it is therefore a waste of time to 
attempt to lay 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 until each of the three known points, (which have 
already been plotted on the plan), lies on a beveled edge of one 
of the three arms of the protractor. When the protractor is in 
this position its center locates the point desired which is then 
marked by a needle point Only one location of this center 
point can be obtained except in the case where the three known 



PROTRACTORS AND PANTOGRAPH 385 

points lie in the circumference of a circle which passes through 
the center. 

422. There are several other types of protractor made, but 
the principle and use of all of them are much the same as those 
of the simple types which have been explained. It is well in 
purchasing a protractor to test it to see that the center point lies 
on a straight line between the 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. PANTOGRAPH. — 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 A, B, 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 a 



386 DRAFTING INSTRUMENTS [Chap. XIV. 

straight line, the scale of the copy can be made to bear any 
desired relation to the scale of the original drawing. These 
instruments are usually provided with scales marked on the arms 
indicating the proper settings for various reductions or enlarge- 
ments. With a pantograph very accurate results cannot as a 
rule be obtained because there is lost motion in the several joints 
of the instrument. Some of the expensive metal pantographs, 
however, will give fairly good results. 

424. PARALLEL RULER. — This is a beveled rule made 
of metal and mounted on two rollers of exactly the same diameter. 
It is used for drawing parallel lines. This instrument can be 
made to do accurate work, but it must be handled with a great 
deal of care to prevent the rollers from slipping. It is especially 
useful in drafting diagrams of graphical statics in connection 
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* CONTOUR PEN. — This pen is constructed very much 
like an ordinary right-line ruling pen except that it has a metal 
shaft, running through the entire length of the holder, to which 
the pen is attached. The shaft revolves inside of the holder, 
and the pen is so shaped that it drags behind taking a position 
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 litde practice is required before one can use a pen of this type 
accurately. When skill in its use is once acquired, however, a 
plan can be easily made on which the contours all have a uni- 
form weight of line giving a very satisfactory appearance. The 



BEAM COMPASS. — PROPORTIONAL DIVIDERS, 387 

purpose of a contour line is to show the facts as to the surface, 
and this pen should not be used unless it is found by trial that 
it does the work in hand properly. Accuracy is more important 
than appearance. 

427. PROPORTIONAL DIVIDERS. — Proportional dividers are 
substantially an ordinary pair of dividers with both legs pro- 
longed through the pivot-point thereby forming another pair of 
legs above the pivot. The pivot is movable so that it can be 
pushed up and down in a slot in the legs and clamped in any 
desired position, thereby altering the relative lengths of the two 
pairs of legs. The sliding is accomplished in some dividers by 
a rack-and-pinion motion. When the pivot is in the middle 
position the legs are equal, and the space between the two points 
of one pair of legs is equal to the space between the other pair. 
There are marks on the legs showing the proper settings for the 
pivot so that the space between one pair of points will bear 
any desired ratio to the space between the other pair. The 
marks on the legs should not be accepted as correct, but should 
be tested by actual trial. One end of the proportional dividers 
is used to space off the distances from the original map and the 
other end used to plot that distance on the new map. Thus by 
means of this instrument a drawing can be enlarged or reduced 
to a definite scale without the use of the engineer's scale. 

A drawing which is to be made two-thirds the size of the 
original can be readily reduced by scaling the distances from the 
original with a 20-ft. scale and plotting them on the new draw- 
ing by use of a 30-ft. scale. But when the reduction is some 
odd ratio which cannot be readily accomplished by means of the 
engineer's scale proportional dividers are very useful. 

428. RAILROAD CURVES, FRENCH CURVES, FLEXIBLE 
CURVE, AND SPLINE. — For drawing arcs of curves of long 
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 PAPERS. 

432. The drawing papers used by surveyors may be divided 
into four general classes; (1) those used for plotting plans, (2) 
tracing paper or tracing cloth which is used for copying draw- 
ings, (3) cross-section and profile papers, and (4) process papers. 

433. DRAWING PAPER FOR PLANS.* — There are numer- 
ous grades of drawing paper ranging from very cheap "detail " 
to heavy paper mounted on cloth, called "mounted paper." 
For rough plots which are to be copied later or which are for 
temporary use only, a manilla detail paper is frequently used; 
but where the drawing is to be of a more permanent character 
a heavy white or manilla paper is used. Still more permanent 

* See Appendix C for description of how to mount drawing paper on cloth. 



DRAWING PAPERS 389 

plans, such as the plan of a survey of a city, should be plotted on 
heavy mounted paper. There is generally a right and a wrong 
side to all papers, which can be distinguished by the "water- 
mark "; this will read direct when the right side of the paper is 
toward the observer. A paper to be satisfactory for use should 
have a surface not too porous to take ink nicely, and of a fiber 
such that after scratching with a knife or rubbing with an ink 
eraser, the surface will still take ink effectively. No paper, 
however, after scratching can be expected to take bottle red ink, 
which permeates the fiber with extraordinary ease. 

434. TRACING PAPER 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. 

* Erasure 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 tliis 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. 

Trom 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 \" 9 -j^", -jV, or 
3^". 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 cloth form, suitable 
for use in making process prints. Profile papers usually come 
in long rolls 20 inches wide. 

438. PROCESS PAPERS. — Blue-Prints. — The most com- 
mon process paper used in drafting 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 
parts of each of them. 

Solution (1) 

Citrate of Iron and Ammonia, 1 part (by weight) 
Water, 5 parts (" " ) 

Solution (2) 

Red Prussiate of Potash (re- 

crystalized), 1 part (by weight) 

Water, 5 parts (" " ) 

The mixed solution is applied to the paper by means of a 
camel's hair brush or a sponge; this is done in a dark room or 
in subdued light. The paper is coated by passing the sponge 
lightly over the surface three or four times, first lengthwise of 
the paper and then crosswise, giving the paper as dry a coating 



39 2 DRAFTING INSTRUMENTS [Chap. XIV. 

as possible consistent with having an even coating; it is then 
hung up to dry. The above coating will require about 5 minutes 
exposure in bright sunlight; for quick printing paper, use a 
larger proportion of citrate of iron and ammonia. 

The blue-print of a plan is generally made in a printing 
frame, which is merely a rectangular frame holding a piece of 
heavy glass, with a back to the frame which can be lifted from 
the glass. This back is padded so as to firtight 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 blue-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 minutes in direct sunlight. In purchasing, then, 
it is necessary to ascertain from the dealer the " speed " of the 
paper and govern the exposure accordingly. Blue-prints can 
be made in cloudy weather as well as when the sun is visible, the 
only difference being that it requires a much longer time for the 
exposure. In all cases where the time of exposure is doubtful 
the following simple test may be applied. Instead of taking a 



BLUE-PRINTING 393 

print of the entire tracing the first time, use only a small piece 
of the blue- print paper and put it in the frame as explained above 
and expose it a given time. Take it out and wash it, and from 
this 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 off its face. The print should be left in the water 
for 10 to 20 minutes, then it is hung up to dry. It will dry more 
quickly if hung so that one corner is lower than the others. It 
shoulii not be hung where the sun will shine on it as the sun- 
light will fade it. 

In taking prints great care must be exercised not to get the 
tracing wet. When the prints are being washed the tracing 
should always be put in a safe place where the water will not 
spatter on it and it should never be handled with moist hands. 
It is practically impossible to eradicate the effect of a drop of 
water or even the marks made by damp fingers on tracing cloth; 
it is sure to show in every subsequent print which is taken from 
the tracing. 

440. Blue-print cloth is prepared in the same manner as 
the blue-print paper. Its advantage over the paper lies solely 
in the fact that it does not shrink as badly and is much more 
durable. Prints which are to be used on construction work 
where they are sure to get rough usage are sometimes made on 
cloth. 

441. Vandyke Solar Paper. — There has always been a call 
for a sensitive paper which will give positive prints, — a black, 
a brown, or a blue line on a white background. Such effect was 
secured by the old so-called "black print process," but its opera- 
tion was not altogether simple and good results were not reason- 



394 DRAFTING INSTRUMENTS [Chap. XIV. 

ably sure. The Vandyke paper has apparently solved this 
difficulty, and in addition affords other advantages which the 
old "black process " paper did not possess. 

Vandyke paper is a sensitized paper which is printed in the 
same way as a blue- print, except that the tracing is put into the 
frame so that the ink lines will be against the Vandyke paper. 
The exposure is about 5 minutes in direct sunlight or, more defi- 
nitely, until the portion of the Vandyke paper which protrudes 
beyond the tracing is a rich dark tan color. Fresh Vandyke paper 
is light 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 uncertainty of the 
sunlight for making prints has brought forward a printing frame 
in which an artificial light is used. 

One form of electrical printing frame is an apparatus con- 
sisting of a hollow glass cylinder, formed of two sections of glass, 
and resting on a circular base which is rotated by clock work. 
An electric light is suspended in the center line of the cylinder 
where it travels up and down by means of a clock work attach- 
ment. 

The tracing and paper are wrapped around the outer sur- 
face of the glass where they are tightly held against the glass by 
a canvas which is wound around the cylinder by means of a 
vertical roller operated by a handwheel. The cylinder can be 
rotated at any desired speed and the light which travels up and 
down the axis of the cylinder can be moved through any desired 
distance or at any desired speed. These motions are all made 
automatically when the apparatus is once adjusted. 

In another type of electrical machine several horizontal rollers 
are provided, with the light so arranged that as the tracing and 
blue-print paper passes from one roller to another the exposure 
is made. The speed of the machine is controllable and the length 
of the tracing that can be printed is limited only by the length of 
the roll of blue-print paper. With this machine, then, long 
plans or profiles can be printed without the necessity of frequent 
splicing which is required with other types of printing frame; 
furthermore the color of the print is also uniform throughout. 
The machine is driven by an electric motor. There are several 
machines of this general type on the market; some of them are 
provided with an apparatus for washing the prints as fast as they 
come from the machine. 

443. INKS AND WATER-COLORS. — Bottled ink, which is 
prepared in various colors, is used extensively on engineering 
drawings. The so-called "waterproof" inks differ from other 



39 6 DRAFTING INSTRUMENTS [Chap. XIV. 

inks in that a water-color wash can be put over the lines without 
causing them to "run." Bottled inks are satisfactory for most 
drawings, but when very sharp and fine hair-lines are required it 
is well to use the stick india ink. This is made by grinding the ink 
together with a little water in a saucer made for this purpose, 
until the ink is thick and black enough to be used. If the ink 
becomes dry it can be restored to as good condition as when first 
ground by adding water, a drop or two at a time, and rubbing it 
with a piece of cork or a pestle; if the water is added too rapidly 
the ink will flake. 

While the bottled black inks are fairly well prepared, the red 
inks are very unsatisfactory. They will sometimes run on paper 
where only very slight erasures have been made; in fact, on some 
of the cheaper papers red ink will always run. For tracing pur- 
poses red ink is wholly unsatisfactory, as it is impossible to ob- 
tain a good reproduction of a red ink line by any of the 
process prints. Where red lines are needed the use of scarlet 
vermilion water-color will 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 the place of the bottled colored inks, such as 
Prussian blue instead of bottled blue ink, or burnt sienna 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. 

METHODS OF PLOTTING* 

444. LAYING OUT A PLAN. — Laying out a plan requires 
careful work. If a good-looking plan is to be obtained this part 
of the work must be done with not a little judgment. Besides the 
plan of the survey or property the drawing must have a title, and 
sometimes notes and a needle to show the direction of the merid- 
ian. These must all be arranged so that the entire drawing 
when completed will have a symmetrical appearance. Often 
the plot is of such awkward shape that it is very difficult to lay 
out the drawing so that it will look well, and the draftsman's 
artistic instincts are taxed to the utmost to produce a satis- 
factory result. 

445. Scale. — In many cases the scale of the plan as well as 
the general arrangement of its parts must be chosen by the 
engineer. Surveys of considerable extent which do not contain a 
great many details, such, for example, as the preliminary survey 
for a railroad, may be drawn to a scale of 400 ft. to an inch. A 
plan of a large piece of woodland or a topographical map of a 
section of a town may be represented on a scale of from 100 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 jjsiw, but some special maps are 
made on scales as large as *tjW The field maps of the U. S. 
Geological Survey are mostly plotted to a scale of **£&* and 
reduced on the lithograph sheets to rota* or m^nnr- 

* For a brief description of different projections for maps of large areas, such 
as states or counties, see Volume II, Chapter X. 

397 



39 8 METHODS OF PLOTTING [Chap. XV. 

These remarks in regard to scales are not to be con- 
sidered in any sense as hard and fast rules to govern all condi- 
tions. They are suggested simply to give some idea of the exist- 
ing practice in this matter. 

METHODS OF PLOTTING TRAVERSES. 

446. PLOTTIWG 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 off by using the proper scale, and its two ex- 
tremities accurately marked by pencil dots or by means of a 
needle point, and surrounded by a light penciled circle. The line 
should be drawn so that it will extend beyond the next angle 
point a distance greater than the radius of the protractor, this 
extension of line being of use in the manipulation of the pro- 
tractor. 

The protractor is placed so that its center is exactly on the 
second angle point and so that both the 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 

If the traverse is a closed traverse the plot, of course, should 
close 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 off 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 close can be computed and the angle the last line makes with 
the meridian measured. If it checks the computed angle it is 
evident that no error has been made in the angles unless 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 paper 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 laid off or checked as illustrated 
by Fig. 1 74 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 4-OI 

It is evident that the bearings of the lines may be computed 
just as well from any assumed meridian as from the magnetic or 
true meridian ; and that the drawing can be fastened to the board 




Fig. 174. Laying off Bearings by Use of T-Square and 
Protractor. 

in such a way that the T-square can be conveniently used. 
This method is especially applicable to compass surveys as it ob- 
viates the necessity of drawing a new meridian line through each 
angle point. 

This method can be easily applied also as a means of check- 
ing any of the angles of a traverse which have been plotted by 
any of the ordinary methods. 

449. PLOTTING BY RECTANGULAR COORDINATES.— In plot- 
ting by this system all points in the traverse are referred to a 
pair of coordinate axes. For convenience these axes are often 
the same as those used in calculating the area enclosed by the 
traverse. The advantages of this method are, (1) 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 A, marked on the paper. 
Point C is also marked opposite a certain definite part of the 
triangle. Then the triangle is reversed to the dotted position 
and brought so that its edge coincides with point A, and then 
point B is marked opposite point C, as nearly as can be judged. 
A point D is plotted midway between B and C and the line AD 
is then drawn which is perpendicular to the straight-edge. If the 
triangle is accurate point B will fall on point C, so that this 
is a method of testing the accuracy of the right angle of any tri- 
angle. If it is found to be inaccurate it should be sent to an» 
instrument maker and be "trued up." A few cents spent in 
keeping drafting instruments in shape will save hours of time 
trying to locate small errors, which are often due to the inac- 
curacy of the instruments used. 

If the compass is used the right angle may be laid off by 
geometric construction. On account of the difficulty of judging 
the points of intersection of the arcs, very careful work is re- 
quired to obtain good results with the compass. 

Since the accuracy of all of the subsequent work of a coordi- 
nate plot depends upon the accuracy with which the rectangle 
is constructed, great care should be taken to check this part of 
the work. The opposite sides of the rectangle should be equal 
and the two diagonals should be equal, and these conditions 
should be tested by scaling or with a beam compass before con- 
tinuing with the plot. 

452. After the rectangle has been constructed, all points in 
the survey can be plotted by use of the scale and straight-edge. 
To plot any point, lay off its total latitude on both the easterly 
and the westerly of the two meridian lines of the rectangle, be- 
ginning at the southerly line of the rectangle. Draw a line 
through both of these points by means of a straight-edge.* 

* Accurate work, of coarse, 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 



Kt.71 >G 




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 
off. 

455- PLOTTING BY TANGENTS. — The traverse should first be 
plotted approximately on some convenient small scale by use of 
the protractor and scale, to ascertain its extent and shape. The 
importance of this little plot is often overlooked, with the result 
that when the plan is completed it is found to be too close to one 
edge of the paper or otherwise awkwardly located on the sheet. 
It takes only a few moments to draw such a sketch, and unless 
the draftsman is sure of the shape and extent of the plot he 
should always determine it in some such manner before the plan 
is started. 

The directions of all the lines are referred to some meridian 



406 



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, 



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 10-inch base-lines are all laid off by use of the 
engineer's scale. By using the 10-ft. or 100-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 9 the 
length of the first course. On this line produced lay off Bg equal 
to 10 inches and erect a perpendicular, scaling off on the perpen- 
dicular the length gd equal to the tangent of the deflection angle 
at B. This determines the direction of BC from the first course. 
Theremaining 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 De f for example, a 
meridian is drawn through D parallel to Aa, De is scaled off 10 
inches, and a perpendicular ef erected. The distance ef\s scaled 
and from the table of tangents the angle fDe is obtained. If the 
angle that the course makes with the meridian line disagrees 
with the calculated bearing of that course by any considerable 
amount, say, 10 minutes of angle or more, the previous courses 
should be replotted. If the error is less than 10 minutes the 
course which is being checked should be drawn in the correct 
direction so that even the slight error discovered may not be 
carried further along in the plot. Then after the plotting has 
proceeded for three or four more courses the check is again 
applied. 

The bearings of the lines can be checked by use of the pro- 
tractor and this will detect errors of any considerable size, but 
this method will not disclose any small errors ; moreover, if it is 
desired to have the plot when completed as accurate as could be 
expected from the precise method employed, it is entirely incon- 
sistent to check by use of a method which is far less accurate 
than the one used in making the plot. For this reason the 
checks on the direction of the lines are applied with the same 
care and by the same method as was used in the original layout 
of the angles. 

Occasionally it is more convenient to plot the complement of 
an angle rather than the angle itself, as was done in plotting the 
line EF. In this case the right angle erected at E must be laid 



4o8 



METHODS OF PLOTTING 



[Chap. XV. 



off with great care, preferably by the method explained in Art. 
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. PLOTTnTG 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 ABCDEF which has been 




Fig. 178. Plotting by Chords. 

plotted by chords. It is the same traverse that is shown in 
Fig. 177. 

On the meridian line the distance Aa is scaled off equal to 
10 inches and the arc ab swung from A as a center by use of 
the ordinary pencil compass. Then from a table of chords * the 

* Tables of chords can be found in Trautwine's " Civil Engineer's Pocket 
Book," published by John Wiley & Sons, New York. 



PLOTTING BY CHORDS 409 

length of the chord ab is found for the angle aAb. The point b 
is sometimes located by setting the dividers at the distance ab 
and with a as a center intersecting the arc ab at b; but the 
more accurate method is to scale from point a the chord distance 
and mark the point £on the arc. Then the line Ab is drawn and 
AB scaled off on it. With B as a center the arc gd is drawn 
and the chord gd, corresponding to the deflection angle at. B, is 
scaled off. Bd is then drawn and BC scaled off on it. In the 
same way the entire traverse is plotted. 

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

• A 
ab = 2X iox sin — - 
2' 

hence, if a table of chords is not available, a table of sines (always 
easily obtainable) can be used. The sine of half the angle can 
be taken from the tables and multiplied by 20 mentally. Some 
draftsmen use the table of sines and a radius of 5 inches to avoid 
the multiplication. This is not recommended because a base of 
5 inches is not long enough to insure a very accurate drawing. 
The necessity of multiplying by 2 can very easily be done away 
with by laying off the radius with a 20-ft. scale and scaling off 
the sine of the angle with a 10ft. scale. 

With dividers of the ordinary size it is impossible to lay out 
an arc with a 10-inch radius. In such a case either beam com- 
passes must be used or the radius employed must be shorter, so 
short, in fact, that it will frequently be better to resort to the 
Tangent Method. 

459. Checks. — Since this method is usually applied to 
traverses which do not close it is desirable to check every 
fourth or fifth course so that a mistake will not be carried too 
far before it is discovered and thereby cause a waste of time. 
In Fig. 1 78 it is desired to check the calculated bearing of De. 
The meridian Df is drawn through D parallel to Aa, the arc/* 
is swung with D as a center and with a radius of 10 inches, and 
the chord ef\s 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 



41 METHODS OF PLOTTING [Chap. XV. 

that suggested for the Tangent Method in Art. 455, unless the 
beam compass is used. The Tangent Method, especially if the 
right angles are laid off by reversing the triangle, gives more 
accurate results than the Chord Method, for the use of the 
ordinary compass in the Chord Method is a fruitful source of 
error unless it is handled with the utmost care. 



METHOD OF PLOTTING DETAILS. 

460. BUILDINGS, FENCES, STREAMS, ETC. — The previous 
articles have dealt with the plotting of the traverse lines only, 
and these in many cases form merely the skeleton of the final 
plan. In the field the details of the survey are located from the 
transit line ; and, in a similar manner, the details are located on 
the plan from the traverse line which has already been plotted. 

Buildings, fences, shore-lines, streams, etc. are all plotted by 
means of the scale for distances and the protractor for the 
angles. Often a smaller protractor is used for this sort of work 
than for the traverse lines. This is permissible, for the lines 
which locate the details are usually short in comparison with 
the traverse lines and the resulting error is small in any case ; 
furthermore any slight error in the location of a detail will not 
as a rule affect the rest of the drawing, whereas an error in a 
transit line will, of course, have an effect on all of the rest of 
the drawing. The plotting of buildings has been taken up in 
connection with their location. (See Chapter VI.) 

In plotting a set of notes where several angles have been 
taken at one point, such as in stadia surveying, it is well to plot 
all of the angles first, marking them by number or by their 
value, and then to plot the distances with the scale. 

461. CONTOURS. — Where contours are located by the cross- 
section method (Art. 304, p. 278), this cross-section system is 
laid out in soft penciled lines on the drawing. The elevations 
which were taken are written at their respective points on the 
plan and then the contours desired are sketched. The ground 
is assumed to slope uniformly between adjacent elevations, and, 
by interpolation between these points, the location of the con- 
tours on the plan can be made. When the contours have been 



PLOTTING DETAILS 4 1 1 

located, the cross-section lines and elevations are erased unless 
the plan is intended to be used as a working drawing. As a 
rule all useful data, such as construction lines and dimensions, 
are left on a working drawing. 

When the contours are located by any other means the 
principle is the same. The points whose elevations have been 
determined are plotted by scale and protractor, and the contours 
are interpolated between the 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. 

Wh^fe 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, 5 to 20 times the horizontal scale, depending upon 
how close it te desired to read the elevations from the drawing. 
If the horizontal scale of the profile is 80 ft. to an inch its ver- 
tical scale should probably be 20, 10, or 8 ft. to an inch. 

464. In plotting any profile the first step is to lay it out 
properly on the paper, i.e., to decide, from an examination of the 
range of the elevations, where to start it on the paper so that it 
will look well when completed, and so that any additions or 
studies which may subsequently be drawn on it will come within 
the limits of the paper. Station o of the profile should come on 
one of the heavy vertical lines, and the heavy horizontal lines 
should represent some even elevation such as 100, 125, 150, etc 

The profile is plotted by using the rulings of the profile 
paper as a scale ; it is drawn in pencil first and afterward inked 
in. It will be found, if these profile papers are carefully meas- 
ured with a scale, that they are not as a rule very accurate. 
The rulings may be uniform, but owing to the shrinkage of the 
paper the divisions frequently do not scale as long as they should. 
In plotting a profile or section on such paper no attempt is made 
to use a scale ; the scale of the paper is assumed to be correct 
and the intermediate points are plotted by estimation, which can 
almost always be accurately done since the rulings of the paper 
are quite close together. 

The data for a profile of the ground generally consist of 
levels taken in the field at such points that the ground may be 
assumed to run straight between adjacent elevations. For this 
reason, in drawing the profile, the points where the slope of the 
ground changes should not be rounded off. On the other hand, 
however, the ground probably does not come to an actual angle 
at that point. The profile should be plotted therefore as a series 
of free-hand straight lines drawn so that the angles are not 
emphasized. When a profile is made from a contour map, the 
line should be a smooth, rather than an angular line. 

465. Profiles of the surface of the ground are generally 
made for the purpose of studying some proposed construction 



PLOTTING PROFILES 413 

which is represented on the profile by a grade line, consisting 
usually of a series of straight lines. The points where the gra- 
dient changes are plotted and connected by straight ruled lines 
unless the proposed grade should happen to be a vertical curve 
(Art. 268, p. 242). Vertical lines are also drawn from the 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 off the top or bottom of the paper the entire 
surface line is lowered or raised some even number of feet, such 
as 20 or 50 ft., and the plotting continued : the number of feet 
represented between two heavy horizontal rulings of the pro- 
file paper should determine the drop or rise of the grade line. 
This change should be made, when convenient, on one of the 
heavy vertical rulings of the 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. Hot 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. 



««aj& 




uqsstuiwoo Wy ¥W *W 



CHAPTER XVI. 

FINISHING AND FILING DRAWINGS.* 

468. WSAT SHOULD APPEAR ON A DRAWING. — Drawings 
are made for a great variety of purposes, so that the data which 
a plan should contain depend entirely upon the use to which it 
is to be put. There are, however, several important things which 
should appear on every engineering drawing. In the first place, 
it should have a complete title which should be a brief descrip- 
tion of the drawing. The title should state whether the drawing 
is a plan, cross-section, profile, etc. ; what it represents, — a lot 
of land, a sewer, a railroad, etc. ; the name of the owner ; the 
place ; the date ; the scale ; and the name of the surveyor. Be- 
sides the title, some plans, such as land plans, always require the 
names of owners of abutting property, and a meridian. Notes 
are frequently added giving such information as is necessary to 
interpret the plan. All essential dimensions are lettered in their 
proper places. 

Besides these it is well to insert in some inconspicuous place 
(preferably near the border) the number of the note-book and the 
page from which the notes were plotted, and also the initials of 
the draftsman who made the drawing and of the man who 
checked it. 

Fig. 1 79 represents a land plan which contains all of the es- 
sentials ; it is a plot of the land shown in the form of notes in 
Fig. 52, p. 103; its computations are on p. 362; and its working 
plot is illustrated by Fig. 176, p. 404. 

469. TRAVERSE LINES. — The convenient use of a plan some- 
times requires the traverse line to be shown on the completed 
drawing. In such a case it is usually shown as a full colored line, 
each of the angle points being represented by a very small circle 
of the same color, the center of which marks the angle point. 
Sometimes the lines of the traverse are drawn to the angle points 

* For methods of finishing topographic and hydrographic maps see Volume II, 
Chapter XI. 



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 marking transit points. 




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



Tnangulation stations are represented by a small equilateral 
triangle drawn around the station point. Fig. 115, p. 258, con- 
tains several examples of this. 

470. PHYSICAL FEATURES. — The boundaries of property 
and the physical features which are represented on a plan, such 
as streets, buildings, etc., are usually drawn in black ink. Any 
additions or proposed changes are frequently drawn in colored 
ink, usually in red, although water-color is much better for 
the reasons stated in Art. 443, p. 395. 

Shore lines and brooks are represented either in black or in 
Prussian blue. As a rule the shore line should be one of the 
heaviest, if not the heaviest line, on the drawing. Water-lining, 
shown in the topographical signs in Fig. 181, adds materially to 
the prominence and appearance of a shore line. 

471. Topographic Conventional signs. — On topo- 
graphic maps certain physical features are shown by conventional 
signs which have come to be used so generally that they are 
practically standard throughout the country. A few of the 
more common of these symbols are shown in Fig. 181. The 
one representing " cultivated land " and the horizontal lines of 
the "salt marsh" and "fresh marsh " symbols are ruled; the 
rest are executed with an ordinary pen, Gillott's No. 303 being 
a good one for such work. (See also Volume II, Chapter XI.) 

It will be noticed that in the symbol for "grass" the indi- 
vidual lines of a group all radiate from a center below the group, 
and also that they end on a horizontal line at the bottom. This 



CONVENTIONAL SIGNS 



417. 







fc w, » ft rt . ^ ». •* v 



Deciduous Trees (Oak). Deciduous Trees (Round Leaf). 



*m ***• Mi» 


■MM4, 


jHU. 




*»»/*. 






#o*. 4^ 




*lfc* 




*M*». .Ofe 


-*»<*, 


*M*» 


40*. 


41* 


*4*. 




>1fc *ft». ^_ 


« 


,*w* r 




•^Uftk 




<•!/•» 






*|l// 






#'** 










.av*. 


Alfc*. 










v**/*. 

*»•». 




_£a 







Grass. 



Cultivated Land. 





Fresh Marsh. 



Salt Marsh — Sand. 





Waterlining. 



Ledges — Evergreen Trees. 



Fig. 181. Topographic Conventional Signs. 



41 8 FINISHING AND FILING DRAWINGS [Chap. XVL 

horizontal line, in the case of "grass" or "marsh" symbols, 
should always be parallel to the bottom of the map. 

In executing "water-lining " the first line outside the shore 
line should be a light full line drawn just as close to the shore 
line as possible, and should follow very carefully every irregular- 
ity of the shore line. The next water-line should be drawn 
parallel to the first but with a little more space between them 
than was left between the shore line and the first water-line. 
Then the third water-line should be spaced a little farther out, 
and so on ; five to ten lines are sufficient to represent this sym- 
bol properly. As the succession lines are added farther and 
farther from the shore line, the little irregularities of the shore 
gradually disappear until the outer water-line shows only a few 
irregularities opposite the most prominent ones of the shore. 

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. 




*■ -- 



~^-= — r 



■i 1 1 1 1 1 1 1 1 1 
1 1 1 M 1 1 1 1 



=£ 



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. 



^..s 


Trail. 


AV> 


Bridge. 


A 


Triangulation Station. 


Q 


Stadia Station. 


O 


Transit Point. Intersection Point 


an x 1232 


Bench Mark. 0» Triangula.) 



Fig. 183. 



420 



& 




3 




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QRST 


N 
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QRST 


N 

>> 

X 

> 


Oh 


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Q. 


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00 ? - 


O 


CD 


O 


(0 


S 3 


£o3 


tr 1 


Zo8 




^1 NJ 


§Na 


2N 


Q_ 


§* ^ ^ ^ 

3 Cs ^ o 


5 Jl* 


o 
c 


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O 

r 


1 s & * 


3^ 


•2 


°2£ 


E 


? 3 IS *> 


£> 


:;? 


x> 


:r> 


H* 


O 


J 


O 


x 


^t^ 


fc 

H 





L. 
U 







r i 

P 




Q 


o 


5 ■? 


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pQ 


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SI 


S s 


K 


cd 


CD 


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5 s 


< 




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^ 2 



421 




422 



FINISHING AND FILING DRAWINGS [Chap. XVI 



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

473. LETTERING.* — 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 illustrations 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 Gothic. 

All plans should be lettered so as to read from the bottom. 
Unless a draftsman exercises considerable care he will find, 
when the plan is completed, that some of the lettering is upside 
down. Fig. 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. XVL 

Preliminary Survey for cl RallrocLd 
fyonv 

Crescent J3eeic/i to Wboctlawn Cemetery. 

October. J802, 
Scale 400 Deet to / inch. 

COMMONWEALTH OF MASSACHUSETTS. 

METROPOLITAN WATER WORKS. 

WACHUSETT DAM 



UPPER GATE-CHAMBER. 



JULY 9. 1900. 

UNITED STATES 
COAST AND GEODETIC SURVEY 

SKETCH OF GENERAL PROGRESS 

JUNE 30 1897 
Eastern Sheet 



Fig. 187. Titles of Plans. 



TITLES 425 



TRACK ELEVATION. 

C. 6c W I. R. R. 

Cross- Section of Bridge Showing 

Floor Construction. 
Scale i in. » I ft. 



HORIZONTAL SECTIONS 

THROUGH UPPER THROUGH LOWER THROUGH LOWER 

SLUICE-GATE SLUICE-GATE VALVE WELL 

o t t S4jjrt 



PneliminaiyRofile 

for a Railroad from 

Redford Junction to North Liberty 

Sta.0 to Sta.498+08.7 
Alay/906 



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 /s 8 L "Long, 
and comes dfrecf/y under each track. 
Leave amp/e room for bridge-seat 

Note:-The datum plane used for con- 
tours and soundings on this map is 
"Boston City Base? 
Boston City Base is 0.64 ft below 
base known as"Mean Low Water at 
Navy Yard" which is the datum used 
by the U.S. Coast Survey, the U.S. 
Engineers Office, and the Mass. 
Harbor and Land Commission. 

Soundings and Contours confirmed and ex- 
tended by data from map (L-476)on fife with 
Massachusetts 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, 1" 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. 
j 91 are suggested as suitable for ordinary land plans. 

The plan should always be drawn, if possible, so that the 



428 



FINISHING AND FILING DRAWINGS [Chap. XVI. 



meridian will point, in general, toward the top of the drawing 
rather than toward the bottom. Sometimes it is drawn with its 
upper part above and its tail below the drawing. In such a case 




Fig. 191. 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 sufficiently 
long, say, 3 to 10 inches (depending upon the size of the draw- 
ing), so that it will be of use in detecting the amount of shrink- 
age. This, of course, will determine the shrinkage only in the 
direction of the scale. These scales are usually placed directly 
under the title or in one of the lower corners. Fig. 192 gives 
two examples of scales. 

In plotting a coordinate survey, the intersections of the north 
and south with the east and west lines should be marked on the 
finished drawing, as these are of great assistance in plotting addi- 
tions. Moreover the distances between these points give a 
reliable measure of the change in scale of the map due to 
shrinkage. 

479. SHRINKAGE OF DRAWING PAPERS. — All of the papers 
in use will shrink and swell more or less with variations of 



MERIDIANS — SCALES 429 

weather conditions. The heavy mounted papers are affected 
the least, but large drawings even on such paper will be found 
on examination to change in size perceptibly. The fact that they 
do not always shrink the same amount in different directions 

SCALE ^5 

?■-* f "*° 3C IT™ 

? 1 \ \ IT 



SCALE, I INCH - 200 PEET 



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 different sizes of draw- 
ings. In some offices the adherence to standard sizes is very 
rigid* and considerable time is often spent to bring drawings 
within the limits of one of these sizes. When these sizes are 
exceeded the plans are either made in sections of standard size, 
as explained in Art. 480, or they are made as large plans which 
are rolled and filed away in pasteboard tubes. Sometimes very 
large plans are filed flat by hanging them from an overhead 
frame. 

Plans filed flat are marked each with its proper index 
number in one corner, preferably the lower right-hand corner, 
so that as the drawer is opened the numbers can be readily exam- 
ined. In some offices it is required that in returning a draw- 
ing it shall be placed in its proper order in the drawer as well 
as in the proper drawer, while in other offices the plan drawers 
are made very shallow, so as to contain only about 15 or 20 
drawings, and when a plan is returned no attempt is made to 
put it in any particular place in the drawer, there being, at the 
most, only a very few drawings to handle to obtain the one 
desired. 

Rolled drawings are marked on the side of the rolls at each 
end so as to be easily read by one standing in front of the shelf 
on which the plans are stored. Another style of roll is closed 
at one end with a white label on the outside of the closed end. 
When the plan has been put into the tube it is so placed on the 



43 2 FINISHING AND FILING DRAWINGS [Chap. XVL 

shelf that the label on which the plan number is marked is at 
the front edge of the shelf where it can be conveniently read 
When the plan is in use the empty tube is left on the shelf 
with its open end outward so that its number is in the back part 
of the shelf where it cannot be read. 

Large plans which are made in sections are often filed in 
large folios or books in such a way that they can be readily 
taken out and used separately. 

484. INDEXING DRAWINGS. — There are so many systems of 
indexing plans that no attempt will be made to explain them other 
than to suggest a few of the essentials of any good system. 
Every system of numbering the plans should be such that one 
can tell from its number whether the drawing is a sketch, a 
working drawing, a finished drawing, a tracing, or a process 
print. The numbering also should suggest the type of draw- 
ing, as a land plan, a construction plan, etc. • 

For offices where few plans are on file an index book may 
suffice for recording the plans, but in large drafting offices the 
card catalogue system is used extensively. By a judicious use 
of "markers" a card catalogue system can be so devised that 
it will be necessary to examine only a very few cards to find 
the one corresponding to any plan. Frequently it is necessary 
to index a plan by two or three different cards under different 
general headings. 

485. FILING NOTE-BOOKS. — Note-books should always be 
filed in vaults where they will be protected against fire. Too 
frequently through lack of forethought note-books containing 
information which it has cost thousands of dollars to collect are 
carelessly filed on a shelf in the drafting office. In some offices 
the rules require that every note-book and valuable plan shall 
be placed in the vault at the end of the day's work, and this 
appears to be the proper practice. 

Some offices go so far as to require that all notes shall be 
copied in ink and the original notes kept permanently filed in 
the vault to guard against their loss. Whether a copy is made 
or not, the original should be preserved as it has a value, in 
a lawsuit for instance, which any copy does not possess. When 
copies are made of the original notes they are sometimes made 



INDEXING — FILING 43 3 

in a loose-leaf book so that if any notes are taken from the office 
it is not necessary to take more than a very few leaves of the 
copy; the original notes never go from the office except in rare 
cases. 

486. Indexing Notes. — The notes contained in the field note- 
books are often indexed either in a book for this purpose or by 
means of a card catalogue. The method of indexing is similar 
to that used for plans. 

487. Other Records. — Other records, such as borings, sound- 
ings, estimates, computations, etc.. are carefully filed and indexed 
so that it will be easy to refer to them. 



TABLES. 



435 



43 6 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


100 


00000 00043 00087 00130 00173 00217 00260 00303 00346 00389 


1 


0432 0475 0518 0561 0604 0647 0689 0732 077$ 0817 


8 


0860 0003 0045 0088 1030 1072 1115 1157 1199 1242 


8 


1284 1326 1368 1410 1452 1494 1536 1578 1620 1662 


4 


1703 1745 1787 1828 1870 1912 1953 1995 2086 2078 


6 


2119 2160 2202 2243 2284 2325 2366 2407 2449 2490 


6 


2531 2572 2612 2653 2694 2735 2776 2816 2857 2898 


7 


2938 2979 3019 3060 8100 8141 8181 3222 3262 3302 


8 


3342 3383 3423 3463 3503 3543 8583 3623 9663 3703 


9 


3743 3782 3822 3882 3902 3941 3981 4021 4060 4100 


110 


04139 04179 04218 04258 04297 04336 04376 04415 04454 04493 


1 


4532 4571 4610 4650 4689 4727 4766 4805 &44 4883 


8 


4922 4961 4999 5038 5077 5115 6154 5192 6231 6269 


8 


5308 5346 5385 5423 6461 5500 5538 5676 6614 6652 


4 


6690 5729 5767 5805 5843 6881 6918 5956 6994 6032 


6 


6070 6108 6145 6183 6221 6258 6296 6388 6371 6408 


6 


6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 


7 


6819 6866 6893 6930 6967 7004 7041 7078 7115 7161 


8 


7188 7225 7262 7298 7335 7372 7408 7445 7482 7618 


9 


7565 7691 7628 7664 7700 7787 7773 7809 7846 7882 


120 


07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 


1 


8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 


8 


8636 8672 8707 8743 8778 8814 8849 8884 8920 8955 


8 


8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 


4 


9342 9377 9412 9447 9482 9517 9552 9587 9621 9656 


6 


9691 9726 9760 9795 9830 9864 9899 9934 996810003 


6 


10037 10072 10106 10140 10175 10209 10243 10278 10812 0346 


7 


0380 0415 0449 0483 0517 0561 0586 0619 0653 0687 


8 


0721 0755 0789 0823 0857 0890 0924 0958 0992 1025 


9 


1059 1093 1126 1160 1198 1227 1261 1294 1327 1361 


180 


11394 11428 11461 11494 11528 11661 11594 11628 11661 11694 


1 


1727 1760 1793 1826 1860 1893 1926 1959 ,1992 2024 


8 


2057 2090 2123 2156 2189 2222 2254 2287 2320 2352 


8 


2385 2418 2450 2483 2516 2648 2581 2613 2646 2678 


4 


2710 2743 2775 2808 2840 2872 2905 2937 2969 3001 


6 


3033 3066 8098 8130 3162 3194 3226 3258 3290 3322 


6 


3354 3386 3418 8450 3481 3513 3645 3577 3609 3640 


7 


3672 3704 8735 3767 3799 3830 3862 3893 3925 9956 


8 


3988 4019 4051 4082 4114 4145 4176 4208 4239 4270 


9 


4301 4333 4364 4395 4426 4457 4489 4520 4651 4582 


140 


14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 


1 


4922 4953 4983 5014 5045 6076 5106 6137 6168 6198 


8 


5229 5259 5290 6320 6351 6381 6412 6442 6473 6603 


8 


6534 5564 5594 6625 5655 6685 6716 6746 6776 6806 


4 


5836 6866 6897 5927 5957 6987 6017 6047 6077 6107 


6 


6137 6167 6197 6227 6256 6286 6316 6346 6876 6406 


6 


6435 6465 6495 6624 6554 6584 6613 6643 6673 6702 


7 


6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 


8 


7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 


9 


7319 7348 7377 7406 7435 7464 7493 7522 7551 7580 


160 


17609 17638 17667 17696 17725 17764 17782 17811 17840 17869 



437 
TABLE I.— LOGARITHMS OF 1TOMBERS. 



N 


0123456789 


160 


17609 17638 17667 17696 17725 17764 17782 17811 17840 17869 


1 


7898 7926 7965 7984 8013 8041 8070 8099 8127 8166 


2 


8184 8213 8241 8270 8298 8327 8366 8384 8412 8441 


8 


8469 8498 8626 8564 8583 8611 8639 8667 8696 8724 


4 


8762 8780 8808 8837 8865 8893 8921 8949 8977 9005 


6 


9033 9061 9089 9117 9145 9173 9201 9229 9267 9285 


6 


9312 9340 9368 9396 9424 9451 9479 9507 9535 9562 


7 


9590 9618 9645 9673 9700 9728 9756 9783 9811 9838 


8 


9866 9893 9921 9948 9976 20003 20030 20058 20065 20112 


9 


20140 20167 20194 20222 20249 0276 0303 0330 0358 0385 


160 


20412 20439 20466 20493 20520 20648 20575 20602 20629 20656 


1 


0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 


8 


0962 0978 1005 1082 1059 1085 1112 1189 1165 1192 


8 


1219 1245 1272 1299 1325 1352 1378 1405 1431 1458 


4 


1484 1611 1637 1664 1590 1617 1643 1669 1696 1722 


6 


1748 1775 1801 1827 1864 1880 1906 1982 1968 1985 


6 


2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 


7 


2272 2298 2324 2350 2376 2401 2427 2453 2479 2505 


8 


2531 2557 2583 2608 2634 2660 2686 2712 2737 2763 


9 


2789 2814 2840 2866 2891 2917 2948 2968 2994 8019 


170 


23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 


1 


8300 8325 3350 3376 3401 3426 3452 3477 3602 8528 


9 


8563 3578 3603 8629 3654 3679 3704 8729 3764 3779 


8 


3805 8830 8865 3880 3905 3930 3965 3980 4005 4030 


4 


4055 4080 4105 4130 4155 4180 4204 4229 4264 4279 


6 


4304 4329 4353 4378 4403 4428 4452 4477 4502 4527 


6 


4551 4576 4601 4625 4650 4674 4699 4724 4748 4773 


7 


4797 4822 4846 £871 4895 4920 4944 4969 4993 5018 


8 


6042 6066 6091 5115 5139 6164 6188 6212 6237 6261 


9 


6286 6310 6334 5358 5382 5406 5431 5455 5479 5503 


180 


26627 26651 26675 26600 26624 26648 25672 25696 26720 26744 


1 


6768 6792 6816 5840 6864 5888 6912 6935 6069 5983 


2 


6007 6031 6055 6079 6102 6126 6160 6174 6198 6221 


8 


6245 6269 6293 6316 6340 6364 6387 6411 6435 6458 


4 


6482 6605 6529 6553 6576 6600 6623 6647 6670 6694 


6 


6717 6741 6764 6788 6811 6834 6868 6881 6905 6928 


6 


6951 6975 6998 7021 7045 7068 7091 7114 7138 7161 


7 


7184 7207 7231 7264 7277 7300 7323 7346 7370 7393 


8 


7416 7439 7462 7485 7608 7631 7554 7677 7600 7623 


9 


7646 7669 7692 7715 7738 7761 7784 7807 7830 7862 


190 


27875 27898 27921 27944 27967 27989 28012 28035 28068 28081 


1 


8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 


8 


8830 8353 8375 8398 8421 8443 8466 8488 8511 8533 


8 


8656 8678 8601 8623 8646 8668 8691 8713 8735 8758 


4 


8780 8803 8825 8847 8870 8892 8914 8937 8959 8981 


6 


9003 9026 9048 9070 9C92 9115 9137 9169 9181 9203 


6 


9226 9248 9270 9292 9314 9330 9358 9380 9403 9425 


7 


9447 9469 9491 9513 9535 9557 9579 9601 9623 9645 


8 


9667 9688 9710 9732 9764 9776 9798 9820 9842 9863 


9 


9885 9907 9929 9951 9973 9994 30016 30038 30060 30081 


200 


30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 



438 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


200 


90103 80122 80146 90168 30190 90211 80299 90262 30276 80296 


1 


0920 0341 0369 0384 0406 0428 0440 0471 0492 0614 


8 


0696 0667 0678 0600 0621 0649 0664 0686 0707 0728 


8 


07^0 0771 0702 0814 0896 0866 0878 0899 0920 0942 


4 


0969 0984 1006 1027 1048 1069 1091 1112 1199 1164 


6 


1176 1197 1218 1299 1260 1281 1802 1929 1342 1966 


6 


1987 1408 1429 1420 1471 1492 1619 1694 1666 1676 


7 


1697 1618 1689 1660 1681 1702 1729 1744 1762 1786 


8 


1806 1827 1848 1869 1890 1911 1991 1962 1979 1994 


9 


2012 2096 2066 2077 2098 2118 2139 2160 2181 2201 


210 


92222 92249 92269 92284 82902 82826 82946 82366 82387 82408 


1 


2428 2449 2469 2490 2610 2631 2662 2672 2693 2613 


8 


2634 2664 2672 2692 2716 2736 2766 2777 2797 2818 


8 


2838 2868 2879 2899 2919 2940 2960 2980 3001 8021 


4 


8041 3062 3082 3102 3122 8143 3168 3183 8208 8224 


6 


3244 3264 3284 3304 8322 8842 8862 8386 8406 8426 


6 


3446 3466 8486 3606 3626 3646 3666 8686 3606 8626 


7 


3646 3666 8686 3706 3726 3746 3766 3786 3806 8826 


8 


3846 3866 3886 3906 3926 3946 8966 3982 4002 4022 


9 


4044 4064 4084 4104 4124 4143 4163 4183 4208 4223 


220 


84242 34262 34282 34301 34321 84341 34861 34880 34400 84420 


1 


4439 4469 4479 4498 4618 4637 4667 4677 4696 4616 


8 


4636 4662 4674 4694 4713 4733 4763 4772 4792 4811 


8 


4830 4820 4869 4889 4908 4928 4947 4967 4986 6006 


4 


6022 6044 6064 6083 6102 6122 6141 6160 6180 6199 


6 


6218 6238 6267 6276 6296 6312 6384 6363 6372 6392 


6 


6411 6430 6449 6468 6488 6607 6626 6646 6664 6683 


7 


6603 6622 6641 6660 6679 6698 6717 6786 6766 6774 


8 


6793 6813 6832 6861 6870 6889 6908 6927 6946 6962 


9 


6984 6008 6021 6040 6069 6078 6097 6116 6186 6164 


280 


36173 36192 36211 36229 36248 36267 36286 36302 86324 36842 


1 


6361 6380 6399 6418 6436 6466 6474 6493 6611 6630 


2 


6649 6668 6686 6602 6624 6642 6661 6680 6698 6717 


8 


6736 6764 6773 6791 6810 6829 6847 6866 6884 6903 


4 


6922 6940 6969 6977 6996 7014 7033 7061 7070 7088 


6 


7107 7125 7144 7162 7181 7199 7218 7286 7264 7273 


6 


7291 7310 7328 7346 7862 7383 7401 7420 7438 7467 


7 


7472 7493 7611 7630 7548 7666 7682 7603 7621 7689 


8 


7668 7676 7694 7712 7731 7749 7767 7786 7803 7822 


9 


7840 7868 7876 7894 7912 7931 7949 7967 7982 8003 


240 


38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 


1 


8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 


2 


8382 8399 8417 8435 8453 8471 8489 8607 8622 8643 


8 


8561 8678 8596 8614 8632 8620 8668 8686 8703 8721 


4 


8739 8767 8772 8792 8810 8828 8846 8863 8881 8899 


5 


8917 8034 8952 8970 8987 9006 9023 9041 9068 9076 


6 


9094 9111 9129 9146 9164 9182 9199 9217 9232 9262 


7 


9270 9287 9302 9322 9340 9358 9376 9393 9410 9428 


8 


9445 9463 9480 9498 9515 9533 9560 9668 9686 9602 


9 


9620 9637 9652 9672 9690 9707 9724 9742 9759 9777 


250 


39794 39811 39829 39846 39863 39681 39898 39915 39933 39960 



439 
TABLE L— LOGARITHMS OF NUMBERS. 



N 


0123456789 


250 


39794 39811 39829 39846 89863 39881 39898 39915 39933 39950 


1 


9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 


8 


40140 40157 0175 0192 0209 0226 0243 0261 0278 0295 


S 


0312 0329 0346 0364 0381 0398 041$ 0432 0449 0466 


4 


0483 0500 0518 0535 0552 0569 0586 0603 0620 0637 


5 


0654 0671 0688 0706 0722 0739 0756 0773 0790 0807 


6 


0824 0841 0858 0875 0892 0909 0926 0943 0960 0976 


7 


0993 1010 1027 1044 1061 1078 109 5 1111 1128 1145 


8 


1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 


9 


1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 


260 


41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 


1 


1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 


2 


1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 


8 


1996 2012 2029 2015 2062 2078 2095 2111 2127 2144 


4 


2160 2177 2193 2210 2226 2243 2259 2276 2292 2308 


6 


2325 2341 2357 2374 2390 2406 2423 2439 2465 2472 
2488 2504 2521 2537 2553 2670 2586 2602 2619 2635 


6 


7 


2651 2667 2684 2700 2716 2732 2749 2765 2781 2797 


8 


2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 


9 


2976 2991 3008 3024 3040 3056 3072 3088 3104 3120 


270 


43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 


1 


3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 


2 


8457 3473 8489 3505 3521 8537 8553 3569 3584 3600 


S 


3616 3632 8648 3664 3680 3696 3712 3727 3743 3759 


4 


3775 3791 3807 3823 3838 3854 3870 3886 3902 ,3917 


6 


3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 


6 


4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 


7 


4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 


8 


4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 


9 


4560 4576 4592 4607 4623 4638 4654 4669 4685 4700 


280 


44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 


1 


4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 


2 


5025 5040 5056 5071 5086 5102 5117 5133 5148 6163 


8 


5179 5194 6209 5225 5240 6255 5271 6286 6301 5317 


4 


5332 5347 5362 5378 5393 5408 5423 5439 6454 5469 


5 


5484 6500 6515 6530 5545 6561 6576 5591 5606 5621 


6 


6637 6652 5667 5682 5697 6712 6728 6748 6758 5773 


7 


6788 5803 5818 5834 5849 5864 6879 5894 6909 5924 


8 


5939 5954 5969 5984 6000 6015 6030 6045 6060 6075 


9 


6090 6105 6120 6135 6150 6165 8180 6195 8210 6225 


290 


46240 46255 46270 46285 46300 46315 46330 46345 46369 46374 


1 


6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 


2 


6538 6653 6568 6583 6598 6613 6627 6642 6657 6672 


8 


6687 6702 6716 6731 6746 6761 6776 6790 6806 6820 


4 


6835 6850 6864 6879 6894 6909 6923 6938 6953 6967 


6 


6982 6997 7012 7026 7041 7056 7070 7085 7100 7114 


6 


7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 


7 


7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 


8 


7422 7436 7451 7465 7480 7494 7509 7524 7538 7663 


9 


7567 7582 7696 7611 7626 7640 7654 7669 7683 7698 


800 


47712 47727 47741 47756 47770 47784 x 47799 47818 47828 47842 



440 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


800 


47712 47727 47741 47766 47770 47784 47799 47813 47828 47842 


1 


7857 7871 7886 7900 7914 7929 7943 7958 7972 7986 


2 


8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 


8 


8144 8159 8173 8187 8202 8216 8230 8244 8259 8273 


4 


8287 8302 8316 8330 8344 8359 8373 8387 8401 8416 


5 


8430 8444 8458 8473 8487 8601 8515 8530 8544 8658 


6 


8672 8586 8601 8615 8629 8643 8657 8671 8686 8700 


7 


8714 8728 8742 8756 8770 8785 8799 8813 8827 8841 


8 


8855 8869 8883 8897 8911 8926 8940 8954 8968 8982 


9 


8996 9010 9024 9038 9052 9066 9080 9094 9108 9122 


810 


49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 


1 


9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 


8 


9415 9429 9443 9457 9471 9485 9499 9513 9527 9641 


8 


9654 9568 9582 9596 9610 9624 9638 9661 9665 9679 


4 


9693 9707 9721 9734 9748 9762 9776 9790 9803 9817 


5 


9831 984^ 9859 9872 9886 9900 9914 9927 9941 9965 


e* 


9969 9982 9996 60010 60024 60037 6005160065 60079 60092 


7 


50106 50120 60133 0147 0161 0174 0188 0202 0216 0229 


8 


0243 0256 0270 0284 0297 0311 0325 0338 0362 0366 


9 


0379 0393 0406 0420 0433 0447 0461 0474 0488 0501 


820 


50515 50529 50542 50556 60569 60583 50596 50610 60623 60637 


1 


0651 0664 0678 0691 0705 0718 0732 0745 0759 0772 


8 


0786 0799 0813 0826 0840 0863 0866 0880 0893 0907 


8 


0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 


4 


1055 1068 1081 1095 1108 1121 1135 H48 1162 1175 


5 


1188 1202 1215 1228 1242 1265 1268 1282 1295 1306 


8 


1322 1335 1348 1362 1376 1388 1402 1415 1428 1441 


7 


1455 1468 1481 1495 1508 1621 1634 1648 1661 1674 


8 


1587 1601 1614 1627 1640 1664 1667 1680 1693 1706 


9 


1720 1733 1746 1759 1772 1786 1799 1&2 1826 1838 


880 


61851 61865 51878 51891 51904 61917 61930 51943 61957 51970 


1 


1983 1996 2009 2022 2035 2048 2061 2075 2088 2101 


8 


2114 2127 2140 2153 2166 2179 2192 2206 2218 2231 


8 


2244 2257 2270 2284 2297 2310 2323 2336 2349 2362 


4 


2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 


6 


2504 2517 2530 2543 2656 2569 2682 2596 2608 2621 


6 


2634 2647 2660 2673 2686 2699 2711 2724 2737 2760 


7 


2763 2776 2789 2802 2815 2827 2840 2863 2866 2879 


8 


2892 2905 2917 2930 2943 2956 2969 2982 2994 3007 


9 


3020 3033 3046 3058 8071 3084 3097 3110 3122 3136 


840 


63148 53161 53173 53186 63199 63212 63224 63237 63250 63263 


1 


3275 3288 3301 3314 3326 3339 3352 3364 8377 3390 


8 


8403 3416 3428 3441 3453 3466 3479 3491 3504 3617 


8 


3629 3542 3555 3567 3580 3593 3606 3618 3631 3643 


4 


3666 3668 3681 3694 3706 3719 3732 3744 3767 3769 


6 


3782 3794 3807 3820 3832 3845 3857 3870 3882 3895 


6 


3908 3920 3933 3946 3958 3970 3983 3996 4008 4020 


7 


4033 4046 4058 4070 4083 4095 4108 4120 4133 4145 


8 


4158 4170 4183 4195 4208 4220 4233 4246 4268 4270 


9 


4283 4295 4307 4320 4332 4345 4357 4370 4382 4394 


850 


54407 54419 54432 64444 54456 54469 54481 64494 64600 64618 



44T 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 



2 3 4 5 6 



850 

1 
8 
3 
4 
6 
6 
7 
8 
9 

880 

1 
8 
8 
4 
6 
8 
7 
8 
9 

870 

1 
8 
8 
4 
6 
8 
7 
8 
9 

880 

1 
8 
8 
4 
6 
8 
7 
8 
9 

890 

1 
8 
8 
4 
6 
8 
7 
8 
9 

400 



54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 

4531 4543 4655 4568 4580 4593 460$ 4617 4630 4642 

4654 4667 4679 4691 4704 4716 4728 4741 4753 4765 

4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 

4900 4913 4925 4937 4949 4962 4974 4986 4998 5011 

5023 5035 5047 5060 5072 5084 5096 5108 5121 5133 

5145 5157 5169 6182 5194 5206 5218 5230 5242 .6255 

6267 6279 6291 6303 6315 6328 5340 6362 6364 6376 

5388 5400 5413 5425 5437 6449 6461 6473 6485 5497 

5509 5522 6534 5546 5558 5570 6582 5594 5606 5618 

55630 55642 56654 65666 56678 55691 55703 55715 55727 55739 

6751 6763 6775 6787 6799 6811 6823 6835 6847 6869 

5871 5883 5895 6907 6919 6931 5943 6955 6967 6979 

6991 6003 6015 6027 6038 6050 6062 6074 6086 6098 

6110 6122 6134 6146 6168 6170 6182 6194 6206 6217 

6229 6241 6253 6265 6277 6289 6301 0312 6324 6336 

6348 6360 6372 6384 6396 6407 6419 6431 6443 6455 

6467 6478 6490 6502 6514 6526 6538 6549 6561 6573 

6585 6597 6608 6620 6632 6644 6666 6667 6679 6691 

6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 

66820 66832 66844 66855 56867 66879 56891 56902 56914 66926 

6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 

7064 7066 7078 7089 7101 7113 7124 7136 7148 7159 

7171 7183 7194 7206 7217 7229 7241 7252 7264 7276 

7287 7299 7310 7322 7334 7345 7357 7368 7380 7392 

7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 

7619 7530 7542 7553 7565 7676 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 7875 7887 7898 7910 7921 7933 7944 7955 7967 

57978 57990 68001 68013 58024 68035 68047 68058 58070 58081 

8092 8104 8115 8127 8138 8149 8101 8172 8184 8195 

8206 8218 8229 8240 8252 8263 8274 8286 8297 8309 

8320 8331 8343 8354 8305 8377 8388 8399 8410 8422 

8433 8444 8456 8467 8478 8490 8501 8512 8524 8635 

8546 8557 8569 8580 8591 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 

8995 9006 9017 9028 9040 9051 9062 9073 9084 9095 

59106 69118 59129 59140 59151 59162 69173 69184 69196 69207 

9218 9229 9240 9251 9202 9273 9284 9295 9306 9318 

9329 9340 9351 9362 9373 9384 9396 9406 9417 9428 

9439 9450 9461 9472 9483 9494 0506 9517 9528 9639 

9550 9561 9572 9583 9594 9605 9616 9627 9638 9649 

9660 9671 9682 9693 9704 9715 9726 9737 9748 9759 

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 



442 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


400 


60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 


1 


0314 0325 0&36 0347 0358 0369 0370 0390 0401 0412 


8 


0423 0433 0444 0455 0466 0477 0487 0498 0509 0520 


8 


0531 0541 0552 0563 0574 0584 0595 0606 0617 0627 


4 


0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 


5 


0746 0766 0767 0778 0788 0799 0810 0821 0831 0842 


6 


0853 0863 0874 0885 0895 0906 0917 0927 0938 0949 


7 


0969 0970 0981 0991 1002 1013 1023 1034 1045 1055 


8 


1066 1077 1087 1098 1109 1119 1130 1140 1161 1162 


9 


1172 1183 1194 1204 1215 1226 1236 1247 1257 1268 


410 


61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 


1 


1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 


8 


1490 1500 1511 1521 1532 1642 1553 1663 1574 1584 


8 


1596 1606 1616 1627 1637 1648 1658 1669 1679 1690 


4 


1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 


6 


1805 1816 1826 1836 1847 1857 1868 1878 1888 1899 


6 


1909 1920 1930 . 1941 1951 1962 1972 1962 1993 2003 


7 


2014 2024 2034 2045 2055 2066 2076* 2086 2097 2107 


8 


2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 


9 


2221 2232 2242 2252 2263 2273 2284 2294 2304 2315 


420 


62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 


1 


2428 2439 2449 2459 2469 2480 2490 2500 2511 2521 


9 


2531 2542 2552 2562 2572 2583 2693 2603 2613 2624 


8 


2684 2644 2655 2665 2675 2686 2696 2706 2716 2726 


4 


2737 2747 2767 2767 2778 2788 2798 2808 2818 2829 


6 


2839 2849 2859 2870 2880 2890 2900 2910 2921 2031 


6 


2941 2961 2961 2972 2982 2992 3002 3012 3022 8033 


7 


3043 3053 3063 8073 3083 3094 8104 8114 3124 8134 


8 


8144 8155 3165 3175 3185 3195 3205 3215 3225 8236 


9 


8246 8256 8266 3276 3286 3296 8306 3317 8327 3337 


430 


63347 63367 63367 63377 63387 63397 68407 63417 63428 63438 


1 


3448 3458 3468 3478 3488 3498 3608 3518 3528 3538 


2 


3548 3568 3568 3579 8589 3599 3609 3619 8629 3639 


8 


8649 3659 3669 8679 8689 8699 3709 8719 3729 8739 


4 


3749 3759 3769 8779 3789 8799 8809 8819 3829 8839 


5 


3849 3850 3869 8879 8889 3899 3909 8919 8929 8939 


6 


8949 3959 3969 8979 3988 8998 4008 4018 4028 4038 


7 


4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 


8 


4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 


9 


4246 4256 4266 4276 4286 4296 4306 4316 4326 4335 


410 


64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 


1 


4444 4454 4104 4473 4483 4103 4503 4513 4523 4532 


2 


4542 4552 4502 4572 4582 4591 4601 4611 4621 4631 


3 


4640 4650 4000 4670 4080 4089 4699 4709 4719 4729 


4 


4738 4748 4758 4708 4777 4787 4707 4807 4810 4826 


5 


4836 4846 4856 4865 4875 4885 4895 4904 4914 4924 


6 


4933 4943 4953 4963 4972 4982 4992 5002 5011 5021 


7 


5031 5040 5050 5060 5070 6079 6089 5099 6108 6118 


8 


6128 5137 6147 6157 5167 6176 5186 6196 5206 6215 


9 


6225 6234 5244 5254 5263 5273 5283 6292 5302 5312 


460 


65321 65331 65341 65350 65360 65369 66379 65389 65398 65408 



445 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 

460 


0123456780 


66321 05331 65341 6535:) 65360 65369 65379 65389 65398 65408 


1 


5418 5427 5437 5447 5456 5466 6475 5486 6495 6504 


8 


6614 6523 5533 6643 6652 6562 6571 5581 6691 6600 


8 


6610 6619 6629 6639 6648 6658 6667 6677 6686 6696 


4 


5706 6715 6725 5734 6744 5753 6763 6772 5782 6792 


6 


6801 6811 6820 5830 5839 5849 5858 6868 5877 6887 


6 


6896 6906 6916 5925 6935 6944 6964 6963 5973 6982 


7 


6992 6001 6011 6020 6030 6039 6049 6068 6068 6077 


8 


6087 6096 6106 6115 6124 6134 6143 6163 6162 6172 


9 


6181 6191 6200 6210 6219 6229 6238 6247 6257 6266 


460 


66276 66285 66295 66304 66314 66323 66332 66342 66351 66361 


1 


6370 6380 6389 6398 6408 6417 6427 6436 6446 6455 


8 


6464 6474 6483 6492 6602 6611 6621 6630 6689 6549 


8 


6558 6667 6677 6586 6596 6605 6614 6624 6638 6642 


4 


6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 


5 


6746 6755 6764 6773 6783 6792 6801 6811 6820 6829 


6 


6839 6848 6857 6867 6876 6886 6894 6904 6918 6922 


7 


6932 6941 6950 6960 6969 6978 6987 6997 7006 7015 


8 


7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 


9 


7117 7127 7136 7146 7164 7164 7173 7182 7191 7201 


470 


67210 67219 67228 67237 67247 67266 67266 67274 67284 67293 


1 


7302 7311 7321 7330 7339 7348 7357 7367 7876 7385 


9 


7394 7408 7413 7422 7431 7440 7449 7469 7468 7477 


8 


7486 7495 7504 7514 7523 7532 7641 7550 7560 7569 


4 


7678 7587 7696 7606 7614 7624 7633 7642 7661 7660 


5 


7669 7679 7688 7697 7706 7716 7724 7733 7742 7752 


6 


7761 7770 7779 7788 7797 7806 7816 7825 7834 7843 


7 


7862 7861 7870 7879 7888 7897 7906 7916 7925 7934 


8 


7943 7952 7961 7970 7979 7988 7997 8006 8016 8024 


9 


8034 8043 8052 8061 8070 8079 8088 8097 8106 8115 


480 


68124 68133 68142 68161 68160 68169 68178 68187 68196 68205 


I 


8215 8224 8233 8242 8261 8260 8269 8278 8287 8296 


9 


8305 8314 8323 8332 8341 8350 8359 8368 8377 8386 


8 


8395 8404 8413 8422 8431 8440 8449 8468 8467 8476 


4 


8485 8494 8602 8511 8620 8529 8538 8547 8666 8565 


5 


8674 8683 8592 8601 8610 8619 8628 8637 8646 8655 


6 


8664 8673 8681 8690 8699 8708 8717 8726 8735 8744 


7 


8763 8762 8771 8780 8789 8797 8806 8816 8824 8833 


8 


8842 8861 8860 8869 8878 8886 8896 8904 8913 8922 


9 


8931 8940 8949 8968 8966 8976 8984 8993 9002 9011 


490 


69020 69028 69037 69046 69066 69064 69073 69082 69090 69099 


1 


9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 


9 


9197 9205 9214 9223 9232 9241 9249 9268 9267 9276 


8 


9285 9294 9302 9311 9320 9329 9338 9346 9366 9364 


4 


9373 9381 9390 9399 9408 9417 9425 9434 9443 9452 


5 


9461 9469 9478 9487 9496 9504 9613 9522 9631 9639 


6 


9548 9567 9566 9574 9583 9592 9601 9609 9618 9627 


7 


9636 9644 9653 9662 9671 9679 9688 9697 9705 9714 


8 


9723 9732 9740 9749 9768 9767 9776 9784 9793 9801 


9 


9810 9819 9827 9836 9845 9854 9862 9671 9880 9888 


600 


69697 69906 69914 69923 69932 69940 69949 69968 69966 69975 



444 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


600 


69897 69906 69014 69923 69932 69940 69949 69958 69966 69975 


1 


9984 9992 70001 70010 70018 70027 70036 70044 70053 70062 


9 


70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 


8 


0157 0165 0174 0183 0191 0200 0209 0217 0226 0234 


4 


0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 


6 


0329 0338 0346 0355 0364 0372 0381 0389 0398 0406 


6 


0415 0424 0432 0441 0449 0458 0467 0475 0484 0492 


7 


0501 0509 0518 0526 0535 0544 0552 0661 0569 0578 


8 


0586 0595 0603 0612 0621 0629 0638 0646 0665 0663 


9 


0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 


510 


70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 


1 


0842 0851 0859 0868 0876 0885 0893 0902 0910 0919 


8 


0927 0935 0944 0952 0961 0969 0978 0986 0995 1003 


8 


1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 


4 


1096 1105 1113 1122 1130 1139 1147 1155 1164 1172 


5 


1181 1189 1198 1206 1214 1223 1231 1240 1248 1257 


6 


1265 1273 1282 1290 1299 1307 1315 1324 1332 1341 


7 


1349 1357 1366 1374 1383 1391 1399 1408 1416 1425 


8 


1433 1441 1450 1458 1466 1475 1483 1492 1500 1508 


9 


1517 1525 1533 1542 1550 1559 1567 1576 1584 1592 


520 


71600 71609 71617 71626 71634 71642 71650 71659 71667 71676 


1 


1684 1692 1700 1709 1717 1726 1734 1742 1750 1759 


9 


1767 1775 1784 1792 1800 1809 1817 1825 1834 1842 


8 


1850 1858 1867 1876 1883 1892 1900 1908 1917 1925 


4 


1933 1941 1950 1958 1966 1975 1983 1991 1999 2008 


5 


2016 2024 2032 2041 2049 2057 2066 2074 2082 2090 


8 


2099 2107 2115 2123 2132 2140 2148 2156 2165 2173 


7 


2181 2189 2198 2206 2214 2222 2230 2239 2247 2255 


8 


2263 2272 2280 2288 2296 2304 2313 2321 2329 2337 


9 


2346 2354 2362 2370 2378 2387 2395 2403 2411 2419 


580 


72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 


1 


2509 2518 2526 2534 2542 2550 2558 2567 2575 2583 


8 


2591 2599 2607 2616 2624 2632 2640 2648 2656 2665 


8 


2673 2681 2689 2607 2705 2713 2722 2730 2738 2746 


4 


2754 2762 2770 2779 2787 .2795 2803 2811 2819 2827 


6 


2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 


6 


2916 2025 2933 2941 2949 2957 2966 2973 2981 2989 


7 


• 2997 3006 3014 3022 3030 3038 3046 3054 3062 3070 


8 


3078 3086 3094 3102 3111 3119 3127 8135 3148 3161 


9 


3159 3167 3175 3183 3191 3199 3207 8215 3223 3231 


540 


73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 


1 


3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 


2 


3400 3408 3416 3424 3432 3440 3448 3456 3464 3472 


8 


3480 3488 3496 3504 3512 3520 3528 3536 3544 3662 


4 


3560 3568 3576 3584 3592 3600 3608 3616 3624 3632 


5 


3640 3648 3656 3664 3672 3679 3687 3695 3703 3711 


6 


3719 3727 3735 3743 3751 3759 3767 3775 3783 3791 


7 


3799 3807 3815 3823 3830 3838 3846 3854 3862 3870 


8 


3878 3886 3894 3902 3910 3918 3926 3933 3941 3949 


9 


3957 3965 3973 3981 3989 3997 4005 4013 4020 4028 


550 


74036 74044 74062 74060 74068 74076 74084 74092 74099 74107 



445 

TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456780 


550 


74036 74044 74052 74060 74068 74076 74084 74002 74099 74107 


1 


4116 4123 4131 4139 4147 4155 4162 4170 4178 4186 


2 


4194 4202 4210 4218 4225 4233 4241 4249 4257 4 26 5 


8 


4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 


4 


4361 4359 4367 4374 4382 4390 4398 4406 4414 4421 


5 


4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 


6 


4507 4616 4523 4531 4539 4547 4554 4562 4570 4578 


7 


4586 4593 4601 4609 4617 4624 4632 4640 4648 4656 


8 


4663 4671 4679 4687 4695 4702 4710 4718 4726 4738 


9 


4741 4749 4767 4764 4772 4780 4788 4796 4803 4811 


560 


74819 74827 74834 74842 74850 74858 74865 74873 74881 74889 


1 


4896 4904 4912 4920 4927 4935 4943 4950 4968 4966 


8 


4974 4981 4989 4997 5005 5012 5020 5028 6036 6043 


8 


6051 6059 5066 6074 6082 6089 6097 6105 6113 6120 


4 


5128 5136 6143 6151 6159 6166 6174 5182 5189 5197 


6 


6205 5213 6220 5228 5236 5243 6261 5259 6266 6274 


6 


6282 5289 5297 5305 5312 5320 6328 6336 6343 6361 


7 


6358 5366 6374 6381 5389 5397 5404 5412 5420 6427 


8 


6435 6442 6450 6458 5465 6473 6481 6488 6496 6504 


9 


6611 6519 5526 6534 6542 5649 5557 6565 6672 5580 


570 


75687 75696 75603 75610 75618 75626 75633 76641 75648 75666 


1 


6664 6671 6679 6686 6694 5702 6709 6717 6724 5732 


8 


5740 6747 5755 6762 6770 6778 6785 6793 6800 6808 


8 


5815 5823 5831 6838 6846 6853 6861 5868 5876 5884 


4 


5891 5899 5906 5914 5921 5929 6937 5944 5952 5969 


5 


5967 6974 6982 6989 6997 6005 6012 6020 6027 6035 


6 


6042 6050 6067 6065 6072 6080 6087 6095 6103 6110 


7 


6118 6125 6133 6140 6148 6155 6163 6170 6178 6186 


8 


6193 6200 6208 6216 6223 6230 6238 6245 6253 6260 


9 


6268 6275 6283 6290 6298 6305 6313 6320 6328 6336 


580 


76343 76360 76368 76366 76373 76380 76388 76396 76403 76410 


1 


6418 6425 6433 6440 6448 6455 6462 6470 6477 6485 


9 


6492 6500 6507 6515 6522 6530 6637 664$ 6552 6559 


8 


6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 


4 


6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 


5 


6716 6723 6730 6738 6745 6763 6760 6768 6775 6782 


6 


6790 6797 6805 6812 6819 6827 6834 6842 6849 6856 


7 


6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 


8 


6938 6946 6953 6960 6967 6975 6982 6989 6997 7004 


9 


7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 


590 


77086 77093 77100 77107 77115 77122 77129 77137 77144 77151 


1 


7169 7166 7173 7181 7188 7195 7203 7210 7217 7225 


9 


7232 7240 7247 7254 7262 7269 7276 7283 7291 7298 


8 


7305 7318 7320 7327 7335 7342 7349 7357 7364 7371 


4 


7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 


6 


7452 7459 7466 7474 7481 7488 7495 7503 7510 7617 


6 


7525 7532 7539 7546 7554 7561 7568 7576 7683 7690 


7 


7697 7605 7612 7619 7627 7634 7041 7648 7666 7663 


8 


7670 7677 7685 7692 7699 7706 7714 7721 7728 7736 


9 


7743 7750 7757 7764 7772 7779 7786 7793 7801 7808 


600 


77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 



446 





TABLE L— LOGARITHMS OF NUMBERS. 


N 


0123456780 


600 


77816 77822 77830 77837 77844 77861 77869 77866 77873 77880 


1 


7887 7895 7902 7909 7916 7924 7931 7938 7946 7968 


8 


7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 


8 


8082 8039 8046 8063 8061 8068 8075 8082 8089 8097 


4 


8104 8111 8118 8126 8132 8140 8147 8164 8161 8168 


5 


8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 


6 


8247 8264 8262 8269 8276 8283 8290 8297 8305 8312 


7 


8319 8326 8333 8340 8347 8365 8362 8369 8376 8383 


8 


8390 8398 8405 8412 8419 8426 8433 8440 8447 8465 


9 


8462 8469 8476 8483 8490 8497 8604 8612 8619 8626 


610 


78633 78640 78647 78664 78661 78669 78676 78683 78690 78697 


1 


8604 8611 8618 8626 8633 8640 8647 8664 8661 8668 


9 


8676 8682 8689 8696 8704 8711 8718 8725 8732 8739 


8 


8746 8763 8760 8707 8774 8781 8789 8796 8803 8810 


4 


8817 8824 8831 8838 8845 8862 8859 8866 8873 8880 


5 


8888 8895 8902 8909 8916 8923 8930 8937 8944 8961 


6 


8958 8966 8972 8979 8986 8993 9000 9007 9014 9021 


7 


9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 


8 


9099 9106 9113 9120 9127 9134 9141 9148 9156 9162 


9 


9169 9176 9183 9190 9197 9204 9211 9218 9225 9232 


620 


79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 


1 


9309 9316 9323 9330 9337 9344 9351 9358 9365 9372 


9 


9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 


8 


9449 9456 9463 9470 9477 9484 9491 9498 9505 9511 


4 


9618 9525 9532 9539 9546 9553 9560 9567 9574 9581 


5 


9588 9595 9602 9609 9616 9623 9630 9637 9644 9650 


6 


9657 9664 9671 9678 9685 9692 9699 9706 9713 9720 


7 


9727 9734 9741 9748 9754 9761 9768 9776 9782 9789 


8 


9796 9803 9810 9817 9824 9831 9837 9844 9851 9858 


9 


9865 9872 9879 9886 9893 9900 9906 9913 9920 9927 


680 


79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 


1 


80003 80010 80017 80024 80030 80037 80044 80061 80058 80065 


9 


0072 0079 0085 0092 0099 0106 0118 0120 0127 0134 


8 


0140 0147 0154 0161 0168 0175 0182 0188 0196 0202 


4 


0209 0216 0223 0229 0236 0243 0260 0267 0264 0271 


5 


0277 0284 0291 0298 0305 0312 0318 0326 0332 0339 


6 


0346 0353 0359 0366 0373 0380 0387 0393 0400 0407 


7 


0414 0421 0428 0434 0441 0448 0455 0462 0468 0475 


8 


0482 0489 0496 0602 0509 0516 0523 0530 0536 0543 


9 


0550 0557 0564 0570 0577 0584 0591 0698 0604 0011 


640 


80618 80625 80632 80638 80645 80652 80659 80666 80672 80679 


1 


0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 


9 


0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 


8 


0821 0828 0835 0841 0848 0855 0862 0868 0875 0882 


4 


0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 


5 


0956 0963 0969 0976 0983 0990 0996 1003 1010 1017 


9 


1023 1030 1037 1043 1050 1057 1064 1070 1077 1084 


7 


1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 


8 


1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 


9 


1224 1231 1238 1245 1251 1258 1265 1271 1278 1285 


660 


81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 



447 



TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456780 


650 


81291 81298 81305 81311 81818 81322 81331 81338 81345 81351 
1858 1305 1371 1378 1385 1391 1398 1405 1411 1418 


1 


8 


1425 1431 1438 1445 1451 1458 1465 1471 1478 1485 


8 


1491 1498 1505 1511 1518 1525 1531 1538 1544 1551 


4 


1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 


6 


1624 1631 1637 1644 1651 1657 1664 1671 1677 1684 


6 


1690 1697 1704 1710 1717 1723 1730 1787 1748 1750 


7 


1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 


8 


1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 


9 


1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 


660 


81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 


1 


2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 


8 


2086 2092 2099 2105 2112 2119 2125 2132 2138 2145 


8 


2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 


4 


2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 


5 


2282 2289 2295 2302 2308 2315 2321 2328 2334 2341 


6 


2347 2354 2360 2367 2373 2880 2387 2393 2400 2406 


7 


2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 


8 


2478 2484 2491 2497 2504 2510 2517 2523 2530 2536 


e 


2543 2549 2556 2562 2569 2575 2582 2588 2595 2601 


670 


82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 


1 


2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 


8 


2737 2743 2750 2756 2763 2769 2776 2782 2789 2796 


8 


2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 


4 


2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 


5 


29M 2937 2943 2950 2956 2963 2969 2975 2982 2988 


6 


2995 3001 3008 3)14 3)20 3327 8033 3040 3046 3052 


7 


3059 3065 8072 3078 3085 3091 3097 3104 3110 3117 


8 


8123 3129 3136 3142 3149 3155 3161 3168 3174 8181 


9 


3187 3193 3200 3206 3213 8219 8225 3232 3238 3245 


6S0 


83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 


1 


3315 3321 3327 3334 8340 3347 3353 3359 3366 3372 


8 


3378 3385 3391 3398 8404 3410 3417 3423 3429 3436 


8 


3442 3448 3455 3461 3467 3474 3480 3487 3493 3499 


4 


3506 3512 3518 3525 3531 3537 8544 3550 3556 3563 


6 


3569 3575 3582 3588 3594 8601 3607 3613 3620 3626 


6 


3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 


7 


3696 3702 3708 3715 3721 3727 3734 8740 3746 3753 


8 


3769 3766 3771 3778 3784 3790 3797 3803 8809 8816 


9 


3822 3828 3835 3841 3847 3853 3860 3866 3872 3879 


690 


83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 


1 


3948 3954 8960 3967 3973 3979 3986 3992 3998 4004 


8 


4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 


8 


4073 4080 4086 4092 4098 4105 4111 4117 4128 4130 


4 


4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 


5 


4198 4205 4211 4217 4223 4230 4236 4242 4248 4265 


6 


4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 


7 


4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 


8 


4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 


9 


4448 4454 4460 4466 4473 4479 4485 4491 4497 4504 


700 


84510 84516 84522 84628 84635 84541 84647 84663 84669 84666 



448 





TABLE I.— LOGARITHMS OF NUMBERS. 


N 


12345 6780 


700 


84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 
4572 4578 4584 4590 4597 4603 4609 4615 4621 4628 


1 


2 


4634 4640 4646 4652 4658 4665 4671 4677 4683 4689 


8 


4696 4702 4708 4714 4720 4726 4733 4739 4745 4751 


4 


4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 


6 


4819 4825 4831 4837 4844 4850 4856 4862 4868 4874 


6 


4880 4887 4893 4899 4905 4911 4917 4924 4930 4936 


7 


4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 


8 


5003 5009 5016 5022 5028 5034 5040 5046 5052 5058 


9 


5065 5071 5077 5083 5089 5095 5101 5107 5114 5120 


710 


85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 


1 


5187 5193 5199 5205 5211 5217 5224 5230 5236 5242 


8 


5248 5254 5260 5266 5272 5278 5285 5291 5297 5303 


8 


5309 5315 5321 5327 5333 5339 5345 5352 5358 5364 


4 


5370 5376 5382 5388 5304 5400 6406 5412 5418 6425 


5 


5431 6437 5443 5449 6455 5461 *467 5473 5479 5486 


6 


5491 5497 5503 6509 5516 5522 5528 6534 5540 6646 


7 


6552 5658 6564 5570 5576 65C2 6688 6694 6600 6606 


8 


6612 6618 5625 5631 5637 5643 5649 5655 6661 6667 


9 


6673 6679 6685 6691 6697 6703 6709 6715 5721 6727 


720 


85733 85739 85745 86751 85757 85763 85769 86775 85781 86788 


1 


5794 6800 5806 6812 6818 6824 5830 5836 6842 6848 


2 


6854 5860 6866 6872 6878 6884 6890 6896 6902 6908 


8 


5914 5920 5926 6932 6988 5944 5950 5956 5962 6968 


4 


5974 5980 5986 5992 5998 6004 6010 6016 6022 6028 


6 


6034 6040 6046 6052 6058 6064 6070 6076 6082 6088 


6 


6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 


7 


6163 6159 6165 6171 6177 6183 6189 6195 6201 6207 


8 


6213 6219 6225 6231 6237 6243 6249 6255 6261 6267 


9 


6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 


730 


86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 


1 


6392 6398 6404 6410 6415 6421 6427 6433 6439 6445 


2 


6451 6457 6463 6469 6475 6481 6487 6493 6499 6604 


8 


6510 6516 6522 6528 6534 6640 6546 6552 6568 6564 


4 


6570 6576 6581 6587 6593 6599 6605 6611 6617 6623 


5 


6629 6635 6641 6646 6662 6668 6664 6670 6676 6682 


6 


6688 6694 6700 6705 6711 6717 6723 6729 6735 6741 


7 


6747 6753 6769 6764 6770 6776 6782 6788 6794 6800 


8 


6806 6812 6817 6823 6829, 6836 6841 6847 6853 6859 


9 


6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 


740 


86923 86929 86935 86941 86947 86953 86968 86964 86970 86076 


1 


6982 6988 6994 6999 7006 7011 7017 7023 7029 7035 


8 


7040 7046 7052 7058 7064 7070 7076 7081 7087 7093 


8 


7099 7105 7111 7116 7122 7128 7134 7140 7146 7151 


4 


7167 7163 7169 7175 7181 7186 7192 7198 7204 7210 


5 


7216 7221 7227 7233 7239 7245 7261 7266 7262 7268 


6 


7274 7280 7286 7291 7297 7303 7309 7315 7320 7326 


7 


7332 7338 7344 7349 7355 7361 7367 7373 7379 7384 


8 


7390 7396 7402 7408 7413 7419 7425 7431 7437 7442 


9 


7448 7454 7460 7466 7471 7477 7483 7489 7495 7600 


760 


87606 87612 87618 87523 87529 87536 87641 87647 87552 87558 



449 
TABLE I.— LOGARITHMS OF NUMBERS. 



N 


123456780 


750 


87606 87612 87618 87623 87629 87635 87541 87647 87552 87658 


\ 


7664 7670 7676 7681 7687 7693 7699 7604 7610 7616 




7622 7628 7633 7639 7645 7651 7656 7662 7668 7674 




7679 7685 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 




7862 7858 7864 7869 7876 7881 7887 7892 7898 7904 




7910 7915 7921 7927 7933 7938 7944 7950 7966 7961 




7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 




8024 8030 8036 8041 8047 8063 8068 8064 8070 8076 


760 


88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 




8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 




8195 8201 8207 8213 8218 8224 8230 8236 8241 8247 




8252 8258 8264 8270 8275 8281 8287 8292 8298 8304 




8309 8315 8321 8326 8332 8338 8343 8349 8355 8360 




8366 8372 8377 838:) 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 8630 




8636 8542 8547 8563 8659 8564 8570 8676 8581 8587 




8593 8598 8604 8610 8615 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 8766 




8762 8767 8773 8779 8784 8790 8796 8801 8807 8812 




8818 8824 8829 8835 8840 8846 8852 8857 8863 8868 




8874 8880 8886 8891 8897 8902 8908 8918 8919 8925 




8930 8936 8941 8947 8963 8958 8964 8969 8975 8981 




8986 8992 8997 9003 9009 9014 9020 9026 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 89248 89248 89264 89260 




9266 9271 9276 9282 9287 9293 9298 9304 9310 9315 




9321 9326 9332 9337 9343 9348 9354 9360 9366 9371 




9376 9382 9387 9393 9398 9404 9409 9415 9421 9426 




9432 9437 9443 9448 9454 9459 9465 9470 9476 9481 




9487 9492 9498 9504 9609 9515 9520 9526 9531 9637 




9542 9548 9553 9559 9564 9570 9575 9581 9586 9592 




9597 9603 9609 9614 9620 9626 9631 9636 9642 9647 




9653 9658 9664 9669 9675 0680 9686 9691 9697 9702 




9708 9713 9719 9724 9730 9735 9741 9746 9752 9757 


790 


89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 




9818 9823 9829 9834 9840 9846 9851 9856 9862 9867 




9873 9878 9883 9889 9894 9900 9905 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 0157 0162 0168 0173 0179 0184 0189 0195 




0200 0206 0211 0217 0222 0227 0233 0238 0244 0249 




0255 0260 0266 0271 0276 0282 0287 0293 0298 0304 


800 


90309 90314 90320 90325 90331 90336 90342 90347 90852 90368 



45© 

TABLE I.— LOGARITHMS OF NUMBERS. 



N 


0123456789 


800 


90300 00314 00320 90325 90331 90336 90342 90347 90352 90358 




0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 




0417 0428 0428 0434 0439 0445 0450 0455 0461 0466 




0472 0477 0482 0488 0493 0499 0504 0509 0515 0520 




0526 0531 0536 0542 0547 0553 0558 0563 0569 0574 




0580 0585 0590 0596 0601 0607 0612 0617 0623 0628 




0634 0639 0644 0650 0655 0660 0666 0671 0677 0682 




0687 0693 0698 0703 0709 0714 0720 0725 0730 0736 




0741 0747 0752 0757 0768 0768 0773 0779 0784 0789 




0795 0800 0806 0811 0816 0822 0827 0832 0838 0843 


810 


90849 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 1128 1132 1137 1142 1148 1163 1168 1164 




1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 




1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 




1276 1281 1286 1291 1297 1302 1307 1312 1318 1323 




1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 


820 


91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 




1434 1440 1445 1460 1455 1461 1466 1471 1477 1482 




1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 




1540 1545 1551 1566 1661 1566 1572 1577 1582 1587 




1693 1598 1603 1609 1614 1619 1624 1630 1635 1640 




1645 1661 1666 1661 1666 1672 1677 1682 1687 1603 




1098 1703 1709 1714 1719 1724 1780 1735 1740 1746 




1751 1766 1761 1766 1772 1777 1782 1787 1793 1798 




1803 1808 1814 1819 1824 1829 1834 1840 1845 1850 




1855 1861 1866 1871 1876 1882 1887 1892 1897 1903 


880 


91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 




1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 




2012 2018 2023 2028 2033 2038 2044 2049 2054 2059 




2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 




2117 2122 2127 2132 2137 2143 2148 2153 2168 2163 




2169 2174 2179 2184 2189 2195 2200 2206 2210 2215 




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 2350 2355 2361 2366 2371 




2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 


840 


92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 




2480 2485 2490 2495 2500 2505 2611 2516 2521 2626 




2631 2636 2642 2547 2562 2667 2562 2667 2672 2678 




2583 2588 2593 2698 2603 2609 2614 2619 2624 2629 




2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 




2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 




2737 2742 2747 2752 2768 2783 2768 2778 2778 2783 




2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 


6 


2840 2845 2850 2855 2860 2865 2870 2875 2881 2886 


£ 


2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 


850 


92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 



451 



TABLE L— LOGARITHMS OF NUMBERS. 



N 


0123456789 


860 


92942 92947 92962 92957 92962 92967 92973 92978 92983 92988 


1 


2993 2998 3003 3008 3013 3018 3024 3029 8034 3039 


8 


3044 3049 3064 3069 3064 3069 3075 3080 3085 3090 


8 


3095 3100 8106 8110 3116 3120 3125 3131 8136 3141 


4 


3146 3151 3166 8161 8166 3171 3176 3181 8186 8192 


6 


3197 3202 3207 3212 8217 3222 3227 3232 3237 3242 


6 


3247 3252 3258 8263 3268 3273 3278 3283 3288 3293 


7 


3298 3303 3808 3313 3318 3323 3328 3334 3339 3344 


8 


3349 3354 8359 3864 3869 3374 3379 3384 3389 3394 


9 


3399 3404 3409 8414 3420 3425 3430 3435 3440 3445 


860 


93450 93455 93460 93465 93470 93475 93480 98485 93490 93496 


1 


3600 3505 8510 8515 3520 8526 3531 3536 3541 3546 


8 


8551 3556 3561 3566 3571 8676 8681 3686 3691 3696 


8 


3601 3606 3611 3616 3621 3626 3631 3636 3641 3646 


4 


3651 3656 8661 3666 3671 3676 3682 3687 3692 3697 


5 


3702 8707 8712 8717 3722 3727 3782 3737 8742 3747 


6 


8762 3767 8762 8787 3772 3777 8782 8787 8792 8797 


7 


3802 8807 3812 3817 3822 3827 3832 8837 3842 3847 


8 


3852 3857 3862 8867 8872 8877 3882 8887 3892 8897 


9 


8902 3907 3912 3917 3922 8927 8932 8937 3942 3947 


870 


93962 93967 93962 93967 93972 93077 93982 98987 93992 93997 


1 


4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 


8 


4052 4067 4062 4067 4072 4077 4082 4086 4091 4096 


8 


4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 


4 


4161 4166 4161 4166 4171 4176 4181 4186 4191 4196 


5 


4201 4206 4211 4216 4221 4226 4231 4236 4240 4246 


. 6 


4250 4265 4260 4265 4270 4275 4280 4285 4290 4295 


7 


4300 4305 4310 4315 4320 4325 4330 4335 4340 4345 


8 


4349 4354 4369 4364 4369 4374 4379 4384 4389 4394 


e 


4899 4404 4409 4414 4419 4424 4429 4433 4438 4443 


880 


94448 94453 94458 94463 94468 94478 94478 94483 94488 94493 


1 


4498 4503 4507 4612 4517 4522 4527 4532 4537 4642 


8 


4547 4552 4567 4562 4567 4571 4576 4581 4586 4591 


8 


4596 4601 4606 4611 4616 4621 4626 4630 4635 4640 


4 


4645 4650 4655 4660 4665 4670 4675 4680 4685 4689 


5 


4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 


6 


4743 4748 4753 4758 4763 4768 4773 4778 4783 4787 


7 


4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 


8 


4841 4846 4861 4866 4861 4866 4871 4876 4880 4885 


9 


4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 


890 


94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 


1 


4988 4993 4998 5002 5007 5012 6017 6022 6027 5032 


9 


6036 6041 6046 6051 5066 6061 6066 6071 6075 5080 


8 


6085 6090 6095 5100 5105 5109 5114 6119 5124 5129 


4 


6134 5139 5143 6148 5153 5158 6163 5168 5173 6177 


5 


5182 6187 5192 5197 5202 5207 6211 6216 5221 5226 


6 


6231 6236 5240 5245 5250 5255 5260 5265 5270 6274 


7 


5279 6284 5289 5294 5299 5303 5308 6313 6318 5823 


8 


6328 6332 5337 5342 6347 6352 5357 5361 6366 6371 


9 


6376 6381 6386 6390 5395 5400 6405 6410 6415 6419 


900 


96424 96429 95434 95439 95444 95448 95463 96458 96463 96468 



45* 

TABLE L— LOGARITHMS OF NUMBERS 



N 


0*23456780 


900 


95424 96420 95434 96439 96444 96448 96463 96468 96463 96468 


1 


6472 6477 6482 6487 6492 6497 6501 6606 6511 5616 


9 


6521 5525 6630 5536 5540 5545 6650 6654 6669 6664 


8 


6569 6574 6578 5583 5688 6593 5598 5602 6607 6612 


4 


5617 5622 5626 6631 6636 5641 5646 5660 6665 6660 


5 


6665 5670 5674 5679 5684 6689 5694 6698 6703 6708 


6 


6713 6718 5722 6727 6732 6737 6742 6746 6761 6766 


7 


5761 6766 6770 6775 6780 6785 6789 5794 6799 5804 


8 


5809 5813 6818 5823 6828 5832 6837 6842 6847 6852 


9 


6856 5861 6866 5871 5875 5880 5885 5890 6895 5899 


910 


95904 95909 95914 95918 95923 95928 96933 96938 96942 95947 


1 


6962 5967 5961 5966 5971 5976 5980 6986 6990 6995 


2 


5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 


8 


6047 6052 6057 6061 6066 6071 6076 6080 6086 6090 


4 


6095 6099 6104 6109 6114 6118 6123 6128 6133 6137 


6 


6142 6147 6152 6156 6161 6166 6171 6176 6180 6185 


6 


6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 


7 


6237 6242 6246 6251 6256 6261 6265 6270 6275 6280 


8 


6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 


9 


6332 6336 6341 6346 6350 6355 6360 6365 0369 6374 


920 


96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 


1 


6426 6431 6435 6440 6445 6450 6454 6469 6464 6468 


9 


6473 6478 6483 6487 6492 6497 6501 6606 6611 6615 


8 


6520 6525 6530 6534 6539 6544 6548 6553 6558 6562 


4 


6567 6572 6577 6581 6586 6591 6595 6600 6605 6609 


6 


6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 


6 


6661 6666 6670 6675 6680 6685 6689 6694 6699 6703 


7 


6708 6713 6717 6722 6727 6731 6736 6741 6746 6750 


8 


6765 6769 6764 6769 6774 6778 6783 6788 6792 6797 


9 


6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 


930 


96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 


1 


6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 


2 


6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 


8 


6988 6993 6997 7002 7007 7011 7016 7021 7026 7030 


4 


7035 7039 7044 7049 7063 7058 7063 7067 7072 7077 


6 


7081 7086 7090 7096 7100 7104 7109 7114 7118 7123 


6 


7128 7132 7137 7142 7146 7151 7156 7160 7165 7169 


7 


7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 


8 


7220 7225 7230 7234 7239 7243 7248 7263 7267 7262 


9 


7267 7271 7276 7280 7285 7290 7294 7299 7304 7308 


940 


97313 97317 97322 97327 97331 97336 97340 97345 97350 97364 


1 


7359 7364 7368 7373 7377 7382 7387 7391 7896 7400 


2 


7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 


8 


7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 


4 


7497 7502 7506 7511 7516 7520 7625 7629 7534 7539 


5 


7543 7548 7552 7557 7562 7566 7571 7676 7680 7685 


6 


7589 7594 7598 7603 7607 7612 7617 7621 7626 7630 


7 


7635 7640 7644 7649 7653 7658 7663 7667 7672 7676 


8 


7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 


9 


7727 7731 7736 7740 7745 7749 7754 7769 7763 7768 


950 


97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 



453 





TABLE I.— LOGARITHMS OF NUMBERS. 


N 


01234567 80 


9&0 


97772 97777 97782 97786 97791 97795 97800 97804 97809 97818 


1 


7818 7823 7827 7832 7836 7841 7845 7850 7855 7859 


8 


7864 7868 7873 7877 7882 7886 7891 7896 7900 7905 


8 


7909 7914 7918 7923 7928 7932 7937 7941 7946 7950 


4 


7955 7959 7964 7968 7973 7978 7982 7987 7991 7996 


6 


8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 


6 


8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 


7 


8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 


8 


8137 8141 8146 8160 8155 8159 8164 8168 8173 8177 


9 


8182 8186 8191 8195 8200 8204 8209 8214 8218 8223 


960 


98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 


1 


8272 8277 8281 8286 8290 8295 8299 8304 8308 8813 


9 


8318 8322 8327 8331 8336 8340 8345 8349 8354 8358 


8 


8363 8367 8372 8376 8381 8385 8390 8394 8399 8403 


4 


8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 


6 


8453 8457 8462 8466 8471 8475 8480 8484 8489 8498 


6 


8498 8502 8607 8511 8516 8620 8525 8529 8534 8538 


7 


8543 8547 8562 8666 8661 8565 8570 8574 8679 8583 


8 


8588 8592 8697 8601 8605 8610 8614 8619 8623 8628 


9 


8632 8637 8641 8646 8650 8655 8669 8664 8668 8673 


970 


98677 98682 98686 98691 98696 98700 98704 98709 98713 96717 


1 


8722 8726 8731 8786 8740 8744 8749 8753 8768 8762 


9 


8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 


8 


8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 


4 


8856 8860 8865 8869 8874 8878 8883 8887 8892 8896 


6 


8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 


6 


8945 8949 8954 8958 8963 8967 8972 8976 8981 8986 


7 


8989 8994 8998 9003 9007 9012 9016 9021 9025 9029 


8 


9034 9038 9043 9047 9062 9056 9061 9065 9069 9074 


9 


9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 


980 


99123 99127 99131 99136 99140 99145 99149 99154 99168 99162 


1 


9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 


9 


9211 9216 9220 9224 9229 9233 9238 9242 9247 9251 


8 


9256 9260 9264 9269 9273 9277 9282 9286 9291 9295 


4 


9300 -9304 9808 9313 9317 9322 9326 9330 9335 9839 


6 


9344 9348 9352 9357 9361 9366 9370 9374 9379 9383 


6 


9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 


7 


9432 9436 9441 9445 9449 9454 9458 9463 9467 9471 


8 


9476 9480 9484 9489 9493 9498 9502 9506 9611 9515 


9 


9520 9524 9528 9533 9537 9642 9646 9650 9555 9569 


990 


99564 99568 99572 99677 99681 99685 99590 99694 99699 99603 


1 


9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 


9 


9651 9666 9660 9664 9669 9673 9677 9682 9686 9691 


8 


9695 9699 9704 9708 9712 9717 9721 9726 9730 9734 


4 


9739 9743 9747 9752 9756 9760 9765 9769 9774 9778 


5 


9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 


6 


9826 9830 9835 9839 9843 9848 9852 9856 9861 9865 


7 


9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 


8 


9913 9917 9922 9926 9930 9935 9939 9944 9948 9962 


9 


9957 9961 9965 9970 9974 9978 9983 9987 9991 9996 


1000 


00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 



455 

TABLE H.— LOGARITHMIC SINES AND COSINES. 



1 




0° 


1 


• 


«• 


9 


Slue 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





—oo 


10.00000 


8.24186 


9.99998 


8.54888 


9.99974 


60 


1 


0.46878 


00000 


94908 


3)998 


64648 


99978 


69 


2 


76470 


00000 


25609 


W098 


64999 


99978 


68 


8 


04066 


00000 


86304 


99998 


65854 


99979 


57 


4 


7.0(3679 


00000 


86988 


99992 


65706 


99978 


66 


5 


16870 


00000 


27661 


99992 


56064 


9997U 


55 





24188 


ooouo 


28894 


99998 


66400 


99971 


54 


7 


60689 


00000 


88971 


99992 


66748 


99970 


58 


8 


86689 


00000 


29621 


99992 


67084 


99970 


52 


9 


41797 


00000 


80866 


99991 


67481 


mwiflft 
VWUW 


51 


10 


7.46878 


10.00000 


8.80879 


9.99991 


8.67767 


9.99969 


50 


11 


60618 


00000 


81496 


99991 


58089 


99968 


49 


. 18 


64891 


00000 


82108 


99990 


68419 


99968 


48 


18 


67767 


00000 


82708 


99990 


58747 


99967 


47 


14 


60986 


00000 


88298 


99990 


69078 


99967 


46 


18 


68988 


00000 


88876 


99990 


59896 


99967 


46 


16 


66784 


00000 


84450 


99980 


59716 


99960 


44 


17 


69417 


9.99999 


85018 


99989 


60088 


99906 


48 


18 


71 IKK) 


99999 


85578 


99989 


60849 


99965 


48 


18 


74848 


99999 


86181 


99989 


60668 


99964 


41 


80 


7.76476 


9.99999 


8.86678 


9.99988 


8.60978 


9.99964 


40 


81 


78694 


99999 


87217 


99988 


61288 


99963 


89 


88 


80616 


99999 


87750 


99988 


61689 


99963 


88 


88 


82645 


99999 


88278 


99987 


61894 


99962 


87 


84 


84898 


99999 


88796 


99987 


62196 


99908 


86 


86 


86166 


99999 


89810 


99987 


62497 


99961 


86 


86 


87870 


WW 


69618 


99988 


62796 


99961 


84 


87 


89509 


90999 


40480 


99946 


68091 


99960 


88 


£8 


91088 


ftftfmft 
WWW 


40816 


99966 


68386 


99960 


82 


28 


92612 


OOOOQ 


41807 


99986 


68678 


99959 


81 


80 


7.94064 


9.99998 


8.41792 


9.99986 


6.63968 


9.99959 


80 


81 


95508 


99998 


42272 


99982 


64256 


99958 


29 


88 


96887 


99998 


42746 


99984 


64548 


99968 


28 


88 


98228 


99998 


48216 


99984 


64827 


99957 


27 


84 


99520 


99996 


48680 


99984 


65110 


99956 


26 


86 


8.00779 


ooooa 

wnw) 


44139 


99983 


65391 


99956 


25 


86 


02002 


99998 


44594 


99988 


65670 


99955 


24 


87 


08192 


99997 


45044 


99983 


65947 


99965 


23 


88 


04850 


99997 


46489 


99982 


66228 


99954 


22 


88 


05478 


99997 


45980 


99982 


66497 


9U954 


21 


40 


8.06578 


9.99997 


8.46866 


9.99982 


8.66769 


9.99958 


20 


41 


07630 


99997 


46799 


99981 


67080 


99952 


19 


43 


08096 


99997 


47226 


99981 


67808 


99952 


18 


48 


09718 


99997 


47650 


99981 


67575 


99951 


17 


44 


10717 


99996 


48069 


99980 


67841 


99961 


16 


46 


11698 


99996 


48485 


99980 


68104 


99950 


15 


46 


19647 


99996 


48896 


99979 


68367 


99949 


14 


47 


18581 


99996 


49304 


99979 


68627 


99949 


18 


48 


14496 


99996 


49708 


99979 


68886 


99948 


18 


40 


15891 


99996 


50108 


99978 


69144 


99948 


11 


60 


8.16868 


9.99996 


8.60504 


9.99978 


8.69400 


9.99947 


10 


61 


17128 


99996 


60897 


99977 


69654 


99946 


9 


68 


17971 


99996 


61287 


99977 


69907 


99940 


8 


68 


18798 


99996 


51678 


99977 


70159 


99945 


7 


64 


19610 


99995 


52055 


99976 


70409 


99944 


6 


66 


80407 


99994 


52484 


99976 


70658 


99944 


5 


66 


21189 


99994 


52810 


99975 


70906 


99948 


4 


67 


81958 


99994 


53188 


99976 


71161 


99942 


8 


68 


22718 


99094 


58552 


99974 


71895 


99942 


2 


60 


26456 


99994 


58919 


99974 


71688 


99941 


1 


00 


24186 


99998 


54282 


99974 


71890 


99940 





t 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


i 




89° 


88° 


87 






45<> 
TABLE n.— LOGARITHMIC SINES AND COSINES. 



/ 




•• 




4* 


«• 


/ 


Sine 


Cosine 


8ine 


Cosine 


Sine 


Cosine 





8.71880 


9.99940 


8.84858 


9.99894 


8.94080 


9.99884 


00 


1 


78120 


99940 


84589 


98808 


94174 


90688 


59 


s 


728G9 


99938 


84718 


99892 


94817 


99682 


58 


8 


78507 


99988 


84897 


90691 


94461 


99681 


57 


4 


78884 


99988 


85075 


99691 


94608 


99680 


66 


6 


78069 


99967 


85258 


99690 


94748 


99689 


55 


6 


78308 


99986 


85429 


90689 


94887 


99828 


54 


7 


78535 


99906 


85605 


99688 


96089 


99627 


53 


8 


78T67 


99985 


85780 


99687 


95170 


99825 


52 


9 


78997 


99934 


85955 


99686 


05810 


99684 


51 


10 


8.74886 


9.99984 


8.86128 


9.99685 


8.05450 


9.00888 


60 


11 


74454 


99988 


86301 


90684 


06689 


99682 


49 


18 


74680 


90988 


86474 


99683 


95728 


99621 


48 


18 


74906 


99938 


86645 


99662 


95867 


99620 


47 


14 


75180 


99981 


86816 


99681 


96005 


09819 


46 


15 


75858 


99930 


86987 


99680 


96148 


99817 


45 


16 


75575 


999*9 


87156 


99679 


96280 


99616 


44 


17 


75796 


99989 


87825 


99879 


96417 


99616 


48 


18 


76015 


999*8 


87494 


996.8 


96558 


99614 


48 


18 


76SS4 


99987 


87661 


99877 


96689 


99618 


41 


90 


8.76451 


9.99986 


8.87829 


9.99876 


8.96885 


9.90618 


40 


21 


76667 


99986 


87995 


99875 


96960 


99810 


89 


88 


76888 


99925 


88161 


99874 


07005 


99609 


88 


88 


77097 


99924 


88826 


99878 


97229 


99608 


87 


84 


77810 


99988 


88490 


99672 


97868 


99607 


86 


85 


77582 


99938 


88654 


99671 


97496 


99806 


85 


86 


77788 


99922 


88817 


99670 


97629 


99604 


84 


27 


77948 


99981 


88980 


99669 


97762 


08608 


88 


88 


78158 


99920 


89148 


99668 


97894 


99608 


88 


89 


78860 


99920 


89804 


90667 


96086 


99801 


81 


80 


8.78568 


9.99919 


8.89464 


9.99866 


8.96157 


9.99600 


80 


81 


7OT74 


99918 


89685 


99665 


96288 


90798 


89 


38 


78979 


99917 


89784 


90664 


98419 


99797 


88 


88 


79133 


99917 


89948 


99668 


98549 


99796 


27 


84 


79386 


iMfllfl. 
99915 


P0108 


90668 


96679 


99795 


26 


85 


79588 


90260 


99861 


96606 


99798 


85 


86 


79789 


99914 


90417 


09660 


96987 


99792 


84 


87 


79990 


99918 


90574 


99659 


99066 


99791 


88 


88 


80189 


99918 


90780 


00658 


09194 


99790 


88 


89 


80888 


99912 


90885 


99657 


99828 


09786 


81 


40 


8.80585 


9.99911 


8.91040 


9.99666 


8.99450 


9.99787 


80 


41 


8078^ 


99910 


91195 


WsOOO 


99577 


09786 


19 


43 


80978 


99909 


91849 


99654 


99704 


99785 


18 


48 


81178 


99909 


91502 


99858 


99880 


99783 


17 


44 


81367 


99908 


91655 


99658 


99956 


90788 


16 


45 


81560 


99907 


91807 


99661 


9.00068 


99781 


16 


46 


81758 


99906 


91959 


99650 


00207 


99780 


14 


47 


81944 


99905 


92110 


99648 


00388 


99778 


18 


48 


82134 


99904 


92261 


99647 


00456 


09777 


18 


49 


88324 


99904 


92411 


99646 


00581 


99778 


11 


60 


8.88518 


9.99903 


8.98561 


9.99645 


9.00704 


9.99775 


10 


61 


82701 


99902 


92710 


00644 


00828 


99778 


9 


58 


88888 


99901 


98859 


99648 


00951 


99772 


8 


58 


83075 


99900 


93007 


90642 


01074 


99771 


7 


54 


88861 


99899 


93154 


99641 


01196 


99759 


6 


55 


83446 


90898 


98301 


09640 


01818 


99768 


6 


56 


83630 


99898 


93448 


93689 


01440 


99767 


4 


57 


83818 


99897 


93594 


09838 


01561 


09765 


8 


58 


88996 


99896 


98740 


99687 


01682 


99764 


8 


59 


84177 


99896 


98885 


99636 


01808 


99768 


1 


60 


84858 


99694 


94080 


99684 


01928 


99761 





9 


Cosine 


Sine 


Cosine 


8lne 


Cosine 


Sine 


# 


\ 




86* 




85* 




84* 



457 



TABLE II.— LOGARITHMIC SINES AND COSINES. 


/ 


6 


• 




r° 


1 


*• 


/ 


Sine 


Cosine 


Sine 


Cosine 


8ine 


Cosine 





9.01928 


9.99761 


9.08589 


9.99675 


9.14856 


9.99575 


60 


1 


04043 


99760 


08699 


99674 


14445 


99574 


59 


2 


08168 


99759 


08795 


9967* 


14585 


99572 


58 


8 


09288 


99757 


06897 


99670 


14624 


99570 


67 


4 


03409 


99756 


08999 


99669 


14714 


99568 


56 


5 


0*380 


99<55 


00101 


99667 


14808 


99566 


55 


6 


09689 


99758 


09209 


69668 


14891 


90565 


54 


7 


02757 


99759 


09804 


99664 


14960 


99568 


58 


8 


02874 


99751 


09405 


99668 


15069 


90561 


59 





09999 


99749 


09508 


99661 


15167 


99559 


51 


10 


9.08109 


9.09748 


9.09606 


9.99659 


9.15245 


9.99657 


50 


11 


06296 


99747 


09707 


99658 


15838 


99556 


49 


18 


03849 


99745 


09607 


89656 


15421 


99554 


48 


18 


03458 


99744 


09907 


99655 


15508 


99559 


47 


14 


06574 


99749 


10006 


99658 


15596 


99550 


46 


15 


03690 


99741 


10106 


99651 


16688 


99548 


45 


16 


08805 


99740 


10905 


99650 


15770 


99548 


44 


17 


03920 


99788 


10304 


99648 


15857 


99545 


48 


18 


04034 


99787 


10408 


69647 


15944 


99548 


49 


19 


04149 


99786 


10501 


99645 


16080 


99541 


41 


90 


9.04969 


9.99784 


9.10599 


9.99648 


9.16116 


9.99539 


40 


91 


04876 


99788 


10697 


89642 


16208 


99537 


89 


29 


04490 


99781 


10795 


99640 


16289 


89585 


38 


94 


04603 


99780 


10898 


99688 


16874- 


99583 


87 


24 


04715 


99798 


1C990 


89637 


16460 


99582 


80 


95 


048 J8 


99797 


11067 


99685 


16545 


99580 


35 


96 


04940 


99796 


11184 


996A 


16631 


99528 


84 


97 


05059 


09784 


11981 


99682 


16716 


90626 


33 


98 


05164 


99723 


11877 


99630 


16801 


90594 


89 


99 


05275 


99791 


11474 


99629 


16886 


99522 


31 


80 


9.05886 


9.99720 


9.11570 


9.99627 


9.16970 


9.99520 


30 


81 


05497 


99718 


11G66 


99626 


171055 


99518 


29 


89 


06607 


99717 


11761 


99624 


17139 


99517 


28 


88 


05717 


99716 


11857 


99622 


17228 


99515 


97 


84 


05827 


99714 


11959 


99620 


17807 


93518 


96 


85 


05987 


90718 


19047 


99618 


17391 


99511 


95 


86 


06046 


99711 


12149 


99617 


17474 


99509 


94 


87 


06155 


99710 


19J86 


99615 


17568 


99507 


98 


88 


06264 


99708 


1288i 


99618 


17641 


9C505 


29 


89 


06872 


99707 


12425 


90612 


17724 


99003 


91 


40 


9.06481 


9.99706 


9.12519 


(.99610 


9.17807 


9.99501 


90 


41 


06589 


99704 


12612 


99608 


17890 


99499 


19 


42 


06J06 


99709 


12706 


99007 


17978 


99497 


18 


48 


06804 


99701 


12799 


99606 


18055 


90496 


17 


44 


06911 


99699 


12892 


99608 


18187 


99494 


16 


45 


07018 


90698 


12965 


99001 


18220 


99492 


15 


46 


07194 


99696 


18078 


99600 


16409 


99490 


14 


47 


07261 


99695 


18171 


99698 


18883 


99488 


18 


48 


07887 


99608 


18268 


89596 


18465 


99486 


12 


49 


07449 


99699 


18355 


99695 


18547 


99484 


11 


50 


9.07548 


9.99690 


9.18447 


9.99693 


9.18628 


9.90489 


10 


51 


07658 


99689 


18589 


99591 


18709 


99480 


9 


59 


07758 


99687 


18680 


99589 


18790 


99478 


8 


58 


07868 


99666 


18729 


99588 


18871 


99476 


7 


54 


07968 


99684 


18818 


99586 


18952 


99474 


6 


55 


08079 


99688 


181)04 


99584 


19038 


99472 


5 


56 


08176 


99681 


l:.994 


99569 


19118 


99470 


4 


57 


08280 


99680 


14085 


99581 


19193 


99468 


8 


58 


06383 


99678 


14175 


99579 


19-278 


99466 


8 


59 


08486 


99677 


14266 


99577 


10358 


99464 


1 


60 


06589 


99675 


14856 


99575 


19483 


99462 





§ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




88* 




Si 9 




81* 



45« 



TABLE II.— LOGARITHMIC SINES AND COSINES. 


/ 


1 


»• 


10* 


If 


» 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.19488 


9.99468 


9.88967 


9:99885 


9.28060 


9.99195 


60 


1 


19518 


99460 


24089 


99888 


28125 


99198 


59 


s 


19598 


99458 


24110 


89881 


28190 


99190 


58 


8 


19678 


99456 


84181 


99828 


88854 


99167 


57 


4 


19751 


99454 


24858 


99396 


26819 


99185 


56 


5 


19680 


99458 


24884 


99884 


28884 


99188 


55 


6 


19909 


99450 


24895 


09828 


28448 


99180 


54 


7 


19988 


Qfrj43 


24466 


99819 


28618 


99177 


58 


8 


80067 


99446 


24586 


99817 


28677 


99175 


58 


9 


80145 


99444 


24607 


99815 


28641 


99178 


51 


10 


9.80888 


9.99448 


9.24677 


9.99818 


9.26705 


9.99170 


50 


11 


80808 


99440 


24748 


99810 


28769 


99167 


49 


18 


80880 


99488 


24818 


99808 


28888 


99165 


48 


18 


90458 


99486 


24888 


99806 


28898 


99162 


47 


14 


80585 


99484 


24958 


99804 


28960 


99160 


46 


15 


90618 


99488 


25088 


99801 


29084 


99157 


45 


16 


20691 


99429 


28098 


99299 


29087 


99155 


44 


17 


20768 


99487 


25168 


99297 


29150 


99158 


48 


18 


20845 


99425 


25287 


99294 


29814 


99150 


42 


19 


20988 


99438 


25807 


99898 


29877 


99147 


41 


90 


9.80999 


9.99421 


9.25876 


9.99290 


9.29840 


9.99145 


40 


81 


21076 


99419 


25445 


ftftnoo 
W30OO 


89408 


99149 


89 


28 


21158 


99417 


25514 


09285 


99466 


99140 


88 


88 


21889 


99415 


25588 


99288 


29529 


99187 


87 


94 


21806 


99418 


25658 


99281 


29591 


99185 


86 


8ft 


21888 


99411 


25781 


99278 


89654 


99188 


85 


83 


21458 


99409 


25790 


99876 


29716 


99180 


84 


87 


21534 


99407 


25858 


99874 


29779 


99187 


88 


88 


21610 


99404 


25987 


99871 


29841 


99124 


88 


29 


21GS5 


99408 


25995 


99269 


29908 


99128 


31 


80 


9.21761 


9.99400 


9.26008 


9.90267 


9.29966 


9.99119 


80 


81 


21886 


99496 


26181 


99264 


80088 


99117 


29 


88 


21918 


99896 


26199 


99262 


80090 


99114 


88 


88 


21987 


99894 


26267 


99260 


80151 


99113 


27 


84 


29068 


99393 


26385 


99257 


80218 


99109 


26 


85 


28187 


99890 


26408 


99255 


80275 


99106 


25 


86 


28211 


99388 


26470 


99852 


80886 


99104 


24 


87 


2*286 


90385 


26588 


99250 


80898 


99001 


88 


88 


28861 


99383 


26605 


99848 


80459 


99099 


22 


89 


28485 


99881 


26673 


99845 


80581 


99096 


81 


40 


9.28509 


9.99379 


9.26789 


9.99248 


9.80588 


9.99098 


80 


41 


28588 


99877 


26806 


99244 


80648 


99091 


19 


48 


22657 


99875 


26678 


99288 


80704 


99088 


18 


48 


22781 


99878 


26940 


99836 


80765 


99086 


17 


44 


28805 


99870 


27007 


99233 


80886 


99088 


16 


45 


22878 


99868 


27078 


99281 


80687 


99080 


15 


46 


28968 


99366 


27140 


99229 


80947 


99078 


14 


47 


28085 


99364 


97206 


99886 


81008 


99075 


IS 


48 


28098 


99868 


27278 


99884 


81068 


99078 


12 


49 


88171 


99859 


87889 


89881 


81189 


99070 


11 


50 


9.98844 


9.99357 


9.87406 


9.99819 


9.81188 


9.99067 


10 


51 


23817 


99255 


27471 


99817 


81250 


99064 


9 


58 


23890 


99353 


27587 


99214 


81810 


99088 


8 


58 


28468 


99351 


27608 


99218 


81870 


99059 


7 


54 


28535 


99348 


27668 


99209 


81480 


99056 


6 


55 


28607 


99846 


27784 


99807 


81490 


99054 


5 


56 


23679 


99844 


27799 


99204 


81549 


99051 


4 


57 


28758 


9934? 


27864 


99203 


81609 


99048 


8 


68 


23823 


90840 


27980 


99200 


81669 


99046 


3 


59 


88895 


99887 


S7995 


99197 


81728 


99048 


1 


60 


28967 


99385 


28060 


99195 


81788 


99040 





t 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


>_ 




80* 




79° 




78* 



459 



TABLE II.— LOGARITHMIC SINES AND COSINES. 


/ 


1«» 


If 


14» 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.31788 


9.99040 


9.86909 


9.98872 


9.88368 


9.96690 


60 


1 


81847 


99038 


85968 


98869 


88418 


98687 


69 


2 


31907 


09085 


85418 


98867 


88469 


96684 


68 


8 


81966 


99082 


85878 


96864 


88519 


96681 


67 


4 


89045 


99080 


86497 


96861 


88570 


98678 


66 


5 


82084 


99097 


85481 


96868 


88690 


98675 


66 


6 


82148 


99024 


86586 


98855 


88670 


98671 


64 


7 


82202 


99029 


85590 


96869 


88791 


98668 


58 


8 


82261 


99019 


86644 


98849 


88771 


96666 


62 


9 


82819 


99016 


86698 


96846 


88891 


96682 


61 


10 


9.89878 


9.99018 


9.85769 


9.96848 


9.88871 


9.98659 


60 


11 


89487 


99011 


85806 


98840 


88921 


96666 


49 


19 


82495 


99008 


85860 


98887 


88971 


96652 


48 


18 


82558 


99005 


85914 


96834 


89021 


96649 


47 


14 


89612 


99009 


85968 


96881 


89071 


98646 


46 


15 


89670 


99000 


86022 


96828 


89121 


98648 


46 


16 


82728 


98997 


86075 


988 J6 


89170 


96640 


44 


17 


82786 


98994 


86199 


96822 


89220 


98686 


48 


18 


82844 


98091 


86189 


98819 


89270 


96688 


49 


19 


82902 


98989 


86286 


96816 


89819 


96U80 


41 


90 


9.82960 


9.98986 


9.86989 


9.96818 


9.89369 


9.98697 


40 


91 


88018 


98988 


86849 


96810 


89418 


96628 


89 


99 


88075 


96980 


86895 


98807 


39467 


96690 


88 


28 


88188 


98978 


86449 


96804 


89517 


96617 


3 


24 


83190 


96975 


86508 


96801 


89566 


98614 


95 


88948 


98972 


86665 


96796 


89615 


98610 


35 


96 


88305 


98969 


86608 


96796 


89664 


98607 


84 


27 


88862 


98967 


86660 


98799 


89718 


96604 


88 


98 


88420 


98964 


86713 


98789 


89762 


98601 


89 


99 


88477 


98961 


86766 


98786 


89811 


98597 


31 


80 


9.33534 


9.98958 


9.86819 


9.98788 


9.89860 


9.96594 


30 


81 


83591 


98965 


86871 


96786 


89909 


98591 


29 


89 


88647 


98953 


86924 


98777 


89958 


96588 


98 


88 


83704 


98950 


86976 


96774 


40006 


98584 


97 


84 


83761 


98947 


87028 


96771 


40055 


98581 


96 


85 


88818 


98944 


87081 


96768 


40103 


96678 


96 


86 


88874 


98941 


87188 


96766 


40152 


96574 


94 


87 


88981 


98988 


87185 


98769 


40900 


98571 


98 


88 


88967 


96986 


87287 


98760 


40249 


98568 


99 


89 


84048 


96988 


87989 


96766 


40297 


98665 


91 


40 


9.84100 


9.98980 


9.87841 


9.96758 


9.40846 


9.98561 


90 


41 


34166 


98997 


87898 


98750 


40894 


96568 


19 


49 


84219 


98994 


87445 


96746 


40449 


98565 


18 


48 


84968 


98921 


87497 


96743 


40490 


98651 


17 


44 


84824 


96919 


87549 


98740 


40688 


98548 


16 


45 


84880 


98916 


87600 


96787 


40586 


96645 


15 


46 


84436 


98918 


87669 


96734 


40684 


98541 


14 


47 


84491 


98910 


87703 


98781 


40689 


96588 


13 


48 


84547 


98907 


87756 


96798 


40730 


98586 


12 


49 


84602 


96904 


87806 


96796 


40778 


98581 


11 


60 


9.84658 


9.98901 


9.87868 


9.98799 


9.40696 


9.96698 


10 


51 


84718 


96896 


87909 


96719 


40678 


98696 


9 


69 


84769 


98896 


87960 


96715 


40091 


98591 


8 


58 


81824 


98898 


88011 


98719 


40968 


98518 


7 


64 


84879 


96890 


88069 


98700 


41016 


98615 


6 


65 


84934 


98887 


88118 


96706 


41068 


96M1 


5 


66 


84909 


98884 


88164 


96708 


41111 


96508 


4 


67 


85044 


96881 


88215 


98700 


41158 


96506 


8 


68 


85099 


96878 


88266 


98697 


41206 


96501 


2 


69 


85154 


98875 


88817 


96694 


41262 


98498 


1 


60 


85909 


98872 


88868 


98690 


41300 


96494 





t 


Cosine 


Bine 


Cosine 


Sine 


Cosine 


Sine 


/ 




77 « 




78« 




76* 



460 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



» 


16* 


18* 


17- 


t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.41800 


9.98494 


9.44084 


9.98884 


9.46594 


9.98060 


60 


1 


41847 


98491 


44078 


96881 


46085 


96056 


69 


s 


41894 


98488 


44188 


98877 


46076 


98053 


58 


8 


41441 


98484 


44166 


98873 


46717 


96048 


57 


4 


41448 


98481 


44810 


96870 


46756 


96044 


66 


5 


41585 


98477 


44858 


98866 


46800 


98040 


55 


6 


41568 


98474 


44497 


98868 


46841 


98036 


64 


7 


41CU8 


98471 


44841 


98359 


46863 


96083 


68 


8 


41675 


96467 


44885 


98855 


46988 


98089 


63 


9 


41738 


98464 


44488 


98851 


46964 


96085 


51 . 


10 


9.41768 


9.98460 


9.44478 


9.98348 


9.47006 


9.98081 


60 


11 


41815 


96457 


44516 


98844 


47045 


96017 


49 


19 


41861 


98458 


44559 


98840 


47086 


96018 


48 


18 


41908 


98460 


44608 


98887 


47187 


96009 


47 


14 


41954 


98447 


44646 


98288 


47168 


98005 


46 


15 


48001 


96448 


44689 


88889 


47809 


96001 


45 


16 


48047 


98440 


44788 


96886 


47849 


97997 


44 


17 


48098 


98186 


44776 


98833 


47290 


97998 


48 


18 


48140 


9*483 


44819 


96818 


47880 


97989 


48 


19 


48186 


98489 


44868 


98815 


47871 


97986 


41 


80 


9.48888 


9.98486 


9.44905 


9.96811 


9.47411 


9.97983 


40 


81 


48878 


98488 


44948 


98807 


47458 


97978 


89 


88 


48384 


98419 


44998 


96804 


47498 


97974 


88 


& 


48870 


98415 


45085 


98800 


47588 


97970 


87 


84 


48416 


98418 


45077 


98196 


47573 


97966 


86 


85 


48401 


96409 


45180 


96193 


47618 


97968 


85 


86 


48507 


98405 


45168 


96189 


47654 


97958 


84 


87 


48558 


98408 


45806 


96165 


47694 


97954 


38 


88 


48599 


98898 


45849 


96181 


47734 


97950 


88 


89 


48644 


08895 


45898 


96177 


47774 


97946 


81 


80 


9.48690 


9.98891 


9.45834 


9.98174 


9.47814 


9.97948 


30 


81 


48785 


98888 


45877 


98170 


47854 


97988 


39 


88 


48781 


96884 


45419 


98166 


47894 


97984 


36 


88 


48886 


98881 


45468 


96163 


47984 


97980 


37 


84 


48878 


96877 


45504 


98169 


47974 


97986 


36 


85 


48917 


96878 


45547 


98155 


4H>14 


97939 


35 


86 


48968 


96870 


45589 


96151 


48054 


97918 


84 


87 


48008 


96366 


45683 


96147 


48094 


97914 


88 


88 


48058 


98868 


45674 


96144 


48188 


97910 


83 


89 


48098 


98859 


45716 


96140 


48173 


97906 


31 


40 


9.48148 


9.98856 


9.45756 


9.96186 


9.48318 


9.97903 


30 


41 


48188 


98858 


45801 


96188 


48853 


97898 


19 


48 


48888 


96849 


45843 


96139 


48893 


97894 


18 


43 


48878 


96845 


45885 


96135 


48383 


97890 


17 


44 


48888 


96848 


45987 


98131 


48871 


97886 


16 


45 


48367 


96838 


45969 


96117 


48411 


97883 


15 


46 


48418 


96884 


46011 


96118 


48450 


97878 


14 


47 


48157 


98381 


46058 


96110 


48490 


97874 


18 


48 


48508 


96887 


46095 


98106 


48589 


97870 


13 


49 


48546 


96884 


46186 


96108 


48568 


97866 


11 


50 


9.48591 


9.98890 


9.46178 


9.98098 


9.48607 


9.97861 


10 


51 


48635 


98817 


46380 


98094 


46647 


97857 


9 


58 


48680 


98818 


46-J68 


98090 


48686 


97858 


8 


58 


48784 


98809 


46308 


98087 


48735 


97849 


7 


54 


48769 


98806 


46345 


96088 


48764 


97845 


6 


55 


48813 


98808 


46886 


98079 


48808 


97841 


6 


56 


48857 


963)9 


46488 


96075 


48848 


97887 


4 


57 


48901 


98895 


46469 


98071 


48881 


97888 


8 


68 


48946 


96291 


46511 


96067 


48980 


97889 


8 


59 


48990 


98888 


46558 


96068 


48959 


97835 


1 


60 


44084 


98884 


46594 


98060 


48999 


97881 





9 


Cosine 


Sine 


Cosine 


81ne 


Cosine 


8lne 


# 




74» 




If 




7V 



4<5i 



TABLE n.— LOGARITHMIC SINES AND 


COSINES. 


$ 


18* 


19° 


80* 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.48996 


9.97891 


9.51964 


9.97587 


9.58405 


9.97999 


60 


1 


49087 


97817 


51801 


97568 


58440 


97994 


59 


9 


49076 


97819 


51888 


97558 


58475 


97989 


58 


8 


49115 


97808 


61374 


97554 


53509 


97985 


57 


4 


49158 


97804 


51411 


97550 


58544 


97280 


56 


5 


49199 


97800 


51447 


97545 


58578 


97976 


66* 


6 


49281 


97796 


61484 


97541 


58618 


97971 


54 


7 


49969 


97799 


51590 


97586 


58647 


97966 


58 


8 


49808 


97788 


61557 


97588 


58689 


97269 


59 


9 


49847 


97784 


51598 


97598 


58716 


97957 


51 


10 


9.49385 


9.97779 


9.51689 


9.97898 


9.58751 


9.97969 


50 


11 


49494 


97775 


51666 


97519 


58785 


97948 


49 


19 


49468 


97771 


51709 


97515 


58819 . 


97948 


48 


18 


49500 


97787 


51788 


97510 


5.1854 


97988 


47 


14 


49589 


97768 


51774 


97506 


58888 


97884 


46 


15 


49577 


97759 


51811 


97501 


58999 


97999 


45 


16 


49615 


97754 


51847 


97497 


53967 


97994 


44 


17 


49654 


97750 


61888 


97499 


58991 


97990 


48 


18 


49699 


97746 


61919 


97488 


54096 


97915 


49 


19 


49780 


97749 


61955 


97484 


5 059 


97910 


41 


90 


9.49768 


9.97788 


9.61991 


9.97479 


9.54098 


9.97906 


40 


91 


49606 


97784 


59097 


97475 


54197 


97901 


89 


99 


49644 


97799 


59068 


97470 


54161 


97196 


88 


28 


49882 


97795 


59099 


97466 


54195 


97199 


87 


94 


49930 


97791 


59186 


97461 


54999 


97187 


86 


95 


49958 


97717 


59171 


97457 


54968 


97189 


86 


96 


49996 


97718 


59907 


97458 


54997 


97178 


84 


97 


50084 


97706 


59949 


97448 


54881 


97178 


88 


98 


50079 


97704 


59978 


97444 


54865 


97168 


89 


99 


50110 


97700 


59814 


97489 


54899 


97168 


81 


80 


9.50148 


9.97696 


9.59850 


9.97435 


9.54488 


9.97159 


80 


81 


50185 


97691 


58885 


97480 


54466 


97154 


89 


89 


60283 


97687 


59491 


97496 


54500 


97149 


28 


88 


50961 


97688 


59456 


97491 


54584 


97145 


97 


84 


50*98 


97679 


53408 


97417 


54567 


97140 


96 


85 


50886 


97674 


69597 


97419 


54601 


97135 


95 


86 


50874 


97670 


59568 


97408 


. 54685 


97180 


94 


87 


60411 


97666 


59596 


97408 


54668 


97126 


98 


88 


50449 


97669 


59684 


97899 


54709 


97191 


99 


80 


50486 


97657 


59669 


97394 


54786 


97116 


81 


40 


9.50598 


9.97658 


9.59705 


9.97890 


9.54769 


9.97111 


90 


41 


50561 


97649 


59740 


97385 


54809 


97107 


19 


49 


50509 


97645 


59775 


97881 


54886 


97109 


18 


48 


50635 


97640 


59811 


97876 


54889 


97097 


17 


44 


50073 


97686 


69846 


97379 


54908 


97099 


16 


45 


50710 


97689 


59881 


97867 


54986 


' 97067 


15 


46 


50747 


97688 


52916 


97868 


54969 


97068 


14 


47 


50784 


97698 


59951 


97858 


55008 


97078 


18 


48 


50691 


97619 


59966 


97858 


65036 


97078 


18 


49 


50658 


97615 


58091 


97349 


65069 


97068 


11 


50 


9.50696 


9.97610 


9.58056 


9.97344 


9.55108 


9.97068 


10 


61 


50933 


97606 


58099 


97340 


56186 


97059 


9 


69 


50970 


97609 


58196 


97885 


55169 


97054 


8 


58 


61007 


97597 


53161 


97881 


55209 


97049 


7 


54 


51048 


97598 


58196 


97896 


55285 


97044 


6 


55 


51060 


97589 


58981 


97829 


55266 


97089 


5 


56 


51117 


97584 


53266 


97317 


55301 


97035 


4 


57 


61154 


97580 


58801 


97819 


55384 


97080 


8 


58 


51101 


97576 


58886 


97808 


55867 


97095 


8 


59 


51997 


97571 


58370 


97803 


55400 


97090 


1 


60 


51964 


97567 


58405 


97999 


65483 


97015 







Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


§ 




If 




70° 




89* 



462 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



/ 


21° 


22° 


28° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.55438 


9.97015 


9.57358 


9.96717 


9.59188 


9.96408 


60 


1 


55466 


97010 


57389 


96711 


59218 


96897 


50 


2 


55499 


97005 


57420 


96706 


50247 


96892 


68 


8 


55582 


97001 


67451 


96701 


59277 


968*7 


57 


4 


55564 


96996 


57482 


OAJtftt 
VOCVO 


69807 


96881 


66 


5 


55597 


96991 


67514 


96691 


59836 


96376 


55 


e 


55680 


96986 


57M5 


96686 


59366 


96370 


54 


7 


55668 


96981 


57576 


96681 


59396 


96865 


53 


8 


55695 


96976 


•67607 


96676 


59425 


96860 


62 


9 


55788 


96971 


57688 


96670 


59455 


96854 


51 


10 


9.55761 


9.96966 


9.57669 


9.96665 


9.59484 


9.96349 


50 


11 


55793 


9G962 


57700 


96660 


50514 


96348 


40 


1* 


558*1 


96957 


57781 


96655 


59543 


96338 


48 


13 


55858 


96952 


57762 


96650 


59578 


96388 


47 


14 


55891 


96947 


57793 


96645 


59602 


96827 


46 


15 


55928 


96942 


57824 


96640 


59682 


96822 


45 


16 


65956 


96987 


57855 


96684 


59661 


96816 


44 


17 


55968 


96932 


57885 


96629 


59690 


96311 


43 


18 


56021 


96927 


67916 


96624 


69720 


96805 


42 


10 


56058 


96922 


57947 


96619 


59749 


96800 


41 


80 


9.56065 


9.96917 


9.57978 


9.96614 


9.59778 


9.96294 


40 


21 


56118 


96912 


68008 


96608 


59808 


96289 


80 


23 


56150 


90907 


56039 


96608 


59687 


96284 


88 


28 


66182 


96908 


58070 


96596 


69666 


96278 


37 


24 


56215 


96898 


58101 


96598 


59895 


96278 


36 


25 


56247 


96898 


58181 


96588 


59924 


96267 


35 


26 


56279 


96888 


68162 


96582 


59954 


96262 


34 


27 


56811 


96883 


58192 


96577 


59988 


96256 


83 


28 


66848 


96878 


68228 


96572 


60012 


96251 


89 


29 


56S75 


96878 


58253 


96567 


60041 


96245 


81 


80 


9.66408 


9.96868 


9.58284 


9.96562 


9.60070 


9.96240 


80 


81 


56440 


96863 


58814 


(MIMA • 
WOOD 


60089 


96284 


20 


32 


56472 


96858 


58345 


96551 


60128 


96229 


28 


83 


56504 


96853 


58375 


96546 


60157 


96&J8 


27 


84 


56536 


96848 


58406 


96541 


60186 


96218 


26 


85 


56568 


96848 


58486 


96535 


60215 


96212 


25 


86 


56599 


96838 


. 58467 


96530 


60244 


96207 


24 


87 


56631 


96838 


58497 


96525 


60278 


96201 


28 


88 


56668 


96828 


58527 


96520 


60802 


96196 


22 


80 


56695 


96828 


58557 


96514 


60881 


96190 


21 


40 


9.66727 


9.96818 


9.58588 


9.96509 


9.60359 


9.96185 


20 


41 


56759 


96818 


58618 


96504 


60*88 


96179 


19 


42 


56790 


96806 


58648 


96498 


60417 


96174 


18 


48 


56822 


96803 


58678 


96498 


60446 


96168 


17 


44 


66854. 


96798 


58709 


964S8 


60474 


96162 


16 


45 


56886 


96793 


58739 


96483 


60503 


96157 


15 


46 


56917 


96788 


58769 


96477 


60532 


96151 


14 


47 


56949 


96783 


58799 


96472 


60561 


96146 


18 


48 


56960 


96778 


58829 


96467 


60589 


96140 


12 


49 


57013 


96772 


58859 


96461 


60618 


96185 


11 


50 


9.57044 


9.96767 


9.58889 


9.96456 


9.60646 


9.96129 


10 


51 


57075 


96762 


58919 


96451 


60675 


96128 


9 


52 


57107 


96757 


58949 


96445 


60704 


96118 


8 


58 


57188 


96752 


58979 


96440 


60782 


96112 


7 


54 


57169 


96747 


59009 


96435 


60761 


96107 


6 


55 


57201 


96742 


69039 


96129 


60789 


96101 


5 


66 


67282 


96787 


59069 


96424 


60618 


96095 


4 


57 


57264 


96732 


59096 


96419 


60646 


96090 


8 


58 


57295 


96727 


59128 


96418 


60875 


96084 


% 


59 


57826 


96722 


59158 


96408 


60903 


96079 


1 


60 


57358 


96717 


59188 


96403 


60981 


96078 





» 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


8ine 


/ 




68° 




67* 




66* 



4^3 



TALBE H.— LOGARITHMIC SINES AND COSINES. 


/ 


84« 




Jo* 


86° 


# 


Sine 


Coeine 


Sine 


Cosine 


Sine 


Conine 





9.60981 


9.96078 


9.62595 


9.95728 


9.64184 


9.95866 


00 


i 


60960 


96067 


62628 


95728 


64810 


95860 


69 


s 


60968 


96068 


62649 


95716 


64286 


95854 


68 


8 


61016 


96036 


62676 


95710 


64368 


95848 


57 


4 


61045 


96060 


68708 


95704 


64288 


96341 


66 


6 


61078 


96045 


68780 


95698 


64818 


95885 


56 


6 


61101 


96039 


68757 


95698 


64889 


95329 


54 


7 


61189 


96084 


62784 


95686 


64365 


95828 


53 


8 


61158 


96088 


62811 


95680 


64891 


95817 


68 


9 


61186 


96088 


62888 


95674 


64417 


95810 


61 


10 


9.61814 


9.96017 


9.62865 


9.95668 


9.64448 


9.95804 


50 


11 


61848 


96011 


68899 


95663 


64468 


95298 


49 


18 


61870 


96005 


63918 


95657 


64494 


95298 


48 


18 


61X98 


96000 


62945 


95651 


64510 


95286 


47 


14 


61386 


95994 


62978 


95645 


64545 


95879 


46 


15 


61354 


95988 


62990 


95639 


64571 


95278 


46 


1ft 


61888 


95982 


68026 


95688 


64596 


95867 


44 


17 


61411 


95977 


63052 


95637 


64622 


95361 


48 


18 


61488 


95971 


63G79 


95621 


64C47 


96854 


42 


19 


61466 


95965 


63106 


95616 


64673 


95348 


41 


80 


9.61494 


9.95960 


9.68138 


9.95609 


9.64698 


9.95348 


40 


81 


61524 


95954 


68159 


96608 


64784 


95886 


89 


83 


61550 


95948 


68186 


95597 


64749 


95329 


88 


88 


61578 


95942 


63213 


95591 


64775 


95338 


87 


84 


61606 


93937 


63239 


05585 


64800 


95317 


86 


85 


61684 


95981 


63266 


95579 


64836 


95311 


86 


86 


61668 


95925 


68298 


95578 


64851 


95904 


84 


27 


61689 


95920 


63319 


95567 


64877 


95196 


88 


88 


61717 


95914 


63815 


95561 


64908 


95198 


88 


89 


61745 


95908 


63378 


95555 


64937 


95185 


81 


80 


9.61778 


9.95908 


9.63396 


9.95549 


9.64958 


9.95179 


80 


81 


61800 


95897 


63425 


93548 


64978 


95178 


29 


3* 


61828 


95891 


68451 


95587 


65003 


95167 


28 


83 


61856 


96835 


63478 


96531 


65029 


95160 


27 


84 


61883 


95879 


68504 


95525 


65054 


05164 


36 


85 


61911 


95878 


68531 


95519 


65079 


95148 


35 


86 


61939 


95868 


68557 


96513 


65104 


93141 


84 


87 


61966 


95868 


0T388 


95507 


65180 


95185 


38 


88 


61994 


95856 


63010 


95500 


65155 


95189 


83 


89 


62021 


95850 


63086 


95494 


65180 


95138 


31 


40 


9.63049 


9.95844 


9.68663 


9.95488 


9.65305 


9.95116 


80 


41 


68076 


95839 


63689 


95482 


65280 


95110 


19 


43 


68104 


95838 


63715 


95476 


65355 


95108 


18 


48 


68131 


95827 


63741 


95470 


65281 


95097 


17 


44 


68159 


05831 


68767 


95464 


65306 


95090 


16 


45 


68186 


95815 


63794 


95458 


65381 


95084 


15 


40 


68814 


95810 


63820 


95453 


65356 


95078 


14 


47 


62841 


95804 


63846 


95446 


65381 


95071 


18 


48 


68808 


95708 


68S78 


95440 


65406 


95065 


13 


49 


68-296 


95793 


68898 


05484 


65431 


95059 


11 


60 


9.68388 


9.95786 


9.68984 


9.95437 


9.65456 


9.95053 


10 


51 


68850 


95780 


68950 


95421 


65481 


95046 


9 


58 


62877 


95775 


63976 


95415 


65506 


95039 


8 


58 


62405 


95709 


64008 


93409 


65531 


93033 


7 


64 


62488 


95708 


64028 


93403 


65565 


95087 


6 


55 


68459 


95757 


64054 


95897 


65580 


95030 


5 


56 


62486 


93751 


64080 


93891 


65605 


95014 


4 


57 


62518 


95745 


64106 


95884 


65030' 


95007 


8 


58 


62541 


95789 


64138 


95378 


65655 


95001 


8 


59 


62568 


95738 


64158 


95373 


65080 


94995 


1 


60 


62595 


95728 


64184 


95866 


65705 


94988 








Cosine 


Sine 


Cosine 


Sine 


Cosine 


8ine 


/ 




66° 




64° 




68° 



Jfi4 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



9 


27* 


28* 


29° 


$ 


Sine 


Cosine 


8lne 


Cosine 


Sine 


Cosine 





9.65705 


9.94988 


9.67161 


9.94598 


9.68657 


9.94182 


60 


1 


657*9 


94988 


67186 


945T 


68580 


94175 


69 


2 


65754 


94975 


67208 


94580 


68603 


94168 


58 


8 


65779 


94969 


67288 


94578 


6ftEfi 


94161 


57 


4 


65804 


94988 


67256 


94567 


68648 


94154 


66 


5 


65828 


94956 


67280 


94560 


68071 


94147 


55 


6 


65858 


94949 


67808 


94558 


68694 


94140 


54 


7 


65878 


94948 


67887 


94546 


68716 


94188 


58 


8 


65908 


94986 


67850 


94540 


68789 


94126 


52 


9 


65947 


94980 


67874 


94588 


68768 


94119 


51 


10 


9.65958 


9.94928 


9.67898 


9.94586 


9.68784 


9.94112 


50 


11 


65976 


94917 


67481 


94519 


68807 


94105 


40 


12 


66001 


94911 


67445 


94618 


68889 


94098 


48 


28 


66025 


94904 


67468 


94506 


68852 


94090 


47 


14 


66050 


94898 


67493 


94499 


66875 


94088 


46 


15 


60075 


94891 


67515 


94498 


66897 


94076 


45 


16 


66099 


94885 


67589 


94486 


66920 


94089 


44 


17 


66184 


94878 


67582 


94479 


68942 


94062 


48 


18 


66148 


94871 


67586 


94479 


66965 


94055 


42 


19 


66178 


04866 


67609 


94465 


68987 


94048 


41 


80 


9.66197 


9.94858 


9.67688 


9.94458 


9.69010 


9.94041 


40 


81 


66J21 


94668 


67656 


94451 


69088 


94034 


89 


88 


66846 


94846 


67680 


94445 


69055 


94087 


88 


83 


66270 


94889 


67703 


94488 


69077 


94080 


87 


84 


66895 


94882 


67786 


94481 


69100 


94012 


86 


85 


66819 


94886 


67750 


94424 


69128 


94005 


85 


86 


66848 


94819 


67778 


94417 


69144 


98998 


84 


87 


66368 


94818 


67798 


94410 


69167 


98991 


88 


88 


66398 


94808 


67880 


94404 


69189 


98984 


82 


29 


66416 


94799 


67848 


94897 


69212 


98977 


81 


80 


9.66441 


9.94798 


9.67866 


9.94390 


9.69284 


9.98970 


80 


81 


66465 


94786 


67890 


94388 


69256 


98988 


29 


88 


66489 


94780 


67918 


94376 


69279 


98955 


28 


83 


66518 


94778 


67986 


94869 


69801 


98948 


27 


84 


66687 


94767 


67959 


94802 


69828 


98941 


26 


85 


66568 


94760 


67988 


94355 


69846 


98984 


25 


86 


665S6 


94758 


68006 


94849 


69868 


93927 


24 


87 


66610 


94747 


68029 


94343 


69390 


98920 


28 


88 


68634 


94740 


68052 


94835 


69412 


98912 


22 


89 


68668 


94784 


68075 


94828 


69434 


98905 


21 


40 


9.66688 


9.94787 


9.68098 


9.94821 


9.69456 


9.98898 


20 


41 


66708 


94720 


68121 


94814 


69479 


98891 


19 


48 


66781 


94714 


68144 


94807 


69501 


98884 


18 


48 


66755 


94707 


68167 


94300 


69528 


98876 


17 


44 


66779 


94700 


68190 


94298 


69545 


98869 


16 


45 


66808 


94694 


68218 


94286 


69567 


98862 


15 


46 


66887 


94687 


68287 


94279 


69589 


98855 


14 


47 


66851 


94680 


68260 


94278 


60611 


93847 


18 


48 


66875 


94674 


68283 


94266 


69688 


93840 


12 


49 


66899 


94667 


68805 


94259 


69665 


98888 


11 


50 


9.66983 


9.94660 


9.68888 


9.94252 


9.69677 


9.98826 


10 


51 


66946 


94654 


68351 


94245 


69699 


98819 


9 


58 


66970 


94647 


68874 


94288 


69721 


98811 


8 


58 


66994 


94640 


68397 


94231 


69748 


93804 


7 


54 


67018 


94684 


68420 


94224 


69765 


98797 


6 


55 


67043 


94627 


68443 


94217 


69787 


93789 


6 


56 


67066 


94620 


68466 


94210 


69809 


98782 


4 


57 


67090 


'94614 


66489 


94203 


69831 


98775 


8 


58 


67113 


94607 


68519 


94196 


69653 


93768 


9 


59 


67187 


94600 


68534 


94189 


69675 


93760 


1 


60 


67161 


94593 


68657 


94188 


69897 


98758 





/ 


Cosine 


Sine 


Cosine 


Sine 


CoBlne 


8ine 


/ 




B8* 




«• 




W 



465 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



/ 


80° 


•1* 


82° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





0.60607 


9.98758 


9.71184 


9.93807 


9.79491 


9.99849 


60 


1 


60010 


98748 


71205 


98290 


72441 


99884 


59 


s 


60041 


98788 


71296 


98991 


72461 


92896 


58 


8 


60068 


98781 


71847 


93284 


79489 


92818 


67 


4 


60084 


08784 


71968 


98276 


72509 


92810 


56 


8 


70006 


98717 


71289 


98269 


72598 


99808 


55 


6 


70098 


08709 


71810 


08261 


72549 


99795 


54 


7 


70050 


93709 


71881 


98258 


72562 


92787 


S 


8 


70079 


98606 


71859 


93246 


72589 


92779 


9 


70008 


98687 


71878 


98238 


72609 


92771 


51 


10 


0.70115 


'9.99680 


9.71398 


9.98930 


9.79629 


9.92768 


50 


11 


70187 


98678 


71414 


03&>8 


79048 


99755 


49 


1* 


70150 


98665 


71485 


93215 


79668 


92747 


48 


18 


70180 


98668 


71456 


98207 


79688 


99789 


47 


14 


70909 


98650 


71477 


98200 


79708 


99731 


46 


15 


70294 


93648 


71498 


93199 


79798 


99798 


45 


16 


70945 


98686 


71519 


93184 


79748 


99715 


44 


17 


70287 


98688 


71589 


98177 


72763 


99707 


48 


18 


70288 


986*1 


71560 


93169 


72788 


92099 


48 


10 


70810 


98614 


71561 


98161 


79808 


99691 


41 


90 


0.70889 


9:98606 


9.71609 


9.98154 


9.79898 


9.99688 


40 


81 


70858 


98599 


71629 


98146 


79848 


99675 


89 


98 


70875 


98591 


71648 


93188 


72868 


99667 


88 


28 


70806 


98584 


71664 


93181 


.79888 


92659 


87 


24 


70418 


98577 


71685 


98198 


79902 


92651 


86 


95 


70480 


98569 


71705 


93115 


79992 


98648 


85 


26 


70461 


93568 


71796 


98106 


72949 


jbbsl, 


84 


97 


70489 


98554 


71747 


93100 


79969 


88 


88 


70504 


98547 


71767 


98099 


72989 


92619 


89 


99 


70595 


08539 


71788 


98064 


78009 


99611 


81 


80 


9.70547 


9.98589 


9.71809 


9.98077 


9.78099 


9.92608 


80 


81 


70568 


98595 


71899 


98069 


78041 


99595 


89 


89 


70590 


98517 


71850 


98061 


78081 


99587 


3 


88 


70611 


98510 


71870 


93058 


78081 


99579 


84 


70688 


93509 


71891 


98046 


78101 


99571 


96 


85 


70654 


98405 


71911 


93088 


78191 


92568 


95 


88 


70675 


98187 


71989 


93080 


78140 


99555 


94 


87 


70607 


93180 


71952 


98029 


78160 


92546 


98 


88 


70718 


93472 


71978 


93014 


73180 


92588 


99 


80 


70789 


93465 


71994 


98007 


73200 


99580 


91 


40 


0.70761 


9.98457 


9.79014 


9.99999 


9.78919 


9.99599 


90 


41 


70789 


93450 


72034 


92991 


78939 


99514 


19 


49 


70808 


98142 


72055 


92988 


78259 


92506 


18 


48 


70694 


98435 


72075 


99976 


78278 


92498 


17 


44 


70846 


98427 


72006 


92968 


78298 


92490 


16 


45 


70667 


98490 


72116 


92960 


78318 


99489 


16 


46 


70688 


98419 


72187 


92959 


78337 


99478 


14 


47 


70000 


98405 


72167 


92944 


78857 


92465 


18 


48 


70031 


93397 


72177 


92086 


73817 


99457 


19 


40 


70939 


98890 


79196 


92999 


78896 


99449 


11 


50 


9.70978 


9.98889 


9.72918 


9.92991 


9.78416 


9.99441 


10 


51 


70094 


98875 


72288 


92918 


73485 


92438 


9 


5* 


71015 


93367 


72259 


92906 


73455 


92425 


8 


58 


71036 


93360 


72279 


92897 


73474 


024111 
"~W40$ 


7 


54 


71058 


93352 


72299 


92889 


73494 


6 


55 


71079 


93344 


72320 


92881 


78518 


92400 


6 


58 


71100 


98337 


72340 


92874 


73533 


92399 


4 


57 


71121 


93329 


72360 


92866 


78552 


92384 


8 


58 


71149 


03392 


72381 


92858 


73579 


92376 


8 


50 


71168 


93314 


72401 


92H50 


73591 


92367 


1 


80 


71184 


93807 


72421 


92849 


73611 


93359 





i 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


# 




6»° 




68° 




«?• 



4M> 



TABLE II.— LOGARITHMIC SINES AND COSINES. 


/ 


8S» 


•4- 


86* 


§ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.78611 


9.92359 


9.74756 


9.91857 


9.75859 


9.91386 


60 


i 


78080 


92851 


74775 


91849 


75877 


91828 


59 


8 


78030 


92348 


74794 


91840 


75895 


91819 


58 


8 


78609 


92885 


74818 


91882 


75918 


91810 


57 


4 


78689 


92826 


74881 


91828 


75981 


91801 


56 


5 


73708 


92818 


74850 


91815 


75949 


91298 


55 


6 


737,»7 


92810 


74868 


91806 


75967 


91288 


54 


7 


78747 


92802 


74887 


91798 


75986 


91274 


58 


8 


78766 


92898 


74906 


91789 


76008 


91266 


58 


9 


78785 


92885 


74984 


91781 


76021 


91857 


51 


10 


9.78805 


9.92277 
98269 


9.74943 


9.91772 


9.76089 


# 9.91848 


50 


11 


78884 


74961 


91768 


76057 


91289 


49 


12 


78848 


92260 


74980 


91755 


76075 


91280 


48 


18 


78868 


92858 


74999 


91746 


76098 


91281 


47 


14 


78882 


92844 


76017 


91788 


76111 


91818 


46 


15 


78901 


92235 


75086 


91729 


76189 


91208 


46 


16 


78921 


92827 


75054 


91780 


76146 


91194 


44 


17 


78940 


92219 


75078 


91713 


76164 


91185 


48 


18 


78959 


92211 


75091 


91708 


76182 


91176 


48 


19 


78978 


98202 


75110 


91695 


76200 


91167 


41 


SO 


9.78997 


9.92194 


9.75188 


9.9K86 


9.76818 


9.91158 


40 


81 


74017 


92186 


75147 


91677 


76286 


91149 


89 


83 


74086 


98177 


75165 


91669 


76258 


91141 


88 


88 


74065 


93169- 


75184 


91660 


76271 


91188 


87 


84 


74074 


92161 


75202 


91651 


76289 


91128 


86 


25 


74098 


92158 


75281 


91648 


76807 


91114 


85 


96 


74118 


92144 


75889 


916*4 


76884 


91105 


84 


87 


74182 


92186 


76258 


91685 


76843 


91096 


88 


88 


74151 


92127 


75276 


91617 


76860 


91067 


83 


89 


74170 


98119 


75294 


91608 


76878 


91078 


31 


80 


9.74189 


9.92111 


9.75813 


9.91599 


9.76895 


9.91069 


30 


81 


74808 


92102 


75831 


91591 


76418 


91060 


89 


38 


74227 


92094 


75850 


91588 


76481 


91051 


88 


88 


74846 


92066 


75868 


91578 


76448 


91048 


87 


84 


'<4865 


92077 


75886 


91565 


76466 


91088 


86 


85 


74284 


92069 


75405 


91556 


76484 


91088 


85 


88 


74808 


92060 


75428 


91547 


76501 


91014 


84 


8Y 


74882 


92052 


75441 


91538 


76519 


91005 


88 


88 


74341 


92044 


75459 


. 91580 


76587 


90996 


28 


89 


74860 


92085 


75478 


91521 


76554 


90987 


81 


40 


9.74879 


9.98027 


9.75496 


9.91618 


9.76572 


9.90978 


80 


41 


74898 


92018 


75514 


91504 


76590 


90969 


19 


42 


74417 


92010 


75533 


91495 


76607 


90960 


18 


48 


74436 


92008 


76561 


91486 


76626 


90961 


17 


44 


74455 


91998 


75569 


91477 


76642 


90948 


16 


45 


74474 


91985 


75567 


91469 


76660 


90988 


15 


46 


74498 


91976 


75605 


91460 


76677 


90884 


14 


47 


74518 


91968 


75684 


91451 


76695 


90915 


18 


48 


74581 


91959 


75643 


91448 


76713 


90906 


18 


49 


74549 


91951 


75660 


91488 


76780 


90896 


11 


50 


9.74568 


9.91948 


9.75678 


9.91425 


9.76747 


9.90887 


10 


51 


74587 


91984 


75696 


91416 


76765 


90878 


9 


58 


74606 


91925 


75714 


91407 


76788 


90809 


8 


58 


74685 


91917 


75788 


91898 


76800 


90860 


7 


54 


74644 


91908 


75751 


91889 


78817 


90851 


6 


55 


74668 


91900 


75769 


91881 


76885 


90848 


6 


56 


74681 


91891 


75787 


91872 


76853 


90688 


4 


57 


74700 


91888 


75805 


91368 


76870 


90628 


8 


58 


74719 


91874 


75828 


91354 


76887 


90814 


8 


59 


74787 


91866 


75841 


91345 


76904 


90805 


1 


60 


74756 


91857 


75859 


91886 


76988 


90796 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


i 




W 




65* 




54* 



4 07 



TABLE II.— LOGARITHMIC SINES AND COSINES. 



# 


86° 


17 o 


88° 


/ 


Sine 


Cosine 


Bine 


Cosine 


Sine 


Cosine 





9.76929 


9.90796 


9.77946 


9.90985 


9.78984 


9.89658 


60 


1 


76939 


90787 


77968 


90996 


78950 


89648 


09 


9 


78957 


90777 


77980 


90916 


78967 


89638 


68 


8 


76974 


90768 


77997 


90906 


78963 


89624 


57 


4 


76991 


90759 


78018 


90197 


78999 


89614 


56 


8 


77009 


90750 


78080 


90187 


79015 


89604 


55 


6 


770* 


90741 


78047 


90178 


79081 


89594 


54 


7 


77048 


90731 


78068 


90168 


79047 


89584 


53 


8 


77061 


90799 


78080 


90159 


79063 


89574 


59 


9 


77078 


90718 


78097 


90149 


79079 


80564 


51 


10 


9.77095 


9.90704 


9.78118 


9.90189 


9.79095 


9.89554 


50 


11 


77119 


90694 


78180 


90180 


79111 


89544 


49 


1* 


77130 


90685 


78147 


90190 


79198 


89584 


48 


18 


77147 


90676 


78168 


90111 


79144 


89534 


47 


14 


77164 


90667 


78180 


90101 


79160 


89514 


46 


15 


77181 


90657 


78197 


90001 


79176 


89504 


46 


16 


77199 


90648 


78218 


90089 


79199 


89496 


44 


17 


77916 


90689 


78930 


90079 


79908 


89485 


48 


18 


77948 


90630 


78246 


90068 


79324 


89475 


49 


19 


77950 


90690 


78963 


90058 


79940 


89465 


41 


90 


9.77968 


9.90611 


9.78280 


9.90048 


9.79956 


9.89455 


40 


91 


77985 


90609 


78996 


90084 


79979 


89445 


89 


9* 


7730* 


90599 


78818 


90094 


79988 


89485 


88 


x8 


77319 


90588 


78399 


90014 


79804 


89425 


87 


94 


77336 


90574 


78340 


90005 


79819 


89415 


86 


95 


77858 


90565 


78369 


89995 


79385 


89405 


85 


90 


77370 


90555 


78379 


89965 


79851 


89896 


84 


97 


job*- 

77405 


90640 


78395 


89976 


79367 


89385 


88 


98 


90587 


78419 


89966 


79383 


89375 


89 


99 


77499 


90597 


78498 


89950 


79399 


89364 


81 


80 


9.77439 


9.90018 


9.78445 


9.89947 


9.79415 


9.89354 


80 


81 


77456 


90509 


78461 


89937 


79481 


89844 


99 


89 


77478 


90499 


78178 


89997 


79447 


89834 


28 


88 


77400 


90190 


78494 


80918 


79468 


69894 


97 


81 


77507 


90480 


78510 


89908 


79478 


89314 


26 


85 


77594 


.90471 


78537 


89898 


79494 


89804 


95 


88 


77541 


90469 


78548 


89888 


79510 


89994 


94 


87 


77558 


90454 


78560 


89679 


79596 


89984 


98 


88 


77575 


90443 


78576 


89869 


79549 


89974 


29 


89 


77598 


90434 


78503 


89859 


79558 


89964 


21 


40 


9.77609 


9.90494 


9.78609 


9.89849 


9.79578 


9.89954 


90 


41 


77696 


90415 


78695 


89840 


79589 


89944 


19 


42 


77643 


90405 


78649 


89830 


79605 


89988 


18 


48 


77660 


90896 


79658 


89830 


79691 


89228 


17 


44 


77677 


90386 


78674 


89810 


79686 


89218 


16 


45 


77694 


90377 


78691 


89801 


79659 


89908 


15 


40 


77711 


90868 


78707 


89791 


79668 


89198 


14 


47 


77798 


90358 


78798 


89781 


79684 


89188 


18 


48 


77744 


90349 


78789 


89771 


79699 


89178 


12 


49 


77761 


90389 


78756 


89761 


79715 


89169 


11 


50 


9.77778 


9.90330 


9.78779 


9.89759 


9.79781 


9.89159 


10 


51 


77795 


90890 


7R788 


89742 


79746 


89149 


9 


5* 


77819 


90311 


78805 


89732 


79769 


89189 


8 


53 


778* 


90301 


78821 


79778 


89199 


7 


54 


77846 


90299 


78837 


89719 


79798 


69119 


6 


55 


77869 


90283 


78853 


89709 


79609 


69101 


6 


50 


77879 


90373 


78869 


89698 


79825 


89091 


4 


57 


77898 


90263 


78886 


89688 


79640 


89061 


8 


58 


77918 


90354 


78909 


89C78 


79656 


89071 


8 


59 


77930 


90344 


78918 


89663 


79872 


80060 


1 


00 


77946 


90935 


78984 


89653 


79887 


89050 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




58° 




62° 




61° 



468 



TABLE n.— LOGARITHMIC SINES AND COSINES. 



1 


*»• 


40* 


«• 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.79887 


9.89060 


9.80807 


9.88425 


9.81694 


9.87778 


60 


1 


79903 


89040 


80622 


88415 


81709 


87767 


69 


s 


79918 


89080 


80887 


88404 


81798 


87756 


68 


8 


79984 


89020 


80852 


88894 


81788 


87746 


67 


4 


79950 


89009 


80867 


88888 


81752 


87784 


66 


6 


79966 


flonnA 

oowv 


80882 


88379 


81767 


87788 


65 


6 


79981 


' 88089 


80897 


88862 


81781 


87718 


64 


7 


79996 


88978 


80912 


88851 


81796 


87701 


68 


8 


80012 


88968 


80927 


88840 


81810 


87690 


62 


9 


80027 


88958 


80942 


88880 


81825 


87679 


61 


10 


9.80048 


9.88948 


9.80057 


9.88819 


9.81889 


9.87668 


60 


11 


80068 


88987 


80979 


88808 


81864 


87657 


49 


13 


80074 


88927 


80067 


88298 


81868 


87646 


48 


18 


80069 


88917 


81002 


88287 


81882 


87685 


47 


14 


80105 


88906 


81017 


88276 


81897 


87624 


46 


18 


80120 


88896 


81032 


88266 


81911 


87618 


45 


16 


80186 


88886 


81047 


O0XOO 


81926 


87601 


44 


17 


80151 


88875 


81061 


88244 


81940 


87590 


48 


18 


80166 


88865 


81076 


88234 


81955 


87679 


42 


19 


80182 


88855 


81091 


88228 


61969 


87568 


41 


20 


9.80197 


9.88844 


9.81106 


9.88212 


9.81968 


9.87657 


40 


21 


80218 


88884 


81121 


88201 


81996 


87646 


89 


22 


80228 


88844 


81186 


88191 


82012 


87585 


88 


28 


80944 


88818 


81151 


88180 


82088 


87524 


87 


24 


80259 


83803 


81166 


88169 


82041 


87618 


86 


25 


80274 


88798 


81180 


88158 


82055 


87501 


85 


28 


80290 


88782 


81195 


88148 


62069 


87490 


84 


27 


80305 


88772 


81210 


88187 


82064 


87479 


88 


28 


80890 


88761 


81225 


88126 


62098 


87468 


82 


29 


80886 


88751 


81940 


88115 


82112 


87457 


81 


80 


9.80851 


9.88741 


9.81264 


9.88105 


9.82126 


9.87446 


80 


81 


80866 


88780 


81269 


88094 


82141 


87434 


29 


82 


80883 


88720 


81284 


86088 


82165 


67428 


28 


88 


80897 


88709 


81299 


88072 


82169 


87412 


27 


84 


80412 


88699 


61314 


88061 


82184 


87401 


26 


86 


80428 


88668 


81328 


88051 


82196 


87890 


25 


86 


80448 


88678 


81343 


88040 


82212 


87878 


94 


87 


80458 


88668 


81858 


88029 


82226 


87867 


28 


88 


80478 


88657 


81372 


88018 


82240 


87866 


29 


89 


80489 


88647 


81887 


88007 


82955 


87845 


91 


40 


9.80504 


9.88686 


9.81409 


9.87996 


9.82269 


9.87884 


90 


41 


80519 


88626 


81417 


87985 


82288 


87822 


19 


42 


80584 


88615 


81481 


87975 


62297 


87811 


18 


48 


80650 


88605 


81446 


87964 


82811 


87800 


17 


44 


80565 


88594 


81461 


87968 


82826 


87288 


16 


46 


80580 


88584 


81475 


87942 


82840 


87277 


15 


46 


80595 


8*578 


81490 


87931 


82854 


87266 


14 


47 


80610 


88668 


81505 


87920 


82868 


87266 


18 


48 


80625 


88552 


81519 


87909 


82889 


87948 


12 


49 


80641 


88542 


81684 


87898 


82896 


87282 


11 


60 


9.80656 


9.88531 


9.81549 


9.87887 


9.82410 


9.87221 


10 


61 


80671 


88521 


81663 


87877 


82424 


87209 


9 


62 


80686 


88510 


81578 


87866 


82489 


87198 


8 


68 


80701 


88499 


81592 


87866 


82458 


87187 


7 


64 


80716 


88489 


81607 


87844 


82467 


87175 


6 


65 


80731 


88478 


81622 


87888 


82481 


87164 


5 


66 


80746 


88468 


81686 


87823 


82495 


87168 


4 
8 


57 


80762 


88457 


81651 


67811 


82609 


87141 


68 


80777 


88447 


81665 


87800 


82628 


87180 


8 


69 


80792 


88486 


81680 


87789 


82687 


87119 


1 


60 


80807 


88425 


81694 


87778 


82551 


87107 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




W 




49* 




48* I 



4^9 
TABLE II.— LOGARITHMIC SINES AND COSINES. 



# 


49* 


4f 


44» 


* 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





9.88551 


9.87107 


9.88878 


9.86418 


9.84177 


9.85693 


60 


1 


88565 


87096 


88899 


86401 


84190 


85681 


59 


8 


82579 


87085 


88405 


86889 


84208 


85669 


58 


8 


68598 


87078 


88419' 


66877 


84216 


85657 


57 


4 


88607 


87068 


83488 


86866 


84889 


86645 


56 


5 


89681 


87050 


83446 


86854 


84848 


85682 


55 


6 


89685 


87039 


88469 


86848 


64855 


85680 


54 


7 


88649 


87028 


88478 


86880 


84869 


85608 


58 


8 


89668 


87016 


88486 


66818 


84289 


85596 


52 


9 


88677 


87005 


88500 


86806 


84895 


85583 


51 


10 


9.88891 


9.86998 


9.88518 


9.86295 


9.84806 


9.86671 


60 


11 


82705 


86969 


88587 


86888 


84881 


85559 


49 


18 


88719 


86970 


88540 


86871 


84884 


85547 


48 


18 


88788 


86959 


88554 


86859 


84847 


65584 


47 


14 


88747 


86947 


88507 


86847 


84860 


85522 


46 


15 


89761 


86936 


88561. 


86885 


84878 


85510 


45 


16 


88775 


86984 


83594 


86828 


84885 


85497 


44 


17 


89788 


86918 


88608 


86211 


84898 


85485 


48 


18 


83808 


80908 


83681 


86800 


84411 


86478 


48 


19 


89816 


86890 


68684 


66188 


84484 


85460 


41 


90 


9.69880 


9.86879 


9.88648 


9.86176 
86164 


9.64487 


9.85448 


40 


81 


82844 


86867 


88661 


84450 


85436 


89 


29 


88858 


86855 


88674 


86158 


84463 


85483 


88 


88 


82878 


86844 


83688 


86140 


84476 


85411 


87 


94 


89686 


86888 


88701 


86128 


84489 


85899 


86 


85 


89899 


86821 


88715 


86116 


84509 


85886 


85 


86 


89918 


86809 


88788 


66104 


84515 


86874 


84 


87 


82987 


86796 


88741 


66088 


84588 


85861 


83 


88 


89941 


86786 


88755 


66080 


84540 


86849 


88 


89 


89955 


86775 


88768 


66068 


84558 


85887 


31 


80 


9.89968 


9.86763 


9.88781 


9.86056 


9.84566 


9.85884 


30 


81 


82969 


86759 


88795 


86044 


84579 


86319 


89 


88 


89996 


86740 


88808 


86088 


84598 


85299 


88 


88 


88010 


86728 


88821 


86080 


84605 


65267 


87 


84 


88028 


86717 


88884 


66008 


84618 


85274 


86 


85 


88037 


86705 


88848 


85996 


84680 


85862 


85 


86 


89051 


86994 


68861 


ami*** 
ODMH 


84648 


86950 


84 


87 


88065 


86689 


88874 


85979 


84656 


85887 


88 


88 


88078 


86670 


83887 


85960 


84669 


85225 


88 


89 


88098 


86659 


88901 


85048 


84688 


85219 


81 


40 


9.88106 


9.86647 


9.83914 


9.85986 


9.84694 


9.85800 


80 


41 


88190 


86035 


88927 


85924 


84707 


85187 


19 


48 


88188 


86624 


83940 


85912 


84780 


85175 


18 


48 


88147 


86618 


88954 


65900 


84788 


85168 


17 


44 


88161 


86600 


88967 


85888 


64745 


85150 


16 


45 


88174 


86589 


68960 


85876 


84758 


85187 


16 


46 


88188 


86577 


83998 


85864 


84771 


85125 


14 


47 


88908 


86565 


84006 


85851 


84784 


85112 


18 


48 


88215 


86554 


84020 


65839 


84796 


85100 


18 


49 


88829 


8654) 


84088 


85827 


84809 


85087 


11 


50 


9.88848 


9.86530 


9.64046 


9.85815 


9.84822 


9.85074 


10 


51 


88956 


86618 


84059 


85803 


84835 


85069 


9 


58 


88270 


86507 


84078 


85791 


84847 


85049 


8 


58 


88263 


86495 


84065 


85779 


64860 


86087 


7 


54 


88997 


86483 


64096 


85766 


84878 


85024 


6 


55 


88310 


86472 


84119 


85754 


84885 


86012 


5 


66 


83894 


86460 


84185 


85742 


84896 


84999 


4 


57 


83338 


86448 


84188 


85730 


84911 


84986 


8 


58 


83351 


86436 


64151 


85718 


84923 


84974 


8 


59 


88865 


86425 


84164 


85706 


84936 


84961 


1 


60 


83378 


86413 


84177 


85698 


84949 


84949 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




47* 




46* 




45* 



47° 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



1 




<>• 




!• 


2 


• 


/ 


Tan 


Cotan 


Tan 


Ootan 


Tan 


Cotan 


o 


— 00 


00 


8.24192 


11.75808 


8.54806 


11.45692 


60 


1 


6.46878 


18.58627 


24910 


75090 


54669 


45881 


59 


2 


76476 


28524 


95616 


74884 


55027 


44973 


58 


8 


94065 


05915 


96812 


78688 


55882 


44618 


57 


4 


7.06579 


12.93421 


26996 


78004 


55784 


44266 


56 


5 


16270 


88780 


27669 


7288W 


56068 


48917 


55 


6 


24188 


75812 


28332 


71668 


56429 


48571 


54 


7 


80682 


69118 


28986 


71014 


66778 


43227 


63 


8 


86682 


68818 


29629 


70871 


67114 


42886 


62 


9 


41797 


58403 


80268 


69737 


57462 


42548 


61 


10 


7.46878 


12.58627 


8.80888 


11.69112 


8.57788 


11.42212 


50 


11 


50512 


49488 


81505 


68495 


58121 


41879 


49 


18 


54291 


45709 


82112 


67888 


58451 


41549 


48 


18 


57767 


42238 


82711 


67289 


58779 


41221 


47 


14 


60986 


89014 


83302 


66698 


59105 


40695 


46 


15 


68982 


86018 


88886 


66114 


59428 


40572 


45 


16 


66785 


88215 


84461 


65539 


59749 


40251 


44 


17 


69418 


80582 


85029 


64971 


60068 


89982 


43 


18 


71900 


28100 


85590 


64410 


60884 


89616 


42 


19 


74248 


25752 


86148 


63857 


60896 


89802 


41 


90 


7.76476 


12.28524 


8.36689 


11.68811 


8.61009 


11.88991 


40. 


81 


78595 


21405 


87229 


62771 


61819 


88681 


89 


88 


80615 


19385 


87762 


62-288 


61626 


88874 


88 


88 


82546 


17454 


88289 


61711 


61981 


88069 


87 


84 


84394 


15606 


88809 


61191 


62284 


87766 


86 


25 


86167 


13833 


89323 


60677 


62535 


87466 


86 


86 


87871 


12129 


89882 


60168 


62834 


87166 


84 


27 


80510 


10490 


40884 


59666 


63181 


86869 


83 


88 


91089 


08911 


40880 


59170 


63426 


86574 


82 


80 


92618 


07387 


41821 


58679 


68718 


86282 


31 


80 


7.94086 


12.05914 


8.41807 


11.58198 


8.64009 


11.35991 


80 


81 


95510 


04490 


42287 


57718 


64298 


85702 


29 


38 


96889 


03111 


42762 


57288 


64585 


85415 


28 


83 


98225 


01775 


48232 


56768 


64870 


85180 


27 


84 


99522 


00478 


48696 


56304 


65154 


84846 


26 


85 


8.00781 


11.99219 


44156 


55844 


65485 


84565 


25 


86 


02004 


97996 


44611 


55389 


65715 


84285 


24 


8? 


03194 


96806 


45061 


54939 


65998 


84007 


23 


88 


04353 


95647 


45507 


54498 


66269 


83781 


22 


80 


05481 


94519 


45948 


54052 


66548 


88457 


21 


40 


6.06581 


11.93419 


8.46886 


11.58615 


8.66816 


11.88184 


90 


41 


07658 


fc!347 


46817 


53183 


67087 


82918 


19 


42 


06700 


91300 


47245 


52755 


67856 


82644 


18 


48 


09722 


90278 


47669 


52381 


67624 


82376 


17 


44 


10720 


89280 


48089 


51911 


67890 


82110 


16 


45 


11696 


88304 


48505 


51495 


68154 


81846 


15 


46 


12651 


87349 


48917 


61C88 


68417 


81563 


14 


47 


13585 


86415 


49335 


50675 


68678 


81822 


18 


48 


14500 


85500 


49729 


50271 


68938 


81062 


12 


40 


15895 


84605 


50180 


49870 


49196 


80804 


11 


50 


8.16273 


11.88727 


8.50527 


11.49478 


8.69458 


11.80547 


10 


51 


17133 


82867 


50920 


49080 


69708 


80292 


9 


52 


17976 


82024 


51310 


48690 


69968 


80088 


8 


58 


18804 


81196 


51696 


48804 


70214 


29786 


7 


54 


19616 


80384 


52079 


47921 


70465 


29585 


6 


55 


20413 


79.-W7 


52459 


47541 


70714 


29286 


5 


56 


21195 


78805 


52635 


47165 


70962 


29088 


4 


57 


21964 


78086 


53208 


46792 


71208 


28792 


8 


58 


22720 


77280 


53578 


46422 


71458 


2F547 


9 


50 


28462 


76588 


53945 


46055 


71697 


28808 


1 


60 


24192 


75808 


54308 


45692 


71940 


28060 








Ootan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




89° 




88° 




87* 



47i 



TABLE ni.— LOG. TANGENTS AND COTANGENTS. 



/ 




I- 




!• 




6° 


/ 
60 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





8.71910 


11.98060 


8.84464 


11.15586 


8.94195 


11.05806 


1 


78181 


97819 


64646 


15854 


94340 


05660 


69 


8 


72420 


37580 


84836 


15174 


94485 


05515 


58 


8 


72659 


27841 


85006 


14994 


94680 


05870 


67 


4 


78896 


97104 


85186 


14815 


94778 


06997 


66 


ft 


73133 


96668 


85863 


14637 


94917 


05088 


55 


6 


73366 


96634 


85540 


14460 


95080 


04940 


64 


7 


73600 


96400 


85717 


14388 


95909 


04798 


63 


8 


78S38 


26164 


85698 


14107 


95344 


04656 


68 


9 


74068 


95937 


86069 


13931 


95486 


04614 


51 


10 


8.74898 


11.95708 


8.86948 


11.18757 


8.95637 


11.04378 


60 


11 


74521 


95479 


86417 


13583 


95767 


04288 


49 


19 


74748 


95959 


86591 


18409 


95908 


04098 


48 


18 


74974 


95096 


86763 


1*337 


96C47 


08958 


47 


14 


75199 


94801 


66935 


13085 


96187 


08818 


46 


15 


75198 


94577 


87106 


12894 


96335 


08675 


45 


18 


75645 


84355 


87377 


19738 


96464 


03536 


44 


17 


75867 


241&1 


87447 


13553 


96C08 


03898 


48 


18 


76067 


23913 


87616 


19384 


96789 


08961 


43 


10 


76806 


98694 


87785 


19915 


96877 


08138 


41 


80 


876585 


11.93475 


8.87968 


11.19047 


8.97013 


11.09987 


40 


81 


76748 


98958 


88130 


11880 


97150 


09650 


89 


88 


76958 


83043 


68387 


11713 


97985 


09715 


38 


88 


77178 


2*8*7 


68153 


11547 


97431 


09579 


87 


84 


77887 


99613 


88618 


11383 


97556 


03444 


36 


85 


77600 


99400 


88783 


11317 


97691 


03309 


85 


86 


77811 


99189 


8S948 


11053 


9T835 


09175 


84 


87 


78088 


81978 


89111 


10889 


97959 


09041 


33 


88 


78238 


31768 


89374 


10736 


98099 


01908 


82 


89 


78441 


81559 


89437 


10563 


96335 


01775 


81 


80 


8.78649 


11.91351 


8.89598 


11.10403 


8.96358 


11.01649 


80 


81 


78855 


91145 


89760 


10340 


98490 


01510 


99 


88 


79061 


90989 


89990 


10060 


96C39 


01378 


98 


83 


79866 


90734 


90080 


09990 


96753 


01247 


87 


84 


79470 


90530 


90340 


09760 


96884 


01116 


36 


85 


79678 


80337 


90399 


09601 


99015 


00965 


95 


86 


79875 


90195 


90557 


09448 


99145 


00655 


84 


87 


80076 


19934 


90715 


00285 


99275 


00735 


88 


88 


80877 


19738 


90879 


09198 


99406 


00595 


98 


89 


80176 


19534 


91029 


06971 


99534 


00466 


91 


40 


8.80674 


11.19396 


8.91185 


11.08815 


8.99669 


11.00888 


80 


41 


80879. 


19198 


91440 


08660 


99791 


00309 


19 


48 


81068 


18933 


91495 


08505 


99919 


00081 


18 


48 


81864 


18736 


91650 


08350 


9.00046 


10.99964 


17 


44 


81459 


18541 


91803 


08197 


00174 


99636 


16 


45 


81658 


18347 


91957 


08043 


00301 




15 


46 


61846 


18154 


93110 


07890 


00497 


99573 


14 


47 


89088 


17969 


93363 


07788 


00558 


99447 


18 


48 


89930 


17770 


93414 


07586 


00679 


99831 


19 


49 


88480 


17580 


93565 


07485 


00805 


99195 


11 


60 


8.89610 


11.17390 


8.93716 


11.07984 


9.00930 


10.99070 


10 


61 


89799 


17901 


93666 


07134 


01055 


98045 


9 


69 


89987 


17013 


93016 


06984 


01179 


96891 


8 


68 


83175 


16895 


93165 


06835 


01808 


98697 


7 


64 


83861 


16689 


98313 


06t*j87 


01437 


98573 


6 


65 


63547 


16458 


93468 


06538 


01550 


98450 


6 


66 


88738 


16968 


93609 


06391 


01678 


96337 


4 


57 


83916 


16084 


93756 


06344 


01796 


98304 


8 


68 


81100 


15900 


93903 


06097 


01918 


96089 


8 


69 


84989 


15718 


94049 


05951 


03040 


97960 


1 


60 


84464 


15536 


94195 


05805 


03163 


97R88 





$ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


# 




86- 




86» 




84* 



472 



TABLE III.— LOG. TANGENTS AND COTANGENTS 



/ 


6 


• 


T 




8- 


i 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





0.02162 


10.97888 


9.08014 


10.91066 


9.14780 


10.85220 


60 


1 


02288 


97717 


09019 


90961 


14872 


85128 


50 


s 


02404 


97596 


09128 


90877 


14968 


86087 


58 


8 


02525 


97475 


09227 


90773 


15054 


84046 


57 


4 


09645 


97355 


09830 


• 90670 


15145 


84855 


56 


5 


02766 


97284 


09484 


90566 


15286 


84784 


55 


6 


02886 


97115 


09537 


90463 


15327 


84678 


64 


7 


08005 


96995 


09640 


90360 


15417 


84588 


58 


8 


03124 


96876 


09742 


90258 


15506 


84488 


62 


9 


03242 


96758 


09645 


90155 


15598 


84402 


51 


10 


0.03861 


10.96689 


9.09947 


10.90058 


9.15688 


10.84312 


60 


11 


08470 


96521 


10049 


89951 


15777 


84228 


40 


13 


08597 


96403 


10150 


89850 


15867 


64138 


48 


18 


03714 


96286 


10252 


89748 


15956 


84044 


47 


14 


08882 


96168 


10858 


89647 


16046 


83954 


46 


15 


08048 


96052 


10454 


89546 


16135 


88865 


45 


18 


04065 


95985 


10555 


89445 


16224 


83778 


44 


1? 


. 04181 


95819 


10656 


89844 


16812 


88688 


48 


18 


04297 


95708 


10756 


89244 


16401 


88590 


42 


10 


04418 


95587 


10656 


89144 


16480 


88511 


41 


90 


9.04528 


10.95472 


0.10956 


10.89044 


0.16577 


10.68428 


40 


21 


04643 


95357 


11056 


88044 


16665 


88885 


SO 


28 


04758 


95242 


11155 


88845 


16758 


63247 


38 


23 


04878 


95127 


11254 


88746 


16641 


88150 


87 


24 


04987 


05018 


11858 


88647 


16028 


68072 


86 


25 


05101 


04899 


11452 


68548 


17016 


82084 


36 


26 


05214 


94786 


11551 


88449 


17108 


62807 


84 


27 


05328 


94672 


11649 


68851 


17190 


82810 


88 


28 


05441 


94559 


11747 


88258 


17277 


82728 


82 


20 


05553 


94447 


11845 


88155 


17863 


82687 


31 


80 


9.05666 


10.94384 


9.11948 


10.88057 


0.17450 


10.82550 


30 


81 


05778 


94222 


12040 


87960 


17586 


82464 


20 


82 


05890 


94110 


12188 


87862 


17622 


62878 


28 


83 


06002 


93998 


12285 


87765 


17708 


82202 


27 


84 


06118 


98887 


12332 


87668 


17704 


82206 


26 


85 


06224 


93776 


12428 


87572 


17880 


82120 


25 


86 


06385 


S8665 


12525 


87475 


17865 


82085 


24 


87 


06445 


93555 


12621 


87879 


18051 


81040 


28 


88 


06556 


93444 


12717 


87283 


18186 


81864 


22 


80 


06666 


98334 


12813 


87187 


18221 


81770 


21 


40 


9.06775 


10.93225 


9.12909 


10.87091 


0.18306 


10.81604 


20 


41 


06885 


93115 


18004 


86996 


18891 


81600 


10 


42 


06994 


98006 


18099 


80901 


18475 


81525 


18 


43 


07103 


92897 


18194 


86806 


18560 


81440 


17 


44 


07211 


92789 


13289 


66711 


18644 


81856 


16 


45 


07820 


92680 


13884 


8C616 


18728 


81272 


15 


46 


07428 


92572 


13478 


86522 


18812 


81188 


14 


47 


07536 


92464 


18578 


86427 


18886 


81104 


18 


48 


07648 


92857 


13667 


86338 


18079 


81021 


18 


49 


07751 


92249 


13781 


86289 


10068 


80087 


11 


50 


9.07858 


10.92142 


9.13854 


10.86146 


0.10146 


10.80854 


10 


61 


07964 


92086 


13948 


86052 


10220 


80771 





52 


08071 


91929 


14041 


86959 


10312 


80668 


8 


53 


08177 


91828 


14134 


65866 


10805 


80605 


7 


54 


06288 


91717 


14227 


85778 


19478 


80529 


6 


55 


06389 


91611 


14320 


85880 


19661 


80480 


5 


56 


08495 


91505 


14412 


85588 


19648 


80857 


4 


67 


06600 


91400 


14504 


85496 


19725 


80275 


8 


58 


08705 


91295 


14597 


85403 


19607 


80103 


2 


60 


08810 


91190 


14688 


85312 


19880 


80111 


1 


60 


08914 


91086 


14780 


85220 


10071 


80089 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


t 




88* 




8£ # 




%!• 



473 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



$ 




»• 


10° 


11° 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.19971 


10.80039 


9.34639 


10.75868 


9.39805 


10.71135 


60 


1 


80008 


79947 


84700 


75394 


38088 


71C67 


69 


* 


80184 


79868 


81779 


75931 


39000 


71000 


58 


8 


80-310 


70784 


84858 


75147 


30067 


70988 


67 


4 


80897 


79708 


84938 


75074 


89184 


70866 


50 


5 


80378 


79633 


85000 


75000 


89301 


70799 


55 


ft 


80459 


79641 


35073 


74937 


29868 


70788 


54 


7 


90510 


79400 


35140 


74854 


39335 


70665 


,58 


8 


80031 


79379 


35319 


74781 


20409 


70598 


68 





80701 


79399 


85393 


74708 


39468 


70582 


51 


10 


9.80788 


10.79318 


9.35865 


10.74035 


9.89585 


10.70405 


60 


n 


80608 


79188 


85187 


74508 


89601 


70899 


49 


13 


80948 


79058 


35510 


74490 


30668 


70888 


48 


18 


91088 


78978 


85588 


74418 


89784 


70366 


47 


14 


91108 


78896 


35055 


74845 


89800 


70300 


48 


1ft 


81183 


78818 


35737 


74278 


29800 


70184 


45 


1ft 


81881 


78789 


35799 


74301 


39988 


70068 


44 


1? 


31341 


78059 


85871 


74139 


89998 


70008 


48 


18 


81480 


78580 


35948 


74057 


30064 


69936 


48 


10 


31499 


78501 


80015 


73985 


30180 


69670 


41 


80 


9.91078 


10.78483 


9.80060 


10.78014 


9.80195 


10.60805 


40 


81 


91057 


78348 


90158 


73843 


80301 


60788 


89 


88 


81780 


78304 


36239 


78771 


80830 


00674 


88 


88 


91814 


78186 


86801 


73099 


80391 


60600 


87 


84 


81898 


78107 


36873 


73G38 


80457 


69548 


36 


8ft 


91971 


78039 


90443 


73557 


80533 


69478 


85 


88 


89049 


77951 


90514 


73480 


80587 


69418 


84 


87 


88187 


77878 


80585 


78415 


80659 


09348 


38 


88 


88905 


77795 


36655 


73845 


80717 


09388 


83 


89 


88888 


77717 


86736 


73374 


80783 


89818 


31 


80 


9.99801 


10.77689 


9.36797 


10.78308 


9.80846 


10.69154 


80 


81 


88488 


77568 


86S07 


73183 


80911 


69089 


29 


88 


88518 


77484 


30907 


78008 


80075 


69035 


38 


88 


88590 


77407 


37008 


73993 


81040 


68960 


37 


84 


83870 


77880 


87078 


78933 


81104 


08896 


30 


85 


88747 


77358 


37148 


73853 


81168 


08833 


25 


88 


83884 


77176 


87318 


78788 


81388 


08767 


34 


87 


88901 


77099 


37388 


73713 


81397 


68708 


38 


88 


88977 


77038 


37357 


73648 


31361 


08689 


38 


89 


93054 


76946 


37437 


73578 


81435 


68575 


31 


40 


9.88180 


10.76870 


9.87490 


10.73504 


9.31489 


10.68511 


20 


41 


88)00 


70794 


87560 


73484 


31553 


68448 


19 


48 


88483 


76717 


37085 


73365 


81616 


68384 


18 


48 


88859 


76041 


87704 


73396 


81679 


68381 


17 


44 


83485 


70505 


37778 


79887 


81748 


68357 


19 


45 


88510 


70490 


37843 


73158 


81806 


68194 


\l 


40 


88580 


70414 


87911 


73089 


31870 


68180 


14 


47 


88081 


70839 


87980 


73090 


81933 


08067 


18 


48 


88787 


70368 


88049 


71951 


31990 


68004 


12 


49 


88812 


76188 


88117 


71888 


83059 


67941 


11 


50 


9.83887. 


10.76113 


9.38188 


10.71*14 


9.88138 


10.07878 


10 


51 


83902 


76088 


38254 


71740 


83185 


67815 


9 


08 


84037 


75963 


88338 


71077 


83348 


67753 


8 


58 


84113 


75888 


88891 


71609 


83811 


67689 


7 


54 


81188 


75814 


38459 


71541 


82878 


67637 





05 


84301 


75739 


88637 


71473 


83430 


07564 


5 


50 


84335 


75665 


88595 


71405 


83498 


07503 


4 


57 


84410 


75590 


88063 


71838 


88561 


67489 


8 


58 


84484 


75510 


88780 


71370 


82638 


67877 


3 


59 


84558 


75443 


38796 


71303 


82685 


67315 


1 


00 


84088 


75368 


88865 


71135 


38747 


67353 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 






80* 




79° 




78° 



474 



TABLE IIL— LOG. 


TANGENTS AND COTANGENTS. 


t 


18* 


18° 


14* 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.82747 


10.67363 


9.86836 


10.68664 


9.89677 


10.60828 


60 


1 


82810 


67190 


86894 


68606 


89781 


00269 


59 


9 


82872 


67128 


86459 


68548 


89785 


60215 


68 


8 


88988 


67067 


86509 


63491 


88838 


60162 


57 


4 


89996 


67005 


86566 


68484 


89892 


60108 


66 


5 


89067 


66943 


86824 


68376 


89945 


60055 


55 


6 


38119 


66881 


86681 


63819 


89999 


60001 


54 


7 


83180 


66820 


86788 


68263 


40059 


69948 


58 


8 


88-348 


66758 


86796 


63205 


40106 


ftOOOJ 

owovt 


69 





83808 


66697 


86852 


68148 


40159 


59641 


61 


10 


9.88866 


10.66035 


9.88909 


10.63091 


9.40212 


10.59788 


50 


11 


83486 


66574 


oOVOO 


63034 


40266 


69784 


49 


18 


88487 


66618 


87088 


62977 


40319 


59681 


48 


18 


88648 


66458 


87080 


62920 


40872 


59628 


47 


14 


88609 


66891 


87187 


62808 


40425 


69575 


46 


1ft 


88670 


66830 


87193 


62807 


40478 


59522 


46 


i6 


88781 


66269 


87250 


62750 


40581 


OVtOV 


44 


17 


88798 


66208 


87806 


68694 


40584 


59416 


48 


18 


83868 


66147 


87863 


62637 


40686 


69364 


42 


10 


88918 


66087 


87419 


62681 


40689 


69311 


41 


90 


9.83974 


10.66026 


9.87478 


10.62524 


9.40742 


10.59258 


40 


8! 


84034 


65966 


87682 


62468 


40795 


69806 


89 


83 


84006 


65905 


87588 


62419 


40647 


69158 


88 


88 


84165 


65845 


87644 


62856 


40900 


69100 


87 


%J 


84215 


65785 


87700 


62800 


40952 


69048 


86 


84276* 


65784 


87766 


62244 


41006 


68995 


85 


96 


84838 


AKJUL4 
00004 


87812 


62188 


41067 


68943 


84 


87 


84896 


65604 


87868 


62188 


41109 


68891 


88 


88 


84466 


65544 


87924 


62076 


41161 


58889 


88 


89 


34516 


65484 


87980 


62090 


41814 


68786 


81 


80 


9.34576 


10.65424 


9.88085 


10.61965 


9.41266 


10.58784 


80 


81 


84685 


65366 


88091 


61909 


41818 


68682 


29 


38 


84695 


65806 


88147 


61858 


41870 


68680 


28 


88 


84756 


65245 


38208 


61798 


41422 


68578 


27 


84 


84814 


65186 


88257 


61743 


41474 


68526 


26 


85 


848T4 


65126 


86818 


61687 


41826 


68474 


86 


86 


84933 


65067 


88368 


61638 


41578 


68422 


84 


87 


84999 


65008 


88428 


61577 


41629 


68371 


28 


88 


85051 


64949 


88479 


61521 


41681 


68319 


28 


89 


86111 


64889 


88534 


61466 


41788 


58267 


21 


40 


9.85170 


10.64880 


9.88589 


10.61411 


9.41784 


10.68216 


80 


41 


85229 


64771 


88644 


61856 


41836 


66164 


19 


48 


85888 


64712 


88699 


61801 


41887 


68118 


18 


48 


85847 


64658 


88754 


61246 


41939 


68061 


17 


£ 


85405 


64595 


88808 


61192 


41990 


68010 


16 


85464 


64536 


88868 


61187 


42041 


67959 


16 


46 


85588 


64477 


88918 


61082 


42098 


57907 


14 


47 


35581 


64419 


8S972 


61028 


42144 


67856 


18 


48 


85640 


64860 


89037 


60973 


42195 


57805 


18 


49 


85698 


64802 


89082 


60918 


42248 


67754 


11 


60 


9.85757 


10.64248 


9.89186 


10.60864 


9.42297 


10.67708 


10 


61 


85815 


64185 


89190 


60810 


42848 


67652 


9 


68 


85878 


64127 


89245 


60755 


42899 


57601 


8 


68 


85931 


64069 


89299 


60701 


42460 


67550 


7 


64 


85989 


64011 


89353 


60647 


42501 


57499 


6 


66 


86047 


63958 


80407 


60593 


42652 


57448 


6 


66 


86106 


68895 


89461 


60589 


42608 


67397 


4 


67 


86168 


63887 


89615 


60485 


42658 


67347 


8 


68 


86281 


63779 


89569 


60481 


42704 


57296 


8 


69 


36279 


68721 


89623 


60877 


42755 


57245 


1 


00 


86836 


63664 


89677 


60328 


42806 


57195 





» 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


• 




77 . 




76« 




76- 



475 
TABLE m.— LOG. TANGENTS AND COTANGENTS. 



* 


16* 


16* 




17- 





Tan 


Co tan 


Tan 


Cotan 


Tan 


Cotan 





8.4-2805 


10.57185 


0.45750 


10.54250 


0.48684 


10.61466 


60 


1 


42856 


57144 


45707 


64208 


48570 


61421 


69 


2 


42006 


57004 


45845 


54155 


48644 


61876 


68 


8 


42057 


57048 


45882 


64108 


48660 


5188! 


57 


4 


48007 


56008 


46840 


64060 


48714 


61286 


66 


5 


48057 


56048 


45887 


54013 


48750 


51241 


55 


6 


48108 


56882 


46086 


58865 


48804 


51186 


54 


7 


48158 


56842 


46082 


53818 


48840 


61151 


53 


8 


48^)8 


56782 


46180 


58870 


48804 


51106 


52 





48258 


56742 


46177 


58828 


46888 


61061 


51 


10 


0.48808 


10.66602 


0.46224 


10.58776 


9.48884 


10.51016 


50 


11 


48858 


66642 


46271 


58728 


48028 


60971 


49 


18 


48406 


66502 


46810 


68681 


48078 


60927 


48 


18 


48458 


66542 


46866 


53684 


48118 


60882 


47 


14 


48508 


56482 


46418 


68567 


48168 


60887 


46 


15 


48558 


66442 


46460 


58540 


40207 


50798 


45 


16 


48607 


66808 


46507 


58408 


49252 


60748 


44 


17 


48657 


56848 


46554 


58446 


49*86 


60704 


48 


18 


48707 


56208 


46601 


68898 


48341 


60659 


42 


10 


48756 


56244 


46648 


58852 


48885 


60615 


41 


80 


0.43806 


10.56104 


0.46684 


10.58306 


8.48480 


10.60670 


40 


21 


48856 


66145 


46741 


68258 


48474 


50626 


89 


22 


48005 


56085 


46788 


68212 


48519 


50481 


88 


28 


48954 


66046 


46885 


63165 


48568 


60437 


87 


24 


44004 


65886 


46881 


68118 


48607 


60898 


86 


26 


44058 


65847 


46828 


68072 


49662 


50348 


86 


26 


44102 


65808 . 


46075 


68025 


48686 


50804 


84 


27 


44151 


65848 


47021 


52079 


49740 


50260 


88 


28 


44201 


65788 


47068 


62082 


49784 


* 60216 


82 


29 


44260 


65750 


47114 


62686 


48828 


60172 


81 


80 


0.44200 


10.65701 


9.47160 


10.52840 


8.48872 


10.50128 


80 


81 


44348 


56652 


47207 


52798 


48816 


60084 


29 


82 


44807 


65608 


47268 


62747 


49860 


50040 


26 


88 


44446 


65554 


47200 


62701 


60004 


48886 


27 


84 


44405 


55505 


47846 


52654 


50048 


48852 


26 


86 


44544 


55466 


47882 


53608 


50092 


48808 


25 


86 


44502 


65408 


47488 


62562 


60186 


48864 


24 


87 


44641 


65358 


47484 


62616 


60180 


48820 


28 


88 


44600 


55810 


47680 


62470 


50228 


48777 


22 


80 


44788 


65262 


47576 


62424 


60267 


49738 


21 


40 


0.44787 


10.66218 


8.47622 


10.62878 


9.50811 


10.48668 


20 


41 


44886 


65164 


47668 


62882 


60855 


48645 


19 


42 


44884 


66116 


47714 


62286 


50888 


48602 


18 


48 


44038 


55067 


47760 


62240 


50442 


48558 


17 


44 


44881 


65018 


47806 


62184 


60485 


49515 


16 


46 


45020 


54871 


47852 


62148 


50528 


48471 


15 


46 


45078 


54822 


47807 


62108 


50572 


48428 


14 


47 


45126 


64874 


47848 


62057 


50616 


48884 


18 


48 


45174 


64826 


478S8 


62011 


50658 


48341 


12 


40 


45222 


54778 


48085 


51866 


50708 


48297 


11 


60 


0.45271 


10.64720 


8.48080 


10.51820 


9.50746 


10.48254 


10 


51 


45810 


64681 


48126 


61874 


60789 


48211 


9 


62 


45867 


54683 


48171 


51829 


50888 


48167 


8 


68 


45415 


64586 


48217 


51788 


50876 


48124 


7 


64 


45468 


54587 


48262 


51738 


50818 


48081 


6 


65 


45511 


54480 


48807 


51688 


50862 


48088 


6 


66 


45550 


54441 


48858 


51647 


51005 


48885 


4 


67 


45606 


64304 


48388 


61602 


51048 


48852 


8 


68 


46654 


54346 


48448 


51557 


51082 


46808 


* 


60 


45702 


64288 


48488 


51511 


51185 


48*65 


1 


60 


45750 


54250 


48584 


51466 


61178 


48822 


° 


/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


-! 




74- 




78° 




If 



470 



TABLE in.— LOG. TANGENTS AND COTANGENTS. 



» 


18° 


10° 




2©° 


t 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.61178 


10.48822 


9.53697 


10.46803 


9.56107 


10.48893 


60 


1 


51291 


48779 


58788 


46269 


56146 


48854 


69 


9 


51264 


48786 


58779 


46991 


56186 


48815 


58 


8 


51806 


48694 


538X0 


46180 


56294 


48776 


67 


4 


61849 


48651 


58861 


46189 


56264 


48786 


56 


5 


61899 


48608 


63909 


46098 


56808 


48697 


55 


6 


51435 


485C5 


68948 


46057 


56342 


48658 


64 


7 


51478 


48699 


53964 


46016 


56381 


48619 


68 


8 


51590 


48480 


54025 


45075 


56490 


48580 


58 





51663 


48487 


54065 


45985 


56459 


48541 


51 


10 


9.51606 


10.48894 


9.54106 


10.45894 


9.56496 


10.48508 


60 


11 


61G48 


48859 


54147 


46853 


56587 


48468 


49 


19 


51691 


48309 


54187 


45818 


56576 


48494 


48 


13 


61734 


48266 


54928 


45772 


56815 


48885 


47 


14 


51776 


48294 


54269 


45781 


56654 


43846 


40 


15 


61819 


48181 


54309 


45691 


66698 


48807 


45 


10 


61861 


48139 


64350 


45650 


56789 


48968 


44 


17 


51903 


48097 


64390 


45610 


66771 


459*9 


48 


18 


61946 


48064 


54431 


45569 


66810 


48190 


48 


19 


61968 


48012 


64471 


45599 


66849 


48151 


41 


90 


9.69031 


10.47969 


9.64519 


10.45488 


9.66887 


10.48118 


40 


91 


69073 


47927 


54559 


45448 


56996 


48074 


89 


99 


69115 


47885 


54598 


45407 


66965 


48085 


88 


98 


69157 


47848 


54688 


46867 


67004 


49996 


87 


94 


69900 


47800 


54678 


45897 


67049 


42956 


86 


9ft 


62242 


47768 


54714 


45286 


57061 


49919 


85 


98 


52284 


47716 


54754 


45246 


57190 


49860 


84 


87 


62326 


47674 


54794 


45206 


67168 


49648 


88 


98 


62368* 


47639 


54885 


45166 


B7197 


49808 


88 


99 


69410 


47690 


54875 


45196 


67986 


49785 


31 


80 


9.59452 


10.47548 


9.54915 


10.45086 


9.57274 


10.49728 


80 


81 


62494 


47506 


54955 


46045 


67312 


42688 


99 


89 


52536 


47464 


54995 


45006 


57851 


49649 


98 


88 


62578 


47499 


55085 


44965 


57380 


49611 


97 


84 


62620 


47880 


55075 


44926 


67498 


49578 


96 


8ft 


62661 


47389 


65115 


44886 


67466 


49584 


95 


86 


59708 


47C97 


55155 


44845 


67504 


49496 


94 


37 


62745 


47955 


55195 


44805 


67548 


49457 


88 


88 


52787 


47218 


55235 


44785 


57581 


49419 


98 


89 


62820 


47171 


55275 


44TO 


57819 


49881 


81 


40 


9.52870 


10.47180 


9.55815 


10.44665 


9.67658 


10.49848 


90 


41 


62912 


47068 


55335 


44645 


57696 


42804 


19 


49 


52053 


47047 


65896 


44605 


67784 


48966 


18 


48 


69995 


47006 


65434 


44566 


57778 


49298 


17 


44 


63087 


46968 


65474 


445*26 


57810 


48190 


16 


45 


68078 


4C922 


65514 


44486 


57849 


49151 


15 


46 


63120 


46880 


55554 


44446 


67887 


49118 


14 


47 


63161 


46839 


55593 


44407 


67925 


49075 


18 


48 


53202 


46798 


65633 


44867 


57968 


42087 


18 


49 


63244 


46756 


55678 


44327 


58001 


41999 


11 


60 


9.53285 


10.46715 


9.55712 


10.44288 


9.68089 


10.41961 


10 


61 


53827 


46678 


65759 


44248 


58077 


41968 


9 


69 


58368 


46682 


55701 


44209 


58115 


41886 


8 


58 


53409 


46591 


65831 


44169 


56158 


41847 


7 


64 


53450 


46550 


55870 


44180 


68191 


41809 


6 


65 


53492 


46508 


55010 


44090 


58229 


41771 


6 


66 


53533 


46467 


5*949 


44051 


58267 


41788 


4 


57 


53574 


46426 


65989 


44011 


66804 


41696 


8 


68 


53615 


46385 


56028 


48979 


58349 


41658 


8 


59 


53656 


46344 


56067 


48088 


68380 


41620 


1 


00 


53697 


46303 


56107 


43893 


58418 


41562 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


$ 




71* 




70» 




69* 



477 
TABLE HI.— LOG. TANGENTS AND COTANGENTS. 



1 


81° 


88* 


88° 


i 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 







9.58418 


10.41688 


9.60641 


10.39859 


9.68785 


10.87815 


60 


1 


68455 


41545 


60677 


89888 


68830 


87180 


69 


8 


58498 


41607 


60714 


89286 


68865 


87145 


68 


8 


56581 


41460 


60750 


89860 


68690 


87110 


57 


4 


58569 


41481 


60786 


88814 


68986 


87074 


56 


6 


68606 


41894 


60688 


89177 


68961 


87069 


66 


6 


58644 


41866 


60659 


89141 


68996 


87004 


54 


7 


68681 


41819 


60695 


89105 


68031 


86909 


68 


8 


68719 


41881 


60981 


89069 


68066 


86934 


68 





58767 


41848 


60967 


89088 


68101 


86899 


61 


10 


9.58794 


10.41806 


9.61004 


10.86996 


9.68135 


10.86865 


60 


11 


68888 


41168 


61040 


88960 


68170 


86880 


49 


18 


68869 


41181 


61076 


88994 


68805 


36795 


48 


18 


69907 


41098 


61118 


88888 


68840 


86760 


47 


14 


68944 


41066 


61148 


88858 


68875 


86785 


46 


15 


68981 


41019 


61184 


88816 


68810 


36690 


45 


18 


59019 


40961 


61880 


88790 


68845 


86655 


44 


17 


69068 


40044 


61856 


88744 


68879 


86681 


48 


18 


59094 


40906 


61899 


88708 


68414 


86586 


48 


10 


59181 


40669 


61888 


88678 


68449 


86551 


41 


80 


9.59168 


10.40883 


9.61804 


10.88686 


9.68484 


10.86516 


40 


81 


59805 


40795 


61400 


88600 


68519 


86481 


89 


88 


69848 


40757 


61486 


88064 


68658 


86447 


88 


88 


69280 


40780 


61478 


88588 


68588 


86418 


87 


84 


69817 


40688 


61506 


88498 


68683 


86377 


86 


85 


69854 


40646 


61544 


88466 


68857 


86848 


85 


88 


69891 


40609 


61579 


88481 


68698 


86808 


84 


87 


59489 


40571 


61615 


88365 


68786 


86274 


88 


88 


59466 


40534 


61651 


88349 


68761 


86889 


88 


89 


69608 


40497 


61667 


88818 


68706 


88804 


81 


80 


9.69640 


10.40460 


9.61788 


10.88878 


0.68880 


10.86170 


80 


81 


50577 


40488 


61758 


86848 


68865 


86180 


89 


88 


59614 


40886 


61794 


88806 


68899 


86101 


88 


88 


69651 


40849 


61880 


86170 


68934 


86066 


87 


84 


69688 


40818 


61865 


88185 


68968 


86068 


86 


85 


69785 


40875 


61901 


88099 


64008 


85997 


85 


86 


69768 


40888 


61986 


88064 


64087 


35968 


84 


87 


69799 
59S85 


40801 


61978 


88088 


64073 


85988 


88 


<*. 


40165 


68006 


87998 


64106 


86894 


88 


89 


698TO 


40186 


69048 


87957 


64140 


85860 


81 


40 


9.59909 


10.40091 


9.68079 


10.87931 


9.64175 


10.85825 


80 


41 


69946 


40054 


68114 


87886 


64809 


85791 




48 


69968 


40017 


68150 


87650 


64843 


85757 




48 


60019 


89961 


6*3185 


87815 


64876 


85788 




44 


60056 


89944 


68881 


87779 


64318 


86688 




45 


00098 


89907 


62256 


87744 


64846 


85654 




48 


60180 


89670 


68898 


87706 


64881 


86619 




47 


60166 


89684 


68387 


87678 


64415 


85585 




48 


60908 


89797 


68868 


87686 


61449 


86551 




49 


60840 


89760 


68898 


87608 


64488 


86517 




50 


9.60876 


10.89784 


9.68488 


10.87567 


9.64517 


10.85488 




51 


60818 


89687 


68468 


87588 


64558 


86448 


• g • 


58 


60849 


89651 


68504 


87496 


64686 


85414 




58 


60888 


89614 


68589 


87461 


64680 


86860 




54 


604*8 


89576 


68574 


87486 


64654 


86846 




55 


60459 


89541 


68609 


87891 


64688 


86312 




58 


60495 


89505 


68645 


87855 


64788 


85878 




57 


60588 


89468 


68660 


873*) 


64756 


86944 




58 


60568 


89488 


68715 


87885 


64790 


86810 




59 


60605 


89895 


68750 


87850 


64884 


86176 




00 


60641 


89359 


68780 


87815 


64858 


86148 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 


< 


W 




67° 


66* 



478 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



/ 


84* 


96* 


86* 


# 


TaD 


CotAII 


Tan 


Cotan 


Tnn 


Cotan 





9.64858 


10.3514? 


9.66867 


10.83138 


9.68618 


10.81189 


60 


1 


64898 


85108 


66900 


88100 


68850 


81150 


59 


8 


64986 


85074 


66938 


88067 


68689 


81118 


58 


8 


64960 


85040 


66966 


88084 


68914 


81086 


57 


4 


64994 


85006 


66999 


88001 


68946 


81054 


66 


5 


65038 


84978 


67038 


89968 


66978 


81092 


55 


6 


65068 


84938 


67065 


89935 


69010 


80990 


54 


7 


65096 


84904 


67098 


89909 


60049 


80958 


53 


8 


65180 


81870 


67131 


89869 


69074 


80996 


69 





65164 


84886 


67168 


89887 


69106 


80894 


61 


10 


9.65197 


10.84803 


9.67196 


10.89804 


9.69188 


10.80869 


60 


11 


65281 


34769 


67999 


89771 


60170 


80630 


49 


IS 


65865 


84785 


67969 


82738 


69909 


80796 


48 


18 


65999 


84701 


67995 


89705 


69934 


80766 


47 


14 


65883 


34667 


67897 


82673 


69966 


80734 


46 


15 


65366 


8^634 


67860 


89640 
82007 


69996 


80708 


45 


16 


65400 


34600 


67398 


69899 


80671 


44 


17 


65484 


34566 


67496 


89574 


69361 


80689 


a 


18 


65467 


84533 


67458 


88548 


69393 


80607 


49 


19 


65501 


84499 


67491 


89509 


694* 


80575 


41 


90 


9.65585 


10.84465 


9.67594 


10.89476 


9.69467 


10.80548 


40 


81 


65568 


84439 


67556 


89444 


69468 


80518 


89 


89 


65608 


84398 


67589 


89411 


69590 


80480 


88 


88 


65686 


84864 


67698 


89378 


69559 


80448 


87 


84 


65669 


84881 


67654 


88846 


69684 


80416 


86 


85 


65708 


84897 


67687 


89818 


69616 


80885 


86 


86 


65786 


84964 


67719 


88881 


69647 


80858 


84 


87 


65770 


84980 


67759 


89948 


69679 


80891 


88 


88 


66908 


84197 


67785 


89915 


69710 


80990 


88 


89 


65637 


84163 


67817 


88188 


69749 


80958 


81 


80 


9.65870 


10.34180 


9.67850 


10.89150 


9.69774 


10.80996 


80 


81 


65904 


84096 


67889 


82118 


69805 


80195 


99 


38 


65937 


84063 


67915 


89085 


69887 


80168 


88 


88 


65971 


84099 


67947 


88058 


69668 


80189 


97 


84 


66004 


83996 


67960 


89090 


69900 


80100 


96 


85 


66088 


88968 


68018 


81968 


69988 


80068 


85 


M 


66071 


83989 


68044 


81956 


69968 


80087 


84 


87 


66104 


83896 


68077 


81998 


69995 


80005 


93 


88 


66138 


88869 


68109 


31891 


70096 


99974 


98 


89 


66171 


838*9 


68148 


81858 


70058 


99949 


91 


40 


9.66904 


10.88796 


9.68174 


10.81896 


9.70089 


10.29911 


SO 


41 


66888 


88709 


68906 


81794 


70181 


99879 


19 


48 


66971 


83799 


68989 


81761 


70158 


99848 


18 


48 


66804 


88696 


68971 


31799 


70184 


99616 


17 


44 


66887 


88668 


66308 


81697 


70915 


99785 


16 


45 


66871 


83689 


68386 


31664 


70947 


99758 


15 


46 


66404 


83596 


68368 


81638 


70978 


99799 


14 


47 


66487 


38563 


68400 


81600 


70809 


99691 


IS 


48 


66470 


83530 


68439 


81568 


70341 


99659 


19 


49 


66503 


83497 


68465 


81585 


70378 


99698 


11 


50 


0.66637 


10.33463 


9.68497 


10.31503 


9.70404 


10.99596 


10 


* 51 


66570 


38430 


68589 


31471 


70485 


99565 


9 


58 


66603 


33397 


68561 


81489 


70466 


99584 


8 


58 


66636 


88864 


68593 


81407 


70498 


99509 


r 


54 


60689 


83831 


68696 


81374 


70599 


29471 


8 


55 


66709 


83998 


68658 


81348 


70560 


99440 


! 6 


56 


66785 


88265 


68690 


81810 


70599 


89408 


4 


57 


66768 


83989 


68799 


81978 


70698 


99877 


8 


. 58 


66801 


38199 


68754 


81946 


70654 


89346 


' S 


59 


66834 


88166 


68788 


81814 


70685 


99315 


' 1 


60 


66867 


88183 


6S8I8 


81189 


70717 


29888 





i 


C<> tan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


# 


■ 


66° 




64« 




68* 



479 



TABLE m.— LOG. TANGENTS AND COTANGENTS. 



$ 


97- 


28* 


28* 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.70717 


10.99988 


9.79567 


10.37488 


9.74375 


10.95695 


60 


1 


70748 


99859 


73596 


37408 


74405 


95596 


59 


8 


70779 


39291 


72698 


37379 


74485 


35565 


58 


8 


70810 


99190 


72659 


37341 


74465 


35585 


67 


4 


70641 


99159 


79689 


37311 • 


74494 


35506 


56 


5 


70878 


99137 


79720 


97280 


74594 


25476 


55 


6 


70904 


99096 


79750 


97350 


74554 


35446 


54 


7 


70985 


99065 


79780 


37220 


74588 


85417 


58 


8 


70966 


39034 


79811 


37189 


74618 


35387 


58 





70997 


99008 


72341 


87159 


74648 


35857 


61 


10 


9.71098 


10.98079 


9.79873 


10.37198 


9.74678 


10.85397 


50 


11 


71069 


28941 


73902 


37096 


74709 


95296 


49 


12 


71090 


98910 


79989 


37068 


74789 


35368 


48 


18 


71131 


38879 


72968 


37C87 


74769 


35338 


47 


14 


71158 


38847 


73998 


37007 


74791 


35309 


46 


15 


71184 


98816 


78098 


36977 


74891 


35179 


45 


16 


71915 


98785 


78054 


36046 


74851 


35149 


44 


17 


71946 


38754 


78084 


20016 


74880 


35190 


48 


18 


71977 


88798 


73114 


36886 


74910 


85090 


43 


19 


71808 


28699 


78144 


36856 


74939 


35061 


41 


90 


9.71389 


10.98661 


9.78175 


10.36825 


9.74969 


10.35061 


40 


91 


71870 


88680 


78.05 


36795 


74996 


95003 


89 


99 


71401 


28599 


78285 


36765 


75098 


34979 


88 


98 


71481 


38569 


78965 


36735 


75058 


34943 


87 


. 94 


71469 


38588 


78905 


90705 


75087 


84918 


36 


95 


71498 


38507 


73326 


86674 


75117 


34888 


86 


96 


71524 


88476 


78836 


36644 


75146 


34854 


84 


97 


71555 


38445 


78886 


36614 


75176 


84824 


88 


98 


71586 


98414 


78416 


26584 


75905 


34795 


89 


99 


71617 


98888 


78446 


86554 


75885 


24765 


81 


80 


9.71648 


10.98858 


9.78476 


10.36524 


9.75964 


10.34786 


80 


81 


71679 


98891 


78507 


96498 


75994 


84706 


89 


89 


71709 


98991 


78587 


36468 


75898 


34677 


28 


88 


" 71740 


98960 


78567 


86438 


75358 


34647 


37 


84 


71771 


98999 


78597 


36408 


75388 


34618 


26 


85 


71802 


98196 


78637 


36378 


75411 


34589 


85 


86 


71888 


38167 


78657 


36343 


75441 


34559 


34 


87 


71868 


98187 


78687 


36818 


75470 


34580 


38 


88 


71894 


28103 


78717 


36288 


75500 


34500 


89 


89 


71995 


38075 


78747 


86958' 


75539 


84471 


31 


40 


9.71955 


10.28045 


9.78777 


10.36398 


9.75558 


10.34448 


90 


41 


71986 


38014 


78807 


36198 


75588 


84413 


19 


49 


79017 


37968 


78887 


36168 


75617 


34888 


18 


48 


79048 


2T9*>2 


78867 


86188 


75647 


3485.') 


17 


44 


79078 


37999 


73807 


26103 


75676 


34894 


16 


45 


79109 


37891 


78997 


36078 


75706 


84395 


15 


46 


79140 


37860 


73957 


36043 


75786 


84965 


14 


Jl> 


79170 


87880 


78987 
74017 


36018 


75764 


34886 


18 


48 


72901 


37799 


35968 


75798 


84907 


13 


49 


79981 


37769 


74047 


85958 


75899 


34178 


11 ; 


50 


9.79909 


10.97738 


9.74077 


10.95993 


9.75659 


10.34148 


10 


51 


79998 


37707 


74107 


35898 


75881 


24119 


9 


59 


79898 


37677 


74187 


35868 


75910 


84080 


8 


58 


79854 


87646 


74166 


35834 


75939 


84061 


7 


54 


79884 


97616 


74196 


35804 


75969 


34081 


6 


55 


78415 


37585 


74936 


85774 


75996 


34009 


5 


56 


7*445 


87555 


74356 


35744 


76097 


88978 


4 


57 


7*475 


37594 


74986 


35714 


76056 


38944 


8 


58 


79506 


87494 


74816 


35684 


76066 


38914 


8 


59 


79587 


87468 


74345 


35655 


76115 


38885 


1 


60 


79567 


37488 


74875 


85625 


76144 


33856 





./ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


— 




69* 




•!• 




60* 



480 



TABLE III.— LOG. TANGENTS AND COTANGENTS. 



t 


80* 


8f 


88* 





Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.76144 


10.98866 


9.77877 


10.99198 * 


9.79679 


10.90491 


60 


1 


76178 


98897 


77906 


92094 


79607 


90698 


69 


s 


76909 


98798 


77985 


99065 


79686 


90865 


58 


8 


78981 


88769 


• 77968 


92087 


79668 


90887 


67 


4 


76961 


88789 


77998 


99008 


79691 


90809 


56 


5 


76990 


98710 


78090 


81980 


79719 


90981 


65 


6 


76819 


98681 


78049 


91961 


79747 


90958 


64 


7 


76848 


28659 


78077 


91998 


79776 


90994 


68 


8 


76877 


98698 


78106 


91894 


79804 


90196 


69 





76406 


98594 


78185 


91865 


79682 


90168 


61 


10 


9.76485 


10.98565 


9.78168 


10.81887 


9.79860 


10.90140 


60 


11 


76464 


98586 


78199 


81808 


79688 


80118 


49 


18 


76498 


98507 


78220 


81780 


79916 


90084 


48 


18 


76599 


98478 


78249 


81751 


79944 


90056 


47 


14 


76651 


98449 


78977 


91798 


79979 


90098 


46 


15 


76580 


93490 


78306 


81694 


80000 


90000 


46 


16 


76609 


88891 


78884 


91666 


80028 


19979 


44 


17 


76689 


98861 


78868 


91687 


80056 


19944 


48 


18 


76668 


88882 


78891 


81609 


80064 


19916 


49 


19 


76697 


98808 


78419 


91581 


80119 


19688 


41 


90 


9.76796 


10.98976 


9.78448 


10.81559 


9.80140 


10.19860 


40 


81 


76754 


28946 


78476 


91594 


80168 


19889 


89 


98 


76788 


98917 


78505 


21496 


80196 


19805 


88 


98 


76819 


98188 


78688 


91467 


80228 


19777 


87 


94 


76841 


98169 


78569 


91488 


80251 


19749 


86 


95 


76870 


98180 


78590 


81410 


80979 


19791 


86 


98 


76899 


28101 


78618 


91888 


80807 


19698 


84 


97 


76988 


98079 


78647 


81858 


80885 


19665 


88 


98 


76967 


98048 


78675 


81895 


80868 


' 19687 


88 


99 


76986 


98014 


78704 


91996 


80891 


19609 


31 


80 


9.77015 


10.92966 


9.78782 


10.91968 


9.80419 


10.19581 


SO 


81 


77044 


99956 


78760 


21240 


80447 


19558 


99 


89 


77078 


99927 


78789 


91911 


80474 


19596 


98 


88 


77101 


99899 


78817 


91188 


80509 


19498 


97 


84 


77180 


92870 


78846 


21155 


80580 


19470 


96 


85 


77169 


92841 


78874 


91196 


80558 


19448 


96 


86 


77188 


92812 


78909 


91098 


80686 


19414 


94 


87 


77217 


92788 


78980 


91070 


80614 


19386 


98 


88 


77246 


99754 


78959 


91041 


80648 


19858 


99 


89 


77274 


29726 


78987 


91018 


80669 


19881 


91 


40 


9.77808 


10.22697 


9.79015 


10.90986 


9.80697 


10.19808 


90 


41 


77889 


22668 


79048 


20967 


80795 


19976 


19 


48 


77861 


99689 


79079 


90998 


80758 


19947 


18 


48 


77890 


29610 


79100 


20900 


80781 


19919 


17 


44 


77418 


92582 


79128 


90878 


80608 


19199 


16 


45 


77447 


22558 


79156 


90844 


80886 


19164 


16 


46 


77476 


99524 


79185 


90615 


80864 


19186 


14 


47 


77505 


22495 


79218 


90787 


80699 


19108 


18 


48 


77588 


22467 


79241 


90750 


80919 


19081 


18 


49 


77562 


92488 


79969 


90781 


80947 


19068 


11 


60 


9.77591 


10.22409 


9.79297 


10.90708 


9.80975 


10.19095 


10 


.61 


77619 


99881 


79896 


90674 


81008 


18997 


9 


69 


77648 


99859 


79854 


90646 


81080 


18970 


8 


68 


77677 


29898 


79882 


90618 


81068 


18949 


7 


64 


77706 


22294 


79410 


90590 


81086 


38914 


6 


65 


77784 


92966 


79488 


90569 


81118 


18887 


6 


68 


77768 


99287 


79466 


90584 


81141 


16859 


4 


67 


77791 


82209 


79495 


90506 


81169 


18881 


8 


58 


77820 


92180 


79528 


90477 


81196 


18804 


8 


50 


77849 


92151 


79551 


90449 


81994 


18776 


1 


80 


77877 


92128 


79579 


90421 


81258 


18748 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


t 




69° 




68* 




•!• 



481 



TABLE HI 


.—LOG. 


TANGENTS AND COTANGENTS. 


/ 


88* 


84* 




86° 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


o 


9.81868 


10.18748 


9.88899 


10.17101 


9.84528 


10.16477 


60 


1 


81*79 


18781 


QQOMt 
BBWCD 


17074 


81550 


15450 


69 


2 


81607 


18698 


88958 


17047 


84676 


15424 


58 


3 


81886 


18665 


82980 


17080 


84606 


15897 


57 


4 


81668 


18668 


88008 


16998 


84680 


16870 


56 


5 


81690 


18610 


86065 


16965 


84657 


15848 


66 


6 
7 
8 



81418 


18588 


88068 


16988 


84884 


15816 


54 


81445 


18655 


88089 


16911 


84711 


15889 


68 


81478 


18587 


88117 


16888 


84788 


15868 


58 


81600 


18500 


88144 


16856 


84764 


15286 


51 


10 
11 
12 
18 

14 
15 
18 


9.81688 


10.18478 


9.88171 


10.16889 


9.84791 


10.16209 


50 


81666 


18444 


86198 


16802 


84818 


16188 


49 


81688 


18417 


88*286 


16775 


84845 


15155 


48 


81611 


18880 


88258 


16748 


8487S 


15128 


47 


81668 


18668 


86280 


16790 


84899 


15101 


46 


81666 


18884 


88807 


16696 


84925 


15075 


46 


81698 


18807 


88884 


16666 


84952 


15048 


44 


17 
18 
19 


81781 


18879 


86861 


16669 


84979 


15021 


48 


81748 


18858 


888S8 


16612 


85006 


14994 


42 


81776 


18884 


88415 


16585 


85088 


14967 


41 


80 


9.81808 


10.18197 


9.88142 


10.16558 


9.85059 


10.14941 


40 


81 
88 


81881 


18169 


83470 


16660 


85086 


14914 


89 


81868 


18148 


86497 


16508 


85118 


14887 


88 


88 


81886 


18114 


88584 


16476 


85140 


14860 


87 


84 


81918 


18087 


86551 


16449 


85166 


14864 


86 


86 


81941 


18069 
18088 


86578 


16428 


86198 


14807 


85 


86 


81968 


88606 


16895 


85280 


14780 


84 


87 


81996 


18004 


86662 


16868 


85847 


14758 


68 


88 
89 


89098 


17977 


88659 


16841 


85878 


14787 


8* 


88051 


17949 


88686 


16814 


85600 


14700 


81 


80 


9.88078 


10.17988 


9.88718 


10.16287 


9.86887 


10.14678 


80 


81 
88 


82106 


17894 


88740 


16260 


85854 


14646 


29 


88168 


17867 


88768 


16888 


85880 


14620 


28 


66 


88161 


17889 


88795 


16806 


85407 


14598 


27 


84 


88188 


17812 


88888 


16178 


85464 


14566 


26 


66 


88816 


17785 


88849 


16151 


85460 


14540 


26 


66 


88848 


17757 


86876 


16184 


85487 


14518 


84 


87 


88*70 


17780 


86908 


16097 


85514 


14486 


28 


88 

89 


88898 


17708 


88980 


16070 


85540 


14460 


22 


88685 


17675 


88957 


16048 


85567 


14466 


21 


40 


9.88868 


10.17648 


9.88984 


10.16016 


9.85594 


10.14406 


20 


41 


88880 


17680 


84011 


15989 


85620 


14880 


19 


48 


88407 


17598 


84088 


15968 


85647 


14858 


18 


46 


88486 


17565 


84065 


15965 


85674 


14626 


17 


44 


88463 


17588 


84098 


15908 


85700 


14600 


16 


46 

46 


88489 


17511 


84119 


16881 


85727 


14878 


15 


88617 


1748S 


84146 


15854 


85754 


14246 


14 


47 


88644 


17456 


84178 


15897 


85780 


14290 


18 


48 


88671 


17489 


84800 


15800 


85807 


14198 


18 


49 


89699 


17401 


84887 


16778 


85864 


14166 


11 


60 


9.88686 


;0. 17874 


9.84864 


10.15746 


9.85860 


10.14140 


10 


61 


88668 


17647 


84280 


15790 


85887 


14118 


9 


69 


88681 


17819 


84807 


15698 


85918 


14067 


8 


68 


88706 


17898 


84864 


15666 


85940 


14060 


7 


64 


83785 


17265 


84861 


15689 


85967 


14068 


6 


66 


88768 


'17288 


84888 


15612 


85996 


14007 


6 


66 


88790 


17210 


84415 


15585 


86080 


18980 


4 


67 


88817 


17188 


84449 


15558 


86046 


18954 


8 


68 


&J844 


?r56 


84469 


15581 


86078 


18927 


9 


69 


89871 


17i29 


84496 


15504 


86100 


16900 


1 


60 


88899 


17101 


84588 


15477 


86126 


18874 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 


1 


' 1 


16* 




66° 




64° 



48a 



TABLE m.— LOG 


. TANGENTS AND COTANGENTS. 


i 


86* 


87° 


88* 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.86126 


10.18874 


9.87711 


10.18289 


9.88881 


10.10719 


60 


l 


86158 


18847 


87788 


12262 


88807 


10698 


50 


s 


88178 


18881 


87764 


18286 


89888 


10667 


58 


8 


86806 


18794 


87790 


12810 


89858 


10641 


57 


4 


86888 


18768 


87817 


18188 


80885 


10615 


66 


5 


86859 


18741 


87848 


12157 


88411 


10589 


55 


6 


86885 


18715 


87868 


12181 


88487 


10568 


54 


7 


86818 


13688 


87895 


12105 


89468 


10587 


58 


8 


86888 


18668 


87928 


18078 


89489 


10511 


62 





86865 


18685 


87948 


12052 


89615 


10485 


51 


10 


0.86888 


10.18608 


9.87974 


10.12026 


9.89541 


10.10459 


50 


11 


86418 


18588 


88000 


12000 


89567 


10488 


49 


18 


86445 


18555 


88087 


11978 


89598 


10407 


48 


18 


86471 


18589 


88058 


11947 


89619 


10881 


47 


14 


86498 


18508 


88079 


11981 


89645 


10856 


46 


15 


86584 


18476 


88105 


11895 


89671 


10829 


45 


16 


86551 


18449 


88181 


11869 


89697 


10808 


44 


17 


86577 


18488 


88158 


11848 


89728 


10277 


a 


18 


86608 


18897 


88184 


11816 


88749 


10261 


42 


10 


86680 


18870 


88810 


11790 


89775 


10885 


41 


80 


9.86656 


10.18844 


9.88886 


10.11764 


9.89801 


10.10199 


40 


81 


86688 


18817 


88868 


11788 


89887 


10178 


80 


88 


86709 


18891 


88280 


11711 


88658 


10147 


88 


88 


86^6 


18864 


88815 


11685 


89879 


10181 


87 


84 


86768 


18888 


86841 


11659 


89906 


10085 


86 


85 


80789 


18811 


88867 


11688 


89981 


10069 


85 


86 


86815 


18185 


88898 


11607 


89957 


10048 


84 


87 


86848 


18168 


88420 


11580 


89988 


10017 


88 


88 


86868 


18188 


88446 


11554 


90009 


08991 


88 


89 


86894 


18106 


88478 


11528 


90085 


09965 


81 


80 


9.86981 


10.18079 


9.88496 


10.11508 


9.90061 


10.09989 


80 


81 


86947 


iaof* 


88524 


11476 


90086 


09914 


88 


38 


86974 


18086 


88550 


11450 


90118 


00688 


88 


88 


87000 


18000 


88577 


11428 


90188 


09888 


87 


84 


87087 


18978 


86608 


11897 


90164 


09886 


86 


85 


87058 


18947 


88629 


11871 


90190 


00810 


85 


86 


87079 


18981 


88655 


11845 


90816 


00784 


84 


87 


87106 


18894 


88681 


11819 


90848 


00758 


88 


88 


87132 


18868 


88707 


11298 


90268 


00782 


88 


88 


87158 


18848 


88788 


11267 


90894 


00706 


81 


40 


9.87185 


10.18815 


9.88759 


10.11841 


9.90880 


10.00680 


80 


41 


87811 


18789 


88786 


11214 


80646 


00654 


19 


48 


87888 


18768 


88818 


11188 


90871 


00629 


18 


48 


87864 


12786 


88888 


11162 


90897 


09608 


17 


44 


87890 


18710 


88864 


11186 


90488 


09577 


16 


45 


87817 


18688 


88890 


11110 


90449 


00551 


16 


46 


87843 


12657 


88916 


11064 


90475 


00585 


14 


47 


87869 


12681 


88942 


11058 


90501 


09499 


18 


48 


87896 


18604 


88968 


11088 


90587 


09478 


18 


49 


87488 


18578 


88994 


11006 


80558 


00447 


11 


50 


9.87448 


10.18558 


9.89020 


10.10980 


9.90578 


10.00488 


10 


51 


87475 


12525 


89046 


10954 


90604 


00806 


9 


58 


87501 


18499 


89078 


10927 


90680 


00870 


8 


58 


87587 


18478 


89099 


10901 


90656 


00844 


7 


54 


87554 


18446 


89185 


10675 


90688 


09818 


6 


55 


67580 


18480 


89151 


10849 


80708 


09292 


5 


56 


87606 


18894 


89177 


10828 


90784 


00866 


4 


57 


87688 


12867 


89208 


10797 


90759 


00841 


8 


58 


87669 


12841 


89829 


10771 


90785 


00815 


8 


58 


87686 


12815 


89255 


10745 


90811 


00180 


1 


60 


87711 


12889 


89281 


10719 


90687 


00168 





# 


Cotan 


Tan 


Cotan 


Tan 


Ootan 


Tan 


# 




68* 




68* 




•!•• 



483 

TABLE Hi.— LOG. TANGENTS AND COTANGENTS. 






89° 


40° 


41° 


> 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





0.90687 


10.00168 


0.92381 


10.07810 


9.93916 


10.06064 


60 


1 


90868 


09187 


93407 


07593 


98942 


06058 


50 


8 


00880 


09111 


93438 


07587 


93967 


06068 


58 


s 


90014 


00086 


93458 


07648 


93998 


06007 


57 


4 


00040 


09060 


92484 


07616 


94018 


05982 


56 


5 


90066 


09084 


93010 


07490 


94044 


05956 


56 





90008 


09008 


92585 


07465 


94089 


06931 


64 


7 


91018 


08883 


93561 


07489 


94005 


05905 


58 


8 


01043 


08057 


92587 


07418 


94120 


05860 


63 





91000 


08081 


98618 


07888 


94146 


06854 


51 


10 


0.91005 


10.08905 


9.92688 


10.07368 


9.94171 


10.06629 


50 


11 


01181 


08879 


92663 


07387 


94197 


05808 


49 


IS 


01147 


06858 


98689 


07811 


94322 


05778 


48 


18 


91178 


08828 


92715 


07285 


94248 


05758 


47 


14 


91198 


08802 


93740 


07860 


94278 


05787 


46 


15 


91284 


08776 


92766 


07234 


94899 


05701 


45 


16 


91850 


08750 


92792 


07208 


94884 


05676 


44 


17 


91876 


08724 


93817 


07183 


94850 


05650 


48 


18 


91801 


08699 


92843 


07167 


94876 


05625 


42 


19 


91887 


08673 


98868 


07188 


94401 


05599 


41 


90 


9.01858 


10.08647 


9.08894 


10.07108 


9.94436 


10.05574 


40 


81 


01879 


08681 


92990 


07080 


94458 


05548 


89 


88 


01404 


08596 


92946 


07055 


94477 


05638 


88 


38 


01480 


08570 


92971 


07029 


94608 


05497 


87 


84 


91456 


08544 


92906 


07004 


94528 


05472 


86 


85 


9148* 


08518 


98022 


06978 


94564 


05446 


85 


88 


91507 


06493 


98048 


06958 


94579 


06431 


84 


87 


01588 


06467 


98078 


06987 


94604 


05398 


88 


88 


01550 


08441 


98099 


06901 


94630 


05870 


88 


88 


01585 


06416 


98124 


06876 


94655 


05345 


81 


80 


0.01610 


10.06890 


9.98150 


10.06850 


9.94681 


10.05819 


80 


81 


91686 


08864 


98175 


06825 


94706 


05294 


89 


88 


91608 


06338 


98801 


06799 


04782 


05268 


28 


88 


9I68S 


06313 


93287 


06778 


04757 


05348 


87 


84 


91718 


06287 


93252 


00748 


94788 


05317 


36 


85 


91789 


06261 


93378 


06722 


94808 


05192 


85 


86 


91765 


06385 


98808 


06697 


94834 


05166 


34 


87 


91791 


08200 


98329 


08071 


94859 


05141 


88 


88 


91816 


06184 


93354 


06646 


94864 


05116 


33 


80 


91848 


06158 


98880 


08630 


94910 


05090 


81 


40 


9.91868 


10.06182 


0.98406 


10.06594 


9.94985 


10.06065 


80 


41 


91893 


06107 


93481 


06569 


94961 


05089 


19 


48 


91919 


08081 


93457 


06548 


94986 


05014 


18 


48 


91945 


06055 


98463 


06518 


95013 


04968 


17 


44 


91971 


08030 


93506 


06498 


95037 


04988 


16 


45 


91908 


08004 


98588 


06467 


06062 


04938 


16 


46 


92022 


07978 


98559 


06441 


95066 


04912 


14 


47 


08048 


07958 


98584 


06416 


95118 


04887 


18 


48 


03073 


07937 


94610 


06390 


95139 


04861 


18 


49 


02090 


07901 


98686 


06864 


95164 


04836 


11 


50 


0.08125 


10.07875 


9.98081 


10.06389 


9.96190 


10.04810 


10 


51 


92150 


07850 


93087 


06318 


95215 


04785 


9 


58 


98176 


07824 


93712 


06888 


95240 


04760 


8 


68 


98803 


07798 


93738 


06262 


95268 


04784 


7 


54 


98227 


07778 


93768 


06237 


95291 


04709 


6 


56 


93258 


07747 


98789 


06211 


95317 


046** 


5 


56 


98879 


07721 


98814 


06186 


95342 


04658 


4 


67 


02304 


07698 


98840 


06160 


95366 


04682 


8 


58 


92380 


07670 


93865 


06185 


95393 


04607 


8 


50 


98356 


07644 


93891 


06109 


95418 


04682 


1 


60 


02881 


07610 


93916 


08084 


95444 


04556 





/ 


Cotan 


Tan 


Cotan 


Tan 


Cotan 


Tan 


/ 




W 




4»» 




48* 



484 



TABLE EL— LOG. 


TANGENTS AND COTANGENTS. 


/ 


48* 


48* 




44° 


/ 


Tan 


Cotan 


Tan 


Cotan 


Tan 


Cotan 





9.96444 


10.04556 


9.90966 


10.08084 


9.96484 


10.01516 


60 


1 


95469 


04581 


96991 


08009 


96509 


01491 


59 


2 


95495 


04505 


97016 


08964 


96584 


01466 


58 


8 


96620 


04460 


97048 


02958 


96560 


01440 


57 


4 


95545 


04455 


97067 


02988 


96585 


01415 


56 


5 


95571 


04489 


97098 


02906 


96610 


01890 


55 


6 


95596 


04404 


97118 


02868 


96685 


01865 


54 


7 


95688 


04878 


97148 


08857 


96661 


01889 


58 


8 


95647 


04858 


97168 


02688 


96686 


01814 


52 


9 


95678 


04888 


97198 


08807 


96711 


01869 


51 


10 


9.95696 


10.04808 


9.97819 


10.08781 


9.96737 


10.01268 


50 


11 


95788 


04877 


97244 


€8756 


96762 


01888 


49 


18 


95748 


04858 


97269 


03781 


96767 


01818 


48 


18 


95774 


04886 


97295 


02705 


96812 


01188 


47 


14 


95799 


04801 


97880 


02680 


96688 


01162 


46 


15 


95885 


04175 


97845 


02655 


96868 

06866 


01187 


45 


16 


95850 


04150 


97871 


08689 


01118 


44 


17 


95875 


04185 


97896 


02604 


96918 


01067 


48 


18 


95901 


04099 


97421 


08579 


96989 


01061 


48 


10 


95986 


04074 


97447 


09553 




01086 


41 


90 


9.95958 


10.04048 


9.97478 


10.08526 


9.06060 


10.01011 


40 


81 


95977 


04088 


97497 


02508 


90015 


00965 


80 


88 


96008 


08996 


97588 


08477 


00040 


00960 


38 


88 


96088 


08973 


97548 


08452 


00065 


00965 


87 


84 


96058 


08947 


97578 


02487 


00000 


00910 


86 


88 


96078 


06989 


97596 


02408 


00116 


00664 


35 


86 


96104 


06896 


97684 


02876 


00141 


00659 


84 


87 


96189 


08671 


97649 


02851 


00166 


00684 


88 


88 


96155 


08845 


97674 


02826 


09101 


00609 


88 


89 


96180 


08890 


97700 


02800 


00817 


00768 


31 


80 


9.96905 


10.06795 


9.97785 


10.08275 


0.00848 


10.00758 


30 


81 


96881 


08769 


97750 


08250 


00867 


00783 


20 


88 


96856 


08744 


97776 


02884 


00208 


00707 


86 


88 


96881 


06719 


97801 


08199 


09818 


00662 


87 


84 


96807 


08608 


97886 


02174 


90848 


00657 


26 


85 


96888 


08666 


97951 


08149 


09868 


00638 


85 


86 


96857 


06648 


97877 


08128 


09894 


00606 


84 


37 


96888 


08617 


97908 


08098 


99419 


0C581 


88 


88 


96406 


08598 


97927 


02078 


99444 


00556 


28 


89 


96488 


08567 


97958 


02047 


99469 


00581 


81 


40 


9.96459 


10.08541 


9.97978 


10.08088 


9.99495 


10.00505 


80 


41 


86484 


08616 


96008 


01997 


99520 


00480 


•10 


48 


96510 


08490 


96029 


01971 


99545 


00455 


18 


48 


96585 


08465 


96054 


01946 


99670 


00430 


17 


44 


96560 


08440 


96079 


01981 


00506 


00404 


16 


45 


96588 


06414 


96104 


01696 


09681 


00879 


15 


46 


96611 


08889 


98180 


01870 


99646 


00354 


14 


47 


96686 


08864 


98155 


01845 


99678 


00888 


13 


48 


96668 


08338 


98180 


01890 


98607 


00808 


19 


49 


96687 


08818 


96806 


01794 


00788 


00278 


11 


50 


9.96718 


10.08888 


9.96281 


10.01769 


0.00747 


10.00858 


10 


51 


96788 


08868 


96256 


01744 


00778 


00827 


9 


58 


96768 


08287 


96261 


01719 


00796 


00808 


8 


58 


86768 


08812 


98807 


01698 


99688 


00177 


7 


54 


96814 


08166 


96382 


01668 


99648 


00158 


6 


55 


96889 


08161 


98357 


01648 


90674 


00126 


6 


58 


96864 


08186 


96883 


01617 


90699 


00101 


4 


57 


96890 


08110 


96406 


01598 


OOOQL4 


00076 


8 


58 


96915 


08085 


96488 


01567 


99949 


00061 


8 


59 


96940 


08060 


96458 


01548 


99975 


00085 


1 


60 


96966 


08084 


96464 


01616 


104)0000 


00000 





i 


Cotan 


Tan 


OOTUII 


Tan 


Cotan 


Tan 


* 




47° 




48* 




#5* 


i 



485 



TABLE IV.— NATURAL SINES AND COSINES. 



0* 



Sine [Cosin 
[00000 One. 



00029 
00058 
00067 
.00116 
.00145 
.00175 



.00968 
.00291 

.00890 
.00849 
.00878 
00407 
00486 



One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 



.00465 
.00495 
.00524 
.00558 



00588 

.00611 
.00640 



99999 ' 
99999 
.99999 






99998 
99998 



.00698 
.00787 
.00756 
.00785 
.00814 
.00844, 
.00878 

.00908> 
.009811 
.00960 
.00989; 
.01018 1 
.01047 ' 
.01076! 
.01105 
.01184 
.01164 

.01198 
.01888 
.01851 \ 
.01880 
.01809 
.01888 
.01867 
.01896 
.01425 
.01454 

.01468 
.01518 
01548 
.01671 
.01600 
.01689 
.01658 
.01687 
.01716 
JN745i 
Cosin i 



99998 
99998 
99998 
99998 
! 99997 



.99997 



99996 

99996 

99996 



99994 
99994 

99994 



.99998 
.99993 
.99998 
.99998 
.99991 
.99991 
.99991 
.99990 
.99990 
.99969 

99989 
! 99989 



99988 
! 99987 
.99987 
.99986 



Sine 



•9* 



Sine 
.01745 
.01774 
.01808 
.01838 
.01862 
.01891 
.01920 
.01949 
.01978 
.02007 
.08086 



Oogin 
99985 



.08065 
.08094 
.08128 
.02152 
.02181 
.02811 
.08240 



.02387 
.08356 



.08414 
.08448 
.03472 
.08501 



.99984 
.90984 



.08560 
.02589 
.02618 

.09647 
.02676 
.08705 
.08784 
.08768 
.02792 



99988 



.99981 
.99980 
.99980 



99979 
.99979 



.99978 
99977 
.99977 



.99976 
.99975 
.99974 
.99974 
.99973 

.99972 
.99978 
.99971 



.99970 
.99969 
.99969 



.08850 
.02879 
.02908 



.02967 
.02996 
.08085 



.08054 
.08063 
.03118 
.08141 
.08170 
.08199 



.03857 
.08286 
.08316 
.08845 
.08374 
.08403 
.08482 
.03461 
.08490 
Cosin 



.99967 
.99966 
.99966 

.99965 
.99964 
9990o 
.99963 
.99962 
.99961 
.99960 
.99959 
.99959 
.99958 



.99957 
.99956 
.99955 
.99954 
.99953 
.99952 



.99951 
.99950 
.99949 

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



Sine 



Sine 
.08490 
08519 
.08548 
03577 
03606 
03635 



Cosin 



.08664 
.08693 
.03788 
.03758 
.08781 

.03610 



.08897 
.03926 
.03955 
.08984 
.04013 
.04042 
.04071 

.04100 
.04129 
.04159 
.04188 
.04217 
.04246 
.04875 
.04804 
.04383 
.04862 

.04891 
.04420 
.04449 
.04478 
.04507 
.04586 
.04565 
.04594 
.04623 
.04653 

.04682 
.04711 
.04740 
.04769 
.04798 
.04827 
.04856 
.04885 
.04914 
.04943 

.04978 
.05001 
.05080 



05059 
.05068 
.05117 
.05146 
05175 
05205 



.05834 



Cosin 



99937 



.99934 



.99932 
99931 



99929 



99987 



99925 



.99921 
.99910 
.99918 
.99917 

.99916 
.99915 
.99913 
.99912 
.99911 
.99910 
.99909 
.99907 
.99906 
.99905 

.99904 



8» 



.99901 
.99900 



.99697 
.99696 
.99804 
.90898 
.99692 



.99886 



! 99879 
.99878 

.99876 
.99875 
.99873 
.99872 
.99870 



.99867 
.99866 
.99864 
.9 9863 

Sine 



«• 



Sine 



Cosin 



06234 
05268 
05298 
05381 
.06350 



.05379 
,05406 
.05487 
.05466 
.05495 



.05558 



.06568 



.05611 
.05640 
.05669 



.05698 
.05727 
.05756 
.05785 
.05814 

.06S44 
.05878 
.05908 
.05931 
.05960 
.05989 
.06018 
.06047 
.06076 
.06105 

.06184 
.06168 
.06192 
.06221 
.06250 
.06279 
.06308 
.06887 
.06866 



.06494 
.06453 
.06482 
.06511 
.06540 
.06569 
.06598 
.06627 
.06656 
.06685 

.06714 
.06743 
.06773 
.06808 
.06831 
.06860 
.06889 
.06918 
.06947 
.06976 
Cosin , 



99861 
.99860 
.99858 

99857 
.99855 

99654 



99851 
99649 
99847 

99846 
99844 
99842 
,99641 



.99886 
99834 
,998331 
.99881 



99827 



.99819 
.99817 
.99815 
.99813 

.99812 

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

.99793 
.99792 
.99790 
.99788 
.99786 
.99784 
.99782 
.99780 
.99778 
.99776 

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

Sine | 



•6° 



Sine 



Cosin 



.06976 
.07005 
.07084 
.07063 
.07092 
.07181 
.07150 
.07179 
.07808 
.07287 
.07866 

.07995 
.07894 
.07858 
.07888 
.07411 
.07440 
.07469 
.07498 
.07587 
.07556 

.07585 
.07614 
.«T648 
.07678 
.07701 
.07780 
.07759 
.07788 
.07817 
.07846 



.07875 
.07904 



.07983 
.07968 
.07991 
.06080 
.06049 
.06078 
.08107 
.06186 

.06166 
.06194 



.06258 



.06810 



.06868 
.08397 
.06426 

.06455 

.06464 
.06513 
.06542 
.06571 
.06600 



.08687 
.08716 
Cosin 



99756 

99754 

99752 

99750 

99748 
.99746 

99744 
.99748, 53 

99740 52 
.99738 51 
.09786 50 



.99784 
99781 
99729 
.99727 
.99725 
.90783 
.99721 
.99719 
.99716 
.99714 

.99712 
.99710 
.99708 
.99705 
.99708 
.99701 



99696 
99694 



99689 
99667 
99685 



.99680 
.99678 
.99676 
.99673 
.99671 
.99668: 

.99666 19 
.996641 18 



24 



99661 
99659 
.99657 
.99654 
.99652 
.99649 
.99647 
.99644 

.99642 
.99639 
99687 



99685 



99630 
99627 
99625 



.99619 
Sine 



W 



486 



TABLE IV.— NATURAL SINES AND COSINES. 



, 


« 


• 


e° 


7° 1 


8» 


1 9 ° 


/ 


1 Sine ICosin 
708710 .99019 


Sine 
.10453 


Cosln' 
.99452 


Sine 
712187 


Cosln 
799255 


Sine 
.18917 


Cosln 

.99027 


| Sine 
715643 


Cosln 


.98769 


60 


1 .06745 


.99017 


.10482 


.99440 


.12216 


.90251 


.18946 


.99023 


.15672!. 98764 69 


2 .08774 


.99614 


.10511 


.99440 


.12245 


.99248 


.13975 .99019 


.15701 .98760 56 


8 


.08803 


.99612 


.10540 


.99443 


.12274 


.99244 


.14004 


.99015 


.15730 .98756 57 


4 


.08831 


.99609 


.10569 


.99440 


.12302 


.99240 


.14083 


.99011 


.15758 .98751 66 


5 


.08860 


.99607 


.10597 


.99437 


.12331 


.99237, 


.14061 


.99006 


.15787 .98746 55 


.6 


.08889 


.99604 


.10026 


.99434 


.12360 


.99233 


.14090 


.99002 


.15816. 98741 54 


7 


.06918 


99602 


.10655 


.99431 


.12389 


.99230 


.14119 


.98998 


.15845 .98787 58 


8 


.08947 


.99599 


.10084 


.99428 


.12418 


.99226 


.14148 


.98994 


.15873 .98782 52 


9 


.08976 


.99596 


.10713 .99424 


.12447 


.99222 


.14177 


.98990 


.15902 .98728 51 


10 


.09005 


.99594 


.107421.99421 


.12476 


.99219 


.14205 


.98986 


.15931 


.96728 50 


11 


.09084 


.99691 


.10771'. 99418 ' 


.12504 


.99215 


.14284 


.98982 


.15959 


.98718 


49 


18 


.09068 


.99588 


.10600 .99415 


.12533 


.99211 


.14263 


.98978 


.15988 


.98714 


48 


13 


.00092 


.90586 


.10829 


.99412 


.12562 


.99208 


.14292 


.98978 


.16017 


.98709 


47 


14 


.09121 


.99583 


.10658 


.99409 


.12591 


.99204 


.14820 


.98969 


.16046 


.98704 


46 


15 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200, 


.14849 


.98965 


j. 16074 


.98700 


46 


16 


.09179 


.99578 


.10916 


.99402 


.12649 


.99197 


.14878 


.98961 


! .16108 


.98695 


44 


17 


.00208 


.99575 


.10945 


99899 


.12678 


.99193' 


.14407 


.98957 


! .16182 


.96690 


48 


18 


.09287 


.99572 


.10973 


.99396 


.12706 


.99189 


.14486 


.98953 


.16160 


.WOOD 


42 


19 


.09266 


.99570 


.11002 


.99393 


.12735 


.99186 


.14464 


.98948 


: .16189 


.98681 


41 


20 


.09295 


.99567 


.11081 


.99890 


.12764 


.99182 


.14498 


.98944 


j.16218 


.98676 40 


21 


.09824 


.99564 


.11060 


.99886 


.12793 


.99178 


.14522 


.98940 


1.16946 


. 98671 ! 39 


22 


.09353 


.99562 


.11089 


.99383 


.12822 


.99175 


.14551 


.98086 .16276 


.98667.88 


28 


.09882 


.99559 


.11118 


.99880 


.12851 


.99171 : 


.14580 


.989811 .16804 


.98662,87 


24 


.09411 


.99556 


.11147 


.99377 


.12860 


.99167, 


.14608 


.98927! .16888 


.98657 


86 


25 


.09440 


.99558 


.11176 


.99374 


.12908 


.99163, 


.14687 


.98923 i .16861 


.96652 


86 


28 


.09469 


.99551 


.11205 


.99370 


.12987 


.99160, 


.14666 


.98919 .16890 


.98648 


84 


27 


.09498 


.99548 


.11234 


.99367 


.12966 


.99156 


.14695 


.98914 


.16419 


.98643 


83 


28 


.09527 


.99545 


.11263 


.99364 


.12995 


.99152 


.14728 


.98910 


.1644? 


.98688 


82 


29 


.09556 


.99542 


.11291 


.993601 


.13024 


.99148 


.14752 


.98906 


.16476 .98688 


81 


80 
81 


.09585 
.09614 


.99540 
.99587 


.11820 


.99357 


.13053 


.99144 


.14781 


.98902 


.16505J.96629 


80 


.11849 


.99854 


.13081 


.99141 


.14810 


.9689? 


.16588 '.98624 


29 


82 


.09642 


.99584 


.11878 


.99351 


.13110 


.99137 


.14888 


.98898 


.16562 


.96619 


28 


88 


.09871 


.99531 


.11407 


.99347 


.13139 


.99183 


.14867 


98889 


.16591 


.96614 27 


84 


.09700 


.99528 


.11436 


.99344 


.13168 


.99129 


.14896 


'98884 


.16620 


.98609 


26 


85 


.09729 


.99526 


.11465 


.99341 


.13197 


.99125 


.14925 


! 98880 


.16648 


.96604 


25 


86 


.09758 


99523 


.11494 


99337 


13226 


99122 


.14954 


96876 


.16677 


98600 


24 


87 


.09787 


.99520 


.11523 


; 99334 


.18254 


! 9911 8 


!l4982 


.'98871 


.16706 


.98596 


28 


88 


.09616 


99517 


.11552 


.99331, 


.13283 


.99114 


.15011 


.98867 


.16784 


.98590 


22 


89 


.09845 


.99514 


.11580 


.99327 


.13312 


.99110 


.15040 


.98863 


.16768 


.98565 


21 


40 


.09874 


.99511 


.11609 


.99824 


.13841 


.99106 


.15069 


.98858 


.16798 


.98660 


20 


41 


.09908 


.99506 


.11638 


.99820 


.18870 


.99102 


.15097 


.98854 


.16820 


.98675 


19 


42 


.09982 


.99506 


.11667 


.99817' 


.18899 


99098 


.15126 


.98849 


.16849 


.98570 


18 


48 


.09961 


.99503 


.11696 


.99314 


.18427 


.'99094 


.15155 


.98845 


.16878 


.98565 


17 


44 


.09990 


.99500 


.11725 


.99310 


.13456 


.99091 


.15184 


.98841 


.16906 


.96561 


16 


45 


.10019 


.99497 


.11754 


-99307 


.13485 


.99087 


.15212 


.98886 


1.16985 .98556 


15 


46 


.10048 


.99494 


.11783 


.99803 


.18514 


.99063. 


.15241 


.98882 


.16964 .16561 


14 


47 


.10077 


.99491 


.11812 


.99300 


.18543 


.99079 


.15270 


.98827 


!. 10992 .98546 


18 


48 


.10106 


.90488 


.11840 


.99297 


.13572 


.99075 


.15299 


.98823 


i. 17021 .98541 


12 


49 


.10185 


.99485 


.11869 


.99293 


.13600 


.99071 


.15827 


.98818 


.17050 .98586 11 


50 


.10164 


.99482 


.11898 


.99290 


.13629 


.99067 


.15856 


.96814 


j .17078 


.WX»1 


1U 


51 


.10192 


.99479 


.11927 


.99286 


.18658 


.99063 


.15885 


.98809 


1.17107 


.98526 


9 


52 


.10221 


.99476 


.11956 


.99283 1 


.13687 


.99059 


.15414 


.98605 


!. 17188 


.98521 


8 


68 


.10250 


.99473 


.11985 


.99279, 


.18716 


.99055 


.15442 


.98600 


.17164 


.98516 


7 


54 


.10279 


.99470 


.12014 


.99278 


.18744 


.99051 


.15471 


.98796 


! .17193 


.98511 


6 


55 


.10808 


.99467 


.12043 


.99272 


.18778 


.99047 


.15500 .98791 


1.17222 '.98506 


5 


66 


.10387 


.99464 


.12071 


.99269 


.13802 


.99043 


.15529 .98787 


.17250 .98601 


4 


57 


10866 


.99461 


i .12100 


.99265 


.13831 


.99089 


.15657 .98782 


1 .17279 .98496 


3 


58 


.10895 


.99458 


.12129 


.99262 


.13860 


.99085 


.15586 .98778 


.17808 .98491 


2 


59 


.10424 


.99455 


.12158 


.99258 


.13889 


.99031 


.15615 .98778 


.17886:. 98486 


1 


60 


.10453 
Cosln 


.99452 
Sine 


.12187 
Cosln 


.99255 
Sine 


.13917 
Cosln 


.99027 
Sine" 


.15643 .98769 
Cosln | Sine 


1.17365 .98481 


_0 

9 


, Cosln | Sine 


84° 


88° 


82* 


81* 


1 80° 



4»7 

TABLE IV.— NATURAL SINES AND COSINES. 




1 
* 

8 

4 
6 

7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
90 

21 
99 
28 
94 
96 
98 
97 
98 
99 
80 

81 
89 
88 
84 
85 
86 
87 



10° 



Sine Cosin 
.17865 .98481 
.17893 .96476 
.17499 .98471 
.17451 .98466 
.17479 .98461 
.17508 .98455 
.17537 .9MW 
.17565 .9K445 
.17591 .98440 
.17698 .9M--« 
.17651 .9fi--' 



11° 



Sine 



Cosin 



.17680 
.17708 
.17787 
.17766 
.17794 
.17898 
.17859 
.17880 
.17909 
.17987 

.17966 
.17995 
.18098 
.18059 
.18081 
.18109 
.18188 
.18166 
.18195 
.18994 

.18959 
18981 
.18309 



18867 
18895 
18494 
18459 
18481 
18509 

18588 
18567 
18595 
18694 
18652 
18681 
18710 
18788 
18767 
18795 

18824 
18852 
18881 
18910 



18967 
18995 
19094 
19052 
19081 



Cosin 



.9Ri^ 
.98l'J" 
.98411 
.98191 

.9*401 

.98m 

.93994 
.9* I 
.98888 
.983™ 

98873: 
989N 

98303* 
,98 83 
988 I 

9sm 

98341 
9833U 
9633] 
983sJD 

9ft;- ' 
98m . 
98810 
98-; '< 
98 J *■* 
983*1 
98- 
9898 
98277 

98^:j 

9856: 
9801 

98^ 

98u:hj 

98045 

98940 

98-U 

98 l 

98 

98flfl 

98912 1 
98207, 
982011 
98196 
98190 
98185 
98179 
98174 
96168 
98168 



.19061 
.19109 
.19186 
.19167 
.19195 
.19224 

19981 1 

. 1'JCjlft] 

J93i& 
19498 

.I'.tl-YJ 



l!Cyl(» 
, I959Q 

]■'■- "■•.' 



.19TO0 

.io;m 

.19794 
JOHsEH 

.19551 

49880 

.1990* 
,1993; 

i99tir> 

.]■'■• m 
jofltt 

.SOUS I I 

.800791 
. 3i ■ I ■ - 
.90130 

.aniiTi. 

a i] ■ ■ 
.bo&w: 

19950 

90807 
.90331} 
10984 

90898 
aotfjl 
50450 

,ao47H 

20507' 

.20535| 
.20568: 
.20592 



Sine 



7* 



20649 
1.20677 
j .20706 
1 20734 
,.20768 
1.20791 
Cosin | 



.98163 
.98157 
.98152 
.96146 
.98140 
.98135 
Q«129 
9*184; 
H8 
&<M2 
OH 107 
UH101 

9 
3 

, I M h . 7 
5*- "1 

!#m&0 
44 

• :»' 
,98088 

,9909!', 

.08021 

18019 ' 

9*>l0 

9*011 
979W 

■'.."J 

.97987 
,97961 
97V75 
97VAJ9 
979$8 
97958 
97952 
Wi '46 i 
.971*40' 
,971134 

.97998 

■i, *£} 

.u;wtt 

.971*10 
.97906 

F 9789ft' 
.97887; 

.07W1 
.97875 

.97869' 
.978631 
.97857, 
.97851 
.97845 
.97839! 
.97833 
.978271 
.978211 
.97815; 



18* 



Sine 



Sine 
.20791 
.20820 
.20848 
.20677 
.20905 
.20938 
.20962 
.20990 
.21019 
.21047 
.21076 

.21104 
.21188 
.21161 
.21189 
.21218 
.21246 
.21275 
.21803 
.21331 
.21860 



Cosin 



.97815 
.97809 
97803 



.21417 
.21445 
.91474 
.21502 
.91530 
.21559 
.21587 
.21616 
.21644 

,21672 
21701 
21729 
21758 
21786 
21814 
.21843 
21871 
21899 



21956 
21985 
22013 
22041 
22070 



.97797 
.97791 
.97784 
.97778 
.97772 
.97766 
.97760 
.97754 

.97748 
.97742 
.97785 
.97729 
.97738 
.97717 
.97711! 
. 97705 ! 
.97698 
.97692; 

.97886 
.97B80 1 
.97878 
.97667. 
.97661 
. 97655 ' 
.97648! 
,97642 
.97636 
.97680! 

97623 ' 
97617 
97611 
97604 
97598 
97592 
97585 
975791 
97573 1 
97566, 

97560 
97553' 
97547, 
97541 
975341 



Sine O 



22495 .1/7437 



.22552 



.22126 .97521 
.22155 .97515 
.22183 .97506 
.22212 .97502 

.22240 .97496 
.22268 .97489 
.22297!. 97483 
.22325 .97476 
.228531.97470! 
.22382 .97468 
.22410 .97457 
.22488 .97450 
.224671.97444 
■224951.97437 
Cosin | Sine I 



II 



77 o 



.22608 
.22637 



22722 
.22750 
.22778 

.92807 



.22948 



.22977 
.28005 
.23033 



.28090 
.28118 
.28146 
.23175 



28316 
23845 

28873 
23401 
23429 
23458 
23486 
23514 
23542 
23571 



23656 
23684 
23712 
23740 
23769 
,23797 



.23910 



.iiti474 



.24734 

.^4*1 8 
."- a 
94889 

.24897 
,vi.-6 

.21954 

.24UK2 

^138 



.96880 
.96873 
.96866 
.96858 
.96851 
.96844 
.96837 



.96815 

.96807 
.96800 
.96798 
.96786 
.96778 
96771 



.96764 
.96756 
.96749 
.96742 

.96734 
.96727 



96690 



I • 



76° 



Cosin Sine 
I 7lV 



488 
TABLE IV.— NATURAL SINES AND COSINES. 



16- 



Sine |Coein • I Sine ' Coa ln 

.25882:796598' 

25910 .96585 | 



.25988 .90578 
25966 .96570 
259941.98562 
260221.96555 
26050 .96547 



.26079 
.26107 
.26185 
.26168 

.26191 
26219 
26247 
26275 
26808 



.26881 
26859 



.26887 
.26415 
.26448 

.26471 
.26500 



.26556 



.26612 
.26640 



.26696 
.26724 

.26752 
.26780 
.26808 



.26864 
.26892 
.26920 
.26948 
.26976 
.27004 

.27082 
.27080 
.27088 
.27116 
.27144 
.27172 
.27200 
.27228 
.27256 
.27284 

.27812 
.27840 
.27868 
.27896 
.27424 
.27452 
.27480 
.27508 
.27586 
.27564 



Cosln 



.96540 
96582 



.96524 

.965171 

.96509! 
.96502 
.964941 
.96486 
.96479 
.96471 
.96468 
.96456 
.96448 
.96440! 

.96488 
.96425 
.96417 
.96410 
.96402 
.96894 
.96886 
.96879 
.96871 
.96368 

.96855 
.96347 
.96840 
.96382 
.96324 
.96316 
.96806 
.96301 



.96277 
96269 
.96261 
96258 
,96246 
.96288 
96280 



.96214 
.96206 

.96198 
.96190 
.96182 
.96174 
.96166 
.96158 
.96150 
.96142 
.96134 
.96126 
Sine I 



74* 



27564 .96126 
27592 .96118 
27620 ,.98110 
27648 .98102 
27676 .96094 
27704. 96086 
27781 .96078 
27759 -96070 
27787 .96062 
27815 .98054 
27848 .96046 




489 



TABLE IV.— NATURAL SINES AND COSINES. 



90* 



Sine 



Cosin 



.84900 
.84289 
.84257 
.84864 
.84811 
.84889 
.84866 
.84898 
.84421 
.84448 
.84475 

.84508 
.84580 
.84557 
.84584 
.84612 



.84660 
.84694 
.84721 
.84748 

.84775 
.84808 
.84880 
.84857 
.84884 
.84912 
.84989 
.84966 
.84998 
.85021 



.85048 
.85075 
.85102 
.86180 
.85157 
.85184 
.85211 



.85820 
.86847 
.85875 
.85402 
.85429 
.85456 
,85484 
85511 
.85588 



.35619 
.85647 
.85674 
.85701 
.85728 
.85755 



85810 



93959 
93949 



93919 
93909 
93899 



93879 
,98869 



93839 



,98819 
.98809 
,98799 
,93789 
.93779 



98759 
98748 
,98788 
98728 
98718 
,98708 
,98698 



.98677 
.98667 

.9865? 
.98647 
.98637 
.98626 
.98616 
.98606 
93596 
.93685 
.93575 
.98565 



.98555 
.98544 



98514 



98493 
98483 
98472 
98462 



I 

984411 
98481 
98420 
98410! 
,98400 
98889 
,98879 



Cosin 



Sine 



Sine Cosin 
T40674 .91355 

.40700 .91848 59 

.40727 .91881 58 

.40753 .91819 67 

.40780 .91807 56 

.40608 .91295 55 

.40838!. 91263 54 

.40860!. 91272 58 




49° 



TABLE IV.— NATURAL SINES AND COSINES. 



25° 



.42315 
.4234i 
.42367 
.42394 
.42420 



42446 
.42473 
.42499 



.42562 
.42578 
.42604 
.42631 
.42657 



.42709 
.42736 
.42762 
.42786 

.42815 
.42841 
.42867 
.42894 
.42920 
.42946 
42972 



.43025 
.43051 

.48077 
.48104 
.43180 
.48156 
.48182 
.43209 



.48261 
.43287 
.48818 

.48840 
.48366 
.48392 
.48418 
.43445 
.48471 
.48497 



.43549 
.48575 

.48602 



.48654 
.43680 
.48706 
.48783 
.48759 
.43785 
.48811 
.43887 



Costa 

.90631 
90618 
90606 

.90594 



,90569 
90557 
90545 



.90520 
.90507 

90495 



90470 
90458 
90446 
90433 
90421 
90408 
90896 



90871 
90858 
90346 
90334 
90321 
90309 
90298 
90284 
90271 
90259 

90246 



90221 
90208 
90198 
.90183 
.90171 
.90158 
.90146 
.9Q133 

.90120 
.90108 
.90095 
90082 
.90070 
.90057 
.90045 
.90032 
.90019 
.90007 

.89994 
.89981 



.89956 



.89918 
.89905 



.89879 



Coain Sine 
64" 



26 


Sine 


OuAiii 


.43NfT 


. 8uht9 ; 


.4MB 


.syrtc7 


43*80 


89854 i 


439 us 


B8641 


.4994* 


.KIS2S 


.4#*">H 


.WK10 


4-W1 


HtiHil I 


,44i»> 


.wirw 


.4-UUfi 


.89777 


.4407;! 


.H07IU 


.40m 


,80901 


,441'J-t 


.89731) 


.44151 


. 89726 


.4*177 


.H9713 


.44408 


.89700 


.4 i-.-J-' 


,89087 


.4 ;,'-''■ 


.89674 


.44281 


.83662' 


.44807 


.89649 


.44333 


.89636 


.44359 


.89623' 


.4MB 


.89610 


.44411 


H09T 


.44487 


.89584 


.44401 


.89571 , 


.444W 


,89558' 


.44516 


80644 


.44542 


,89632 


.445W* 


s'.r.r.i 


A ;.-■ • s 


nj*:hmi 


.44422U 


.89493 


.4rll54G 


80480 


.44672 


.89467 


.44496 


,8U451 


.44724 


.89441 


.44750 


.89428 


.44776 


80415 


.44801 


.89402 


.44088 


, K1WN9 


A,-* 


89376 


.44880 


80363 


.44006 


,80350 ! 


.4493? 


.89387 


.44068 


.81*324 ' 


MW4 


.89811 


.45910 


8929H 


.45036 


.86985 


.4 ■■•"■ : 


. H'^JTU 1 


.45088 


<:#:& 


.4 !li 


B9945 


.45140 


,89889 


.45166 


.89219 


.45192 


,80908 


.4018 


.WI93 


.45243 


s'.ijm 


.45269 


.89107 


.4521*5 


,89158 


.45321 


,89140 


.42847 


.K9 I 27 


.45373 


HUM I 


.48399 


B8101 


C* '"ML 


Sine 1 


ft 


s- 



87* 



.45310 
.45125 
.45151 
.15177 

I.V.* 

!9 

. 4 

. 4 '■'HO 

.451V 16 

.406*2 

.45058 

.45084 
.4,-10 

l.'.T-* 

.457H2 

1.-7H7 

i >l8 

19 

15 

.45891 

.45017 

.45942 
,45908 
.45994 

K 20 
.413016 
.40072 
.48097 

.46123 
.46149 
.46175 

.46201 
.46226 
>2 
,46278 
.48894 
.46330 
.46335 
.40381 
,46407 
.46i:» 

.46458 

,46484 

.46510 

16 

■i' '-H 
,413587 
. -i ■■ 8 
.48889 
,46664 
.46600 

.46716 
46742 
.46767 
.46798 

1-.-I.9 
.40*44 
.46870 
.46896 
,46921 

16917 
■ '.--,La 



Costa 

;89101 
.89087 
.89074 
.89061 
.89048 



.89021 
.89008 
.88995 
.88981 



.88955 

.88942 



.88915 
.88902 



.68875 
.88862 
.88848 



.88795 
.88782 
.88768 
.88755 
.88741 



.88715 
.88701 



.88674 
.88661 
.88647 
.88634 
.88620 
.88607 
.88593 
.88580 
.88566 



.88512 
.88499 
.88485 
.88472 
.88458 
,88445 
.88481 

,88417 
.88404 



.88377 



.88349 



Sine 



68* 



28* 



Sine 
.46947 
.46973 
.46999 
.47024 
.47050 
.47076 
.47101 
.47127 
.47153 
.47178 
.47204 

.47229 
.47255 
.47281 
.47806 
.47882 
.47858 
.47888 
.47409 
.47484 
.47460 

.47486 
.47511 
.47587 
.47562 
.47588 
.47614 
.47689 
.47665 
.47690 
.47716 

.47741 
.47767 
.47798 
.47818 
.47844 
.47869 
.47895 
.47920 
.47946 
.47971 

.47997 



.48048 
.48073 
.48099 
.48124 
.48150 
.48175 
.48201 



.48877 



.48854 

.48879 
.48405 
.48480 
.48456 
.48481 



Costa 



Cosin 
.W293 

.8^*1 

.8 r 

.8 L 
.8 ) 
.8 I 
.8*211 

.8 ■■) 

.8*1 72 
.8*158 

.8S144 
.8si:n) 
.8*117 
.fr*lii:l 
.8 > 
.8*175 
.8*>62 
.88048 
.8 L 
.88020 

.86006 
.8 J 
.87979 
.8 S 
.8 L 

.8 r 

.8 J 

.8 ) 

.8 J 

.8 -I 

.89888 

.8 I 

.87*40 

.8 J 
.87M* 
.8779* 
.877*1 
.87770 
.87786 
.87743 

.87989 
.8 i 
.87701 
.87687 
.87TO 
.8 ) 
.8 S 
.8 I 
.87617 
.87wrl 

.875*1) 
.8 > 
.8*881 
.8 1 
.875,12 
.8 \ 
.87M*i 
.871l«) 
.87476 

•A *' 
SiniT 



61° 



89° 



Sine 
.48481 
.48506 



.48557 



.48608 
.48634 
.48659 
.48684 
.48710 
.48785 

.48761 
.48786 
.48811 
.48887 
.48802 



.48913 
.48988 
.48964 



.49014 
.49040 
.49065 



.49090 
.49116 
.49141 
.49166 
.49192 
.49217 
.49242 



49298 
!49318 
.49844 



.49894 
.49419 



.49470 
.49495 

.49821 
.49546 
.49571 
.49596 



Costa 
.87462 
.87448 
.87434 
.87420 
.87406 
.87391 
.87377 
.87363 
.87349 
.87835 
.87321 

.87806 
.87292 
.87278 
.87264 
.87250 
.87235 
.87221 
.87207 
.87198 
.87178 

.87164 
.87150 
.87186 
.87121 
.87107 
.87098 
.87079 
.87064 
.87050 
.87086 

.87021 
.87007 
.86998 
.86978 
.86964 
.86949 
.86985 
.86921 
.86906 



.49647 
.49672 
.49697 
.49728 
.49748 

.49773 
.49798 
.49824 
.49849 
.49874 
.49899 



.49950 
.49975 
.50000 



Costa 



.86878 



.86849 
.86884 



.86805 



.86791 
.86777 
.86762 
.86748 

.86788 
.86719 
.86704 
.86690 
.86675 
.86061 



.86617 
.86608 



Sine 



491 

TABLE IV.— NATURAL SINES AND COSINES, 



o 
l 

2 

8 
4 
6 
6 
7 
8 

10 

11 
18 
18 
14 
15 
16 
17 
18 
19 
90 

21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
85 
86 
87 
88 
89 
40 

41 
42 
48 
44 

45 
46 
47 
48 
49 
50 

51 
52 
58 
54 
55 
56 
57 
58 
50 
60 



80* 



Slne^ 

rsoobo 

.50025 
.50050 
.50076 
.50101 
.50126 
.50151 
.50176 
50201 
.50227 



Coain 
86608 



,50277 
,50802 



.50327 



.50377 
50408 
.50428 
.50458 
.50478 
.50503 



.50553 



50578 
.50603 



.50654 

.50879 
.50704 
.50729 
.50754 

.50779 
.50804 



.86573 



86544 
86530 



.86515 
86501 
86486 

.86471 
86457 

86442 
,86427 
86418 



.50854 

.50879 
.50904 
.50929 
.50954 
.50979 
.51001 

.51029 
.61054 
.51079 
.51104 
.51129 
.61154 
.61179 
.51204 
.51229 
.51254 

.61279 
.51804 
.51829 
.51854 
.51879 
.51404 
.51429 
.51454 
.51479 
.51504 



Cosin 



86869 
86354 
86340 



86310 



,86281 



,86251 



.86207 
.86192 
.86178 
.86163 

.86148 
.86183 
.86119 
.86104 
.86089 
.86074 
.86059 
.86045 
.86080 
.86015 

.80000 



*!• 



.85970 
.85956 
.85941 



.85911 
.85898 
.85881 
.85866 

.85851 



.85806 
.85792 
.85777 
.85762 
.85747 
.85782 
.85717 
Sine 



Sine 

rsisoi 

.51529 
.51554 
.61579 
.51004 



Cosin. 



.51653 
.51678 
.61708 
.51728 
.51753 

.51778 
.51803 
.51828 
.51852 
.51877 
.51902 
.51927 
.61952 
.51977 
.52002 



69° 



.52051 
.52076 
.52101 
.52126 
.52151 
.52175 
.52200 



,52250 



.52349 
.52374 



,52428 



.52478 
.52498 



.85717 
85702 



85672 
85657 
85642 



.85612 
85597 



.52547 
.52572 
.52597 
.52621 
.52646 
.52671 
.52696 
.52720 
.52745 

.52770 
.52794 
.52819 
.52814 



Coain 



,85567 
.85551 



.52808 



.52948 

.52987 



.85521 
.85506 
.85491 
.85476 
.85461 
.85446 
.85431 
.85416 

.85401 
.85385 
.85870 
.85355 
.85340 



.85310 



,85279 
.85264 



.85218 



.85188 
.85173 
.85157 
.85142 
.85127 
.85112 

.85096 
.85081 
.85066 
.85051 
.85035 
.85020 



.85005 
84989 
84974 
84959 

84943 



82° 



,84913 
,84897 



.84866 
.84851 
.84836 



.84805 
Sine 



Sine I Cosin 
.52992 .81805 



88° 



53017 
.53041 

53066 
.53091 



84789 
.84774 
84759 
84748 



.53115 .84728 
53140 .84712 



.53164 
.58189 
.63214 



.53812 
.53837 
.53361 



.53411 
.58485 
.58460 

.58484 

.58509 
.53534 
.53558 



.84697 
84681 



,84650 



.84619 
.84604 
.81588 
.81573 
.81557] 
.84542 
.84526 
.84511 
.84496 

.84480 
.84464 
.84448 
.84438 
.84417 
.84402 
.81886 
.8*370 
.84355 



.53607 
.58632 
.53656 
.58681 
.68705 
.58730 

.58754 .84324 
.53779 .84308 
.58804 .81292 
.53828 .84277 
.84261 



.53877 



68* 



.53951 
.53975 

.54000 
.54024 
.54049 
.54078 
.54097 
.54122 
.54146 
.54171 
.54195 
.54220 

.54244 
.54269 
.54293 
.54817 
.54842 
.54366 
.54891 
.54415 
.54440 
.54464 
Cosin 



.81245 



.84214 
.84198 
.84182 

.84167 
.84151 
.84135 
.84120 
.84104 
.84088 
.84072 
.84057 
.84041 
.84025 

.84009 



.88978 



.88915 



Sine 



Sine Cosin 



.544641.83867 
.54488 .88851 
.54513;. 83835 
.54587 .88819 
.54561 .88801 
.545861.88788 
.54610 .83772 
.54635 .83756 
.54659!. 88740 
.54688 .83724 
.54708 I.837u8 

.54782 1 . 88692 



.54756 
.54781 
.54605 



88670 
.83660 
.83645 



84* 



Sine Cosin 



.83618 
.88597 
.83581 
.88565 

.88549 



.83517 
.83501 



.54854 
.54878 
.54902 
.54927 
.54951 

.54975 
.54990 
.55024 
.55048 
.55072 
.55097 
.55121 
.55145 
.55109 
.55194 

.55218 .88878 
.55242 .88856 
.55266 .83340 
.55291 
.55315 



.88469 
,83453 
,83437 
.88421 
.88405 



.55863 



.88276 



i 

.55412 .83244 
.55436 ,.88228 



,55919 
,55948 
,55968 
,55992 
.56016 
.56040 
.56064 
.56088 
.56112 
.56130 
.50160 

.56184 
.56208 



.56256 



.56805 
.66329 
.56858 
.56877 
.56401 

.56425 
.56449 
.56473 
.56497 
.56521 
.56545 
.56569 
.56598 



.56617 
.56641 

.56665 
.56689 
.56713 
.56786 
.56760 
.56784 
.56808 



.56856 
.56880 



.55460! 
.55484 
.55509 
.56533 
.55557 
.55581 
.55605 
.65680 
.55654 
.55678 

.55702 
.55726 
.55750 
.55775 
.55799 
.55828 
.55847 
.65871 
.55895 
.559J9 
Cosin 



82904 



.88050 
.88034 
.88017 
.83001 



.82904 
Sine 



57- 



I' 



66° 



.83212 ". 56904 
.88195 .56 
.83179 .66952 
.88163; .56976 
.88147! .57000 
.88181' .57024 
.88115 .57047 
.57071 
.57095 
.57119 

.67148 
.57167 
.57191 
.57215 



82790 
82778 
82757 
82741 

82724 
82708 
82692 
,82675 



82648 



82610 



.82446 



.57262 
.67286 
.57810 
.57884 
.57858 
Cosin 



60 
69 
58 
67 
66 
66 
64 
58 
52 
61 
50 

49 

48 
47 
46 
46 
44 
48 
42 
41 
,82577 40 

82561189 
82544 88 
82528, 87 
82511 86 
.82495 85 
.82478,84 
88 
82 
81 
80 

29 



.82418 



82297 



82264 



82214 
82198 
,82181 
,82165 
82148 
,82182 
,82115 



.89066 



.82015 
.81999 



.81965 
.81949 



.81915 
Sine i 



56* 



492 



TABLE IV.— NATURAL SINES AND COSINES. 



35» 



Sine jOoainl 

.81915 



.57S68 
.57881 
.57406 
.57429 
.57458 
.57477 
.57501 
.57524 
.57548 
.57572 
.57598 

.57819 
.57543 
.57667 
.57891 
.57715 
.57788 
.57762 
.57786 
.57810 




1 
2 
8 
4 
6 
6 
7 
8 
9 
10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

21 
22 

28 
24 

25 
26 

27 
28 
29 
80 

81 .58094 
.58118 
88 I .58141 

84 .58165 

85 i .58189: 

86 : .58212' 

87 .58236 
88 
89 
40 1.56307 



.57857 
.57881 
.57904 
.57928 
.57952 
.57976 
.57999 



.58047 
.58070 



81899 
.81882 
.81865 
.81848 



41 

42 .58854' 

.58878! 

.58401 

45 .58425 

46 .58449 

47 .58472 

48 .58496 

49 . .58519 

50 .58548 



60 



.58567 
.58590 
.58614 
.58687 

55 .58661 

56 .58664 
67 .58708 
58 .58781 

.58755 
.58779 
Cosin 



.81815 
.81798 
.81782 
.81765 
.81748 

.81781 
.81714 
.81698 
.61681 
.81664 
.81647 
.81681 
.81614 
.81597 
.81560 

.81568 
.81546 
.81580 
.81518 
.81496 
.81479 
.81463 
.81445 
.81428 
.81412 

.81896 
.81878 
.81361 
.81844 
.81827 
.81810 
81293 
.81276 
.81259 
.81242 

.81225 
.81208 
.81191 
.81174 
.81157 
.81140 
81128 
.81106 
.81089 
.81072 

.81055 
.81088 
.81021 
.81004 
.80987 
.80970 
.80953 



.80919 
.80902 
Sine 



64* 



86* 



JMn?- 

.58779 
.58802 
.58826 
.58849 



.58896 



.58948 
.58967 
.58990 
.59014 

.59087 
.59061 
.59084 
.59108 
.59181 
.59154 
.59178 
.59201 
.59225 
.59248 



.80902 
.808851 
.80867 
.80850, 
.80833! 
.80816 
.80799 1 

.8om 

.80765 
.80748 
.80730 

.80718 
.80696 
.80679 



.59272 



.59818 
.59842 
.59365 



.59412 
.59436 
.59459 
.59482 

.59506 
.59529 



.59552 
.59576 
.59599 
.59622 
.59646 
.59669 



,80644 
80627 
,80610 
80593 
,80576 
.80558 

80541 ! 
.80524! 
,805071 
,80489 ' 
,80472 
,80455 
,80488 
804201 
,80408 



,80868 
,80851 
80834 
.80316 



.802641 
, 80247 i 



.59716!. 80212; 



.59789 
.59763 
.59786 
.59809 



.59856 
.59679 
.59902 
.59926 
.59949 

.59972 
.59995 
.60019 
.60042 
.60085 
.60069 
.60112 
.60135 
.60158 
.60182 
Cosin 



.80195 
.80178 
.80160, 
.80143 
.801251 
.80108' 
.60091! 
.80073; 
.800561 
.80038 

80021 ! 

80008 
.79966 
.79968 
.79951! 
.79934 
.79916 
.79899 
.79881 
.79864 
Sine I 



W 



Sine 

.60182 
.60205 
.60228 
.60251 
.60274 
.60298 
.60321 
.60344 
.60367 
.60390 
.60414 

.60487 
.60460 
.60488 
.60506 
.60529 
.60558 
.60576 
.60599 
.60622 
.60645 

.60668 
.60691 
.60714 
.60738 
.60761 
.60794 
.60807 
.60630 
60858 
.60676 

.60699 
.60922 
.60945 
.60968 
.60991 
.61015 
.61088 
.61081 
.61084 
.61107 

.61180 
.61158 
.61176 
.61199 
.61222 
.61245 
.61268 
.61291 
.61314 
.61887 



Cosin 



,79494 
.79477 
.79459 
.79441 
.79424 
.79406 
.79888 
79871 
,79853 
79885 

,79818 
.79800 



.79083 
.79016 
76998 
78980 

78962 
76944 
78926 
78906 
78891 



78878 
78855 
.78887 
.78819 
.78801 
Sine 



88° 



Slne^ 
.61566 
.61569 
.61612 
.61635 
.61658 
.61681 
.61704 
.61726 
.61749 
.617TO 
.61795 



.62251 



.62842 



.62615 



.62706 



Cosin 



.78604 
.78586 
.78568 
.78550 
.78582 
.78614 
.78496 
.78478 
.78460 
.78442 



.73869 
.78851 
.76888 

.78315 
.78297 
.76279 
.76261 

.78248 



.78225 
.78206 
.76188 
.78170 
.78152 
.78184 
.78116 
.76098 
,76079 



.76061 
.78043 
.78025 
.76007 
.77988 
.77970 
.77952 
.77984 
.77916 
.77897 

.77879 
.77861 
.77848 
.77824) 
.778061 
.77788 1 
. 7776 9 
. 7775 1 
.77783 
.77715 
Sine 



•!• 



Sine Cosin 



.62955 
.62977 
.63000 



.68045 
.68068 

.68090 



.63118 
.68185 
.63168 

.68180 
.68208 



.68948 
.68271 



.68816 



.68861 



.68406 



.68451 
.68478 
.68496 
.68518 
.68540 
.68568 



.68680 



.68675 
.68696 

.68720 
.68742 
.68766 
.68787 
.68810 



.68854 

.68877 
.68899 



.68944 
.68966 



.64011 
.64088 
.64056 

.64078 
.64100 
.64128 
.64146 
.64167 
.64190 
.64212 



.64256 
.64279 



Cosin 



. 77715 
. 77896 
.77679 
.77660 
.77641 
.77628 
77606 
.77586 
.77568 
. 77560 
.77581 
.77518 
.77494 
.77478 
.77456 
.77489 
.77421 
.77402 
.77884 



.77847 

.77829 
.77810 
.77282 
.77278 
.77266 
.77286 
.77218 
.77199 
.77181 
.77162 

.77144 
.77126 
.77107 
.77086 
.77070 
.77051 
.77088 
.77014 
.76896 
.76977 

.78969 
.76940 
.76921 



.76864 
.76666 
.76647 
.76826 
.76810 
.76791 

.78772 
.78754 
.76786 
.76717 
.76698 
.76679 
.76661 
.76642 
.78628 
.76604 



Sine 



50* 



60 
69 
58 
57 
56 
56 
64 
68 
68 
51 
60 

49 
48 
47 
46 
45 
44 
43 
48 
41 
40 

89 
88 
87 
86 
86 
84 
88 
88 
81 
80 

29 
26 
27 
26 
26 
24 
28 
22 
21 
20 

19 

18 
17 
16 
15 
14 
18 
12 
11 
10 

9 

8 
7 
6 
5 

4 
3 
2 

1 




493 

TABLE IV.— NATURAL SINES AND COSINES. 




1 
2 
8 

4 
5 
6 
7 
8 

10 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

21 
22 
28 
24 
25 
26 
27 
28 
29 
80 

81 
82 
88 
84 
85 
86 
87 
88 
89 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 

51 
52 
58 
54 
55 
56 
57 
58 
59 
60 



Cosin 
.74314 
.74205 
.74276 
.74256 
.74237 
.74217 
.74198 
.74178 
.74159 
.74189 
.74120 




48* 



Sine 
T68200 
.68221 
.68242 
.68204 
.68285 
.68806 
.68327 
.68349 
.68870 
.68391 
.68412 

.68484 
.68455 


Cosin 
.78135 
.78116 
.78096 
.78076 
.78056 
.78036 
.73016 
.72996 
.72976 
.72957 
.72987 

.72917 
.72807 


.68476 


.72877 


.68497 


.72857 


.68518 
.68589 
.68561 
.68582 
.68003 
.68624 

.68645 
.68666 


.72687 
.72817 
.72797 
.72777 
.72757 
.72787 

.72717 
.72807 


.68688 
.68709 
.68730 
.68751 
.68772 
.68793 
.68814 
.68835 

.68857 
.68878 
.68899 
.68920 
.68941 
.68962 
.68983 
.69004 
.69025 
.69046 

.69067 


.72677 
.72857' 
.72687! 
.72617, 
.72597. 
.72577 
.72557. 
.72587i 

.72517 
.72497 
.72477 
.72457 
.72487 
.72417 
.72397 
.72877 
.72857 
.72887 

.72817 


.69088 


.72297 


.60109 
.69130 
.69151 
.69172 
.69193 
.69214 
.69285 
.69256 

.69277 
.69298 
.69319 
.69340 
.69361 
.69382 
.69403 
.60424 
.69445 
.69466 


.72277 
.72257 
.72236 
.72216 
.72198 
.72176 
.72156 
.72186 

.72116 
.72095 
.72075 
.72055 
.72035 
.72015 
.71995 
.71974 
.71954 
.71934 


Cosin 


Sine 


44 


5* 



44* 



Sine ! Cosin 
.69466 771934 



69487 
.69508 
69529 



.69549 
.09570 
.60591 
.69612 



.69654 
.69675 



.69717 
69TC7 



71914 
71894 
.71878 
.71858 
.71838 
.71818 
.71792 
.71772 
.71752 
.71782 

.71711 
.71691 
.71671 
.71650 
.71680 
.71610 



.69004 
.60925 
.60946 



.69758 
.69779 

.69800 

698211.71590 
69842 .71569 
.71549 
.71529 

.71508 
.71488 
.71468 
.71447 
.71427 
.71407 
.71886 
.71866 
.71845 
.71825 

.71805 
.71284 
.71264 
.71248 
.71228 
.71208 
.71182 
.71162 
.71141 
.71121 

.71100 
.71080 
.71059 
.71039 
.71019 
.70998 



.69987 
.7W08 
.70029 
.70049 
.70070 
.70091 

.70112 
70132 
.70153 
.70174 
.70195 
.70215 



.70257 



.70277 
.70298 

70819 
70639 
70360 
70381 
,70401 
70422 
70443 
70463 
70484 
.70505 

70525 
70546 
,70567 
,70587 
70608 
.70628 
.70649 
.70670 
.70690 
.70711 
Oosin 



70978 
.70957 
.70987 
.70916 

.70896 
.70875 
.70855 
.70684 
.70618 
.70798 



.70772 
.70752 
.70781 
. 70711 
Sine 



45* 



60 
50 
58 
57 
56 
55 
54 
58 
52 
51 
50 

49 
48 
47 
46 
45 
44 
48 
42 
41 
40 

89 
88 
87 
86 
85 
84 
88 
82 
81 
80 

29 
28 
27 
26 
25 
24 
28 
22 
21 
20 

19 
18 
17 
16 
15 
14 
18 
12 
11 
10 

9 
8 
7 
6 
5 
4 
8 
2 
1 




494 



TABLE V.— NATURAL TANGENTS AND COTANGENTS. 





0* 


*• 


I *• 


8» 




Tang 

.00000 


Ootang 


Tang 

.01746 


Cotang 


Tang 


Cotang 


Tang 1 


Cotang 




Infinite. 


67.2900, 


.03492 


28.6868 


.05241 


19.0811 


60 


1 


.00029 


8487.75 


.01775 


56.8506 


.03521 


28.8994 


.05270 


18.9755 59 


8 


.00058 


1718.87 


.01804 


55.4415 


.03550 


28.1664 


.05299 


18.8711 56 


8 


.00087 


1145.92 


.01888 


54.5618 


.03579 


27.9372 


.05328 


18.7678 57 


4 


.00116 


859.486 


.01862 


53.7086 


.03609 


27.7117 


.05357 


18.6656 156 


6 


.00145 


687.549 


.01891 


52.8821. 


.03638 


27.4899 


.05387 


18.5645 55 


6 


.00175 


672.957 


.01920 


52.0807 


.03667 


27.2715 


.05416 


18.4645 54 


7 


.00204 


491.108 


.01949 


61.8082 


.03696 


27.0566 


.05445 


18.8655 58 


8 


.00233 


429.718 


.01978 


50.5485 


.08725 


26.8450 


.05474 


18.2677 


52 


9 


.00262 


881.971 


.02007 


49.8167 


.03754 


26.6367 


.05508 


18.1708 


51 


10 


.00291 


848.774 


.02086 


49.1089 


.03788 


26.4316 


.05588 


18.0750 


50 


11 


.00820 


812.521 


.02066 


48.4121 


.08812 


26.2296 


.05562 


17.9802 


49 


12 


.00849 


286.478 


.02095 


47.7895 


.08842 


26.0307 


.05591 


17.8868 


48 


IS 


.00878 


264.441 


.02124 


47.0858 


.08871 


95.8348 


.05620 


17.7984 


47 


14 


.00407 


245.552 


.02158 


46.4489 


.03900 


25.6418 


.05649 


17.7015 


46 


15 


.00436 


229.183 


.02182 


45.8294 


.03929 


25.4517 


.05678 


17.6108 


45 


16 


.00465 


214.858 


.02211 


45.2261 


.03958 


25.2644 


.05708 


17.5905 


44