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REPAIRED BY
WISCONSIN
W P A
BOOK REPAIR
PROJECT NO.
.11.^1:
Date..|^^^.ii
%■ A*
IN PREPARATION
ADVANCED SURVEYING
C. B. BREED AND G. L. HOSMER
THE
PRINCIPLES AND PRACTICE
OF
SURVEYING
BY
CHARLES B. BREED and GEORGE L. HOSMER
N
INSTRUCTORS IN CIVIL ENGINEERING, MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
First Edition
NEW YORK
JOHN WILEY & SONS
London : CHAPMAN & HALL, Limited
1906
Copyright, 1906
BY
CHARLES B. BREED and GEORGE L. HOSMER
Stanbope prcM
r. H. OILSON COMPANT
BOSTON. U. •. A.
105155
APR 2 31907
SUB
PREFACE
In the preparation of this volume, it has been the authors'
chief purpose to produce a text-book which shall include the
essentials of a comprehensive knowledge of practical surveying
and at the same time be adapted to the use of teachers and
students in technical schools. In this book, which is essentially
an elementary treatise, such subjects as stadia, plane table,
hydrographic and geodetic surveying, are entirely omitted, these
sabjects being left for a later volume.
Considerable stress is laid upon the practical side of sur-
veying. The attempt is made not only to give the student a
thorough training in the fundamental principles and in approved
methods of surveying, computing, and plotting, but also to
impress upon him the importance of accuracy and precision in
all of his work in the field and the drafting-room. In carrying
out this purpose it has seemed necessary to lay particular stress
upon some points which to the experienced engineer or the
advanced student may appear too obvious to require explanation,
but which teaching experience has shown to be most helpful to
the beginner. The most common errors and mistakes have
therefore been pointed out and numerous methods of checking
have been explained. Every effort has been made to inculcate
right methods even in minor details, and for this purpose a large
number of examples from actual practice have been introduced.
In arranging the subject matter of the work, the four parts
are presented in what appears to be a logical sequence. First,
the use, adjustment, and care of instruments are taken up ; then
the next three parts, surveying methods, computations, and
plotting, are taken in the order in which they are met in the
daily practice of the surveyor. To show more clearly the steps
in the process, the notes which are used as illustrations in sur-
veying methods are calculated in the computation section, and
iv PREFACE
are treated again under the methods of plotting, finally appearing
as a completed plan.
While the authors recognize fully their indebtedness to those
who have preceded them in this field, they hope that they have
made some useful contributions of their own to the treatment
of the subject. Thus in the section on Surveying Methods,
many practical suggestions have been inserted which they have
found of value in their own work and which, so far as they are
aware, now appear in a text-book for the first time. On the sub-
ject of Computations, much emphasis is laid upon the proper use
of significant figures and the arrangement of the work, matters
which heretofore have not been adequately treated in books on
surveying. The section on Plotting contains many hints refer-
ring particularly to surveying drafting, which are not given in the
published books on drawing and lettering. It is hoped also that
the complete set of original illustrations which have been in-
troduced throughout the book will aid materially in making the
text clear. ^
A comprehensive cross-reference system giving the page as
well as the article number has been adopted : this, together with
the complete index at the end of the book and the many practi-
cal hints throughout the volume will, it is hoped, render it tise-
ful to the practical surveyor as a reference book.
The authors desire to acknowledge their indebtedness to their
various associates in the teaching and engineering professions
who have kindly responded to requests for information and as-
sisted in the preparation of this work, particularly to Blamey
Stevens, M. Sc, of Ellamar, Alaska, who supplied the entire
chapter on Mining Surveying, They are also under obligations
for the use of electrotype plates of tables : to W. H. Searles for
Tables IV, V, and VI ; to Professor J. C. Nagle for Tables II
and III ; and to Professor Daniel Carhart for Table I ; all of these
plates were furnished by John Wiley & Sons. The authors are
under special obligation to Professors C. F. Allen, A. G. Rob-
bins, and C. W. Doten of the Massachusetts Institute of Tech-
nology, and to H. K. Barrows, Engineer U. S. Geological Sur-
vey, who have read the entire manuscript and who have offered
many valuable suggestions in preparing the work for the press.
PREFACE V
The authors also desire to express their appreciation of the ex-
cellent work of W. L. Vennard, who made the drawings for illus-
trations.
No pains has been spared to eliminate all errors, but the
authors cannot hope that their efforts in this line have been com-
pletely successful, and they wiirconsider it a favor if their atten-
tion is called to any which may be found.
Boston, Mass., September, 1906. C. B. B.
G. L. H.
CONTENTS.
PART I.
USE, ADJUSTMENT, AND CARE OF INSTRUMENTS.
Chapter I. — General Definitions. — Measurement of Lines.
Art. Paob
1. Definition 3
2. Purposes op Surveys 3
3. Horizontal Lines 3
instruments for measuring lines.
4. The Chain 3
5 Metric Chain 5
6. The Tape 5
7. Steel Tapes 5
8. The Stadia 6
9. Other Instruments; — Wooden Rods, Two-Foot Rule, Odome-
ter 6
MEASUREMENT OF LINES.
^ 10. Measurement of a Horizontal Line with a Chain 7
> II. Measurement of a Horizontal Line wtth a Cblain on Slop-
ing Ground 9
^ 12. Measurebcent with a Steel Tape 10
13. Common Sources of Error in Measurement of Lines .... 11
14. Common Mistakes in Reading and Recording Measurements ii
15. Avoiding Mistakes 11
16. Accuracy Required 12
17. Amount of Different Errors 12
18. Pull 12
19. Temperature 13
20. AuGNMENT 13
21. Sag 13
22. Effect of Wearing on Length of the Chain 14
23. Accuracy of Measurements Z4
Problems 15
Chapter II. — Measurement of Direchon.
24. The Surveyor's Compass 16
25. The Pocket Compass 18
96. Method of Taking a Magnetic Bearing x8
vii
vm CONTENTS
Akt. Paob
27. The Earth's Magnetism, — Dip op the Needle 19
28. Decunation of the Needle 19
29. Variations in Declination ao
3a IsoGONic Chart 25
31. Observations for Declination 25
ADJUSTMENTS OF THS COMPASS.
33. Adjustment of the Bubbles 25
34. Detecting Errors in Adjustment of the Needle 25
35. To Straighten the Compass-Needle 26
36. To Center the Pivot-point 27
37. To Remagnetize the Needle 28
38. Common Sources of Error in Compass Work 28
39. Common Mistakes 28
40. Detecting Local Attractions op the Needle 29
41. Calculating Angles from Bearings 30
Problems 30
Chapter III. — Measurement of Angles.
THE transit.
42. General Description of the Transit 31
45. The Telescope 34
46. The Objective 35
47. Cross-Hairs 36
48. Eyepiece 37
49. Magnifying Power 37
51. Field of View 38
52. The Vernier 38
53. Verniers Used on Transits • 39
54. Eccentricity 44
use of the transit.
56. Setting up the Transit 45
58. To Measure a Horizontal Angle 47
59. To Measure an Angle by Repetition 48
61. To Lay Off an Angle by Repetition 50
62. Running a Straight Line — One Point Visible from the Other 51
63. Neither Point Visible from the Other 51
64. Prolonging a Straight Line 52
65. Methods of Showing Sights 52
66. Signals 53
67. To Measure a Vertical Angle 54
68. Precautions in the Use op the Transit 55
adjustments of the transit.
70. Adjustment of the Plate Bubbles 56
71. Adjustment of the Cross-Hairs 57
CONTENTS ix
AsT. Pacb
73. Adjustment op the Standards 59
74. Adjustment of the Telescope Bubble 59
75. Adjustment op the Auxiliary Level on the Vernier op the
Vertical Arc 59
76. Adjustment of the Vernier of the Vertical Circle .... 60
77. Adjustment of the Objective Slide 60
78. Shop Adjustments 61
79. How TO Eliminate the Effect of Errors of Adjustment in
THE Transit 61
80. Care op Instruments 61
81. Common Sources of Error in Transit Wore 63
82. Common Mistakes in Transit Work 62
THE SOLAR ATTACHMENT.
83. Description of Solar Attachment 65
84. The Celestial Sphere 65
85. Observation on the Sun for Meridian with Solar Attachment 66
86. Computation op Declination Settings 68
87. Comstock's Method op Finding the Refraction 69
89. Mistakes in Using the Solar Attachment 70
ADJUSTMENTS OF THE SOLAR ATTACHMENT.
90. Adjustment of the Polar Axis 70
91. Adjustment of the Cross-Hairs 70
92. Adjustment of Telescope Bubble 71
Problems 71
Chapter IV. — Measurement of Differences of Elevation.
93. Level Surface 72
94. The Spirit Level 72
95. Angular Value of One Division of the Level Tube .... 73
THE LEVEL.
97. The Wye Level 73
98. The Dumpy Level 74
99. Comparison of Wye and Dumpy Levels 77
100. The Locke Hand Level 77
LEVELING rods.
102. Boston Rod 78
103. New York Rod 80
104. Philadelphia Rod 81
105. Special Self-Reading Rods 81
106. Tape Rod 81
107. Precise Level Rod 82
108. Advantages op the Self-Reading Rod 82
109. Attachments to the Rod for Plumbing 83
no. Effect of Heat and Moisture 83
X CONTENTS
USE OF THE LEVEL AND ROD.
AsT. Faob
112. To Level the Instrument ... 83
113. To Take a Rod-Reading 84
114. Waving the Rod 84
115. Signals 85
116. Differential Leveling 85
117. The Proper Length of Sight 87
118. Effect of the Earth's Curvature and of Refraction on
Leveling 87
119. Precautions in Level Work 88
adjustments of the level.
I. adjustments of the wye level.
120. Adjustment of the Cross-Hair 88
122. Adjustment of the Level Tube 89
123. Adjustment of the Level Tube by Indirect Method ... 89
125. Adjustment of the Wyes 90
II. adjustments of the dumpy level.
126. Adjustment of the Cross-Hairs 91
127. Adjustment of the Bubble Tube 91
128. The Direct or "Peg" Adjustment 91
129. Adjustment of the Locke Hand Level 93
130. Common Sources of Error in Leveling 94
131. Common Mistakes in Leveling 94
Problems 94
PART II.
SURVEYING METHODS.
Chapter V. — Land Surveying.
132. Surveying for Area 99
133. Surveying for Area with Compass and Chain 99
SURVEY OF FIELD WITH TRANSIT AND TAPE.
134. Survey of a Field by a Traverse loi
137. Irregular Curved Boundaries 105
138. Survey of a Field with a Single Set-up of the Transit ... 105
139. Survey of a Field with a Tape only 106
140. Selecting the Corners 106
141. Method of Procedure 107
142. Ties 107
143. Measurement of the Angles of the Traverse 108
144. Measurement of Azimxtth Angles 108
145. Checking the Fieldwork 109
CONTENTS xi
AiT. Page
146. Accuracy Required m
147. Organization op Transit Party m
148. Note-keeping 112
149. Survey of a Field for a Deed 114
150. Deed Description ^ 115
151. JuDiCTAL Functions of the Surveyor 116
152. Rerunning Old Surveys from a Deed 118
153. How TO Look up a Recorded Deed 120
THE UNITED STAIBS SYSTEM OF SURVEYING THE PUBLIC LANDS.
154. The System 121
155. Initial Points .• 124
156. Base Line 124
157. Principal Meridian 126
158. Standard Parallels 126
159. Guide Meridians 126
160. Township Exteriors 127
161. Method of Subdividing 128
162. Meandering 134
163. Summary of Objects and Data Intersected by the Line or
IN Its Vicinity, to be Noted 139
164. Prescribed Limits for Closings and Lengths of Lines ... 141
165. Field Notes 142
166. Marking the Corners 147
167. To Establish a Parallel of Latitude 148
168. The Secant Method 148
169. Tangent Method 150
170. Convergence of the Meridians 154
Chapter VI. — Traverse Lines — Location of Buildings —
Miscellaneous Surveying Problems,
traverse lines.
171. Traverses which do not Form Closed Figures 156
172. Method of Checking Traverses which do Not form
Closed Figures, — Checking by Astronomical Method . 156
173. Checking by Cut-Off Lines 157
174. Checking by Angles to a Distant Object 158
175. Checking by Connecting with Tri angulation Points .... 158
location of buildings from transit line.
176. Methods of Locating Buildings 159
177. Geometric Principles 159
178. Ties, Offsets, Swing Offsets, and Range Lines 159
179. General Suggestions 160
180. Typical Cases 160
181. Building near Transit Line and Nearly Parallel to it. . . 160
Plotting 161
182. Building near Transit Line, Making a Slight Angle with
IT 161
Plotting 162
xii CONTENTS
An. Page
183. Building Located Entirely by Direct Ties 162
Plotting 163
184. Building Located at a Considerable Skew to the Transit Line 163
Plotting 164
185. Buildings at a Long Distance from the Transit Line ... 164
Plotting 164
186. Buildings Located from Other Buildings 165
187. Buildings of Irregular Shape 166
188. Large City Buildings 166
189. Location of Buildings by Angles and Distances 167
190. Location of Buildings and Fences from Transit Line ... 167
miscellaneous surveying problems.
191. Random Line 169
192. Obstacles on Line 170
193. Offsetting Transit Line 170
194. Short Transit Sights 172
195. Bisection Method 172
196. Measuring Around a Small Obstacle 173
197. Equilateral Triangle Method 174
198. Inaccessible Distances , . . , 174
199. By Right Triangle Method 174
200. Intersecting Transit Lines 175
201. By Swing Offset Method 175
202. By Tangent Offset Method 176
203. By Oblique Triangle Method 177
204. To Obtain the Distance Between Two Inaccessible Points
BY Observation from Two Accessible Points 178
205. To Obtain the Inaccessible Distance Between Two Accessible
Points by Observation on Two Inaccessible Points of
Known Distance Apart 179
Chapter VII. — Observation for Meridian and Latitude.
observations for meridian.
206. To Establish a True Meridian Line by Observation on Polaris
WITH THE Transit 180
207. Observation for Meridian on Polaris at Elongation . . . 182
208. Observation for Meridian on Polaris at Culmination ... 185
209. To Find the Standard Time of Culmination and Elongation. 186
210. Meridian Observations on Polaris with the Compass ... 187
211. Meridian Observation on Polaris AT ANY Time WITH THE Transit 188
212. Solar Observations 190
213. Observation for Meridian by Equal Altitudes of the Sun in
THE Forenoon and Afternoon 190
214. Observation for Meridian by a Single Altitude of the Sun 193
215. Observation for Meridian by Means of the Solar Attachment 195
CONTENTS xiii
OBSERVATIONS FOR LATITUDE.
Akt. Page
216. By the Altitude of Polaris at Upper or Lower Culmination 196
217. By the Altitude of the Sun at Noon 196
Problems 197
Chapter VIII. — Leveling.
218. Definitions 198
219. Leveling to Establish Bench Marks 198
222. Double Rodded Lines 201
224. Bench Marks and Turning Points 202
225. Leveling for Profile 203
227. Cross-Sectioning 206
228. Use of the Tape Rod in Cross-Section Work 206
229. Cross-Sectioning for Earthwork 207
230. Road Cross-Sections 207
231. Setting Slope Stakes 208
232. Earthwork Notes for Road Cross-Sections 209
233. Cross-Sections for Borrow-Pits 210
234. Cross-Sections for Trench Excavation 210
235. Leveling to Establish a Grade Line 210
236. "Shooting in" a Grade Line 211
237. To Establish a Datum Plane by Means of Tidal Observations 211
238. The Staff Gauge 212
239. Leveling Across a River 213
Problems 215
Chapter IX. — City Surveying
240. Instruments Used 216
241. Tapes and Tape Measurements 216
242. Transits and Levels 217
243. City Standard 218
CITY LAYOUTS.
245. Streets 219
246. Location op Streets 220
247. Size of Blocks and Lots '. 222
248. Width of Streets 222
249. Street Grades 224
250. The Datum Plane 226
251. Establishing Bench Marks 226
252. Water and Sewer Systems 227
staking out aTY WORK.
253. Staking out a New District 227
254. Monuments 228
255. Setting Stone Bounds 230
xiv CONTENTS
Art. Pack
256. Curved Layouts 232
257. Elements of a Circular Curve 233
258. Staking Out Circular Curves 234
259. Deflection Angles 234
260. Keeping the Notes 237
261. When the Entire Curve Cannot be Laid Out from One End 238
262. First Method 238
263. Second Method 239
264. Curved Street Corners 239
265. One Street Line Straight, the Other Curved 239
266. Both Street Lines Curved 240
367. Staking Out Street Grades 241
268. Vertical Curves 242
269. Cross-Section of Street 243
270. Gutters at Same Elevation 244
271. One Gutter Higher than the Other 244
273. Irregular Shaped Blocks 247
274. Staking out City Lots 248
275. Staking Out Curb Lines and Grades 249
276. Staking Out Sewers 251
277. Staking Out Street Railway Tracks 251
278. Rerunning Street Lines and Grades 251
279. Revising Street Lines 252
280. Revising Street Grades 252
281. Setting Batter-Boards for a Building 253
282. City Plans and Records 255
RECTANGULAR COORDINATE SYSTEM OF SURVEYING CITIES.
283. General Description 255
284. Triangulation Scheme 256
285. Measurement of Base-Line 261
286. Measurement of Angles 261
287. Adjustment of the Angles 262
288. Azimuth 262
289. Secondary and Tertiary Triangulation 263
290. Traverses 263
291. Method of Locating Property Lines and Buildings .... 263
Chapter X. — Topographical Surveying.
293. Triangulation for Control 264
294. Location of Points from the Transit Line 265
295. Contour Lines 265
296. Characteristics of Contours 268
297. Rel.\tion Between Contour Map and Profile 270
298. Relation Between Contour Map and Side Elevation or Pro-
jection 271
299. Drainage Areas 273
300. Sketching Contours from Streams and Summits 273
301. Sketching Contours from Known Elevations 276
CONTENTS XV
Abt. Page
302. Mistakes in Sketching Contours 276
303. Locating Contours 277
304. Locating Contours by Cross-Sections 278
305. Locating Contours by Profiles 278
306. Locating Points on Contours 278
307. Locating Contours by Means of the Hand Level .... 278
308. Location of Streams and Shore Lines 280
309. Contour Problems 280
310. Intersection of Plane with Surface of Ground 280
311. Intersection of Curved Surface with Surface of Ground . 281
31 2. Intersection of Side Slopes of Road with Surface of Ground 283
Chapter XI. — Mining Surveying.
313. General Remarks 285
314. Definitions of Mining Terms 285
MINING INSTRUMENTS.
315. Mining Transits 287
316. Side Telescope 287
317. Top Telescope 287
318. Adjustments of Side Tei^^scope 287
319. Adjustment of Top Telescope 290
320. Interchangeable Side and Top Telescope 290
321. Eccentric Bearing Telescope 290
322. Combined Solar Attachment and Top Telescope 292
324. Use of the Ordinary Transit in Mining Surveying .... 292
325. Compasses used in Mines 293
UNDERGROUND SURVEYING.
326. Transferring a Meridian into a Mine by use of the Transit 294
327. Plumbing the Meridian down a Shaft 296
328. Transferring a Meridian into a Mine when there are Two
Shafts 298
329. Underground Traverses 299
330. Establishing Station Points 300
331. Notes of a Mine Traverse 305
332. Plotting a Mine Traverse 305
333- Underground Leveling 306
334. Mine Maps and Charts 306
335. Laying out Mining Work 307
336. Underground Surveying Problems 307
337. Vertical Angle Correction for Eccentricity of the Top
Telescope 308
338. Vertical Angle Correction for Eccentric Bearing Tele-
scope 308
339. To Establish a Boundary Ln^E of the Claim Underground 309
340. To Lay Out a Connection in a Mine 309
341. Hydraulic Surveying for Mines 310
342. Testing for Ore by Electric Currents 311
xvi CONTENTS
SURFACE SURVEYING.
Art, Paoe
343. SUIUFACE StJRVEYINO IN RuGGED MOUNTAIN REGIONS 311
344. Mine Boundaries. Appropriations Under United States Laws 312
345. Surveying for Patent 314
346. The Surveying of Boreholes 315
347. Staking out the Probable Apex of a Vein 316
348. Economic Principles 316
Problems 318
PART III.
COMPUTATIONS.
Chapter XII. — General Principles — Miscellaneous Prob-
lems — Earthwork Computations.
349. General Remarks 323
354. Logarithmic or Natural Functions 327
355. Short Cuts 328
357. Arrangement of Computations 330
358. Checks 330
359. Slide Rule 330
360. Thacher Slide Rule 333
361. Reducing the Field Notes for Computations 333
362. Curved Boundary by Offsets 333
363. Trapezoidal Rule 334
364. Simpson's One-Third Rule 334
365. Straightening Crooked Boundary Lines 336
366. Area by Triangles 336
367. Area of a Quadrilateral by Triangles 337
368. Area of a Curved Corner Lot 338
369. Rough Checks on Areas 339
370. Planimeter 339
371. Deflection Angles and Chords for a Circular Curve ... 341
372. Computations of Observations 342
computation of volume.
373. BoRROw-Prrs 342
374. Volume of Prismoid 344
375. End Area Formula , 344
376. Prismoidal Formula 344
378. Estimates for Grading 345
382. Rough Estimates 349
Problems 350
CONTENTS xvii
Chapter XIII. — Area by Double Meridian Distances —
Coordinates.
Art. Pace
383. Computation of Ajkea 352
384. AiiEA BY Double Meridian Distance Method 352
386. Computation or Area of Compass Survey . . •. 356
387. Balancing a Chain and Compass Traverse 358
390. Double Parallel Distance 359
391. Error op Closure 360
392. Computation of Area of a Transit and Tape Survey. . . . 360
395. Balancing a Transit and Tape Survey 364
396. Fractional Areas 365
397. Supplying Missing Data 366
401. Detecting Mistakes 368
402. The Subdivision of Land 368
40 jl To Cut Off from a Traverse a Given Area by a Straight
Line starting from a Known Point on the Traverse . . . 369
404. To Cut Off from a Traverse a Given Area by a Line Running
in a Given Direction 369
405. To Find the Area Cut Off from a Traverse by a Line run-
ning in a Given Direction from a Given Point en the Trav-
erse 370
CALCULATIONS RELATING TO TRAVERSES WHICH DO NOT CLOSE.
406. To Calculate the Total Distance between End Points . . 371
407. Cut-Off Lines 371
408. Computation of Azimuths when Checking Angles to a Dis-
tant Object 372
409. Calculation of Triangulation 372
4ia Coordinates 373
411. To Determine the Area of a Field by Rectangular Co-
ordinates 374
Problems 376
PART IV.
PLOTTING.
Chapter XIV. — Drafting Instruments and Material.
engineering drafting instruments.
415. Straight Edge . 381
416. Engineer's Scale 382
417. Protractor 382
418. Semictrcular Protractor 382
419. Full-Circle Protractor 384
421. Three-Armed Protractor 384
xviii CONTENTS
Aet. Pjm»
423. Pantcx>raph . 385
424. Parallel Ruler 386
425. Beam Compass 386
426. Contour Pen 386
427. Proportional Dividers 387
429-31. Railroad Curves, French Curves, Flexible Curves, akd
Spline 387
DRAWING PAPERS.
433. Drawing Paper por Plans 388
434. Tracing Paper and Tracing Cloth 389
436. Cross-Section and Profile Papers 390
438. Process Papers, — Blue-Prints 391
441. Vandyke Solar Paper 393
442. Electrical Printing Frames 395
443. Inks and Water-Colors 395
Chapter XV. — Methods of Plotting.
444. Laying Out a Plan 397
445. Scale 397
methods of plotting traverses.
446. Plotting by Protractor and Scale 398
447. Checks 398
448. Protractor and T-Square 400
449. Plotting by Rectangular Coordinates 401
454. Checks 405
455. Plotting by Tangents 405
456. Checks 407
457. Plotting by Chords 408
458. Use of the Sine 409
459. Checks 409
method of plotting details.
460. Buildings, Fences, Streams, Etc. 410
461. Contours 410
462. Cross-Sections 411
463. Profiles 4"
467. Checks 413
Problems 413
Chapter XVI. — Finishing and Filing Drawings.
468. What Should Appear on a Drawing 415
469. Traverse Lines 415
470. Physical Features 416
471. Topographic Conventional Signs 416
• CONTENTS xix
Akt. Page
473. Lettering 422
474. Titles 423
475. Notes 426
476. Border Lines 427
477. Meridians 427
478. Scales 428
479. Shrinkage of Drawing Papers 428
480. Maps op Large Extent 429
481. Inking in a Profile 430
482. Cleaning Drawings 430
483. Filing Drawings 431
484. Indexing Drawings 432
485. Filing Note-books 432
486. Indexing Notes 433
487. Other Records 433
TABLES.
I. Logarithms of Numbers 437
11. Logarithmic Sines and Cosines 455
III. Logarithmic Tangents and Cotangents 470
IV. Natural Sines and Cosines 485
V. Natural Tangents and Cotangents 494
VI. Lengths of Circular Arcs: Radius — i 506
VII. Mean Refractions in Declination 507
Vni. Trigonometric and Miscellaneous Formxtlas 511
IX. Circular Curve Formulas 513
X. Geometric Formulas 513
XI. Linear Measure 514
XII. Square Measure 514
Xin. Linear Measure — Metric System 514
XIV. Square Measure — Metric System 514
XV. Constants 515
Greek Alphabet 516
THE PRINCIPLES AND PRACTICE OF
SURVEYING.
PART I.
USE, ADJUSTMENT, AITD CASE OF INSTRUHENTS.
CHAPTER I.
OBNBRAL DEFINITIONS. — MEASXTRISltfENT OF LINES.
1. Definition Surveying is the art of measuring and
locating lines and angles on the surface of the earth. When
the survey is of such limited extent that the effect of the earth's
ciu-vature may be safely neglected it is called Plane Survey-
ing. When the survey is so large that the effect of curvature
of the earth must be taken into account as, for instance, in the
survey of a state or a country, it is called Geodetic Surveying,
2. Purposes of Surveys, — Surveys are made for a variety
of purposes such as the determination of areas, the fixing of
boundary lines, and the plotting of maps. Furthermore, engi-
neering constructions, such as waterworks, railroads, mines,
bridges, and buildings, all require surveys.
$• Horizontal Lines. — In siuveying, all measurements of
lengths are horizontal or else are subsequently reduced to hori-
zontal distances. As a matter of convenience, measurements
are sometimes taken on slopes, but the horizontal projection is
afterward computed. The distance between two points as
shown on a map' then is always this horizontal projection.
INSTRUMENTS FOR MEASURING LINES.
4. The Chain. — There are two kinds of chain in common use,
the Surveyor^ s {or Guntet's) Chain, and the Engineer's Chain
(Fig. i). Gunter's chain is 66 feet long, and its use is confined
chiefly to land surve)dng on account of its simple relation to the
acre and to the mile.
I Gunter's Chain = 4 Rods= 100 Links.
I Mile =80 Chains.
I Acre =10 Square Chains.
Evidently each link is ^^ of a foot (or 7.92 inches) long.
The inch, however, is never used in surveying fieldwork.
3
MEASUREMENT OF LINES
[Chap. I.
The engineer's chain is icx) feet long and is divided into one
hundred hnks of one foot each. Each end hnk is provided with a
handle, the outside of which is the zero point, or end, of the chain.
In these chains, every tenth link counting from either end is
marked by a brass tag having one, two, three, or four points
corresponding to the number of tens which it marks. The
middle of the chain is marked by a round tag. In the en-
gineer's chain then the lo-ft. and 90-ft. points, the 20-ft. and
80-ft. points, etc., are marked alike ; hence it is necessary to ob-
Engineer's Chain. Gunter's Half-Chain.
Fig. 1.
serve on which side of the 50ft. point a' measurement falls in
order to read the distance correctly. Distances measured with
the surveyor's chain are recorded as chains andlinks^ (or in chains
and decimals) ; while those measured with the engineer's chain
are recorded zsfeet and decimals.
On account of the large number of wearing surfaces and the
consequent lengthening with use, the chain should be frequently
compared with a standard of length (Art. 243, p. 218). It may
be adjusted to agree with the standard, by means of a nut at the
THE CHAIN AND TAPE 5
handle, which allows the length of the chain to be altered by
lengthening or shortening the end link.
5. Metric Chain. — The Metric Chain is usually 20 meters
long and is divided into one hundred links, each 2 decimeters
long.
6. The Tape. — There are three kinds of tape in common
use, — cloth, metallic J and steel. Cloth tapes stretch so easily
that they are of little use in surveying. The so-called metallic
tapes are cloth tapes havmg very fine brass wires woven into
them to prevent stretching. They are usually graduated into
feet, tenths, and half -tenths and are made in lengths of 25 ft.,
50 ft., and 100 ft. When precise results are required a steel
tape should be used. While a steel tape varies a slight
amount in length with the temperature and with the pull, it is
possible to determine the amount of these variations and hence
to arrive at accurate results.
7. Steel Tapes. — Steel tapes may be obtained in lengths
up to 500 ft., but the most common in use are the 50-ft. and
loo-ft. lengths. While the shorter tapes are usually made of
thin steel ribbon the longer ones are of sufficiently large cross-
section to withstand hard usage. These heavy tapes are gen-
erally marked every 10 ft. by a brass tag, the lo-ft. length at
one end of the tape being marked at every foot, and the last
foot divided into tenths. Some of these tapes are marked every
foot throughout their entire length. The light tapes are divided
throughout their entire length into feet, tenths, and hundredths,
each line being etched on the steel. The numbering is contin-
uous from o ft. to icx) ft. These tapes are more convenient* to
handle than the heavy ones, but are not suited to very rough
work as they are easily kinked and broken. They can be readily
mended, however, by riveting to the back of the tape a piece of
tape of the same width.
Since the surveyor's measurements are usually in feet and
decimals, they are not in convenient form for use by mechanics
in construction work. It is therefore often necessary to con-
vert decimals of a foot into inches and vice versa. The follow-
ing table shows the general relation between these two and is
sufficiently close for most work
6 MEASUREMENT OF LINES [Chap. I.
TABLE 1.
Decimals of Foot in Inches.
Dbcimal op Foot. Inchbs.
.01 - i-
.08 := I-
.17 = 2 -f
.25 « 3 (exact)
.50 — 6 (exact)
.75 - 9 (exact)
Decimals of a foot can easily be converted mentally into
inches, by use of the equivalents in the above table, for example,
0.22 ft. = .25 — .03 = 3'' — |"= 2f".
In surveying farms, timber lands, or other property of low
value, chain measurements are usually of sufficient accuracy and
the chain is well adapted to work in rough country. In city
surveys, and in fact in all surveys where great accuracy is de-
manded, the steel tape is indispensable. In preliminary railroad
surveys the engineer's chain, which formerly was *used exclusively,
is gradually being replaced by the long heavy tape which, while
adapted to rough work, will at the same time give accurate
results.
8. The stadia. — Where it is desired to measure distances
with great rapidity but not with very great accuracy the stadia
method is coming to be very generally used. The distance is
obtained by simply sighting with a transit instrument at a grad-
uated rod held at the other end of the line and noting the space
on the rod included between two special cross-hairs set in the
instrument at a known distance apart. From this observed in-
terval on the rod the distance from the transit to the rod can be
easily calculated.
9. OTHER Instruments. — Wooden Rods^xQ used in certain
kinds of work for making short measurements, usually less than
15 ft.
The Two-Foot Rule divided into tenths and hundredths of a
foot is very convenient for short measurements.
The Odometer is an instrument which may be attached to a
carriage in such a manner as to register the number of revolutions
of one of the wheels. The circumference of the wheel being
known the approximate distance traversed is easily determined.
CHAINING A HORIZONTAL LINE ^
MEASUREMENT OF LINES.
10. MEASUREMENT OF A HORIZONTAL LINE WITH A CHAIN.
— This work is done by two chainmen using a chain and a set of
eleven steel marking pins. One man, called the head-chainman,
carries ten of the marking pins and the front end of the chain.
The rear-chainman takes the eleventh pin and the other end of
the chain. The head-chainman then goes forward keeping as
nearly on the line as he can. The rear-chainman holds his end
of the chain just to one side of the initial point, as in Fig. 2, so
that any jerking of the chain will not disturb the pin at which he
is holding. The rear-chainman, with his eye over the point,
places the head-chainman in line with some object, such as a
Head-chainman. Rear-Chainman.
Fig. 2. Measuring a Horizontal Line with a Chain.
sighting-rod, which marks the other end or some point on the
line. When the head-chainman is nearly in line he takes a pin
and, standing to one side of the line, holds it upright on the
ground a foot or so short of the end of the chain and the rear-
chainman motions him to the right or left until his pin is on the
line. When the head-chainman has the pin in line he stretches
the chain taut, seeing that there are no " kinks ** and that no ob-
structions cause bends in the chain. The rear-chainman at the
same time holds his end of the chain at his pin and when he
calls out, "All right here," the head-chainman stretching the
8 MEASUREMENT OF LINES [Chap. I.
chain past his line pin, removes this line pin, places it at the end
of the chain, as in Fig. 2, and presses it vertically into the
ground. When the chainmen are experienced the pin may be
set for both line and distance at the same time. When the pin
is in place the head-chainman calls, "All right," the rear-chain-
man takes the pin left at his end of the line and they proceed to
the next chain-length. The pin that the rear-chainman has is a
record of the first chain-length. Just before reaching the second
pin the rear-chainman calls out,- "Chain," to give the head-chain-
man warning that he has nearly reached a chain-length. The
process of lining in the head-chainman and measuring a
chain-length is then repeated. When the third pin is stuck in
the ground the rear-chainman pulls the second pin ; in this way
the number of pins the rear-chainman holds is a record of the
number of chain-lengths measured. There is always one pin
in the ground which simply marks the distance and is not
counted.
When lo chains have been measured the head-chainman will
be out of pins and calls to the rear-chainman, who brings for-
ward lo pins. The pins are then counted by both chainmen.
Every time lo chains are measured a record of it is made in
note-books kept by both men and the process is repeated until the
end of the line is reached.
In measuring the fraction of a chain the head-chainman holds
his end of the chain at the required point and the fractional
distance is read by the rear-chainman at the last pin. In some
kinds of work, however, it is more convenient to draw the chain
ahead past the end point and, while the rear-chainman holds his
end of the chain at the last pin, the head-chainman reads the
fractional measurement. The links are read by counting from
the proper tag and the tenths of a link are estimated. Great
care should be taken to count the tags from the proper end of
the chain since the loft. points each side of the center, as has
been explained, are marked alike.
It can be easily shown that if a pin is placed a few tenths of
a foot to the right or left of the line the resulting error in the
distance is very small and consequently " lining in " by eye is
accurate enough, so far as the distance is concerned. But when
MEASURING ON SLOPING GROUND 9
any side measurements or angles are to be taken the points
should be set accurately on line by means of a transit instrument.
The chain should always be kept stretched out full length ;
it should never be doubled back on itself as it may become
tangled and the links bent.
Much time can be saved if the head-chainman will pace the
chain-length and then place himself very nearly in the line by
means of objects which he knows to be on line as, for example,
the instrument, a pole, or the last pin. The beginner should
pace, several times, some line of known length so as to deter-
mine approximately how many steps he takes in lOO ft. In
doing this he should take his natural step and avoid any attempt
to take steps just 3 ft. long.
II. Measurement of a Horizontal Line with a Chain on Slop-
ing Ground. — If the measurement is not on level ground the
chain must be held horizontal and the distance transferred to
the ground by means of a plumb-line. This is difficult to do
accurately and is a fruitful source of error. Beginners usually
hold the downhill end of the chain too low. Horizontal lines
on buildings are very useful in judging when the chain is level.
Since it is supported only at the ends its weight will cause
it to sag so that the distance between the ends is less than
a chain-length. The pull exerted on the chain should be such
that it will stretch enough to balance as nearly as possible the
shortening due to sag.
Whenever a slope is so steep that the chainman on the lower
end cannot plumb high enough to keep the chain horizontal the
measurement must be made in sections, 50-ft., 20-ft., or even 10-
ft lengths being used. Mistakes will be avoided if the rear-chain-
man comes forward at each measurement and holds the same
fractional point on the chain that the head-chainman held, and
so on until a whole chain-length has been measured. In this
way it will be unnecessary to count the fractional distances, but
care should be taken that these pins which marked the inter-
mediate points are returned to the head-chainman so that the
count of the chain-lengths will not be lost. Chaining downhill
will, in general, give more accurate results than chaining uphill,
because in the former case the rear end is held firmly at a point
lO MEASUREMENT OF LINES [Chap. I.
on the ground so that the head-chainroan can pull steadily on
the chain and transfer the distance to the ground by means of
the plumb-line ; in the latter case the rear-chainman is plumbing
his end of the chain over the point and it is difficult to hold it
steady. The result is that the head-chainman cannot easily judge
where the pin should be placed.
12. MEASUREMENT WITH A STEEL TAPE. — In measuring
with the steel tape the process is similar to that described for
the chain. As the tape is used for more precise work than the
chain it is necessary to employ more exact methods of marking
the intermediate points. In some cases stakes are driven into
the ground and tacks or pencil marks used to mark the points.
A small nail pressed into the ground so that the center of the
head is in the proper position makes a good temporary mark, but
of course is easily lost. In measuring on the surfaces of hard
roads, spikes are used for permanent marks.
Measurements of important lines which are not checked by
some geometric test should be checked by repeating the meas-
urement, and in such a way as not to use the same intermedi-
ate points taken in the first measurement.*
Where distances are to be measured continuously from the
initial point of a line without regard to angles in the line, as in
railroad surveys, it is. customary to establish the loo-ft. points.
Mistakes will often be avoided by setting the lOO-ft. points as fol-
lows:— suppose an angle to occur at 870.1 ft. from the point of
beginnmg; this would be called "Station 8 + 70.1." To set
" Station 9 " the 70. i-ft. point of the tape should be held on stake
8 -h 70.1 and the stake at station 9 placed at the lOO-ft. point
of the tape. This is preferable to making a measurement of
29.9 ft. from the zero end of the tape.
* In measuring with the tape some prefer to make a series of measure-
ments between points set in the ground a little less than lOo ft. apart, summing
up the partial measurements when the end of the line is reached. This guards
against the mistake of omitting a whole tape-length. Another advantage is that
it is easier to read the distance to a fixed point than to set a point accurately at
the end of the tape ; this is especially true in measurements where plumbing is
necessary. This method takes less time than the usual method, but it is not
applicable when it is necessary to mark the loo-ft. points on the line.
ERRORS AND MISTAKES IN MEASURING II
13. COHMOir SOURCES OF ERROR IN MEASUREMEUT OF
LDIES. —
1. Not pulling chain or tape taut.
2. Careless plumbing.
3. Incorrect alignment.
4. Effect of wind.
5. Variation in temperature.
6. Erroneous length of chain or tape.
14. COHMOir MISTAKES IN READING AND RECORDING
MEASUREMENTS. —
1. Failure to observe the position of the zero point of the tape.
(In some tapes it is not at the end of the ring.)
2. Omitting a whole chain- or tape-length.
3. Reading from wrong end of chain, as 40 ft. for 60 ft., or in
the wrong direction from a tag, as 47 ft. for 53 ft.
4. Transposing figures, e.g., 46.24 for 46.42 (mental) ; or read-
ing tape upside down, e.g., 6 for 9, or 86 for 98.
5. Reading wrong foot-mark, as 48.92 for 47.92.
15. Avoiding mistakes. — Mistakes in counting the tape-
lengths may be avoided if more than one person keeps the tally.
Mistakes of reading the wrong foot-mark may be avoided by
noting not only the foot-mark preceding, but also the next fol-
lowing foot-mark, as, "46.84 ... 47 feet," and also by holding
the tape so that the numbers are right sidC' up when being read.
In calling off distances to the note keeper, the chainman
should be systematic and always call them distinctly and in such
terms that they cannot be mistaken. As an instance of how
mistakes of this kind occur, suppose a chainman calls, " Forty-
nine, three ;" it can easily be mistaken for "Forty-nine feet."
The note keeper should repeat the distances aloud so that the
chainman may know that they were correctly understood. It
is frequently useful in doubtful cases for the note keeper to use
different words in answering, which will remove possible ambi-
guity. For example, if the chainman calls, "Thirty-six, five,"
fhe note keeper might answer, "Thirty-six and a half." If the
12 MEASUREMENT OF LINES [Chap. L
chainman had meant 36.05 the mistake would be noticed
The chainman should have called in such a case, "Thirty-six
naught five." The following is a set of readings which will
be easily misinterpreted unless extreme care is taken in calling
them off.
40.7 — " Forty and seven."
47.0 — " Forty seven naught."
40.07 — " Forty, — naught seven."
All of these might be carelessly called off, " Forty-seven.**
In all cases the chainmen should make mental estimates of
the distances when measuring, in order to avoid large and ab-
surd mistakes.
16. ACCURACY REQUIRED. — If, in a survey, it is allowable
to make an error of one foot in every five hundred feet the chain
is sufficiently accurate for the work. To reach an accuracy of i
in 1000 or greater with a chain it is necessary to give careful
attention to the pull, the plumbing, and the deviation from the
standard length. With the steel tape an accuracy of i in 5000
can be obtained without difficulty if ordinary care is used in
plumbing and aligning, and if an allowance is made for any con-
siderable error in the length of the tape. For accuracy greater
than about i in 10,000 it is necessary to know definitely the tem-
perature and the tension at which the tape is of standard length
and to make allowance for any considerable variation from these
values. While the actual deviation from the U. S. Standard
under ordinary conditions may be i in 10,000, still a series of
measurements of a line taken under similar conditions may
check themselves with far greater precision.
17. Amount of different errors. — The surveyor
should have a clear idea of the effects of the different errors on
his results. For very precise work they should be accurately
determined, but for ordinary work it is sufficient to know
approximately the amount of each of them. A general idea of
the effect of these errors will be shown by the following.
* i8. Pull. — At the tension ordinarily used, the light steel
tape will stretch between 0.0 1 and 0.02 ft. in 100 ft. if the pull
is increased 10 pounds.
AMOUNT OF DIFFERENT ERRORS 1 3
19. Temperature. — The average coefficient of expansion for
a sted tape is nearly o.ocx>oo63 for i° F. Hence a change of
temperature of 15® produces nearly 0.01 ft. change in the length
of the t2q>e. Tapes are usually manu&ictured to be of standard
length at 62^ F., with a pull of 12 lbs. on them while supported
throughout their entire length.
20. Alignment. — The error in length due to poor alignment
can be calculated from the approximate formula
f^ •
^— tf = —
2C
where h is the distance of the end of the tape from the line, c is
the length of the tape, and a is the distance along the straight
line. For example, if one end of a loo-ft. tape is held i ft. to
one side of the line the error produced in the length of the line
wiD be — — = 0.005 ft-> (about -^ inch). The correction
to be applied to the distance when the two ends of the tape are
not at the same level is computed in the same way.
21. Sag. — If a tape is suspended only at the ends it will hang
in a curve which is known as the " catenary." On account of this
curvature the distance between the end points is evidently less
than the length of the tape. The amount of this shortening,
called the effect of sag, depends upon the weight of the tape, the
distance between the points of suspension, and the pull exerted
* In the right triangle,
{c + fl) (r - a) - k\
assoming c ^ a and applying it to the first parenthesis only,
ze {c — a) ^ k^ (approximately) ^--^^
i. tf — tf— —(approximately) ^^^00^"'^'^ A-j^
Similariy ^ ~" ^ "" 2tf (approximately)
It is evident that the smaller ^ is in comparison with the other two sides the
more exact will be the results obtained by this formula. This formula is even cor-
rect to the nearest ^ ft. when A -^ 14 ft and tf — 100 ft, or when A — 30 ft.
and a — 300 ft.
14 MEASUREMENT OF LINES [Chap. I.
at the ends of the tape. With a I24b. pull on an ordinary loo-
ft. steel tape supported at the ends the effect of sag is about
O.OI ft.
22. Effect of Wearing on Length of the Chain. — When a
chain is new it is very nearly the standard length. During
its first use the links become bent and the chain thus shortened
But there are nearly six hundred wearing surfaces and before
long the small amount of wear on each surface lengthens the
chain an appreciable amount. It is very common to find chains
which, after considerable use, have lengthened 0.3 ft. or more.
23. ACCURACY OF MEASUREMENTS. — In surveying we are
dealing entirely with measurements. Since absolute accuracy
can never be attained, we are forced to make a careful study of
the errors of measurement. Extremely accurate measurements
are expensive, and the cost of making the survey usually limits
its accuracy. On the other hand, if a given degree of accuracy
is required, the surveyor must endeavor to do the work at a
minimum cost. In most surveys certain measurements are far
more important than others and should therefore be taken With
more care than the relatively unimportant measurements.
The surveyor should distinguish carefully between errors
which are of such a nature that they tend to balance each other
and those which continually accumulate. The latter are by far
the more serious. Suppose that a line 5000 ft. long is measured
with a steel tape which is 0.0 1 ft. too long and that the error in
measuring a tape-length is, say, 0.02 ft., which may of course be a
+ or a — error. There will then be 50 tape-lengths in the scxx>
ft. line. A study of the laws governing the distribution of ac-
cidental errors (Method of Least Squares) shows that in such a
case as this the number of errors that will probably remain uncom-
pensated is the square root of the total number of opportunities
for error, Le., in the long run this would be true. Hence
the total number of such uncompensated errors in the line is 7 ;
and 7 X 0.02 = 0.14 ft., which is the total error due to inaccuracy
in marking the tape-lengths on the ground. Since the error due
to erroneous length of tape increases directly as the number of
measurements, and since these errors are not compensating, the
total error in the line due to the fact that the tape is 0.0 1 ft. too
ACCURACY OF MEASUREMENTS 1 5
long is 50 X 0.01 = 0.50 ft. The small (0.0 1) accumulative
error is therefore seen to have far greater eiSect than the larger
(0.02) compeiisating error.
PROBLEMa
1. A distance is measiired with an engineer's chain and found to be 7964 ft.
The chain when compared with a standard is found to be 0.27 ft too long. What
is the actual length of the line ?
2. A metallic tape which was originally 50 ft. is found to be 50.14 ft. long.
A house 26 ft. X 30 ft. is to be laid out. What measurements must be made,
using this tape, in order that the house shall have the desired dimensions ?
5. A steel tape is known to be 100.000 ft. long at 62° F. with a pull of 12 lbs.
and supported its entire length. Its coefficient of expansion is 0.0000063 for i^ F.
A line was measured and found to be 142.67 ft. when the temperature was Bf*
below zero. What is the true length of the line ?
4. In chaining down a hill with a surveyor's chain the head-chainman held
his end of the chain 1.5 ft. too low. What error per chain-length would this
produce ?
5. In measuring a line with a loo-ft. tape the forward end is held 3 ft. to the
side of the line. What is the error in one tape-length ?
CHAPTER IL
BCBASURBMENT OF DIRBCTZON.
24. The surveyor's compass. — -The surveyor's compass
(Fig. 3) is an instrument for determining the direction of a line
with reference to the direction of a magnetic needle. The needle
is balanced at its center on a pivot so that it swings freely in
a horizontal plane. The pivot is at the center of a horizontal
circle which is graduated to degrees and half -degrees, and num-
bered from two opposite zero points each way to 90*^. The zero
points are marked with the letters N and S, and the 90® points
are marked E and W. The circle is covered with a glass plate
to protect the needle and the graduations, the part enclosed
being known as the compass-box, A screw is provided for
raising the needle from the pivot by means of a lever. The
needle should always be raised when the compass is lifted or
carried, to prevent dulling the pivot-point ; a dull pivot-point
is a fruitful source of error. Both the circle and the pivot
are secured to a brass frame, on which are two vertical sights
so placed that the plane through them also passes through
the two zero points of the circle. This frame rests on a tripod
and is fastened to it by means of a ball-and-socket joint. On
the frame are two spirit levels at right angles to each other,
which afford a means of leveling the instrument. This ball-and-
socket joint is connected with the frame by means of a spindle
which allows the compass-head to be revolved in a horizontal
plane, and to be clamped in any position.
The magpietic needle possesses the property of pointing in a
fixed direction, namely, the Magnetic Meridian, The horizontal
angle between the direction of this meridian and of any other
line may be determined by means of the graduated circle, and
this angle is called the Magnetic Bearing of the line, or simply
its Bearing, By means of two such bearings the angle between
two lines may be obtained. Bearings are reckoned from o® to 90%
16
THE SURVEYOR'S COMPASS
17
the o® being either at the N or the S point and the 90® either
at the E or the W point. The quadrant in which a bearing
falls is designated by the letters N.E., S.E., S.W., or N.W.
For example, if a line makes an angle of 20^ with the meridian
and is in the southeast quadrant its bearing is written S 20® E.
Sometimes the bearing is reckoned in a similar manner from
Fig. 8. Surveyor's Compass.
the geographical meridian, when it is called the true bearing.
In general this will not be the same as the magnetic bearing.
True bearings are often called azimittfts^ and are commonly
reckoned from the south point right-handed (clockwise) to 360° ;
i.e., a line running due West has an azimuth of 90°, a hne due
North an azimuth of 180° Sometimes, however, the azimuth
i8
MEASUREMENT OF DIRECTION
[Chap. n.
is reckoned from the north as in the case of the azimuth of the
Pole-Star (Art. 206, p. 180). •
25. The Pocket Compass. — The pocket compass is a small
hand instrument for obtaining roughly the bearing of a line.
There are two kinds, the plain and the prismatic. The former
is much like the surveyor's compass, except that it has no sights.
In the prismatic compass the graduations, instead of being on
the compass-box, are on a card which is fastened to the needle
(like a mariner's compass) and which moves with it. This com-
pass is provided with two short sights and the bearing can be
read, by means of a prism, at the same instant that the compass
is sighted along the line.
26. METHOD OF TAKING A MAGNETIC BEARING. — The
surveyor's compass is set up (and leveled) at some point on the
line whose bearing is desired. The needle is let down on
the pivot ; and the compass is turned so that the sights point
along the line. While looking through the two sights the sur-
Bcorlng of AB Bearing cf AB Bearing of AB
seo'E seo'w n45'w
Diagram Illustrating Reading of Bearings.
veyor turns the compass-box so that they point exactly at a
lining pole or other object marking a point on the line. The
glass should be tapped lightly over the end of the needle to be
sure that the latter is free to move. If it appears to cling to
the glass this may be due to the glass being electrified, which
condition can be removed at once by placing the moistened fin-
ger on the glass. The position of the end of the needle is then
read on the circle and recorded. Bearings are usually read to
the nearest quarter of a degree.
Since the needle stands still and the box turns under it, the
letters E and W on the box are reversed from their natural
position so that the reading of the needle will not only give the
TAKING A MAGNETIC BEARING 1 9
angle but also the proper quadrant. Reference to Fig. 4 will
show the following rule to be correct : — When the north point
of the compass-box is toward the point whose bearing is desired,
read the north end of the needle. When the south point of the
box is toward the point, read the south end of the needle. If a
bearing of the line is taken looking in the opposite direction it
is called the reverse bearing.
Since iron or steel near the instrument affects the position
of the needle, great care should be taken that the chain, axe, or
marking pins are not left near the compass. Small pieces of
iron on the person, such as keys, iron buttons, or the iron wire in a
stiff hat, also produce a noticeable effect on the needle. Electric
currents are a great source of disturbance to the needle and in
cities, where electricity is so common, the compass is practically
useless.
In reading the compass-needle, the surveyor should take
care to read the farther end of the needle, always looking along
the needle, not across it. By looking at the needle sidewise it is
possible to make it appear to coincide with a graduation which is
reaUy at one side of it. This error is called parallax.
27. The Earth's Magnetism. — Dip of the Needle. — The
earth is a great magnet. On account of its magnetic influence
a permanent magnet, such as a compass-needle, when freely
suspended will take a definite direction depending upon the
direction of the lines of magnetic force at any given place and
time. If the needle is perfectly balanced before it is magnetized
it will, after being magnetized, dip toward the pole. In the
northern hemisphere the end of the needle toward the north
pole points downward, the inclination to the horizon being slight
in low latitudes and great near the polar region. In order to
counteract this dipping a small weight, usually a fine brass wire,
is placed on the higher end of the needle at such a point that
the needle assumes a horizontal position.
28. DECLINATION OF THE NEEDLE. — The direction which
the needle assumes after the counterweight is in position is
called the magnetic meridian and rarely coincides with the true
meridian. The angle which the needle makes with the true me-
ridian is called the declination of the needle. When the north
20 MEASUREMENT OF DIRECTION [Chap. IL
end of the needle points east of the true, or geographical, north
the declination is called east; when the north end of the needle
points west of true north it has a west declination.
29. Variations in Declination. — The needle does not con-
stantly point in the same direction. Changes in the value of the
declination are called variations of the declination,'* The prin-
cipal variations are known as the Secular^ Daily^ Annualy and
Irregular.
The Secular Variation is a long, extremely slow swing. It
is probably periodic in character but its period covers so many
years that the nature of it is not thoroughly understood. The
following table shows the amount of secular variation as observed
in Massachusetts during two centuries.
1750
r 13' w.
1800
e*' 28' w.
1850
9*» 10' w.
1900
12'' 00' w.
In the United States all east declinations are now gradually
decreasing and all west declinations are gradually increasing, at
an average rate of about 2 minutes per year.
The Daily Variation consists of a swing which averages
about 7 minutes of arc from its extreme easterly position at
about 8 A.M. to its most westerly position at about 1.30 P.M.
It is in its mean position at about 10 A.M. and at 5 or 6 P.M.
The amount of daily variation is from 3 to 12 minutes according
to the season and the locality.
The Annual Variation is a periodic variation so small (about
one minute a year) that it need not be considered in surveying
work.
*The angle called Declination by surveyors is usually called Variation by
navigators.
t See p. 107 of U.S. Coast and Geodetic Survey special publication entitled
<«U. S. Magnetic Declination Tables and Isogenic Chart for 1902, and Principal
Facts Relating to the Earth's Magnetism," by L. A. Bauer, issued in 1^2.
TABLE 2. I
Observed Declinations of Needle in Eastern MASSACHusBTTS.t '
Year. Declination.
1700 10° 31' w. !
Fig. 5> IsoGONic Chart of the Vsn
(From the U. S. Coast and Geodetic Surrey tpedal pablication entitled " U. S. Magnetic Da
by L. A. Bi
► States for the Epoch January, 1902
in Tables and laogonic Chart for 1909, and Principal Facts Relating to the Earth's Magnetism,"
iliMitJ in 19M.)
DECLINATION OF THE NEEDLE 2$
Irregular Variations in the declination are due chiefly to
magnetic storms. They are uncertain in character and cannot
be predicted. They are, however, usually observed whenever
there is a display of the Aurora Borealis. Such storms often
cause variations of from ten to twenty minutes in the United
States and even more in higher latitudes.
30. Isogenic Chart. — If lines are drawn on a map so
as to join all places where the declination of the needle is the
same at a given time, the result will be what is called an isogenic
chart, (See Fig. 5.) Such charts have been constructed by
the United States Coast and Geodetic Survey. While they do
not give results at any place with great precision they are very
useful in finding approximate values of the declination in differ-
ent localities.
An examination of the isogenic chart of the United States
shows that in the Eastern States the needle points west of north
while in the Western States it points east of north. The line of
no declination, or the agonic line, paisses at the present time (1906)
through the Carolinas, Ohio and Michigan.
31. OBSERVATIONS FOR DECLINATION. — For any survey
where the value of the present declination is important, it should
be found by special observations. The value found at one place
may be considerably different from that of a place only a few
miles distant. The method of finding the declination by ob-
servation on the Pole-Star (Polaris) is described in Art. 2 10, p. 187.
ADJUSTMENTS OF THE COMPASS.
32. The three adjustments which need to be most frequently
made are (i) adjusting the bubbles, (2) straightening the needle,
(3) centering the pivot-point.
33. Adjustment of the bubbles. — To make the Plane
of the Bubbles Perpendicular to the Vertical Axis. — Level the in-
strument in any position. Turn 180° about the vertical axis
and, if the bubbles move from the center, bring each half-way
back by means of the adjusting screws; and repeat the process
until the desired fineness of adjustment is secured.
34. detecting errors in adjustment of the Needle.
— If the readings of the two ends of the needle are not 180®
26
MEASUREMENT OF DIRECTION
[Chap, n
Fig. 6.
Bent Compass-
Needle.
apart, this may be due to the needle being bent, to the pivot-
point not being in the center of the graduated circle, or to both.
If the difference of the two readings is the same in whatever
direction the compass is turned, it
follows that the needle is bent but
the pivot-point is in the center of the
circle. (See Fig. 6.) The bent needle
is represented by the line AOB and
the position of a straight needle shown
by the line A OC. In the two positions
shown it is seen that the difference in
readings will be the same, i.e., arc
CB = arc C'B\ If the difference of
the readings varies as the compass
is turned around it follows that the
pivot-point is not in the center, and the needle may or may
not be bent. Suppose the needle is straight but the pivot is
not in the center, then the effect in different parts of the circle
is shown in Fig. 7. When the needle is in the position ADy
perpendicular to CO, (where C is
the true center and C is the
position of the pivot-point), then
the error is a maximum. If B is
a point 180® from A then the dif-
ference of the two readings is BD,
When the needle is at A^D' the
error is less than before and equals
B'ly. When the needle is in the
line CC, i.e., in the position yi"Z>",
the ends read alike.
In making these adjustments
it is better to first straighten the
needle, because the error due to
the needle being bent can be detected independently of the
error of the pivot.
35. TO STRAIGHTEN THE COMPASS-HEEDLE. — Level the
instrument and let the needle down on the pivot. Remove the
glass cover. By means of a brass wire or alight stick of wood
Fig. 7.
Pivot-Point
Center.
OUT of
ADJUSTMENTS OF THE COMPASS 2/
Steady the needle so that one end of it, say the south end, is
opposite some graduation on the circle as A in Fig. 8. Note
the position of the north end of the needle C Now, without mov-
ing the compass itself, turn the needle around so that the north
end is at the graduation A. Hold it in this position with the
brass wire and read the position of the south end C\ One-
half the difference of the readings, or, the distance C^D is the
^ c' amount by which the needle is bent.
^ ^^^ Carefully remove the needle from the
^^ pivot and bend it by the amount CD
\ in the direction which will move the
\ south end half-way back from C
Q I toward C. It is better not to touch
\ T j the needle with the hands more than
\ I / is absolutely necessary as this weakens
\ I y the magnetism. Instrument makers
V^^^^ \ ^^ usually leave the central part of the
^ needle quite soft so that it can be
Fig. 8. Straightening the easily bent in making this adjustment.
Compass-Needle. Since the amount by which the
needle is bent is a matter of estimation it should be replaced
on the pivot and the test repeated until it is found that reversing
the needle does not change the readings.
36, To CEHTER the PiVOT-POnrr. — If the difference of
readings of the two ends of the needle varies in different parts
of the circle it is due to the pivot-point being out of center.
Take readings of the two ends of the needle in various positions
of the compass and find the position of the needle in which the
difference of the two readings is greatest (Art. 34, p. 25). The
pivot is to be bent at right angles to this direction an amount
equal to half this difference. Remove the needle and bend the
pivot by means of a pair of small flat pliers. Replace the needle
and see if the difference of end readings is zero. If not,
the pivot must be bent until this condition is fulfilled. As the
pivot may become bent somewhat in a direction other than that
intended, a complete test for adjustment must be made again,
and the process continued until the difference in the readings of the
ends of the needle is zero in all positions of the compass. The
28
MEASUREMENT OF DIRECTION
[Chap. n.
metal at the base of the pivot is left soft so that it can be easily
bent.
37. To REMAGNETIZE THE NEEDLE. — Rub each end of
the needle from the center toward the end several times with a
bar-magnet, using the N end of the magnet for the S end of the
needle and vice versa. (The N end of the magnet attracts the
S end of the needle and repels its N end.) When the magnet
is drawn along the needle it should move in a straight line,
parallel to the axis of the needle. When returning the bar from
the end of the needle toward the center, lift it several inches
above the needle as indicated in Fig. 9.
Fig. 9. ReMagnetizing the Compass-Needle.
38. CcmMON SOURCES OF ERROR IN COMPASS WORK* —
1. Iron or steel near compass.
2. Parallax in reading needle,
39* COlfMON MISTAKES.—
1. Reading wrong end of needle.
2. Not letting needle down on pivot.
3. Reading the wrong side of the loth degree,
viz., reading 61° instead of 59°.
DETECTING LOCAL ATTRACTION 29
40. DETECTING LOCAL ATTRACTION OF THE NEEDLE. — As
the needle is always affected by masses of iron near the compass
it is important that the bearings in any survey should be checked.
This is most readily done by taking the bearing of any line from
both its ends or from intermediate points on the line. If the two
bearings agree it is probable that there is no local magnetic dis-
turbance. If the two do not agree it remains to discover which
is correct.
In Fig. 1 1 suppose that the compass is at A and that the
Fig. 11. Diagram Illustrating Local Attraction at A.
bearing of AB is N 50^^ E, and with the compass at B the
bearing BA is found to be S 49® W. It is evident that there is
local attraction at one or both points. In order to ascertain the
correct magnetic bearing, turn the compass toward a point C which
is apparently free from magnetic disturbance, and observe the
bearing of BQ which is, say, S 72° E. Now move the compass
to Cand observe the bearing CB. If this is N 72^ W it indi-
cates that there is no local attraction at C or B, hence S 59° W
is the correct bearing of line BA, and there is i°J error in all
bearings taken at A, If the bearings of BC and CB had not
agreed it would have been necessary to take the bearing and
reverse bearing of a new line CD. This process is continued
until a line is found whose bearing and reverse bearing differ by
exactly 180**.
30
MEASUREMENT OF DIRECTION
[Chap. IL
41. CALCULATING ANGLES FROM BEARINGS.— In calcu-
lating the angle between two lines it is necessary only to
remember that the bearing is in all cases reckoned from the
meridian, either N or S, toward the E and W points. In Fig. lo,
AOB =s difference of bearings.
AOC = 1 80® — sum of bearings.
AOD = 180® — difference of bear- w
ings.
AOF = sum of bearings.
Fig. 10.
PROBLEMS.
I. Compute the angle AOB from the given bearings in each of the following
(a) OA. N39«iE. (c)
OB, N76<»iE.
(b) OA, N 3S« IS' E. (d)
OB, S 88° 00' W.
OA, N 15° E.
OB, S 36° E.
OA, N 40° 15' E.
OB, N 66° 45' W.
. 2. The bearing of one side of a field in the shape of a regular hexagon is
S \<f\ E. Find the bearings of the other sides taken around the field in order.
3. (a) In 1859a certain line had a bearing of N 21° W. The declination of
the needle at that place in 1859 was 8° 39' W. In 1902 the declination was
10° 58' W. What was the bearing of the line in 1902 ?
(b) In 1877 a line had a bearing of N 89° 30' E. The declination was
0° 13' E. In 1902 the declination was 1° 39' W. Find the bearing of the line
in 2902.
(c) At a certain place the declination was 4° 25' W in 1700, 1° 39' W in 1750,
0° 2x' E in 1800, 1° 03' W in 1850, 4° 00' W in 1900. If a line had a bearing of
S 65°i W in 1900, what was its bearing in 1700, 1750, 1800, and 1850"?
4. The following bearings were observed with a compass : AB, N 27°} E ;
BA, S 25°i W ; ^C. S 88° W ; C^, N 87°! E ; CZ?, N 47°i W ; DC, S 47°^ E.
Find the true bearing of AB. Where is the local attraction ? Which way is the
needle deflected at each point, and how much ?
CHAPTER III.
MBASUREMIINT OF ANaiiBS.
THE TRANSIT.
42. GENERAL DESCRIPTION OF THE TRANSIT. — The en-
gineer's transit is an instrument for measuring horizontal and
vertical angles. A section of the transit is shown in Fig. 1 2.
Fig. 12. Section of Transit.
A^ inner spindle ; By outer spindle ; C, upper plate ; Z>, lower plate ; Ey graduated
circle ; Fy vernier ; Gy magnetic needle ; Hy upper clamp (turned 90° from its nor-
mal position so as to show in section, corresponding tangent screw not shown) ;
Jy lower clamp ; Ky lower tangent screw ; Z, leveling screws ; My ball-and-socket
joint ; Ny shifting head ; O, base of transit.
Two spindles, one inside the other, are each attached to a hori-
zontal circular plate, the outer spindle being attached to the
lower plate and the inner one to the upper plate. Except in
some older instruments, the lower plate carries a graduated
circle and the upper plate carries the verniers for reading the
circle. On this upper plate are two uprights or standards
31
32 MEASUREMENT OF ANGLES [Chap. m.
supporting a horizontal axis. The length of the telescope and
the height of the standards are commonly such as to allow the
telescope to make a complete rotation on its horizontal axis.
The motion of this axis is usually controlled by a clamp and a
slow-motion screw called a tangent screw. In older instruments
this often consisted of two opposing screws ; in modern instru-
ments it usually consists of a single screw with an opposing
spring. At the center of the horizontal axis is a telescope
attached at right angles to it.
For leveling the instrument, there are two spirit levels on the
upper plate, one parallel and the other at right angles to the hori-
zontal axis. The spirit level which is parallel to the axis is the
more important one because it controls the position of the hori-
zontal axis of the telescope ; it should be and generally is made
more sensitive than the other. In the transit, the leveling is
done by means of four (sometimes three) leveling screws.
The upper plate is usually provided with a magnetic needle
and a graduated circle so that the transit may be used as a com-
pass. The lower spindle is attached to the base of the instru-
ment by means of a ball-and-socket joint the same as in the
compass. Both the upper and lower plates are provided with
clamps for holding them in any desired position and with tan-
gent screws for making exact settings.
At the center of the ball-and-socket joint is a ring to which the
plumb-line may be attached. The plumb-bob used with the tran-
sit is generally heavier than that used in taking tape measure-
ments. Modern transits are so made that the entire head of the
instrument can be shifted laterally \vith reference to the tripod
and can thus be readily placed exactly over a point on the
ground.
The horizontal circle is usually graduated either to half-degrees
or to 20-minute spaces. The graduations are often numbered
from o° to 360° by two rows of figures running in opposite direc-
tions. In some transits they are numbered from 0° to 360° in a
right-hand direction and, by a second row of figures, from o*^
each way to 180®; and still others (older types) are numbered
from o® to 90"^ in opposite directions, like a compass circle.
Transits are all provided with two opposite verniers.
THE TRANSIT
33
Fig. 18. Engineer's Transit.
34
MEASUREMENT OF ANGLES
[Chap. IIL
M
o
u
u
.J
H
c/)
55
H
O
o
H
O
»
<
Q
D
H
o
o
43. The normal or ^/V^?^/ position of
the transit is with the upper clamp and its
tangent screw nearest the observer and the
focusing screw of the telescope on the right-
hand side (in some instruments, on top) of
the telescope. When the instrument is
turned 180° in azimuth from the direct
position and the telescope is inverted (turned
over about the horizontal axis) it is said to
be in the reversed position.
44. .If the telescope is provided with a
long level tube and a vertical circle, or arc,
it is called an Engineer's Transit^ or Sur-
veyor's Transit. (Fig. 13.) If it does not
have these attachments it is called a Plain
Transit.
45. The TELESCOPE. — The essential
parts of the telescope are the objective^ the
cross-hairs, and the eyepiece. (See Fig. 14.)
The line of sight, or line of collimation,
is the straight line drawn through the op-
tical center of the objective and the point
of intersection of the cross-hairs. When
light from any point A falls on the objective,
the rays from A are bent and brought to a
focus at a single point B called the image.
The only exception to this is in the case
when A is on the optical axis ; the ray
which coincides with the optical axis is not
bent. The cross-hairs are placed in the
telescope tube near where the image is
formed, as shown in Fig. 14. The objective
is screwed into a tube, which is inside the
main tube and which can be moved by means
of a rack-and-pinion screw so as to bring the
plane of the image of the object into coinci-
dence with the plane of the cross-hairs. The
instrument is so constructed that the motion
THE TRANSIT TELESCOPE 35
of this tube is parallel to the line of sight. The eyepiece is
simply a microscope for viewing the image and the cross-hairs.
When the plane of the image coincides with the plane of the
cross-hairs, both can be viewed at the same instant by means of
the eyepiece. The adjustment of the eyepiece and the objective,
to enable the cross-hairs and the image to be clearly seen at
the same time, is QsH^&di focusing.
In focusing, first the eye-piece tube is moved in or out until the
cross-hairs appear distinct ; then the objective is moved until the
image is distinct. If it is found that the cross-hairs are no longer
distinct after moving the objective the above process is repeated
until both image and cross-hairs are clearly seen at the same
instant. The focus should be tested for i)arallax by moving the
eye slightly from one side to the other ; if the cross-hairs appear to
move over the image the focus is imperfect. In focusing on ob-
jects at different distances it should be remembered that the
nearer the object is to the telescope, the farther the objective must
be from the cross-hairs ; "and that for points near the instrument
the focus changes rapidly, i.e., the objective is moved consider-
ably in changing from a focus on a point lo ft. away to one
20 ft. away, whereas for distant objects the focus changes very
slowly, the focus for 200 ft. being nearly the same as that for
20CXD ft. An instrument can be quickly focused on a distant
object if the objective is first moved in as far as it will go and
then turned out slowly until the image is distinct. The objec-
tive should not be turned too rapidly as it may pass the correct
position before the eye can detect the distinct image. If an in-
strument is badly out of focus it may be pointing directly at an
object and yet the image may not be visible.
46. The Objective. — The objective might consist of a simple
bi-convex lens, like that shown in Fig. 15, which is formed by
the intersection of two spheres. The line 0(y joining the
centers of the two spheres is called the optical axis. If rays
parallel to the optical axis fall on the lens those near the edge of
the lens are bent, or refracted, more than those near the center,
so that all the rays are brought to a focus (nearly) at a point F
on the optical axis called ^^ principal focus. If light falls on
the lens from any direction there is one of the rays such as
36
MEASUREMENT OF ANGLES
[Chap. HI.
AC or BD which passes through the lens without permanent
deviation, i.e., it emerges from the other side of the lens parallel
to its original direction. All such rays intersect at a point JTon
the optical axis which is called the optical center,
A simple bi-convex lens does not make the best objective
because the rays do not all come to a focus at exactly the same
point. This causes indistinctness and also color in the field of
Fig. 15. Bi-CoNVEX Lens.
view, particularly near the edges. This difficulty is overcome
by using a combination of lenses, consisting of " crown ** and
"flint** glass as shown in Fig. 14, which very nearly corrects
these imperfections.
The position of the image of any point is located on a straight
line (nearly) through the
point and the optical center ;
hence it will be seen that
the image formed by the
objective is inverted.
47. Cross-Hairs. — The
cross-hairs consist of two
very fine spider threads
stretched across a metallic
ring at right angles to each ^ ,« ^
t. f r J i_ Fig. 16. Cross-Hair Ring.
other and fastened by means
of shellac. The cross-hair ring (Fig. 16) is held in place by
four capstan-headed screws which permit of its being moved
MAGNIFYING POWER 37
vertically or horizontally in the telescope tube. The holes in the
tube through which the screws pass are large enough to allow
some motion of the ring in adjusting.
48. Eyepiece. — The eyepiece of the ordinary transit tele-
scope may be either of two kinds, that which shows an inverted
image or that which shows an erect image. An erecting eyepiece
requires two more lenses than the inverting eyepiece, which add
to its length and also absorb light ; but in spite of these disad-
vantages the erecting eyepiece is generally used on ordinary
transits. It will be seen, however, that with the same length of
telescope a greater magnifying power and a clearer, definition of
the image can be obtained by the use of the inverting eyepiece.
These advantages are so important and the disadvantage of see-
ing objects inverted is so slight that inverting eyepieces should
be used more generally than they are at present.
49. Magnifying Power. — The magnifying power is the
amount by which an object is increased in apparent size. It is
tan ^ A A
equal to - — ^ — ,(or nearly equal to — ), A being the angle sub-
tended by an object as seen through the telescope and a the
angle as seen by the unaided eye.
50. The magnifying power may be measured in two ways,
(i) The dimensions on a graduated rod will appear magnified
when viewed through a telescope. If, with one eye at the tele-
scope, the rod is viewed directly with the other eye it will be
noticed that one space as viewed through the telescope will
appear to cover a certain number of spaces as seen with the naked
eye. This number is approximately the magnifying power of the
telescope.
(2) Viewed through a telescope wrong-end-to, an object is re-
duced in apparent size in the same ratio that it is magnified when
seen through the telescope in the usual manner. Measure with
a transit some small angle A between distant points and then
place the telescope to be tested in front of the transit, with its
objective next the objective of the transit. Measure the angle a
between the same points ; this new angle will be smaller. Then
tan -i A
the Magnifying Power =——t—- The magnifying power
38
MEASUREMENT OF ANGLES
[Chap. m.
of the ordinary transit telescope is between twenty and thirty
diameters.
51. Field of View. — The field of view is the angular space
that can be seen at one time through the telescope. It is the
angle subtended at the optical center of the objective by the
opening in the eyepiece. In the ordinary transit this angle is
about one degree, but in some instruments it is considerably
more.
_— 9
—7
-K)
&
—4
—3
-I
c^
0
— 10
—9
—6
—7
i :
-5-
-*''
—3
— ^2
5-
(BUUO
6-
—9
■—8
■7
3
5-
5-
—3
—Z
I
Fig. 17.
Fig. 18.
Fig. 19.
52. THE VERlflER. — The vernier is a device for determin-
ing the subdivision of the smallest division of a scale more accu-
THE VERNIER 39
lately than can be done by simply estimating the fractional part.
It depends upon the fact that the eye can judge much more
exactly when two lines coincide than it can estimate a fractional
part of a space.
A simple form of vernier, shown in Fig. 17, is constructed
by taking a length equal to 9 divisions on the scale and dividing
this length into 10 equal parts. One space on the vernier is
then equal to -^^ of a space on the scale, i.e., it is -^ part
shorter than a space on the scale, hence ad = -j^ of a space
on the scale, cd = -j^^ of a space, etc. Now if the vernier is
raised until a coincides with 6, i.e., until the first line on. the
vernier coincides with the next higher line on the scale, then
the index line has moved over ^ of a space and the reading
will be 501. If the vernier is moved -^ space higher then
line 2 coincides with the next higher line on the scale and
the reading is 502, as shown in Fig. 18. Similarly Fig. 19
shows reading 526. Thus it is seen that the number of the line
on the vernier which coincides with a line on the scale is the
number of tenths of the smallest division of the scale that the
index point (zero) lies above the next lower division on the scale.
Furthermore it will be seen from its construction that it is im-
possible to have more than one coincidence at a time on a single
vernier. The type of vernier just described is used on leveling
rods.
53. Verniers used on Transits. — In transits, since angles
,may be measured in either direction, the verniers are usually
double, Le., there is a single vernier on each side of the index
point, one of which is to be used in reading angles to the right,
and the other in reading angles to the left.
The vernier most commonly found on the transit reads to
one minute of arc (Fig. 20). When this vernier is used the
circle is divided into degrees and half-degrees. The vernier
scale is made by taking a length equal to 29 of the half-degree
spaces and subdividing it into 30 equal parts. Each space on
the vernier is then equal to f^ X 30' = 29'. Therefore the differ-
ence in length of one division on the circle and one division on
the vernier is equal to the difference between the 30' on the
circle and the 29' on the vernier, or one minute of arc. In
40
MEASUREMENT OF ANGLES
[Chap. III.
Fig. 20 the zero of the vernier coincides with the 0° mark on
the circle. The first graduation on the vernier to the left of the
zero fails to coincide with the o*^ 30' line by just i' of arc.
The second line on the vernier falls 2' short of the i*^ mark,
the third line 3' short of the i*" 30' mark, etc. If the vernier
should be moved one minute to the left the first line would coin-
30
VERNIER
^
^
\0 ^ /O
*
fc
1 ' 1
Yiwv
360
CIRCLE
Fig.
ao.
One-Minute Vernier Set
AT
o^
3?
cide and the reading would be o*^ 01'. If the vernier were
moved one minute more the second line would coincide and
the reading would be o°02', etc. Therefore the number of the
line on the vernier which coincides with some line on the circle
is the number of minutes to be added to o*^. After the vernier
has moved beyond the point where the 30' line coincides, it
begins subdividing the next space of the circle, and we must
then add the vernier reading to o® 30'.
The following figures show various types of vernier com-
monly used on transits.
TRANSIT VERNIERS 4 1
Fig. 21. — Double vernier reading id i\ Circle divided
into 30' spaces. 29 divisions of the circle divided into 30 parts
to make one division of the vernier.
Reading, inner row of figures, 9® 16'.
Reading, outer row of figures, 350° 44'.
Since the vernier moves with the telescope, read the angle
on th3 circb in the same direction that the teldScop3 has moved-
Read the number of degrees and half-degrees the index has
passed over and estimate roughly the number of minutes beyond
the last half-degree mark. Then follow along the vernier in the
same direction and find the coincidence. The number of this
line is the number of minutes to be added to the degrees and
half-degrees which were read from the circle. An estimate of
the number of minutes should always be made as a check against
large mistakes in reading the vernier or in reading the wrong
vernier.
Fig. 22. — Double vernier reading to 30". Circle divided
Fig. 22.
into 20' spaces. 39 divisions of the circle divided into 40 parts
to make one division of the vernier.
Reading, inner row of figures, 31® 17' 30".
Reading, outer row of figures, 328° 42' 30".
42
MEASUREMENT OF ANGLES
[Chap. IIL
Fig. 23. — Single vernier reading to 20". Circle divided
into 20' spaces. 59 divisions of the circle divided into 60 parts
to make one division of the vernier.
Reading, 73° 48' 40".
CIRCLE
Fig. 28.
On account of the length of this vernier it is impracticable
to use a double vernier. Where it is desirable to read the
angles in either direction the circle has two rows of figures as
shown in Fig. 24.
Fig. 24 — Reading, inner row of figures, 73*" 48' 40".
Reading, outer row of figures, 266"* 31' 20".
CIRCLE
Fig. 24.
It is evident that if angles are to be read " clockwise " the
index at the right end of this vernier should be set at o^ If
TRANSIT VERNIERS
43
angles are to be measured in the opposite direction the index at
the left end should be set at o^ To avoid this inconvenience of
resetting, some surveyors set the middle line (ic/ line) of the
vernier on o*^ and disregard the numbering on the vernier,
reading it as explained under Fig. 26.
Fig. 25. — Single vernier reading to 10". Circle divided
into i& spaces. 59 divisions of the circle divided into 60 parts
to make one division of the vernier.
Readmg, 59° 15' 50".
Fig. 26. — Single vernier reading in either direction to i'.
CIRCLE 5
5
ijrm]r\
^ 9
VERNIER
Fig. 26.
Circle divided into $<y spaces. 29 divisions of the circle
divided into 30 parts to make one division of the vernier.
Reading, 2° 23'.
44
MEASUREMENT OF ANGLES
[Chap. HI.
This vernier is read like the ordinary i' vernier except that
if a coincidence is not reached by passing along the vernier in
the direction in which the circle is numbered, it is necessary to
go to the other end of the vernier and continue in the same
direction, toward the center, until the coincidence is found.
This vernier is used on the vertical circle of transits when the
space is too small for a double vernier.
There is another type of transit vernier, which is occasionally
used, in which the degree is divided into hundredths instead of
minutes.
54. Eccentricity. — : If the two opposite verniers of a tran-
sit do not read exactly alike it is usually due to a combination
of two causes, (i) because the center of the vernier plate does
not coincide with the center of the graduated circle, (2) because
the vernier zeros have not been set exactly 180*^ apart. The
first cause produces a variable difference while the second
produces a constant difference.
It will be noticed that the effect of these errors is similar
to that described in Art. 34, p. 25, on Adjustments of the Com-
pass ; the eccentricity of the circles of the transit corresponding
to the bent pivot of the compass
and the error in the position of
the verniers of the transit corre-
sponding to the bent needle of the
compass.
With reference to the eccen-
tricity of the plates, let C in
Fig. 27 be the center of the
vernier plate and C the center of
the circle. Let GF be a line
through the two centers. When
one vernier is at F and the
other is at G the vernier readings
will be the same as though C
and C were coincident, since the displacement of the center of
the circle occurs in the direction of the lines of graduation at F
and G. If the telescope is then turned at right angles to its
former position, the verniers then being at D and E^ the readings
g D
Fig. 27.
Eccentricity of
Circle.
ECCENTRICITY OF CIRCLES 45
of opposite verniers will differ by the maximum amount. Suppose
that the graduations are numbered from o** right-handed to 360°.
When the vernier is at an intermediate position, as at Ay it will
be seen that it reads too much by the amount AA\ The opposite
vernier at B reads too little by the amount BB\ Since AB and
A'ff are parallel, BB* and AA' are equal. Consequently the
mean of the two vernier readings will be the true reading and
the eccentricity is in this way eliminated. Since the effect of
eccentricity is never more than a very few minutes it is cus-
tomary to read the degrees and minutes on one vernier and the
minutes only on the other.
55. In spite of the fact that the two verniers are not 180°
apart no error is introduced provided; (i) that the same vernier
is always used, or (2) that the mean of the two vernier readings
is always taken. But if vernier A is set and the angle is read
on vernier B an error does enter. Where only one vernier is
read always read the vernier that was set at O^.
In good instruments both of these errors are very small,
usually smaller than the finest reading of the vernier.
USE OF THE TRAIVSrr.
56. SETTIlfG UP THE TRANSIT. — In setting the transit
over a point, place one leg of the tripod in nearly the right posi-
tion on the ground, then g^sp the other two and move the in-
strument in such a way as to bring the head over the point and
at the same time keep the plates of the instrument approximately
level, giving the tripod sufficient spread to insure steadiness.
The tripod legs should be pressed firmly into the ground. The
nuts at the top of the tripod legs should be tight enough so that
the legs are just on the point of falling of their own weight
when raised from the ground. If they are loose the instrument
is not rigid ; if they are too tight it is not in a stable condition
and may shift at any moment.
If the point is on sloping ground it is often convenient, and
usually insures greater stability, to set two legs on the down-
hill side and one leg uphill. When the center of the instrument
is over the point but the tripod head is not nearly level it can be
46 MEASUREMENT OF ANGLES [Chap. in.
leveled approximately without moving the instrument away from
the point by moving, one, sometimes two, of the tripod legs in an
arc of a circle about the point. Nothing but practice will make
one expert in setting up the transit.
It is desirable to bring the instrument very nearly level by
means of the tripod ; this is really a saving of time because under
ordinary conditions it takes longer to level up by the leveling
screws than by the tripod. It also saves time on the next set-
up to have the leveling screws nearly in their mid position. If
the transit is set by means of the tripod, say, within o.oi or 0.02
ft. of the point, the exact position can be readily reached by
means of the shifting head^ which may be moved freely after
any two adjacent leveling screws are loosened. When the tran-
sit has been brought directly over the point, the leveling screws
should be brought back to a bearing. In the first (rough) setting
the plumb-bob should hang, say, an inch above the point, but
when the shifting head is used it should be lowered to within
about \ inch or less of the point.
57. In leveling the instrument, first turn the plates so that
each plate level is parallel to a pair of opposite leveling screws.
Fig. 28. Cut Showing How Fingers Move in Leveling.
Each level is therefore controlled by the pair of leveling screws
which is parallel to it. Great care should be used in leveling.
The screws must not be loose as this will cause the plates to tip
and perhaps to move horizontally which would change the posi-
tion of the plumb-bob over the point. On the other hand they
SETTING UP THE TRANSIT 47
must not be too tight as this will not only injure the instrument
but will cause errors due to strains in the metal. To level
the instrument, grasp one pair of opposite screws between the
thumbs and forefingers and turn so that the thumbs move either
toward each other or away from each other, as illustrated in
Fig. 28. In this way one screw is tightened as much as the
other is loosened. The motion of both screws must be uniform ;
if they bind, the one which is being loosened should be turned
faster. If this does not appear to remedy matters then the
other pair of screws is binding and should be loosened slightly.
Only experience will teach one to level an instrument quickly
and correctly. It may be convenient for beginners to remember
that in leveling the instrument the bubble will move in the same
direction as the left thumb moves. After one bubble has been
brought nearly to the center of its tube the other bubble is
centered in a similar manner by its pair of leveling screws. In-
stead of tr)dng to cent'er one bubble exactly before beginning on
the second one it is better to get both of them approximately
level, after which first one bubble and then the other may be
brought exactly to the centfer. After the instrument is leveled
the plumb-bob should be examined to see that it has not been
moved from over the point during the process of leveling.
58. To MEASURE A HORIZONTAL ANGLE. — After setting
the instrument up over the point, first set the zero of one of the
verniers opposite the zero of the circle. This is done by turning
the two plates until the two zeros are nearly opposite, clamping
the plates firmly together with the upper clampi and then bring-
ing the two into exact coincidence by means of the tangent screw
which goes with the upper clamp. If a line on the vernier is
coincident with a line on the circle then the two adjacent hnes
on the vernier will fail to coincide with the corresponding lines
on the circle by equal amounts (Art. 53, p. 39). Hence the coin-
cidence of any line on the vernier with a line on the circle can
be more accurately judged by examining also the adjacent di-
visions and noting that they are symmetrical with respect to
the coincident lines. A pocket magnifier, or "reading glass,'*
is generally used for setting and reading the vernier. Never
touch the clamp after a setting has been made by means of the
48 MEASUREMENT OF ANGLES [Chap. HI.
tangent screw. In setting with the tangent screw it is better
to do this by a right-hand turn, i.e., by turning the screw in
the direction which compresses the spring against which it
works. If the screw needs to be turned back, instead of turn-
ing it to the exact setting turn it back too far and then bring
it up to the accurate setting with a right-hand motion, thereby
insuring a firm bearing of the spring against the screw. The
two plates which are now clamped in proper position are free to
turn together about the vertical axis. Turn to the first object
and point the telescope at it approximately by looking over the
top of the telescope. When turning the instrument so as to
sight the first point it is good practice to touch the lower plate
only. Focus the telescope by moving the eyepiece until the
cross-hairs are distinct and then moving the objective until the
image is distinct. It is sometimes convenient to point the tele-
scope at the object when focusing the cross-hairs so that they
can be readily seen.* Test for parallax by moving the eye
slightly from one side to the other. Move the telescope until
the vertical cross-hair is very nearly on the point. It is better
to use that part of the cross-hair which is near the center of the
field of view. Clamp the lower plate by means of the lower
clamp, and set exactly on the point by the lower tangent 5crew.
The line of sight is now fixed on the first object. To measure
the angle loosen the upper clamp, turn the telescope to the
second point, and focus the objective if necessary. Set nearly
on the point, clamp the upper plate, and set the vertical cross-
hair exactly on the point by means of the upper tangent screw.
The angle is then read on the vernier which was set at 0^.
The tangent screws should not be used to move the plates
over large angles. Acquire the habit of setting closely by hand
and using the tangent screw for slight motions only.
59. TO MEASURE AN ANGLE BY REPETITION. — The
eyepiece magnifies the image so much thnt it is possible to set
the cross-hair on a point much more closely than the vernier will
* If the eyepiece is focused on the cross-hairs with the telescope pointing at
the sky, as is frequently done, they will be found to be approximately in locos
when looking at the object ; but for accurate work the eyepiece should be focused
on the cross-hairs when the objective is in focus on the object.
MEASURING ANGLES BY REPETITION 49
read. The graduation of the circle is very accurate and can be
depended upon closer than the vernier can be read, consequently
the full value of the instrument is not utilized by single readings
of an angle. To obtain the value of an angle more accurately
proceed as follows. After the first angle has been measured
leave the two plates clamped together, loosen the lower clamp
and turn back to the first point. Set on the first point, using
the lower clamp and its tangent screw. Then loosen the upper
clamp and set on the second point, using the upper clamp
and its tangent screw, thus adding another angle, equal to the
first one, to the reading on the circle. Repeat this operation,
say, six times. The total angle divided by six will give a more
precise result than the first reading. Suppose that the angle is
actually 18® 12' 08' ; if a "one-minute " instrument is being used
it is impossible to read the 08" on the vernier, so the reading
will be 18° 12'. Each repetition will add 08" (nearly) and after
the 6th repetition, the amount wiU be 48" which will be read as
i'. After the 6th pointing the total angle will then be read 109°
13' which divided by 6 gives 18° 12' 10", a result in this case
correct to the nearest 10". To eliminate errors in the adjust-
ment of the transit the above process should be repeated with
the instrument reversed and the mean of the two values used.
(See Art. 79, p. 61.) It is customary to take only the ist and
6th readings, but as a check against mistakes it is well for the
beginner to examine the vernier reading after each repetition
and see that ^ the second reading, J the third, etc., nearly
equals the first reading.
Repetition has also the advantage of eliminating, to a great
extent, errors of graduation. If an angle is about 60® and is
repeated 6 times it will cover a whole circumference. If there
are systematic errors in the graduations the result is nearly free
from them. The effect of accidental, or irregular, errors of
graduation is decreased in proportion to the number of repeti-
tions. In the best modern instruments the errors of graduation
seldom exceed a few seconds.
Little is gained by making a very large number of repetitions
as there are systematic errors introduced by the action of the
clamps, and the accuracy apparently gained is really lost on this
50 MEASUREMENT OF ANGLES [Chap. HI.
account. Three repetitions with the telescope normal and three
with the telescope inverted are sufficient for anything but very
exact work.
It is desirable that as little time as possible should elapse
between pointings, as the instrument cannot be relied upon to
remain perfectly still. As a matter of fact it is vibrating and
"creeping" nearly all the time from numerous causes. For
example, when the instrument is set up on frozen ground, it will
quickly change its position on account of the unequal settlement
of the tripod legs. Changes of temperature, causing expansion
or contraction of the metal of the instrument, and the effect of
wind introduce errors. The more rapidly the measurements can
be made, consistent with careful manipulation, the better the
results will be. If the transit is set up on shaky ground
the transitman should avoid walking around his instrument.
60. Repetition is useful not only to secure precision, but
also as a check against mistakes. If a mistake is made on the
first reading of an angle the vernier, on the second reading,
falls in a new place on the circle so that the mistake is not
likely to be repeated. It is common practice to repeat, or
"double," all important angles and divide the second reading
by 2 simply as a check on the first reading.
61. TO LAY OFF AN ANGLE BY REPETITION. — There is
no direct method of laying off an angle by repetition as in the
case of measuring an angle, therefore the following indirect
method is used. With the vernier set at 0° and the telescope
sighted on the first point the angle is carefully laid off on the
circle and the second point set in line with the new position of
the telescope. Then this angle which has been laid off is
measured by repetition as precisely as is desired as described in
Art. 59. The resulting angle obtained by repetition is a more
precise value than the angle first set on the vernier. The
difference between this value and the angle desired is the cor-
rection which should be made at the second point. This can be
readily done by measuring approximately the distance from the
instrument to the second point, and computing the perpendicular
offset to be laid off at the second point. (The offset for an
angle of one minute at a distance of 100 ft. is nearly 0.03 ft.)
LAYING OFF AN ANGLE BY REPETITION 5 1
62. RUNNING A STRAIGHT LINE — One Point Visible from
the Other. — There are several ways in which a straight line
may be fixed on the ground, depending upon the existing con-
ditions. If the line is fixed by the two end points one of which
is visible from the other, the method of setting intermediate
points would be to set the transit over one point, take a " fore-
sight " on the other and place points in line. For very exact
work the instrument should be used in both the direct and re-
versed positions (Art. 79, p. 61). This will eliminate errors of
adjustment such as failure of the telescope to revolve in a true
vertical plane, or failure of the objective tube to travel parallel
to the line of sight.
63. RUNNING A STRAIGHT LINE — Neither Point Visible
from the Other. — If neither point can be seen from the other
then it is necessary to find some point, by trial, from which the
terminal points can be seen. The transit is set up at some point
estimated to be on the line, a "backsight" is taken on one of
the points and the instrument clamped. The telescope is then
reversed on its horizontal axis. If the vertical cross-hair strikes
the second point the instrument is in line ; if not, then the error
in the position of the instrument must be estimated (or meas-
ured) and a second approximation made. In this way, by suc-
cessive trials, the true point is attained. The final tests should
be made with the instrument in direct and reversed positions to
eliminate errors of adjustment of the line of sight and the hor-
izontal axis. To eliminate errors in the adjustment of the plate
bubbles the plate level which is perpendicular to the line should
be relevded just before making the second backsight and while
the telescope is pointing in that direction. This can be more
readily done if, when the transit is set up, one pair of opposite
leveling screws is turned so as to be in the direction of the line ;
then the other pair will control the level which is perpendicular
to the line of sight. After one point has* been found by this
method other points may be set as described in the previous
article.
Another method of running a line between two points one of
which is not visible from the other would be to run what is called
a random line as described in Art. 191, p. 169.
52 MEASUREMENT OF ANGLES [Chap. m.
64. Prolonging a Straight Line. — If a line is fixed by two
points A ^nd B and it is desired to prolong this line in the direc-
tion AB, the instrument should be set up at A, a sight taken
on B and other points set in line beyond B. When it is not
possible to see beyond B from Ay the transit should be set up at
B and points ahead should be set by the method of backsighting
and foresighting as follows. With the transit at -5 a backsight
is taken on A and the instrument clamped. The telescope is
inverted and a point set ahead in line. The process is repeated,
the backsight being taken with the telescope in the inverted
position. The mean of the two results is a point on the line AB
produced. The transit is then moved to the new point, a back-
sight is taken on B, and another point set ahead as before.
In this last case, if a line is prolonged several times its own
length by backsighting and foresighting, there is likely to be a
constantly increasing error. In the first case, where the line is
run continually toward a point known to be correct, the errors
are not accumulating.
65. Methods of Showing Sights. — If the point sighted is
within a few hundred feet of the instrument, a pencil may be used
and held vertically in showing a point for the transitman to sight
on. Sighting-rods are used on long distances.* Where only
the top of the rod or pole is visible a considerable error is intro-
duced if it is not held plumb. A plumb-line is much more
accurate for such work but cannot be easily seen on long sights.
Under conditions where the plumb-line cannot be readily seen
some surveyors use for a sight an ordinary white card held with
one edge against the string or held so that the center of the
card is directly behind the string. If the edge of the card is
held against the string, the transitman must be extremely careful
that he is sighting on the proper edge.f
* It is desirable that the foresight should be of a color such that the cross-
hair is clearly seen, and of a width such that the cross-hair nearly (but not quite)
covers it.
t It is common among some surveyors to use a two-foot rule for a sight.
The rule is opened so that it forms an inverted V (A). The plumb-string is
jammed into the angle of the A by pressing the two arms of the rule together.
The rule is then held so that the plumb-string as it hangs from the rule appears to
bisect the angle of the A.
Another device b to attach to the plumb-line an ordinary fish-line float (shaped
RUNNING A STRAIGHT LINE 53
Whenever the instrument is sighted along a line which is to
be frequently used or along which the transit is to remain sifted
for any considerable time the transitman should if possible select
some well-defined point which is in the line of sight, called a
"foresight." If no definite point can be found one may be
placed in line for his use. By means of this " foresight " the
transitman can detect if his instrument moves off the line, and
can set the telescope exactly " on line " at any time without re-
quiring the aid of another man to show him a point on the line.
66, Signals. — In surveying work the distances are frequently
so g^eat that it is necessary to use hand signals. The following
are in common use.
''Right'' or ''Left:' — The arm is extended in the direc-
tion of the motion desired, the right arm being used for a motion
to the right and the left arm for a motion to the left. A slow
motion is used to indicate a long distance and a quick motion a
short distance.
"Plumb the Pole.'* — The hand is extended vertically above
the head and moved slowly in the direction it is desired to have
the pole plumbed.
" All Right," — Both arms are extended horizontally and
moved vertically.
" Give a Foresight." — The transitman, desiring a foresight,
motions to the rodman, by holding one arm vertically above his
head.
" Take a Foresight." — The rodman desiring the transit-
man to sight on a point, motions the transitman by holding one
arm vertically above his head and then he holds his lining-pole
vertically on the point.
" Give Line." — When the rodman desires to be placed "on
line " he holds his lining-pole horizontally with both hands over
his head and then brings it down to the ground in a vertical
position. If the point is to be set carefully, as a transit point,
like a plumb-bob). This may be fastened so that its axis coincides with the
string and so that it can be raised and lowered on the string. It should be painted
with such colots that it can be seen against any background.
The man showing the sight for the transitman should always try to stand so
that the sun will shine on the object he is holding ; on long sights it is difficult
(sometimes imposaUe) to see an object in a shadow.
54 MEASUREMENT OF ANGLES [Chap. III.
the rodman waves the top end of pole in a circle before bringing
it to the vertical position.
''Pickup the Transit:' -^V^en the chief of the party de-
sires to have the instrument set at another point he signals to
the transitman by extending both arms downward and outward
and then raising them quickly.
All signals should be distinct so as to leave no doubt as to
their meaning. Care should be taken to stand so that the back-
ground will not prevent the signals being distinctly seen. The
palms of the hands should be shown in making the signals, and
for distant signals a white handkerchief is often used Where
much distant signaling is to be done flags are attached to the
lining-poles. Special signals may be devised for different kinds
of work and conditions.
67. TO MEASURE A VERTICAL ANGLE. — In measuring a
vertical angle with a transit, first point the vertical cross-hair
approximately at the object, then set the horizontal cross-hair
exactly on the point by means of the clamp and tangent screw
controlling the vertical motion. Next read the vertical arc or
circle. Then, without disturbing the rest of the transit, unclamp
the vertical arc, and bring the telescope to the horizontal position
by means of the level attached to the telescope, and the clamp
and tangent screw of the vertical arc. When the telescope
bubble is in the center read the vertical arc again. This gives
the index correctiony to be added or subtracted according to
whether the two readings are on opposite or on the same side of
zero. In some forms of transit the vernier is on a separate arm
which also carries a level. By bringing this level to the center
of the tube by means of its tangent screw the index correction
is reduced to zero each time and the true angle read directly.
Instruments provided with this form of level have no level
attached to the telescope.
If the transit has a complete vertical circle errors in the ad-
justment of the bubble and the horizontal cross-hair may be eli-
minated by inverting the telescope, turning it through 180°
azimuth, and remeasuring the angle. The mean of the two
results is free from such errors. If the transit is provided with
only a portion of a circle the vernier will be off the arc when
PRECAUTIONS IN USE OF TRANSIT 55
the telescope is inverted, consequently with a transit of this type
the elimination cannot he effected.
68. PRECAUTIONS IN THE USE OF THE TRANSIT. — In
the preceding text several sources of error and also precautions
against mistakes have been mentioned, but in order that the be-
ginner may appreciate the importance of handling the instru-
ment carefully he should make the following simple tests.
1. Set the transit up with the three points of the tripod
rather near together so that the instrument will be high and un-
stable. Sight the cross-hair on some definite object, such as the
tip of a church spire, so that the slightest motion can be seen.
Take one tripod leg between the thumb and forefinger and twist
it strongly; at the same time look through the telescope and
observe the eflfect.
2. Press' the tripod leg laterally and observe the effect on
the level attached to the telescope ; center the bubble before
testing.
3. Step on the ground about i or 2 inches from the foot of
one of the tripod legs and observe the effect on the line of sight.
4. Breathe on one end of the level vial and observe the mo-
tion of the bubble.
5. Press laterally on the eyepiece and observe the effect on
the line of sight.
These motions, plainly seen in such tests, are really going on
all the time, even if they are not readily apparent to the observer,
and show the necessity for careful and skillful manipulation.
The overcoat dragging over the tripod, or a hand carelessly rest-
ing on the tripod, are common sources of error in transit work.
Before picking up the transit center the movable head bring,
the leveling screws back to their mid position, loosen the lower
dampi and turn the telescope either up or down.
56
MEASUREMENT OF ANGLES
[Chap. HI.
ADJUSTMENTS OF THE TRANSIT.
69. If an instrument is badly out of adjustment in all re-
spects, it is better not to try to completely adjust one part at a
time but to bring the instrument as a whole gradually into ad-
justment. If this is done, any one process of adjusting will not
disturb the preceding adjustments, the parts are not subjected
to strains, and the instrument will be found to remain in adjust-
ment much longer than it would if each adjustment were com-
pleted separately.
Nearly all adjustments of the transit, in fact of nearly all
surveying instruments, are made to depend on the principle of
reversion. By reversing the position of the instrument the effect
of an error is doubled.
70. ADJUSTMENT OF THE PLATE BUBBLES. — To adjust
the Plate Levels so that Each lies in a Plane Perpendicular to the
Vertical Axis of the Instrument. Set up the transit and bring
— il
Fig. 29. Adjustment of the Plate Bubbles.
the bubbles to the center of their respective tubes. Turn the
plate 180° about its vertical axis and see if the bubbles remain
in the center. If they move from the center, half this distance
is the error in the adjustment of the tube. (See Fig. 29.) The
adjustment is made by turning the capstan-headed screws on the
ADJUSTMENTS OF TRANSIT $7
bubble tube until the bubble moves half-way back to the center
as nearly as this can be estimated Each bubble must be ad-
justed independently. The adjustment should be tested again
by releveling and reversing as before, and the process continued
imtil the bubbles remain in the center when reversed. When
both levels are adjusted the bubbles should remain in the
centers during an entire revolution about the vertical axis.
7z. Adjustmeut of the Cross-Hairs. — ist. To put the
Vertical Cross-Hair in a Plane Perpendicular to the Horizontal
Axis. Sight the vertical hair on some well-defined point, and,
leaving both plates clamped, rotate the telescope slightly about
the horizontal axis (see Fig. 30).
The point should appear to travel on the vertical cross-hair
throughout its entire length. If it does not, loosen the screws
Fig. 80* Adjustment of the Cross-Hairs (First Part).
holding the cross-hair ring, and by tapping lightly on one
of the screws, rotate the ring until the above condition is
satisfied. Tighten the screws and proceed with the next
adjustment*
58
MEASUREMENT OF ANGLES
[Chap. III.
7a. 2nd. To make the Line of Sight Perpendicular to the
Horizontal Axis.* (See Fig. 31.) Set the transit over a point
Fig. 81. Adjustment of the Cross-Hatrs (Second Part).
A. Level up, clamp both plates, and sight accurately on a
point B which is approximately at the same level as A.
Reverse the telescope and set C in line with the vertical cross-
hair. By A, and C should be in a straight line. To test this,
turn the instrument about the vertical axis until B is again
sighted. Clamp the plate, reverse the telescope, and observe if
point Cis in line. If not, set point D in line just to one side
of C and then the cross-hair ring must be moved until the
vertical hair appears to have moved to poinf E, one-fourth the
distance from D toward C, since, in this case, a double reversal
has been made.
The cross-hair ring is moved by loosening the screw on one
side of the telescope tube and tightening the opposite screw.
If D falls to the right of C then the cross-hair ring should be
moved to the left ; but if the transit has an erecting eyepiece
the cross-hair will appear to move to the right when viewed
through the telescope. If the transit has an inverting eyepiece
the cross-hair appears to move in the same direction in which
the cross-hair is actually moved.
The process of reversal should be repeated until no further
adjustment is required. When finally adjusted, the screws
should hold the ring firmly but without straining it.
* In making the adjustment in the shop with collimators instrument makers
seldom level the transit carefully. In field adjustments it is desirable, although
not necessary, to level the instrument. The essential condition is that the veiticaJ
axis shall not alter its position.
ADJUSTMENTS OF TRANSIT
59
73. Abjustmeut of the STAHDARDS. — To make the Hori-
zontal Axis of the Telescope Perpendicular to the Vertical Axis of
the Instrument. (See Fig. 32.) Set up the transit and sight
the vertical cross-hair on a high point A,
such as the top of a church steeple.
Lower the telescope and set a point B in
line, on the same level as the telescope.
Reverse the telescope, turn the instru-
ment about its vertical axis, and sight
on B, Raise the telescope until the point
A is visible and see if the cross-hair comes
on A. If not, note point C in line and
at same height as A. Then half the dis-
tance from C to -4 is the error of adjust-
ment. Loosen the screws in the pivot
cap and raise or lower the adjustable end
of the horizontal axis by means of the
capstan-headed screw under the end of
the axis. Repeat the test until the high
and the low points are both on the cross-
hair in either the direct or reversed posi-
tions of the transit. The adjusting screw
should be brought into position by a right-
hand turn, otherwise the block on which
the horizontal axis rests may stick and
not follow the screw. The cap screws
should then be tightened just enough to
avoid looseness of the bearing.
74. Adjustment of the Telescope Bubble. — This is adjusted
by the "/^g"" mefAod, or direct method^ as explained in Art. 128,
p. 91. This consists in first determining a level line by using
the instrument in such a way as to eliminate the error of the
bubble, and then centering the bubble while the line of sight is
horizontal.
75* Adjustment of the Auxiliary Level on the Vernier of the
Vertical Arc— (See Art. (yy^ p. 54.) To adjust the Level
80 that it is in the Center of the Tube when the Line of Sight
18 Level and the Vernier reads 0^ This is adjusted by the " peg
Fig. 32. Adjustment
OF THE Standards.
6o MEASUREMENT OF ANGLES [Chap. IIL
method" (Art. 128, p. 91). The bubble is first brought to
the center of the tube by means of its tangent screw. Then the
telescope is moved until the vernier of the vertical arc reads o^
The instrument is then in condition to be used as a leveling
instrument and is adjusted by the "peg method."
If the telescope is provided with an attached level the auxiliary
level could be adjusted by comparing it with the telescope level
as follows. Level the telescope by means of its attached level,
make the vernier read o by means of the tangent screw of the
vernier, and then bring the bubble of the auxiliary level to the
center by means of its adjusting screws.
76. Adjustment of the Vernier of the Vertical Circle. — To
make the Vernier read 0^ when the Telescope Bubble is in the
Center of the Tube. If there is any index error (Art. (>y, p. 54)
bring the bubble to the center, loosen the screws holding the
vernier, and tap lightly until the zeros coincide. Tighten the
screws and test again. In some instruments the vernier is con-
trolled by a slow-motion screw for setting the index at the zero
of the circle.
77. Adjustment of the Objective Slide. — To make the
Objective Slide move Parallel to the Line of Sight. If the tube
holding the objective.is adjustable it must be placed so that the
direction of the line of sight will not be disturbed when the
telescope is focused. The adjustment may be made as follows.
Adjust tlie line of sight as in Art. 72, using very distant
points. This will require the objective to be drawn in nearly
as far as it will go and hence the position of the objective will
be changed but little by any subsequent lateral adjustment of
the tube. Next repeat the test for the adjustment of the line of
sight by using two points which are very near the instrument.
In sighting on these points the objective must be run out and
any error in its adjustment will change the direction of the line
of sight so that it is no longer perpendicular to the horizontal axis
of the instrument. In case the instrument fails to stand this
test the objective slide does not move parallel to the line of
sight. The adjustment is made by moving the adjustment
screws of the objective slide so as to apparently increase the
error making, by estimation, one-quarter the correction required.
HOW TO ELIMINATE ERRORS 6l
The adjustment of the line of sight should be again tested on
two distant points and the cross-hairs moved in case the second
adjustment appears to have disturbed the first.
78. Shop Adjustments. — The adjustment of the objective
slide and other adjustments such as centering the eyepiece tube
and centering the circles are usually made by the instrument
maker.
79. HOW TO ELIMINATE THE EFFECT OF ERRORS OF
ADJUSTMEUT in THE TRANSIT. — Errors of adjustment in the
plate bubble may be avoided by leveling up and reversing as
when adjusting. Then, instead of altering the adjustment,
simply move the bubble half-way back by means of the leveling
screws. This makes the vertical axis truly vertical. Then the
bubbles should remain in the same parts of their respective
tubes as the instrument revolves about its vertical axis.
Errors of the line of sight and errors of the horizontal axis
are eliminated by using the instrument with the telescope in the
direct and then in the reversed position and taking the mean of
the results whether the work is measuring angles or running
straight lines.
Errors of eccentricity of the circle are completely eliminated
by reading the two opposite verniers and taking the mean.
Errors of graduation of the circle are nearly eliminated by
reading the angle in different parts of the circle or by measur-
ing the angle by repetition.
80. Care of Instruments. — A delicate instrument like the
transit requires constant care in order that the various parts
may not become loose or strained. Care should be taken that
the tripod legs do not move too freely, and that the metal shoes
on the feet of the tripod do not become loose. The transit
should be securely screwed to the tripod. In caring for the
lenses a camel's hair brush should be used for dusting them and
soft linen with alcohol for cleaning them. The objective should
not be unscrewed except when absolutely necessary, and when
replaced it should be screwed in to the reference mark on the
barrel of the telescope. Grease should never be used on exposed
parts of an instrument, as it collects dust. Care should be
taken not to strain the adjusting screws in making adjustments.
62 MEASUREMENT OF ANGLES [Chap. HI.
The instrument should be protected as much as possible from
the sun, rain, and dust. If the instrument is carried in the box
it is less likely to get out of adjustment than when carried on
the shoulder, but the former is often inconvenient. It is cus-
tomary in traveling by carriage or rail to carry the transit in its
box. While being carried on the shoulder the lower clamp
should be left undamped so that in case the instrument strikes
against anything, some parts can give easily and save the instru-
ment from a severe shock. When the transit is in use, be care-
ful not to clamp it too hard, but clamp it firmly enough to in-
sure a positive working of the tangent screws and so that no
slipping can occur.
8i. Common sources of error in Transit work. —
1. Nonadjustment, eccentricity of circle, and errors of gradu-
ation. «
2. Changes due to temperature and wind.
3. Uneven settling of tripod.
4. Poor focusing (parallax).
5. Inaccurate setting over point.
6. Irregular refraction of atmosphere.
82. Common MISTAKES IN TRANSIT WORK. —
1. Reading in the wrong direction from the index on^ a
double vernier.
2. Reading the opposite vernier from the one which was set
3. Reading the circle wrong, e.g., reading 59° for 61°. If
the angle is nearly 90^ reading the wrong side of the 90
point, e.g., 88° for 92°.
4. Using the wrong tangent screw.
64
SOLAR ATTACHMENT
[Chap. UI.
Fig. 33. Solar Attachment to Transit.
(The authors are indebted to C. L. Berger & Son
for the photograph from which this cut was made.)
SOLAR ATTACHMENT 65
THE SOLAR ATTACHMBIIT.
83. DESCRIPTIOH OF SOLAR ATTACHMEHT. — One of the
most important auxiliaries to the engineer's transit is the solar
attachment, one form of which is shown in Fig. 33. This is a
small instrument which may be attached to the telescope and by
means of which a true meridian line can be found by an observa-
tion on the sun. In the form here shown the principal parts are
the polar axis^ which is attached to the telescope perpendicular
to the line of sight and to the horizontal axis, and a small
telescope which is mounted on the polar axis. This telescope
can be revolved about the polar axis and can be inclined to it
at any desired angle. The polar axis is provided with four ad-
justing screws for making it perpendicular to the line of sight
and to the horizontal axis.
Another form of attachment has the solar telescope replaced
by a lens and a screen on which the sun's image can be thrown.
This defines a line of sight and is in reality the equivalent of a
telescope. This instrument is provided with the arc of a circle
known as the declination arc, the use of which will be explained
later.
Still another form consists of a combination of mirrors (simi-
lar to those of a sextant) which can be placed in front of the
objective. In this form the telescope of the transit serves as the
polar axis.
While these various solar attachments differ in the details
of construction, they all depend upon the same general prin-
ciples.
84. THE Celestial sphere. — In order to understand the
theory of this instrument it will be necessary to define a few
astronomical terms. Fig. 34 represents that half of the celes-
tial sphere which is visible at one time to an observer on the
surface of the earth. For the purposes of this problem the
celestial sphere may be regarded as one having its center at
the center of the earth and a radius equal to the distance of the
sun from the earth. The sun in its apparent daily motion
would then move around in a circle on the surface of this
66
MEASUREMENT OF ANGLES
[Chap. HI.
sphere. The circle NES IV is the observer's horizon and is the
boundary between the visible and invisible parts of the celestial
sphere. The point Z is the zenith and is the point where a
plumb-line produced would pierce the celestial sphere. The
circle SZPN is the observer's meridian and is a vertical circle
through the pole. The circle EQ W is the celestial equator.
The circle AMB^ parallel to the equator, is a parallel of decli-
nation^ or the path described by the sun in its apparent daily
Fig. 34. Diagram of the Celestial Hemisphere.
motion from east to west. The sutis declination is its angular
distance from the equator, or the arc OT. The declination is
considered positive when north and negative when south. The
polar distance of the sun is the complement of the declination
represented by the arc OP
85. OBSERVATION ON THE SUN FOR- MERIDIAN WITH
SOLAR Attachment. — If the polar axis of the instrument is
made to point to the celestial pole, i.e., made parallel to the
earth's axis, then the small telescope can be made ' to follow
the sun in its daily path by simply giving it an inclination to the
polar axis equal to the sun!s polar distance and revolving it
about the polar axis.
OBSERVATION FOR MERIDIAN WITH SOLAR 6/
(i) To find the true meridian by an observation on the sun
first make the angle between the polar axis and the solar tele-
scope equal to the sun's polar distance at the time of the obser-
vation. This is done by turning the solar telescope into the
same plane ss the main telescope by sighting both on some dis-
tant object, and then making the angle between the two telescopes
equal to the sun's declination. Some instruments are provided
with a declination arc upon which the declination angle can be
laid off directly. Others have a small spirit level attached to
the small telescope, in which case the vertical circle of the tran-
sit is used for laying off the declination angle. Incline the main
telescope until the reading of the vertical circle equals the de-
clination, and clamp ; then level the solar telescope by means of
the attached level. The angle between the polar axis and the
solar telescope is then 90° plus or minus the reading of the
vertical circle.
(2) By means of the vertical cfa-cle of the transit incline the
polar axis to the vertical by an angle equal to the co-latitude of
the place, which is 90° minus the latitude. The polar axis now
has the same angle of the elevation as the celestial pole.
(3) If the observation is in the forenoon, place the solar
telescope on the left of the main telescope (on the right if in the
afternoon); then, by moving the whole instrument about the
vertical axis and the solar telescope about the polar axis, point
the solar telescope at the sun. The sun's image is brought to
the center of the square formed by foifr cross-hairs, or ruled
lines, in the solar telescope. The final setting is made by the
tangent screw controlling the horizontal motion of the transit
and the one controlling the motion of the solar about the polar
axis. Only one position can be found where the solar telescope
will point to the sun. In this position the vertical axis points
to the zenith, the polar axis to the pole, and the solar telescope
to the sun. The instrument has thus solved mechanically the
spherical triangle having these three points (Z, /*, O) as vertices.
The horizontal angle between the two telescopes is equal to the
sun's true bearing. Since the solar telescope is pointing to the
sun the main telescope must be in the plane of the meridian.
If all of the work has been correctly done it will be observed
68 MEASUREMENT OF ANGLES [Chap.HI.
that the sun's image will remain between the cross-hairs set
parallel to the equator, and therefore the sun can be followed
in its path by a motion of the solar telescope alone. If it is
necessary to move the instrument about the vertical axis to
point the solar telescope again at the sun this shows that the
main telescope was not truly in the meridian.
After the meridian has been determined the main telescope
may then be lowered and a point set which will be due north or
due south of the instrument.
86. Computation of Declination Settings. — The sun's polar
distance may be obtained from the " American Ephemeris and
Nautical Almanac/' published by the Government. The polar
distance is not given directly, but its complement, the sun's
apparent declination^ is given for each day and for the instant
of Greenwich Mean Noon, The rate of change of the declina-
tion, or the difference for I hour^ is also given. In order to use
this for any given locality, it is first necessary to find the local
or the standard time corresponding to mean noon of Greenwich.
In the United States, where standard time is used, the relation
to Greenwich time is very simple. In the Eastern time belt
the time is exactly 5 hours earlier than at Greenwich ; in the
Central^ 6 hours earlier ; in the Mountain, 7 hours earlier ; in
the Pacific, 8 hours earlier. If a certain declination corresponds
to Greenwich mean noon, then the same declination corresponds
to 7 A.M. in the Eastern belt or 6 A.M. in the Central belt,
etc. The declinatioif for any subsequent hour of the day may
be found by adding (algebraically) the difference for i hour mul-
tiplied by the number of hours elapsed. Declinations marked
North must be regarded as positive and those marked South as
negative. An examination of the values of the declination for
successive days will show which way the correction is to be
applied. It will be useful also to remember that the declination
is 0° about March 21, and increases until about June 22, when
it is approximately 23*^ 27' North ; it then decreases, passing the
o*^ point about September 22, until about December 21 when it
is approximately 23° 27' South; it then goes North until
March 2 1 when it is o® again.
After the correct declination is found it has still to be cor-
OBSERVATION FOR MERIDIAN WITH SOLAR 69
rected for refraction of the atmosphere. The effect of refrac-
tion is to make the sun appear higher up in the sky than it
actually is. In the northern hemisphere, when the declination
is North this correction must be added, when South, subtracted ;
or algebraically it is always added.
The refraction correction may be taken from Table VII,
p. 507.
The co-latitude which must be set off on the vertical circle
may be obtained from a map or may be determimed by an ob-
servation which is made as follows. Set off the sun's .declination
for noon, as for any other observation, the two telescopes being
in the same vertical plane, and point the small telescope at the
sun. By varying the angle of elevation of the main telescope,
keep the solar telescope pomting at the sun until the maximum
altitude is reached. The angle read on the vertical circle is the
co-latitude (see also Art. 217, p. 195).
Example.
Latitude 40*»N. Longitude 4b 45m W.
Jan. 10, 1900.
Declination for Greenwich mean noon 21^ 59' 04^'
Difference for ih + 22''.25
TiMB. DBCX.INATION. RbPRACTION. SbTTINO.
7h. A.M. 21° 59' 04"
8 58 42 S'40" 21° 53' 02"
9 58 20 2 51 21 55 29
10 57 57 2 07 21 55 50
11 57 35 I 51 21 55 44
12 M. 57 13 (I 47) (21 55 26)
1 P.M. 56 51 I 51 21 55 00
2 56 28 2 07 21 54 21
3 56 06 2 51 ^ 21 53 15
4 55 44 5 40 21 50 04
87. "^ Comstpck's Method of finding the Refraction. — Set
the vertical cross-hair on one edge (or limi) of the sun and note
the instant by a watch. Set the vernier of the plate 10' ahead
and note the time when the limb again touches the cross-hair.
* See Bulletin of the University of Wisconsin, Science Series, VoL I, No. 3.
70 MEASUREMENT OF ANGLES [Chap. m.
Call the number of seconds between these observations n. Read
the altitude A. Then the refraction in minutes will be nearly
, ^ 2000
equal to — ;
An
88. Observation for meridian should not be made when the
sun's altitude is less than about lo^ because the refraction cor-
rection will be unreliable. Observations near noon are to be
avoided because a slight error in altitude produces a large error
in the resulting meridian. For good results therefore the obser-
vation should be made neither within an hour of noon nor near
sunrise or sunset.
89. MISTAKES m USING THE SOLAR ATTACHMEHT. —
1. Solar on wrong side of main telescope.
2. Refraction correction appUed wrong way.
ADJUSTMENTS OF THE SOLAR ATTACHMENT.
go. Adjustment of polar axis. — To make the Polar
Axis Perpendicular to the Plane of the Line of Sight and the Hori-
zontal Axis. Level the transit and the main telescope. Bring the
bubble of the solar telescope to the center of its tube while it is
parallel to a pair of opposite adjusting screws which are at the
foot of the polar axis. Reverse the solar telescope 1 80° about
the polar axis. If the bubble moves from the center position,
bring it half-way back by means of the adjusting screws just
mentioned and the other half by means of the tangent screw
controlling the vertical motion of the solar. This should be
done over each pair of opposite adjusting screws and repeated
until the bubble remains central in all positions.
91. ADJUSTMENT OF THE CROSS-HAIRS. — To make the
Vertical Cross-Hair truly Vertical. Sight on some distant point
with all the clamps tightened and, by means of the tangent
screw controlling the vertical motion of the solar, revolve the
solar telescope about its horizontal axis and see if the vertical
cross-hair remains on the point. If not, adjust by rotating the
cross-hair ring, as described in Art. 71, p. 57.
ADJUSTMENTS OF SOLAR ATTACHMENT J I
92. adjustmeut of Telescope bubble. — To make the
Axis of the Bubble Parallel to the Line of Sight. Level the main
telescope and mark a point about 200 ft. from the instrument in
line with the horizontal cross-hair. Measure the distance be-
tween the two telescopes and lay this off above the first point
which will give a point on a level with the center of the solar
telescope. Sight the solar at this point and clamp. Bring the
bubble to the center by means of the adjusting screws on the
bubble tube.
PROBLEMS.
1. Is ^ necessary that the adjustments of the transit should be made in the
order given in this chapter ? Give your reasons.
2. A transit is sighting toward B from a point A, In setting up the transit at
A it was carelessly set 0.0 1 ft. directly to one side of A, as at A'. What would
be the resulting error, i.e., the difference in direction (^in seconds) between AS and
A'B, (i) when AB — 40 ft., (2) when i^^ — 1000 ft.?
3. An angle of 90° is laid ofif with a ** one minute " transit, and the angle then
determined by six repetitions, the final reading being 179° 58'+ 360®. The point
sighted is 185 feet from the transit. Compute the offset to be laid off in order to
correct the first angle. Express the result in feet and also in inches.
4. An angle measured with a transit is 10° 15' 41'^^ The telescope of a level-
ing instrument is placed in front of the transit (with its objective toward the
transit) and the angle again measured and found to be o^ 18' 22^'. What is the
magnifying power of this level telescope ?
5. Compute the declination setting for every hour when observations on the
sun for meridian can be made at Boston ( Lat. 42^ 21' N, Long. 71^ 04' 30'' W)
on each of the following dates.
January i, 1906.
DecL S 23° 03' 27".9
Dift. for I hour, +ii ''.70
April 16, 1906.
Decl. N 9*> sy 34".2
Diff. for I hour, + 53''.44
July 2, 1906.
Decl. N 23° 05' 49''.5
Diff. for 1 hour, — io''.39
Sept. 25, 1906.
Decl. S o*> 35' 49".4
Diff. for I hour, -58''.5i
CHAPTER IV.
MBASURBMBNT OF DIFFERIINCB OF BLBVATION.
93, LEVEL SURFACE. — A level surface is a curved surface
which at every point is perpendicular to the direction of gravity
at that point, such, for example, as the surface of still water.
Any line of sight which is perpendicular to the direction of
gravity at a given point is therefore tangent to the level sur-
face at that point and is called a horizontal line.
94. The Spirit LeveL — In nearly all instruments the direc-
tion of gravity is determined by means of either a plumb-line or
a spirit level. A spirit level is a glass tube, the inside of which
is ground to a circular curve longitudinally, and nearly filled
with a liquid such as alcohol or ether, leaving enough space to
form a bubble. The grinding is usually done only on the inside
upper surface of the tube. The radius of the curve varies accord-
ing to the use which is to be made of the level ; a very short ra-
dius makes a slow moving bubble while a long radius makes a
very sensitive bubble. It is important that the curve should be
exactly circular so that equal distances on the tube should sub-
tend equal angles at the center. The level is provided with a
scale of equal parts, which may be either a metallic scale screwed
to the brass case holding the glass bubble tube, or it may con-
sist of lines etched on the glass itself. A point near the middle
of the tube is selected as the zero point and the graduations are
numbered both ways from that point. The straight line tangent
to the curve at the zero point of the scale is called the axis of
the bubble. The position of the bubble in the tube is deter-
mined by noting the positions of both ends. The bubble will
change its length with changes in temperature, consequently the
reading of one end is not sufficient to determine the position of
the bubble. On account of the action of gravity the bubble will
always move toward the higher end of the tube ; hence, when
the bubble is central the axis of the tube is horizontal.
72
THE WYE LEVEL 73
95. Angular Value of One Division of the Level Tube. — The
angular value of one division of a level tube is the angle, usually
expressed in seconds, through which the axis of the tube must
be tilted to cause the bubble to move over the length of one di-
vision on the scale. The simplest way of finding this in the
field consists in moving the bubble over several divisions on the
scale by means of the leveling screws and observing the space on
a rod passed over by the horizontal cross-hair, the rod being
placed at a known distance from the instrument. The space on
the rod divided by the distance to the rod gives the natural tan-
gent of the angle through which the line of sight has moved.
Since the angle is very small its value in seconds of arc may be
obtained by dividing its tangent by the tangent of one second,
(log tan i" = 4.6855749 — 10). Dividing the angle found by
the number of divisions of the scale passed over on the bubble
tube, gives a result which is the average number of seconds
corresponding to a single division.
In a properly constructed leveling instrument the value of
one division of the level should have a definite relation to the
magnifying power of the telescope. The smallest angular move-
ment that can be detected by the level bubble should correspond
to the smallest movement of the cross-hairs that can be detected
by means of the telescope.
THE LEVEL.
96. The instruments chiefly used for the direct determina-
tion of differences of elevation are known as the Wye Levels the
Dumpy Level, and the Hand Level, The Precise Level differs
in its details from the others but does not really constitute a
diflferent type ; it is essentially a wye level or a dumpy level,
according to the principle of its construction. The engineer's
transit, which has the long level attached to the telescope, is
frequently used for direct leveling. All of these instruments
are so constructed that the line of sight is horizontal when the
bubble of the attached spirit level is in the middle of its tube.
97. THE WYE LEVEL. — In the wye level (Figs. 35 and 36)
the spirit level is attached to the telescope tube which rests in
74
MEASUREMENT OF ELEVATION
[Chap. IV.
two Y shaped bearings from
which it derives its name. Those
parts of the telescope which bear
on the wyes are made cylin-
drical and are called fi'n^s or
pivots. The telescope is held in
the wyes by means of two clips.
The level is attached to the
telescope by means of screws
which allow vertical and lateral
adjustments. The two wye sup-
ports are secured, by means of
adjusting screws, to a horizontal
bar which is attached rigidly at
right angles to a spindle, or
vertical axis, similar to that of
a transit. The instrument is
provided with leveling screws,
clamp, and tangent screw, but
has no shifting head nor plumb-
line attachment. The whole
upper portion of the instrument
is screwed to a tripod in the
same manner as a transit. The
characteristic feature of the wye
level is that the telescope can
be lifted out of its supports,
turned end for end and replaced,
each ring then resting in the
opposite wye.
98. THE DUMPY LEVEL.—
In the dumpy level (Fig. 37)
the telescope, the vertical sup-
ports, the horizontal bar and the
vertical spindle are all made in
one casting or else the parts are
fastened together rigidly so as to
be essentially one piece. The
75
3
5
H
X
w
r
w
<
w
76
Q
c
THE DUMPY LEVEL jy
spirit level is fastened to the horizontal bar and can be adjusted
in the vertical plane ; there is no other adjustable part except
the cross-hair ring.
99. Comparison of Wye and Dumpy Levels. — The wye level
has long been a favorite in this country, chiefly on account of
the ease with which it can be adjusted, which depends upon the
fact that when the telescope is reversed in the wye supports the
line through the centers of the pivots is exactly coincident with its
first position. While this feature of the wye level is of practical
advantage in adjusting the instrument it is based on the assump-
tion that both pivots are circular and of exactly the same diame-
ter, which may or may not be true. For, even supposing the
pivots to be perfect when new, they" soon wear, and perhaps
unevenly, and consequently the method of adjusting by reversal
will then fail and the '^peg'' adj'tistmenty or direct method,
must be used. (See Art. 128, p. 91.) It is not uncommon to find
a wye level of excellent manufacture which, after being adjusted
by reversals, jfails to stand the test by the direct method, but
which is capable of excellent work when adjusted by the latter
method.
The dumpy level has very few movable parts, and conse-
quently it does not easily get out of adjustment even when
subjected to rough usage.* Furthermore the recent work of the
United States Coast and Geodetic Survey with a new precise
level, which is really a dumpy level with certain refinements,
indicates the superiority of the dumpy form for the most precise
work.
Fig. 88. The Locke Hand Level.
100. The Locke HAITO level. — The hand level (Fig.
38) has no telescope, but is simply a metal tube with plain glass
• See Reports of the Superintendent of the U. S. Coast and Geodetic Sur-
vey for the year i89S-99» p. 351, and the year 1900, p. 525.
78 MEASUREMENT OF ELEVATION [Chap. IV.
covers at the ends and with a spirit level on top. When look-
ing through the tube one sees the level bubble on one side
of the tube in a mirror set at 45° with the line of sight,
and the landscape on the other side. In order that the eye may
see the bubble and the distant object at the same instant the
instrument is focused on the bubble by means of a lens placed
in a sliding tube. The level line is marked by a horizontal wire,
which can be adjusted by means of two screws. The instrument
is held at the eye and the farther end is raised or lowered until
the bubble is in the center of the tube. At this instant a point
in line with the horizontal wire is noted. In this way approxi-
mate levels may be obtained.
LEVELING RODS.
loi. According to their construction rods are either Self-
reading or Target rods, or a combination of the two. Self -read-
ing rods are those which can be read directly from the instrument
by the levelman whereas target rods can be read only by the
rodman. The commonest forms of leveling rods are known as
the Bostofty the New Yorky and the Philadelphia rods. (See
Fig. 39.)
102. BOSTON ROD. — The Boston rod (Fig. 39) is a target
rod of well seasoned wood about 6 J ft. long, made in two strips,
one of which slides in a groove in the other. A target is
fastened rigidly to one of these strips about 0.3 ft. from one
end. Clamps are provided for holding the two parts in any
desired position. There is a scale on each side of the rod, one
starting from either end, graduated to hundredths of a foot and
each with a vernier placed about the height of the eye and read-
ing to thousandths of a foot. When the rod-reading is less than
5.8 ft. the rod is first placed on the ground with the target near
the bottom. Then the strip carrying the target is raised to the
proper height while the bottom of the other strip rests on the
ground, as shown in Fig. 39. For readings over 5.8 ft. the rod
is turned end for end so that the target is at the top and can be
moved from 5.8 to 11. 4 ft., the limit of the rod. The terms
■4
i
6
i
4
1
7
^
J
h
e
«
I i
Fig. 89. Leveling Rods.
8o
MEASUREMENT OF ELEVATION
[Chap. IV.
^^ short rod'' and '^ long rod'' are used to distinguish these two
positions.
The common form of target used on the Boston rod is shown
in Fig. 40. Instead of this target one of a design similar to
that in Fig. 41 is sometimes used, in which the white strip in
the center may be bisected by the horizontal cross-hair.
Bisection is more precise under all conditions than setting on a
Fig. 40. Boston Rod Target.
Fig. 41. Bisection Target.
single line or on the division line between two surfaces of
different color.
A serious objection to the Boston rod is that in reversing it
(changing from long to short rod) any error in the position of
the target with reference to the scale is doubled by the reversal,
and such an error is not readily eliminated.
103. NEW YORK ROD. — The New York rod (Fig. 39)
consists of two strips of wood, arranged similarly to those of the
Boston rod. Unlike the latter the target on the New York
rod is movable. For "short rod" the target is moved up or
down on the rod until the proper height is reached. The face
of the rod is graduated to hundredths of a foot. The vernier is
on the target itself and reads to thousandths of a foot. The
graduations on the rod cannot be read from the instrument except
at short distances. For "long rod*' the target is set at the
highest graduation, usually 6.5 ft., and clamped to one of the
sliding strips which is then raised until the target is in the right
position. A clamp is provided for holding the two strips
together. The reading for "long rod'* is found on the side of
the strip that is raised, and opposite the vernier which is on the
LEVELING RODS "* 8 1
Other strip, the scale reading downward In this case the rod
cannot be read directly from the instrument.
104. PHILADELPHIA ROD. — This rod has the graduations
plainly painted on its face so that it can be used as a self-reading
rod (Fig. 39). It has also a target the arrangement of which is
similar to that of the New York rod. The target has no vernier
but is graduated directly to 0.005 ft- 5 ^^^ thousandths can be
readily estimated. The rod is extended in the same manner as
the New York rod, and it can be read to 0.005 ft. and estimated
to o.ooi ft. by means of a scale fastened on the back of the rod.
When the rod is fully extended, the graduations on the front
face are continuous and the readings can be made directly by
the levelman if desired.
105. SPECIAL SELF-READING RODS. — There are a large
number of self-reading rods of special design. One of the com-
monest types shown in Fig. 39, is similar to the Philadelphia
rod except that it has no target and is not graduated closer than
tenths. The figures on the face of the rod are made of definite
height (0.06 or 0.08 ft.) and of definite thickness (0.0 1 or 0.02
ft.) so that it is easy for the levelman to estimate the readings
to hundredths of a foot. These rods are usually constructed
so that they can be extended for "long rod " readings.
106. Tape Rod.* — The tape rod (Fig. 39) is a self-reading
rod of decidedly different design from the Philadelphia rod. It
is a wooden rod made in one piece with a metal roller set in it
near each end. Passing over these rollers is a continuous steel
band 20 ft. long and o. i ft. wide, on the outside of which for
its entire length is painted a scale graduated to feet, tenths, and
half-tenths, with the details of the numbers so designed that
readings to the nearest 0.0 1 ft. can readily be made. Unlike the
other rods mentioned the scale reads down on the face of the
rod instead of up. It is provided with a clamp so that the metal
band, or tape, can be set at any desired reading and held firmly
in that position. The use of this type of rod is limited to cer-
• This rod was invented by Thomas F. Richardson and is used extensively
by the Metropolitan Water and Sewerage Board of Boston, Mass.
82
MEASUREMENT OF ELEVATION
[Chap. IV.
tain kinds of work, its advantagjB being the time saved in calcu-
lations as explained in Art. 228, p. 206.
107. Precise Level Rod — The self-reading rod used by
the U. S. Coast and Geodetic Survey is made of a single piece
of wood, soaked in paraffin to prevent changes in length due to
moisture. Metal plugs are inserted at equal distances so that
changes in length can be accurately determined. It is divided
into centimeters, painted alternately black and white. The bot-
tom of the rod carries a foot-plate. The meters and centimeters
are read directly and the millimeters estimated. This rod has
attached to it a thermometer, and a level for plumbing.
108. Advantages of the Self-Reading Rod. — While the ad-
vantage in the speed with which leveling can be accomplished
by use of the self-reading rod is well understood, it is also true
Fig. 42. Rod Levels.
although not so generally recognized that very accurate results
can be obtained. For any single reading the error may be
larger than with the target rod, but the errors of estimating
fractional parts are compensating, so that in the long run the
results are found to be very accurate. Precise leveling carried
on by the U. S. Coast and Geodetic Survey and by European
surveys has demonstrated the superiority of such rods. The
SELF-READING RODS 83
self-reading rod might to advantage be more generally used than
it is at present.
109. Attachments to the Rod for Plumbing. — In accurate
work it will be convenient to use some device for holding the
rod plumb. Spirit levels attached to brass ** angles*' which may
be secured to a comer of the rod are very convenient. Two
patterns are shown in Fig. 42. In some rods the levels are set
permanently into the rod itself.
no. Effect of Heat and Moisture. — Changes of temperature
do not have a serious effect on rods since the coefficient of ex-
pansion of wood is small. The effect of moisture is greater,
however, and consequently if very accurate leveling is to be
done the rod should be compared frequently with a standard.
Rods soaked in paraffin are less affected by moisture than those
which have not been so treated.
USE OF THE LEVEL AlTD ROD.
111. In order to obtain the difference in elevation between
two points, hold the rod at the first point and, while the instru-
ment is level, take a rod-reading. This is the distance that the
bottom of the rod is below the line of sight of the level. Then
take a rod-reading on the second point and the difference
between the two rod-readings is the difference in elevation of
the two points.
112. To LEVEL THE INSTRUMENT. — Set up the instru-
ment in such a position that the rod can be seen when held on
either point and at such height that the horizontal cross-
hair will strike somewhere on the rod. In setting up the level,
time will be saved if the habit is formed of doing nearly all of
the leveling by means of the tripod legs, using the leveling
screws only for slight, motions of the bubble in bringing it to the
middle of the tube. Turn the telescope so that it is directly
over two opposite leveling screws. Bring the bubble to the
center of the tube approximately; then turn the telescope until
it is over the other pair of leveling screws and bring the bubble
exactly to the center. Move the telescope back to the first
position and level carefully, and again to the second position if
84 MEASUREMENl OF ELEVATION [Chap. IV.
necessary. If the instrument is in adjustment and is properly
leveled in both directions, then the bubble will remain in the
center during an entire revolution of the telescope about the
vertical axis. The instrument should not be clamped ordinarily,
but this may bie necessary -under some circumstances, for ex-
ample, in a strong wind.
113. To TAKE A ROD-READING. — The rodman holds the
rod on the first point, taking pains to keep it as nearly plumb as
possible. The levelman focuses the telescope on the rod, and
brings the bubble to the center while the telescope is point-
ing at the rod, because leveling over both sets of screws will not
make the bubble remain in the center in all positions unless the
adjustment is perfect. If a target rod is used, the target should
be set so that the horizontal cross-hair bisects it while the bubble
is in the center of the tube. It is not sufficient to trust the
bubble to remain in the center; it should be examined just
before setting the target and immediately afterward, at every read-
ing. The levelman signals the rodman to move the target up or
down. When the center of the target coincides with the hori-
zontal cross-hair the levelman signals the rodman, " All right "
(Art. 115), and the rodman clamps the target and reads the
rod. This reading is then recorded in the note-book. In ac-
curate work the levelman should check the position of the target
after it has been clamped to make sure that it has not slipped
in clamping. For readings to hundredths of a foot it is not neces-
sary to clamp the target ; the rodman can hold the two parts of
the rod firmly together while he reads it.
While the levelman is sighting the target, the rodman should
stand beside the rod so that he can hold it as nearly vertical as
possible in the direction of the line of sight. The levelman can
tell by means of the vertical cross-hair whether it is plumb in .
the direction at right angles to the Ime of sight. It is extremely
important that the rod be held plumb. Vertical lines on build-
ings are a great aid to the rodman in judging when his rod is
plumb. If the wind is not blowing the rodman can tell when
the rod is plumb by balancing it on the point.
114. Waving the Rod. — In careful work when the "long
rod '* is used it may be plumbed in the direction of the line of
USE OF LEVEL AND ROD . 85
sight by *^'waving the rod'' To do this the rodman stands
directly behind the rod and inclines it toward the instrument so
that the target will drop below the line of sight. He then slowly
draws it back, causing the target to rise. It will be highest
when the rod is plumb. If at any point the target appears
above the cross-hair it should be lowered. If, while the rod is
being waved, the target does not reach the cross-hair the target
must be raised and the process repeated until as the rod is
waved there appears to be just one place where the target coin-
cides with the horizontal line of sight. Whenever close results
are desired it will be well to take several readings on each point
and use the mean.
115. Signals. — While the rodman is seldom very far away
from the levelman in this work still it is often convenient to use
hand signals. The following are commonly used in leveling.
" Up'' or ^^ Down." — The levelman motions to the rodman
by raising his arm above his shoulder for an upward motion and
dropping his arm below his waist for a downward motion. A
slow motion indicates that the target should be moved a consid-
erable amount and a quick motion indicates a short distance.
"-4// Right." — The levelman extends both hands hori-
zontally and waves them up and down.
** Plumb the Rod." — The hand is extended vertically above
the head and moved slowly in the direction it is desired to have
the rod plumbed.
^^ Take a Turning Point." — The arm is swung slowly in a
circle above the head.
^^ Pick up the Level." — When a new set-up of the level is
desired the chief of party signals the levelman by extending
both arms downward and outward and then raising them quickly.
Some surveyors use a system of signals for communicating
the rod-readings, but mistakes are liable to be made unless great
care is used.
116. DIFFERENTIAL LEVELING. — Differential leveling is the
name given to the process of finding the difference in eleva.
tion of any two points. In Art. 1 1 1 the simplest case of differ-
ential leveling is described. When the points are far apart the
instrument is set up and a rod-reading is taken on the first point.
86
MEASUREMENT OF ELEVATION
[Chap. IV.
This is called a backsight ox plus sight and is usually written B, 5.
or + 5. Next the rod is taken to some well-defined point which
will not change in elevation (such as the top of a firm rock)
and held upon it and a reading taken. This is called ?l foresight
or minus sight and is written F. S, or — 5. The difference be-
tween the two readings gives the difference in elevation between,
this new point and the first point. This second point is called a
turning point and is written T, P. The level is next set up in
a new position and a backsight taken on the turning point. A
Diagram Illustrating Differential Leveling.
new turning point is then selected and a foresight taken upon it.
This process is continued until the foresight is taken on the final
point. The elevation of the last point above the first is equal to
the sum of all the backsights minus the sum of all the foresights.
If the result is negative, i.e., if the sum of the foresights is the
greater, then the last point is below the first. The form of notes
for this work is shown below, and the fieldwork is illustrated by
Fig. 43-
DIFFERENTIAL LEVELING
87
POIMT
B. S.
F.S.
Remarks
A.
T.P.
T. P.
T.P.
B.
8.160
7.901
9.446
8.005
2.404
3.070
6.906
2.107
Highest point on stone bound, S. W. cor. X and Y Sts.
N. £. cor. stone step No. 64 M St.
33.512
14.487
14.487
Diff. 19.025 B above A.
117. The Proper Length of Sight. — The proper length of
sight will depend upon the distance at which the rod appears
distinct and steady to the levelman, upon the variations in read-
ings taken on the same point, and upon the degree of precision
required. Under ordinary conditions the length of sight should
not exceed about 300 ft. where elevations to the nearest o.oi ft. are
desired. " Boiling " of the air due to irregular refraction is fre-
quently so troublesome that long sights cannot be taken accurately.
If the level is out of adjustment the resulting error in the rod-
reading is proportional to the distance from the instrument to the
rod. If the level is at equal distances from the rod the errors are
equal and since it is the difference of the rod-readings that gives
the difference in elevation, the error is eliminated from the final
result if the rodman makes the distance to the point where the
foresight is taken equal to the distance to the backsight by count-
ing his paces as he goes from one point to the other.
118. Effect of the Earth's Curvature and of Refraction on
Leveling. — Since the surface of the earth is very nearly spherical,
Fig. 44. Diagram Illustrating Effect of Earth's
AND OF Refraction.
Curvature
any line on it made by the intersection of a vertical plane with the
earth's surface is practically circular. In Fig. 44 the distance
AA' varies nearly as AJV (see foot-note, p. 339). The effect of
88 MEASXmEMENT OF ELEVATION [Chap. IV.
the refraction of the atmosphere is to make this offset from the
tangent appear to be A'B which is about one-seventh part smaller
than A' A, This offset, corrected for refraction, is about 0.57
ft. in one mile ; for 300 ft. it is 0,002 ft. ; for 500 ft., 0.005 ft. ;
for 1000 ft., 0.020 ft. If the rod is equally distant from the
instrument on the foresight and backsight the effect of curva-
ture and refraction is eliminated from the result.
119. PRECAUTIONS IN LEVEL WORK. — Nearly all of the
precautions mentioned in Art. 68, p. 55, for the transit instru-
ment, are also applicable to the level. Care should be taken
not to strike the rod on the ground after it has been clamped
and before it has been read.
ADJUSTMENTS OF THE LEVEL.
I. ADJUSTMENTS OF THE WYE LEVEL.
120. ADJUSTMENT OF THE CROSS-HAIRS. — (a) To make the
Horizontal Cross-Hair truly Horizontal when the Instrument is
Leveled. This may be done by rotating the cross-hair ring as in
the case of the transit (Art. 71, p. 57), if the instrument is so
constructed that the telescope cannot be rotated in the wyes.
In many instruments the telescope can be rotated in the wyes.
In some levels the telescope is always free to rotate in the
wyes, while others are provided with a stop regulated by an ad-
justing screw, which prevents the telescope from rotating beyond
a certain point.
The instrument is leveled and some point found which is
covered by the horizontal cross-hair. The telescope is turned
slowly about the vertical axis so that the point appears to traverse
the field of view. If the point remains on the cross-hair the ad-
justment is perfect. If it does not, then an adjustment must be
made, the manner of doing this depending upon the construction
of the instrument. If the telescope cannot be rotated in the
wyes the adjustment is made by rotating the cross-hair ring,
similar to the adjustment described in Art. 71, p, 57. If the tele-
scope has a stop-screw this must be moved until the instrument
ADJUSTMENTS OF THE WYE LEVEL
89
satisfies this test. If the telescope can rotate freely in the wyes
it can be turned by hand until it satisfies the test. Since there
is nothing to hold the telescope in this position the adjustment
in the last case is likely to be disturbed at any time.
121. (b) When the above adjustment is completed the Line
of Sight should be made to Coincide with the Axis of Pivots, or
Parallel to it. (See Fig. 45.) Pull out the pins which hold the
clips on the telescope and turn the clips back so that the telescope
is free to turn in the wyes. Sight the intersection of the cross-
hairs at some well-defined point, using the leveling screws for the
vertical motion and the clamp and tangent screw for the hori-
zontal motion. Then rotate the
telescope 180° in the wyes, so that
the level tube is above the tele-
scope. The intersection of the
cross-hairs should still be on the
point. If not, move the horizontal
cross-hair half-way back to its first
position by means of the upper and
lower adjusting screws of the cross-
hair ring. Then move the vertical
cross-hair half-way back to its first
position by the other pair of screws.
Repeat the test until the adjust-
ment is perfect.
122. Adjustment of the level Tube. — To make the
Line of Sight and the Level Tube Parallel to Each Other. Two
methods are used, — the direct ^ or '^peg,'* method and the in-
direct method. While the former is the only one applicable to
the dumpy level either one can be used for the wye level,
although the indirect method is the simpler.
123. ADJUSTMENT OF THE LEVEL TUBE BY INDIRECT
HETHOD. — (a) To put the Axis of the Bubble Tube in the Same
Plane with the Line of Sight. Bring the bubble to the center of
the tube and rotate the telescope in the wyes for a few de-
grees (very little is necessary) ; if the bubble moves toward one
end of the tube that end must be the higher, which indicates
the direction in which the adjustment should be made. Move
/ £nd.ta
|WI_
Cro
k-holr \
1 ThifPb,
»ton
Ciw
i-Mr 1
I let PbtJi
on
Crw
W 1
\
■~7
*• POINT 1
X t
J
v^
lJ
^
Fig. 46. Adjustment of
THE Cross-Hairs (Sec-
ond. Part).
90
MEASUREMENT OF ELEVATION
[Chap. IV.
the screws controlling the lateral movement of the tube until the
bubble returns to the center. Test the adjustment by rotating
the telescope each way.
124* (b) To make the Axis of the Bubble Tube and the Line of
Sight Parallel to Each Other. First damp the instrument (over a
pair of leveling screws), then bring the bubble to the center of
the tube, lift the telescope out of the wyes, turn it end for end
and set it down in the wyes, the eye end now being where the
objective was originally. (See Fig. 46.) This operation must
be performed with the greatest care, as the slightest jar of the
instrument will vitiate the result. If the bubble returns to the
center of the tube, the axis of the tube is in the correct position.
If it does not return to the center, the end of the tube provided
LIHE or SIGHT
Fig. 46. Adjustment of the Bubble Tube by Indirect Method.
with the vertical adjustment should be moved until the bubble
moves half-way back to the center. This test must be repeated
to make sure that the movement is due to defective adjustment
and not to the jarring of the instrument.
125. ADJUSTMENT OF THE WYES. — To make the Axis of
Fig. 47. Adjustment of the Wyes.
ADJUSTMENTS OF THE DUMPY LEVEL 9 1
the Level Tube Perpendicular to the Vertical Axis of the Instrument.
Bring the two dips down over the telescope and fasten them.
Level the instrument, bring the bubble precisely to the middle
of the tube over one set of leveling screws, and then turn the
telescope 180° about the vertical axis. If the bubble moves
from the center bring it half-way back by means of the adjusting
screws at the foot of one of the wye supports. (See Fig. 47.)
Since the bubble is brought to the center of the tube each
time a rod-reading is taken this last adjustment in no^ way affects
the accuracy of the leveling work but is a convenience and a
saving of time.
ZZ. ADJUSTMEIVTS OF THE DUMPY LEVEL.
126. ADJUSTMENT OF THE CROSS-HAIR. — If the horizon-
tal cross-hair is not truly horizontal when the mstrument is level
it should be made so by rotating the cross-hair ring as described
in the adjustment of the transit, Art. 71, p. 57.
227. Adjustment of the bubble Tube. — To make the
Axis of the Bubble Tube Perpendicular to the Vertical Axis. Owing
to the construction of the dumpy level it is necessary to make
this adjustment before making the line of sight parallel to the
bubble tube. It is done by centering the bubble over one pair
of leveling screws, and turning the instrument 180° about the
vertical axis. If the bubble does not remain in the center of the
tube, move it half-way back to the center by means of the adjust-
ing screws on the level tube.
128. The direct, or « Peg,»» Adjustment. —To make the
Line of Sight Parallel to the Axis of the Bubble. (See Fig. 48.)
Select two points A and B, say, 200 ft. or more apart. Set up
the level close to A so that when a rod is held upon it the eye-
piece wDl be only about a quarter of an inch from the rod.
Look through the telescope wrong end to at the rod and find the
reading opposite the center of the field. After a little experience
it will be found that this can be done very accurately. From
the fact that only a small portion of the rod is visible it will be
foimd convenient to set a pencil-point on the rod at the center of
92 MEASUREMENT OF ELEVATION [Chap. IV.
the small field of view. Turn the telescope toward B and take
a rod-reading on it in the usual way, being certain that the bub-
ble is in the middle of the tube. The difference between these
two rod-readings is the difference of elevation of the two points
plus or minus the error of adjustment. The level is next taken to
B and the above operation is repeated. The result is the differ-
ence in elevation minus or plus the same error of adjustment.
The mean of the two results is the true difference in elevation of
points A and J5. Knowing the difference in elevation between
the two points and the height of the instrument above B the rod-
reading at A which will bring the target on the same level as
the instrument may be computed. The bubble is brought to the
center of the tube and the horizontal cross-hair raised or lowered
by means of the adjusting screws on the cross-hair ring until the
line of sight strikes the target. In this method the small error
due to curvature of the earth (nearly o.ooi ft. for a 200-ft. sight)
has been neglected.
Example. (See Fig. 48.)
Instrument at A.
Rod-reading on A a 4.062
Rod-reading on B— 5.129
Diff . in elev. of A and B » i .067
Instrument at B.
Rod-reading on B « 5.076
Rod-reading on A « 4.127
Diff. in elev. of B and A « 0.949
Mean of two diff. in elev. = — '-^^ — 1.008 true diff. in elev.
2
Instrument is now 5.076 above B.
Rod-reading at A should be 5.076 — 1.008 » 4.068 to give a level sight.
The peg method may be used for adjusting the wye level or
the transit, the difference being that in the dumpy level the axis
of the bubble tube is first made horizontal and then the line of
sight is brought parallel to it, while in the wye level and in the
transit the line of sight is first made horizontal and then the axis
of the bubble tube is made parallel to it. Consequently, in the
former case the cross-hair ring is moved in adjusting whereas in
the latter case the adjustment is made in the bubble tube. This •
THE PEG ADJUSTMENT
93
adjustment in its simplest form is described in the following
article.
Fig. 48. Peg Adjustment.
129. Adjustment of the Locke haitd level. — In ad-
justing the hand level the principle of the peg adjustment is
used. The level is placed at a mark A (Fig. 49) and another
mark B in line with the cross-hair is made, say, 100 ft. away.
51?^-^
Fig. 49. Peg Adjustment for Hand Level.
when the bubble is in the middle. The level is then taken to
By held so that its center is at the height of this mark, and
sighted toward the first point. A third point C is marked in
94 MEASUREMENT OF ELEVATION [Chap. IV.
line with the cross-hair when the bubble is in the middle. The
point midway between A and C is at the same level as B. The
adjustment is made by screws which move the horizontal wire.
130. COMMON SOURCES OF ERROR IN LEVELING.—
1. Improper focusing (parallax).
2. Bubble not in middle of tube at instant of sighting.
3. Rod not held plumb.
4. Foresights and corresponding backsights on turning
points not equally distant from the instrument.
5. Poor turning points selected. (See Art. 224, p. 202.)
131. COMMON MISTAKES.—
1. Foresight and Backsight not taken on exactly the same
point.
2. Neglecting to set target accurately when " long rod " is
used.
3. In the use of the self-reading rod neglecting to clamp the
rod at the proper place when "long rod '* is used.
4. Reading the wrong foot-mark or tenth-mark.
5. In keeping notes, — getting F. S. in B. S. column or
vice versa,
6. In working up notes, adding F. S. or subtracting B. S.
PROBLEMS.
1. A wye level was tested for the sensitiveness of the bubble, as follows :
the rod was held on a point 200 ft. away ; the bubble was moved over 13.6 divi-
sions of the scale ; the rod-readings at the two extreme positions of the bubble
were 4.360 and 4.578. Compute the average angular value of one division of the
level.
2. A dumpy level was tested by the peg method with the following results.
Instrument 2X A\ — Instrument at i9 : —
B. S. on Ay 4.139 B. S. on B^ 3.900
F. S. on B, 4.589 F. S. on A^ 3.250
Find the rod-reading on A to give a level line of sight, the instrument remaining
3.900 above B, Was the line of sight inclined upward or downward ? How
much ?
ERRORS AND MISTAOIS IN LEVELING 95
3. The target on a Boston rod has been disturbed and it is desired to find oat
if the target is in the correct position with reference to the scale. Describe a
method by which the amount of this error can be determined.
4. A New York rod is found to be 0.002 ft. short, due to wear on the brass
foot-plate. Explain what effect this will have in finding the difference in eleva-
tion between two points.
5. (a). A level is set up and a B. S. of 5.0^8 is taken on a point 400 ft. away,
then a F. S. of 3.260 is taken on a point 900 ft. away. What is the curvature and
refraction correction ? What is the difference in elevation of the two points ?
(b). In another case a B. S. of 8.266 was taken on a point 100 ft. away
and a F. S. of 6.405 taken on a point 600 ft. away. What is the curvature and
refraction correction ? What is the difference in elevation of the two points ?
PART II.
SURVEYING METHODS.
PART II.
SURVEYING METHODS.
CHAPTER V.
LAND SXTRVBTINO.
132. SURVEYING FOR AREA. — In surveying a field for the
purpose of finding its area the instruments and methods
used will be determined largely by the degree of accuracy
required. If it is permissible to have an error in the area of,
say, 0.5 per cent then the compass and chain may be used. If
accuracy much greater than this is required it will be necessary
to use the transit and the steel tape. At the present time,
however, in nearly all work except surveys of farms and wood-
lands, the transit is used even under conditions where the
compass would give the required accuracy.
In surveying a field all the angles and lengths of the sides
are determined consecutively, the survey ending at the point
from which it was started. Then by trigonometry the position of
the final point or of any other point with relation to the starting
point can be readily calculated. If the survey were absolutely
accurate the last point as calculated would coincide with' the
first, but this condition is never attained in practice. The
calculated distance between the two, divided by the perimeter
of the field, is usually called the error of closure ;* it is often
expressed in the form of a fraction in which the numerator is
unity. In surveying with a compass and chain the error of
closure expected is about i part in 500, expressed as
133. SURVEYING FOR AREA WITH COMPASS AND CHADI. —
If the area alone is desired the surveyor's 4-rod chain will be
* The term error of closure more properly applies to the actual distance by
which the survey fails to close, but as this is generally expressed in the form of a
fraction the term has commonly been applied to the latter.
99
lOO
LAND SURVEYING
[Chap. V.
convenient on account of the simple relation existing between
the square chain and the acre (Art. 4, p. 3). In making a survey
enclosing an area it is customary to begin at some convenient
comer and to take the bearings and the distances in order
around the field. As the measurements are made they are
recorded in a field note-book. It is not necessary to take the
sides in order, but since they must be arranged in order for the
purpose of computing the area it will be convenient to have them
so arranged in the original notes. If the length and bearing
of any side are omitted the area is nevertheless completely
determined (Art. 397, p. 366), but as these two measurements
furnish a valuable check on the accuracy of all the measurements
(Lbpt-Hand Pagb)
(Right-hand Pagb)
r Sry^cfmKxtLarofJB^Snm, Abrmttm^Akes.
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MXlkW
17.1s
$t^9i^Shms atrM^mM,,8Jmifm m^ l.^khardlutr.
B
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asj
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C
Nl'^E
MTiw
XJ6
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/axT
Cedar 6M.S'S£ «• 4«y* oa*.
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safitr
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Fig. 60. Notes op Chain and Compass Survey.
they* never should be omitted if they can be taken. It is of the
utmost importance in every survey that check measurements
should be taken. Even a few rough checks taken in the field
which will require only a little extra time often prove to be of
great value in detecting mistakes. Both a forward bearing and
a back (or reversed) bearing should be taken at each comer ;
from these the angle at a corner can be obtained free from error
due to any local attraction of the needle. The' above process
gives a series of connected straight lines and their bearings
(or the angles between them), which is called a traverse.
It is often impossible to set the compass up at the comers
of the property, and in such cases assumed lines running
parallel or approximately parallel to the property lines can be
SURVEY BY A TRAVERSE lOI
surveyed as described in Art. 134, and its area determined
In some cases the compass can be set on the property line at
an intermediate point and the bearing obtained, but the sur-
veyor must be sure that there is no local attraction of the needle
at this point. All points where the compass is set should
be marked and described so that they can be found again. If
any instrument point is not otherwise defined it may be tem-
porarily marked by a small stake and several reference
measurements made from this stake to prominent objects
nearby, so that its position can be relocated if the stake is lost.
These measurements are called ties.
Notes of the traverse are usually recorded as shown in
Fig. 50.
SURVEY OF FIELD WITH TRABSIT AND TAPE.
134. SURVEY OF A FIELD BY A TRAVERSE. — Surveying
a field for area can usually be done in one of the three following
ways.
(i). By setting up the transit at the corners of the property
and measuring the angles directly ; the distances being measured
directly along the property lines.
(2). When the property lines are so occupied by buildings
or fences that the transit cannot be set up at the corners, but
the distances can still be measured along the property lines, then
the angles at the comers are obtained by measuring the angles
between lines which are parallel to the property lines.
(3). If the boundaries of the property are such that it is not
practicable to set the transit up at the comers nor to measure
the distance directly on the property lines, a traverse is run
approximately parallel to the property lines and these lines con-
nected with the traverse by means of angles and distances.
135. In the case (2) the parallel lines are established in
the following manner. Set the transit up at some point E
I02 LAND SURVEYING [Chap. V.
(Fig. 51) within 2 or 3 ft. of the comer A. Establish the line EF
parallel to AD by making DF = AH by trial. Point H cannot
be seen through the telescope, but it is so near the instrument
that by means of the plumb-line on the transit it can be accu-
rately sighted in by eye. Similarly EG is established parallel to
AB. Then the angle FEG is measured ; and this is the property
f7K
« H r
Fig. 61. Transit Lines Parallel to the Sides of Field.
angle at^^. It is evident that the values of Alfsnd DFznd of
/4/and BG are of no permanent use and are therefore not recorded
in the notes. When practicable it is advisable to choose the
transit point, K for example, on one of the property lines or its
prolongation. Fig. 52 is a set of notes illustrating either
case (I) or (2).
136. In the case (3) the transit can be set up at an arbi-
trary point marked by a stake and chosen far enough from one
of the corners so that the telescope can be focused on it. In
this way all the comers of the traverse are chosen so that the
traverse will be approximately parallel to the sides of the field.
The angles and distances of this traverse are then measured.
To connect the property lines with this traverse, angles and dis-
tances are measured to the respective corners of the property
before the instrument is moved to the next point. Fig. 53 is a
set of notes illustrating this case. Time can be saved in the
computations and a good check on the work may be obtained if
the property lines are also measured when possible. These are
not only useful as checks on the accuracy of the survey, but the
SURVEY BY A TRAVERSE
103
Kitkr ,
J.H.Brtiolley£skrie<tinhn,/V.Y ^nty ^
Bearing cfCF iromphn ofFbrk Com.CiSvtMericlian)
Z^oe .oosiooshorf:
IboihjH ^
^^ 207,42(f> "
■/
/ihwYorkfbrkCom.
t
L.FBrown
L.F.Brown
Fig. 62. Notes of Survey with Transit and Tape.
ro4
LAND SURVEYING
[Chap. V.
Survey o/Land o/S/'/as Cokm/yr, Ai/rcrt//^ Af///Sj Ak.
kVeJ/s
yiUloyN
Fig. 68. Notes of Survey with Transit and Tape.
SURVEY BY SINGLE SET-UP IO5
length of the sides will be needed in giving a description of the
property.
These three methods which have been described may be
combined in any survey according to circumstances.
137. Irregular Curved Boundaries. — When a tract of land is
bounded by an irregular curved line such as a brook it is custom-
ary to run the traverse line near it, sometimes crossing it several
times, and to take perpendicular offsets to the brook. If it is a
winding brook with no distinct turns in it, offsets at regular in-
tervals are measured from the transit line as in the portion near
point A of Fig. 53. Near point B in this figure the brook has
practically a direct course between its turns, in which case the
proper measurements to make are the offsets to those points
where the course of the brook changes and the distances
along the transit line between these offset lines. Since they
are usually short the right-angle offset lines are laid oflf by
eye.
138. SURVEY OF A FIELD BY A SINGLE SET-UP OF
TBE TRANSIT. — When it is necessary to economize time in the
field at the expense of accuracy and of the time required to calculate
the survey the following method may be used. If possible set up
at a point within the field, preferably near the middle, from which
all the corners can be seen, and measure the angles and distances to
each corner. In this way the field is divided into several oblique
triangles in each of which two sides and the included angle have
been measured and from these the area and third side (property
line) can be computed. As a check on the measured angles
their sum should be 360° ; there is no check on the property
lines unless they are measured directly.
This method of surveying a field may be employed as a check
on one of the other methods which have already been described,
but is not recommended as a method to be used by itself except
in emergencies. The weak point in it is the low degree of pre-
cision with which the angles are usually measured. Here the
effect of an error of, say, 30 seconds in an angle may often be
much larger than the errors in the measured distances (Art. 352,
P- 325). The additional measurement of the property line gives
the length of all three sides of the various triangles into which
I06 LAND SURVEYING [Chap. V.
the field is divided. If the area is calculated from the three
sides of the triangles, using the measured angles as checks only,
an accurate result may be obtained, but at the expense of con-
siderable office work.
139. SURVEY OF A FIELD WITH A TAPE ONLY. — Some-
times it may be necessary to survey a field when a transit is not
at hand. This can be done by dividing the field into several tri-
angles and measuring all their sides. To insure accuracy of re-
sults the triangles should be so chosen that there are no angles
in them less than 30*^ or greater than 1 50*^. This method will
require a large amount of computation if the angles as well as
the area of the field are desired. Lining in by eye will give ac-
curate results in distances along the line, but only approximate
side measurements can be obtained from such a line.
140. Selecting the Comers. — If a corner is marked by a
stone bound the exact point may be easily found ; but where it is
simply defined as the intersection of stone walls or fences the
surveyor will have to examine all evidence as to its position and
use his judgment in deciding where the true corner is located
(Art. 151, p. 116). When the property is bounded by a public
way or a town boundary such data relating to the location of these
lines must be obtained from the proper local authorities. After
determining the position of the corner points, the surveyor should
use precisely the same points in all distance or angle measure-
ments. If stakes are used the exact point is marked by a small
tack driven into the top of the stake.
In deciding upon the location of the boundary lines from an
examination of artificial features it should be borne in mind that
it is customary to build fences or walls along highways entirely
on private property so that the face of the wall or fence is on the
side line of the highway. In cities the base-board of a fence is
usually built so that its face is on the street line, but the location
of the fences has no weight when the street line is defined by
stone bounds or other permanent marks (Art. 253, p. 227). For
boundaries between private lands the legal line is, in the case of
a stream, the thread (not necessarily the center) of the stream ;
the center of the stone wall or Virginia rail fence ; the line be-
tween the bottom stringer and the boarding or pickets of an
TIES 107
ordinary fence, the fence-posts being entirely on one side of the
boundary line. Not infrequently woodland is marked off by
blazing the trees on one or both ^des of the boundary line, the
blazing being done on the side of the tree nearest the boundary
line. If a tree comes directly on the line it is blazed on both
sides where the line strikes it. A small pile of stones, sometimes
with a stake in the center of the pile, is often used to mark the
corners of such land.
141. Method of Procedtire. — In deciding where the traverse
shall be run the surveyor should keep in mind both convenience
in fieldwork and economy in office work. Frequently a method
of procedure which shortens the time spent in the field will
greatly increase the amount of the labor in the office. Circum-
stances will determine which method should be used. If there
is no special reason why the time in the field should be shortened,
the best arrangement of the traverse will be th e one that will
make the computation simple, and hence mistakes will be less
liable to occur. If the lines of the traverse coincide with the
boundary, as in cases (i) and (2), the amount of office work will
be the least. If in case (3) the traverse lines are approximately
parallel and near to the boundaries of the property the computa-
tion of the small areas to be added to or subtracted from the
area enclosed by the traverse is simplified to some extent.
142. Ties. — All important points temporarily marked by
stakes should be " tied in," i.e., measurements should be so taken
that the point may be readily
found or replaced in the future.
There should be at least three \ $c^.6'Wrch
horizontal ties which intersect at ^ .t^^^
angles not less than 30®. They
should be taken from easily rec-
ognized definite points, such as
blazed trees, stone bounds, fence
posts, or buildings. All such .
measurements should be care- Xu^ujj,
fuDy recorded, usually by means p,^ ^ approximate Ties.
of a sketch Fig. 54 shows a
stake located by ties measured to tenth of a foot ; these are taken
io8 LAND SURVEYING [Chap, V.
2^^^ simply to aid in finding the
stake.
It is often desired to take the
ties so that the exact point can
be replaced. In such cases the
^\^3,^»^ surveyor should mark carefully
"^•"•^ • by tack or crow-foot the exact
Fig. 66. Exact Ties. points from which measure-
ments (taken to -j-Jtt ^^0 ^^^
made, and record the entire information in the notes as shown
in Fig. 55.
143. Meastirement of the Angles of the Traverse. — The '
angles of the traverse may be measured in any one of three ways;
by measuring the interior angle ^ by measuring the deflection angle^
which is the difference between the intierior angle and i8o^ or
by measuring the azimuth angle.
In practice the deflection angle is measured directly by sight-
ing back on the previous point with the vernier at o® and the
telescope inverted, then revolving the telescope about its hori-
zontal axis to the direct position and turning the upper limb to
the right or left until the next point is sighted. The deflection
angle as recorded in the notes is marked /? or Z to indicate
whether the telescope was turned to the right or left. It is evi-
dent that a single measurement of the deflection angle is affected
by any error in the adjustment of the line of sight as well as of
the standards. If the deflection angle is "doubled " by turning
to the backsight with the instrument direct and the angle re-
peated a check on the angle is obtained and the errors of adjust-
ment are also eliminated (Art. 79, p. 61). Where this procedure
is followed it will be convenient to make the first backsight with
the instrument direct so that when the second foresight is taken
the instrument will again be in the direct position and ready for
lining in.
144. Measurement of Azimuth Angles. — By the azi-
muth method the angles are measured as follows. The transit
is set up at a point A (Fig. 56), the vernier set at o^ the tele-
scope turned until it points to the south, and the lower plate
clamped. Either the true or the magnetic south may be used.
MEASURING ANGLES OF TRAVERSE
109
but if neither is known any arbitrary direction may be assumed.
The upper clamp is loosened and the telescope sighted on B,
The angle read on the vernier is the azimuth of ABy the circle
being read in a clockwise direction (Art. 24, p. 16). The tran-
sit is next moved to B,
The azimuth of BC may be obtained in one of two ways.
( I ) Invert the telescope and backsight on Ay the vernier remain-
ing at the reading it had at A ; then clamp the lower plate, turn
the telescope to its direct
position, and sight on C.
The angle on the vernier is
the azimuth of BC referred
to the same meridian as the
azimuth of AB, The disad-
vantage of this method is
that the error of collimation
enters the azimuth angle each
time. (2) (Add 180° to the
azimuth of AB^ set this off
on the vernier, and sight on
A, The telescope may then
be turned directly to C (with-
out inverting) and the azi-
muth of BC can be read directly on the vernier. The disad-
vantages of this method as compared with the former are that
the error of eccentricity of the circle enters, that time is con-
sumed in setting the veniier at each set-up of the instrument,
and that there is an opportunity for mistakes in calculating and
in making the setting on the vernier.
In the azimuth method the angles of the traverse are checked
by the fieldwork. After point D has been occupied, the transit
is again set up at A and the azimuth of AB determined from a
backsight on D. This azimuth of AB should agree with the
original azimuth of this line. In ordinary land surveying the
azimuth method of measuring the angles is little used.
145. Checking the Fieldwork. — The transit is set over the
selected points and the angles between the adjacent lines
measured. If the work is not to be of unusual precision a
Fig. 66. Azimuth Angles.
I lO LAND SURVEYING [Chap. V.
transit reading to one minute will be sufficient. A single
measurement will give the angle with sufficient precision, but
as it is important in all cases to have a check on the work it is
advisable to "double the angle** (Art 60, p. 50), even though
it is not necessary to use this method for the purpose of pre-
cision. Referring to Fig. 52, p. 103, it will be seen that the
angles were quadrupled where the sides were long, and doubled
where they were short. In this case the angles were repeated
to obtain greater precision.
As an additional check against large errors in the angles, the
magnetic bearing of each line should be read, thus enabling one
to detect mistakes greater than a quarter of a degree and to
guard against reading Right for Left in deflection angles. These
bearings also show the approximate directions of the lines of the
survey. This check should always be applied in the field so that
any mistake in reading the angles can be rectified before leaving
the work. This may be done by calculating each angle from the
observed bearings of the adjacent sides ; or by starting with one
observed bearing (assumed to be correct), calculating the other
bearings in succession by means of the measured angles, and
noting whether the observed bearings agree approximately with
the calculated bearings.
After the angles have been measured, the accuracy of the
transit work may be tested by adding them together. The sum
of the interior angles of the field should equal («-2) X 180°,
where n is the number of sides in the field. If the deflection
angles are used the sum of all the rigbt deflections should differ
from the sum of all the left deflections by 360**, or in other
words, the algebraic sum of the deflection angles should be
3600.
It is frequently important to check the distances before
leaving the field. If there is any doubt as regards the correctness
of the measurement of a line it should be remeasured, preferably
in the opposite direction, so that the same mistake will not be re-
peated. (See line AB in Fig. 52, p. 103.) If the traverse lines
do not coincide with the boundaries, an independent check is
obtained by measuring along the boundaries as well as on the
traverse line, as in Fig. 53, p. 104. This furnishes at once a
CHECKING FIELD WORK III
rough check on the distances in the field and a close check after
the survey has been calculated. It is often advisable to run a
line across the traverse, especially when there are many sides
to the field, thus dividing the field into two parts, as in
Fig. 52, p. 103. If any mistake has been made it is then
possible to tell in which portion of the traverse it occurred
(Art. 407, p. 370-
146. Accuracy Required. — In order that the accuracy m the
measurement of distances shall be consistent with that of the angles
it is necessary that great care should be exercised in holding the
tape horizontal, in the plumbing, in the aligning, and in securing
the proper tension.
If the angles are measured to the nearest minute and the dis-
tances to the nearest tenth of a foot, it will be sufficiently accu-
rate to use sighting-rods in "giving line." The error of closure
of such a survey should be not greater than j^j^^, but would sel-
dom be less than j-^^^-f^ (Art. 132, p. 99).
If the property is very valuable, as in the case of city build-
ing lots, it is well to use a transit reading to 30" or 20". The
angles should be repeated, not only as a check against mistakes,
but to increase the precision of the measurement (Art. 59, p. 48).
The tape measurements should be made with special care, and
should be taken to the nearest hundredth of a foot. In the best
work the temperature correction should be applied, a spring bal-
ance should be used to give the right pull on the tape, the cor-
rection to the standard distance should be determined (Art. 241,
p. 216), the alignment given with the transit, and great care
taken in plumbing. Sights are given by holding a pencil vertically
on top of the tack on the stake or by plumb-line (Art. 65, p. 52).
In this work it is important that the property line should be fol-
lowed, when possible, to insure the most accurate results. In
such work an error of closure of -^-qI-q^ or better is expected
It is customary on most city work to neglect the effect of tempera-
ture and to omit the use of the spring balance, the pull being
carefully judged. This sort of work should give results as close
as ^-a^^rs* ^^^ ^^ accuracy of :j^^^j is sometimes reached.
147. Oi'ganization of Transit Party. — Transit surveys can be
readily carried on by a party of three men. The note keeper
112 LAND SURVEYING [Chap. V.
who is in charge of the party directs the entire work ; the transit-
man who has the instrument always in his care sets It up where
directed by the note keeper, reads the angles and gives line when
desired; the chainman generally acting as head-chainman and the
note keeper as rear-chainman, measure all distances.
148. Note Keeping. — All measurements should be recorded
in a special note-book as soon as they are made and never left to
be filled in from memory. The notes should be neat and in clear
form so that there will be no doubt as to their meaning. Great
care should be taken so that they shall not be susceptible of any
interpretation except the right one. They are generally re-
corded in pencil, but they should always be regarded as
permanent records and not as temporary memoranda. As other
persons who are not familiar with the locality will probably use
the notes and will depend entirely on what is recorded, it is
very important that the notes should contain all necessary data
without any superfluous information. If the note keeper will
bear in mind constantly how the survey is to be calculated or
plotted it will aid him greatly in judging which measurements
must be taken and which ones are unnecessary. Clearness is of
utmost importance in note keeping, and to attain it the usual
custom is not to attempt to sketch to scale ; and yet in surveys
where considerable detail is desired it is sometimes well to
carry out the sketches in the note-book approximately to scale.
Care should be taken not to crowd the notes, — paper is
cheap, — and an extra page of the note-book devoted to a
survey may save hours of time in the office consumed in trying
to interpret a page of crowded data. Too much stress cannot
be laid on the importance of being careful not to lose the note-
book ; not infrequently a note-book contains data which thou-
sands of dollars could not replace.
Although sufficient fulness to make the notes clear is
desirable, it is customary to abbreviate the names of the
artificial features most commonly met with by the surveyor.
To properly understand a set of notes one must be familiar
with these abbreviations, some of the more common of which
are enumerated.
NOTE KEEPING II3
9M, Stone bound.
M«n- Monument.
-^ Triangulation Station.
SHU Stake.
Ik Tack.
«*• Nail.
•fk. Spike.
^ Drill-hole.
cii Crow-foot (a mark like this \| or >^).
cciS Cut crow-foot (cut into wood or stone).
^ Center.
* Center line.
ci^ Curb.
C8. Catch basin.
MX Manhole.
Tki Telegraph pole.
Fence.
Fence, showing on which side the posts are.
Base-board of fence.
Line of building; the outside line is the base-
board, the cross-hatched part is the line of the
stone or brick underpinning.
Distances should always be recorded in such a way as
to indicate the precision with which they were taken. For
example, if they were taken to hundredths of a foot and a
measurement happened to be just 124 ft. it should be re-
corded as 124.00, not as 124. The two zeros are of as much
consequence as any other two digits which might have come in
their places. Angles which have been read to the nearest half-
minute, however, are recorded as follows : 6® 47' 30". It will
be seen that this is not consistent with the foregoing. A more
114 LAND SURVEYING [Chap. V.
proper way of reading this angle would be 6® 47 Y, but this is not
common practice.
In addition to the measurements every set of notes should
contain the following information: — the kind of work, the
locality, the date, and the names of members of the field party.
It is well to also state the names or numbers of the instruments
used and their errors. Where a survey is continued for several
pages the date may be placed at the top of every page ; other
data need not be repeated. Fig. 50, p. 100, Fig. 52, p. 103, and
Fig. 53, p. 104, are good examples of field notes.
149. SURVEY OF A FIELD FOR A DEED. — In this case
the lengths and bearings of all the boundaries are desired. The
traverse lines should therefore follow the property lines, if
possible. The bearings desired are not the observed magnetic
bearings, but are those calculated by means of the transit angles
as explained in Art. 145, p. no, and therefore are relatively as
accurate as the angles themselves. In case a true meridian is
found by observation (Chapter VII) the bearings should be re-
ferred to this and marked true bearings by a note on the plan,
and this information should also be contained in the deed.
A plan which is to accompany a deed should show such
features as watercourses, highways, buildings, and adjoining
property lines, as well as stone bounds, stakes, fences, walls, or
other artificial objects which mark the boundaries of the property.
This plan should contain the following information.
(i) Lengths of all property lines together with their calcu-
lated bearings or the angles at the corners.*
(2) Location and description of corner bounds.
(3) Conventional sign or name on walls, fences, etc.
* It is customary with many surveyors to omit from the plan certain data
such as the angles or bearings, so that, while it may answer the purpose for which
it was made, it does not contain all the data and frequently not enough to enable
another surveyor to relocate the property by means of it. This is done, of course,
so that when the tract is to be resurveyed or plotted it will be necessary to em-
ploy the same surveyor who has in his possession data for which the owner has
paid and which the surveyor should have turned over to him. For a valuable
paper on this subject see " The Ownership of Surveys, and what Constitutes a
Survey and Map," by Professor William G. Raymond, published in TAe Polytechnic^
the student journal of the Rensselaer Polytechnic Institute, Troy, N. Y., January.
1894.
DEED DESCRIPTION II5
(4) Names of highways, streams or ponds, and names of
adjacent property owners.
(5) Scale of drawing and direction of the meridian used (true
or magnetic). It is better to refer all bearings to the true
meridian when possible, and in such a case the direction of the
magnetic needle should also be shown.*
(6) The title should include a simple and complete state-
ment giving the name of owner, place, date, and name of sur-
veyor. An explanatory note such as a statement as to whether
bearings refer to true or magnetic meridian may also be neces-
sary. (See Art. 468, p. 415.)
150. Deed Description. — The written description of the prop-
erty which is recorded in the deed should be given by bearings
(or angles) and distances, stating in every case how the sides of
the property are marked and whether bounded by a highway,
stream, or private property, giving the name of- the present
owner of the adjacent property. The following is an example
of a deed description of the property shown in the form of notes
in Fig, 53, p, 104.
" Beginning at a point in the northerly line of Willow Road
in the town of Bancroft Mills, Maine, at an iron pipe sunk in
the ground at the S.E. corner of land now or formerly belonging
to Nathan H. Barnes, and running along the said northerly line
N 85** 34' E a distance of two hundred ninety-seven and seven-
tenths (297.7) feet to the thread of channel of Stony Brook at
land now or formerly belonging to James F. Hall; thence turn-
ing and running in a northerly direction, by thread of channel of
said Stony Brook and land of said Hall, a distance of about three
hundred and eight (308 ±) feet to a stone wall at land now or
formerly belonging to Hiram Cole; thence turning and running
along the middle of said stone wall and by land of said Cole
* As magnetic bearings are unreliable (Art. 28, p. 19) true bearings should be
used wherever their adoption does not entail too much additional expense. In
those parts of the country which have been subdivided by the U. S. General Land
Office true meridians can be readily obtained from the government surveys ; in
many of the older (Eastern) states true meridians have been established by local
authorities. If the survey can be connected with any triangulation system such
as that of the United States or state surveys then, since the true bearings of all
of the triangulation lines are known, the bearings of the traverse lines can be
obtained.
Il6 LAND SURVEYING [Chap. V.
N 86® 45' W a distance of two hundred and five and eight-tenths
(205.8) feet to the middle of another stone wall at land of said
Barnes; thence turning and running by latter stoiie wall and land
of said Barnes S o** 53' E a distance of one hundred and seventy-
seven and two-tenths (177.2) feet to a fence ; thence turning and
running by said fence and land of said Barnes N 87** 09' W a
distance of ninety-three and three-tenths (93.3) feet to an iron
pipe sunk in the ground ; thence turning and running by a fence
and land of said Barnes S i*^ 51' W a distance of one hundred
and sixty-nine and four-tenths (169.4) feet to the point of begin-
ning ; all the bearings being magnetic and the parcel containing
a calculated area of 79,270 square feet more or less.**
It is unfortunate that the description of the property in deeds
in the vast majority of cases, does not define the property in
such a manner that it can be plotted from the description. Some
deeds are so loosely written as to contain only the names of the
owners of adja^cent property, no bearings or distances being given.
151. JUDICIAL FUNCTIONS OF THE SURVEYOR. — In rerun-
ning old property lines which have been obliterated, the surveyor
is called upon to set aside temporarily his strict adherence to the
mathematical side of surveying and must endeavor to find if pos-
sible where the lines originally ran. He should therefore be fa-
miliar with the relative importance of various evidence regarding
the location of the property lines, as determined by court deci-
sions. It is distinctly his duty to find the position of the original
boundaries of the property and not attempt to correct the original
survey even though he may be sure that an error exists in it.
Very often it is true that, owing to the cheapness of land, the
original survey was roughly made with little thought of the effect
it would have when the land became valuable.
The surveyor therefore must first of all hunt for all physical
evidence of the location of the boundaries * and failing in this he
* It must not be assumed that a boundary is missing because it is not at once
visible. Stone bounds are often buried two or three feet deep ; the top of a
stake soon rots off, but evidences of the existence of the stake are often found
many years after the top has disappeared, and the supposed location should be
carefully dug over to find traces of the old stake. The shovel and common sense
are of as much use as the transit and tape in relocating an old corner.
..'"^^^'^
DISPUTED BOUNDARIES 11/
will base his judgment on any other reliable evidence such as
occupancy or the word of competent witnesses. It is obvious
that this is along equitable lines, since the property was originally
purchased with reference to the actual or visible bounds which
vest the owner with rights to the property bounded by these
lines.
If there is a dispute between adjoining owners over the loca-
tion of a boundary line this presents a question which must be
settled by the courts unless the parties can come to an agree-
ment themselves. In such cases the surveyor acts simply as an
expert in judging where the line originally ran and has no power
to establish a new line. He can, however, be employed by the
disputing parties as an arbitrator to decide on the equitable line,
but they are not necessarily obliged to accept his judgment.
If they come to an agreement between themselves, however,
regarding the location of the line and occupy to that line, this
agreement is binding even though no court has intervened in the
matter.
It is to be assumed that the deed was drawn by the grantor
with honest intent to convey the property to the grantee. It is
intended then that it shall be interpreted if possible so as to
make it effectual rather than void. The deed should also be
construed in the light of what was known at the time when the
title was transferred.
In the interpretation of a deed it is assumed that it was in-
tended to convey property the boundaries of which will form a
closed traverse. Therefore it is within the jurisdiction of the
surveyor to reject any evident mistake in the description when
running out the property line, e.g., a bearing may have been re-
corded in the opposite direction or an entire side omitted. Where
artificial features are mentioned as boundaries, these always take
precedence over the recorded measurements or angles, but these
marks must be mentioned in the deed in order to have the force
or authority of monuments. When the area does not agree with
the boundaries as described in the deed the boundaries control.
All distances unless otherwise specified are to be taken as straight
lines ; but distances given as so many feet along a wall or high-
way are supposed to follow these lines even if they are not
Il8 LAND SURVEYING [Chap. V.
Straight. When a deed refers to a plan the dimensions on this
plan become a part of the description of the property.
Where property is bounded by a highway the abutters usu-
ally own to the center line, but where it is an accepted street
each abutter yields his portion of the street for public use ; if,
however, the street is abandoned the land reverts to the original
owners. If a street has been opened and used for a long period
bounded by walls or fences, and there has been no protest re-
garding them, these lines hold as legal boundaries. In the case
of a line between private owners acquiescence in the location of
the boundary will, in general, make it the legal line. But if
there is a mistake in its location and it has not been brought to
the attention of the interested parties or the question of its po-
sition raised, then occupancy for many years does not make it a
legal line.
Where property is bounded by a non-navigable stream it ex-
tends to the thread of the stream. If the property is described
as running to the bank of a river it is interpreted to mean to the
low water mark unless otherwise stated. Where original owner-
ship ran to the shore line of a navigable river and the water has
subsequently receded the proper subdivision is one that gives to
each owner along the shore his proportional share of the channel
of the river. These lines will therefore run, in general, perpen-
dicular to the channel of the stream from the original intersec-
tion of division lines and shore lines.
A more complete statement of the principles mentioned above
particularly with reference to the U. S. Public Land Surveys
will be found in an address on " The Judicial Functions of Sur-
veyors," by Chief-Justice Cooley of the Michigan Supreme Court,
read before the Michigan Association of Engineers and Survey-
ors, and published in the proceedings of the society for 1882,
pp. 1 12-122.
152. RERUNNING OLD SURVEYS FROM A DEED. — The vis-
ible marks which are mentioned in a deed are of primary impor-
tance in determining the extent of a piece of property ; the
lengths of the sides and the bearings (or angles), which should
agree with the boundaries, are of secondary importance. It
sometimes occurs, however, that all evidences of artificial bound-
RERUNNING OLD SURVEYS 1 19
aries of the property or of portions of it are missing, and the
surveyor must then fall back on the dimensions given in the
deed as the best informatipn available (Art. 150, p. 115). Fur-
thermore it is sometimes necessary to "run out '' an old deed
to determine which of two lines is the correct boundary, or in
some cases to find how close the actual boundaries of a property
agree with the original deed.
If the directions of the boundaries are defined in the deed by
the magnetic bearings, as was formerly the usual custom, it is
necessary first to find the declination of the needle at the date of
the original survey as well as the present declination of the needle
and to correct all the bearings accordingly (Art. 29, p. 20).
The declination of the needle should appear on the original deed
or plan ; but unfortunately it seldom does, and the year the sur-
vey was made must then be obtained either from the deed, the
old plan, or from witnesses, and the declination of the needle at
that time computed. Observations at different places arid times
have been compiled by the U. S. Coast and Geodetic Survey,
and these results may be found in convenient form for calculation
in the annual Reports of the Superintendent, particularly the 1886
report.* From these observations the approximate change in
declination may be obtained. In this way the magnetic bearings,
corrected to date, can be determined as closely probably as the
original bearings were taken. It is evident that the change in the
declination of the needle between the date of the original survey
and the present time is what is desired. If there exists there-
fore one well-defined line which is known to be one of the original
boundary lines, a bearing taken on this Une and compared with
that given in the deed will determine directly the change in
declination. There may be more than one well-defined line
whose bearings can be obtained and a comparison of the results
on these different lines will give an idea of the reliability of the
original survey as well as a more accurate determination of the
change in declination.
* In 1902 the U. S. Coast and Geodetic Survey issued a special publication
entitled, '* Magnetic Declination Tables and Isogenic Charts for 1902/' in which
IS given a very complete list of declinations for various places in the United States.
I20 LAND SURVEYING [Chap. V.
Not infrequently in attempting to rerun old compass surveys
it is found that the traverse as described in the deed does not
•* close/' i.e., the last point does not coincide with the first. If
this error of closure is small it may be due to the difference in
length between the chain used for the original survey and the
one being used. Before any attempt is made to run out the old
survey this difference should be determined by measuring one or
more of the well-defined lines of the property, if any can be found,
and comparing the measurements obtained with the recorded
distances.
Occasionally it is found that the traverse will not close by a
large amount owing to a mistake in the original survey. Often
in such cases the deeds of adjacent property will show what
the mistake was, and in such cases it is allowable to make a cor-
rection if it will give a description that is consistent. For ex-
ample, it occasionally happens that a bearing has been recorded in
the reverse direction so that no area is enclosed by the boundaries*
Sometimes an entire chain-length has been omitted in one of
the lines and by supplying this the description is made consistent.
Other inconsistencies are to be dealt with in the same general
manner, or as suggested in the preceding article.
153. How to Look Up a Recorded Deed. — In all the states of
the Union the transfer of real property must be recorded in the
respective county Registry of Deeds or in the office of the city
or town clerk. At the Registry of Deeds is kept an exact copy
of the deed, which can be examined by any one. It is fre-
quently necessary for the surveyor to make use of these copies
when it is not convenient to obtain the deed from the owner of
the property or when it is necessary to look up the deed of ad-
jacent property or previous transfers of any of them.
In every Registry of Deeds an index of the deeds is kept,
which is divided into two parts, th^ grantor index and the £^rantee
index ; the grantor being the party who sells the land and the
grantee the one who buys it. These indexes are frequently
divided by years and for this reason the surveyor should know
not only the name of the party who bought or sold the property
(both if convenient to get them), but also the approximate date
of the transaction. With this information he can readily find
SURVEY OF PUBLIC LANDS 121
in the proper index the name of the party, opposite which will
appear the date of the transaction and the number of the deed
book and page on which the copy of the deed is recorded. He
then finds the deed book, from which he can copy whatever data
he desires from the deed ; usually the description of the property
is all that concerns the surveyor. In the deed book is usually a
reference number in the margin or in the text of the deed which
refers to the next preceding transfer of the same property or
to any attachments, assignments, and the like which may have
been made on it. This method of indexing and filing deeds is
used in the New England States and in many of the other states ;
in fact the general principles are the same throughout the coun-
try although the details may differ to some extent.
THE UAITED STATES SYSTEM OF SURVEYING THE PUBLIC LANDS.*
154, THE SYSTEM. — The United States System of Sur-
veying the Public Lands, which was inaugurated in 1784, and
modified since by various acts of Congress, requires that the
public lands "shall be divided by north and south lines run
according to the true meridian, and by others crossing them at
right angles so as to form townships six miles square," and that
the comers of the townships thus surveyed "must be marked
with progressive numbers from the beginning.** Also, that the
townships shall be subdivided into thirty-six sections, each of
which shall contain six hundred and forty acres, as nearly as
may be, by a system of two sets of parallel lines, one governed
by true meridians and the other by parallels of latitude, the
latter intersecting the former at right angles, at intervals of a
mile.
Since the meridians converge it is evident that the require-
* The vork of surveying the government lands is carried on under the direc-
tion of the Commissioner of the General Land Office. In each of the districts
where such surveys are made is a Surveyor General, appointed by the President.
The work is usually done under contract by experienced surveyors, called Depu-
ties. The Deputies are paid by the mile, according to classified rates. All surveys,
before being accepted, are inspected by a corps of Examiners of Surveys, who are
appointed especially for this duty.
122
LAND SURVEYING
[Chap. V.
ment that the lines shall conform to true meridians and also
that townships shall be six miles square, is mathematically im-
possible.
In order to overcome this difficulty the subdivision is carried
on as follows: — (See Fig. 57.)
J
LlSTANDAROI ' ' "* "
T4H
Kit
T3N
TEH
RIE
TIN
RIE
T4M
REE
T3N
RZE
TEN
REE
TIN
REE
T4N
R3E
T3H
R3E
TEN
R3E
TIN
R3E
Znd.
1st.
T4N
R4E
T3H
R4E
T2.N
R4E
TIM
R4E
5
SrAMDARD
FV^R/
LLEL^RH
W^RALULNOmH
It
"B"
J
UJ
J.
Fig. 67.
BASE-UNt
Diagram Illustratinq Meridian, Base-Line, Standard
Parallels, Ranges, and Townships.
First. The establishment of a principal meridian conform-
ing to the true meridian, and at right angles to it, a base-line
conforming to a parallel of latitude, as is described in Art. 1 56,
p. 124, and Art. 157, p. 126.
Second. The establishment of standard parallels conform-
ing to parallels of latitude, initiated from the principal meridian
at intervals of 24 miles and extended east and west of the same.
Third. The establishment of guide meridians conforming
to true meridians, initiated upon the base-line and successive
standard parallels at intervals of 24 miles, resulting in tracts of
land 24 miles square, as nearly as may be, which shall be subse-
quently divided into tracts of land 6 miles square by two sets of
lines, one conforming to true meridians, crossed by others con-
forming to parallels of latitude at intervals of 6 miles, containing
23,040 acres, as nearly as may be, and designated townships.
METHOD OF SUBDIVISION
123
Such townships are divided into 36 tracts, called sections^
each of which contains 640 acres, as nearly as may be, by two
sets of parallel lines, one set parallel to a true meridian and the
other conforming to parallels of latitude, intersecting at intervals
of I mile, and at right angles, as nearly as may be, as shown
in Fig. 57.
Any series of contiguous townships or sections situated
north and south of each other constitutes a range, while such
a series situated in an east and west direction constitutes
a tier.
Section lines are surveyed from south to north, and from east to
west, in order to place the excess or deficiency, according to the
requirement of the law, on the north and west sides of the town-
ships. #
The tiers of townships are numbered, to the north or south,
commencing with No. i at the base-line ; and the ranges or
townships, to the east or west, beginning with No. i at the prin-
cipal meridian of the system.
The thirty-six sections into which a township is subdivided
are numberec^ commencing with No, i at the northeast angle of
the township, and proceed-
ing west to No. 6, and then
proceeding east to No. 12,
and so on, alternately, to No.
36, in the southeast angle as
illustrated by Fig. 58. In all
cases of surveys of frac-
tional townships the sections
will bear the same numbers
they would have if the town-
ship were complete.
Standard parallels (form-
erly called correction lines)
are established at intervals of
24 miles, north and south of
the base line, and guide
meridians at intervals of 24
miles, east and west of the principal meridians ; thus confining
6
5
4
3
2
1
♦7
8
9
10
11
12
18
17
16
15
14
13
19
20
21
22
??>
24
30
29
28
27
26
25
31
3Z
33
34
a.")
36
Fig. 58. Diagram of a Town-
ship Illustrating Method of
Numbering the Sections.
124 LAND SURVEYING tCHAP. V.
the errors resulting from convergence of meridians and inaccu-
racies in measurement within comparatively small areas.
" 155. Initial Points.* — Initial points from which the lines
of the public surveys are to be extended will be established
whenever necessary, under such special instructions as may be
prescribed in each case by the Commissioner of the General
Land Office. The locus of such initial points will be selected
with great care and due consideration for their prominence and
easy identification, and must be established astronomically.
"An initial point should have a conspicuous location, visible
from distant points on lines ; it should be perpetuated by an in-
destructible monument, preferably a copper bolt firmly set in a
rock edge ; and it should be witnessed by rock bearings, without
relying on anything perishable like wood.
" The initial point having been established the lines of public-
land surveys will be extended therefrom. They are classified as
follows :
"Class I. Base lines and standard parallels.
"Class 2. Principal. and guide meridians.
" Class 3. Township exteriors (or meridional and latitudinal
township boundaries).
" Class 4. Subdivision and meander lines.
" Only the base line and principal meridian can pass through
the initial point.
" 156. Base Line. — From the initial point the base line will
be extended east and west on a true parallel of latitude, [Art.
168, p. 148,] by the use of transit or solar instruments, as may
be directed by the surveyor general in his written special instruc-
tions. The transit will be used for the alinement of all import-
ant lines.
"The direction of base lines will conform to parallels of
latitude and will be controlled by true meridians ; consequently
the correct determination of true meridians by observations on
Polaris at elongation is a matter of prime importance.
" Certain reference lines, called tangents and secants, having
* These instructions are taken from the *' Manual of Surveying Instructions
for the Survey of the Public Lands of the United States," prepared by the Com-
missioner of the General Land Office in 1902.
INITIAL POINTS 12$
a known position and relation to the required parallel of latitude,
will be prolonged as straight lines. Two back and two fore sights
are taken at each setting of the instrument, the horizontal limb
being revolved i8o^ in azimuth between the observations, in one
method, taking the mean of observations. Another method,
called double back and fore sights, is still more exact, and there-
fore preferable. In this process the vertical cross-wire is fixed
upon two transit points at some distance apart, in the rear, and
then reversed to set one or two new points in advance. This
not only insures a straight line, if the transit is leveled, but also
detects the least error of collimation.
" Where solar apparatus is used in connection with a transit,
the deputy will test the instrument, whenever practicable, by
comparing its indications with a meridian determined by Polaris
observations ; and in all cases where error is discovered he will
make the necessary corrections of his line before proceeding
with the survey. All operations will be fully described in the
field notes.
"The proper township, section, and quarter-section corners
will be established at lawful intervals, and meander corners at
the intersection of the line with all meanderable streams, lakes,
or bayous.
" In order to detect errors and insure accuracy in measure-
ment, two sets of chainmen will be employed ; one to note dis-
tances to intermediate points and to locate topographical features,
the other to act as a check. Each will measure 40 chains, and
in case the difference is inconsiderable, the proper corner will
be placed midway between the ending points of the two measure-
ments ; but if the discrepancy exceed 8 links on even ground, or
25 links on mountainous surface, the true distance will be found
by careful re-chaining by one party or both.
" The deputy will be present when each corner is thus estab-
lished, and will record in the body of his field notes the distances to
the same, according to the measurement by each set of chainmen.
"To obviate collusion between the sets of chainmen, the
second set should commence at a point in advance of the begin-
ning comer of the first set, the initial difference in measurement
thus obtained being known only to the deputy.
126 LAND SURVEYING [Chap. V.
Z57. '* Principal Meridian. — This line shall conform to a
true meridian [Chapter VII] and will be extended from the initial
point, either north or south, or in both directions, as the condi-
tions may require, by the use of transit or solar instruments, as
may be directed by the surveyor general in his special written
instructions. The methods used for determination of directions,
and the precautions to be observed to secure accuracy in measure-
ment, are fully stated above under the title " Base Line," and will
be complied, with in every particular.
"In addition to the above general instructions, it is required
that in all cases where the establishment of a new principal meri-
dian seems to be necessary to the surveyor general, he shall
submit the matter, together with his reasons therefor, to the
Commissioner of the General Land Office, and the survey of
such principal meridian shall not be commenced until written
authority, together with such special instructions as he may deem
necessary, shall have been received from the Commissioner.
158. " Standard Parallels. — Standard parallels, which are also
called correction lines, shall be extended east and west from the
principal meridian, at intervals of 24 miles north and south of
the base line, in the manner prescribed for running said line,
and all requirements under the title * Base Line ' will be care-
fully observed.
" Where standard parallels have been placed at intervals of 30
or 36 miles, regardless of existing instructions, and where gross
irregularities require additional standard lines, from which to
initiate new, or upon which to close old surveys, an intermediate
correction line should be established to which a local name may
be given, e.g., ' Cedar Creek Correction Line ' ; and the same
will be run, in all respects, like the regular standard parallels.
159. " Guide Meridians. — Guide meridians shall be extended
north from the base line, or standard parallels, at intervals of 24
miles east and west from the principal meridian, in the manner
prescribed for running the principal meridian, and all the pro-
visions for securing accuracy of alignment and measurement,
found or referred to under the titles Base Line and Principal
Meridian, will apply to the survey of said guide meridians.
" When existing conditions require that such guide meridians
PRINCIPAL MERIDIAN 12/
shall be run south from the base or correction lines, they will
be initiated at properly established comers on such lines,
marked as closing comers.
"Where guide meridians have been improperly placed at
intervals greatly exceeding the authorized distance of 24 miles,
and standard lines are required to limit errors of old, or govern
new surveys, a new guide merijdian may be run from a standard,
or properly established closing corner, and a local name may be
assigned to the same, e.g., 'Grass Valley Guide Meridian.'
These additional guide meridians will be surveyed in all respects
like regular guide meridians.
160. " Township Exteriors. — Whenever practicable, the town-
ship exteriors in a block of land 24 miles square, bounded by
standard lines, will be surveyed successively through the block,
beginning with those of the southwestem township.
" The meridional boundaries of townships will have prece-
dence in the order of survey and will be run from south to
north on true meridians, with permanent comers at lawful
distances ; the latitudinal boundaries will be run from east to
west on random or trial lines, and corrected back on true lines.
" The falling of a random, north or south of the township
corner to be closed upon, will be carefully measured, and, with
the resulting tme return course, will be duly recorded in the
field notes.
"Should it happen, however, that such random intersects
the meridian of the objective comer, north or south of said
comer, or falls short of, or overruns the length of the south
boundary of the township by more than three chains (due
allowance being made for convergency), said random, and, if
necessary, all the exterior boundaries of the township, will be
retraced and remeasured to discover and correct the error.
" When running random lines from east to west, temporary
corners will be set at intervals of 40.00 chains, and proper
permanent comers will be established upon the true line, cor-
rected back in accordance with these instructions, thereby
throwing the excess or deficiency against the west boundary of
the township, as required by law.
"Whenever practicable, the exterior boundaries of town-
128 LAND SURVEYING [Chap. V.
ships belonging to the west range, in a tract or block 24 miles
square, will first be surveyed in succession, through the range,
from south to north ; and in a similar manner, the other three
ranges will be surveyed in regular sequence.
" In cases where impassable obstacles occur and the fore-
going rules cannot be complied with, township corners will be
established as follows :
" In extending the south or north boundaries of a township
to the west, where the southwest or northwest corners cannot
be established in the regular way by running a north and south
line, such boundaries will be run west on a true line, allowing
for convergency on the west half mile ; and from the township
corner established at the end of such boundary, the west
boundary will be run north or south, as the case may be. In
extending south or north boundaries of a township to the east,
where the southeast or northeast comer cannot be established
in the regular way, the same rule will be observed, except that
such boundaries will be run east on a true line, and the east
boundary run north or south, as the case may be. Allowance
for the convergency of meridians will be made whenever
necessary.
161. " Method of Subdividing. — The exterior boundaries of
a full township having been properly established so far as pos-
sible, the subdivision thereof will be made as follows :
"At or near the southeast corner of the township, a true
meridian will be determined by Polaris or solar observations,
and the deputy's instrument will be tested thereon ; then from
said corner the first mile of the east and south boundaries will
be retraced, if subdivisions and survey of the exteriors have
been provided for in separate contracts ; but, if the survey of
the exterior and subdivisional lines are included in the same
contract, the retracements referred to will be omitted. All dis-
crepancies resulting from disagreement of bearings or measure-
ments will be carefully stated in the field notes.
" The meridional sectional lines will be made parallel to the
range line or east boundary of the township, by applying to the
bearing of the latter a small correction, dependent on the lati-
tude, taken from the following table, which gives, to the
METHOD OF SUBDIVIDING
129
nearest whole minute, the convergency of two meridians 6
miles long and from i to 5 miles apart ; and supplies directly
the deviation of meridional section lines west of north, when
the range line is a true meridian. Add the corrfection to the
bearing of the range line, if the same is west of north, but
subtract when it bears east of north.
TABLE 3.
Corrections for Convergency within a Township.
Latitude.
30 to 35
35 to 40
40 to 45
45 to 50
50 to 55
55 to 60
60 to 65
65 to 70
Correction to be applied to bearing of range lines
at a distance of —
I mile, a miles. 3 miles. 4 miles. 5 miles.
3
3
4
I
7
8
10
^'Example. — Latitude, 47®. Range Ime bears N. 0° 2' E.
then parallel meridional section lines will be run as follows:
From the corner for sections —
35 and 36, N.o^ i' E.
34 and 35, north.
33 and 34, N. o® i' W.
32 and 33, N. o^ 2' W.
31 and 32, N. 0° 3' W.
" After testing his instrument on the true meridian thus deter-
mined, the deputy will commence at the corner to sections 35
and 36, on the south boundary, and run a line parallel to the
range line, establishing at 40.00 chains, the quarter-section corner
between sections 35 and 36, and at 8Q.00 chains the corner for
sections 25, 26, 35, and 36.
I30 LAND SURVEYING [Chap. V-
" From the last-named corner, a random line wDl be run east-
ward, without blazing, parallel to the' south boundary of section
36, to its intersection with the east boundary of the township,
placing at 40.00 chains from the point of beginning, a post for
temporary quarter-section corner. If the random line intersects
said township boundary exactly at the corner for sections 25 and
36, it will be blazed back and established as the true line, the
permanent quarter-section comer being established theron, mid-
way between the initial and terminal section comers.
"When the objective corner is in sight from the starting
comer, or the deputy has evidence of its location to prove that a
different random course would fall closer to the corner, he may
use such changed course for his random. A line may be run as
a "randoni for distance only," when the course is certain.
"If the random intersects said township boundary to the
north or south of said corner, the falling will be carefully mea-
sured, and from the data thus obtained, the tme return course
will be calculated, and the true line blazed and established and
the position of the quarter-section corner determined, as directed
above.
The details of the entire operation will be recorded in the
field notes.
"Having thus established the line between sections 25 and
36, from the comer for sections 25, 26, 35, and 36, the west and
north boundaries of sections 25, 24, 13, and 12, will be established
as directed for those of section 36 ; with the exception that the
random lines of said north boundaries will be run parallel to the
established south boundary of section 36 ; e.g., the random line
between sections 24 and 25 will be run parallel to the established
south boundary of section 25, etc.
"Then, from the last established section corner, i.e., the
corner of sections i, 2, 11, and 12, the line between sections i
and 2 will be projected northward, on a random line, parallel
to the east boundary of the township, setting a post for tempo-
rary quarter-section corner at 40.00 chains, to its intersection
with the north boundary of the township. If the random inter-
sects said north boundary exactly at comer for sections i and 2,
it will be blazed back and established as the true line, the tem-
SUBDIVISION OF TOWNSHIP 131
porary quarter-section comer being established permanently in its
original position, and the fractional measurement thrown into
that portion of the line between said comer and the north boun-
dary of the township.
" If, however, said random intersects the north boundary of
the township, to the east or west of the corner for sections i
and 2, the consequent falling will be carefully measured, and
from the data thus obtained the true return course will be calcu-
lated and the true line established, the permanent quarter-section
corner being placed upon the same at 40.00 chains from the
initial corner of the random line, thereby throwing the fractional-
measurement in that portion lying between the quarter-section
comer and the north boundary of the township.
"When the north boundary of a township is a base line or
standard parallel, the line between sections i and 2 will be nm
parallel to the range line as a true line, the quarter-section cor-
ner will be placed at 40.00 chains, and a closing corner will be
established at the point of intersection with such base or stand-
ard line ; and in such case, the distance from said closing corner,
to the nearest standard comer on such base or standard line, will
be carefully measured and noted as a connection line.
" Each successive range of sections progressing to the west,
until the fifth range is retained, will be surveyed in a similar
manner ; then, from the section corners established on the west
boundary of said range of sections, random lines will be projected
to their intersection with the west boundary of. the township,
and the true return lines established as prescribed for the survey
of the first or most eastern range of sections, with the exception
that on the true lines thus established the quarter-section corners
will be established at 40.00 chains from the initial comers of
randoms, the fractional measurements being thereby thrown
into those portions of the lines situated between said quarter-
section corners and the west boundary of the township.
" The following general requirements are reiterated for em-
phasis :
** The random of a latitudinal section line will alwa3rs be nm par-
allel to the south boundary of the section to which it belongs, and
132
LAND SURVEYING
[Chap. V.
with the true bearing of said boundary; and when a section has
no linear south boundary, the random will be run parallel to the
south boundary of the range of sections in which it is situated,
and fractional true lines will be run in a similar manner.
H«f^^
>4 6Sr53W
H.S3.SSW.
S.e?3BL.
5B5'saE
Fig. 69. Portion of Township Illustrating Subdivision of
Sections.
" The deputy is not required to complete the survey of the
first range of sections from the south to north before commenc-
ing the survey of the second or any subsequent range of sections,
but the corner on which any random line closes shall have been
previously established by running the line which determines its
SUBDIVISION OF SECTION 1 33
position, except as follows : Where it is impracticable to estab-
lish such section comer in the regular manner, it will be estab-
lished by running the latitudinal section line as a true Une, with
a true bearing, determined as above directed for random lines,
setting the quarter-section comer at 40.00 chains and the section
corner at 8o,oo chains.
" Quarter-section comers, both upon meridional and latitudi-
nal section lines, will be established at points equidistant from
the corresponding section comers, except upon the lines closing
on the north and west boundaries of the township, and in those
"situations the quarter-section corners will always be established
at precisely forty chains to the north or west (as the case may
be) of the respective section corners from which those lines re-
spectively start, by which procedure the excess or deficiency in
the measurements will be thrown, according to law, on the
extreme tier or range of quarter sections, as the case may be.
(See Fig. 59.)
" Where by reason of impassable objects only a portion of the
south boundary of a township can be established, an auxiliary base
line (or lines, as the case may require) will be run through the
portion which has no linear south boundary, first random, then
corrected, connecting properly established corresponding section
comers (either interior or exterior) and as far south as possible ;
and from such line or lines, the section lines will be extended
northwardly in the usual manner, and any fraction south of said
line will be surveyed in the opposite direction from the section
comers on the auxiliary base thus established.
"Where by re^on of impassable objects or other reasons no
part of the south boundary of a township can be regularly es-
tablished, the subdivision thereof will proceed from north to
south and from east to west, thereby throwing all fractional
measurements and areas against the west boundary, and the
meanderable stream or other boundary limiting the township
on the south.
"If the east boundary is without regular section comers
and the north boundary has been run eastwardly as a true line,
with section corners at regular intervals of 80.00 chains, the
subdivision of the township will be made from west to east, and
1 34 LAND SURVEYING [Chap. V.
fractional measurements and areas will be thrown against the
irregular east boundary.
"When the proper point for the establishment of a town-
ship or section comer is inaccessible, and a witness comer can
be erected upon each of the two lines which approach the same,
at distances not exceeding twenty chains therefrom, said witness
comers will be properly established, and the half miles, upon
which they stand will be recognized as surveyed lines.
" The witness comer will be marked as conspicuously as a
section comer, and bearing trees will be used wherever possible.
" The deputy will be required to furnish good evidence that
the section corner is actually inaccessible.
" Where impassable precipices, deep canyons, or lands other-
wise quite unsurveyable, prevent the extension of regular lines,
deputies are not authorized to set meander comers, nor to
meander the line separating lands that can be traversed from
those that cannot. In place of meandering, they are to set
witness corners on line, near the intersection of section lines with
the brink or foot of the impassable cliffs, or at the margin
of the impracticable marsh, to represent an inaccessible regular
section or quarter-section corner if within twenty chains. Such
quarter sections thus marked may be platted as surveyed.
" Where a large or desirable track is found to have its acces-
sible section lines too short to justify the erection of such witness
comers, and to render it regularly surveyed, offset lines may
be run on lines of legal subdivision, far enough to show, by
necessary witness comers, the 40-acre tracts that would other-
wise have been excluded from survey.
" The topographic sketches of mesas and impassable canyon
regions, returned by deputies, will show as nearly as practicable
the location of these features and their margins ; and where
possible the comers on opposite sides of a canyon should be
connected by triangulation at least once in each township.
162. ''Meandering. — The running of meander lines has
always been authorized in the survey of public lands fronting on
large streams and other bodies of water, but does not appear to
have been proper in other cases. The mere fact that an irregular
or sinuous line must be mn, as in case of a reservation bound-
MEANDER LINES 135
ary, does not entitle it to be called a meander line except where
it closely follows a stream or lake shore. The legal riparian
rights connected with meandered lines do not apply in case
of other irregular lines, as the latter are strict boundaries.
" Lands bounded by waters are to be meandered at mean
high-water mart This term has been defined in a State deci-
sion (47 Iowa, 370) in substance as follows: High water mark
in the Mississippi River is to be determined from the river-bed ;
and that only is river-bed which the river occupies long enough
to wrest it from vegetation.
"In another case (14 Penn. St. 59) a bank is defined as the
continuous margin where vegetation ceases, and the shore is the
sandy space between it and low-water mark.
" Numerous decisions in State and U. S. Supreme Courts,
assert the principle that meander lines are not boundaries defin-
ing the area of ownership of tracts adjacent to waters. The
general rule is well set forth (10 Iowa, 549) by saying that in a
navigable stream, as the Des Moines River in Iowa, high-water
mark is the boundary line. When by actibn of the water the
river bed changes, high-water mark changes and ownership of
adjoining land changes with it. The location of meander lines
does not aflfect the question.
" Inasmuch as it is not practicable in public land surveys to
meander in such a way as to follow and reproduce all the minute
windings of the high-water line, the U. S. Supreme Court has
given the principles governing the use and purpose of meander-
ing shores, in its decision in a noted case (R. R. Co. v, Schurmeier,
7 Wallace, 286-7) as follows :
" In cases where the deputy finds it impossible to carry his
meander line along mean high-water mark, his notes should state
"Meander lines are ran in surveying fractional portions of the public lands
bordering on navigable rivers, not as boundaries of the tract, but for the purpose
of defining the sinuosities of the banks of the stream, and as the means of ascer-
taining the quantity of land in the fraction subject to sale, which is to be paid for
by the purchaser. In preparing the official plat from the field notes, the meander
line is represented as the border line of the stream, and shows to a demonstration
that the water-course, and not the meander line as actually ran on the land, is
the boundary.
I
136 LAND SURVEYING [Chap. V.
the distance therefrom, and the obstacles which justify the
deviation.
"Proceeding down stream, the bank on the left hand is
termed the left bank and that on the right hand the right bank.
These terms will be universally used to distinguish the two
banks of a river or stream.
" Navigable rivers, as well as all rivers not embraced in the
class denominated 'navigable,' the right-angle width of which
is three chains and upwards, will be meandered on both banks,
at the ordinary mean high-water mark, by taking the general
courses and distances of their sinuosities, and the same will be
entered in the field book. Rivers not classed as navigable will
not be meandered above the point where the average right-angle
width is less than three chains, except that streams which are
less than three chains wide and which are so deep, swift, and
dangerous as to be impassable through the agricultural season,
may be meandered, where good agricultural lands along the
shores require their separation into fractional lots for the benefit
of settlers. But such meander surveys shall be subject to re-
jection if proved unnecessary by field inspection.
" Shallow streams, without any well-defined channel or per-
manent banks, will not be meandered ; except tide-water streams,
whether more or less than three chains wide, which should
be meandered at ordinary high-water mark, as far as tide-water
extends.
"At every point where either standard, township, or section
lines intersect the bank of a navigable stream, or any meandera-
ble shore, corners will be established at the time of running
these lines. Such corners are called meander corners, and the
deputy will commence at one of these corners, follow the bank
or boundary line, and take the bearing and measure the length
of each course, from the beginning corner to the next meander
corner.
" All courses reported are to be compass courses, taken or
counted from the meridian, and not from a latitudinal line ; and
* transit angles ' showing only the amount of deviation from the
preceding course, are not allowed in field notes of meanders.
"For convenience of testing by traverse, the courses of
MEANDER LINES 1 37
meander lines should be given by the nearest quarter degree.
As meandered lines are not strict boundaries, this method will
give results with approximate accuracy for good closings within
the limits of a section. Meander lines will be examined in the
field as well as rectangular lines, before acceptance.
" All meanders should be traversed before leaving the vicin-
ity, and if misclosure is found, indicating error in measurement
or in reading courses, the lines must be re-meandered.
"The crossing distance between meander corners on same
line, and the true bearing and distance between corresponding
meander corners, will be ascertained by triangulation or direct
measurement, in order that both shores may be protracted. The
particulars will be given in the field notes.
" For convenience of platting and computation, the deputy
is required to use in meanders distances having whole chains,
or multiples of ten links, with odd links only in closing distances.
" The meanders of all lakes, navigable bayous, and deep ponds
of the area of twenty-five acres and upwards, will be commenced
at a meander corner and continued, as above directed for navi-
gable streams; from said corner, the courses and distances of
the entire margin of the same, and the intersections with all
meander corners established thereon, will be noted.
" All streams falling into the river, lake, or bayou will be
noted, and the width at their mouths stated ; also, the position,
size, and depth of springs, whether the water be pure or mineral ;
also, the heads and mouths of all bayous ; all islands, rapids, and
bars will be noted, with intersections to their upper and lower ends,
to establish their exact situation. The elevation of the banks of
lakes, bayous, and streams, the height of falls and cascades, and
the length and fall of rapids will be recorded in the field notes.
" To meander a lake or deep pond lying entirely within the
boundaries of a section, two lines will be run from the two
nearest corners on different sides of such lake or pond, the
courses and lengths of which will be recorded, and if coincident
with unsurveyed lines of legal subdivisions, that fact will also be
stated in the field notes, and at each of the points where said
lines intersect the margin of the pond or lake, a special meander
comer will be established as above directed.
138 LAND SURVEYING [Chap. V.
"A special meander comer is one established on a line of
legal subdivision, not a standard, township, or section line.
"The relative position of these points being thus definitely
fixed in the section, the meandering will commence at one of
them and be continued to the other, noting the intersection, and
thence to the beginning. The proceedings are to be fully en-
tered in the field notes.
"Meander lines will not be established at the segregation
line between dry and swamp or overflowed land, but at the ordi-
nary high-water mark of the actual margin of the rivers or lakes
on which such swamp or overflowed lands border.
"The precise relative position of an island, in a township
made fractional by a river or lake in which the island is situated,
will be determined by triangulation from a special and carefully
measured base line, initiated upon the surveyed lines, on or near
the lake or river bank on the mainland, so as to connect by
course and distance on a direct line, the meander comer on the
mainland with the corresponding point on the island, where the
proper meander corner will be established.
" In making the connection of an island lying entirely within
a section, with the mainland, a special base will be measured
from the most convenient meander comer, and from such base,
the location of an auxiliary meander comer (that is, one not on
a line belonging to the system of rectangular surveying) will
be determined by triangulation, at which the meanders of the
island will be initiated.
" In the survey of lands bordering on tide waters, meander
corners may be temporarily set at the intersection of the sur-
veyed lines with the line of mean high tide, but no monument
should be placed in a position exposed to the beating of waves
and the action of ice in severe weather. In all such cases, the
rule given in section 90 must be observed, by establishing a wit-
ness corner on line at a secure point near the true point for the
meander comer.
" The field notes of meanders will show the dates on which
the work was performed
The field notes of meanders will state and describe
the corner from which the meanders commenced, and upon
DATA TO BE NOTED 1 39
which they closed, and will exhibit the meanders of each frac-
tional section separately; following, and composing a part of
such notes, will be given a description of the land, timber, depth
of inundation to which the bottom is subject, and the banks,
current, and bottom of the stream or body of water meandered.
The utmost care will be taken to pass no object of topography,
or change therein, without giving a particular description thereof
in its proper place in the notes of the meanders.
163. <^ Summary of objects and data intersected by the line
or in its vicinity, to be noted. — i. The precise course and
length of every line run, noting all necessary offsets therefrom,
with the reason for making them, and method employed.
" 2. The kind and diameter of all bearing trees, with the
course and distance of the same from their respective corners ;
and the precise relative position of witness comers to the true
corners.
" 3. The kind of materials of which corners are constructed.
"4. Trees on line. The name, diameter, and distance on
line to all trees which it intersects.
" 5. Intersections by line of land objects. The distance at
which the line intersects the boundary lines of every reservation,
town site, donation claim, Indian allotment, settler's claim, im-
provement, or rancho ; prairie, bottom land, swamp, marsh, grove,
and windfall, with the course of the same at all points of inter-
section ; also, the distances at which the line begins to ascend,
arrives at the top, begins to descend, and reaches the foot of all
remarkable hills and ridges, with their courses, and estimated
height in feet, above the level land of the surrounding country,
or above the bottom lands, ravines, or waters near which they
are situated. Also, distance to and across large ravines, their
depth and course.
" 6. Intersections by line of water objects. All rivers, creeks,
and smaller streams of water which the line crosses ; the distances
measured on the true line to the bank first arrived at, the course
down stream at points of intersection, and their widths on line.
In cases of navigable streams, their width will be ascertained
between the meander comers, as set forth under the proper
head.
I40 LAND SURVEYING [Chap. V.
" 7. The land's surface — whether level, rolling, broken, hilly,
or mountainous.
" 8. The soil — whether rocky, stony, sandy, clay, etc., and
also whether first, second, third, or fourth rate.
" 9. Timber — the several kinds of timber and undergrowth,
in the order in which they predominate.
" 10. Bottom lands — to be described as wet or dry, and if
subject to inundation, state to what depth.
"II. Springs of water — whether fresh, saline, or mineral,
with the course of the streams flowing from them.
"12. Lakes and ponds — describing their banks and giving
their height, and whether it be pure or stagnant, deep or shallow.
• "13. Improvements. Towns and villages ; houses or cabins,
fields, or other improvements with owners' names; mill sites,
forges, and factories, U. S. mineral monuments, and all corners
not belonging to the system of rectangular surveying ; will be
located by bearing and distance, or by intersecting bearings from
given points.
" 14. Coal banks or beds ; peat or turf grounds ; minerals and
ores ; with particular description of the same as to quality and ex-
tent, and all diggings therefor ; also salt springs and licks. All
reliable information that can be obtained respecting these objects,
whether they be on the line or not, will appear, in the general
description.
"15. Roads and trails, wifh their directions, whence and
whither.
" 16. Rapids, cataracts, cascades, or falls of water, with the
estimated height of their fall in feet.
" 1 7. Precipices, eaves, sink holes, ravines, remarkable crags,
stone quarries, ledges of rocks, with the kind of stone they
afford.
" 18. Natural curiosities, interesting fossils, petrifactions,
organic remains, etc. ; also all ancient works of art, such as
mounds, fortifications, embankments, ditches or objects of like
nature.
" 19. The magnetic declination will be incidentally noted
at all points of the lines being surveyed, where any material change
in the same indicates the probable presencie of iron ores ; and
LIMITS FOR CLOSING I4I
the position of such j)oints will be perfectly identified in the
field notes.
164. ** Prescribed Limits for Closings and Lengths of Lines.
— If in running a random township exterior, such random
exceeds or falls short of its proper length by more than three
chains, allowing for convergency, or falls more than three chains
to the right or left of the objective point (or shows a proportion-
ate error for lines of greater or less length than six miles),
it will be re-run, and if found correctly run, so much of the
remaining boundaries of the township will be retraced, or
resurveyed, as may be found necessary to locate cause of
misclosure.
"Every meridional section line, except those which ter-
minate upon a fractional side of a township, will be 80 chains
in length, without allowance of 50 links per mile for difference
of measure, or any other allowance beyond a small reasonable
discrepancy according to the nature of the surface, to be deter-
mined after examination.
"The random meridional or latitudinal lines through a tier
or range of fractional sections shall fall within 50 links of the
objective comers, and a greater falling will indicate negligence
or error.
" The actual lengths of meridional section lines through a
fractional north or south tier of sections shall be within 1 50 links
of their theoretical length. The latter will be determined from
the given lengths of meridional boundaries on the east and the
west range lines.
" Each latitudinal section line, except in a fractional east or
west range of sections, shall be within 50 links of the actual dis-
tance established on the governing north or south boundary of
the township for the width of the same range of sections.
" The north boundary and the south boundary of any section,
except in a fractional range, shall be within 50 links of equal
length.
"The meanders within each fractional section or between
any two successive meander corners, or of an island or lake in
the interior of a section, should close by traverse within a limit
to be determined by allowing five-eighths of a link for each chain
142 LAND SURVEYING [Chap. V.
of such meander line. This rule does not apply to irregular bound-
aries of reservations or private claims, except as far as the same
are natural water boundaries. The total misclosure of meanders
will not be permitted to exceed 150 links, except in large pri-
vate land claims, which are governed by a different rule and
limit.
"In closing upon accepted surveys, when irregularities be-
yond the allowable limits are developed, either in the length or
direction of the closing lines, closing comers will be set, with
quarter-section corners at 40 chains from the last interior sec-
tion corner ;
" And, in general, when conditions are met which result in a
random line being defective, either in length or direction, such
procedure will be adopted as will secure the greatest number of
new rectangular legal subdivisions, without disturbing the con-
dition of accepted surveys.
165. ** Field Notes. — The proper blank books for original
field notes will be furnished by the surveyor general, and in
such books the deputy surveyor will make a faithful, distinct,
and minute record of everything done and observed by himself
and his assistants, pursuant to instructions, in relation to run-
ning, measuring, and marking lines, establishing corners, etc.,
and present, as far as possible, full and complete topographical
sketches of all standard and exterior lines, drawn to the usual
scale of township exteriors. These 'original field notes' are
not necessarily the entries made in the field, in the deputy's
pocket note books called tablets ; but they are to be fully and
correctly written out in ink, from such tablets, for the permanent
record of the work. Tablets should be so fully written as to
verify the original field notes whenever the surveyor general
requires them for inspection.
" A full description of all comers belonging to old surveys,
from which the lines of new surveys start, or upon which they
close, will in all cases be furnished the deputy from the sur-
veyor general's office, when authority is given for commencing
work; then, if the old comers are found to agree with said de-
scriptions, the deputy will describe any one of them in this form,
'which is a firmly set, marked and witnessed as described
FIELD NOTES 1 43
by the surveyor general ; ' but, should a comer, not answer the
description supplied, the deputy will give a full description of
such corner and its accessories, following the proper approved
form given in these instructions.
" A full description of each comer established under any one
contract will be given once only ; subsequent reference to such
comer will be made in the form, 'heretofore described,' or *the
comer of sections 2, 3, 10, and 11,' as the case may require.
" In all cases where a comer is reestablished, the field notes
will describe fully the manner in which it is done.
" The field notes of the survey of base, standard, and meri-
dian lines will describe all corners established thereon, how es-
tablished, the crossings of streams, ravines, hills, and mountains ;
character of soil, timber, minerals, etc. ; and after the description
of each township comer established in running such lines, the
deputy will note particularly in the * general description' the
character of townships on each side of the lines run.
"The field notes of the survey of exterior boundaries of
townships will describe the corners and topography, as above
required, and the 'general description' at the end of such
notes will describe the townships as fully as possible, and also
state whether or not they should be subdivided.
♦'The field notes of the subdivisional survey of townships
will describe the comers and topography as above required,
and the 'general description' at the end of such notes will
state minutely the character of the land, soil, timber, etc., found
in such townships.
'* The topography will be given on the true line in all cases,
and will be taken correctly, not estimated or approximated.
" With the field notes of the survey of base lines and stand-
ard parallels, and principal and guide meridians forming a tract
24 miles square, including those of the township exteriors therein,
the deputy will submit a diagram of the lines surveyed, drawn to
a scale of half an inch to one mile, upon which will be written
the true bearings and lengths of all surveyed lines, except
the lengths of those which are actually 40.00 or 80.00 chains.
These diagrams will exhibit all water courses, with the direction
of each indicated by an arrow head pointing down stream ; also.
144 LAND SURVEYING [Chap. V.
the intersection of the lines with all prairies, marshes, swamps,
ravines, lakes, ponds, mountains, hills, and all other natural or
artificial topographical features mentioned in the field notes, to
the fullest extent possible.
" With the special instructions for making subdivisional sur-
veys of townships into sections, the deputy will be furnished by
the surveyor general with blank township diagrams drawn to a
scale of one inch to forty chains, upon which the true bearings
and lengths of the township and section lines, from which the
surveys are to be projected, or upon which they are to close, will
be carefully marked ; and on such diagrams the deputy who sub-
divides will make appropriate sketches of the various objects
of topography as they occur on his lines, so as to exhibit not
only the points of intersection therewith, but also the direc-
tions and relative positions of such objects between the lines,
or within each section, as far as practicable, so that every
topographical feature may be properly completed and connected
in the showing.
" Triangulations, offsets, or traverses, made to determine dis-
tances that cannot be directly measured, such as those over
deep streams, lakes, impassable swamps, caftons, etc., will be
made on the random lines, when random lines are run. All
particulars will be fully stated in the field notes.
*'The exhibition of every mile of surveying, whether on
standard, township, or subdivision lines, and the meanders in
each section, will be complete in itself, and will be separated
from other records by a bkck line drawn across that part of the
page containing the body of notes. The description of the sur-
face, soil, minerals, timber, undergrowth, etc., on each mile of line
will follow the notes of survey of such line, and not be mingled
with them.
" Particular care will be taken to record at the end of each
mile the number of chains of mountainous land, heavily timbered
land, or land covered with dense undergrowth.
" The date of each day's work will immediately follow the
notes thereof.
" Near the end of the field notes of exteriors and immediately
before the 'general description,* the deputy surveyor will add,
FIELD NOTES 145
in the form shown in the specimen field notes, a tabular state-
ment of the latitude and departure of all boundary lines of the
township, derived from a traverse table, and will give the totals,
and the errors in latitude and departure ; said errors shall in no
case exceed three chains, the prescribed limit for the falling of
the random north boundary of a township. If a part or the
whole of one or more boundaries is made up of meander lines,
the northings, southings, eastings, and westings of the full
section lines, nearest said meanders, will replace the missing
N., S., E., or W. township lines, as the case may require,
thereby presenting the errors of said boundaries of a closed
survey.
" If all the exterior lines have been surveyed by the deputy,
the bearings and distances for the table will be taken from
his own notes. In a case where some of the boundaries have
been surveyed under another contract, the deputy will use the
bearings and distances supplied by the surveyor general, in con-
nection with those of his own lines ; and, if errors exceed the
allowance of three chains, specified in paragraph i of the " Pre-
scribed Limits," the deputy will determine by retracement where
the error occurs, correct the same before he leaves the field, and
place the table in his original field notes.
" Besides the ordinary notes taken on line (and which will
always be written down on the spot, leaving nothing to be sup-
plied by memory), the deputy will subjoin, at the conclusion of
his book, such further description or information touching any
matter or thing connected with the township (or other) survey
which he may be able to afford, and may deem useful or neces-
sary to be known — with a general description of the town-
ship in the aggregate, as respects the face of the country, its
soil and geological features, timber, minerals, waters, settle-
ments, etc.
'* Following the general description of the township will be
placed * A list of the names of the individuals employed to
assist in running, measuring, and marking the lines and corners
described in the foregoing field notes of township No. of
the base line of range No. of the meridian, show-
ing the respective capacities in which they acted."
146
LAND SURVEYING
[Chap. V.
Specimen of Field Notes.
Subdivision of T. 15 N., R. 20 E.
Chains. I commence at the cor. of sees, i, 2, 35, and 36, on the S. bdy. of the
Tp., which is a sandstone, 6 X 8 x 5 ins. above ground, fiimly set, and
marked and witnessed as described by the surveyor general.
Thence I run
N. 0° 01' W., bet. sees. 35 and 36.
Over level bottom land.
4 .50 Wire fence, bears £. and W.
20.00 Enter scattering cottonwood timber, bears £. and W. F. G. Alexan-
der's house bears N. 28** W.
29 .30 Leave scattering cottonwoods, bearing £. and W.; enter road, bears N.
30.00 S£. cor. of F. G. Alexander's field ; thence along west side of road.
39.50 To crossroads, bears £. to Mound City; N. to Lake City. F. G.
Alexander's house bears S. 40® W. The } sec. cor. point will
fall in road ; therefore
Set a cedar post, 3 ft. long, 3 ins. sq., with quart of charcoal, 24 ins.
in the groundi, for witness cor. to \ sec. cor., marked W C ^ S 35
on W. and 36 on £. face ; dig pits, 18 x 18 X 12 ins. N. and S. of
post, 3 ft. dist. ; and raise a mound of earth, 3) ft. base, i) ft. high.
W. of cor.
40.00 Point for \ sec. cor. in road.
Deposit a marked stone, 24 ins. in the ground, for i sec. cor.
The SE. cor. of Pat. Curran's field bears W., 5 Iks. dist.
40.50 Set a limestone, 15x8x6 ins. 10 ins. in the ground, for witness cor.
to i sec. cor., marked W C } S on W. face; dig pits, 18 x 18 x 12
ins. N. and S. of stone, 3 ft. dist. ; and raise a mound of earth, 3} ft.
base, i^ ft. high, W. of cor.
Thence along £. side of field.
50 . 50 NE. cor. of Pat. Curran's field, bears W. 4 Iks. dist.
51 . 50 Leave road ; which turns to N. 70° W., leads to ferry on Yellowstone
River ; thence to Lake City.
57 . 50 Enter dense cottonwood and willow undergrowth, bears N. 54° E. and
S. 54** W.
72 . 50 Leave undergrowth, enter scattering timber, bears N. 60^ E. and S.
60° W.
80.00 Set a locust post, 3 ft. long, 4 ins. sq., 24 ins. in the ground, for cor. of
sees. 25, 26, 35 and 36, marked
T 15 N S 25 on NE.,
R 20 E S 36 on SE.,
S 35 on SW., and
S 20 on N W. face ; with i notch on S. and E. faces ; from which
An ash, 13 ins. diam., bears N. 22^ E., 26 Iks. dist., marked T 15
N R 20 E S 25 B T.
A sycamore, 23 ins. diam., bears S. 71}° E., 37 Iks. dist., marked
T15N R20ES36BT.
A walnut, 17 ins. diam., bears S. 64° W., 41 Ucs. dist., marked
T 15N R20ES35 BT.
A cottonwood, 13 ins. diam., bears N. 21 J® W., 36 Iks. dist.
marked T 15 N R 20 E S 26 B T.
Last 20.00 chs. of this mile subject to overflow, 2 to 4 ft. deep.
Land, level bottom.
Soil, alluvial ; ist rate. No stones were obtainable.
Timber, scattering cottonwood, sycamore, ash, and walnut ; under-
growth, cottonwood and willow.
Dense undergrowth, 15.00 chs.
MARKING THE CORNERS I47
x66. Marking the Comers. — After the positions of the cor-
ners are determined they are marked according to instructions
issued by the Land Office. The character of the monuments
set will depend upon the kind of comer to be marked, the charac-
ter of the country, and the existing conditions. There are four-
teen different classes of comers, as follows :
1. Standard township comers.
2. Closing township corners.
3. Comers common to four townships.
4. Comers common to two townships only.
5. Comers referring to one township only.
6. Standard section comers.
7. Closing section corners.
8. Corners common to four sections.
9. Comers common to two sections only.
10. Comers referring to one section only.
1 1. Quarter-section corners.
12. Standard quarter-section corners.
13. Meandered comers.
14. Comers on reservation or other boundaries not conforming
to the regular system.
There are eight different classes of monuments allowed, de-
pending upon the character of the country and the difficulty of
transportation.
1. Stone, with pits and mounds of earth.
2. Stone, with mounds of stone.
3. Stone, with bearing trees.
4. Post, with pits and mounds of earth.
5. Post, with bearing trees.
6. Mound of earth, with deposit, and stake in pit.
7. Tree comer, with pits and mounds of earth.
8. Tree corner, with bearing trees.
There are many details in regard to the proper marking of
corners which can only be learned by experience in this kind of
surveying.
148 LAND SURVEYING [Chap. V.
167. To ESTABLISH A PARALLEL OF LATITUDE.— A paral-
lel of latitude on the surface of a sphere is a curved line. This
may be understood from the facts that the meridians converge
toward the pole, and that a parallel is at every point at right
angles to the meridian at that point. If vertical lines are drawn
through every point on a parallel of latitude they will form a
conical surface, the apex of the cone being at the center of the
sphere. In the case of a straight line all of the verticals would
lie in the same plane, and this plane would intersect the sphere
in a great circle.
A parallel of latitude may be run out by means of the solar
attachment to the transit, since by means of this instrument the
direction of the meridian may be quickly found whenever the sun
is visible (Art. 85, p. 66). A line which at every point is at
right angles to the meridian will be a true parallel of latitude.
This method, however, is found to give results less accurate than
are required, chiefly on account of the errors in the adjustment
of the solar attachment.
A better method of establishing a parallel is by taking offsets
from a straight line. Two methods of doing this, known as the
Secant Method and the Tangent Method^ are used in the Public
Land Surveys.
168. The Secant Method. — (Fig. 60.)' " This method consists
of running a connected series of straight lin^s, each six miles long,
T.13N,lt21E.
3* »5
^:::sA^ 7,.^<L .^^Z ,S^CAffr line __ — Y:
' 5 I s s ? » ! 5 * .
^ 71 ^ * [Offsers-To^ 4]
Fig. 60. Secant Method for Establishing a Parallel of
Latitude.
on such courses that any one of the lines will intersect the curve of
the parallel of latitude in two points, separated by an interval of
four miles ; and from this line thus established, measuring north
TABLE 4.
Azimuths op the Secant, and Offsets, in Feet, to the Parallel.
Ladtode in le£t4iaod colnmn and distance from starting point at top or bottom of the table.
Lati-
tude
30
31
3a
33
34
35
36
37
38
39
40
41
43
43
44
45
46
47
48
49
Azlmnths and oflfsets at —
80^ W A
X.93 N.
89»58'.4
a.oz N.
a.og N.
890 68'.3
3.Z7 N.
89«68'.2
3.35 N.
89<»68'.2
a.33 N.
a8«68M
3.43 N.
89«58'.0
3.51 H.
89o58'.0
3.61 H.
«9«»57'.9
3.70 N.
89» 57'.8
3.79 N.
890 67'.7
a.89 N.
89«67'.7
3.00 N.
89° 67'.6
3.ZZ N.
89*67'.6
3<33 N.
89« 57'.4
3.33 H.
89<»67'.3
3.44 N.
89«B7'.2
3.57 N.
89» 57M
3.70 N.
89* 67'.0
3.8a N.
890 68'.9
3.96 N.
lAti.
6 miles.
imUe.
89»68'.7
0.87 N.
89»68'.6
0.91 N.
89059.6
0.94 N.
89°68JS
0.97 N.
89«68'J5
z.oz N.
89<»58'J>
Z.05 N.
89*68'.4
Z.09N.
89°68'.3
I.Z3 N.
890 68'.3
X.17 N.
89° 68' .2
Z.3Z H.
89«58'.l
Z.35N.
89« 58^.0
1.30 N.
89«68'.0
X.35 N.
89»68'.0
Z.40N.
89*67'.9
1.4s N.
89« 67'.8
Z.50N.
89«67'.7
X.55 N.
890 57'.6
i.6z N.
89» 67'.5
1.66 N.
89«67'J5
1.73 N.
89» 67'.4
Z.78N.
similes.
89«09'.O
0.00
89*58'.9
0.00
89*68'.9
0.00
89«58'.8
0.00
89°68'.8
0.00
89«»68'.8
0.00
890 58'.7
0.00
89°68'.6
0.00
89«»58'.6
0.00
89° 58' .6
0.00
89«»68'.5
0.00
89« 68^.4
0.00
89° 58' .4
0.00
89° 68'.4
0.00
89°58'.d
0.00
89°58'.3
0.00
89°58'.2
0.00
89°68'.l
0.00
89°58'.0
0.00
89°68'.0
0.00
89° 67'.9
0.00
zi miles. 3 iziiles. 3^ miles.
89° 59' .2
0.67 S.
89° 59' .2
0.70 S.
89°69'.2
0.73 s.
89° 59'.!
0.76 S.
89°59M
0.79 S.
89° 59'.!
0.83 S.
89°89'.0
0.8s S.
89°68'.9
0.88 S.
89°58'.9
0.9Z S.
89°58'.9
0.94 s.
89°58'.9
0.98 S.
89°58'.8
Z.03S.
89° 68'.8
z.05 s.
89°58'.8
Z.08S.
89° 58'.7
Z.Z3S.
89°58'.7
z.z6 S.
89° 58'.6
Z.3Z S.
89°58'.6
Z.35 S.
89°58'J>
Z.30S.
89°58'.5
Z.34 S.
89°58'.4
Z.39S.
5 miles.
4i miles.
89°59'J{
Z.Z5S.
89°59'.5
Z.30 S.
89°69'.5
z.35 s.
89°59'.4
z.30 S.
89°69'.4
z.35 s.
89°59'.4
Z.40 S.
89°59'.4
Z.46 S.
89°69'.3
Z.5Z S.
89° 59^.8
Z.56S.
89°69'.3
Z.63S.
89°59'.3
1.68 S.
89°69'.2
Z.74S.
89»59'.2
Z.80 8.
89° 69' .2
Z.86S.
89°69'.2
Z.93 s.
89°69'.l
3.00 S.
89° 59'.!
3.07 S.
89° 69'.!
3.Z4S.
89°59'.0
3.33 S.
89°59'.0
3.30 S.
89°69'.0
3.38 S.
4 miles.
89°59'.7
Z.44S.
89° 59'.7
Z.50 S.
89°69'.7
Z.56 S.
89°69'.7
Z.63 S.
89° 69' .7
Z.69 S.
89° 59' .7
Z.75 s.
89° 69' .7
Z.83 S.
89°59'.7
Z.89 S.
89°59'.7
1.95 s.
89°59'.7
3.03 S.
89° 59' .7
3.Z0S.
89°59'.6
3.Z7 S.
89°59'.6
3.35 S.
89°59'.6
a.J3S.
89°69'.6
3.4Z S.
89°59'.6
3.49 S.
89° 59'.5
3.59 s.
89° 69^.5
3.67 S.
89°59'.6
3.78 S.
89°59'.5
3.87 S.
89° 59'.5
3.97 S.
similes.
3 miles.
90° (E. or W.)
Z.54S.
90°(E.orW.)
Z.60 S.
90° (E. or W.)
Z.67 S.
90° (E. or W.)
Z.73S.
90° (E. or W.
Z.80S.
90° (E. or W.
Z.87S.
90° (E. or W.
Z.94S.
90° (E. or W.)
3.0Z S
90° (E. or W.
3.08 S.)
90° (E. or W.)
3.z6 S.
90° (E. or W.)
3.34 S.
90° (E. or W.)
3.33 S.
90° (E. or W.)
3.40 S.
90° (E. or W.)
3.48 S.
90° (E. or W.)
3.57 S.
90° (E. or W.)
3.6^ S
90° (E orW.)
3.76 S.
90° (E. or W.)
3.86 S.
90° (E. or W.)
3.96 S.
90° (E. or W.)
3.06 S
90° (E. or W.)
3.17 S
Defleo-
tion Angle
and nat.
tan. to
Rad.eeft.
3' 00".2
0.69 i~s.
3' 07".4
0.73 ins.
3'16".0
0.75 ins.
3' 22".6
0.78 ins.
3'30".4
0.8 z ins.
3' 88".4
0.84 ins.
3'4e".4
0.87 ins.
3' 66".0
0.90 ins.
4' 08".6
0.93 ins.
4' 12".6
0.97 ins.
4' 21 ".6
z.ooins.
4' 31". 2
Z.04 ins.
4'40".8
z.o8ins.
4^60" .8
Z.X3 ins.
6' 01".0
z.z6 ins.
5' 11" .8
z. 30 ins.
6' 22" .8
z.34 i-^s.
6' 34".2
Z.38 ins.
5' 4«".2
Z.33 ins.
5' 58".6
z.38 ins.
8' 11".4
Z.43 ins.
3 miles.
Azimuths and offsets at-
Deflec-
tion Angle
and nat.
tan. to
Rad. 66 ft.
ISO LAND SURVEYING [Chap. V.
or south, as the case may be, to attain other required points on
the latitude curve." The o and 6 mile points of a parallel will
be north of the secant, and the 2, 3, and 4 mile points will be
south of the secant.
The instrument is set up south of the township corner where
the survey is to begin, the distance from the comer being found
in Table 4 in the column headed "o miles.*' For example, in
latitude 40° the transit would be set 2.79 ft. south of the comer.
The direction of the first secant at its initial point is found by
observing on Polaris (Chapter VII) to obtain the true meridian
and then laying off the azimuth angle found in Table 4 under
"o miles." (See Fig. 60.) This angle should be repeated sev-
eral times to determine accurately the direction of the secant.
This direction is then prolonged 6 miles. At each mile and half-
mile point an offset is measured to establish a point on the curve,
the distance and direction of the offset being shown in Table 4.
When the 6-milfe point is reached the direction of a new secant
is found by turning off to the north the deflection angle given in
the right-hand column of Table 4. The offsets are then measured
from this line as from the preceding one. The chief advantage
3. *
^ _ V an >< E^»*""^ ', I
t'. *'. ^
^«-™^ lit
[0ffaet3--nibte 6] • ^'
Fig. 61. Tangent Method for Establishing a Parallel of
Latitude.
M
of this method is that the offsets are short and hence much cut-
ting is saved in wooded regions.
169. "Tangent Method. — This method consists in laying
off from a tme meridian, established by observations on Polaris
at elongation, an , angle of 90®, producing the direction thus
TANGENT METHOD 151
determined, a distance of 6 miles, in a straight line, and
measuring north therefrom, at half-mile intervals, distances of
correct length, taken from Table 6 (interpolated if necessary),
for the given latitude, to attain other points on the latitude
curve passing through the tangential or initial points.
"The azimuth or bearing of the tangent at successive mile
points will be taken from Table 5 to the nearest whole minute
only, and will be inserted in the field notes, no interpolation
being required, except when test sights are taken. The true
bearing between two points on a standard parallel will be,
derived from Table 5 by taking it in the column headed with
one-half the distance between said points. The offsets at
intervals of one mile are inserted in Table 6 ; to obtain the
length of offsets at the half-mile points, take one-fourth of the
oflFset corresponding to twice the distance of the half-mile point
from the tangential point.
"This method is suitable for running standard parallels and
latitudinal township Unes in a level open coimtry, where no
intersections with topographical features will be required ; but,
in all cases the secant method will be found most convenient."
IS2
LAND SURVEYING
[Chap. V.
TABLE 5.
Azimuths op the Tangent to the Parallel.
[The azimuth is the smaller angle the tangent makes with the true meridian and always
measured from the north andtowards the tangential pomts.]
Lati-
tude.
I mile.
a
mUes.
3
miles.
4
miles.
S
miles.
6 miles.
0
0
/
ff
0
/
//
0
,
//
0
/
f*
0
/
//
0
/
//
90
3a
88
89
89
69
69
69
30.0
28.8
27.6
89
89
89
68
68
68
69.9
67.6
66.0
89
89
89
68
68
68
29.9
26.3
22.6
89
89
89
67
67
67
69.9
66.0
60.0
89
89
89
57
67
67
29.9
23.8
17.5
89
89
89
56
66
66
698
62 6
45.0
33
34
35
89
89
89
69
60
69
28.2
24.9
23.6
89
89
89
68
68
68
62.6
49.9
47.2
89
89
89
68
68
68
18.7
14.8
10.8
89
89
89
67
67
67
44.9
39.7
34.4
89
89
89
67
67
66
11.2
01.6
58.0
89
80
89
66
66
66
37.4
296
21.6
36
89
89
89
69
69
69
22.2
20.8
19.4
89
89
89
68
68
68
44.4
41.6
38.8
89
89
89
68
68
67
06.8
02.6
68.2
89
89
89
67
67
57
28.9
23.3
17.5
89
89
89
66
66
56
61.1
44.1
36.9
89
89
89
66
66
66
13.4
06.0
66.3
39
40
4«
89
89
89
69
69
69
17.9
18.4
14.8
89
89
89
68
68
68
36.8
32.8
29.6
89
89
89
67
67
67
63.7
49.2
44.4
89
89
89
67
67
56
11.6
05.6
60.3
89
89
89
56
56
56
29.6
21.9
14.1
89
89
89
66
55
66
47.6
38.3
28.9
4a
43
44
89
89
89
69
69
69
13.2
11.6
00.8
89
89
89
68
68
68
26.4
23.1
19.6
89
89
89
67
67
67
39.6
34.6
29.5
89
89
89
66
66
66
62.8
46.2
39.3
89
89
89
56
66
65
06.0
67.7
49.1
89
89
89
66
66
54
19.2
09.2
68.9
47
89
89
89
60
69
69
06.0
06.2
04.3
89
89
89
68
68
68
16.1
12.4
08.6
89
89
89
57
67
67
24.1
18.6
12.9
89
89
89
66
66
66
32.1
24.8
17.1
89
i
65
66
66
40.2
31.0
21.4
89
89
89
64
64
64
48.2
37.2
26.7
4S
49
50
89
89
89
60
69
68
02.3
00.2
68.1
89
89
89
68
68
67
04.6
00.6
66.2
89
89
89
67
57
66
06.9
00.7
64.3
89
89
89
66
66
66
09.2
00.9
62.6
89
89
89
66
65
54
11.5
01.2
60.5
89
89
89
64
64
63
13.8
01.4
48.6
LaU-
tade.
7
miles.
8 miles.
9
miles.
10 miles.
I]
miles.
xa
miles.
0
0
/
„
0
,
„
0
,
ft
0
/
„
0
f
//
0
/
/»
30
31
3»
89
89
89
66
56
56
29.8
21.3
12.6
89
89
89
55
55
66
68.8
50.0
40.0
88
88
88
56
55
55
29.8
18.8
07.6
89
89
89
64
54
64
68.7
47.6
36.1
89
89
89
64
64
64
29.7
16.3
02.6
89
89
89
63
53
63
69.7
45.1
90.1
33
34
35
89
89
89
66
66
66
03.6
54.5
46.2
89
89
89
66
56
66
28.8
18.4
08.8
88
88
88
54
54
54
56.1
44.4
32.3
89
88
89
64
54
53
22.3
08.3
65.8
89
89
89
63
63
53
48.5
34.2
19.6
89
89
88
53
52
62
14.8
69.1
43.1
36
89
89
89
65
65
55
36.6
25.8
15.7
89
89
89
54
64
54
67.8
46.6
35.1
88
88
89
64
54
63
20.0
07.4
54.5
89
89
89
53
53
53
42.3
28.2
13.8
89
89
89
53
52
52
04.5
49.1
33.2
89
89
89
62
52
61
26.7
09.9
62.6
39
40
41
89
89
89
55
54
54
05.4
54.7
43.7
89
88
89
54
54
53
23.3
11.1
68.5
88
88
88
53
53
53
41.2
27.5
13.4
89
89
89
52
62
52
58.1
43.8
28.2
89
89
89
52
52
51
17.0
00.2
43.0
89
89
89
51
61
60
34.9
16.6
57.8
4a
43
44
89
89
89
54
54
54
32.4
20.8
08.7
89
88
89
53
53
53
45.6
32.3
18.5
88
88
88
52
52
52
58.8
43.8
28.4
89
89
89
62
61
51
12.0
65.4
38.2
89
89
89
51
51
60
25.2
06.8
48.0
89
89
89
50
60
49
38.4
18.5
57.8
47
89
89
89
53
63
53
56.3
43.4
30.0
89
88
88
63
52
52
04.3
49.5
34.3
88
88
88
52
51
51
12.3
66.7
38.6
89
89
89
51
51
60
20.4
01.8
42.8
89
89
89
60
60
48
28.4
06.1
47.2
89
89
89
49
49
48
96.4
14.3
51.4
48
49
50
89
89
89
53
63
62
16.1
01.7
46.6
88
88
88
62
52
51
18.4
01.8
44.7
89
89
89
51
51
60
20.7
02.1
42.8
89
88
89
60
60
48
23.0
02.4
40.8
89
89
89
49
49
48
26.3
02.6
38.0
89
89
89
48
48
47
27.6
02.8
37.1
TANGENT METHOD
TABLE a
Offsets, in Chains, from Tangent to Parallel.
»S3
Lati-
tnde.
imile.
a miles.
3 miles.
4 miles.
smiles.
6 miles.
o
CAaim.
CJkaiMs.
CAanu.
C/kaiMS.
Chains,
ChaiMs.
30
3a
0.006
0006
0.006
0.023
0.024
0.025
0.063
0.065
0.067
0.09
0.10
0.10
0.14
0.16
0.16
0.21
0.22
0.23
33
34
35
0.007
0.007
0.007
0.026
0.027
0.028
0069
0.061
0.064
0.10
0.11
0.11
0.16
0.17
0.18
0.24
0.26
0.26
36
12
0.007
0.008
0.008
0.029
0.031
0.032
0.066
0.068
0.071
0.12
0.12
0.13
0.18
0.19
0.20
0.28
0.27
0.28
39
40
4X
0.008
0.008
0.009
0.033
0.034
0.036
0.074
0.076
0.079
0.13
0.13
0.14
0.20
0.21
0.22
0.29
0.30
0.32
4a
43
44
0.009
0.009
0.010
0.036
0.038
0.039
0.082
0.086
0.088
0.14
0.16
0.16
0.23
0.24
0.24
0.38
0.34
0.36
47
0.010
0.010
0.011
0.040
0.042
0.044
0.091
0.094
0.097
0.16
0.17
0.17
0.26
0.26
0.27
0.36
0.37
0.39
48
49
50
0.011
0.012
0.012
0.046
0.046
0.048
0.101
0.104
0.108
0.18
0.19
0.19
0.28
0.29
0.30
0.40
0.42
0.43
Lati-
tude.
7 miles.
Smiles.
9 miles.
xo miles.
xz miles.
xa miles.
0
CkaiTU,
Chains,
Chains.
Chains.
Chains,
3D
31
38
0.29
0.30
0.31
0.37
0.39
0.40
0.47
0.49
0.51
0.68
0.60
0.63
0.71
0.74
0.76
0.84
0.88
0.91
33
34
35
0.32
0.33
0.36
0.42
0.43
0.45
0.53
0.66
0.57
0.66
0.68
0.70
0.79
0.82
0.86
0.96 .
0.98
1.02
36
12
0.36
0.37
0.38
0.47
0.48
0.50
0.69
0.61
0.64
0.73
0.76
0.78
0.89
0.91
0.95
1.06
1.10
1.14
39
40
41
0.40
0.41
0.43
0.62
0.64
0.56
0.66
0.68
0.70
0.81
0.84
0.87
0.99
1.02
1.06
1.18
1.22
1.26
4a
43
44
0.44
0.46
0.48
0.58
0.60
0.62
0.73
0.75
0.79
0.90
0.93
0.97
1.09
1.14
1.18
1.31
1.36
1.40
47
0.49
0.51
0.63
0.64
0.66
0.68
0.81
0.84
0.87
1.00
1.04
1.07
1.22
1.26
1.31
1.45
1.60
1.66
48
49
30
0.56
0.67
0.69
0.71
0.74
0.77
0.91
0.93
0.97
1.12
1.16
,20
1.36
1.40
1.46
1.61
1.67
1.73
154
LAND SURVEYING
[Chap. V.
170. CONVERGENCE OF THE
MERIDIANS. — The angular con-
vergence of the meridians, given
in Table 3, may be computed as
follows. In Fig. 62 AB is an
arc of a parallel of latitude and
EQ the arc of the equator
intercepted by the meridians
through 4 and B. ATand BT
are lines tangent to the meri-
dians at A and B, meeting the
earth's axis, prolonged, at T. It
will be seen that the angle
BTO equals the angle BOQ,
which is the latitude of points
A and B. The angle AO'B
is the difference in longitude
of points A and B, The angle
Fig. 62.
between the meridians at A and B is the angle A TB
In the triangle B00\
B(y = BO cos O'BO
= BO cos BOQ
In the sector AO'By
In the sector A TB,
AB
-^^= angle A TB (approximately)
BT
But
BT=
BO'
BC
sin BTO sin BOQ
.'. angle ATB =4^, sin BOQ
BU
=^ngle AO'B sin BOQ,
i.e., the angular convergence equals the difference in longitude
times the sine of the latitude.
CONVERGENCE OF MERIDIANS
^SS
Example. — To find the angular convergence between two
meridians 6 miles apart in latitude 37®. The length of 1° of
longitude in latitude 37° is 55.30 miles (Table 7).
55.30
X sin 37^ X 6o = 3'.9.
TABLE 7.
Length of a Degree in Longitude.
Dqree of Longi-
Degree of Longi-
Dqnree of Longi-
Lat.
tude
Lat.
tude.
Lat.
tude.
Statute MUtt.
Statute MUee.
Statute Miles.
0
69.160
30
59-944
60
34.666
I
.150
31
0-334
61
33-615
2
•"?
32
58.706
62
32.553
31 .481
3
.066
33
.060
f3
4
68.992
34
57.396
64
30.399
1
"^■^
35
56.715
65
29.308
36
.016
66
28.208
- 1
.647
37
55- 300
67
27.100
.491
38
54.568
53-819
68
25 983
9
•3M
39
69
24.857
10
68.116
40
53-053
70
23 723
II
67.898
41
52.271
71
22.582
12
.659
42
51 -473
72
21 .435
20.282
13
.400
43
50.659
73
14
.120
44
49 -830
74
19.122
•;i
66.820
•499
t
48.986
.126
M
:^:?i!
17
.158
47
47.251
77
15.607
18
65.797
48
46 .362
78
14.425
19
.416
49
45 -459
79
13.238
20
65.015
50
44 542
80
12 .047
21
64.594
51
43 .611
81
10.853
22
, .'54
52
42 .667
82
9.656
23
"'■%
53
41.710
l^
8.456
24
54
40.740
84
7.253
^1
62.718
55
39-758
85
6.048
.201
56
38.763
37.756
86
4.841
27
61.665
57
87
3-632
28
.110
58
36.737
88
2.422
29
60.536
59
35 -707
89
I .2X1
CHAPTER VI.
TRAVERSE IJNB8. — LOCATION OF BUILDINaS. — MISCEL-
LANBons suRVxnriNa problems.
TRAVERSE LINES.
171. TRAVERSES WHICH DO NOT FORM CLOSED FIGURES. —
A great many surveys, such, for example, as the preliminary sur-
veys for railroads or pipe lines, call for traverses which do not
return to the starting point. In this work the line is usually
measured continuously from one end to the other, and the form
of notes is commonly as follows. The starting point of the tra-
verse is called ** Station o," the next station 100 ft. away is
"Station i,'* the next "Station 2," etc. Every loo^ft. length is
2^ full station and any fractional distance is called Ih^plus. The
distance from Station o to any point, measured along the traverse
line, is the station of that point and is recorded always by the
number of the last station with the plus station in addition, e.g.,
the station of a point at 872.4 ft. from Station o is 8 + 72.4,
At the angle points it is customary to measure the deflection
angles rather than the interior angles because the former are
usually the smaller. These should be checked in the field by
"doubling*' the angles (Art. 143, p. 108).
The notes are kept so as to read up the page. The left-hand
page is for the traverse notes and the right-hand page for the
sketch, the stations in the sketch being opposite the same station
in the notes. Fig. 63 is a set of notes illustrating this type of
traverse. Frequently no notes are kept in tabular form, all of
the data being recorded on the sketch.
172. METHODS OF CHECKING TRAVERSES WHICH DO NOT
FORM CLOSED FIGURES. — Checking by Astronomical Methods. —
The angles of any traverse can be checked by determining the
azimuth of the first and last lines by astronomical methods.
(See Chapter VII.) But since the meridians converge it is neces-
156
CHECKING TRAVERSES
157
sary to make proper allowance for this convergence, the amount
of which can be obtained from Table 3, p. 1 29.
173. Checking by Cut-Off Lines. — The angles may also be
checked in some cases by cutting across from one point on the
traverse to another at a considerable distance ahead, and measur-
ing the angles from the traverse line at each end of this cut-off
(Lbpt-Hano Pack.) (Right-Hand Paob.)
PMimnary5uryefFcrX^YfUt,Knmhci^m
^.W'^
^
Sfa.
fhi/rt
DeH.Aryli
%!^
OoMTtns
1
9
•♦0«2
4f/7L
Nif£
NO*06£
{
)
6
*4^
« -^tto,
IviffSrSt
7
• t^Uf
kMl
Af$0*9f
AfJoVw
f <
1
6
s
4,
•M?7
MTrr/r
/fz/iw
Af/n»W
~"^^i^-^
1
3
^
-!"«
ft$^
^
z
.^
Kf\
1
0
0
MifXw
HifUsn
•-
-
v
J
Fig. 63. Traverse Notes.
line, thereby obtaining all the angles of a closed traverse in which
the length of one side only (the cut-off line) is missing. Some-
times the angle at only one end of the cut-off line can be meas-
ured, in which case the calculations for checking are not so simple
as in the former case. When both angles have been measured
the check consists in simply obtaining the algebraic sum of the de-
flection angles, while in the latter case the traverse must be
computed.
1 58 TRAVERSE LINES [Chap. VI.
174. Checking by Angles to a Distant Object. — A practical
and very useful method of checking the azimuth of any line of the
traverse is as follows. At intervals along the line, measure carefully
the angle from the traverse line to some well-defined distant ob-
ject, such as a distinct tree on a hill or the steeple of a church.
If the survey is plotted and it is found by laying off the angles
taken to the distant object that these lines do not meet at one
point on the plan there is a mistake in the angles, and a study of
the plot will show the approximate location of the mistake. If
convenient, an angle to the distant object should be taken at
every transit point. When plotted, if these lines meet at the
same point in one section of the traverse and in another section
meet at another point, then there is a mistake in the line which
connects these two parts of the traverse. Frequently this dis-
tant point is so far away that it cannot be plotted on the plan.
In this case as well as when it is desired to check more accu.
rately than by plotting, the location of the distant point with
reference to the traverse hne can be computed by using these
-measured angles, as explained in Art. 408, p. 372. Plotting will'
not disclose minor errors of a few minutes only.
175. Checking by Connecting with Triangulation Points. —
An accurate and practical method of checking both the angles
and distances of a traverse is to connect the traverse with reli-
able triangulation points which can be easily identified. (See
Art. 283, p. 255.) The latitude and longitude of these triangula-
tion points and the distances between them can be obtained from
the proper authorities. Sometimes the distances between them
are not known but they can be computed. Then by connecting
the traverse lines with these triangulation points by angles and
distances a closed traverse is obtained, which serves as a good
check.
Many surveyors fail to appreciate the value of this method of
checking and do not realize how many such points are available.
The information concerning such triangulation points can be
obtained from The U. S. Coast and Geodetic Survey, The U. S.
Geological Survey, State surveys, and frequently from City or
Town surveys.
GENERAL PRINCIPLES 159
LOCATION OF BUILDINGS FROM TRANSIT LINE.
176. METHODS OF LOCATINO BUILDINGS. — Many objects,
such as buildings, are plotted directly from the survey line. In
this case the measurements taken should be such as will permit
the most accurate and rapid plotting. Sometimes where it is
desirable to shorten the amount of fieldwork, the methods used
are such as to gain time at the expense of accuracy or of sim-
plicity in plotting. The accuracy with which such locations are
made will depend upon the purpose of the survey. In city plans
the accurate location of buildings is of g^eat importance, while
in topog^phic maps a rough location is often sufficient. There
are so many different cases which will arise that this work
requires considerable skill and judgment on the part of the
surveyor.
177. GEOMETRIC PRINCIPLES, — Whether the locations are
accurate or only rough, the principles involved are the same.
In order to make clear the various methods used in the location
of buildings it will be well to enumerate the geometric principles
involved before giving particular cases occurring in practice.
A point may be located : —
(i) By rectangular coordinates, i.e., by its station
and perpendicular offset.
(2) By two ties from known points.
(3) By an angle and a distance from a known point.
(4) By an angle at each of two known points.
(5) By a perpendicular swing offset from a known
line and a tie from a known point.
(6) By perpendicular swing offsets from two known
lines.
A line may be located : —
(i) By two points on the line.
(2) By one point on the line and the direction of
the line.
178. Ties, offsets, swing Offsets, and range lines. —
In the above, the word tie is used as meaning a direct horizon-
tal measurement between two points.
l60 LOCATION OF BUILDINGS [Chap. VI.
An offset is the distance from a line, usually at right angles.
A swing offset is the perpendicular distance to a line and is
found by trial. The zero end of the tape is held at the point to
be located and the tape is swung in a short arc about the point
as a center, the tape being pulled taut and kept horizontal.
The tape is read from the transit in various positions, and the
shortest reading obtainable is the perpendicular distance desired.
A range line is a line produced to intersect the transit line
or some other line.
179. GENERAL SUGGESnoilS. — By whatever method the
buildings are located the following suggestions should be carried
out.
fi) All the sides of the building should be measured and
checked by comparing the lengths of opposite sides.
(2) Other things being equal, a long side of a building
should be located in preference to a short side.
(3) Taes should intersect at an angle as near 90° as practi-
cable, and never less than 30°.
(4) One or more clieck measurements should be taken in
every case.
(5) In order to secure the best location the surveyor should
keep constantly in mind how the building or other object which
is being located is to be plotted.
In most work of this character it is customary to record the
measurements to tenths of a foot. How precisely the measure-
ments should be taken, however, depends upon the scale to
which they are to be plotted.
180. Typical Cases. — Although each case will have to
be dealt with according to circumstances there are certain typi-
cal cases which will ser\^e as guides. These are illustrated by
the following examples.
181. Example I. Building Near Transit Line and Nearly
Parallel to it. — As will be seen in Fig. 64 swing offsets are
taken at the two front comers which, together with the tie from
A to station 1 and the length of the front of the building locate
points A and B. Then the general dimensions of the building
are sufficient to plot and check the remaining sides. It is as-
sumed that the corners of the building are square unless it is
TYPICAL CASES l6l
obvious that they are not. The tie from C to station J? is a
check against an error in the other measurements.
Plotting. — This building would be plotted thus : — scale
the distance AX perpendicular (estimated) to the transit line
and draw a line with triangles parallel to the transit line ; then
scale Al from station 1 to this parallel line. Point A is then
located. Point B is located in the same way, AB being used as
the tie from A, Then by means of triangles and scale the
building is completed and the distance C2 scaled and compared
with the notes. Another way to plot point A would be to set
on the compass the distance lA and swing an arc about i as a
center ; then, keeping the scale perpendicular to the transit line,
find where the distance XA will cut this arc, thus locating point
A, Point B can be similarly located after A has been plotted.
For the same degree of accuracy distances can be measured
more rapidly with a scale than they can be laid off with a com-
pass, therefore the former method is usually more practicable.
This building might have been located by four ties AO^ Al,
Bl, and B^. The plotting in this case would be slow because
at least two of the ties must be swung by use of a compass, and
inaccurate because the intersections would be bad.
182. Example II. Building Near Transit Line and Making
a Slight Angle with it. — Fig. 65 illustrates two ways of locating
, a building in such a position that the intersection of the transit
1 line by the long side (produced) can be readily obtained.
The left-hand building is located by the method of Example I.
The tie Bl could have been taken instead of BS. It would
have given a better intersection at B, but since it is a longer tie
than B2 the fieldwork necessary is slightly greater. If B2 is
1 62 LOCATION OF BUILDINGS [Chap. VL
taken Bl might be measured as a check tie although Al would
make a better check tie since it will also check the measurement
of the side AB.
The right-hand figiu-e illustrates another method of locating
such a building. The front and side of the building are ranged
out by eye, a method which is thoroughly practical and suffi-
ciently precise for all ordinary purposes, and the plus station of
points E and /'are measured. The range lines CE and DFare
also measured and the check tie C3. C2 could have been taken
as a check tie ; it would have given a better intersection at C
than the tie C3^ but it is much longer.
6j
TRANSfT LINE ^ v
Stal Z C
Fig. 66.
Plotting. — The left-hand building is plotted as described
in Example I. In plotting the right-hand building the plus sta-
tions on the transit line are first scaled. Then with the com-
pass set at the distance EC an arc is swung from £ as a center.
From F the distance FC is scaled to intersect the arc, which
locates point C and the direction of the side CD, The building
is then plotted with triangles and scale. The check tie C3
should scale to agree with the notes and the line GC produced
should strike point E,
There is little difference between these two methods in the
amount of fieldwork, there being only one more measurement in
the right-hand than in the left-hand figures, but one extra check
is thereby obtained. In plotting, the method used in the right-
hand figure is shorter.
183. Example III. Building Located Entirely by Direct
Ties. — Any building not far from the transit line can be located
and checked by four ties as in Fig. 66. This method has the
advantage of being very simple and direct, especially in the field,
but the plotting of the building calls for the use of the compass
in two of the ties and hence is less rapid and accurate than
where swing offsets or ranges can be used.
TYPICAL CASES 1 63
Plotting. — The plotting of this building is done by swing-
ing the tie from one station to a corner of the building and scal-
ing from the other station the tie to the same comer. Then the
9ta2
Fig. 66.
other comer is plotted in the same way or by using the side of the
building as one of the ties in case it gives a better intersection.
184. Example IV. Building Located at a Considerable Skew
to the Transit Line. — A building which is at a considerable skew
to the transit line can best be located by range ties as illus-
trated in Fig. 67. The range ties through A are sufficient to
Fig. 67.
locate the buflding, provided AE and AF are not too short in
comparison with the sides of the building. If the seranges are
long enough, then BS is a check tie ; but if the ranges are short,
B3 must be depended upon to determine the position of point
B and in this event one of the range ties becomes a check. But
if A is within two or three feet of the transit line it will be well
to omit one of the ranges and take the additional tie ^C or the
range tie DC produced.
1 64 LOCATION OF BUILDINGS [Chap. VI.
Plotting. — If the ranges are of fair length the building is
plotted as explained for the right-hand building in Art. 182, but
if the range ties are short point B is located either by swinging
the arc with radius EB and scaling B3 or by arc SB and scaling
EB. Then the direction of AB is determined and the building
is plotted. CA produced should strik^ at F^ and AF should scale
the measured distance.
185. Example V. Buildings at a Long Distance from the
Transit Line. — It is evident that in this case (Fig, 68) the tape
A.
T1W<5IT /" %^ UNE,,
SlAl 2 3 4. 5 6
Fig. 68.
is not long enough to allow the use of swing offsets. Range
ties may be used provided the building is not so far away that
the eye cannot judge the range line with reasonable accuracy.
Sometimes the only methods available are long ties or angles or
a combination of the two. In any specific case there may be
some objections to any of these methods, and the surveyor will
have to decide according to circumstances which method he will
use. For example, where there are obstacles to the measure-
ment of ties, the corners of the building may have to be located
entirely by angles from two points on the transit line. Loca-
tion by angles is objectionable because it is difficult to plot an
angle quickly and at the same time accurately. It often hap-
pens, however, that when a building is at a considerable distance
from the transit line its accurate position is not required, since
as a rule the features near the transit line are the important
ones. This method of "cutting in " the comers of the building
by angle is often used in rough topographic surveying and is
decidedly the quickest of all methods so far as the fieldwork is
concerned.
Plotting. — The angles are laid off from the transit line
TYPICAL CASES 165
with a protractor and the proper intersections determine the
corners of the buildings. If the building is measured the side
between the comers located will be a check tie.
In some cases, e.g., in making a topographic map on a small
scale, the buildings are not measured at all, their corners being
simply " cut in " by several angles from different transit points,
and the shape of the building sketched in the notes.
186. Example VI. Buiidymgs Located from Other Buildings. —
Buildings which cannot be conveniently located from the transit
line on account of intervening buildings may be defined by ties
from the ones already located. Fig. 69 shows several ways
I.
f
"-y^^
.^i
TWANsrr ^^ ^i j)r^ ^'^'^ '-'^t
Slu4
Fig. 69.
in which such buildings may be located. Any of the preceding
methods are applicable, using the side of the house as a base-line,
but it will be found that range ties are almost always prefer-
able. For example, the bam is located by the distance BK, the
range tie KC and the tie BC, and checked by the tie BE, An-
other location of the barn is the distance AK or BK, the range
tie KC, and the two range ties A/ and CJ, By this latter method
the directions of both sides of the barn are checked. Still an-
other location of the point C would be to substitute in the place
of the range tie CK a swing offset from C to the house. The
shed is located by the range ties AFzxid FG and by the tie AG,
The check tie HD in general checks the location of both the
bam and the shed. If the side HL is ranged out instead of
the opposite side it will be seen that the tie AL will give a
i66
LOCATION OF BUILDINGS
[Chap. VI.
poorer intersection at Z. If convenient a tie from Z to ^ or
the range GF continued to the transit line may be measured as
a check.
187. Example Vn. Buildings of Irregular Shape Occa-
sionally a building of irregular shape has to be located. For
example, the shop in Fig. 70 is located on the front by ties and
swing offsets like Example I ; then the direction of AB is deter-
mined by the range tie BC, The back corner E is determined
by the ranges FH^nd EH, and by the dimensions of the build-
ing; FA is assumed parallel to GB. If the angle F \sk right
angle the tie EF may be taken instead of the range ties FH and
EH^ but even when F is a right angle it will be well if time will
permit to take these range distances as they give valuable checks
on the other measurements which the single tie EF does not
furnish. ED is scaled along HE produced and the rest of the
building plotted by its dimensions and checked by AD.
The ties shown on Fig. 70 to locate the factory chimney
will locate its sides even if they do not form a regular polygon.
If such a structure is situated at a considerable distance from
the transit line probably the best way to locate it is by angles
and 'distances to the corners, by the measurements of the sides,
together with a few such ranges as NJ or KJ.
188. Example Vm. Large City Buildings. — Fig. 71 illus-
trates the location of several buildings in a city block where
the transit line runs around the block. The fronts of the build-
ings are located from the transit line and the rear comers are
tied together. The range ties are shown by dotted lines and
other ties by dashes. The angles measured are marked by
TYPICAL CASES
167
arcs. At the curve ^B, the side lines of the building are ranged
out to point C which is located from the transit line by an angle
TRANSIT LINE
Fig. 71.
and distance and checked by a swing offset ; CD is also measured
to locate point D on the curve.
Frequently large buildings have their walls reinforced by
pilasters, and care should be taken in such cases not to confuse
the neat line of the wall with the line of the pilasters.
189. Example IX. Locatioii of Buildings by Angles and
Distances. — It will be seen from Figs. 71 and 72 that some of
the buildings have been located by angles and distances from
transit points. Any of the buildings in the above examples
could be located by this method, and on account of the rapidity
with which the work can be done in the field many surveyors
prefer to use it almost exclusively.
190. Location of Buildings and Fences from Transit Line. —
Fig. 72 is a sample page from a note-book illustrating the above
principles. It will be noticed that in the field notes the letter
R appears where the lines are ranges.
1 68
LOCATION OF BUILDINGS
[Chap. VI.
Fig 72.
RANDOM LINE I69
MISCELLANEOUS SURVEYING PROBLEMS*.
191. Random line. — Not infrequently in attempting to
run a straight line between two points A and B (Fig. 73) it is
impossible to see one point from the other or to see both points
A and B from an intermediate set-up on a straight line between
them. When this condition exists it is necessary to start at one
point, e.g., Af and run what is called a trial, or random, line AC
by the method explained in Art. 64, p. 52, in the direction of
the other end of the line as nearly as can be judged.
Where the random line passes the point B the perpendicular
offset YB is measured and also the distance to point Y along AC.
Unless the random line is very close, say, within about two feet of
the line AB^ the point Y where a perpendicular to i4C will pass
through B cannot be accurately chosen by eye. The method
resorted to in this case is one which has very general application
in all kinds of surveying work, and is as follows.
Fig. 73.
With the transit at A point X is set carefully on the line
AC and as nearly opposite point B as possible. Then the instru-
ment is set up at X and 90° turned off in the direction XZ. If
this line does not strike B (and it seldom will exactly) the dis-
tance BZ is carefully measured by a swing offset as described in
Art. 178, p. 159. The distance BZ is equal to the distance XY
which is added to AX giving the length of the long leg i4 F of the
right triangle A YB. The distance YB is then measured, and
AB and angle YAB are easily calculated.
Angle DA Y has been measured from some previous course
I70 MISCELLANEOUS SURVEYING PROBLEM [Chap. VL
such zsAD and the addition of the angle YAB together with
the known distance AB makes the traverse complete to the
point B without any further fieldwork. If the transit is now
moved to B with a view to carrying on the survey it will be
found that, since A cannot be seen from B^ there is no point on the
line BA to use as a backsight. But any point such as E can be
AM
readOy set on the line AB by making the offset ME = ^^ -j—.
Another point can be similarly set on AB as a check on the back-
sight.
Fig. 73.
This random line method is sometimes employed when AB
is a boundary which is covered with shrubs. In such cases, al-
though the view from AtoB may not be obstructed, it may be so
difficult to measure the line AB that its length can be more
easily obtained by the use of the random line while the angle
DAB may be measured directly at i4. If it is desired to mark
the line ABhy several intermediate points these may be estab-
lished by means of perpendicular offsets calculated as described
above.
192. Obstacles on line. — When an obstacle of limited
extent, such as a building or a small pond, lies on the transit
line various methods are resorted to for prolonging the line
through such obstructions; the most useful of these methods
will be explained.
193. Offsetting Transit Line. — This method is illustrated by
Fig. 74. It is desired to produce the line AB beyond the house.
Point B is set on line and as near as is practicable to the house.
OFFSETTING AROUND BUILDING I71
The instrument is then set up at B and a right angle ABF laid
off with the transit. BFv& made any convenient distance which
will bring the auxiliary line beyond the building. Similarly
point E is set opposite point A^ and sometimes a second point E^
opposite^', points A and -4' being exactly on the transit line.
These points E and E' need not be set by means of a transit
set up at A and at A' unless AE is quite long.
The instrument is then set up at F and backsighted on -£", the
sight is checked on E\ the telescope inverted, and points G, IT,
and H set on line. Leaving the telescope inverted, another
backsight is taken on Ay and the process repeated as described
in Art. 64, p. 52. Then the transit is moved to point G, and a
right angle turned off, and point C set on the right angle line,
the distance GC being made equal to BF,
Fig. 74.
Then by setting up at Cand sighting ahead on D, {DH= GC),
and checking on point i?', {lyiT =^GC), the transit line is again
run forward in its original location. The distance FG is care-
fully measured which gives the distance BC, and thus it appears
why it is so necessary that the lines BF and GC shall be laid off
at rigth angles by means of the transit. The other offsets AE,
A'E\ DHy and D'H' are not in any way connected with the
measurement along the line; they simply define the direction of
the line so that if convenient it is often only necessary to show
these distances as swing offsets for the transitman to sight on.
From what has been said it will be seen that offsets AE' and
VIF are not absolutely necessary, but they serve as desirable
checks on the work and in first-class surveying they should not
be omitted. For obvious reasons the offsets AE^xrid DH should
be taken as far back from the obstacle as is practicable.
Should the house be in a hollow so that it is possible to see
over it with the instrument at A^ the point D, or a foresight of
some sort (Art. 64, p. 52) should be set on line beyond the house
1 7^ MISCELLANEOUS SUrVeVING fROBLEiA^ fCHAP. Vl.
to be used as a foresight when the transit is set up again on the
original line. The distance may be obtained by an offset line
around the house or by slope measurements to the ridgepole.
Sometimes it is possible to place exactly on line on the ridgepole
of the house a nail or a larger wooden sight which gives an
excellent backsight when extending the line on the other side of
the building.
If the building has a flat roof it may not be out of the ques-
tion to set a point on the roof exactly on line, move the instru-
ment to this point on the roof, and prolong the line in this way.
Under these conditions the transitman will have to be extremely
careful in the use of his instrument as it will be set up on an in-
secure foundation. If he walks around the transit he will find that
it affects the level bubbles and the position of the line of sight ;
it is therefore well for him if possible to stand in the same tracks
while he backsights and foresights. Sometimes two men, one in
front and one behind the transit, can carry on the work under these
conditions more accurately and conveniently. This method in-
sures an accurate prolongation of the line, but the distance
through the building must be measured by an offset method,
unless it can be done by plumbing from the edge of the flat roof.
194. Short Transit Sights. — Sometimes the offset BF
(Fig. 74) does not need to be more than 2 or 3 feet. The
shorter this offset line can be made, and still clear the build-
ing, the better. But to lay off the short line BF will require a
method somewhat different from any that has been hereto-
fore explained. As the ordinary transit instrument cannot be
focused on a point much less than about 5 ft. distant it is im-
possible to set point F directly. The method employed is to
set a temporary point, say 10 ft. distant, on which the transit
can be focused, and on a line perpendicular to the original
transit line. From the transit point to this auxiliary point a
piece of string may be stretched and the point F set at the re-
quired distance from B and directly under the string.
195, Bisection Method. — A method which is economical in
field work but not very accurate is the following. In Fig. 75
the instrument is set up at A, backsighted on the transit line,
and equal angles turned off on each side of the transit line pro-
MEASURING AROUND OBSTACLES
173
duced. Points B' and C are carefully set on one of these lines
and at convenient distances from Ay and on the other line points
Fig. 75.
B' and L" are set at the same distances from A, Then point B
is placed midway between W and B"y and similarly point C is
set midway between C and C. The line EC is the prolonga-
tion of the transit line. Of course the distance B^C should be
made as long as practicable. The inaccuracy in this method
lies entirely in laying off the two angles. (See Art. 61, p. 50.)
In this case the distance AB can be computed from the formula
~BB*^
(approximately). (See foot-note, p. 339.)
AF ^AB=
196.
2AB
Measuring Around a Small Obstacle.
In Fig. 76 the
line AB runs through a tree. Point D is set with the transit at
A^ and DE is made equal to some convenient short distance and
laid off at right angles to the transit line by eye. Then AE
and EB are measured. The distance
AB = AE
DE"
+ EB-
DE"
(See foot-note, p. 339.)
2AE • 2EB
When DE is taken as some whole number of feet the compu-
tation of the above is extremely simple.
This method of measuring around a small obstacle might be
applied much more generally than it is at present if its accuracy
and its simplicity were more fully realized by surveyors.
174 MISCELLANEOUS SURVEYING PROBLEM [Chap. VI.
197. Equilateral Triangle Method. — While this method re-
quires much less fieldwork than the offset method described
above it is at the same time less accurate. Point B (Fig. ^y) is set
on the transit line as near the building as practicable but so that a
line BC at 60° with the transit line can be run out. The instru-
ment is set up at B, backsighted on A, and an angle of 120® laid
off ; the line BC is made long enough so that wh^n the instru-
ment is set up at Cand 60® is laid off from it, CjD will fall outside
the building. BC is measured and CD is made equal to BC
If the instrument is set up at D and angle CDE laid ofif equal to
1 20** the line D£ is the continuation of the original transit line,
and the line BD^BC> This method is subject in three places
to the errors incident to laying off angles and, when BC and CD
are small, it has in two of its intermediate steps the disadvan-
tages due to producing a short line.
108. INACCESSIBLE DISTANCES. — If the obstruction is a
pond, points on the far side of it can be set and these should be
used in producing the transit line. When the line can be pro-
duced across the obstacles the following methods may be used.
igg. Inaccessible Distance by Right Triangle Method. —
In Fig. 78 the line AB is made any convenient length and at
any convenient angle to the transit line. The line BC is
laid off at 90° to BA and is intersected with the transit line and
the distance BC measured, which distance should check with its
computed value. Also the angle CAB can be measured which
will check the transit work.
INACCESSIBLE DISTANCES
175
Fig. 78.
200. Intersecting Transit Lines. — In many kinds of
surveying work it is necessary to put in points at the intersection
of two transit lines. It would be an easy matter to set the
point if two transits could be used, one on each line, and the
sight simultaneously given by each transitman. As it is seldom
practicable to use more than one transit in a surveying party the
following method is resorted to.
An estimate is made by eye where the lines will cross each
other and temporary points not more than 10 ft. apart are set on
one of the transit lines by means of the instrument, enough points
being marked to make sure that the second line will cross some-
where among this set of temporary points. A string is then
used to connect two of these temporary points and the transit
is set up on the other transit line and the point where the
second line cuts the string is the intersection point. Some-
times when the lines cross each other at nearly 90° the intersec-
tion point can be estimated so closely that only two temporary
points need be placed on the first line. In other cases, where
the two transit lines cross at a very small angle, it is impossible
to tell by eye within several feet where the lines will intersect
and a number of points must be used because in practice the
stretching line is seldom applicable for distances much over
15 ft. For short distances the plumb-line can be used as a
stretching line.
201. Inaccessible Distance by Swing Offset Method, — If the
distance across a pond or river is not great the following method
1/6 MISCELLANEOUS SURVEYING PROBLEM [Chap. VL
may be used. It has the advantage of requiring the min-
imum amount of fieldwork. With the instrument at A
(Fig. 79) point C is set on the transit line on the far side of the
river. The instrument is then set up at C and the angle ACB
measured between the transit line and a loo-ft. swing offset
from point A.
A pencil is held vertically at the lOO-ft. mark of the tape
and while the zero point is held firmly at A the tape, which
is constantly kept horizontal and taut, is swimg slowly in
an arc ad. The transitman, using the tangent screw, can fol-
low the pencil with the vertical cross-hair of the transit, stop-
ping the cross-hair when the pencil is in its farthest position
Fig. 79.
from A. Then as the tape is swung the second time he can
check his setting and when this is established the angle ACB is
read. The distance AC then is very easily calculated. It
should be noted, however, that if AC is several times as long as
AB the resulting error in ACmzy be so great as to prohibit the
use of this method where very precise results are required.
There is no reason why the swing offset could not be made at
C with the instrument at /I if more convenient.
202. Inaccessible Distance by Tangent Offset Method. — In the
method described above the distance across the pond may be so
great that 100 ft. will be too short a base to use, or point A may be
situated on ground sloping upward towards B so that a swing offset
INACCESSIBLE DISTANCES
177
cannot be made. In such cases the line AB (Fig. 80) can be laid
off at right angles to the transit line and of any convenient length.
TRANSfT i
\[fr m
UNE
\
A i
11 lit
^y^^
^"%.
1
& i
^•^H
N
1^
'
Fig. 80.
Then the angle ACB is measured and the line AC computed.
By another set-up of the instrument the angle B can be meas-
ured as a check, and if the line BC does not cut across the pond
its length can also be measured as a further check.
203. Inaccessible Distance by Oblique Triangle Method. —
Often the shores of a stream are covered with trees so that none
of the above methods are applicable. It may be convenient to
measure a line AB (Fig. 81) in but one direction along the shore.
In this case the point C is first carefully set on the opposite side,
the line AB measured along the shore, and the angles at A and
178 MISCELLANEOUS SURVEYING PROBLEM [Chap. VI.
•
at C are measured. The distance AC can then be computed.
It will be well also to set up at B and measure the angle £ as a
check on the work. At the time when point C is set it is also
good practice to set a point further ahead on the line, to use as a
foresight to check the transit line when the instrument is moved
across the river.
204. To Obtain the Distance Between Two Inaccessible Points
by Observation from Two Accessible Points. — In Fig. 82 the pomts
A and B are inaccessible and it is desired to obtain the distance
AB and the angle that AB makes with the transit line. From
the point D the distance DC and the angles BDA and ADC are
measured, and similarly at C the angles ACB and BCD are meas-
ured. AB can then be calculated as follows : — in the triangle
CBD compute CB\ in triangle ACD compute AC\ and in the
triangle ACB calculate AB^ the inaccessible distance. In the tri-
FlG. 82.
angle ACBi angle ABC can be computed, which, together with
the measured angle BCD^ will give the difference in direction
between AB and CD. It is not at all necessary that DC should
have, been measured as one straight line in the traverse ; the trav-
erse might have run as indicated by the dotted lines, but in such
an event the distance CD and the necessary angles could have
been easily figured so that it could be reduced to the above prob-
lem.
INACCESSIBLE DISTANCES 1 79
This problem occurs when the distance between two triangu-
lation stations, A and Bt and the azimuth of AB are desired and
when it is inconvenient or impossible to measure the line AB or
to occupy the points with the transit.
20$. To Obtain the Inaccessible Distance Between Two Ac-
cessible Points by Observations on Two Inaccessible Points of
Known Distance Apart. — In this case (Fig. 82) i4 and B are the
two accessible points and C and D are the two inaccessible points
but the distance DC is known ; the distance i45 is required.
With the transit at A, the angles CAD and DAB are measured ;
at B the angle CBD and ABC are measured. The length of
the line CD is 'known. While it is simple to obtain CD in terms
of i4J5, it is not easy to directly determine AB in terms of CD \
it will be well therefore to use an indirect method. Assume
i4£ as unity. Then by the same process as described in the
preceding problem the length of CD can be readily found.
This establishes a ratio between the lengths of the lines AB and
CA and the actual length of CD being known the distance AB
can be computed.
A problem of this sort would occur under the following
circumstances. If the distance CD between two church spires
were accurately known (from a triangulation system) and it is
desired to use this line CD as a base-line for a survey, two points
A and B could be assumed, and the distance between them and
the azimuth ot AB could be found by this method.
CHAPTER VII.
OBBSRVATIONB FOR MBRIDIAN AND LATXTX7DB.
O^SERVATIOHS FOR HERXDIAN.
LMHni Clony#ien
3o6. To ESTABLISH A TRUE MERIDIAlf LIRE BY OBSERVA-
TION ON POLARIS WITH THE TRANSIT. — On account of the
earth's daily rotation on its axis all heavenly bodies appear to
revolve once a day around the earth. Stars in the south appear to
revolve in large circles parallel
to the daily path of the sun.
As we look farther north the
apparent size of the circles
grows smaller. The center of
these circles is the nortA pole of
the celestial sphere^ a point in
the sky in the prolongation of
the earth's axis. The pole-star
(Polaris) revolves about the
pole in a small circle whose
radius is less than a degree and
a quarter (Fig. 83). This an-
gular distance from the pole to
a star is called its polar distance.
When the star is directly
above the pole its bearing is
the same as that of the pole
itself and the star is said to be
at upper culmination. At this
instant it is in the true meri-
dian. About twelve hours later
it will be below the pole at
lower culmination and will be
again in the true meridian. About half-way between these
two positions the star reaches its greatest east or west bearing,
i5q
Fig. 88.
MERIDIAN OBSERVATIONS ON POLARIS
I8l
and at such times is said to be at its greatest elongation. At
either eastern or western elongation the star's bearing is not
changing perceptibly because it is moving almost vertically, a
»T3
O
to
H
o
o
g
n
o
H
S
CO
S
n
2
►
w
en
H
W
I«
W
r
o
5:
o
>
H
o
a;
NOIiVNIWiriD H3AV01 iv sisiHvaoj
NaHM SNOixviiaxsNOO anx ^o NOixisod
2;
o
H
<
o
o
2;
CCS
M
H
CO
<
H
O
H
CO
O
u
M
S
H
O
Fig. 84.
Ursa Major ^
* I
Position of the Constellations when
Polaris is at UPPER CULMINATION.
Relative Position of the Constellations near the
North Pole.
o
1 82 OBSERVATIONS FOR MERIDIAN [Chap. VLL
condition which is most favorable for an accurate observation.
At cuhnination the star is changing its bearing at the maximum
rate, and therefore this is not as good a time to make an accurate
observation as at elongation. This star moves so slowly, how-
ever, that even at culmination its bearing can be obtained with
sufficient accuracy for determining the decUnation of the needle.
Polaris can be easily found by means of two conspicuous con-
stellations near it, Cassiopeia and Ursa Major, The seven most
conspicuous stars of the latter form what is commonly known as
the " Great Dipper'' (Fig. 84). The two stars forming the part
of the bowl of the Dipper farthest from the handle are called the
^^ pointers " because a line through them points almost directly
at the pole. On the opposite side of Polaris is Cassiopeia, shaped
like the letter W. A line drawn from h * Cassiopeia^ the lower
left-hand star of the W, to f Ursa Majoris, the middle star of the
Dipper handle, passes very dose to Polaris and also to the pole
itself.
207. OBSERVATION FOR MERIDIAN ON POLARIS AT
Elongation. — When the Dipper is on the right and Cassiopeia
on the left, Polaris is near its western elongation ; when the dipper is
on the left Polaris is near eastern elongation. When the constella-
tions are approaching one of these positions the transit should be
set over a stake and leveled, and the telescope focused upon the
star.f Unless the observation occurs at about sunrise or sunset
it will be necessary to use an artificial light to make the cross-hairs
visible. If the transit is not provided with a special reflector for
throwing light down the tube a good substitute may be made by
cutting a small hole in a piece of tracing cloth or oiled paper and
then fastening it over the end of the telescope tube by a rubber
band. If a lantern is then held in front and a little to one
side of the telescope the cross-hairs can be plainly seen. The
star should be bisected by the vertical wire and followed by
means of the tangent screw in its horizontal motion until it no
• The Greek Alphabet will be found on p. 516.
t It is difficult to find a star in the field of view unless the telescope is focused
for a very distant object. The surveyor will find it a convenience if he marks on
the telescope lube the position of the objective tube when it is focused for a distant
object.
OBSERVATIONS AT ELONGATION 1 83
longer changes its bearing but moves vertically. (It will be seen
from Fig. 83 that when the star is approaching eastern elonga-
tion it is moving eastward and upward ; when approaching western
elongation it is moving westward and downward.) As soon as
this position is reached the telescope should be lowered and a
point set in line with the vertical cross-hair at a distance of sev-
eral hundred feet from the transit. Everything should be ar-
ranged beforehand so that this can be done quickly. Immediately
after setting this point the instrument should be reversed and
again pointed on the star. A second point is then set at one
side of the first. The mean of these two points is free from the
errors of adjustment of the transit. If the instrument is in ad-
justment, of course, the first and second points coincide. On
account of the great difference in altitude between the star and
the mark the elimination of instrumental errors is of unusual
importance (Art. 79, p. 61). For 10 minutes of time on either
side of elongation the bearing of the star does not change more
than 5 seconds of arc and therefore there is sufficient time to
make these two pointings accurately.
After the direction of the star at elongation has been found,
the meridian may be established by laying off an angle equal to
the azimuth, or true bearing of the star. Since this angle to be
laid off is the horizontal angle between the star and the pole, it
is not equal to the polar distance but may be found from the
equation : —
o- o^ » T- -D • Sin Polar Distance of Star *
Sm Star s True Beanng = p; — = — : —
Cos Latitude
The mean polar distances for the years 1906 to 1920 m^y be
* This equation may be derived as follows ; in Fig. 83, let P represent the
pole, Z the zenith, and £ the position of the star at elongation. Then by spherical
trigonometry,
sin PZE sin P£
sin ZEP^^n ZP'
But PZE is the angle between the two vertical circles and equals the bearing.
ZEP^go^ because ZE is tangent to the circle WUEL^ which represents the path
of Polaris. PE is the polar distance and ZP may be shown to be equal to
90^ - latitude.
Hence, sin /'Z-ff-?!?^.
cos lat.
1 84
OBSERVATIONS FOR MERIDIAN [Chap. VII.
TABLE 8.
Mean Polar Distances of Polaris.*
Ytttf.
Mean Polar Distance.
Year.
Mean Polar Dbtance.
0 * //
0 / /.
1906
I II 41.05
1914
I 09 12.07
1907
I II 22.37
1915
1916
1 08 53.51
1908
I II 03.71
I 08 34.97
1909
I 10 45.07
I 10 26.44
1917
I 08 16.45
1910
1918
I 07 57.94
1911
I 10 07.82
1919
I 07 39.45
I 07 20.98
1912
I 09 49.22
1920
1913
I 09 30.64
. .
■;
found in Table 8. The latitude may be obtained from a reliable
map or by observation (Arts. 216-17, p. 196).
When the transit is set up at the south end of the line the
^^^^_^ angle thus computed must be laid off to the right if
the elongation is west, to the left if the elongation is
east. A convenient and accurate way of laying off
the angle is by measuring the distance between the
two stakes A and B (Fig. 8$), and calculating the per-
pendicular distance BC which must be laid off at the
north stake B to give a meridian AC.
* The above table was derived from data furnished by the Sup>
erintendent of the United States Coast and Geodetic Survey. The
Mean Polar Distance is the average value of the polar distance for
the entire year.
In taking the polar distance from the table for the purpose of
looking up its sine the student should keep in mind the degree of
precision desired in the computed azimuth. If the azimuth is to be
within about one minute of the true value the polar disunce need be
taken only to the nearest minute, but if the azimuth is to be correct
within a few seconds the polar distance should be taken to the nearest
second. It should be noted however that since the values ^ven in
the table are only the average values for the year there will in general
be an error of a few seconds due to neglecting the variation of the
polar distance during the year. The exact value for every day in the
year may be found in the "American Ephemeris and Nautical
Almanac," published by the Bureau of Equipment, Navy Depart-
FlG. 86. ment.
OBSERVATIONS AT CULMINATION 1 8$
208. OBSERVATION FOR MERIDIAN ON POLARIS AT CULMI-
NATION.— At the instant when Polaris is above the pole the
star f Ursce Majoris will be almost exactly underneath Polaris.
When Polaris is below the pole h Cassiopeia will be almost di-
• rectly below Polaris (Fig. 84). In order to know the instant
when Polaris is exactly on the meridian it is necessary first to
observe the instant when one of these two stars is vertically
below Polaris. From this the time when Polaris will be on the
meridian can be calculated by adding a certain interval of time,
and the meridian line can thus be directly established. This
interval of time was, for f Ursa Majoris^ about 2"36» in the year
1900, and it increases about 2 1 » per year. The intervals computed
by this rule are only approximate, but are sufficiently accurate for
many purposes and, as the change is very slow, the rule is good for
many years. It may also be used for any latitude in the United
States. When f Ursa Majoris cannot be used, as is the case in
the spring of the year, especially in northern latitudes, a sim-
ilar observation can be made on h Cassiopeia, The interval
for this star was 3"24' for 1900, with an annual increase of
about 20».
The observation to determine when the two stars are in the
same vertical plane is at best only approximate, since the instru-
ment must be pointed first at one star and then at the other ;
but since Polaris changes its azimuth only about i minute of
angle in 2 minutes of time, there is no difficulty in getting fair
results by this method. The vertical hair should first be set
on Polaris, then the telescope lowered to the approximate
altitude of the other star to be used. As soon as this star
comes into the field the vertical hair is again set carefully on
Polaris. As it will take the other star about 2 minutes to reach
the center of the field there will be ample time for this pointing.
Then the telescope is lowered and the instant when the star
passes the vertical hair is observed by a watch. This will be the
time desired, with an error of only a very few seconds. The time
of culmination should then be computed as described above and
the vertical hair set on Polaris when this computed time arrives.
The telescope is then in the meridian which may be marked on
the ground.
1 86
OBSERVATIONS FOR MERIDIAN
[Chap, VII.
It will be seen that in this method the actual error of the
watch has no effect on the result since it is used only for meas-
uring the interval of a few minutes. The error in the meridian
obtained by this method will seldom exceed one minute of angle.
209. To Find the Standard Time of Culminatioii and Elonga-
tion. — The approximate times of culmination and elongation of
Polaris for the ist and isth of each month in the year 1907 may
be found in Table 9.
TABLE 9.
Approximate Times of Culmination and Elongation of Polaris
Computed for the 90TH Meridian West of Greenwich,
for the Year 1907.
Date.
7907
Jan. I .
" IS •
Feb. I .
". IS .
Mar. I .
a" '5-
Apr. I .
*• 15 .
May I .
" 15 .
Jun. I .
" IS .
Jul. I .
" 15.
Aug. I .
Sep. I .
" IS .
Oct. I .
" 15 .
Nov. I . ,
Dec. I .
" 15 •
ulminatioa.
Western
Lower
Eastern
Elongation.
Culmination.
Elongation.
A m
h m
h m
k m
6 44
12 39
18 42
0 49
5 49
II 44
17 47
23 50
4 41
10 36
16 39
22 42
3 .46
2 ^l
15 44
21 47
2 51
8 46
14 49
20 52
I 56
7 51
13 54
19 57
0 49
6 44
12 47
18 50
23 SO
5 40
II 52
17 55
22 47
4 46
10 49
16 52
21 52
3 51
9 54
15 57
20 45
2 44
8 47
14 50
'2 51
I so
7 53
13 56
18 48
0 47
6 50
12 53
II 58
17 53
23 48
5 55
16 47
22 42
4 49
10 52
15 52
21 47
3 •?4
I 57
14 45
20 40
2 47
8 50
13 SO
19 45
I 52
• 7 55
12 47
18 42
0 49
6 52
II 53
17 48
23 51
5 58
10 46
16 41
22 44
4 51
2 51
15 46
21 49
3 56
8 47
14 42
20 45
2 52
7 52
13 47
19 50
I 57
To find the time for any other date interpolate between the
values given in the table, the daily change being about 4 min-
utes.
STANDARD TIME 1 8/
In order to find the exact time of culmination or elongation
for any observation it would be necessary to take into account
the latitude and longitude of the place and the exact date of the
observation. The times given in Table 9 are only approximate
in any case and are to be regarded merely as a guide so that the
surveyor may know when to prepare for his observations.
The times are computed for mean local astronomical time at
the 90th meridian west of Greenwich and for the year 1907.
These numbers increase about J minute each year so that this
table may be used to obtain approximate results for subsequent
years. Astronomical time begins at noon of the civil day of the*
same date and is reckoned from Oh to 24^ e.g., 18** would mean
6^ A.M. The tabular numbers are nearly correct for the Stand-
ard Meridians, i.e., the 7Sth, 90th, 105th, and 120th west of
Greenwich. All watches keeping "railroad time,'* or "standard
time," are set to the local mean time of one of these four meri-
dians (Art. 86, p. 6S). To find the watch time of culmination
or elongation for any other meridian, first find the difference
in longitude in degrees between the place of observation and
the standard meridian, and then convert this into minutes and
seconds of time by dividing by 15, since 15° of longitude are
equivalent to one hour of time. The standard, or watch, time
of the observation is then obtained by adding this correction to
the time taken from the table if the place is west or by sub-
tracttng it if the place is east of the standard meridian.
210. MERIDIAN OBSERVATIONS ON POLARIS WITH THE
Compass. — In determining a meridian with the compass the
observations are made as described for the transit except that
the following modifications will be necessary. Suspend a long
plumb-line a few feet away from the point where the instru-
ment is to be set. Since the rear sight is the only part of the
compass to be used in the observation it may be unscrewed
from the compass and fastened to a piece of board. This board
should be placed on a table. The compass sight may then be
shifted to the right or left to bring it in line with the star and
the plumb-line. The plumb-line should be illuminated by means
of a lantern. ' The direction of the star may be marked by set-
ting stakes in line. If the observation is made at elongation the
1 88 OBSERVATIONS FOR MERIDIAN [Chap. VII.
meridian should be laid out as described in Art. 207. In finding
the declination of the needle the compass is set up over one of
the meridian stakes and sighted at the other, when the declina-
tion can be read off directly. In order to obtain as nearly as
possible the mean value, of the declination this should be done
at about 10 A.M. or 5 to 6 P.M. because at these times the
needle is in its mean position for the day.
211. Meridian Observation on Polaris at any Time
WITH THE Transit. — in order to make this observation, it is necessary to
know the local time very closely. As in most cases the time which the surveyor
canies is ** standard time *' it is assumed that such is the case here. The observa-
tion itself consists in either marking the direction of the star, as previously de-
scribed, and noting the time by the watch when the star is sighted ; or in repeating
the angle between the star and some reference mark, the time of each pointing on
the star being noted. In the latter case, take the average of the observed
times and assume that it corresponds to the average angle. This is very nearly
true if the observations extend over a few minutes of time only.
After finding the standard time of the observation, the next step is to compute
the hour angle of the star at the time of the observation. Take from the Nautical
Almanac : (i) the right ascension of Polaris for the date; (2) the right ascension
of the ** mean sun " for the date ; (3) the increase in the sur^s right ascension since
Greenwich noon, which is found in Table III in the Appendix to the Nautical
Almanac. Remember that the dates in the Almanac are in Astronomical time
(Art. 209, p. 186). Reduce the standard time to local time by adding or subtract-
ing the difference in longitude expressed in hours, minutes, and seconds, remem-
bering that if the place is west of the standard meridian the local time is earlier
than standard time and vice versa. To the local time add the sun's right ascen-
sion and the correction from Table III, Appendix, Nautical Almanac. The result
is the sidereal time. From this subtract the star's right ascension, and the result
is the hour angle of the star reckoned from the meridian from oh to 24h in the di-
rection of the star's apparent motion. Convert this angle into degrees, minutes,
and seconds. The azimuth of the star may now be computed from the formula,
^ „ sin /
• tan Z =-
cos L tan Z> — sin Z cos /
where Z ^ the azimuth, or true bearing ; / — the hour angle ; Z i>« the latitude ;
D <-i the declination » go'' — the polar distance. If the hour angle is between oh
and I2h the star is west of the meridian ; if between I2h and 24h it is east of the
meridian (see Example below).
In the " Manual of Surveying Instruction " issued by the General Land Office
a set of tables is given which will enable the surveyor to perform all of the above
work by simple inspection and without the aid of the Nautical Almanac.
* See Hayford's Geodetic Astronomy, p> an, Art. 193.
AZIMUTH OF POLARIS AT ANY HOUR 1 89
Example.
Observation on Polaris for azimath April 15, 1908. Latitude 38^ 58^. Longi.
tude 92® 25^. Angle between a mark (approximately N.W.) and Polaris is
repeated 6 times* Watch im 138 fast. The times are
8h 35m
8 37
8 38
8 39
8 41
8 43
40»
20
50
59
30
00
Mean of 6 readings 8 39
Watch fast i
True Central time 8 38
Longitude of Standard Meridian
26.2
13
»3
6
Greenwich time 14^ 38™ 13'
From Nautical Almanac, Right Ascension of " Mean Sun ** at Greenwich Mean
Noon a- ih 32™ 57S.82 ; Right Ascension of Polaris =» ih 25m ois.47 ; Declination
of Polaris — + 88** 48' 52'' ; Correction from Table III (Nautical Almanac) for
Greenwich Time = I4h ^S^ =» 2^ 24s. 2
92°
25' =
6h ogm ^ffi
.'. longitude correction »
09m 40*
Mean of observed times
8h 38m 13s
Longitude correction
9 40
Local time
8 28 33
Right Ascension " Mean Sun
>»
I 32 58
Correction (Table III)
2 24
Sidereal time
10 03 55
Right Ascension Polaris
I 2$ 01
Hour Ang^e Polaris
r»
8h 38^548
/-
129^3' 30"
log cos Z = 9.89071
log!
sin Z « 9.79856
log tan Z> » 1.68413
log<
cos/- 9.80558 (n)*
1.57484
9.60414 (n)
37.570
- .4019
.402
37.972
log sin / s-
9.88600
log denominator —
1.57946
log tan Z B
8.30654
Z«
i<>09'
37" W. of N.
* The n after the logarithm indicates that the number corresponding is negative
190 OBSERVATIONS FOR MERIDIAN [Chap. VU.
212. Solar OBSERVATIOIVS. — Where great accuracy is not
required many surveyors prefer solar observations because
they can be made without much additional work, while star
observations have to be made at night and require special
arrangements for illuminating the field of view and the mark.
If it is sufficient for the purpose in view to obtain the azimuth
within i minute of angle solar observations will answer. In
making these observations with the ordinary transit it is neces-
sary to have some means of cutting down the sun's light so
that it will not be too bright for the eye while making point-
ings. This is usually effected by placing a dark glass over the
eyepiece. A dark glass in front of the objective wiU introduce
error into the pointings unless the faces of this glass have been
made plane and exactly parallel. If the instrument is not pro-
vided with a dark glass the observation may be made by hold-
ing a white card back of the eyepiece while the telescope is
pointing at the sun. If the eyepiece tube is drawn out the
sun's disc and the cross-hairs can both be sharply focused on the
card. By this means pointings can be made almost as well as
by direct observation. It is also well to cut down the amount
of light entering the objective by having a cap with a hole in
the center or by using a piece of tracing cloth as explained in
Art. 207, p. 182.
213. OBSERVATION FOR MERIDIAN BY EQUAL ALTITUDES
OF THE SUN IN THE FORENOON AND AFTERNOON. — This obser-
vation consists in measuring in the forenoon the horizontal angle
betweeii the sun and some reference mark at the instant when
the sun has a certain altitude, and again measuring the angle
when the sun has an equal altitude in the afternoon. If the
distance of the sun from the equator were the same in the two
cases the horizontal angles between the sun and the meridian
would be the same in both observations, hence the mean of the
two readings of the horizontal circle would be the reading for
the meridian. But since the sun is changing its distance from the
equator the measured angles must be corrected accordingly.
The correction is computed by the equation
COS L sin /
MERIDIAN BY SOLAR OBSERVATIONS I9I
in which X = the correction to the mean vernier reading,
d^ the hourly change in declination of the sun taken from
Table 10 and multiplied by half the number of hours between
the two observations, L = the latitude, and / = half the elapsed
time converted into degrees, minutes, and seconds. Since the
hourly change for any given day is nearly the same year after
year an almanac is not necessary but the table given below is
sufficient.
TABLE 10.
Hourly Change in the Sun's Declination.
ist. loth. 20tb. 30th.
January +12" + 22" +32" +41"
February +43 +49 +54
March +57 +59 +59 +58
April +58 +54 +49 +46
May +45 +39 +39 +23
June +21 4-12 +02 —09
July —10 —19 —28 —36
August -38 —44 -49 -54
September —54 —57 —58 —59
October -58 -57 -54 -49
November —48 —42 —34 —25
December —23 —14 —02 +10
The observation is made as follows : — * at some time in the
forenoon, preferably not later than 9 o'clock, the instrument is
set up at one end of the line the azimuth of which is to be
found, and one vernier is set at 0°. The vertical cross-hair is
then sighted at the other end of the line and the lower plate
clamped. The upper clamp is loosened and the telescope turned
until the sun can be seen in the field of view. The horizontal
cross-hair is to be set on the lower edge of the sun and the
vertical cross-hair on the left edge. Since the sun is rising and
also changing its bearing it is difficult to set both of the cross-
hairs at once and it will be found easier to set the horizontal
hair so that it will cut across the sun's disc leaving it clamped
in this position while the vertical hair is kept tangent to the left
edge of the sun by means of the upper tangent screw. When
the sun has risen until the lower edge is on the horizontal hair
* The nearer the sun is due East or due West, the better the result.
192 OBSERVATIONS FOR MERIDIAN [Chap. VIL
the instrument is in the desired position and after this position
is reached the upper tangent screw should not be moved. As
soon as this position is reached the time is noted. Both the
vertical and the horizontal circles should now be read and the
angles recorded.
In the afternoon, when the sun is found to be nearly at the
same altitude as at the forenoon observation, the instrument
should be set up at the same point and again sighted on the
mark. The observation described above is repeated, the point-
ings now being made on the lower and right edges of the disc.
The telescope is inclined until the vernier of the vertical circle
reads the same as it did at the forenoon observation. When the
sun comes into the field the vertical hair is set on the right edge
and kept there until the lower edge is in contact with the hori-
zontal hair. The time is again noted and the verniers are read.
If desired, the accuracy may be increased by taking several pairs
of observations. The mean of the two circle readings (suppos-
ing the graduations to be numbered from o° to 360® in a clock-
wise direction) is now to be corrected for the sun's change in
declination. The correction as obtained by the formula given on
p. 190 is to be added to the mean vernier reading ii d is minus,
and subtracted if d is plus, i.e., if the sun is going south the mean
vernier reading is east of the south point, and vice versa.
When the circle reading of the south point is known the true
bearing of the mark becomes known and the bearings of other
points may be found (see Example below).
The disadvantage of this method is that it is necessary to be
at the same place both in the forenoon and afternoon, whereas in
many cases the surveyor might in the afternoon be a long distance
from where he was working in the forenoon.
Example.
Latitude 42® 18^ N. April 19, 1906.
A.M. Observation. P.M. Observation.
Reading on Mark, o^oc/oc/' Reading on Mark, o^oq'oc/'
Pointings on Upper and Left Limbs. Pointings on Upper and Right Limbs.
Vertical Arc, 24^*58' Vertical Arc, 24**58'
Horizontal Circle, 357**i4'i5" Horizontal Circle, i62®28'oo''
Time 7hi9ni3oe Time 4I&1 21^158
MERIDIAN BY SOLAR OBSERVATIONS
193
it elapsed time <- 41^26111228
- 66°3S'3o-
log sin / 9.96270
log cos L 9.86902
983172
log 230".9 2.36342
2.53170
correction 340". 2 — 5'4o''.2
Increase in declination in 4li26°^22B«i
52'' X 444 - 230"-9
Mean circle reading ■- 79^51 '08''
5 40
S 79<»45'28'' E
Azimuth of mark » 280^14^32'^
' 214. OBSERVATION FOR MERIDIAN BY A SINGLE ALTI-
TUDE OF THE SUN. — The azimuth of a line may be obtained
by measuring a single altitude of the sun with the transit and
computing the azimuth by spherical trigonometry. The instru-
ment is set at 0° and pointed at a mark. The upper clamp is
loosened and pointings made as follows. First, the cross-hairs
are set on the left and lower limbs of the sun and both circles
are read ; the time is also noted. If desired several sets of
observations may be made. Second, the cross-hairs are set on
the right and upper limbs, and the reading of the circles and
the time are again recorded. The mean of the vertical circle
readings is taken, and corrected for atmospheric refraction by
subtracting the correction given in Table 11. This corrected
mean is called A in the formula given below.
TABLE 11.
Refraction Correction.
Altitude.
Refraction.
10°
_/
19"
II
51
12
27
13
07
14
49
15
16
34
20
17
18
»9
2
2
08
57
48
Altitude.
Refraction.
20*»
2' 39''
25
2 04
30
I 41
35
1 23
40
I 09
45
0 58
50
0 49
60
0 34
70
0 21
So
0 10
194 OBSERVATIONS FOR MERIDIAN [Chap. Vn.
In order to compute the azimuth it is necessary to know the
latitude of the place. This may be obtained from a reliable map
or from an observation as described in Art. 2x6, p. 196. It is
also necessary to know the declination of the sun at the instant
of the observation ; this is found as described in Art. 86, p. 68.
If Z represents the azimuth of the sun's center from the south;
Z, the latitude ; A, the altitude ; /, the distance from the north
pole to the sun (or 90°-declination) ; and j = |^ (Z + A +/) ;
then
cot' \Z ^ si" (-y - L) sin {s - h)
^ cos s cos {s — /)
Five place logarithms will give the value of Z within 10 seconds
of angle, which is amply accurate for this observation.
When the true bearing of the sun is known the bearing of
the mark from the instrument can be foimd.
OBSERVATION FOR AZIMUTH
195
Example.
Observation on Sun for Azimuth.
Latitude 42*' 21' N. Longitude 4^ 44m i8« W
Time, Nov. a8, 1905, A.M.
Horizontal Circle
Vertical Circle
Watch
Vernier A B
Mark 238^ 14' 14'
A.M.
Right and Lower Limbs 311 48 48.5
14*^41'
8h 39m 428
M It u .. J, 2 20 20
IS 00
8 42 19
The inst. reversed
Left and Upper Limbe 312 27 26.5
15 55
8 45 34
« « u « 3,2 52 51.5
16 08
8 47 34
Mark 238 14 14
Mean reading on Mark » 238^ 14^0
« « •* San — 312 21 .7
Mark N. of Sun — 74. 07'.7
Observed Altitude 15° 26'.o
Mean - 15° 26'
Mean -i 8h 4311
5
4/«
Greenwich Time — 13^ 43m 478
Sun's apparent declination at
Greenwich Mean Noon — — 21® 14' 54''.4
Refraction 3.5
Difference for i hour — — 26".8i
True Altitude 15® 22'.5 « h
- 26''. 81 X 13^.73 - - 6'o8'' I
Declination — — 21® 2i'o2".5
Polar Distance — 1 1 1 ® 2 1 ' 02''.5
Z- 420 2I'.0
log sin (j - Z) - 9.82708
h - 15'' 22^5
log sin (j — A) — 9.97062
P^ III*>2l'.0
log sec J — 1. 02132
t - 84^32'.2
log sec (j — />) — 0 04940
*-Z - 42*> Il'.2
2) 0.86842
s^h -69«09'.7
cot } Z » 0.43421
/ - -P - - 260 48^.8
1 Z - 20*> I2'.4
Z -40° 24'.8 East of South
Mark N. of Sun 74° o7'-7
ii4^32'-5
MarkN 65** 27^.5 E
215. OBSERVATION FOR MERIDIAN BY MEANS OF THE
SOLAR ATTACHMENT. — This observation has been described
in detail in Art. 85, p. 66.
196 OBSERVATIONS FOR LATITUDE [Chap. VII.
OBSERVATIOIIS FOR LATXTUDE.
216. (I) BY THE ALTITUDE OF POLARIS AT UPPER OR LOWER
CULMINATION. — When Polaris is approaching either culmination
(see Art. 206, p. 180, and Fig. 83) set up the transit and point the
horizontal hair on the star. Keep the cross-hair pointed on the
star until the culmination is reached. Read the vertical arc and
determine the index correction. The altitude is to be corrected
for refraction by Table 1 1, p. 193. This gives the true altitude.
If Polaris is at upper culmination subtract from the true altitude
the polar distance of the star at the date of the observation (Table
8, p. 184). If the star is at lower culmination the polar distance
is to be added. The result is the latitude of the place of obser-
vation.
217. (2). BY THE ALTITUDE OF THE SUN AT NOON.— The ob-
servation consists in finding the greatest altitude of the sun's
lower limb. This will occur when the sun is on the meridian
(very nearly). Begin the observation a little before apparent
noon, remembering that this differs sometimes as much as 17""
from mean noon.* Furthermore it should be remembered that
standard time may differ a half hour or so from inean time.
When the maximum altitude is found the following corrections
are to be made: first, the refraction correction is to be sub-
tracted, (Table II, p. 193); second, the sun's semi-diameter
(found in the Nautical Almanac) is to be added; third, the
sun's declination is to be subtracted if plus or added if minus.
The result, subtracted from 90°, is the latitude.
* Apparent noon occurs when the sun is on the meridian. Mean nqon is the
instant when the sun would be on the meridian if it moved at a uniform rate along
the equator. The difference between the two is known as the Equation of time
and may be found in the Nautical Almanac. For example, on November ist, the
sun ]>asses the meridian i6m i8s before mean noon, i.e., when it is izh com cos
apparent time it is i ih 43m 42s mean time.
OBSERVATIONS FOR LATITUDE I97
Example.
Observed 'mazimiim altitude of the sun's lower limb on
Jan. 8, 1906. «
Observed altitude
Index Correction
Refraction
25** 06'
2S« od'.o
i'.o
25« 0/.0
2'.0
2500^0
i6'.3
25^ 2I'.3
22« l8'.0
Index Correction — +
Declination of sun at
Greenwich mean noon
l'
■ 22"i9'35''
+ 1 33
(S)
Sun's semi-diameter
Altitude of son's center ~
Declination -
220 l8'02''
(S)
LaUtude
47"
42"
39'.3
Diff. ih - 4. 19
+ i9"-58 X 4^.
r. 58
74 - + 1'
33"
PROBLElfS.
1. (a) What was the azimuth of Polaris at its greatest western elongation at
Boston when the polar distance of the star was i^ 14' 1 2'^ ? The latitude of Boston
is42*'2i'N.
(b) In making an observation for meridian two stakes were set 329 feet apart,
marking the direction of the star at elongation. Compute the length of the per-
pendicular offset to be laid off at one end of the line to obtain the true meridian.
2. What is the approximate Eastern Standard Time of the eastern elongation
of Polaris on August loth at a place in longitude 72^ 56' West ?
3. Observation on May 15, 1906, for determining the azimuth of a line from
an altitude of the sun. Reading of vernier A of the horizontal circle while point-
ing on the azimuth mark « 0° 00^. At first pointing on sun, lower and right limbs,
vernier A, horizontal circle read 168^ 59' ; vertical arc read 43® 36'; the Eastern
Standard Time was 2h 52m 45s P.M. At second pointing on the sun, upper and
left limbs, vernier A, read 168** 52'; vertical arc, 42® 33'; time, 2h 55m 379 P.M.
The second pointing on the mark -> o® 00', the mark being to the left of the sun.
The sun's declination at Greenwich Mean Noon was +18® 42' 43".6 (North). The
change for i hour was -f- 35^'.94 (sun going north). The latitude of the place was
42** 1/ N. ; The longitude was 71^ 05' W. Find the azimuth of the mark.
4. Observation for latitude. The observed altitude of Polaris at upper cul-
mination was 43° 27'. The polar distance of the star was i^ 12''. What was the
latitude of the place ?
5. Observation for latitude. The observed maximum altitude of the sun's
lower limb on August loth, 1906, was 66^ 29'. The Eastern Standard Time was
approximately iil^ 50m A.M. The semi-diameter of the sun was 15' 48^^7. The
declination of the sun at Greenwich Mean Noon was North 15^46' ^3" '3 ( + ). The
difference for i hour was— 43''.46 (sun going south). What was the latitude of
the place ?
CHAPTER VIII.
LEVELZNO.
2i8. DEFnfinoilS. — Leveling consists in ascertaining differ-
ences in elevation; there are two kinds, Direct Leveling^ and
Trigonometric Leveling, The former alone will be considered
in this book, as trigonometric leveling is used only in advanced
surveying work.
Wherever extensive leveling operations are to be carried on
it is necessary to have a system of reference points called bench
marks (B.Ms.), the relative heights of which are accurately known.
These heights are usually referred to some definite zero plane,
such, for instance, as mean sea-level or mean low water^ and the
height of a point above this plane is called its elevation. This
plane is called the datum. (See Art. 237, p. 211, and Art. 250,
p. 226.) Strictly speaking it is not a plane but a level surface, ie.,
it is at every point perpendicular to the direction of gravity. If
mean sea-level is not known a datum can be arbitrarily assumed.
219. LEVELING TO ESTABLISH BENCH MARKS. — When it is
necessary to run a line of levels to establish new bench marks
the rod is first held on some bench mark the elevation of which
is accurately known, and a backsight taken (Art. 1 16, p. 85). If
this backsight is added to the known elevation of the bench
mark it gives the height of the instrument {H. /.) above the
datum. A turning point is then selected ahead on the route (to
be traversed), and a foresight taken on it. (See Art. 224, p. 202.)
If the foresight is subtracted from the height of the instrument
the elevation of the turning point is obtained. When a target
rod is used it is customary to take readings on bench marks and
turning points to thousandths of a foot, and in this case often
more than one rod-reading is taken on each point. If the first
and second readings agree within 0.002 ft. it is unnecessary to
take more readings ; if they differ by a greater amount it may
be necessary to take three or four or even more readings to
properly determine the correct value. The object of taking
198
BENCH MARK LEVELING I99
more than one reading is not so much to increase the precision
as to check the former readings.
When it is desired to establish a bench mark a suitable point
is selected and used as a turning point. The elevation of this
bench mark could be obtained by simply taking a foresight upon
it and not using it as a turning point, but by making the bench
mark also a turning point it becomes a part of the line of levels
and if the levels check, the elevation of the bench mark is also
checked. Each bench mark established should be carefully re-
corded by a description or a sketch, or both. The elevations
of the remaining turning points are as accurate as the elevations
of the bench marks themselves, so that any of the turning
points might be used as a bench mark. Consequently it is
advisable to describe those turning points which can be readily
identified so that they may be used when it is not convenient or
possible to use one of the established bench marks.
In leveling up or down slopes the levelman should be able to
judge quickly where to set his instrument in order to have it the
desired height above the turning point. In going downhill the
rod-reading of the backsight should be as small as possible in order
to overcome the height with the minimum number of set-ups of
the level. But while the levelman may waste much time by
having large backsights necessitating additional set-ups, it is
also possible for him to waste quite as much time in attempting
to place his instrument so as to get very small backsights. The
proper way to handle the instrument is as follows. Set up
roughly (without pressing the tripod legs into the ground), turn
the telescope toward the rod and then level it, approximately,
in that direction. By sighting along the outside of the telescope,
the approximate place where the line of sight will strike the rod
can be noted and the distance the instrument should be moved
up or down the slope can readily be estimated. Then move to
the new position, level up carefully, and proceed to take the back-
sight. This general procedure should be followed whether level-
ing up or down a slope.
220. In this work it is very important to eliminate as far as
possible errors of adjustment in the instrument. If at every set-
up of the level the foresight and its corresponding backsight are
200
LEVELING
[Chap. VIII.
taken at points which are equally distant from the instrument such
errors will be eliminated. If the level is not in perfect adjustment
the resulting error in any reading is proportional to the distance.
At equal distances from the instrument the errors are equal, and,
since it is the difference of the rod-readings that gives the dif-
ference in elevation, the error is eliminated from the final result
by this method. By making the length of foresights and back-
sights equal on turning points it is possible to eliminate not only
the error due to non-adjustment of the bubble but also any error
due to non-adjustment of the objective tube, since this will occupy
the same position in the telescope in each sight. The distance
to the backsight is determined by the place where the instrument
is set up, and the rodman, as he passes from one turning point to
the next, can by pacing make the foresight distance approximately
equal to that of the backsight. The line of levels should be "closed"
by continuing the leveling until the original bench mark, or some
other bench mark whose elevation is well established, is reached.
221. The notes for this work may consist of five columns, as
shown in Fig. 86. The height of instrument is obtained by add-
ing the backsight to the elevation of the point on which it is taken.
The elevation of any point is found by subtracting the foresight
for that point from the height of the instrument. Notice that the
Lbpt^Hakd Pack.
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DOUBLE RODDED LINES
201
calculations maybe checked by adding the foresights and the back-
sights. The difference of these sums should be the same as the
difference in elevation between the first and last points.
222. Double Rodded Lines. — A good check on the line of
levels may be secured by running a double line of turning points.
Instead of taking a foresight on a single turning point, foresights
may be taken on two different points near together, from the
same set-up of the instrument. When the level is set up again
a backsight is taken on each turning point and two independent
values of the new height of mstrument are obtained. In
ordinary bench mark leveling these two values should not differ
by more than 0.002 or 0.003 ft. from the previous difference,
i.e., if the two heights of instrument differed by 0.013 at a
certain set-up they should not differ by more than 0.016 nor
less than 0.0 10 at the next set-up. If the two turning points
of a pair are so chosen that their diflference in elevation is more
than a foot then any mistake of a foot in the computations or
in reading the rod will be immediately detected.
In this way, by little additional work the accuracy of the levels
may be checked as the work progresses. This method of using
double turning points is particularly useful in running long lines of
levels where no established bench marks are available for checking.
223. A set of notes illustrating double turning points is
shown in Fig. 87. It will be noticed that the higher and lower
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202 • LEVELING [Chap. Vni.
turning points of a pair are arranged in a systematic order. The
readings in this case have been taken on the lower turning point
first at each set-up. It is very important that some definite
system shall be followed so that the two lines of levels will not
be confused.
224. Bench Marks and Turning Points. — Both the bench
marks and the turning points should be such that their elevations
will not change during the time they are needed. The only
difference between the two is that turning points may be of use
for only a few minutes while bench marks may be needed for
many years. Bench marks should be very carefully and accu-
rately described, and their heights should be checked before
being accepted as correct. They are frequently taken on such
points as these : — stone bounds, tops of boulders, spikes in
trees, and on sills, stone steps, or underpinning of buildings.
Curb stones or tops of hydrants are also used but are not so
permanent. As it is often impossible in a new country to find
existing points where bench marks can be established, it is usual
in such cases to set stone monuments or iron rods and to care-
fully determine their elevation. The U. S. Geological Survey,
for example, sets an iron pipe with a cap on the top of it ; or in
some cases a plate with a horizontal line across it in the masonry
wall of a building. Some of the bench marks of the U. S.
Coast and Geodetic Survey and of the Missouri River Com-
mission consist of stones buried 3 or 4 ft. under ground. The ,
exact bench is the top of a spherical headed bolt set in the top
of the stone. This is reached by lowering the rod through an
iron pipe which extends to the surface of the ground.
Bench marks should be established at frequent intervals for
convenience in dependent work. Some surveyors consider it
advisable to have two bench marks in the same locality to ser\'e
as checks on each other. In choosing a bench or a turning
point it is best to select a point which is slightly raised so that
the rod will always rest on exactly the same point. A rounded
surface is better than a sharp point, especially when it is on a
rock, as the rod may chip off a small piece and alter the eleva-
tion. If a turning point is taken on a flat surface it is difficult
to get the rod at exactly the same height each time. Bench
BENCH MARKS AND TURNING POINTS 203
marks are, however, sometimes established on flat level surfaces
such as the coping stone of a masonry structure, because per-
manence is of more importance than great precision. Bench
marks are not only described in the notes, but are themselves fre-
quently marked by red chalk, by chisel marks, or drill-holes.
225. LEVELINO FOR PROFILE. — Profile leveling is for the
purpose of determining the changes in elevation of the surface of
the ground along some definite line. The line is first " stationed,*'
i.e., marked at every hundred feet or such other interval as is
desired. The level is set up and a backsight taken on a bench
mark to determine the height of the instrument. Foresights
are then read on as many station points on the line as can be
conveniently taken from the position of the instrument. Inter-
mediate sights are taken at any points where marked changes
of slope occur, and the plus stations of these intermediate points
are recorded with the rod-readings. It will be noticed that here
the terms foresight and backsight do not refer to the forward
and backward directions. A backsight is a reading taken on a
point of known elevation for the purpose of obtaining the height
of the instrument. A foresight is a reading taken on a new point
to determine its elevation. For this reason backsights are fre-
quently called //m^ sights ( + 5), and foresights are called mintis
sights (— 5). When it is necessary to move the level to a new
position in order to take readings on stations ahead, a turning
point is selected and its elevation determined. The level is
then taken forward and its new height of instrument determined
by taking a backsight on the turning point. This general pro-
cess is continued until the end of the line is reached.
A line of levels should be checked by connecting with some
reliable bench mark if possible. If there are any bench marks
along the line of levels they should be used as turning points if
convenient, or at least check readings should be taken on them
in order to detect mistakes. In such a case it is evident that
the reading taken on the bench mark is really a foresight since
its elevation is being found anew from the height of instru-
ment. Readings on bench marks and turning points should be
taken to thousandths or to hundredths of a foot, depending upon
the accuracy desired. If the elevations of the profile are de-
204
LEVELING
(Chap. VUI.
sired to the nearest hundredth of a foot, as in the case of a
raikoad track, the turning points should be taken to thousandths
of a foot. Elevations on the surface of the ground will not usu-
ally be needed closer than to tenths in which case the T. Ps. are
taken only to hundredths. In calculating the elevations the re-
sults should not be carried to more decimal places than the rod-
readings themselves, otherwise the results will appear to be more
accurate than they really are.
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226. Profile notes are kept as shown in Fig. 88. In this
case also the heights of instrument and the elevations of turn-
ing points may be checked by means of the sums of the fore-
sights and backsights, provided only the sights on turning points
and the initial and final benches are included. If it seems
desirable the elevations of stations may be checked by means of
PROFILE LEVELING
20S
differences in foresights. The difference between the eleva-
tions of any two points, which are obtained at the same set-up
of the instrument, is equal to the difference between the fore-
sights taken on these points. For example, if the difference be-
tween the foresights on stations 4 and 5 is 3 ft. this should also
be the difference between their elevations. In these notes the
elevations of B. Ms. and T. Ps. are put in a different column
from the surface elevations simply for the sake of clearness, but
many surveyors prefer to put all the elevations in the same
column. Another arrangement of columns which will be found
convenient when plotting the notes is to place the station col-
umn immediately to the right of the elevation column.
T^i
PLAN
SIKO
PROFILL
Fig. 89.
Fig. 89 represents a rough plan and profile of the line of
levels shown by the notes in Fig. 88. Angle points in the
transit line are shown in the plan, but they do not appear in the
profile of the line. It will be noticed that the T. Ps. and B. Ms.
are not on the transit line in plan, and that they consequently
do not appear on the profile. It is not customary to introduce
206
LEVELING
[Chap. VIII.
any sketches into the profile notes except those used in describ-
ing bench marks or turning points.
227. Cross-Sectioning. — If it is desired to know the shape
of the surface of a piece of ground, the area may be divided
into squares and the elevation taken at each corner of these
squares and at as many intermediate points as seem necessary
to determine the changes of slope. These surface elevations
are obtained to tenths of a foot. The squares which may be
anywhere from 10 ft. to 100 ft. on a side are laid out with the
transit and tape, stakes being driven at the comers. It is well
to choose some long line of the traverse as the primary line
from which the cross-section system is to be laid out. The
points are usually designated by a system of rectangular coor-
dinates, one set of parallel lines being marked by letters and the
other by numbers, as shown in Fig. 90. For example, the
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point / would be called (C, 7) ; the point j, {D, 5) ; the point
r, {B f 80, 4 -h 35) ; etc. The notes are kept as in profile leveling
except as to designation of points.
228. Use of the Tape Rod in Cross-Section Work. — In this
work, where there are a large number of elevations to be cal-
culated, it will save much time to use a tape rod (Art. ic6,
p. 81), which is so arranged that no elaborate figuring is re-
quired In this rod the numbers increase from the top toward
the bottom, the opposite way from ordinary rods. The level is
CROSS-SECTION LEVELS 20/
set up at a convenient point and the rod held on a bench mark.
The tape, or band, on the rod is then moved up or down as di-
rected by the levelman until he reads the feet, tenths, and hun-
dredths which are the same as those of the elevation of the bench
mark, e.g., if the elevation of the B. M. is 195.62, the tape will be
moved until it reads 5.62. If the rod is then held on a point
1.61 ft. lower than the bench, the rod-reading will be 4.01, since
with this rod the readings decrease as the rod is lowered. The
elevation of the point is then 194.01 ft., or sufficiently precise for
topographic work, 194.0 ft. In this way the elevations are read
directly on the rod to feet and decimals of feet, the tens and hun-
dreds of feet being supplied mentaUy. Obviously the only notes
kept are the columns of stations and elevations.
229. CROSS-SECTIONING FOR EARTHWORK. — Whenever it
is desired to ascertain the quantity of earthwork in an excava-
tion or an embankment, it is necessary to take levels to determine
the vertical dimensions, and to obtain the horizontal dimensions
by means of the transit and tape. The three general cases where
the quantity of earthwork is to be estimated by the engineer
are : (i) an excavation or embankment having a known base and
side slopes as in the construction of a railroad or a highway, (2)
an irregular excavation from a bank of earth called a borrow-pit^
(3) a trench excavation such as is used for sewer construction.
230. ( i) Road Cross-Sections. — Cross-sections for estimating
the earthwork in highways or railroads are usually taken at full
station points (sometimes oftener) and at right angles to the
center line of the road.* By this method is obtained a section
of the general shape shown in Figs. 91 and 92. These cross-
sections are taken in the field before the construction begins so
that a proper record of the surface heights can be obtained be-
fore the ground is disturbed.
From the plan of the proposed road its alignment is staked
out and a profile is taken along the center line, which is subse-
quently plotted (Art. 225, p. 203). On this profile the grade line
is drawn, which corresponds to the finished surface of the road.
Roads are usually first finished to sub-grade^ which is below the
* For a more complete treatment of this snbject see ** Railroad Carves and
Earthwork," by Professor C. F. Allen, published by Spon & Chamberlain,
New York.
208 LEVELING [Chap. VIIL
completed surface by an amount equal to the thickness of the
road covering, i.e., the pavement of a highway or the ballast in
the case of a railroad. The width of the base of the road and
the inclination of the side slopes are known. For ordinary
gravel the slope is usually i^ ft. horizontal to i ft. vertical, called
"a slope of I J to i."
For construction work the engineer sets grade stakes at
every full station or oftener on the center line and at both sides
where the finished slope intersects the surface of the ground,
e.g., at points A, B and C on Figs. 91 and 92. All of these
Fig. 91. Excavation. Fig. 92. Embankment.
stakes are marked, giving the amount of "cut" or "fill" to be
made at these points. The cut or fill marked on the stakes at
B and C is the vertical distance from the base of the road to the
surface of the ground at these points, e.g., the distance 6C
These cuts and fills are determined in the field by the follow-
ing method. The level is set up and the height of instru-
ment obtained from some convenient bench mark. Then, the
elevation of the finished grade being known (from the profile
prepared in the office), the difference between the height of
instrument and the elevation of the finished road will give what
is called the rod-reading for grade^ i.e., the rod-reading which
would be obtained if the foot of the rod could be held on the
finished surface of the road. Then the rod is held on the surface
of the ground at the center stake and a reading is taken (to the
nearest tenth of a foot), and the difference between the rod-
reading for grade and the rod-reading on the surface will give
the cut or fill at that point, and this is marked on the center
grade stake thus, C5.2 or F4.7.
231. Setting Slope Stakes. — The points where the
side slopes intersect the surface of the ground are found by
trial as follows. Hold the rod at a point where it is estimated
that the side slope will cut the surface, and take a rod-reading.
The difference between this rod-reading and the rod-reading for
ROAD CROSS-SECTIONS
209
grade will give the cut or fill at this point, from which the
distance out from the center of the section to the point on the
side slope having this cut can be computed. This distance out
equals (J base -f cut x slope). Then the distance is measured
from the center to the rod, and if the measured distance
equals the computed distance the rod was held at the right
place and the stake should be driven and marked with the cut
or fill at that point (distance bC^ Fig. 92). If the measured
distance does not agree with the calculated distance a second
trial must be made by holding the rod at another point and
repeating the operation. The difference between the measured
and calculated distances is an aid in judging where the rod
should be held at the second trial. After a little practice it
will be possible to set the slope stake at the second or third trial.
232. Earthwork Notes for Road Cross-Sections. —
The notes for this work will contain the cut or fill at the
center, the cut or fill at either side, and the corresponding
distances out. A cut is usually written in the notes as a plus
( + ) height and a fill as a minus (— ) height; but the stakes
Lbft-Hand Pagb.
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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
210 LEVELING [Chap. Vm.
base of the road, as in Sta. lo, the section is called a Level
Section. Where the surface of the ground is not parallel to
the base and where three cuts or fills only are recorded, as at
Sta. 1 1, the section is called a Three Level Section. If, besides
the three readings which are taken for a three level section, two
more intermediate readings are taken one directly over each end
of the base, as at Sta. 1 1 + 50, the section is called a Five
Level Section. If intermediate readings (one or more of them)
are taken anywhere except over the ends of the base, as in Sta.
12, the section is called an Irregular Section, For methods of
computing the amount of earthwork see Chapter XII.
It will be noticed that in the column of the notes headed
" Cross-Sections " the distances out appear above and the corre-
sponding cuts below the lines. Besides this set of notes there is
a simple set of level notes similar to Fig. 86, p. 200, from which
the height of instrument is determined. This is conveniently kept
in another part of the note-book, often at the back of the book.
233. (2) Cross-Sections for Borrow-Pits. — The ground is
first staked out in squares or rectangles and the elevation at each
corner and at every change in slope is determined as explained
in Art. 227, p. 206. Then the work of excavating is carried on,
and when it is desired to determine the amount that has been
excavated, the same system of cross-sections is again run out
and the new elevations at the corners and at the necessary
intermediate points are determined.
The notes are kept as shown in Fig. 90, p. 206. For methods
of computing the earthwork in borrow-pits see Art. 373, p. 342.
234. (3) Cross-Sections for Trench Excavation. — The sur-
face elevations are determined by making a profile of the
line. The grade of the bottom of the trench is obtained either
from the plan or by direct leveling. The width of the trench is
measured wherever it changes and the stations of these places
noted. For methods of computing the quantity of earthwork
see Chapter XII.
235. LEVELING TO ESTABLISH A GRADE LINE. — The level
may be used for setting points at desired elevations as, for
example, in establishing the grade line of a sewer. To set any
point at a given elevation, set up the level and take a backsight
ESTABLISmNG A GRADE LINE 211
on a bench mark, thus determining the height of instrument.
Subtract the given elevation from the height of instrument and
the result is the rod-reading for grade. Raise or lower the rod
until the horizontal cross-hair indicates this reading. The foot
of the rod is then at grade. This is usually set for construction
work to hundredths of a foot ; for some purposes tenths of a
foot will be sufficiently exact. If a target rod is used the target
is set at the proper reading, and the bottom of the rod is at
grade when the cross-hair bisects the target:
If the grade line comes beneath the surface of the ground
and cannot be reached a point may be set a convenient whole
number of feet above grade and the depth marked on a stake>
or vice versa if the grade line comes far above the surface.
236. " Shooting in " a Grade Line. — To save time and to
diminish the liability of mistakes, grades are often set by a
method known as " shooting in '* the grade. First set a point
at the proper elevation at each end of the straight grade line.
The instrument (usually a transit with a telescope bubble) is
set up 6 or 8 inches to one side of the first point, and the dis-
tance from the top of the first stake to the axis of the telescope
is measured with the tape or rod.* Then the rod, which is set at
this reading, is carried to the last point on the straight grade line,
and, whDe it is held vertical on this point, the instrument man
raises or lowers the telescope until the horizontal cross-hair is on
the target, clamping the instrument in this position. If. a level
is used the horizontal cross-hair is set by means of the leveling
screws ; but if the transit is used the cross-hair is set by means
of the clamp and tangent screw of the vertical motion. The
line of sight is then along an inclined line parallel to the grade
line. All intermediate points on the grade line are then set by
raising or lowering the rod until the target coincides with the
horizontal cross-hair.
237. To ESTABLISH A DATUM PLAITE BY MEANS OF TTOAL
OBSERVATIONS. — Whenever it is necessary to establish a datum
from tidal observations it may be determined as follows. Set up
* Where the grade is flat some surveyors prefer to set the instrument just
behind the point instead of to one side of it.
212
LEVELING
[Chap. vm.
a vertical staff, graduated to feet and tenths, in such a manner
that the high and low water can be read. Read the positions
of high and low water for each day for as long a period as prac-
ticable. The mean value obtained from an equal number of
high and low water observations will give the approximate value
of mean sea-level. If the observations extend over just one lunar
month the result will be fairly good, whereas in less than one
month a satisfactory result cannot be obtained ; to determine this
accurately will require observations extending over several years.
The proper location of the gauge is an important factor in
obtaining the true mean sea-level. The place chosen for setting
up the gauge should be near the open sea, so that local conditions
will not influence the tide. It should be somewhat sheltered
against bad weather. The water should be deep so that at the
lowest tide the water will stand at same height on the gauge.
At the beginning of the series the zero of
the staff and some permanent bench marks
should be connected by a line of levels. This
should be tested occasionally to see if the staff
is moved. After the reading of the rod for
mean sea-level is found the elevation of the
bench mark can be computed.
238. The Staff Gauge. — This is a form
of gauge (Fig. 94) which can be easily con-
/l ^H structed, and which is sufficient where only a
I ^^1 short series of observations is to be made. If
!l (5% niade in sections not over 3 feet long, as de-
I ^N scribed below, it can easily be packed in a box
^, ^^ for transportation. Each section consists of
I two strips of wood about i^ inches square,
■ and 3 feet long, fastened together at the
I ends by strips of brass, leaving a space
I between them of about i inch. In this
B space is placed a glass tube of about J inch
I 4(ll!l!ii diameter and held in place by brass hooks.
M^l| On one side of the tube is a red strip blovm
into the glass. When the gauge is set up
for observations the sections are screwed to
Fig.
94. Staff
Gauge.
LEVELING ACROSS A RIVER 213
a long vertical piece of joist. The ends of the tube are nearly
closed by corks, in which small glass tubes of approximately i
mm. (inside) diameter have been inserted. When the water rises
in the main tube, the red strip appears to be much wider than it
really is on account of the refraction of light by the water. Above
the water surface the strip appears its true width. By observing
the position of the wide strip the height of the water surface can be
read within a hundredth of a foot. The heights are read on a scale
of feet painted on the wooden strips. If the size of the small
glass tube is properly chosen, the fluctuations of the water sur-
face outside will not disturb the water in the tube, so that the
reading is a fair average of the water surface. A gauge of this
sort may be read by means of a transit telescope or field glass
at a distance of several hundred feet.
When a long series of observations is to be made a self-regis-
tering tide gauge should be used. A description of such a gauge
may be found in the Reports of the U. S. Coast and Geodetic
Survey.*
239. LEVELING ACROSS A RIVER. — While the effect of cur-
vature and refraction (Art. 118, p. 87) is usually negligible in
leveling operations, it may in certain special cases become of
great importance to eliminate this error. For example, it is
sometimes necessary to carry a line of levels across a river of
considerable width, say, half a mile. In this distance the correc-
tion for curvature and refraction amounts to about 0.143 ft. un-
der normal conditions, which in a line of bench levels is too large
a quantity to neglect. If the correction as derived from formu-
las could be depended upon under all circumstances it would be
sufficient to compute and apply it to the rod-reading. But the
amount of the refraction correction is so variable that the actual
value often differs considerably from the computed value.
If it is desired to obtain the difference in elevation between
two distant points with great accuracy it will be necessary to use
a method which will eliminate the effects of curvature and re-
fraction no matter what their actual amount may be. In Fig.
95 suppose a backsight were taken on T. P.j with the instrument
• Report for 1897, pp. 315-320 and pp. 480-489.
Report for 1853, pp. 94-96.
214 LEVELING [Chap. Vm.
at A and then a foresight taken on T. P., The elevation of T. P.,
as computed from T. P.^ will be too low by the amount ab^
since the foresight on T. P., is too great by this amount. If
the difference in elevation is determined by the instrument at
B the backsight on T. P.j is too large by the amount cd. Hence
the H. I. of the instrument at ^ is too great, and consequently
Fig. 96. Leveling Across a River.
the elevation of T. P., too great by the amount cd. The mean
of the two determinations would give the true elevation of T. P.,
if ab^cd, but this occurs only when the two sights are taken
under the same atmospheric conditions. Therefore it will be
seen that the two sights must be taken simtdtaneously. In
order to eliminate the errors of adjustment * in the instrument
it is necessary to use the same instrument at both ends of the
line. To accomplish both of these results at once it is neces-
sary to take simultaneous readings with two instruments and then
to repeat the operation with the instruments interchanged. The
magnifying powers of the two telescopes and the sensitiveness of
the two spirit levels should be about equal in order to g^ve the
best results. It will be noticed that this process is similar to
that of the peg adjustment (Art. 128, p. 91).
* Errors due to non-adjustment are of unusual importance because the sight is
much longer than that used in adjusting the instrument.
LEVELING PROBLEMS
215
PROBLEMS.
X. Compute fhe following set of level notes.
Sta.
B.S.
H. L
F. S.
Elev.
B.M.,
B. M.,
T. P.j
B.M..
T.P..
B. M.,
4.702
11.846
7.276
8.760
0.687
1.607
6.727
9.689
4.726
11.000
8.496
16^27
2. Compute the elevations in the following set of level notes.
Sta.
B.S.
H. L
F. S.
Elev.
B.M.^
6.427
4.273
62.473
20
6.2
21
7.4
+4*
5-2
22
4.7
T. P.„
4.724
9.976
23
11.2
+63
10.4
B. M.]!
oj^09
7.482
24
11.2
3. Compute the elevations in the following set of level notes.
Sta.
+s.
H.L
-s.
Elev.
B.M...
6.214
84.238
T. P., L.
3-515
9.280
T. P.j H.
2.152
7.919
T. P., L.
2.971
8.263
B.M«,H.
2.338
7.629
T. P., L.
4.278
7.529
T. P.,H.
2.646
5894
B.M.^L.
5721
6.072
T. P.4 H.
4.837
5.187
B.M.„
5817
4. Make up a set of cross section notes for road construction which shall be
consistent with the following data: width of road, 50 ft., slopes i J to i ; grade
elevation of Sta. o » 107.20; grade, I- i 4. Show complete notes from Sta. o to
Sta. 3 inclusive as follows: Sta. o« a level section; Sta. i, a three level section;
Sta. 2, a five level section ; Sta. 3, an inegnlar section.
CHAPTER IX.
CITY SURVBYINO.
240. INSTRUMENTS USED. — Owing to the comparatively
high value of land in cities and to the fact that a large propor-
tion of city surveying is the establishing of lines and grades for
construction work, the chain and compass are discarded entirely
and the steel tape and transit are used.
241. Tapes and Tape Measurements. — The tape most com-
monly employed is the light loo-ft. steel tape, graduated to
hundredths of a foot, described in Art. 7, p. 5. All ordinary
measurements are taken in the usual manner, the pull and the
horizontal position of the tape being judged by the men taking
the measurements. But frequently it is necessary to obtain
results with a greater degree of accuracy than is afforded by
the ordinary method of measurement. For example, in measur-
ing the base-line for triangulation work or in the survey of the
valuable portions of large cities, there is call for an accuracy of
measiu-ements which can only be obtained by using a method
which will insure a uniform pull on the tape, a careful align-
ment, little or no sag in the tape, and some means by which the
temperature of the tape can be taken and its correction applied
to the results. In such cases the pull is measured by use of a
tension handle (ordinary spring balance) which can be attached
by a clamp to any part of the tape, the alignment is given with
the transit, and, where feasible, just enough pull is given so that
the stretch in the tape equals the shortage due to sag. The
correction for temperature can be computed from the difference
between the temperature of the tape taken in the field and the
temperature at which it is standardized /"Art. 19, p. 13). The
tape should be compared with the City Standard (Art. 243, p.
21 8 \ at a definite tension, and the temperature noted at the
time. From this information all of the field measurements can
216
INSTRUMENTS USED 21/
be reduced to agree with the City Standard and very accurate
results may be obtained.
Where the ground is not level and there is call for frequent
plumbing it is impossible to obtain accurate results unless the
plumbing is carefully done by experienced tapemen. For very
accurate work it may be desirable to entirely eliminate the
plumbing. This is sometimes done by measuring directly on
the surface (on the slope) from point to point, and by means of
the level instrument and rod the relative elevations of these
points are obtained and the horizontal projection of the slope
distances computed. Instead of measuring the difference in
elevation between the two ends of the line, the angle of inclina-
tion of the slope line is often measured on the vertical arc of a
transit which is set up over one of the end points.
The government Bureau of Standards at Washington will,
for a nominal charge, standardize tapes ; and city and private
engineers frequently avail themselves of this opportunity. This
Bureau will give the exact length of the tape at a given temper-
ature or the temperature at which the tape is of standard length,
whichever is desired by the engineer. It is well to have the
tape also tested at a few intermediate points, e.g., the 25 ft., 50
ft., and 75 ft. marks. One tape which has been standardized
should be kept in reserve, with which tapes in service can be
compared both when new and after being mended.
Besides the ordinary steel tape, steel or metallic tapes
reading to tenths of a foot are used in taking measurements for
making approximate estimates of construction and for measuring
earthwork, paving, and the like.
242. Transits and Levels. — The transits usually employed
in city work read to 30" or to 20" ; and for most city work no
finer graduation is necessary. With these instruments the re-
quired precision in reading angles on triangulation work
can be obtained by repeating the angles as explained in Art.
59, p. 48. In such work, however, it will be of advantage to
have an instrument reading to 10". It is well also to have
one or more transits equipped with stadia hairs for use on
rough surveys.
Much of the city work, such as the staking out of new streets,
2l8 CITY SURVEYING [Chap. IX.
paving, sewers, or curbs, requires the establishment of both lines
and grades. Since this class of work does not as a rule call for
very precise results, the measurements and rod-readings are usually
taken to hundreidths of a foot. It is not convenient, for the ordi-
nary surveying party of three men, to carry both a transit and a
level instrument in addition to the ordinary equipment of sight-
ing-rods, level-rod, stakes, tape, etc., so the engineer's transit,
>yith a level attached to the telescope, is extensively used in set-
ting grades as well as in establishing lines. For this reason sev-
eral of the transits in a city office ,should be equipped with telescope
levels and some of them with vertical arcs. The degree of pre-
cision possible with an engineer's transit is entirely satisfactory
for all ordinary leveling.
Where leveling work alone is to be done the ordinary wye or
dumpy level instrument is used together with target or self -reading
rods. (See Chapter IV.) For bench leveling it is customary, in
large cities at least, to use a precise levels an instrument which is
similar in principle to the ordinary level but which has a more
delicate bubble and a telescope of higher power, and is therefore
capable of yielding more accurate results.
243. CITY STANDARD.* — It is customary in all large cities to
have a standard of length, usually 100 ft. long, established in
some convenient place, often near the office of the City Engineer.
It sometimes consists of two brass plugs set in a stone pavement,
or it may be a long steel rod supported on rollers on the side of
a wall or building in such a way that the rod can expand or con-
tract freely. The end points and the so-ft. points are so marked
that they can be readily found and used by any surveyor who
desires to test his tape.
A city standard is often established by carefully transferring
the length of some other standard, by means of different tapes
and under different weather conditions ; or it can be established
by means of a tape which has been standardized by the U. S.
Bureau of Standards (Art. 241, p. 216). The City Standard is
• See a paper entitled "The 100 -foot Standard of Length of the Boston
'Water Works at Chestnut Hill Reservoir," by Charles W. Sherman, published in
the Jour. Assoc. Eng. Soc., Vol. XVIII, Ko. 4, April, 1897.
STANr)ARD OF LENGTH 219
generally placed where it will not be exposed to the direct rays
of the sun, and with this end in view it is sometimes covered with
a wooden box.
When a tape is tested it should be stretched out at full length
beside the standard and left there until it acquires the same
temperature as the standard before the comparison is made, to
avoid the necessity of applying a temperature correction.
CITY LAYOUTS.
244. In la)ring out or extending a city it is the duty of the
surveyor to consider the future needs of its population and to
design the general plan of the city accordingly. Nearly all of
our large cities show examples of lack of forethought relative
to future growth, which have necessitated the outlay of millions
of dollars for revision of street lines, sewer systems, water
works, and the like.
Occasionally the engineer is called upon to plan a new city
or to' design the general layout of the suburbs of an existing
city. The basis for such work should be a top)ographic map of
the entire area, for the topographic features of a locality will
influence its development to a marked degree.
245. STREETS. — In planning the arrangement of the streets
for a city such features as a water front, a river or lake, the loca-
tion of an existing railroad, or the probable location of some
projected railroad line will determine to a large degree where
the business section of the city will be located. This section
should then be so divided as to yield the greatest convenience
for business purposes. Other sections will be reserved for
residential districts, and their design will be of a different
character. Easy access should be provided from the business
"to the residential districts and to outlying towns or adjacent
cities.
The streets must be of the proper width to accommodate
the traffic they are to carry, and their alignment and grades
must be carefully studied with the topographic map as a guide.
Adequate drainage' of the streets is, of course, one. of the most
220 CITY SURVEYING [Chap. DC
important features, for which ample provision must be made
in establishing the alignments and grades.
In the business section the traffic will move in certain
directions, e.g., to and from important points such as a river,
railroad station, or freight yard, and this traffic must be pro-
vided for by wide streets with easy grades. In the residential
portions, narrower streets and steeper grades are permissible
when made necessary by the topography of the district.
246. Location of Streets. — In establishing the location of
city streets in hilly districts it is probable that to obtain the
essential requisites of easy grades and good drainage the topog-
raphy will govern the street layout. Whereas in a practically
level country, with no steep grades in any direction, the street
layout can be such that the most direct communication between
diflferent parts of the city is secured.
Fig. 96 shows the location of a rectangular system of streets
laid out without reference to the topographic features. The
lower portion is on rolling ground where this system may be
properly applied ; but from a study of the contours it will be
seen that in the upper portion this method introduces very steep
grades on all of the streets which cross the valley and also leaves
a hollow in these streets which is difficult to drain. Fig. 97
shows a layout which will obviate this difficulty to some extent,
the diagonal streets being located in the valleys to take the sur-
face drainage of surrounding property. It is obviotis that the
construction of a sewer through these diagonal streets will be
much more economical than through the streets as laid out in
Fig. 96, for a sewer must have a continual drop toward its out-
let, and cannot be laid uphill and downhill like a water pipe.
With reference to directness of communication between
different parts of a city the two general systems which have
been used in this country are the rectangular block system and
a combination of rectangular blocks with diagonal streets, run-
ning in the direction of the greatest traffic.
The rectangular system gives the maximum area for private
occupation and is consistent with the general style of rectangu-
lar building construction. Where the topography admits of it,
this system of streets is advisable. Many of our large cities,
LOCATION OF STREETS
Fig. 06. Layout of Streets without Regard to Topography.
Fig. 97. Layout of Streets with Regard to Topography.
222 CITY SURVEYING [Chap. IX.
like Philadelphia, for example, have been laid out in this manner.
The streets frequently run parallel and perpendicular to the
shore of a lake or river. More often, however, they are laid out
in north and south, and east and west directions. When diagonal
streets also are introduced they should connect the points be-
tween which the traffic is the heaviest. Indianapolis is planned
in this manner, having four broad diagonal avenues running from
a central park ; but the city of Washington (Fig. 98) is the best
example of this system in the United States.
247. Size of Blocks and Lots. — No definite size of blocks
and lots can be prescribed which will fit all conditions. Experi-
ence has shown that the depth of lot most convenient for both
business and residential districts is from 100 to 150 feet. In
business districts particularly, it is well to provide an alley from
15 to 25 ft. wide running lengthwise through the block. This
makes the width of blocks from 215 to 325 feet, which is about
the range in existing cities.
The length of the blocks should be in the direction of great-
est travel, and this dimension will therefore depend upon the
necessity for cross-streets to accommodate the traffic which
moves at right angles to the principal line of traffic. In busi-
ness districts then the cross-streets should be much more fre-
quent than in residential p)ortions of the same city.. The length
of blocks therefore varies considerably in different cities and in
different parts of the same city ; ranging all the way from 400
to 900 feet. In New York the typical blocks are 200 X 900
ft., and 200 X 400 ft. ; in Boston they vary in width from 125
to 252 ft. and in length from 200 to 700 ft., depending upon the
locality.
The frontage of lots is frequently 25 ft. in business and
congested residential districts and 50 feet or more in sub-
urban districts, but these dimensions are by no means
universal.
248. Width of Streets. — The widest streets should in gen-
eral be the ones which have the greatest traffic. Important
business streets should be from 100 to 150 ft. in width, while
streets of secondary importance in business districts may be
from 60 to 80 ft. wide. In residential districts the main streets
223
224 CITY SURVEYING [Chap. DC
should be 60 to 80 ft. wide, but those of lesser importance are
often made 50 ft. These widths, however, are more liberal than
have been used in many of our older cities, e.g., such cities as
Boston, Baltimore, and New York which are especially afflicted
with narrow streets.
The alleys which are run through the middle of city blocks
should be made from 15 to 20 ft. wide. If they are made nar-
rower than 1 5 ft. two teams cannot pass each other unless cer-
tain parts Qi the alley are widened for this purpose. Alleys
furnish a convenient place for the location of water pipes and
sewers.
The width of sidewalks varies greatly with the locality. In
business districts, where there is usually a necessity for ample
width, some cities devote two-fifths of the entire width of the
street to sidewalks; while in residential districts, the sidewalks
are frequently much narrower in proportion to the width of the
street. In Boston the general rule is to make each sidewalk
one-sixth the width of the street. Sidewalks 8 ft. wide are
ample for most residential districts. In some localities walks
as narrow as 4 ft. are laid out with a liberal grass-plot between
the sidewalk and the roadway, which not only gives a pleasing
appearance to the street, but also lessens the width of sidewalk
and of roadway to be paved and maintained, thereby decreasing
the burden of taxation and leaving room for an increase in width
of roading if afterwards needed.
249. STREET Grades. — In connection with the layout of
anew city or suburb the grade of the streets is of quite as
much importance as the street alignment. While, in the
residential districts of some cities, street grades as steep as 10
and 15 per cent, are not uncommon, still it is considered
advisable, if possible without excessive cost, to keep the grades
down to about 5 or 6 per cent., especially those which extend
for any considerable distance. In business districts, where
heavy loads are to be hauled, it is desirable that the grades
should not exceed 3.5 or 4 per cent. In any case where one
street crosses another the grade should be flattened between
curb lines to 3 or 4 per cent, if the grade of either street is
greater than this amount.
STREET GRADES 225
On account of drainage it is well to build a street with a
slight grade rather than level. A grade of 6 inches in loo feet
is a good working minimum for proper drainage, and if the
street does not have this gradient the gutters must be made of
varying depth so as to properly carry ofiF the water. Other
elements which govern the rate of grades are the cost of earth-
work and the proper balancing of the excavation and embank-
ment in the construction, the effect on abutting property, and
the general appearance of the street.
At points where there is a decided change in grade it is
customary to introduce a parabolic vertical curve. (Art.. 268,
p. 242.)
For the purpose of establishing the grades, profiles are
made of each street. Levels taken for the purpose of making
a profile should include elevations at the center of the street and
along both side lines, and it is often desirable to have a cross-
section plan of the entire area of the vicinity where the street is
to be located. A description of the street grade is written up
for acceptance by the proper municipal authorities. When this
description has been formally accepted by an order of the City
Government the grade is said to have been "established."
Such an order may refer to the profile by title or recorded
number, instead of a description of the grade. The profile of
each street should contain one or more cross-sections on which
is indicated to what part of the cross-section the profile refers,
i.e., whether the profile grade is the grade of the center of the
street, the curb, or the sidewalk at the property line.
The following is an example of a description of an estab-
lished street grade : —
"Beginning at Station 146 (Maple St.) at the junction of
the center lines of Maple St. and Ocean Ave., at grade * 52.00,
the grade line falls 0.50 per 100 for 726 ft. to grade 48.37—
thence rises 0.82 per 100 for 322 ft. to grade 51.01 — thence
• The word grade U frequently used to mean the elevation of a point. In
such a case care should be taken not to confuse the meaning of grade with rate of
grade. The latter is sometimes called gradients a word which has some advan-
tages but is not entirely satisfactory.
226 CITY SURVEYING [Chap. IX
falls 0.50 per 100 for 122 ft. to grade 50.40 — thence falls by a
vertical curve for 100 ft. as follows :
Sta. EleT.
157+60 S<MO
157 + 85 4990
158+ 10 49-30
158 + 35 48.55
158 + 60 47.70
thence falls 3.60 per 100 for 239 ft. to Station 160 + 99
(Maple St.), grade 39.10."
250. The Datum Plane. — One of the first tasks of the
surveyor in laying out a town site is to establish a datum plane to
which all elevations may be referred. It is customary to choose
a datum that bears an intimate relation to the topography of the
locality. For example, if the town is located on the seashore a
series of tidal observations may be taken to determine the mean
sea-level or mean low water either of which is often used as a
datum (Art. 237, p. 211). The mean level of lakes is used as
a datum for many inland cities. Frequently the elevation of
some point not far from the town site has been established by
the U. S. Geological Survey, the U. S. Coast and Geodetic Sur-
vey, or by the line of levels of a railroad ; and by careful level-
ing the elevation of some permanent point in the town site can
be established which will serve as the starting point for all the
elevations in the town. Where nothing of this sort is available,
the elevation of some point is found by barometer so that the
recorded elevation may approximate the actual height above sea-
level.
251. ESTABLISHING BENCH MARKS. — When the datum has
been determined, bench marks are established by the method
explained in Art. 219, p. 198. The establishment, at the start,
of a reliable system of bench marks is of utmost importance, in
order that the elevations of all parts of the city shall refer to the
same datum. In laying out construction work it is absolutely
necessary that bench marks which can be relied upon shall be
available and sufficiently numerous to be of use in any section
of the city without requiring several set-ups of the level to con-
nect a bench mark with the level work that is to be done.
WATER AND SEWER SYSTEMS 22/
Another advantage in having them close together is that they may
serve as ready checks on each other as well as on the work at
hand. It is not uncommon for a bench mark to be disturbed,
and, if the level work is not occasionally checked on some other
bench mark, an error will surely enter into all of the level work
which was started from that bench.
252. WATER AND SEWER SYSTEMS. — The water and sewer
systems of any community are of vital importance and provis-
ion for them must be made in the layout of every town site.
The location of the water supply and the storage and distributing
reservoirs is a matter of such magnitude that it cannot be dis-
cussed in this short treatise.* The conditions essential to an
economical water or sewer system will sometimes radically affect
the alignment and grades of many of the streets. The gradi-
ents of water pipes are of little importance since the water is
working under pressure, and the pipes can be laid uphill and
downhill so long as there is sufficient " head " to force the water
through the pipes.
In a sewer system the problem is far different ; every sewer
must have proper g^dients, and the entire system must fall
gradually from the most remote points to the main sewer outlet.
The topographic map therefore is of utmost importance as a basis
for a study of this problem, f
STAKING OUT CITY WORK.
253. STAKING OUT A NEW DISTRICT. — In staking out
a new district the information at hand is usually a plan of the
proposed layout of the streets which has been studied out in
the office from a map of the district. If this layout has been
approved by the municipal authorities the street lines as they
appear on the plan are the "established lines."
It is the surveyor's duty to stake out these lines on the
ground, connecting them properly with the street lines of the
* See Public Water Supply, by Toumeaure and Russell, published by John
Wiley & Sons, New York.
t See Sewerage, by Professor A. P. Folwell, published by John Wiley &
Sons, New York.
228 CITY SURVEYING [Chap. DC
older portion of the city, and in short, to produce on the ground
a layout exactly like that on the plan. Sometimes the angles
and distances necessary for the layout have been computed m
the office, but more frequently these are not determined until
the lines are laid out on the ground. In reproducing these lines
on the ground the surveyor will often find that the exact dimen-
sions given on the plan do not correspond with his fieldwork
owing probably to the fact that' his tape differs in length from
that used by the surveyor who made the original plan. In such
a case he must distribute the discrepancies (unless ihey are large
enough to indicate that a mistake has been made) in the proper
manner in his work.
Not infrequently the entire work is staked out from a plan
which has been made in the office, and the exact angles and
distances as determined in the field are recorded on this plan
which then goes to the proper authorities to be put in the form
of a city order. As soon as the plan is accepted the street lines
should be marked by monuments (Art. 254), so that there may
be no difficulty in retracing the lines as they were originally laid
out and accepted. If considerable grading work is to be done
in building the new streets it may not be practicable to set
many of the corner bounds at first on account of the likelihood
of their being disturbed. In such cases it is the duty of the
surveyor to properly reference the points by cross transit lines
or otherwise before construction work begins ; for it is important
that the layout, as recorded in the city order, shall be accurately
and definitely defined so that when the streets are brought to
the proper grade and the monuments are finally set they will
mark the exact position of the original layout.
254. MONUMENTS. — It is important and at the same time
customary to define street lines by setting stone bounds, often
called monuments^ at the street corners and at angles in the
street lines. The bounds are set sometimes on the side lines,
sometimes on the center lines, and sometimes in the sidewalks.
At street intersections, one monument at the intersection of
the center lines will suffice to mark both street lines, but since
this point will come in the center of the road pavement where it
is likely to be disturbed by traffic or by street repairing it is sel-
MONUMENTS 22$
dom placed there. The more practicable method is to defini
the street lines by marking the side lines at the angles or, in
the case of rounded corners, at the beginning and end of the
curves. It is not necessary that all four corners of a street
intersection shall be marked, as a bound on one corner will define
the side lines of the two streets and, the width of the streets
being known, the other sides can easily be determined. Nor
is it necessary to place a bound at one of the corners of every
street intersection, provided a street is straight for several blocks,
although it is good practice to do so. On account of the liability
of bounds which are placed on the side lines of the street being
disturbed by building operations, some surveyors prefer to place
them on an offset line, say 2 ft. from the street line. All mon-
uments should be placed with extreme care as regards both
their accuracy of position and their stability. If any bounds
are set with more care than others, they should be the ones
which occur at angle points in the street lines rather than the
intermediate bounds which are set along a straight line.
Monuments are usually roughly squared stone posts about 4
to 8 inches square and 3 to 4 feet long, the length depending
upon the severity of the climate, e.g., in New England a monu-
ment less than 4 ft. long is likely to be disturbed by frost action.
They are carefully squared on top and a drill-hole in this end
marks the exact point. This drill-hole may be made before the
stone is set in place, or after it has been placed so that its
center is about in position the exact point may be defined by
drilling a hole in the top of the bound. Frequently the hole is
filled with lead and a copper nail set in the lead is used to mark
the exact point. For nice definition of the point, a copper bolt
is inserted and two lines scratched across it ; the intersection
marks the exact point. When the stone bound is placed at the
intersection of the side lines of the streets it is sometimes
located entirely in the sidewalk in such a way that its inside
corner is exactly on the intersection of the street lines. In such
a case the three other corners of the bound are usually chipped
off so that there may be no mistake as to which corner defines
the line, but the line comer frequently becomes worn oflf and
this practice is therefore not recommended. Some surveyors
230
CITY SURVEYING
[Crap. DC
use, in the place of stone bounds, a piece of iron pipe or iron
plug with a punch-hole in the top of it, driven into the ground
or embedded in cement concrete. Long heavy stakes are
employed to temporarily define intermediate points or points of
secondary importance.
255. Setting Stone Bounds. ^— When the street lines are
laid out the corners are marked by tacks in the top of ordinary
wooden stakes. The monuments which are to take the place of
the stakes should be set before the frost has entered the ground
or before any other disturbance of the stakes has taken place.
When the bound is ready to be set the first thing to do is to
drive four temporary stakes around the corner stake about two
feet from it and in such a way that a line stretched from two
opposite stakes will pass over the tack in the head of the corner
stake (Fig. 99). Then tacks are carefully set in the tops of
these temporary stakes in
such positions that a stretch-
ing line running from the
tack on one stake to the
tack on the opposite stake
will pass exactly over the
tack in the corner stake.
Then the comer stake is
removed and the hole dug
for the stone bound. Care
should be taken not to
dig the hole any deeper
than is necessary so that the
bound may be set on firm
earth. As to the position of the top of the bound with refer-
ence to the surrounding ground, surveyors disagree. Some
prefer that the monument should stick out of the ground so
that it can be readily found ; while others claim that if it pro-
jects above the surface the bound is likely to become misplaced
by traffic, and therefore that it is better to set it just flush with
the ground or slightly below the natural surface. If any grad-
ing is to be done in the vicinity the bound should be set so that
it will conform to the proposed grade. When the hole for the
SHi-B-
Shntrching
I
B
E3r
Line
-BSHu
Stone Dound
Hk.
Fig. 99. Setting a Stone Bound.
SETTING STONE BOUNDS
231
w:^A
bound has been dug to the proper depth it is well to stretch the
strings across between the temporary stakes and plumb down
roughly into the hole to determine where the center of the
bound will come, so that when the monument is dropped into
the hole it can be placed so that it will set plumb.
The bound having been set in the hole, the next operation is
to fill around it. This should be done with considerable care,
the material being properly rammed as the filling proceeds and
the bound kept in such a position that the drill-hole in the top
of it, if there is one, shall be exactly under the intersection of
the strings. It is sometimes desirable to put in a foundation
of concrete and to fill with concrete around the monument to
within a foot of the surface, as shown in Fig. 100, where a
very substantial bound is re-
quired, or where the ground is
so soft as to furnish an insecure
foundation. If the top of the
bound is plain and the hole is
to be drilled after the bound
is in place, care should be taken
to place the monument so that
this hole will come practically
in the center of the top in order
that it may present a workman-
like appearance. After the
bound is set exactly in place the
temporary stakes are removed.
Some surveyors prefer to use only two opposite stakes and
one stretching line, the position of the monument being deter-
mined by a measurement along the stretching line from one or
both of the temporary stakes. Still another method of tempo-
rarily tying in the stone bound, and one which many surveyors
use, is to set two stakes such as A and B in Fig. 99, and
either measure the distance from them to the bound or set
them at some even distance from the bound. This process
of using temporary stakes and the stretching line is employed
also in setting other types of bounds such as gas pipes or
iron rods. .
W3^:
i»l
W^M
Fig. 100. Stone Bound with
Concrete Foundation.
232
CITY SURVEYING
[Chap. DC
In the construction of buildings or fences, monuments axe
frequently distCirbed and too often they are reset by the owner
of the property without the services of a surveyor. In rerunning
a street line, therefore, a surveyor should be on the lookout
for such conditions, and he should be cautious in the use of
any monument which he has any reason to suspect may have
been misplaced.
256. CURVED LAYOUTS. — It is not unusual for streets to
be laid out with curved lines. In the design of boulevards,
parks, and residential sections a landscape architect is often
called in and the plan he presents is sometimes almost devoid of
any straight street lines. (See Fig. loi.) The surveyor must
Fig. 101. Curved Layout for REsmENXiAL Part of a City.
take this plan and from the design there given stake out the
layout and obtain the necessary dimensions to definitely locate
all parts of it.
As a rule the landscape architect simply draws on the topo-
graphic map his scheme of layout with very few dimensions and
leaves the rest to be worked out by the surveyor. Occasionally
CIRCULAR CURVES 233
the radii of the curves are noted on the plan, but the street
widths are often the only dimensions given. If the radii are
not given the surveyor must determine from the plan either
these radii or some other distances, such as the tangent lengths,
so that he can go into the field, and, beginning with some known
street line, run out the new street lines in such a way that when
the data he determines are plotted the lines will coincide with
those on the plan prepared by the landscape architect. As a
rule these curved lines can be made up of a combmation of cir-
cular curves.
257. ELEMENTS OF A CIRCULAR CURVE.— Before consider-
ing how to stake out a curve it will be well first to refer to the
elements of a simple circular curve. In Fig. 102 which repre-
sents a simple circular curve
05=- Radius =/?
A HB = Length of Arc = L^
AB^ Long Chord =C
VA = F5 = Tangent Distance = T
VH = External Distance = E
HF == Middle Ordinate = M
I = Intersection Angle, or
Central Angle
V = Vertex
P.C. = Point of Curvature
P.r. = Point of Tangency fig. 102. Circular Curve.
From simple geometric and trigonometric relations,
2 K
T = i?tan -
2
Exsec ^ = £
2 R
E = R exsec -
2
Vers^ = ^
2 R
M = R vers-
2
Sinl=£.
2 2R
C= 2/2 sin -
2
L^—RX Circular measure of /.*
• The curves used in railroad engineering are usually very flat, so that there is
little difference between the chords and their corresponding arcs. This fact
234
CITY SURVEYING
[Chap. DC
258. STAKING OUT CIRCULAR CURVES In Fig. I02 the
two lines BC and EA are produced in the field and a point is set
at their intersection V, as described in Art. 200, p. 175. The
instrument is then set up at V
and the central angle / carefully
measured, or if point / is inacces-
sible other angles such as VEC
and VCE may be measured from
which / can be easily computed.
Then the radius R which is deter-
mined from the plan being known,
the tangent distance T is obtained
by the formula, T' = i? tan ^ /.
Points P.T. and P,C. are then set
and the curve is usually laid out
by the method of deflection angles as explained in the following
article.
259. DEFLECTION ANGLEa — A deflection angle is usually
referred to as an angle between a tangent and a chord, e.g., in
Fig. 103 angles VAh, VAc, etc., are deflection angles. Since
Fig. 102. Circular Curve.
Fig. 103. Deflection Angles.
makes it possible to compute the length 0/ curve by a simple approximate method,
which, however, is sufficiently exact for most railroad work.
The Degree of Curve^ which is the angle at the center subtended by a chord of
100 ft., 18 an element of the circular curve which is used extensively in railroad en-
DEFLECTION ANGLES 235
the angle between a tangent and a chord is measured by half the
included arc these deflection angles must be equal to half the
angle at the center subtended by the same chord or arc.
If the total length of the curve is divided into an even num-
ber of parts, «, the angle at the center under each of these arcs
will be -, and the deflection angle for one chord will be — ,
n 2n
which in Fig. 103 is the angle VAh, Angle hAc = angle VAh^
both being measured by one-half of equal arcs. It follows then
that the deflection angle to point
/ /
^ = 2 X — = -
2n n
2n 2n
I 2/
^ = 4 X — = —
2« n
etc.
Evidently, after the first deflection VAb is found, the
other ddSections can be obtained by simply adding the incre-
ment — to the preceding deflection angle, and this is the
2H
method which should be used. The deflection angle from the
P. C. to the P. T, should be equal to - , and this check should
always be applied to the computations before they are used in
laying out the curve.
The chords Ab, be, cdy etc. are equal since their arcs are
equal. With the radius and the central angle [ - for one chord]
given, the chord length can readily be found from the formula,
gineering. The central angle divided by the degree of curve will give the number
of loo-ft. chords in the length of the curve, i-c.-^ = Z ( in loo-ft. stations).
Therefore L (in feet) -■ -^ • For a complete discusaon of railroad curves see
** Railroad Carves and Earthwork/' by Professor C. F. Allen, published by Spon
& Chamberlain, New York.
^26 CITY SURVEYING [Chap. DC
c = 2 sin — Since the angle at the center is usually small
and the radius large the angle will have to be carried out in
some instances much closer than to the nearest minute in order
that the length of the chord may be obtained to hundredths of
a foot (Art. 371, p. 341). An approximate value for the chord
length corresponding to a given arc may be obtained by the
approximate formula,
^ C *
, or =
24i?«' 24/e*
in which 4 is the length of the arc, c is the chord length, and
R the radius.
The fieldbooks in use by most surveyors contain tables of
chords and corresponding arcs for curves of different radii,
which assist greatly in shortening these computations.
When the deflection angles have been computed and checked
and the chord length found, the instrument is set up at A,
(Fig. 103) a foresight taken on the vertex with the vernier
reading o^ and the point 6 set by measuring Ab and placing d
on line by means of the transit on which the first deflection
angle VAb has been laid off. Point c is set by measuring 6c
and placing c on line with the transit on which the second
deflection angle has been laid off, and so on, until the last point
(P. T.) has been set.
It is evident that with the transit at the P.C. the curve
could have been laid out just as well by taking the measurements
from the P. T, end, and some surveyors prefer to do it this way.
Similarly the instrument might just as well have been set up at
the P.T. instead of the P.C. and the measurements started from
the P.C. if it were found to be more convenient.
* The following will give some idea of the accaracy of this formula.
With R -■ 100 and /• — 25, the formula gives c — 25.065, (correct value
is 25.066).
With R « 100 and /• — 50, the formu a gives c — 50.521, (correct value
is 50.536).
With R « 1000 and /« » 100, the formula gives c « 100.042, (correct value
is 100.042).
. . .This formula will be found very useful if a slide rule is employed for the
computation.
LAYING OUT CIRCULAR CURVES
^37
It is sometimes necessary to set definite station points on the
curve rather than to cut the curve up into several equal parts as
suggested above. The principle is exactly the same as described
above ; but in figuring the deflection angles and the chord lengths
to be used the computations are not quite so simple. No trouble
will be experienced, however, if it is borne in mind that the total
deflection angle to any point is equal to half the central angle to
that point from the P.C.y and that the central angle for any arc
bears the same relation to the entire central angle that the arc
does to the entire length of curve.
260, Keeping the Notes. — In a curved street the notes of
alignment generally refer to the center line, the two side lines
being parallel to the center line. All three of these lines have
to be nm out by the use of chords and deflection angles ; Fig.
104 is an example of a concise form of notes for this work. In
^CMTWI
Station
Oishmee
(An:)
Chords
UA- I Ufiher I ^y>»/
DthbcfiMt
Ait^MS
nuKfrK
ft*200
T'96j66
t8*SZj50
l7t22M
/6^72AZ
30jOB
Jif,oo
SOjOO
S>jOO
S&59
.S3J9
SSS9
3(kQS
49,87
49.87
70f€t.
24J9
4/.I4
4StS7
4IM
4/.f4.
2SU7'-4Cr
7'09^40
i?7
/?C.
Fig. 104. Notes of a Circular Curve.
the first column is a description of the curve, which refers to
the center line of the street. This particular curve is marked
" To Right " meaning that it deflects to the right while passing
around it in the direction in which the stations run! In the
third column are the distances measured on the actual arc along
the center line. The next three columns headed " Chords " are
the chord measurements across the curve from station to station
on the left side line, the center line, and the right side line of
the street, the terms left and right meanmg left and right look-
ing in the direction in which the stations run. In the column
headed"*' Deflection Angles '* are the total deflections to be laid
off with the instrument set up at theP.C. These same deflection
238
CITY SURVEYING
[Chap. DC
angles are used in running out the side lines for the chords which
have been computed for the side lines run between points which
are radially opposite the corresponding points on the center line.
The computation of these notes will be found in Art. 371, p. 341.
261. When the Entire Curve Cannot be Laid Out from One
End. — It is often impossible to see from the P.C* to the P, T,
of a curve on account of intervening obstructions. In such a
case the curve is run from the P.C. as far as is practicable and a
point is carefully set on the curve ; then the transit is brought
forward and set up at the point thus fixed, and the curve extended
beyond. There are two different methods employed in this case.
262. First Method. — Assume the circular curve in Fig.
105 to be laid out from A to d as described above. Point d is
Fig. 106. Intermediate Set-up on Curve,
carefully set and the instrument then taken to that point and
set up. The vernier is turned back to o*^ and beyond 0° by the
value of the deflection angle VAd, Then by using the lower
clamp and tangent screw the telescope is sighted on point A.
The upper plate is then undamped and, if the telescope is turned
so that the arc reads 0° the instrument will be pointing along
the direction of an auxiliary tangent df, for angles VAd and Adf
are equal. It is well to note whether the instrument appears
to point in the direction of the tangent. Then reverse the tele-
scope, set off on the vernier the angle gde = — , and lay out the
2H
INTERMEDIATE SET-UPS.
239
curve from dtoB just as though it were an independent curve
beginning at d and ending at B.
263. Second Method. — When the transit has been set
up at dy the vernier is set at 0° and a backsight taken on A,
Then an angle equal to the deflection angle VAe is laid off on
the arc; this will cause the telescope to point in some such
direction as dA. The line of sight is reversed and point e set on
//^/ produced, making the chord de of the proper length. Then
point B is set by laying off on the vernier an angle equal to VAB
and measuring the chord eB. This method is correct for
VAe^ VAd+dAe
^fdA + kdft being measured by half of equal arcs.
This second method is sometimes to be preferred since the
original deflection angles figured can be used throughout the
curve. The first method calls for the calculation of a few more
angles ; but this is so simple a process that there is probably
little choice between the two methods.
264. CURVED STREET CORNERS. — It is the practice in many
cities to curve the corners of the streets by introducing a cir-
cular curve of short radius. Where both street lines are straight
the problem is handled as explained in Art. 258, p. 234.
265. * One Street
Line Straight, the Other
Curved. — In Fig. 106
the curved street line
DEF intersects the
straight street line
A V and at this point
the circular curve whose
center is C and with a
given radius r is to be in-
troduced ta round off the
corner. It is required
to stake out the curve
GE on the ground. In
Fig.
• The authors are indebted to I. T. Famham» City Engineer of Newton,
Mass., for the solution of the problems given in Arts. 265-6,
240
CITY SURVEYING
[Chap. DC
the field any tangent line, such as FV, is run ofF from some
known point on the curve and intersected with A V, and the
angle fi and the distance FV^lvq measured. In the right triangle
CT^Fin which R and T'are known, compute angle a and distance
CV. In the right triangle CAVyC Fand 7 = ^ — a being known,
compute CA and AV\ CA' = CA -r; CC ^ R^ r. In the
right triangle CA'C\ CA' and CC being known, compute A'C
and A'CC = GCE.
Angle ilCF - 1 80° — p.
Angle ECF^ACF —
A'CC\ from which the
length of the arc FE
can be readily com-
puted, which locates the
point E. VG^ A F—
A'C J which locates point
G of the curve GE^ and
any intermediate points
can be located as ex-
plained in the previous
articles.
As the radius OE is often quite short the center of the
curve can be located from either its P,C, or P.T. or both, and
any intermediate points on the curve can be easily swung in
from its center.
266. Both Street Lines Curved. — In Fig. 107 the two curved
street lines ABD and A^B'T/ intersect each other and the curve
whose center is E and with a g^ven radius r is introduced at the
intersection of the two street lines. It is required to locate the
curve B*B on the ground. In the field the tangent DV
is run off from some known point D on the curve ABD and
intersected with a tangent jy Ffrom the curve A'SU and angle
a and distances T and T' are measured. In the right triangle
CDVy R and T being known, compute angle C7Z) and distance
CV. Similarly in the triangle CVD' compute angle C'VIV and
distance C^. In the oblique triangle CYC, C7* CV and
angle CVC = 360° — (a + CVD + CVD') being known, com-
pute CC and the angle CC'F and CCV, In the oblique tri-
FiG. 106.
ROUNDING STREET CORNERS
241
angle CCE, CE -:- R + r, CE ^ R' -- r, and CC being known,
compute the angle CCE, CCE and CEd which is the sup-
plement of the central angle of the curve BB. Angle
DCB = DCV+ VCC — CCE, from which arc DB can be com-
Fig. 107.
puted. Similarly angle lyCW = D'CV + VCC - CCE, from
which arc D'B' is computed. These locate the P.C. and P.T.
of the small curve whose center is E.
267. STAKING OUT STREET GRADES. — The fieldwork neces-
sary in setting grade stakes is explained in Arts. 235-6, p. 210.
When new streets are constructed the excavation or embankment
is first brought to sub-grade, i.e., to the grade of the bottom of the
road covering or pavement. The grade stakes set for this work are
usually the center and the two side slope stakes, properly marked
with the cut or fill, as described in Arts. 230-2, pp. 207-10.
As the work progresses the center stake is dug out or covered
up and when the construction has progressed nearly to the sub-
242
CITY SURVEYING
[Chap. DC.
grade it is customary to set stakes at the elevation of the sub-
grade along the center line and on each side line of the street.
268. Vertical Curves. — Where the rate of grade of a street
changes, in order to avoid an abrupt transition from one grade to
the other, a vertical curve is introduced which is tangent to both
grade lines. The simplest curve to locate for this purpose is the
parabola.
In Fig. 108 L f^and KAf represent two grade lines intersect-
ing at V. The parabola AHB is tangent to these lines at A and
B. It is often customary to set the grade stakes on a vertical
Fig. 108.
Vertical Parabolic Curve.
curve at every 25-ft. station ; in such a case, then, Fig. 108 rep-
resents a vertical curve 200 ft. long on which the elevation of
nine points must be determined. The equation of the parabola is
^2 _ 4 p^^ or j' = (constant) Xy
(0
the jr dimensions being parallel to VK (vertical) and the^ dimen-
sions being along A V, From the equation it is readily seen that
the offsets from the tangent vary as the squares of the distances
along the tangent, oxx^'.x^^^ y'^ y}. The lines VP and NQ are
vertical and AQ is horizontal. Since the curve extends an equal
distance each side of V, AP =^ PQ ; and therefore AK =^ KB.
NB^aVH; VIf=4CC; CC = 4DD'; etc. (from equation i.)
Let ^ and g-^ represent the rate of grade of LV and VM,
and n the number of 25-ft. stations (in this case 4) on each side
of the vertex V, then
NB^-U-hg,)
4
KV =
NB
(from similar triangles)
VERTICAL CURVES 243
but NB = 4 VH (from above)
therefore KV = 2HV,
or point ff is midway between V and K.
The elevation of f^is determined from the established grade.
The number of 25-ft. stations will determine the distance VA
and VB. The elevation of A and of B can be readily computed
along their respective straight grade lines.
Elev./r = ?l?X-^+Elev.B
Elev. If =
2
Elev. V+ Elev. /i:
2
Fif == Elev. /T- Elev. F.
Elevations of all the other intermediate points along the curve can
be computed by finding the elevation of the points D\ C\ £',
Fy G', and J/ and by adding to these elevations the ordinates
D'A CCET £, etc.
VH
lyn = rj =
CC = GG =
EE =^ FF =
16
VH
4
16
269. CROSS-SECTION OF STREET. — On account of the ne-
cessity for draining the surface of a road the center is raised or
" crowned" above the grade of the gutters by an amount depend-
ing on various conditions. The shape of the road surface is
sometimes two planes, running straight from the gutter to a
summit or ridge in the center of the street, this ridge being
rounded off by rolling ; but more frequently it is a curved sur-
face in the form of a parabola or a circle. The ordinary width
and crown of streets are such that the parabola and the circle
are practically coincident.
When a street is to be paved the curbstones are first set to
proper line and grade, then stakes are set for the finished grade
of the roadway. The center grade stake is frequently the only
244 CITY SURVEYING [Chap. DC
grade given and va templet, or form, which can be set on the
curbs and on this center stake is used to give the form of the
cross-section. The form of the teftiplet for this work is laid
out by the surveyor. If no templet is used he should put
in intermediate grade stakes betVeen the center and the curb
lines. In either case the surveyor, triust compute the necessary
ordinates to give the proper shape to the surface.
Usually the mean transverse slope of the pavement is given
either in the form of a ratio thus :
•hM T CI Crown I
Mean Transverse Slope =
Half the Width of Carriageway 30
or. Mean Transverse Slope = f ' per ft.
270. Gutters at Same Elevation. — Fig. 109 represents the
"< w >
Fig. 109. Cross-Section of Pavement ; Gutters at same Elevation
cross-section of a pavement and sidewalks. The crown AB is
computed from the mean transverse slope and the width of the
pavement.
The ordinate Diy at any other point on the parabola
= CC X -TTTzr since in a parabola the offsets from a tangent
Q
vary as the square of the distance out along the tangent (Art.
268, p. 242). But CC = AB ; hence, if D' is half-way from
AB
the center to the curb, DD^ = .
4
271. One Gutter Higher than the Other. — When one gutter
is higher than the other the following application of the parabola
CROSS-SECTION OF PAVEMENT 245
can be used In Fig. 1 10 the maximum ordinate .r is at a dis-
FiG. 110. Cross-Section of Pavement: One Gutter Higher
THAN THE OTHER.
tance a from the lower gutter. The first step is to find this
distance a and then x is readily found from the mean transverse
X
slope since - = Mean transverse slope. When x is found the
other offsets can be computed as explained in the previous article.
At A the offset = -
4
at 5 = JT— *
^ x-b
at C=
4
The width of pavement, the difference in elevation of the
gutters, and the mean transverse slope being given, the formula
for a is derived as follows.
In Fig. no, W ^ width of pavement.
R -■ radius of the circular curve DACB.
a ^ distance from the line of the lower gutter to the highest
point of the pavement.
/ i« distance from the line of the highest gutter to the highest
point of the pavement
b mm difference in elevation between the two gutters.
/ » mean transverse slope, expressed as a ratio of crown to half
the width of pavement.
X i* difference in elevation between the lower gutter and the
highest point on the pavement.
x - £_ (i) (See (I) in foot-note, p. 339.)
2 R
and X ^b ^ — (2) (See (i) in foot-note, p. 339.)
2R
2R
246
CITY SURVEYING
[Chap. IX.
Combiiiing (i) and (2),a>
-/»-
zRb
{a+/)(a-
-/)-
zRb
a
-/-
2Rb
But a
+ /-
W
.*. a
-/-
zRb
W
(tf + /) + (a .
-/)-
'^+'^
2a «
^+'^
a «
IV Rb
T'^ W
From (i),
2-ff-
a*
(3)
But
- i- Mean transverse slope ■■ j
a
From (3),
.-. aJ? - eL . «
as s
ab
W Ys
a ^ — + _
2 ^
\ 2Ws]
2
2
2lVs
272. If, instead of assuming the mean transverse slope of
the pavement, the elevation of the center of the pavement D
(Fig. Ill) with respect to the elevation of A and B is assumed.
CROSS-SECTION OF PAVEMENT
247
then DG is readily found and the elevation of such points as E
or F, which are midway between D and the gutters, are com-
puted from the method explained in Art. 270, eE and fF both
being equal to -—1 .
4
Similarly, Elevation E = Elevation G +
Elevation F = Elevation H -h
Elevation J = Elevation K +
Elevation L == Elevation M +
4
4
7DC
16
tdc
16
etc.
373. IRREGULAR SHAPED BLOCKa — There is a wide vari-
ance of practice in the method of cutting up irregular shaped
blocks into lots. One good general rule in such cases is to give
J'
60
eo
60
60
60
60
./>^
8
^
^
k
r>J
^^
U^ -
s X>
>
y^^
l^
i^
Fig. 112. Arrangebcent of Lots in a Wedge-Shaped Block.
each lot as much street front as is possible consistent with mak-
ing the side lines of the lots at right angles to the street lines.
If the side lines do not run at right angles to the street there
will be portions of the lot which are not available for the custom-
ary rectangular style of building construction and which are
therefore not so desirable for business purposes. This is not of
248
CITY SURVEYING
[Chap. DC
SO much importance in residential districts where the rectangu-
lar system is often purposely avoided to some extent, to obtain
a layout which has an attractive appearance, as illustrated by
Fig. loi, p. 232.
Fig. 112 is an example of an irregular shaped block in which
rectangular lots have been planned, the wedge-shaped remnants
being thrown into the comer lots.
274. STAKING OUT City lots.— In staking out the lots of
a rectangular block, the comers of which have been established,
the most direct method is as follows. The transit is set up on
the S. B. at A, (Fig. 1 1 3)> a sight is taken on B, and the front
_ji
J
9
'»
f
0
90'
JO
M
•
ao
9-
SO
■
ae
M
M
tt
M
n
JO
n
90
V
t
i
4
9
«
7
•
»
»
II
«c
A
«>
SO
M
»•_
»
*»
ae
ao
30
-iUS
t
~i r
Fig. 113. Rectangular City Block.
comer stakes of lots i, 2, 3, 4, etc., are set, with a tack, exactly
on line, in the top of each stake. All such work should be done
to the nearest 0.0 1 ft. It will be well first to measure the line
ABf to see that it is just 600 ft. long. Since it is assumed that
considerable care was used in setting the S. Bs. exactly in the
correct position, if it. is to be found to be a few hundredths over
or under 600 ft., it is probable that this discrepancy is due to
the difference between the length of the tape on the present
.work and that, used in the original layout. In such a case the
twelve lots m\ist be laid out with equal frontages. For example.
STAKING OUT CITY LOTS 249
it may be a hot day when the lots are to be staked out and the
tape may give a distance from ^4 to 5 of 599.88 ft. In this case
each lot should measure 49.99 f^- wide.
With the instrument still at A and sighted on C, point D is
set by measuring 66 ft. from C, and then point E v% placed mid-
way between A and D. Whatever slight discrepancy there may
be in the distance between the S. B. at A and that at C is thrown
into the depth of the lots rather than the width of the street.
By setting up the instrument at B and sighting on H^ points
F and G are set. Then by setting up at F and sighting on D
the front comers of lots 1 3, 14, 1 5, etc., are determined. Another
set-up of the transit at G with the line of sight on E will allow
the "back bone" to be run out and the back corners of all the
lots established. The check on the lines ABy EG^ and DF is
their total length. The depth of the lots can easily be checked
by taking direct measurements from the front to their rear corner
stakes. If a further check is desired the transit can be set up
at each of the front corner stakes of the lots on one street and
a right angle turned off to check the position of the rear corner
stakes and the front corner stakes of the lots on the street be-
yond.
By the method suggested above the street lines are made
straight and the slight inaccuracies which may occur in the field-
work are put into the back and side lines of the lots.
Some surveyors prefer, after the front stakes on both streets
are located, to set up the instrument at each front comer and
locate the back corner stakes by tuming a 90® angle and laying
out the depth of the lot, at the same time checking the position
of the front stakes on the street on the other side of the block.
Then the distances along EG are measured to check this field-
work.
275. STAKING OUT CURB* LINES AND GRADES. — If the
line stakes which are set for the curbstones are placed directly
on the line of the curb they will be disturbed when the trench
is excavated. For this reason they are usually set in the side-
walk on an offset line, say, 3 ft. from the outside edge of the
* Called edgestones in some localities.
250 CITY SURVEYING [Chap. IX.
curb, arid at intervals of about 25 ft. The grade stakes are set at
about the sarae interval, with their tops at grade or at some even
distance (6 inches or i foot) above or below the grade of the
curb. Sometimes the grade stakes are not driven so that their
tops bear any relation to the finished grade, but a horizontal
chalkmark is made on the side of the stake marking the proper
grade. A stake can be marked much more quickly than the
top can be driven to the exact grade.
When new curbstones are being set in an old street, stakes
cannot as a rule be used. The sidewalks are too hard to permit
the driving of stakes, and even if they could be driven those
projecting above the surface of the sidewalk would be a source
of danger to pedestrians. In such cases it is customary to use
heavy spikes about 6" long. These are driven into the side-
walk on the offset line and the elevation of their tops determined
by leveling. The difference between the elevation of each spike
and the grade of the curb opposite it is calculated. A list of the
stations and the distances the spikes are above or below the curb
is given to the foreman in charge of the work. These distances
should always be transposed into feet and inches (to the nearest
J") before being given to the foreman, as it is seldom that the
men employed to lay the curbstones have any conception of the
meaning of tenths and hundredths of a foot. (See Art. 7,
p. 5.)
Where there are trees growing in line with the curbs, a nail
can sometimes be set in the side of a tree on the line of the curb
as well as at its grade. Points like these, of course, should be set
in preference to offset stakes or spikes wherever possible, as
there is little liability of the workmen misinterpreting such
marks. They can fasten their string directly to the nail and set
the curb to agree with it.
Before the curbstones are ordered the surveyor usually meas-
ures the distances between trees and locates driveways, and
then makes out a list of the lengths of straight, of curved, and of
chamfered stones (opposite driveways) to be used on the job.
This list is used in ordering the stones, and when they are de-
livered they should be found to fit the conditions without the
necessity of cutting any of them.
STAKING OUT CURBS AND SEWERS 25 1
276. STAKING OUT SEWERS. — The lines and grades of
sewers are sometimes run out in the same way as those described
for curbstones. The stakes or spikes (in bard paving) are set on
an offset line and the grades figured as described in Art. 275.
Another method which is extensively used is to spike out the
center line of the sewer and, from the profile of the street,
determine the depth of digging. When the excavation is com-
pleted the surveyor again runs out the center line and places batter-
boards at the proper grade and line. This eliminates the errors
which are likely to creep in during the leveling over from the
ofifset spikes as is done in the previous method.
277. STAKING OUT STREET RAILWAY TRACKS. — The
lines and gfrades for street railway tracks are given usually by the
use of an offset line of spikes. The spikes are frequently placed
on an offset line 5 ft. from the center, or on a line 3 ft. from the
gauge of the nearer rail, and at every 50-ft. station or oftener.
The diflFerences between the desired elevation of the track and
the spikes is calculated, and this information is given to the fore-
man in charge, usually in the form of printed "grade sheets."
278. RERUNNING STREET LINES AND GRADES. — There is
a constant call for lines and grades of streets. All kinds of
work, such as the construction of fences, buildings, and street
improvements, call for rerunning the street lines and grades.
The work of running out the line is simple enough if the
original S. Bs. are in place. It is not uncommon, however, to
find that in excavating a cellar on a corner lot the corner bound
has been disturbed or that it has been removed entirely ; and
before the line can be properly staked out it may be necessary
to begin at some reliable S. B. farther down the street or even
on some other nearby street line.
When the line has finally been rerun it is customary to take
and record swing offsets from the corners of the underpining of
several of the buildings located along the street and near to the
line. By this record of offsets, then, this street line can very
easily and quickly be run out at any future time, and any dis-
turbance of the S. Bs, at the comers can readily be detected.
Several offsets to substantial buildings are often of more perma-
nent valne than stone bounds. In some offices these offsets to
252 CITY SURVEYING [Chap. DC
buildings are recorded directly on the street plans. Whenever
a street line or grade is rerun full note should be made showing
all measurements taken for determining the lines or grades.
Sometimes the original street lines have been so completely
obliterated that it is necessary to resurvey them and make a new
record plan and description of them and have these new lines
"established" by a city ordinance. Such work, for example,
has been done by the City of Providence since 1857 when a state
law was passed requiring that accurate street lines be marked
where the adjacent land was about to be built upon. To
properly carry out this law the resurvey of a number of the
principal streets was required and the policy then originated has
been continued.
When a new building is to be constructed the owner generally
requests the City Engineer to define the street grade in front of
his property. The surveyor who has charge of this work goes
to the place and levels from the nearest B. M. to the site of the
new building. He has in his possession the established grade
of the street and its cross-section. From these he can compute
the elevation of the sidewalk grade at those points along the
street line where the grades are desired. On the fence or on
stakes set on the side line of the street he marks the grade of
the sidewalk at the property line, usually to a hundredth of a foot.
279. REVISHfG STREET LUfES. — In older cities much is
being done toward straightening some of the crooked streets,
and widening the narrow streets. A survey of existing struc-
tures is made and plotted, and the new street lines are then
studied with reference to existing conditions. Several proposed
lines are sometimes considered and run out on the ground. The
line finally selected is carefully run out and offsets to existing
structures determined so that it may be definitely located, and
the areas of all property taken from each abutter are then
surveyed, computed, and described. This layout is then accepted
by city ordinance and the necessary construction is made in
accordance with the revision.
280. REYISIHG STREET GRADES. — Sometimes the estab.
lished grades of city streets have been laid down in the early
days of the city, and it is subsequently found that these grades
BATTER-BOARDS FOR BUILDINGS 253
need revision. In such a case the surveyor will make a profile
of the center line of the street, of each curb (if there are any)
and sometimes along the side lines of the street. He will also
take all necessary elevations on the steps of buildings which lie
near the street lines, and a few levels in the front yards of abut-
ting property. From a study of these grades together with a
plan of the street the new grade line is laid out so as to affect
existing property as little as possible. When this grade line has
been accepted it is run out in the usual manner and the street
regraded. Stakes for final grading are set to hundredths of a foot.
281. SETTING Batter-Boards for a BuiLDmG. — One
of the most common tasks of the surveyor is to set the batter-
boards for the excavation and construction of the cellar of a new
building. The dimensions of the building and the elevation at
which to set it are usually obtained from the architect, although
sometimes the elevation of the ground floor of the building is
recorded on the plan itself. In a brick or stone building the
lines to be defined are the outside neat lines of the building, and
the elevation desired is usually the top of the first floor. In the
case of a wooden building the line usually given is the outside
line of the brick or stone imderpinning and the elevation given
is the top of this underpinning on which the sill of the house is
to rest. Sometimes the outside line of the sill is desired instead
of the outside line of the underpinning. There should be a
definite understanding in regard to these points before the work
of staking out is begun.
Generally there is no elevation marked on the plan and the
surveyor is simply told to set the top of underpining a certain
distance above the sidewalk or above the surface of some portion
of the lot. If there is an elevation referred to City Datum
marked on the plan, he should level from the nearest B. M. and
set the batter-boards at the grade given.
The location of the building on the lot is given either by
plan or by orders from the architect or owner. Not infre-
quently the surveyor receives the directions to place the build-
ing so tliat its front line is on line with the other buildings on
the street and so that it will stand a certain number of feet from
one of the side lines of the lot.
254
CITY SURVEYING
[Chap. DC
His first work is to stake out the location of the building by
accurately setting temporary stakes at all of the corners of the
building, e.g., in Fig. 114, at A^ B, C, A ^> and F, A stake
should be set at G also
so that the entire work
can be checked by meas-
uring the diagonals AG
and FB, and GD Ind EC.
These checks should
always be applied where
possible. Then the posts
for the batter-boards are
driven into the ground 3
or 4 ft. outside the line of
the cellar so that they
will not be disturbed
when the-walls are being
constructed. On these
posts, which are usually
of 2" X 4" scantling, i"
boards are nailed. These
boards are set by the sur-
veyor so that their top
edges are level with the
grade of the top of the
underpinning or for whatever other part of the building he is
giving grades. After the batter-boards are all in place they
should be checked roughly by sighting across them ; they should
all appear at the same level. Sometimes, however, on account
of the slope of the ground some of them have to be set a definite
number of feet above or below grade.
Then the lines are to be marked by nails driven in the top
of these batter-boards. The transit is set up on one of the
comer stakes of the house at A (Fig. 1 14), for example, and a
sight is taken on F. This line is then marked on the batter-
board beyond (at/) and on the one near the transit (at a). If
the batter-board is so near the transit that the telescope cannot
be focused on it, then point a can be set within a hundredth
Setting Batter-Boards for
A Building.
BATTER-BOARDS FOR BUILDINGS 255
of a foot by eye if the surveyor will stand outside of the
batter-board and sight point ^i in a line determined by point
/and the plumb-line on the instrument. Then a sight is taken
along AB and this line is produced both ways and nails set on
the batter-boards at h and/. In a similar manner all of the lines
are marked on the batters. These points should be marked
with nails driven in the top edges of the batter-boards and
there should be some lettering on the boards to make clear
which lines have been given. It is well for the surveyor also
to show these marks to the builder or inspector and have it
clearly understood just what parts of the structure these lines
and grades govern.
It is customary to set batters for the jogs in the building as
well as for the main corners ; but small bay windows of dwell-
ings are not usually staked out, but are constructed from
wooden patterns made and set by the builder.
As soon as the excavation is begun the corner stakes are
dug out and the building lines are then obtained by stretching
Unes between the nails in the opposite batter-boards. These
batter-boards are preserved until the sills or first floor are in
place, when they may be removed.
382. dry Plans AIID records. — Every city has a large
number of valuable plans and records in its possession. Too
frequently these are not kept with anything like the care con-
sistent with the amount of money that has been expended to
obtain them. For suggestions regarding the filing and index-
ing of pkns and records see Arts. 483-7, pp. 431-3.
RECTANGULAR COORDHf ATE SYSTEM OF SURVEYING CITIES.
283. GENERAL DESCRIPTION. — It is customary to disregard
the effect of curvature of the earth in the survey of a city on
account of its limited extent, and to use a system of rectangular
coordinates based upon plane surveying. In a coordinate system
two arbitrary lines are chosen for coordinate axes, one usually
coinciding with some meridian and the other at right angles to it.
All points in the city are located by distances from these two
axes, these distances being known as X's and Y's, or sometimes
B56 CITY SURVEYING [Chap. K.
as latitudes and longitudes. The axes are sometimes chosen
entirely outside the area to be surveyed, and where they meet
(their origin) is designated as (o, o.). Sometimes they are taken
through some conspicuous point, such as the tower of the city
hall, and are considered as being certain distances from the zero
lines as (lo ooo, lo ooo). By either of these arrangements
negative values for coordinates are avoided. The coordinates are
usually considered positive toward the north and the east, in
accordance with the custom of analytic geometry, as is the case
in ordinary land surveying. The convergence of the meridians
is neglected and all points having the same X coordinate there-
fore lie on a straight line parallel to the initial meridian and
are not all on the same true meridian line.
In the survey of the city of Baltimore (Fig. 115) the origin
of coordinates was taken through the Washington Monument
in the central part of the city, and the map divided into squares
1000 feet on a side. Each square mile is shown on a separate
page of the atlas of the city and these squares are designated
by their number north or south, and east or west of the origin,
as 1S2W, 3N4E, etc. Any point is designated by the distance
in feet north or south, and east or west, as (icxx) E, 2000 N).
One of the chief advantages of any coordinate system is that
if any point is lost it can be exactly replaced by means of the
known coordinates. This would be especially true in case a
large section of the city were destroyed by fire.
284. TriaNGULATION scheme. — The principal points of
the survey are usually located by a system of triangulation.
Prominent points are selected in such positions that the lines
joining them form well shaped triangles, i.e., preferably triangles
which are not far from jequilateral. These points may be signals
on tops of hills, church spires, and the like. If the cupola of the
city hall, or some such point is chosen as the origin of coordi-
nates it should also be one of the triangulation points. Points
which can be occupied by an instrument are in general to be
preferred. Such points as steeples or flag poles are definite
enough, but where no definite object exists on which to sight
the instrument signals are erected for this purpose. Such a
signal usually consists of a pole placed carefully over the exact
FlO. 116. TrI ANGULATION SCHEHB f
(Printed by pennission of Major Joseph W. Shirley, Chief E&g
Survey op the city of Baltimore.
tr of the Topographical Survey Commission, Baltimore, Md.)
MEASUREMENT OF BASE-LINE 26 1
point and braced in a vertical position by other poles forming
a tripod.
The system of triangles should cover the entire area but
should not contain more lines than are necessary to establish
a sufficient number of points to control the subsequent work
of the survey.
285. MEASUREMENT OF BASE-LINE. — At least one line in
the system must be chosen where its length can be very ac-
curately measured; this is called the base-line. The lengths of
all the other lines are to be computed from this line by means
of the measured angles, hence it will be seen how important
it is that this line should be measured with great accuracy,
and that it should also form a well shaped triangle with the
connecting triangulation stations.
It should be chosen if possible in some level spot where there
are no serious obstacles to the measurement. It is sometimes
an advantage to have the ends of the base-line slightly elevated
above the general level. The base should be measured with a
steel tape the exact length of which is known. The tension
should be kept constant by means of a spring balance, and the
temperature carefully taken. If the work is done on a cloudy or
rainy day the thermometer readings will represent the tempera-
ture of the tape much more nearly than when taken in sunshine.
The points should be lined in with a transit and the tape held
horizontal, or, if the measurements are taken directly from stake
to stake, the slope should be determined, by means of a leveling
instrument. There should be at least two independent measure-
ments of the line.
286. MEASUREMENT OF ANGLES. — If possible all of the
angles of each triangle should be measured by repetition. An
"inverting" instrument reading to 20" or to 10" is to be pre-
ferred for this work. The angles are repeated at least six times
with the telescope direct and the same number of times with
the telescope inverted. Several of these sets of readings are
made beginning each time with a different initial setting on the
circle. For example, if the first setting was at 0° and four sets are
to be taken the second would begin with a setting of 90®, and
so on. In each case both verniers should be read and the mean
262 CITY SURVEYING [Chap. DC.
of the two taken. Sometimes the direction of the measurement
is changed during the set, the first six repetitions being taken
from left to right, and the second six from right to left. In this
work it is important that the instrument should be carefully
centered over the point, and that the signals are also carefully
centered It is also important to keep the instrument carefully
leveled, especially if there is great difference in the angular ele-
vation of the points sighted.
287. Adjustment of the Angles. — The test of the accuracy
of the angle measurements is in the " closure " of the triangles.
In good work the sum of the angles of a triangle should not
differ from 180® by more than about 5 seconds, under fair con-
ditions. After the angles have been measured the errors in the
closure of the triangles should be distributed equally among the
angles, thus making the sum of the angles in each triangle ex-
actly equal to i8o<*. If the best results are desired all of the
discrepancies due to errors of measurement can be removed by
adjusting the system in accordance with the " Method of Least
Squares." In ordinary work, however, where the errors have
been kept small, the expense of such a computation is not war-
ranted. After all of the angles have been corrected the sides
of the triangles may be computed.
288. Azimuth. — If the coordinate lines are to run N and
S and E and W it is necessary to know the astronomical azimuth
of at least one line of the triangulation system before the
coordinates can be computed. This may be determined by
observation on Polaris as described in Chapter VII, or, in case
there are other triangulation points already established in the
vicirifity, the new system can be connected with them and the
azimuths computed from one of these lines. Azimuths are
reckoned in this work from lines parallel to the initial meri-
dian, from the south point right-handed, i.e., in the direction
S-W-N-E, and from 0° to 360®. When the azimuth of one
line is known all of the others may be computed. With the
azimuth and length of each line known the difference of the
latitudes and departures, i.e., the difference of the Xs and Fs of
the ends can be found, and with the coordinates of some one
SECONDARY TRIANGXJLATION 263
point gi>^en, or assumed, the coordinates of all of the other
points can be computed as explained in Art. 410, p. 373.
289. SECONDARY AND TERTIARY TRIANGULATION.— After
the prmcipal triangles have been completed, forming a system of
control, smaller triangles are selected, locating a system of points
of lesser importance so far as the survey is concerned. This is
called the secondary system. Sometimes a third {or tertiary)
system is introduced, the triangles being still smaller. The ter-
tiary triangles are the ones that would be used for locating the
city boundaries, street comers, and important monuments.
It frequently happens that, owing to the large number of
angle measurements and the consequent accumulated error, the
lengths of the sides of the small triangles become much less
accurate than they would be if measured directly; and since
many of these lines naturally lie in places where the distance
can easily be measured, this measurement should be made
as a check, in which case this line becomes a secondary
base-line. It is a good plan to introduce these measurements
frequently, where it can be conveniently done without great
expense, in order to prevent the errors of the survey from
accumulating unnecessarily.
290. Traverses. — After all of the triangulation is com-
pleted the system is extended by running traverses with the
transit and tape, from one known point to another. The tri-
angulation points are regarded as fixed and the errors of closure
of the traverses are assumed to be entirely in the traverse sur-
veys, the traverses being made to fit in exactly between the
triangulation points.
All street lines, or parallel offset lines, are connected with
the coordinate system so that the azimuth of every street line in
the city may be known, and the coordinates of all important
points, such as street corners and lot comers, are computed.
291. METHOD OF LOCATraO PROPERTY LINES AND BUILD-
INGS. — Since the coordinates of the property cqmers are to be
computed it is advisable to locate them by angle and distance
from the transit points, for with these data the calculation of
the coordinates is simple. The buildings are located from the
transit line by methods explained in Chapter VI.
CHAPTER X.
TOPOaRAPHICAL SUKVlSYJma.
292. In making a survey for a topographical map the methods
used will depend upon the purpose for which the map is made
and the degree of accuracy which is required. But whatever the
purpose of the map may be it is not necessary to locate points
in the field more accurately than they can be represented on
paper, whereas in surveying for an area measurements are made
with far greater precision than would be necessary for the pur-
pose of plotting.
While most of the details of topographical surveying can be
filled in more economically by the use of the transit and stadia
or by the plane table it is thought best to describe here only
the more elementary methods, and to reserve the complete
treatment of the stadia and plane table for an advanced work.
293. TRIANGULATION for Control. — In all cases where the
area is large it will be advisable to use a system of triangulation
to control the survey, as this is the cheapest method of accurately
determining the relative position of a few points whfch are a con-
siderable distance apart. The details of this triangulation work
have already been described under the head of " RectanguIarCo-
ordinate System of Surveying Cities," Chapter IX. One line
of the survey, the base-line, must be carefully measured. The
precision with which the angles of all the triangles must be
measured depends upon the use to be made of the map. After
the principal triangulation points have been established then
positions are plotted on the map. This may be done conven-
iently by the method of rectangular coordinates described in
Art. 283, p. 255. The extension of the system to smaller systems
of triangles, called secondary and tertiary, may be made if neces-
sary. After the triangulation system has been extended far
264
TRIANGULATION 265
enough to furnish a sufficient number of points for controlling
the accuracy of the map, traverses, may be run wherever con-
venient or necessary for locating topographic details. In all
cases the traverses should be connected with the triangulation
points at frequent intervals in order that the relative positions
of all points may be kept as nearly correct as possible. Where
a high degree of accuracy is necessary these traverses should
be run with a transit and tape ; if, however, errors of a foot or
two would not be appreciable on the map it will be sufficiently
accurate to use the stadia method of measuring the distances
and thus save time.
294. LOCATION OF POnVTS FROM THE TRANSIT LINE.—
Where a tape is used for measuring the distances, such objects
as fences, walls, and buildings may be located as described in
Chapter VI, but it will not be necessary to make the measure-
ments with as great precision. Fig. 116 is a sample page of
notes of a topographical survey where the transit and tape were
used. On city plans, which are frequently drawn to a scale of
40 feet to an inch, a fraction of a foot can easily be shown. On
a topographic map the scale is often such that an error of a frac-
tion of a foot becomes insignificant in the side measurements
from the transit line, where such errors cannot accumulate. In
some cases it may be sufficient to obtain the distances by pacing,
and the angles or directions by means of a pocket compass. Lo-.
cations may frequently be checked by noting where range lines
intersect the transit line. In making a series of measurements
it is well to take each measurement with a little more precision
than is actually needed for plotting, in order to be sure that the
accumulated errors do not become too large.
In taking measurements the surveyor should constantly keep
in mind how the notes can be plotted ; this will often prevent
the omission of necessary measurements. No matter whether
an accurate or only a rough survey is desired check measure-
ments should be taken on all important lines.
29s. CONTOUR LINES. — There are two general systems of
representing on paper the form of the surface of the ground.
266
TOPOGRAPHICAL SURVEYING
[Chap. X.
]Jr-
Fig. lie. Field Notes of a Portion of Topographical Survey
WITH Transit and Tape. ,
HACHURES AND CONTOURS
267
In one of these systems (Fig. 117) slopes are represented by
hachure lines, i.e., lines which always run in the direction of the
steepest slope of the ground. In the other system (Fig. 118)
contour lines, lines joining points of equal elevation, are used.
In the latter system elevations may be read directly from the
map, and for this reason it is much more used by surveyors.
Fig. 117. Hachure Lines.
Fig. 118. Contour Lines.
A contour line is the intersection of a level surface with the
surface of the ground. A clearer conception of a contour line
may be obtained from the following. Imagine a valley, or de-
pression in the surface of the ground, partly filled with water.
The shore line of this body of water will then be a contour line,
since it is the intersection of a level surface with the surface of
the ground. If the water stands at an elevation of 50 feet the
shore line is the 50-ft. contour. If the surface of the water
were raised 5 feet the new shore line would then be the S5-ft.
contour. Contour lines if extended far enough will therefore
be closed curves, and all of the points on any one contour will
have the same elevation above the datum. It is customary to
take contours a whole number of feet above the datum, spacing
them in regard to height, so as to make the contour intervals
equal, e.g., a contour may be taken at every 5 feet or every 10
feet of elevation. Since the contours are equidistant in a verti-
cal direction their distance apart in a horizontal direction shows
the steepness of the slope.
268 TOPOGRAPHICAL SURVEYING [Chap. X.
Fig. 1 19 illustrates contour maps of simple solids.
CLEVAHON
Fig. 119.
cornouR map
Contour Maps of Simple Solids.
296. Characteristics of (Contours. — The chief characteristics
of contours are illustrated in Fig. 121, and may be summed up
as follows.
1. All points on anyone contour have the same elevation, as
at A.
2. Every contour closes on itself, either within or beyond
the limits of the map. In the latter case the contour line will
not end within the limits of the map but will run to the edge of
the map, as at B-
3. A contour which closes within the limits of the map in-
dicates either a summit or a depression. In depressions there
will usually be found a pond or a lake; but where there is.no
water the contours are usually marked in some way to indicate
a depression, as at C.
4. Contours can never cross each other except where there
is an overhanging cliff, iij which case there must be two inter-
sections, as at D. Such cases as this seldom occur.
CHARACTERISTICS OF CONTOURS 269
5. On a uniform slope contours are spaced equally, as at E.
6. On a plane surface they are straight and parallel to each
other, as at F.
7. In crossing a valley the contours run up the valley on one
side and, turning at the stream, run back on the other side, as
at G. Since the contours are always at right angles to the lines
of steepest slope they are at right angles to the thread of the
stream at the point of crossing.
8. Contours cross the ridge lines (watersheds) at right
angles, as at H.
Fio. 121. Illustrating Characteristics op Contours.
270
TOPOGRAPHICAL SURVEYING
[Chap. X.
Fig. 120, Contour Crossing a
Street.
Fig. 1 20 shows a contour across an ordinary city street with
sidewalks and curbstones,
the street being located on
a steep grade. In order to
trace out the position of a
contour it is necessary to
keep in mind that it is a
line all points on which are
at the same elevation. It
will be noticed that the con-
tour from A to B crosses
the sidewalk in a straight
line but not perpendicular
to the street line because the
sidewalk is sloped toward
the gutter. Turning at B
it runs straight along the face of the curbstone until it strikes
the gutter at Z?, and returns on the other side of the gutter
along the surface of the road, the point E being where it swings
around and travels back toward the other gutter. The other
half of the street is similar. If the center of the road is at
the same elevation as the top of the curb opposite, then E will
be opposite B. This illustrates how contours run around val-
leys (gutters) and ridges (crown of street).
If the side of the street to the right {HF) were at a lower
elevation than the left side then the contour at the point where
it crosses the gutter, /s would be farther up the road irom E^
le., the contour would be unsymmetrical, EF being longer
than DE,
297. RELATION BETWEEN CONTOUR MAP AND PROFILE. —
If a line is drawn across a contour map the profile of the surface
along that line may be constructed, since the points where the
contours are cut by the line are points of known elevation and
the horizontal distances between these points can be scaled or
projected from the map. The profile shown in Fig. 122 is con-
structed by first drawing, as a basis for the profile, equidistant
lines, corresponding to the contour interval, and parallel to AB.
From the points where AB cuts the contours lines are projected
PROFILE FROM A MAP
271
to the corresponding line on the profile. Conversely, if the pro-
files of a sufficient number of lines on the map are given it is pos-
sible to plot these lines on the map, mark the elevations, and
from these points to sketch the contours as described in Art.
301, p. 276.
PROFILE ON.LINLAB
Fig. 122. Profh^e Constructed from a Contour Map.
298. RELATION BETWEEN CONTOUR MAP AND SIDE ELE-
VATION OR PROJECTION. — A photograph of a landscape
represents approximately a side elevation of the country. To
272 TOPOGRAPHICAL SURVEYING [Chap. X.
construct such a projection from a contour map (Fig. 123), lines
projection on, plane ad
Fig. 123. Side Elevation Constructed prom a Contour Map.
are drawn perpendicular to AB, the plane of projection, and tan-
gent to the contours. These tangent points show the limits
between the visible and invisible portions of the landscape, the
observer being assumed to stand on the line AB and to look in
a direction perpendicular to ^45.
DRAINAGE AREAS 273
299. DRAINAGE AREAS. — The drainage area that supplies a
stream or pond is limited by the divide line which is a line
drawn on the ridges surrounding a depression as indicated by
the dotted line on Fig. 124. Since the perpendicular to the
contout at any point is the direction of steepest slope the direc-
tion in which water will flow at any point can be determined at
once by examining the contours. On the ridge there is a line
(its summit) on one side of which water will flow down one of
the slopes and on the other side of .which it will flow down the
other slope. This line is the divide line or watershed line.
If a dam were built as shown in Fig. 124, its elevation being
960 ft., the area actually flooded by the water at full height of
dam is the area included within the 960 ft. contour, which is in-
dicated by the shaded section. The drainage area for the por-
tion of the stream above the dam is the area included within the
heavy dotted line, which follows the line of the divide.
300. SKETCHING CONTOURS FROM STREAMS AND SUM-
MITS.— The present topography of some parts of the country
is due almost entirely to erosion by streams. Consequently the
position and fall of the streams give more information regarding
the position of the contours than any other topographic features.
If a definite position of the contours is desired it will be neces-
sary to obtain the elevation of a few governing points on the
ridges as well as the location and elevation of the streams, as
shown in Fig. 126. '
In sketching in contours from these data it should be borne
in mind that the contours cross the stream at right angles to its
thread and that they curve around from the hill on either side so
as to represent the valley of the stream. The contours are
farther apart at the top and bottom of the slope of an eroded
hill than near the middle, because in these portions the slope
is somewhat flatter. A stream is usually steeper near its source
than in the lower portion and therefore the contours are closer
together near the source. This is true of most cases but the
shape of the contours in any particular case will depend upon the
geological formation. Fig. 127 represents the same country as
Fig. 126 but with the contours sketched on it, following out the
general suggestions which have just been mentioned.
274
TOPOGRAPHICAL SURVEYING [Chap. X.
Fig. 124. Illustrating Flooded Area and Drainage Area.
CONTOUR SKETCHING
275
Fig. 126. Map Showing the Location and Elevation of Streams
AND Summits.
Fig. 127. Contours Sketched from the Data given in thk
m.ap above.
2y6
TOPOGRAPHICAL SURVEYING.
3ai. SKETCHING CONTOURS FROM KNOWN ELEVATIONS.^
— A portion of the country can be cross-sectioned as described
in Art. 227, p. 206, or profiles can be run on any desired lines
as explained in Art. 225, p. 203. From these known elevations
contours can be sketched by interpolation. This is usually done
by estimation and the principle involved is the same whether the
elevations were obtained by cross-sectioning or by profiles.
Fig. 125 illustrates how contours can be sketched from cross-
TRAVERSE UNC
Fig. 126. Contour Sketched for Cross-Section Notes.
section notes. The points at which elevations are taken in the
field should be so chosen that the slope of the ground is practi-
cally uniform between any two adjacent points. Then by simple
interpolation the contours may be accurately sketched. This
interpolation may be done by geometric construction, but for
most topographic work it is accurate enough to interpolate by
eye.
302. MISTAKES IN SKETCHING CONTOURS. — Fig. 1 28
shows several examples of impossible and incorrectly sketched
contours ; the streams are assumed to be correctly located. The
numbers on the figure refer to the tabulation made in Art. 296,
p. 268, and will assist in detecting the type of error present.
SKETCHING CONTOURS FROM CROSS-SECTIONS 277
Fig. 128. Contours INCORRECTLY Sketched.
303. Locating CoirroURS. — Contours are often most eco-
nomically located by means of the transit and stadia or by an
instrument called the plane table* In this chapter, however,
only those methods will be considered which call for the use of
the transit and tape.
* A discussion of the Stadia and the Plane Table does not come within the
province of this book.
278 TOPOGRAPHICAL SURVEYING. [Chap. X.
304. Locating Contours by Cross-Sections. — A very com-
mon as well as expensive method of locating contours is that of
taking cross-sections. Elevations on the surface of the ground
are usually taken to tenths of a foot. From these elevations
the contours may be sketched by interpolating between these
known elevations as explained in Art. 301. The accuracy may
be increased by taking a larger number of intermediate points.
The size of the squares used should depend upon the roughness
of the surface.
305. Locating Contours by Profiles. — In some cases where
the ground is fairly smooth it is sufficient to take a few profiles
on known lines, not necessarily at right angles to each other.
These lines are stationed and elevations are taken at every full
station and at the points of marked change in slope. From
these data the contours are sketched on the map by interpola-
tion as described in Art. 301.
306. Locating Points on the Contours. — Where the contour
interval is small, say one or two feet, and the topography is to
be determined with considerable accuracy, it is advisable to find,
in the field, points actually on the contours and thus avoid the
errors of interpolation. The rodman moves up or down the
slope until the rod-reading indicates that the foot of the rod is
on a contour. The position of the rod may then be located by
an angle and a distance from some known line, the distance
being taken with a tape.
307. Locating Contours by the Hand LeveL— A more rapid
but less accurate way of putting in contours is by means of the
hand level. The work is done by making profiles of lines whose
positions on the map are known. A point on some contour is
found in the following manner.
The first step to take is to measure to the nearest tenth of
a foot the distance from the ground to the eye of the leveler,
which may be, say, 5.4 ft. If the B. M. is at elevation 143.43
and it is desired to locate a point on the 140-ft. contour, the
rodman holds the rod (or a tape) on the B. M. while the leveler
attempts to place himself on the 140ft. contour. When he is
on the 140-ft. contour the elevation of his eye (H.I.) is 145.4
LOCATING CONTOURS BY THE HAND LEVEL 279
and the rod-reading at the B. M. must be 145.4 — 143*43 "=
1.97, or 2.0 to the nearest tenth of a foot. The leveler there-
fore travels along the line on which the point is to be located
until he reads 1.97 on the rod. His feet are then on the 140-
ft. contour, the position of which is located from some known
point on the line. Sometimes this is done by measurement and
sometimes by pacing. A point on the 145-ft. contour could
have been located first by applying the same principle, but if
the 140-ft. contour is established it is very easy to locate a
point on the i4S-ft. contour as follows. The distance from
the leveler's feet to his eye being 5.4 ft., if he stands on the
140-ft. contour and reads 0.4 ft. on the rod, the bottom of the
rod must be on the 145-ft. contour. By trial then the point is
found where the rod reads 0.4 ft.* Then the leveler walks up
the hill and, standing on the point just found, places the rodman
on the next higher contour by the same process.
In working down the hill to locate the 135-ft. contour, if the
leveler is standing on the 140-ft. contour, the rod will be on the
135-ft. contour when it reads 10.4 ft. Or, when the 140-ft.
contour has been found by the leveler the rodman comes forward
and holds the rod on this spot and the leveler backs down the
hill until he reads 0.4 ft. on the rod ; he is then standing on the
135-ft. contour. Some surveyors prefer to cut a stick just 5 ft.
long and hold the hand level on the top of it in taking sights.
The points thus found at regular contour elevations are then
plotted on the corresponding lines and the contours sketched by
joining points of equal elevation. Where the lines which are
profiled are far apart or where the country is very rough it is
frequently necessary to obtain the correct position of the con-
tours, to locate extra points on them between these profiled lines.
The extra points are located by right-angle offsets from the
lines. Most oi this work is plotted in the field upon paper ruled
in small squares to facilitate sketching. Where practicable it is
always well to sketch the contours in the field rather than in the
office.
* For very rough work sometimes the rod is not used, the leveler simply es-
timating where the rod-reading will come on the rodman*s body and placing him
so that his feet will be on the proper contour.
280 TOPOGRAPHICAL SURVEYING [Chap. X.
308. LOCATION OF STREAMS AND SHORE LINES. — Streams
or shore lines of ponds may be very rapidly located by stadia
measurements. If the shore lines are to be located by tape
measurements, however, a convenient way is to run a transit
line aproximately parallel to the general direction of the shore
line, and to take perpendicular offsets at regular intervals and
at all points where there is a marked change in the direction of
the shore line, as was done in the notes in Fig. 53, p. 104.
309. Contour problems. — There are many surveying
problems involving earthwork which can be worked out approxi-
mately by use of a contour map. As a rule the smaller the
contour interval, the more accurate will be the result of such
work. Contour studies occur in a variety of problems, so
numerous that it would be useless to attempt to cover the sub-
ject fully. Three typical problems, however, are illustrated and
explained ; and these contain the essential principles applicable
to practically all contour studies.
310. Example i. — (Fig. 129). Given a contour map, the
surface being represented by contours shown by full lines, a
plane (extended indefinitely) is passed through the straight lines
AB and CD, which are level and parallel, AB being at elevation
12.5 and CD being at elevation 40. It is required to find where
this plane intersects the surface, and to shade the portion which
is above the plane.
Since the proposed surface is a plane, contours on it will be
parallel to AB and CD, The elevations of AB and CD being
known, other contours, such as ef and gh, can be interpolated
between AB and CD. Their interval is made 5 ft. the same as
the contour interval for the original surface. Evidently the
point where any of these parallel lines crosses an original con-
tour of the same elevation, as 7, k, /, my or «, is a point on the
intersection of the plane with the surface. Joining these points
gives the line of intersection of the plane with the original sur-
face, which is indicated by the heavy full line on the figure.
Such points as y, j, or / are determined by interpolation. Inter-
mediate contours are drawn at one-foot intervals between the
original surface contours ; corresponding lines are interpolated
between the straight contours which show the plane ; additional
CONTOUR PROBLEMS
281
intersections obtained, and in this way the point / is determined.
Again it will be seen that point /, with reference to the parallel
straight contours, is at about 18.5; with reference to the original
Fig. 129.
contours, it will be seen that wt is about three-tenths of wr^ the
distance between contours, and this makes the elevation of point
/ equal to 18.5.
311. Example 2. — (Fig. 130.) Given a contour map which
ncludes a road, and on which the original contours are represented
by full lines. It is desired that all of the road between A and B
shall be visible from the ground at point C. Sketch on the map
and shade the portions which will have to be cut down to fulfill
this requirement. *
The general method of solving this problem is to sketch a
new set of contours on the map, which will represent a uniform
282
TOPOGRAPHICAL SURVEYING
[Chap. X.
slope from C to the nearer edge of the road. Everything that
is above the surface represented by these new contours must be
cut away.
First draw lines, such as Cay Cb^ and Cr, the points a, by and
c being points on the upper side of the road between which it
may be assumed that the slope is uniform (Art. 301, p. 276).
Along these lines interpolate points which will lie on the uniform
slope from C to the road and also on the regular 5 ft. intervals
which correspond to the contours. For example along the line Ca
Fig. 130.
from the summit which is at elevation 89 to the road at ^ which
is at elevation 55, there is a drop of 34 ft., or a little less than
7 contour intervals. Points ^, /, gy //, etc. are therefore plotted so
as to divide Ca into 9 equal parts. Similarly points /, 7, ky etc.
are plotted along the line C6, but the point by being at elevation
56, is plotted so that the distance ib is four-fifths of the other
distances ijyjky etc. When these points have been plotted on all
of the necessary diagonal lines, the contours representing a uni-
form slope from C to the road are sketched on the map as shown
by the dotted lines on the figure. The points, such as w, «, or
Ty where the new contours cut the old contours of equal elevation,
are points of " no cut and no fill.'* A line connecting these
CONTOUR PROBLEMS
283
points encloses portions of either cut or fill. The shaded por-
tions or the figure, where the new contours are nearer C than
the corresponding old ones, represent the portions where it
will be necessary to excavate to the surface represented by the
dotted contours. In the central portion of the figure, from point
c to /, the road can already be seen.
312. Example 3. — (Fig. 131.) Given a contour map on
which are shown the two side lines of a road, the contours being
represented by full lines. The road is to be built on a 4%
down grade starting at ^4 at elevation 55. Scale i inch =150
Fig. 131.
feet Side slopes of road to be i^ horizontal to i vertical. It
is desired to sketch the new contours on the slopes of the road,
to sketch on the map the top and foot of slopes, and to designate
the portion in embankment and the portion in excavation.
First, the new contours which are to cross the road are
plotted at abf cd, ef, gh. These will be 125 ft. apart, as a 4 %
grade falls 5 ft. in a distance of 125 ft. If the road is assumed
to be level on top, then these lines will cross the road at right
angles to its general direction as shown in the figure. From
points a and by on either edge of the road, the new contour
284
TOPOGRAPHICAL SURVEYING
[CttAP. x:
lines will follow along the slope, e.g., the line ao represents the
new 50 ft. contour. Where this contour ao passes point c it is
just 5 ft. above the road. Since the slope of the cut is ij to i,
then the distance cut from c must be U x 5 = 7.5 ft.; opposite
e it is 10 ft. below the road and similarly the distance out from
e must be 15 ft. Where this new 50 ft. contour meets the old
Fig. 131.
50 ft. contour at o, is a point at the top of the slope. Similarly
all of the new contour lines, which are represented on the
figure by dash lines are plotted and their intersections with the
corresponding contours of the original surface give points of " no
cut" or "no fill,'* or top of slope (in excavated portions) and
foot of slope (in embankment portions). These lines are shown
in the figure by heavy dotted lines. Where this heavy dotted
line crosses the road it marks a " no cut " and " no fill " line,
i.e., the road bed cuts the surface of the ground.
CHAPTER XL
MININa STTRVETINa'.*
313. GENERAL REMARKS. — In this chapter the limitations
and difl&culties met with in surveying a mine will be pointed out
and some of the instruments and methods generally used will
be described. As rocky and precipitous mountain regions are
more the home of metal mining than of any other industry, the
special diflSculties of surface' surveying in such localities will also
be considered. Lastly, the methods of establishing the bound-
aries of mining claims in United States territory will be briefly
described.
Two of the principal objects to be accomplished in accurate
mine surveying are the locating of the ownership boundaries
underground and the laying out of passageways so as to connect
with one another, thereby facilitating the working of the mine.
Such passages are usually highly inclined and while under con-
struction are called connections.
314. DEFINITIONS OF MINING TERMS. — The following
terms are in common use in mining surveying.
Adit. A horizontal underground passageway running from the
surface and used only for drainage and ventilation.
Apex. The trace of the intersection of the vein with the sur-
face of the undisturbed rock formation.
Compartment. One of the smaller passageways of a large shaft,
divided by timber partitions.
Connections. Passageways which are being driven from one ac-
cessible part of a mine to another.
Cross-cut. A horizontal passageway at right angles to or across
the direction of the deposit.
* This chapter was written by Blarney Stevens, M. Sc, Mining Engineer,
Ellamar, Alaska.
285
286 MINING SURVEYING [Chap. XI.
Dip. The inclination of the plane of the deposit to the
horizon.
Drift. A horizontal passageway along, or parallel to, the trend
of the deposit.
Heading. Any preliminary passageway driven to explore the
mine or to facilitate future operations.
Levels. Horizontal passageways run at regular intervals (verti-
cally) along the deposit for working the mine.
Manhole. A small passage from one level into the next level
above or below, or into stopes.
Mill-hole. A passage between a stope and a level through which
the ore is conveyed.
Outcrop. The portion of the vein where it intersects the surface
of the ground.
Pitch. The direction of an ore body (called a chimney or
chute) in an ore bearing body, sometimes expressed as an
azimuth.
Raise. A passage leading upwards from any portion of the
mine.
Shaft. A vertical or steeply inclined passage used in working the
mine.
Stopes. Rooms excavated, within the walls of the deposit and
above or below the leyels, for exploiting the mines.
Strike. The direction (bearing) of a horizontal line in the plane
of the deposit. The strike is always at right angles to the
dip.
SPuU. Timber running crosswise between the side walls of a
passageway.
Tunnel. A horizontal passageway from the surface to the mine.
Wall. The boundary between a highly inclined vein and the
rock each side of it. The upper wall is called the " hang-
ing wall " and the lower one the " foot wall."
Winze. A subsidiary shaft not starting from the surface.
MINING TRANSITS 287
MmuiO mSTRUMEinS.
Owing to the confined nature and steep inclination of many
of the passages through which survey lines have to be carried,
specially constructed instruments are necessary.
315. Mining transits. — in modem mining, all the
accurate angle measurements are taken with a transit, the details
being filled in with a miner^s dial or other light compass instru-
ment. Several forms of transit are designed for mining and
mountain work. The essentials are lightness and capability of
measuring accurate azimuths of nearly vertical or of very short
sights.
With an ordinary transit one cannot take a downward sight
more steeply inclined than 55® or 60° to the horizon. For
taking highly inclined sights various devices have been used by
which telescopic sights may be taken over the edge of the horizon-
tal circle of the instrument. This is commonly done by attach-
ing an auxiliary telescope, usually smaller than the main tele-
scope, to the side or to the top of the ordinary engineer's transit
so that the instrument will afford all the advantages of the ordi-
nary transit and also allow vertical sights to be taken.
316. SIDE TELESCOPE. — Fig. 132 shows a mining transit
in which the auxiliary telescope is attached to an end extension
of the horizontal axis. When this instrument is used the azi-
muths which are measiu-ed by means of the side telescope have
to be corrected for the eccentricity of this telescope. A striding
level is used to adjust the horizontal axis. This is a sensitive
spirit level having two V-shaped bearings so that it can be set on
top of the horizontal axis; it can be lifted and turned end for end.
317. Top Telescope. — in this type of mining transit the
auxiliary telescope is mounted on top of the main telescope.
Since this telescope is directly over the main telescope, azimuths
measured with the auxiliary telescope will be the same as though
they were measured by the use of the main telescope. But if
vertical angles are measured by means of the top telescope it will
be necessary to allow for the distance between the two telescopes.
318. Adjustments of Side Telescope. — It is assumed that
all ordinary adjustments of the transit have been made; in
288
MINING SURVEYING
[Chap. XL
Fig. 132. Mining Transit with Side Telescope.
(From the catalogue of C. L. Berger & Sons, by permission.)
ADJUSTMENT OF SIDE TELESCOPE 289
mining work the adjustment of the objective slide (Art. 77, p. 60)
is of unusual importance. The side telescope is generally ad-
justed by first making the line of sight parallel to the axis of the
telescope tube. This is done by the cross-hair adjustment and
the aid of a pair of fixed wyes in which the tube is rotated; it is
the same adjustment as for the level, Art. 121, p. 89. It is as-
sumed that the instrument maker has made the optical axis
parallel to the axis of the tube.
Secondly, the line of sight is made parallel to that of the main
telescope. It is first brought into a vertical plane parallel to the
vertical plane of the main telescope by means of the adjusting
screws on the vertical trivet plate of the side telescope, the sight of
each telescope being taken to the same very distant object. If it
is not convenient to sight on a distant object, a piece of paper
with two vertical marks connected by a horizontal line may be
used, the distance between the marks being equal to the distance
between the telescopes. This piece of paper should be set at
right angles to the line of sight and not too near the instrument.
The vertical cross-hair of the main telescope is sighted at one
point by means of the clamp and tangent screw of the plates and
then the vertical cross-hair of the side telescope is sighted at the
other point by means of the trivet plate adjustment on the side
telescope.
The side telescope and main telescope are then brought into
the same plane at right angles to the vertical plane as follows: —
the horizontal cross-hair of the main telescope is sighted at some
point, preferably a distant one; then the horizontal cross- hair of
the side telescope is sighted at the same point by means of the
tangent screws on the side telescope.
With this form of attachment a correction for eccentricity
of the side telescope is necessary in both azimuth and altitude
readings. The necessity for correction in azimuth may be elimi-
nated by using the instrument in both the direct and reversed
positions; reversing brings the side telescope to the other side of
the main telescope, thereby eliminating the errors of the line of
sight. The striding level should be used in both of its positions,
i.e., with the main telescope direct the striding level is used in
both positions and two azimuths are read, and with the telescope
290 MINING SURVEYING [Chap. XI.
inverted the striding level is again used in both positions and two
more azimuths are read. The mean of the two readings of each
pair gives two mean lines of sight which are symmetrically related
to the vertical plane passing through the two station points and
the correct azimuth reading is therefore the mean of these two
azimuths. There is a small correction to be applied to the mean
of the altitude readings but this is not usually of any importance.
319. Adjustment of Top Telescope. — The top telescope is
adjusted in much the same manner as the side telescope. No
corrections have to be apphed for single azimuths readings, if the
instrument is in adjustment; but the altitude readings require a
correction on account of eccentricity of the telescope. This
form of instrument will not reverse so that errors cannot be elimi-
nated in that way, and it is not so well adapted to the use of the
striding level.
320. IWTERCHANGEABLE SIDE AND TOP TELESCOPE.—
Some instruments are made with an interchangeable telescope
which can be attached at either the top or the side of the main tele-
scope, according to whether horizontal or vertical angles are being
measured. In such an instrument no correction for eccentricity
of the auxiliary telescope is necessary, and it is arranged so as
not to require readjustment when changed from side to top or
vice versa,
321. Eccentric bearing telescope. — Another form
of instrument (Fig. 133) known as the "eccentric bearing"
transit, is regarded by many as the most accurate type. This
instrument has an extra pair of supports for the horizontal axis
of the telescope, which are so arranged that the axis can be dis-
placed horizontally by a fixed amount along the course sighted.
When the telescope is set in the eccentric supports vertical sights
can be taken. A striding level is used to adjust the horizontal
axis. With this instrument the foresight and backsight should
both be taken with the horizontal axis in the same pair of bear-
ings, as these two pairs of bearings are not so adjusted that the
horizontal axis is exactly parallel in the two positions. All
errors of adjustment may be eliminated by taking four readings,
two with the horizontal axis in one position and two with it
tinned end fd|^ end in the eccentric supports, leveling up with the
291
ts?
6 i
H
>
P g
r H
» O
H
w
r
CD
rs
o
P3
-^^
J I
292 MINING SURVEYING [Chap. XI.
striding level each time the sight is taken, the striding level being
used in both its positions in each of the two positions of the hori-
zontal axis.
322. COMBINED SOLAR ATTACHMEIfT AND TOP TELESCOPE.
— A special top telescope is sometimes made to do the duty of
a solar attachment; but it is now generally admitted that better
meridian determinations can be made by direct, single observations
with the main telescope, and the surveyor is advised not to get any
such complex attachment for mining work.
323. In comparing the relative merits of the various forms of
attachment it must be remembered that the object to be accom-
plished is to transfer the meridian accurately from one station to
another, these stations being close together in plan and distant
in elevation. All other virtues of any attachment are of minor
importance. Therefore, in addition to the ordinary adjust-
ments of the transit, special care must be taken to get the hori-
zontal axis of the telescope truly horizontal and the line of sight
exactly perpendicular to it. A high power telescope is more
necessary than for ordinary surveying, as a small variation of the
line of sight means a large error in the azimuth.
324. USE OF THE ORDINARY TRANSIT IN MINING SUR-
VEYING. — Where a special attachment is not to be obtained,
or when the auxiliary telescope is too small for accurate work,
the ordinary transit can be used in such a manner as to ac-
complish the same result as the eccentric bearing instrument
(Art. 321). The instrument, firmly screwed on to the tripod, is
inclined over the shaft at an angle just sufficient for the line of
sight to clear the horizontal plate. It is then braced in position
by such rigid supports as the circumstances afford, and the head
of the instrument is rotated so that the horizontal axis of the tele-
scope becomes truly horizontal, as determined by a striding level,
while the telescope is sighting in the desired azimuth. One or
more station points are then set out down the mine and one each
way on the surface, all in the same azimuth, and these are re-
spectively connected with the mine and surface surveys. All
errors of adjustment may be eliminated by repeating sights with
the telescope in the direct and the reversed positions and by re-
COMPASSES USED IN MINES 293
«
versing the striding level each time and taking the mean posi-
tion of the four points so set.
An attachment which is very necessary in performing some
of the work required of mining transits is the reflecting or pris-
matic eyepiece. This makes it possible to take any sight what-
ever above the horizon, and being a handy instrument to use and
not requiring any adjustment, it shotild be carried by every mine
surveyor.
325. Compasses used IW mines. — The transit has taken,
to a great extent, the place of the old miner's dial in which the
compass was the mam feature. This is partly because, in mod-
ern mines, so much heavy machinery is used that the compass
needle cannot be depended upon, even to its ordinary degree of
accuracy.
Compasses, however, serve a useful purpose in general min-
ing work. They are made in many sizes and of different design.
A compass with a plain needle is to be preferred to one with a
swinging card, since the former can be brought to a central posi-
tion more quickly and is more accurate by reason of the lesser
amount of weight on the center bearing. Compasses may be
used for reconnoissance surveys and also for filling in the details
of a mine from the main stations. A mining compass should be
capable of sighting fairly high altitudes above or below the hori-
zon, and a sighting clinometer* attachment for measuring
vertical angles is very convenient as it obviates the use of any
other instrument. A small modem mining dial mounted on a
light tripod fulfills all these conditions. The hanging compass
and clinometer is made so as to be hung from a wire stretched
between two station points thus rendering sighting unnecessary,
but it is not much used.
A mounted compass is more accurate than one simply held
in the hand, but any hand compass may always be mounted when
\
* The Abney hand level and clinometer consists of a modification of the hand
level described in Art. 100, p. 77. On top of the instrument is a level tobe pivoted
ai the center of a graduated arc, and seen by reflection in a mirror placed inside
the telescope tube. The instrument is pointed along the line whose inclination is
desired and the level turned until the bubble is in its mid position, when the angle
of inclination may be read on the graduated arc.
294 MINING SURVEYING [Chap. XI
the conditions permit. Perhaps the best form of hand compass
is one in which the observer looks down on the instrument and
the line of sight is reflected upward towards him by a hinged
mirror so that the object and the compass box are seen simul-
taneously.
Ore of a magnetic nature has often been discovered by local
variations of the compass needle and by the dipping needle, a
special self- plumbing form of which is made for the use of miners.
A combination compass, or clinometer of special form, is
also useful in taking local strikes and dips of formation. One
of the straight edges of the instrument is f)ut against the ledge
of rock and turned in contact with it until the level line is reached
as shown by an attached spirit level. The instrument is then
folded up or down about this edge as a hinge until the compass
needle is horizontal and the strike is read. The dip is always
at right angles to the strike, but it is not of great importance to
set out this right angle accurately.
UirDERGROUND SURVEYING.
326. TRANSFERRING A MERIDIAN INTO A MINE BY USE
OF THE Transit. — Only a moment's thought will convince
the student that some difficulty must be experienced in ac-
curately transferring the meridian to the bottom of a narrow
shaft several hundred feet in depth. The ordinary method of
transferring a meridian into a mine is to set up the transit at a
station fixed at the mouth of the shaft and, after taking a back-
sight on the previous station on the surface, to take a foresight
down the shaft, the line of sight being made as much inclined to
the vertical as possible. Having ascertained the intervening
distance, the transit is set up at the bottom station, a backsight
taken on the top station, and the survey then carried into the gal-
leries of the mine. The top and bottom stations are not always
the surface and bottom of the shaft, although for simplicity,
they may be referred to as such in this chapter.
In sighting from both ends of the same highly inclined line it
will be found that errors due to the line of sight not being per-
UNDERGROUND SURVEYING 295
pendicular to the horizontal axis are eliminated if the readings
are made with the telescope in the same position at both sights,
whereas errors due to inclination of the horizontal axis are elim-
inated if the readings are made with the telescope direct when at
the top and reversed when at the bottom of the shaft, or
vice versa.
When it is impossible to sight up a shaft on account of its
being too wet, two or more points can be set in line at the bot-
tom of the shaft by means of the instrument when at the top,
and these will determine a line of known azimuth at the bottom
of the shaft.
In some cases a wire is stretched horizontally across the
bottom of the shaft and as far back into the workings as possible,
the wire being carefully aligned by the instrument at the top.
This method may admit of even more accuracy than that of
taking a backsight to the surface from a station established on
the bottom of the mine. Errors due to a slight inclination of
the horizontal axis are not important when this method is used
and for that reason it is also useful in cases where a sensitive
striding level is not to be had. The effect of a slight inclination
of the horizontal axis is simply to shift the line slightly to one
side but parallel to the true position.
When no extra telescope or eccentric bearings are to be had,
an ordinary transit with a prismatic eyepiece attached may be
used to drop the meridian down a vertical or highly inclined
shaft, provided it is not so wet as to prevent sighting upward
from below. To accomplish this a thin wire is stretched hori-
zontally across the top of the shaft at a known azimuth; the wire
should be prolonged one or both ways in order to give a good
base-line. Two points may be fixed at the top of the shaft if
preferred. The transit is then set up on the bottom and it is
brought by trial into the same vertical plane as the wire. The
striding level is used in both positions and the transit is used in
both the direct and reversed positions to eliminate errors*.
It is to be noted that in mining and mountain work slight
errors occur in sighting up steep inclines owing to the refraction
of the atmosphere, but this is so slight that it does not aflFect the
transfer of the meridian and is never taken account of.
296 MINING SURVEYING [Chap. XI.
327. PLUMBING THE MERIDIAN DOWN A SHAFT.— To
the mine surveyor the plumb-line is an instrument of precision,
excelling even the transit, and under most conditions, the work
of transferring the meridian down a mine can be accomplished
more accurately by means of the plumb-line than by any other
method accessible to the surveyor.
The method usually followed is to suspend two bobs from the
staging above the mine so that a horizontal line in their plane can
be sighted both from above and from below. The transit is
set up both above and below on this line and thus an azimuth
connection is established between the surface and the workings.
Sometimes a much longer base-line than can be directly sighted
can be obtained by plumbing down at the comers of a shaft as
shown in Fig. 134. Points A and B have been plumbed down
and, by the triangulation method
there indicated, a connection with
the underground traverse can be
established. In this triangle the
angles shoidd be chosen so as
to give good intersections.
All kinds of drafts in the shaft
should be avoided during the
alignment at the bottom. No
cages or skips should be run and
Fig. 134. Triangulating at the the passages leading to the shaft
Bottom of Shaft. u * u j j -^l
may have to be damped with
sheets of canvas. No lateral streams of water should impinge
on the plumb-lines; in fact it is desirable that no water at all
should drop in their vicinity.
The best plumb-line for this work is one made of wire. An-
nealed copper wire is most flexible, but soft steel or piano wire
being thinner will be less afi^ected by drafts and will also stretch
less. The plumb-bob should not weigh less than five pounds
and should be heavier for a deep shaft. A good working weight
is one-third of the load at which the wire will break.
The plumb-bob is hung in a bucket or a barrel of viscous
liquid so as to bring it to a standstill in the shortest possible
time. The shape of the plumb- bob is of importance in this respect
PLUMBING DOWN A SHAFT 297
r%l
and the form shown in Fig. 135 is a good one,
since it prevents rotary as well as lateral oscilla-
tions. It should hang near the top of the ves-
sel as the wire will be in a high state of tension
and will stretch considerably. A mark should
also be made on the wire showing how far the
bob is above the bottom of the vessel.
The liquid must be a true one (not a mud
or slime) and it must be neither too limpid nor
too viscous; for in the former case it will not
stop the oscillations within a reasonable period,
and in the latter the bob may not reach the cen-
tral position quickly enough. The amplitude fig. 136. Type
of the vibrations of the plumb-bob decreases in of Plumb-Bob
a fixed ratio with equal increments of time, and j^^^ MERmil^
the viscosity of the fluid should be such as to Down a Shaft.
make each oscillation, say, about one-quarter
of the preceding. The ratio of decrease during equal incre-
ments of time is independent of the length of the plumb-Une
and of the amplitude of the oscillations if the resistance is
purely viscous. This law makes it possible to select the fluid
above ground, with the aid of a short length of wire attached to
the bob; it applies only when the bob swings through a very small
arc so that the resistance is wholly viscous. It may be noted
that the period of oscillation varies approximately as the square
root of the length of the plumb-hne, the same as for a pendulum
swinging in air.
If the shaft is wet the vessel should be covered with a sloping
lid having a hole in it of an inch or so in diameter so that the
wire can swing freely. In order to obtain as long a base-line as
possible the wire should be hung as near to the casing of the shaft
as is consistent with the precaution that it shall be perfectly
plumb. It should be carefully examined along all its length to
make sure that there are no obstacles to interfere with it. In
some cases it may be sufiicient to pass a lighted candle around
the wire at the bottom and observe any obstacles by sighting
from the top.- The distance between the wires at the bottom
and top of the shaft should always be measured and compared.
298 MINING SURVEYING [Chap. XL
as this gives the best test of the accuracy of the plumbing opera-
tion. If four lines one in each comer of the shaft are hung
instead of two an accurate check or measure of the errors is
possible.
When once the plumb-lines are hung the meridim may be
transferred to all the levels of the mine once and for all time,
so that a little extra precaution and time given to this operation
are worth while. The surveyor should always keep in mind the
fact that in plumbing the meridian down the mine the direction
of the meridian is of much more importance than the actual posi-
tion of the points themselves, because an error due to an incorrect
direction of the meridian may be multiplied many hundreds of
times in carrying the traverse through the mine (Art. 348, p. 316).
328. TRANSFERRING A MERIDIAN INTO A MINE WHEN
THERE ARE TWO SHAFTS. — The above methods presuppose
that the mine has so far been opened only by one shaft. If
there is a second shaft or an adit, it is, of course, only necessary
to plumb or otherwise transfer the position down each shaft; the
computed distance between these points then becomes a base-line
of substantial length. In Fig. 136 the traverse ^4 J5 C-D is run
B TRAVERSE
^^
\.
A ^^ -
_Closinq Line >^ -.
A' """■- -^G'
/o*
UNDERGROUND TRAVERSE E
PLAN
Fig. 136.
out on the surface to connect the two shafts at A and D. The
points A and D are plumbed down the shafts and the correspond-
ing points A' and D' established at the bottom. A.i underground
traverse A'OF'E'U is then run out. In the surface traverse
the length and azimuth of AD and in the underground traverse
the length and azimuth of A'D' are missing. The horizontal
UNDERGROUND TRAVERSES 299
length and azimuth of each of these lines can be determined from
their respective traverses as explained in Art. 397, p. 366. The sur-
face traverse is referred to the true meridian, and, since nothing is
yet known in regard to the direction of the meridian in the mine,
the underground traverse is referred to an assumed meridian.
The true azimuth of A' D' is the same as the azimuth of AD^ pro-
vided the plumbing down the shaft has been accurately done.
The diflference between the true and assumed azimuths of A'ly is
a correction to be applied to the azimuths of all of the lines of this
underground traverse.
329. UNDERGROUKD TRAVERSES. — Surveying in a mine
is necessarily a process of traversing, for only the working passages
are available for lines of survey. The line of traverse is not always
in the center of the passage but is often varied from it in order
that the longest possible sight may be taken. In the tortuous
passages of a mine it is frequently necessary to take very short
sights on the main traverse and since the azimuth is transferred to
distant connections through these short lines great care should
be exercised. The positions of the walls of the passages are noted
as the work proceeds and are sketched in approximately on
the plot. After the main traverses have been run, the surface
boimdaries, if touched, may be accurately estabUshed and the
stopes and working places surveyed by more convenient and less
accurate methods, from the stations already established.
It is often very convenient in underground work to take the
azimuth from an estimated general direction (or strike) of the vein ;
for the diriection of the meridian is of no importance in the actual
working of a mine, while the direction of most of the passages will
usually vary only a few degrees from the strike, and thus all trav-
erse calculations are simplified.
A speedy and convenient manner of running an underground
traverse is to use three tripods having leveling heads and centering
plates like those of the transit. The transit fits on to any of
these heads and while it is attached to one of them the other two
are surmounted by lamp targets in* which the sighting center has
exactly the same position as the sighting center of the transit
would have if set on the same tripod. These tripods are placed
vertically over or under the stations and the transit is attached to
300 MINING SURVEYING [Chap. XI.
the middle one. When the transit head is moved from the
middle to the foremost tripod a target takes its former place and
the hindmost tripod is brought ahead of the transit and set up on
the new forward station. The lamp behind the plumb-target or
plumb-line should give a diflFused illumination of considerable
area so that it may be easily found with the telescope and so that
it may render the cross-hairs of the telescope plainly visible. In
cases where the illumination of the object is such that the hairs
cannot be distinguished, a light is thrown obliquely into the
telescope tube in front of the hairs, preferably by a tube reflector
(Fig. 133) in front of the object glass.
Sometimes a brass lamp with a small central flame, called a
plummet-lamp, is suspended in place of a plumb-line and the
flame is sighted at, but this is too small a target for quick work
and the surveyor may also mistake other lights, such as miner's
lamps or candles, for it when sighting through the telescope.
330. Establishing Station Points. — The station point is es-
tablished either on the floor or the roof, according to the character
and condition of the mine: the chief object sought is permanence
of position rather than convenience in getting at the point for
future use, which is of secondary importance. In a vein mine a
timber in the roof, especially a stuU, is often more permanent than
the floor or rock roof, but any timber is likely to be moved by the
miners. The hanging wall- is a good place for the station, but if
the inclination is small, as in a coal vein, the foot wall or floor is
best.
To establish a station, get a miner to make a drill hole about
six inches deep, more or less, according to the hardness of the
rock. Cut a wooden plug to fit this hole tightly when ham-
mered in dry, and do not let any more of the plug project than is
necessary. Small screw eyes make good roof station points
from which to suspend the plumb-line, but where the lines are
short a finishing nail bent to a sharp angle is better as the plumb-
line will then always hang in exactly the same position. For
measuring between stations a hundred-foot steel ribbon tape,
divided to hundredths of a foot is used; but for long straight
tunnels and shaft work, a longer steel wire tape is more con-
venient.
NOTES OF A MINE SURVEY
301
Survey of Bear Creek Mine, West Boulder, Montana.
Sta.
Bearing.
Distance.
Vert.
Angle.
Back,
•ighton
May 17, 1906.
Pvty : Keene, Chase, Holbrook.
0
NSS^lCE
650.8
- 1*17'
San
To top of sdr shaft extending
to I at level ; C, of S. edge
of air shaft, 4' x 4'. Sta. 2.
0
N 2°lo'W
"74
-8o*>io'
Sun
To Sta. 1 01 at ist level.
Line runs 3' from S. side
and 4.5' from IV. side of
shaft; shaft 8' X 8'.
1 01
N 87^45' E
230-8
+ o«45'
0
To Sta. 102 in ist level.
lOl
N 2°io'W
112. 6
-8o*>io'
0
To Sta. 201 in 2nd level.
102
N89*»io'E
750
+ o-sy
lOI
To top of center of raise ex-
tending to 2nd level, raise
4' X 4'- Sta. 107.
102
N89*»io'E
153 5
+ o°53'
lOI
To Sta. 103 in 1st level.
103
S 89^15' E
105 7
+ o*'39'
102
To Sta. I04 in ist level.
104
S 88°i2' E
162. 1
+ o*'48'
103
To Sta. 105 at foot of S,
side of air shaft extending
to surface.
lOS
S 9°55'W
92.3
+88*25'
104
To top of air shaft, Sta. 2.
105
S 88«i2' E
15.9
level
Compass
To Sta. 106 at breast of ist
level.
201
N 2^10' W
iiS.S
-80*10'
lOI
To Sta. 301 in 3rd level.
201
N 85*»52' E
167.4
+ 0*50'
lOI
To Sta. 202 in 2nd level.
201
S 85<'46' W
196.0
+ 0*47'
lOI
To Sta. 205 in 2nd level.
202
N 88^20' E
138.0
+ 0*44'
201
To C. of raise extending to
1st and 3rd levels, 4' x 4'
Sta. 208.
202
N 88^20' E
106.3
+ 0*44'
201
To Sta. 203 in 2nd level.
203
S 89*»05' E
176.9
+ 0*42'
202
To Sta. 204 at breast of 2nd
level.
208
S 3^14' E
113.7
+ 77^19'
202
To Sta. 107.
205
S 86*10' W
216.8
+ 0*48'
201
To Sta. 206 in 2nd level.
302
MINING SURVEYING
[Chap. XI.
Survey of Bear Creek Mine, West Boulder, Montana. (Cont'd.)
su.
Bearing.
Distance.
Vert.
Angle.
Back-
sight on
^
206
S 87°I4' W
118.0
+ 0°4I'
205
To top center of winze ex-
tending to 3rd level, 4' X
4'. Sta. 209.
206
S 87^14' w
152.0
+ o°4l'
205
To Sta. 207 at breast of 2nd
level.
301
N 86«2o' E
304.0
+ 0^46'
201
To Sta. 302 at C. of raise ex-
tending to 2nd and 4th
levels, 4' X ^\
301
N 86^20' E
316.0
+ 0^46'
201
To Su. 303 in 3Td level.
301
S 86°4o' W
195.0
+ 0^50'
201
To Sta. 305 in 3rd level.
301
N 2°io'W
116. 8
- 8o°io'
201
To Sta. 401 at 4th level.
302
S 5^35' E
116. 5
+ 78^29'
301
To Sta. 208.
303
S 89^07' E
289.0
+ 0O39'
301
To Sta. 304 at breast of 3rd
level.
305
S 88^52' W
186.2
+ 0^46'
301
To Sta, 306 in 3rd level.
306
S 89^48' W
150.0
+ o°43'
305
To Sta, 307 at C. of bottom
of winze extending to 2nd
level, 4' X 4'.
307
S 2°4i'E
120.5
+ 7i°ii'
306
To Sta. 209.
307
S 89^48' W
10.9
level
Compass
To Sta. 308 at breast of 3rd
level.
401
N 85^48' E
219.7
+ 0^48'
301
To Sta. 402 in 4th level.
401
S 88«io' W
116. 4
+ 0^52'
301
To Sta. 406 at breast of 4th
level.
401
N 2°io'W
49-7
^88°io'
301
To bottom of shaft, 3' from
S, side and 4' from E side.
Shaft 8' X 8'.
402
N 89^56' E
85.0
+ 0O45'
401
To Sta. 403 to C of raise ex-
tending to 3rd level, 4' X 4'.
402
N 89'»s6' E
92.6
+ o°45'
401
To Sta. 404 in 4th level.
403
S o°o6'E
116.2
+ 8i°46'
402
To. Sta. 302.
404 S 87*'2o' E
217.6
+ 0O43'
402
To Sta. 405 at breast of 4th
level.
lOb
MAP OF
BEAR CREEK MINE
WEST BOWLDER. MONT.
Scale. I in. "SO ffr.
Scale of Reduced Plate »
I in. « 150 ft.
TRANSVERSE SECTION
LOOKING WeST
\
PLOTTING A MINE TRAVERSE 305
331. Notes of a Mine Traverse. — As a rule the notes of mine
surveys are kept in the form of sketches, especially the details,
such as the location and extent of the stopes. These details are
plotted on to the skeleton survey which is simply a traverse, the
notes for which may be kept in the following form.
The different station points of this survey are numbered for
identification only, their numbers bearing no relation to the
distances between them. For convenience the stations on the
first level are numbered loi, 102, etc.; on the second level 201,
202, and so on. In larger ard more complex mines the
system of numbering and lettering stations is carried out still
farther.
332. Plotting a Mine Traverse. — This survey can be plotted
by computing three sets of coordinates which give all the data
needed for showing the mine in plan, longitudinal section, and
transverse section. If the three coordinate planes are the meri-
dian plane, the east and west plane, and the horizontal plane, then
the vertical and horizontal distances from each slope measure-
ment are first obtained by multiplying the measured distance by
the sine and the cosine respectively of their vertical angles. The
vertical distance is the difference in elevation between the two
points. From the horizontal projection and the azimuth, or
bearing, the latitude and departure of the course can be computed
as usual (Art. 384, p. 352). A plot of these notes will be seen in
Fig- 137-
It is assumed in plotting these notes that all the transit lines
in the galleries run 2 ft. below the roof and in the center of the
galleries, which are 6 ft. high and 4 ft. wide; conditions which
are more uniform than would occur in actual practice. The
measurements which locate the walls of the galleries have been
purposely omitted from the foregoing notes for the sake of
simplic'ty.
If it is desired to substitute for the meridian plane a vertical
plane through the strike and for the east and west plane one
which is at right angles to the strike, then all of the true bearings
or azimuths must be corrected by an amount equal to the strike.
After these bearings have been corrected the three coordinates
are calculated in the same way as described above. The ad-
306 MINING SURVEYING [Chap. XL
vantage of this latter method is that the levels are shown in their
full length in the longitudinal section and the shaft is shown in
its true length in the transverse section.
333. UlfDERGROUITD LEVELING. — The drainage of a
mine is usually toward the shaft (Fig. 137), and the grade of the
levels is such as will make the tractive force of a full car going
towards the shaft equal to the pull required to move an empty
car in the opposite direction. When connections, other than ver-
tical ones, have to be made the grades must be taken into consid-
eration. In this work an ordinary surveyor's level is generally
used in conjunction with a short leveling rod about five or six
feet high.
334. MINE MAPS AND CHARTS. — The galleries of a mine
are often so nearly over one another that confusion is liable to
arise in charting, unless some special means of identifying them
is employed. As these galleries or working passages appertain
to definite levels or strata, a different color may be assigned to
each level or strata and adhered to throughout. The lines of
survey are in a colored ink and the passages or workings are of a
fainter tint of the same color. These colors can also be adhered
to on the elevations, of which there are usually two, one along the
strike and the other at right angles to it. (See Fig. 137.)
Some surveyors use large scale plots and simply mark the
position of the stations on them so that when a course has to be
set out its distance and direction can be scaled directly from the
map.
Another method is to use a small scale map and mark on it,
in figures, the exact coordinates of every station point. The
origin, or point of reference, is usually the plumb-line of the
shaft, and the two vertical planes of reference may conveniently
be taken through the estimated general strike and dip of the vein.
The true course of the survey lines may also be marked and all
the exact data can be clearly kept in a minimum space.
The progress of work in the stopes or rooms of the mine is
generally represented on different plans from those used to show
the main headings. These working plots may be either vertical,
horizontal, or parallel to the vein or seam. In any case, the
thickness of the deposit is recorded at frequent intervals together
MINE MAPS 307
with other particulars, such as thickness of waste or value of ore.
These thicknesses are all measured at right angles to the plane
of the working plan, so that when multiphed by the area on the
plot, the cubic capacity of any section is obtained. Where the
ore occurs in irregular masses, not conforming particularly to
any one plane, the above system does not apply and some other
method must be devised by the surveyor.
The best way of estimating amounts not mined is to sketch
their probable extent on such a chart from the data available and
to make use of the area and thickness method as suggested in the
preceding paragraph. Ground explored by bore-holes but not
opened by headings may be best shown by plotting in plan the
positions of both of the walls, where struck in the bore-holes,
marking the elevations in figures on the plan. Contours may
then be sketched in colors, and a very good idea of the shape and
trend of the ore body obtained, and the quantities of ore may
also be calculated therefrom. (See Computation of Volume,
Chapter XII.)
335. LAYING OUT MUflHG WORK. — Drifts or cross-cuts are
laid out by putting in two nails or hooks in the roof, not too
near together, from which the miner can hang two plumb-lines
and sight the center of the heading he is to rim.
Vertical shafts are carefully plumbed on the inside of the
frames, and frame by frame, as these are put in. It is best to
hang the plumb-Une from several frames above the bottom one,
as these upper ones are more likely to have ceased to move.
Hang the line an even fraction of an inch each way from the true
position of the comers and note any accidental variation in the
last frame set, so that in future work, if it is desired to hang the
plumb-line from this frame, its error of position can be allowed
for. The dimensions of a shaft or drift are given either "in
the clear," meaning net measurements inside all timbers, or
"over all" meaning gross measurement outside all timber and
lagging.
336. UlfDERGROUND SURVEYING PROBLEMS. — In the prac-
tice of mine surveying, problems are constantly arising which
tax the ability and ingenuity of the surveyor, although the actual
3o8
MINING SURVEYING
[Chap. XL
solution of most of them is quite simple. A few of the common
problems met with in such work are given below.
337. Vertical Angle Correction
for Eccentricity of the Top Tele-
scope. — As has been stated in Art.
317, all vertical angles taken by
means of the top telescope must be
corrected for the eccentricity of this
attachment. In Fig. 138 the vertical
angle has been taken to a point C in
the bottom of a shaft. The distance
AC was measured, A being the hori-
zontal axis of the main telescope.
Since the transit is set up over a siu*-
face station at JE, the distances de-
sired are EC and AD. HB and
H'A are both horizontal,
V - ACB.
JK \ ' Am -- ^^ — Distance between telescopes
AC Distance measured
AD = AC sin F,
and DC = AC cos 7'.
The height of instrument above the datum being known the ele-
vation of C can be readily calculated.
Had the distance BC been measured instead oi AC then
DC=CF + FD = BCcos V + .45 sin F.
Similarly i4Z) = BC sin V - AB cos V.
338. Vertical Angle Correction for
Eccentric Bearing Telescope. — In
Fig. 139, A is the central bearing for
the telescope and B is the eccentric
bearing in which the telescope rested
when the vertical angle V and the
distance BC were measured.
Fig. 188.
then V
DC = FC '\- AB = BC cos V + AB
iD ^ BF ^ BC sin V.
Fig. 189.
ESTABLISHING BOUNDARY UNDERGROUND 309
339. To Establish a Boundary Line of the Claim Underground.
— In Fig. 140 points A and B are on the boundary of the claim.
The shaft is located at 5, and it is desired to prolong the
underground working in the drift ilV to a point K' vertically
under the boundary line. The surface traverse BADCS is run
out, point 5 is plumbed down to 5', and the meridian trans-
ferred into the mine. Then the underground traverse
^EF'G'H'r is run out. The horizontal projections of all the
measured lines on both traverses are computed (or measured),
and the length of the level line AH' and its bearing can be cal-
culated as described in Art. 398, p. 367. In the horizontal triangle
AH'K'y AH' and all the angles being known, the line H'K' can
readily be computed. If the drift H'J is not level the distance
from H' along the drift to the boundary plane will be equal to
the horizontal distanc : H'K divided by the cosine of the vertical
angle.
340. To Lay Out a Connection in a Mine. — Here the prob-
lem is to determine the bearing (or azimuth) and the vertical
angle and the distance to run from point -4 in a mine to point B
in another portion of the mine. A traverse can be run from A
to B through the passages already cut in the mine, and all the
distances reduced to horizontal distances which, together T/vdth
the azimuths, form a traverse in which the length of the closing
line AB (horizontal projection) and its azimuth are missing.
These can easily be computed by the method explained in Art.
398, p. 367. The difference in elevation between the actual points
3IO MINING SURVEYING [Chap. XI.
A and B together with the length of the horizontal projection of
AB will give the vertical angle; from these data the direct dis-
tance between the points A and B can be computed.
341. HYDRAULIC SURVEYING FOR MDfES. — The miner's
unit for measuring water is the miner^s inch. By an inch of
water was originally meant such continuous flow as will go
through a one inch square hole, the head of water behind it
being usually six to nine inches. This very loose definition has
been done away with but the name still applies, being defined
more exactly as ninety cubic feet of water per hour (ij cubic
feet per minute). In spite of all criticism, the miner's inch has
become by custom the standard unit for the flow of water in most
mining districts. It no doubt retains its hold on the practical
mind because no good definite time or capacity units are in gen-
eral use, seconds, minutes, hours, and days, or gallons and cubic
feet with their clumsy relations to one another, being used ac-
cording to the whim of the individual. To get an idea of the
magnitude of a standard miner's inch, it may be remembered
that it is equivalent to a stream one inch square running at a
uniform rate of 3.6 feet per second. This is about a medium
speed for small mountain streams; and, with a little practice,
the flow of such a stream in miner's inches may be calculated
mentally, after rough measurements have been made of the cross-
section of the stream and the speed of flow of the water at the
surface. The accuracy of this process is within the ordinary
limits of fluctuation of the stream from day to day. If the flow
has to be recorded over a long period it is well to put in a weir.
For estimating the flow of larger and more important moun-
tain streams, a portion of the stream where the width and flow
are comparatively uniform may be chosen and the length of this
portion measured and marked by flags. A cross-section of the
bottom of the stream is obtained at each flag and at intermediate
points if necessary by measuring the depth at equal intervals
across the stream; from these a mean cross-section is obtained.
Floats are started at intervals across the stream opposite the up-
stream flag, and timed with a stop watch while running to the
down-stream flag; the speed of each float represents the velocity
of the stream in its respective longitudinal strip. Each velocity
SURFACE SURVEYING 3II
is multiplied by the area of the corresponding portion of the
cross-section of the stream, and from the total flow so computed
a certain percentage is deducted for the excess of surface over
mean flow; this, for ordinary mountain streams, is approximately
twice the percentage of the grade of the channel.
The surveying and staking out of mining ditches, flumes, and
pipe lines follow the general practice for this work in other fields
of engineering.
342. Testing for Ore by Electric Currents. — Methods of
testing the earth for ores by means of electrical currents and
waves are being experimented upon, and the working out, record-
ing, and plotting of the results are likely to become a part of the
mine surveyor's work.
SURFACE SURVEYIlfG.
343. SURFACE SURVEYING Df RUGGED MOUITrAIN REGIONS.
— In accurate work, such as the surveying of mining
claims for patent,* the ordinary mining transit may be used.
Measurements are made with a steel wire tape, 300 to 500 feet
long and marked every 10 feet (or 20 feet) so as to be used with
a short auxiliary steel ribbon tape which is divided to hun-
dredths of a foot. The measurements are taken from the center
of the instrument to the object at which it is pointed, care being
taken not to overstretch the tape nor to kink it. The most
accurate work is done by stretching the tape with a tension
handle (a spring balance) which can be attached by a clamp to
any part of the tape. Where it is feasible, just enough tension
is given so that the stretch of the tape compensates for the short-
age due to sag. In many cases assistants will have to hold the
middle point or the points at one-third and two-thirds the length
of the tape up to the line of sight, giving at the same time enough
puU to make the sag equal in the different sections of the tape.
There are several systems of traversing. The most common
is to measure the height of the center of the instrument above the
* By patent proceedings is meant the proceedings necessary to obtain from
the government a fee simple deed to the mining claim.
312 MINING SURVEYING [Chap. XI.
station point, and then to sight an equal height on a graduated
staff held on the back and forward stations, recording the azi-
muth, vertical angle, and distance. Another method is to sight
and measure to targets set at a fixed height above the stations,
recording the vertical angle only at alternate stations. If the
vertical angles are read at every station there wi.l be two sets of
vertical angle and distance measurements. The three tripod
method may also be used as described for underground work;
and lastly two transits and instrument men may be em-
ployed, each sighting to the other's telescope and measuring the
distances between them. Each of these methods has its advan-
tages and disadvantages, and the best one to use depends upon the
conditions of the work to be done. In some cases there will be
twice as many altitudes and in some cases twice as many dis-
tance readings as are actually needed, but these extra readings
may be used as a check available in the field.
In making general maps of a mining district, only monu-
ments and important locations need be accurately shown. This
accurate work which is the first to be done forms a skeleton on
which to make a general map. The topography can be filled in
by a transit fitted with fixed stadia wires and a compass.
The best topographical data in mountainous country are ob-
tained by running traverses along the ridges and valleys; these
are also usually the best places to travel. Much sketching is
necessary and the work should be plotted by the surveyor him-
self each day as the work proceeds. In this work a rough deter-
mination of the topography is sufficient, since the plans are usu-
ally plotted to the scale of i^^^^i^ or smaller, and therefore such
instruments as the hand compass, clinometer, and aneroid baro-
meter can be used. With such instruments one man can do the
entire work. The plane table cannot be used to advantage in
mountain or mine surveying, but photographic surveying may
orten prove useful in filling in details of topography.
344. MINE BOUNDARIES. — ^APPROPRIATIONS UNDER UNITED
STATES LAWS.* — In most countries mineral rights are defined
• For further information with regard to this subject see the Manual of
Instructions for the Survey of the Mineral Land of the United States, issued in
1895 by the Commissioner of the General Land Office, Washington, D. C.
MINING CLAIMS 3^3
by vertical planes through lines marked out on the surface.
Title to metalliferous lands, however, as granted by the United
States, conveys the right to all minerals included in the down-
ward prolongation of the portions of veins cut off by the verti-
cal end bounding planes, i.e., a vein can be worked in the
dip indefinitely, but in the direction of the strike it is limited by
the end bounding planes of the claim. This law has given rise to
much litigation and there are still many unsettled points involved.
The Federal law allows a claim to cover 1500 feet located
along the direction of a vein and 300 feet of surface ground on
each side of it. These dimensions which constitute the maxi-
mum can be reduced by local laws. The ordinary method of
locating a claim is shown in Fig. 141. The discovery being
WW _____^.___ ^^
yi — k^ — £ — ^c
sw : SE
Fig. 141. Plan of Lode Claim.
made at D the center line WC-D-EC is run and then the end
lines SE-NE and SW-NW are put in, being made parallel with
each other and straight. The side lines must not be 6ver 300
feet apart, measured at right angles from the center line.
A monument with explanations is placed at each of the seven
points marked. If in a timbered country, the lines run should
be blazed, and squared trees may be used as monuments. At
D (Fig. 141) a location notice is posted, defining the boundaries
of the claim and containing such explanation as would identify
the claims in case of dispute. The miner usually makes the loca-
tion survey himself, using approximate courses and distances.
There is legally no objection to this work being done roughly,
but when a patent survey comes to be made, neither the dimen-
sions specified in the location notice nor the limits of the claim
as marked off on the ground can be exceeded. So when the
location survey is roughly made certain "fractions " of ground
314 MINING SURVEYING [Chap. XL
are not included, and these may cause much trouble, especially
when "groups" of claims are located.
In such preliminary surveying, traverses may be run along
coiu'ses where the sights can be conveniently taken and the azi-
muth taken from the direction of the vein; this may save much
time and considerably simplify the work, (specially in thickly
timbered regions (Art. 329, p. 299). In the description it is
sufficient to state the approximate compass bearings of the
boundaries. The center line and side lines need not be straight or
parallel, but are assumed to be so unless marked with additional
monuments. If, on account of the crookedness of the vein, it
is advisable to make the center line of the claim a series of straight
lines (like a traverse), this can be done, but the above condi-
tions must be fulfilled with regard to the length and breadth of
the claim and the two end lin6s must be parallel. In order to
guard against troublesome litigation, an effort is sometimes
made to surround a valuable claim with others, thus forming a
"group." The more valuable claim is then protected as regards
all " extralateral rights."
Flat deposits, such as coal and placer, are subject only
to vertical bounding planes, and, provided the boundaries are
marked plainly on the ground and the legal dimensions are
not exceeded, no difficulty need be encountered. The Federal
law allows 20 acres to be taken for a placer claim but fixes no
limits in regard to breadth or length. Local laws can regulate the
size, provided the 20 acre limit per claim is not exceeded. The
coal lands law is made subject to the general system of public
land surveys for agricultural lands.
345. SURVEYING FOR PATENT. — The surveying of claims
for patent from the United States Government can only be
obtained by those who have received appointment of United
States Deputy Mineral Surveyor and they must have an order
from the Surveyor General of the state or territory in which
the claims are located before making any such survey.
In survejdng for patent, much more accurate work has to be
done than when merely locating a claim. After the shape of
the claim as originally staked has been determined, the positions
of the new corners and other boundary marks are computed and
PATENT SURVEYING 315
laid out on the ground. The original claim cannot anywhere be
exceeded and usually has to be cut down so as to make the end
lines parallel and bring the dimensions of the claim within
statutory limits. All this must be done accurately, the limit of
error allowed being one in. two thousand. Besides 'the marking
of the boundaries on the ground, the position of at least one of
the comers of each claim must be determined with reference to
permanent monuments recognized by the government. The
true meridian must also be determined by observations of the
sun and all courses must be referred to it. The position of all
buildings and surface improvement must be found and shown
on the plot, and also the position of all comers of other claims
for which a patent has already been applied. The surveyor
must also make an estimate of the value of and describe all
improvements, such as tunnels, shafts, open-cuts and other
mining work done on the ground, and these should amount to
not less than $500.00 worth per claim. The Manual of Instruc-
tions describes in detail the character of the comers required to
be established, and a great many other details which must be
known to the Deputy Mineral Surveyor before his survey will
be accepted, and defines the penalties attached to poor or dis-
honest work. Patented claims may overlap, and in fact do, in
all mining districts, but in making application for patents to
claims which lap on ground previously patented, the exact rights
desired on the area of intersection must be defined.
Placer claims may be taken in twenty acre tracts, the bound-
ing lines of which must conform with the general system of
survey lines established by the Government, but if such survey
has not been extended to the district, they must be bounded by
true meridian and east and west lines. The survey of coal land
is subject to somewhat similar rules.
346. THE SURVEYING OF BOREHOLES. — Boreholes,
whether made by a rotary or a percussion drill, are never per-
fectly straight and unless the ground is remarkably homo-
geneous, are not amenable to any mathematical law. Means
have been devised, however, of measuring the strike and dip of
a hole at any particular distance from its mouth. The trend of
the borehole can thus be plotted with some degree of approxi-
3l6 MINING SURVEYING [Chap. XL
mation and the position of any particular body or strata struck
in the borehole determined. One method depends in principle
upon the conversion from liquid into jelly, by cooling, of a
solution of gelatin, contained in a small vessel together with a
compass needle and a plumb-bob and of such a shape as to
align itself with any part of the hole in which it may be placed
Another instrument takes a photographic record of the
position of the compass needle and plumb-bob, after the lapse of
such an interval of time as is necessary to place the instrument
in proper position and allow the needle and plumb-bob to come
to rest. The position of points in any plane stratum, as found
by three boreholes, determines it. If, however, the angle at
which a borehole cuts this stratum is known, only two boreholes
are necessary and if the strike and dip of the stratum is known,
one borehole is sufficient to determine it.
347. STAKING OUT THE PROBABLE APEX OF A VEDf. —
It is often required to prolong the course of an inclined vein on
the rugged surface, either for exploration purposes or to locate
a claim. This may be accomplished by setting up on the vein a
transit fitted with a solar attachment, the main telescope being
inclined at the angle of dip of the vein in altitude and pointed at
right angles to the strike The solar attachment, when set for
the zero declination, will sight points only in the plane of the
vein.
As veins are usually somewhat irregular, the survey need not
necessarily be made with a transit. Instead, such instruments
as the compass, clinometer, or a small improvised plane table may
be used.
348. ECONOMIC PRINCIPLES. — The surveying of mine
workings is strictly an economic problem and the surveyor must
study it as such. The accuracy attained must be such that the
cost in obtaining it and the saving of expense in mining oper-
ations through it together efTect the maximum of economy. The
surveyor bearing this in mind will be neither too careless nor too
exact. He will not, for example, close down the mine in order
to carry a meridian into it when no important connections are
needed, and there are no boundary disputes. On the other
hand, in some cases accuracy of a survey is of such prime impor-
ECONOMIC PRINCIPLES
317
tance that a temporary interference with the working of the mine
may be warranted.
In any given case the surveyor must make a scientific meas-
ure of the accuracy re-
quired. There is no
better method of ob-
taining accurate results
than to go over the lines
several times with vary-
ing conditions, but this
is not always good econ-
omy, especially in such
work as plumbing a
shaft which necessitates
a temporary stoppage of
all hoisting operations.
Often the controlling
error will be the error in
plumbing of the meri-
dian. In Fig. 142, D is
the horizontal distance
in a straight line from
the shaft to the connec-
tion, d is the distance be-
tween the plumb-lines,
and e its error as ascertained by measuring or other means.
The controlling error at the connection \s E = ^^and is in a
direction perpendicular to that in which D was measured on the
chart. This is obvious, for the surveys of the galleries are con-
sidered accurate, the error being one of relative rotation around
the shaft as an axis. Where there are many angular errors of
the same degree of magnitude, such as occur when a number of
short sights are included in the traverse, the distances may be
measured from these short lines to the connection and their
respective errors E^ E^, E^, etc. found at these localities. These
are then resolved according to their respective latitudes and
departures into S„ .S„ S,, etc., and W^, W,^, W^, etc. The greatest
Shorr
Fig. 142.
3l8 MINING SURVEYING [Chap. XI.
possible error is then 5i + 5, + 5, + etc. to the north or south
and W^ + W^ + W^ -}- etc., to the east or west, these summa-
tions being made without regard to any sign.
Likewise the mean probable error (by method of least
squares), is V5i^ + S^ + S^ + etc. to the north or south and
VWi' 4- W,* + Wg^ + etc. to the east or west. Errors due to
the measurement of distances, which are not likely to be great,
may be divided into latitudes and departures directly and com-
pounded with those due to angular error.
Besides being of immediate service to the surveyor, the prac-
tice of computing possible and probable errors gets him in the
habit of thinking along the most business-like lines instead of
drifting into a rut or losing interest in his work.
The surveyor should keep his plans up to date and see that
the men in charge of the mining operations fully understand
their instructions; for many mining "bosses" of considerable
experience get entirely wrong notions of the shape of their work-
ings and are often too proud to ask for information. The sur-
veyor, without assuming a "know-it-all '' attitude, can, from the
specialized nature of his work, often make useful suggestions
in regard to the exploration of a mine. By working always in
harmony with the other officials of the mine, he can further the
interests of all concerned, both employers and employees.
PROBLEMS.
1. From a monument at the mouth of a tunnel a line is run in the tunnel, azi-
muth 37° 24', slope distance 424 ft., vertical angle 4- 2° 10'; thence azimuth 62° 42',
slope distance 278.5 ft., vertical angle + 2° 18' to breast. From the same monu-
ment a line is run on the surface, azimuth 98° 33', slope distance 318.5 ft., vertical
angle — 3° 22'; thence azimuth 38° 02', slope distance 647 ft., vertical angle -f 14° 13'
to the center of a vertical shaft. How deep must the shaft be to meet a connect-
ing drift run on a grade of + 2.4 % from the breast of tunnel, and what is the
slope length and azimuth of this drift?
2. The strike of a certain vein at point of outcrop is N 43° E and the dip is
71° 5o'» pitch S.E. From this point of outcrop a surface line is run, N 83° 15' E,
slope distance 248 ft., vertical angle — 12° 34'; thence S 2^54' E, slope distance
208.5, vertical angle — 14° 34' to a point from which the tunnel is to be driven in
the direction N 71° W and with a grade of 4- 3.8^ until it intersects the vein.
ECONOMIC PRINCIPLES 319
(a) What would be the slope length of such a tunnel?
(b) What would be the slope length and bearing of the shortest possible tun-
nel run on a + 1.3% grade to intersect the vein ?
3. A vein has a pitch of S 67® W and its dip is 55®. What is the azimuth of
an incline on the vein having a slope of 44®?
4. From the bottom of vertical shaft No. i a horizontal traverse was run in the
mme to the bottom of vertical shaft No. 2 as follows: Assumed azimuth 0°, distance
243 ft; thence azimuth 340°, distance 121 ft.; thence southeasterly a distance of 473
ft. along a vein which shows a pitch of 60** (azimuth) and a dip of 35®; thence azi-
muth 42°, distance 25 ft. to the center of shaft No. 2. From a point vertically
above the last point a line is run on the surface with true azimuth 116^20', dis-
tance 41 1 ft. (horizontal) to a point A from which the center of shaft No. i is sighted
at azimuth 71® 3c/.
(a) How much deeper will shaft No. 2 have to be sunk to reach the vein?
(b) What is the true strike of the vein ?
5. A vertical winze has been sunk below the level of a tunnel. It is desired
to sink a vertical shaft from the surface to connect with the winze. The monu-
ment X is established at the mouth of the tunnel and the monument Y is near the
site of the proposed shaft. Y bears S 88*58' 56" W, 896.796 ft. from X, The
following are the notes of the survey connecting X and the winze comers A, B, C,
and D: —
Station. Mean Deflection. Horizontal Distance. Station.
X .. 0° oo' 896.796 Y
403.080 I
587.208 2
3 32* 23' 43" ^ 67.000 3
3 54** 43' 47" R 44.803 4
4 39** 51' 57" ^ 41.075 '5
45° 05'
34"
R
74° OS'
06"
L
32° 23'
43"
L
54° 43'
47"
R
39° 51'
57"
R
31° 10'
10"
R
31° 10'
10"
R
31° 43'
40"
R
24° 02'
40"
R
5 31° 10' 10" R 19-573 Cor. A
27.240 Cor. B
21.477 Cor. C
25773 Cor. D
Required the location of the shaft comers on the surface.
6. From a monument M at the mouth of a tunnel a traverse is run in the tunnel,
azimuth 20® 35', distance 352 ft., vertical angle -f- 1° to point A ; thence azimuth 61®,
distance 528 ft., vertical angle -h i® 40' to point B at the breast of the tunnel
From M a surface traverse is run, azimuth 11® 10', distance 578 ft., vertical angle
4- 4° 25' to point C; tbence azimuth 11**, distance 407 ft., vertical angle -f- 14** 20'
to point D, which is the center of a vertical shaft 120 ft. deep. Find the length
and grade of a connecting incline from the bottom of the shaft to the breast of the
tunnel.
7. The course of Tunnel il is N 34° 45' 10" W., the grade 0.42%, and the ele-
vation of the mouth 2570 ft. The course of Tunnel B b N o® 45' 00" W, the
grade 0.33%, and the elevation of the mouth 2608 ft. The following traverse con-
320 MINING SURVEYING [Chap. XL
nects the mouths of the two tunnels: — from mouth of Tunnel B, N o® 45' 10" W,
100 ft.; thence N 19® if 30" E, 381.60 ft.; thence S 10® 21' 20" E, i 30.60 ft;
thence N 74*14' 30" E, 3662.01 ft.; thence N ^s"* 4$' 30^' E, 1547.21 ft; thence
N 73° 48' 00" E, 1455.00 ft.; thence S 12® oo' 00" E, 205.40 ft; thence
S 70® oo' 10" E, 205.00 ft. to the mouth of Tunnel A . Where and how far must one
upraise vertically in order to connect the tunnels? Do not consider the dimen-
sions of the tunnels.
8. Assuming the transit to be in perfect adjustment what is the error in hori-
zontal angle in sighting down a 500-ft shaft, 5 ft in breadth, when the telescope
cannot be sighted closer than 3 seconds along the inclined line?
PART III.
COMPUTATIONS.
3*1
PART III.
COMPUTATIONS.
CHAPTER. Xn.
GXSNISRAI. PRINdPLES.— MISCBLZJLNXIOnS PROBLEMS.—
EARTHWORK COMPX7TATIONS.
349. GENERAL REMARKS. — The ultimate purpose of many
surveys is to obtain certain numerical results to represent
quantities such as areas or volumes. In the section on Survey-
ing Methods it has been pointed out that in all surveys there
should be a proper relation between the precision of measure-
ment of the angles and distances. To secure final results to any
given degree of precision, the measurements in the field must be
taken with sufficient precision to yield such results. In com-
puting from a given set of field notes the surveyor should first
determine how many places of figures he should use in the com-
putations, the aim being to obtain all the accuracy which the
field measurements wiU )deld without wasting time by using
more significant figures than are necessary. Professor Silas W.
Holman* in the preface to his "Computation Rules and Loga-
rithms" says: — "It would probably be within safe limits to
assert that one-half of the time expended in computations is
wasted through the use of an excessive number of places of fig-
ures, and through failiu-e to employ logarithms."
Final results should be carried to as many significant figures
as the data will warrant and no more. In order to insure the
desired precision in the last figure of the result it will usually be
necessary to carry the intermediate work one place further than
is required for the final result.
350. The number of significant figures in the result of an
observation is the number of digits which are known. For in-
stance, if a distance is recorded as 24,000 ft. when its value was
* See " Computation Rules and Logarithms," by Professor Silas W. Holman,
published by Macmillan & Co., New York.
323
324 COMPUTATIONS [Chap XH.
obtained to the nearest thousand feet only, it contains but two
significant figures. The zeros are simply put in to show the
place of the decimal point. If, however, the distance has been
measured to the nearest foot and found to be 24,000 ft. there
are five significant figures, for the zeros are here as significant as
the 2 or 4. Similarly a measurement such as 0.00047 contains
but two significant figures, the zeros simply designating the posi-
tion of the decimal point, for, had this same value been recorded
in a unit jj^-^^j^-^jj^ as large the result would have been 47.
Again, if a series of rod-readings are taken on different points
to thousandths of a foot and three of the readings are 4.876,
5.106, and 4.000 it is evident that each of these readings contains
four significant figures; if each of them is multiplied by 1.246 the
respective results are 6.075, 6.362, and 4.984. But had the
results been measured to the nearest tenth of a foot and found to
be 4.9, 5.1, and 4.0 these values when multiplied by 1.246 should
appear as 6.1, 6.4, and 5.0. This illustration indicates the proper
use of significant figures. Since the rod- readings 4.9, 5.1, and 4.0
are reliable only to about 1.5 to 2 per cent, the multiple 1.246
should be used in this computation as 1.25. Similarly in the use
of such a constant as 7r=3.i4i5927 it is a waste of time to use any
more significant figures in the constant than exist in numbers with
which the constant is to be combined in the computation.
351. In deciding how many places of decimals to use in the
trigonometric functions the student should examine the tabular
differences and determine what percentage error is introduced
by any error in an angle. For example, suppose an angle of a
triangle to have been measured in the field to the nearest minute.
There may be an error of 30 seconds in this angle, and it will be
seen from the table of natural sines that the tabular difference
for one minute in the fourth decimal place varies from 3 for a
small angle to less than i for a large angle, and that the vari-
ation is about the same for cosines, and for tangents and cotan-
gents of angles under 45°. Then for half a minute the differ-
ence will be, on an average, about i in the fourth place. There-
fore, in general, four places will be sufficient when the angles
have been measured to the nearest minute only. But if there
are several steps in the computations it may be advisable to use
SIGNIFICANT FIGURES $2$
five-place tables. Similarly it can be seen that five-place tables
of functions will, in general, give angles to the nearest lo seconds,
and six-place tables to the nearest second. These are only aver-
age results and are intended to give the student a suggestion as
to how to decide for himself whether to use four, five, or six-
place tables. It is obviously a great saving of time to use four-
place tables where foiir places are needed rather than to use six
or seven-place tables and drop oflF the last two or three digits.
The alnount of labor increases about as the square of the number
of places in the tables, i.e., work with 6-place tables: work with
4-place table = 36 : 16.
352. The following simple examples illustrate the useless-
ness of measuring the distances with a precision which is incon-
sistent with that of the angles, when the angles are to be used in
the computation of other distances. Given the measurements
shown on Fig. 143. If the angle B was measured to the nearest
minute only there may be an error of
30 seconds in this angle and the tab-
ular difference for 30 seconds for the
sine and cosine of this angle in four-
place tables is o.oooi; therefore use
four-place tables. In this case it is
evident that the 0.02 on the hypo-
tenuse distance is of no value what-
ever in determining the length of the
other two sides a and 6, that the 0.6
being the fourth significant figure
should be retained, and that the resulting length of a or 6 will not
be reliable to more than four significant figures.
log 773.6 = 2.8885 log 773.6 = 2.8885
log cos 44''i2' = 9.8555 log sin 44^12' = 9.8433
log a = 2.7440 log b = 2.7318
a = 554.6 b = 539.3
If it is assumed, however, that the angle B is measured by
repetition and found to be 44°i2'25'^ the error in the original
angle then was about 25^^. By using the same value for the hypo-
326 COMPUTATIONS [Chap. XIL
tennse (773.6) and six-place tables to secure greater precision the
value of a is 554.5 and of b 539.4. Comparing these results with
those obtained above will give a good idea of the error in length
of these lines due to reading the angle to the nearest minute only
and also a proper conception of the fallacy of computing with
tables of more than four places when the angles are read to the
nearest minute only. The difference between the values of a
and b obtained by use of the angle 44° 12' and similar results by
use of 44° 12' 25^^ is due entirely to the 25^^ and not to the fact that
four-place tables were used in the former case and six-place
tables in the latter, for in both cases the result has been obtained
to four significant figures only.
It is also evident that when the angle B was measured to the
nearest minute it was inconsistent to measure the hypotenuse
closer than to the nearest tenth of a foot. But if angle B was
measured to the nearest 10 seconds the line AB should have been
measured to the nearest hundredth. It should not, however,
be assumed that in all cases where 'angles are only measured to
the nearest minute the sides should be recorded to tenths of a
foot. It is the percentage error in the measurement of the sides
which must be the same as the percentage error in the angles.
If the sides are very short, they should be measured to hun-
dredths of a foot to be consistent with angles to the nearest min-
ute. In general, when the angles are read to nearest minute
only, the sides should be measured to four significant figures;
with angle to nearest 10 seconds they should be measured to
five significant figures; and with angles measured to i second the
sides should be measured to six significant figures. All the sides
of a triangle of considerable size might be measured to hun-
dredths of a foot, the angles being recorded to the nearest minute
only, and the distances used for the computations, the angles
serving merely as checks; this, of course, is practicable at times.
353. In Fig. 144 the angle is measured to the nearest minute,
but the distance is measured to him-
dredths of a foot. In this case we
are to determine the length of a long
line from a short one and the error
in the short Une is therefore miilti-
LOGARITHMIC OR NATURAL FUNCTIONS 327
plied several times. The same degree of precision should be
secured in the measured line BC as is desired in the computed
lines AC or ABy which, it is assumed in this case, is required
to four significant figures. In order that the measurements of
line BC and angle A may be consistent with the precision of
the required result, BC should be taken to the nearest hundredth
of a foot and angle A to the nearest minute. In this computa-
tion four- place tables should be used and the value obtained for
AC ox AB should be recorded only to four significant figures.
log 12.34 = 1. 0913
log tan 3°56' = 8.8373
log AC = 2.2540
AC = 179.5
If i4C is desired to the nearest hundredth of a foot the angle
A might be determined closely by repetition, but this will not
give the length ACio the nearest hundredth unless BC has been
measured closer than to the nearest hundredth; for, suppose
there is an error of 0.005 ft. in the measurement of 5C, then the
line AC being about 15 times as long as BC will have an error of
0.075 ft- ^^ matter how exact the angle at A may be measured.
In other words, if -4 C is desired correct to five significant figures
BC should contain five significant figures. Evidently the prac-
tical way of obtaining an exact value for the inaccessible dis-
tance i4C is to measure AB io the nearest hundredth, and to
compute AC from AB and BC, using the angle at -4 as a check
on the measured distances. In both of the above examples it
is assumed that the 90*^ is exact.
354. LOGARITHMIC OR NATURAL FUNCTIONS. — The ques-
tion as to whether logarithmic or natural functions shall be
used will depend upon the computation in hand. Many sur-
veyors have become so accustomed to using naturals that they
will often use them when logarithms would require less work
and offer fewer opportunities for mistakes. Each method has
its proper place, and the computer must decide which will be the
better in any given case. The use of logarithms saves consid-
erable time spent in actual computation because the process is
328
COMPUTATIONS
[Chap. XIL
simpler, but, on the other hand, looking up the logarithms con-
sumes time. The result is in many cases, however, a saving of
time over that required to do the arithmetical work of multi-
plying or dividing. While the multiplication of two numbers
of three or four digits each can possibly be done directly more
quickly than by logarithms, still it takes more mental effort and
there is more opportunity for making mistakes ; but in case sev-
eral such multiplications are to be made logarithms are almost
always preferable. Furthermore when there are several multi-
plications of the same number logarithms will save time since the
logarithm of this common number has to be taken from the table
but once. Frequently, however, the computation is so simple
that the use of logarithms would be almost absurd, e.g., the multi-
plication of any number by a simple number like 20, 25, 150, or
500. If a function of an angle is to be multiplied or divided by
any such number the natural
function should of course be
used.
355. SHORTCUTS.— The so-
lution of a right triangle, when
one of the angles is small, involv-
ing the use of the cosine of this
small angle, can often be more
easily obtained by the use of the
versed sine or external secant of
Fig. 146. the angle. In Fig. 145
Vers A -
CD
AB
ExsecA«^
AB = 207.42
A = f 32'
AC =207.42 cos 7^32'
But AC = AB - CD
= 207.42 — 207.42 vers ^^ 32'
= 207.42— 207.42 X 0.00863
(207.42 X 0.00863 = 1.79. by slide rule.)
= 207.42 — 1.79
- 205.63
(I)
(2)
SHORT CUTS 329
Obviously, when the angle is quite small, the result of the multi-
plication indicated in ( 2) can be taken from the table to the near-
est hundredth of a foot with much less efiort than is required for
the computation called for in (i). In fact, the computation in
(2) can often be done more quickly by the use of natural num-
bers than by logarithms, and in most cases the slide rule will
give results suflSciently exact (Art. 359, p. 330).
Had AC been given (205.63) and the angle -4, (7° 32') then
COS 7*^ 32
But AB =- AE + EB
= 205.63 + 205.63 exsec 7° 32'
= 205.63 + 205.63 X 0.00871
(205.63 X 0.00871 = 1.79, by slide rule.)
= 205.63 + 1.79
= 207.42
356. There are many "short cuts" in arithmetical work
which are of great value to the computer, and the student should
endeavor to learn the most common and simple ones. The fol-
lowing are a few illustrations.
247 X 100 24700
247X25 = -^^— = -^
4 4
68200
682 X so =
694 X 150 = 69400 -f 34700
927 X 62.5 = 92700 X I
672 X 1002.3 = 672000 -f 1344 -h 201.6
547 X .9968 = 547 (i - .0032) = 547 - 5.47 X .32
-^ == -^ (reducing minutes to decimals of a degree)
60 o
-^ = 8.43 X 8
12.5 ^^
330 COMPUTATIONS [Chap. Xn.
The student should cultivate the habit of performing men-
tally as much of the work as can be done without fatigue, delay,
or danger of mistakes. No hard and fast rule can be laid down
in this matter, as some persons have more aptitude than others
for work of this kind. Such subtractions as i8o**— 36^47' iS*'
should always be performed mentally. Also in taking the co-
logarithm of a number from a table of logarithms the result
should be written down directly.
357. ARRANGEMEIfT OF COMPUTATIONS. — All survey-
ing computations should be kept in a special computation
book. At the head of the page should appear the title of the
work, the number and page of the field note-book from which
the data are copied, the names of the computer and checker,
and the date. The work should be arranged neatly and system-
atically so that every part of the computations can be traced
by any one who is familiar with such work. Where possible the
work should be so arranged that numbers will have to be written
but once. Each important value, each column, etc. should be
labeled so that it can be readily found.
358. CHECKS. — It is very important that all calculations
should be checked, not merely at the end of the computation
but also at as many intermediate steps as possible. In this way
a great waste of time may be prevented and serious mistakes
avoided. One good method of checking is to perform the oper-
ations when possible by two independent methods, for example,
by the use of logarithms and by natural functions. Very often
two men do the computing, one man's work acting as a check
on that of the other. The two may each work by the same or
by different methods, and the results may be compared at inter-
vals. Every part of the work should be done independently, from
the copjring of data out of the npte-book to the final results. It
is not uncommon to find two men computing the same area
where only one of them looks up the logarithms. In case a mis-
take is made in looking up the logarithms the results may check
but both are wrong. The computer should also check his work
roughly by estimating approximately what the result should be.
359. SLIDE RULE. — A valuable aid in checking calculations is
an instrument known as the slide rule, which enables the computer
SUDE RULE 331
to multiply and divide numbers by logarithms by a purely mechan-
ical process. It is really the equivalent of a table of logarithms.
It consists of a wooden rule, usually about 10 inches long, having
a groove in one side in which runs a small wooden strip called the
slide. On one face of the rule are placed two scales, A and J9,
Fig. 146, one above and one below the slide which is indicated by
iHilmiDinN I [i [iti[i[iti[iiii|iin[mi|im[ I [l[i[i[iDD[t
)l]l]l]l]l3l]l]l]l[llll|lill[lllj|llll[lMl|llll[ I [ I [I [I [l[s
Fig. 146.
B and C. These are constructed by plotting logarithms of num-
bers by subdividing a unit of some convenient length, say 10
inches. For example, the log of i is o, so this is taken as the
left end of the scale and the number i placed at this point. The
log of 2, to three significant figures, is 0.301, and a line is placed
therefore at a distance equal to ^^ of the 10 inches, or 3.01
inches, and marked with the number 2. Similarly at 4.77 (log
3 = 0.477) a line is marked 3. In this way the logarithms of
other numbers are plotted. The space between i and 2 is
subdivided by plotting log i.i, log 1.2, etc. The subdivision is
continued until the spaces are as small as will admit of rapid
and accurate reading of the scale.
It is customary to make the spacing on the upper scale just
half that on the lower, i.e., if 10 inches is chosen as the unit for
the lower scale, then the unit for the upper scale will be 5 inches.
Since the length of this upper sca\e is only half the length of the
rule there ^re usually two scales exactly alike marked on the upper
part of the rule, the right end of one coinciding with the left end
of the other.
On the slide are two scales, B and C, exact duplicates of those
on the rule and so placed that when the end line of the scale B
on the slide is placed opposite the end line of the scale A on the
rule, every line on the slide is exactly opposite its corresponding
line on the rule. A runner is usually attached to the rule for
convenience in setting and reading the scales. This runner is
a small metal slide which fits over the face of the rule in such
332 COMPUTATIONS [Chap. Xfl.
a way that it can be slid along the rule and set at any reading of
the scale. It is usually provided with a fine line running cross-
wise of the rule which is used in marking the exact setting.
Multiplication or division of numbers is performed by add-
ing or subtracting the scale distances corresponding to these
numbers. The scale distance is the logarithm of the number.
Adding two scale distances is, in effect, adding two logarithms,
and the resulting scale distance is the logarithm of the number
marked opposite on the scale. For example, if the left end of
scale C, Fig. 147, is set opposite the number 2 of the scale Z),
then opposite the number 3 on scale C, is found the product, 6,
on scale D. The distances which have been added are those
corresponding to log 2 and log 3 respectively. The sum of
these distances is the distance corresponding to log 6. Division
is performed by placing the divisor on scale C over the di\adend
on scale D and reading the result, opposite the end of the scale
C on the scale D.
Fig. 147 shows the position of the scales for dividing 6 by 3.
I
1I 1I illlilllilif.l.l.i.i.l I I.I I |*ii.iilM.nrM.M|ir.nr^/
iimfiitiliiiir I. f .l.f.ifj.f.
^■j.j.j,j.j.jg.j.>tniiiiiiiliiniiiiiC.i.iihi.|,'v'C'va'V'Ova'v^^^
Fig. 147.
The scales A and B may be used in a like manner. It is evident
that, by setting the runner on the result of one operation and
then moving the slide so that one of its ends coincides with the
runner setting, continued multiplication and division can be
performed without the necessity of reading intermediate results.
Scale D may be used in connection with scale A for obtain-
ing squares or extracting square roots. Since the spaces on scale
A are one-half those on scale D the number 4 on scale A is oppo-
site number 2 on scale D, 9 is opposite 3, and so on, every num-
ber on scale A being the square of the corresponding number on
scale D. Other scales, generally log sines and log tangents, are
placed on the reverse side of the slide, so that trigonometric cal-
culations can also be performed with this instrument. Results
REDUCING FIELD NOTES 333
obtained with the ordinary 10 inch slide rule are usually correct
to 3 significant figures, so that this slide rule is the equivalent
of three-place logarithm tables.
360. Thacher Slide Rule. — The Thacher slide rule consists
of a cylinder about four inches in diameter and eighteen inches
long woi king within a framework of triangular bars. On these
bars is fastened a scale corresponding to the scale on an ordi-
nary slide rule, and on the cylinder is marked another scale like
that on the bars. The cylinder is the slide and the triangular
bars form the rule. This rule is operated in a manner similar
to the one explained above. Results can be obtained with it
which are correct to four and usually to five significant figures.
361. REDUCIIIG THE FIELD NOTES FOR COMPUTATIONS.
— Before any of the computations are made the measure-
ments taken in the field frequently have to be corrected on
account of erroneous length of tape. This correction can usu-
ally be made mentally when the distar ces are transcribed into
the computation book. The errors in the angles are balanced
by altering the value of those angles which were taken from short
sights since the angular errors are most Ukely to occm: in these.
In some cases, where it has been found desirable to take meas-
urements on a slope, these distances are reduced to horizontal
distances by multiplying them by the versed sine of the vertical
angle and subtracting the result from the corrected slope dis-
tance; the correction for error in the tape being made before this
is done. Sometimes instead of a vertical angle the slope dis-
tance and the difiFerence in elevation between the points are the
data contained in the field notes. In this case the formula given
in Art. 20, p. 13, should ordinarily be used.
362. Curved boundary by offsets, — The offsets to the
brook (Fig. 53, p. 104) were taken at regular intervals in one
portion of the survey and in another portion offsets were taken
at the points where the direction of the brook changes. The
offsets which were taken at regular intervals give a series of
trapezoids with equal altitudes the area of which can be obtained
by one computation. Although there are several approximate
rules for this computation the two most common are what are
known as the Trapezoidal Rule and Simpson^ s One-Third Rule.
334 COMPUTATIONS [Chap. XII.
363. Trapezoidal Rule. — If the figure is considered as made
up of a series of trapezoids their area can be found by the follow-
ing rule: —
Area = d (^ + 2 A + ^)
where d = common distance between offsets,
Ae and h\ = end offsets of the series of trapezoids,
and 2 A = sum of the intermediate offsets.
364. Simpson's One-Third Rule. — In the development of
this formula the curved line is assumed to be a parabolic curve.
It is claimed by some that this afiFords results more nearly correct
than the Trapezoidal Rule, although for most problems of this
kind, where the offsets at best can give but an approximate
location of the boundary, frequently a brook or crooked wall the
center of which must be estimated, it is quite probable that the
Trapezoidal Rule is sufficiently exact. Simpson's One-Third
Rule is as follows: —
Area = - (A. + 2S Aodd + 4S Aeren + A'e)
where d = common distance between offsets,
Ae and h\ = end offsets of the series,
2 2 hodd = twice the sum of all the odd offsets
(the 3d, 5th, 7th, etc., from the end)
4 2 Aevcn = twice the sum of all the even offsets
(the 2d, 4th, 6th, etc., from the end).
For this rule to apply there must be an even number of
trapezoids; if there is an odd number, an even number of them
may be computed by this rule and the extra trapezoid must be
computed separately. Or, if there is a triangle or trapezoid at
the end of this series, which has a base greater or less than d,
it must also be computed separately.
TRAPEZOIDAL RULE
335
Fig. 148 shows the computation of a series by both methods
and also the computation of several trapezoids and triangles at
the ends of the series. The data are taken from the field notes
in Fig. 53, p. 104.
I
z
B K V
-ByStmpsoi7''S Offe^/ref fii/M
r
m
38.S
3cs'.5*
0/ Jftfp^zo/cfa/ /fuM
if
6jO
AS
4.0
^f^MS^niit
\0I9
230
/03
383
/03A-
&6
f?
pArg
/6,ZK.9Z7s/3j^^Z
aA>.7xZ7Za Z9/ ¥S/XZU
ZSowxB*^
Fig. 148.
336 COMPUTATIONS [Chap. XII.
365- STRAIGHTENING CROOKED BOUNDARY LINES. — In
Fig. 149, AEFGH represents a curved boundary between two
Fig. 149. Straightening a Crooked Boundary.
tracts of land, and it is desired to run a line from A so as to make
the boundary a straight line and to leave each tract of the same
area as before.
The trial line ABh first run, and the distance AB^ the angles
at A and 5, and the necessary offsets to the curved boundary are
measured in the field. Then the areas of the property between
this trial line and the curved line are computed as explained in
the previous articles. The sum of the fractional areas on one
side of the trial line and the sum of the areas on the other side of
it should be equal. If not made so by the trial line, the difference
between these sums is the area of a correction triangle ABC
which must be taken from one tract and added to the other.
The area and the base AB being known the altitude dC can tc
computed. Then in the triangle ABC, the lines BC and AC
and the angle at A are calculated; and the line AC is staked out,
its calculated length being checked by measuring the line AC
in the field and the angle at A being checked by the measured
distance BC.
366. AREA BY TRIANGLES. — If the field has been sur-
veyed by setting the transit in the middle of the field and taking
angles between the corners (Art. 138, p. 105), the areas of the
triangles may be foimd by the trigonometric formula:
Area = J a 6 sin C,
where C is the angle included between the sides a and h.
If all three sides of any of the triangles have been measured
STRAIGHTENING CROOKED BOUNDARY
337
or if the fidd has been surveyed with the tape alone (Art. 139,
p. 106), the area of the triangles can be found by the trigono-
metric formula: —
Area = V5 (5 - a ) (^ - 6 ) ( j — c )
a + b + c
where a, 6, and c are the sides and 5 = -
367. AREA OF A QUADRILATERAL BY TRIANGLES.—
Most city lots have four sides, and while the Double Meridian
Distance Method (Art. 384, p. 352) is often employed in com-
puting their areas, it is not at all uncommon in computing such
quadrilateral lots to divide them into triangles, checking the field-
work and computations, and computing the areas by triangles.
In Fig. 150, A BCD represents an ordinary city lot in which
all the sides and angles have
been measured. It is evident
that the diagonal BC can be
computed either from BDy
CD, and the angle Z), or from
AB, AC, and the angle A.
These two determinations of
BC should check each other.
Similariy two independent de-
terminations oi AD can be
found. These evidently check
all the fieldwork and calcula-
tions as far as they have gone.
In computing these triangles
the best way is to resolve all
the work into right triangle
calculations, as suggested by
the dotted lines on the figure.
Not only is this method more simple than to use the oblique
triangle formulas, but it gives at the same time altitude distances
which are useful in computing the area of the lot. The area
can be obtained by calculating the area of one pair of triangles
and readily checked by calculating the other pair.
338 COMPUTATIONS [Chap. XII.
368. AREA OF CURVBD
CORNER LOT. — In Fig. 151,
ABFHGDE is the boundary
of a corner lot, all the angles
and distances of which have
been determined in the field. The
area of ABCDE can be easily
computed by the method ex-
plained in Art. 384, p. 352.
Then the area of FCGH must be
subtracted from the traverse
^^®- ^^^- area. The angle / is known and
the radius KF of the curve is given or can be computed from data
such as CH or CF obtained in the field (Art. 257, p. 233),
KFHG = EmjiHK ^ P X O.OI74533* X HK ^^^^
2 2
VI, p. 506.)
KFCG ^ FC X FK
FCGH = KFCG - KFHG
The area of FCGH could have been calculated by comput-
ing the area of the triangle FCG and then subtracting the area of
the segment FHG from it. The area of this segment, however,
cannot be calculated accurately by any short formula. An
approximate formula for the area of a segment is
2
Area of Circular Segment = - MC (approximate), where M is
3
the middle ordinate and C is the chord length.
M =
Expressed in terms of C and R,
Area of Circular Segment = (approximately).
I2/v
* The length of the arc of curve whose radius is i and whose central angle
is I® is 0.0174533, which will give results to six significant figures, provided /and
/^ are correct to six significant figures.
t In Fig, 152, OB =» Radius of circular curve.
C/f «= Middle Ordinate for chord AB,
CD is drawn tangent to the curve.
M = —— ' (approximately).
AREA OF CURVED CORNER LOT 339
These formulas are fairly accurate when M is very small a&
compared with C. They are most useful, however, as a check
on computations made by the prec ding method.
369. ROUGH Checks on areas. — if the traverse has
been plotted to scale, it can be easily divided into simple figures
such as rectangles or triangles, their dimensions scaled from the
plan, and their areas computed, thereby giving an independent
rough check on the area.
A piece of tracing cloth divided into small squares can be
placed over the plan of the traverse and the number of squares
counted and the fractional parts estimated, generally to tenths
of a square, by inspection. Then the area of one square being
known an approximate area of the traverse may be obtained.
370. Plardme er. — One of the commonest ways of checking
the area of a traverse is to obtain its area by means of an instru-
DB « Tangent Offset for chord CB.
OE is drawn perpendicular to CB,
In the two similar triangles OEB and CBD^
DBiCB^BEi OB
DB',CB ^ — lOB
2
DB^^-^
2 OB
Offset from Tangent - J^^^^
a X Sadins
But DB « CH, and AB ^zxCB (approximately) (1)
\ 2 I AB*
•• ^^" 2 OB " 8aS (approximately)
Middle Ordinate - 3 ^^^^1^, (approximately) (2)
The following will give some idea of the accuracy of this formula:
When radius — 20 ft. and chord » 10 ft, M -s 0.625, (correct value is 0.635).
When radius — 100 ft and chord » 25 ft., M » 0.781, (correct value is 0.784).
When radius — 100 ft. and chord ■■ 100 ft., M ■■ 12.500, (correctvalue is 13.397).
When radius ^ 1000 ft. and chord «» 100 ft., M a 1.250, (correct value is I.251).
It is evident from the above that this formula will not give accurate results
when the chord is large in comparison with the radius.
34^ COMPUTATIONS [Chap. XIL
ment called the planimeier^ which is in principle a mechanical
integrator. It is a small instrument consisting of an arm, carry-
ing a tracing point, which is fastened to the frame of the instru-
ment; the arm can be adjusted to any desired length. The
frame touches the paper at only two points; one, the anchor
point, and the other, the circumference of a small wheel which is
free to revolve. On the rim of this wheel is a scale which is read
by means of a small vernier. The length of the arm can he
regulated by setting it at the proper reading on a scale which is
marked on the arm, so that a unit on the wheel scale will repre-
sent any desired unit area such as a square inch or a square
centimeter.
In using the instrument the anchor point is set at some con-
venient position on the drawing outside of the area to be measured
and then the tracing point is run around the perimeter of the area
to be determined. The reading on the wheel is recorded when
the tracer is at the starting point. The tracer, in passing around
the perimeter, should be kept as closely as possible on the bound-
ary line and should return exactly to the starting point. Then
the scale is again read, and the difference between the two read-
ings is the area which has been traced out, expressed in some unit
depending on the length of the arm. The result can be easily
transposed into the unit of the scale of the map.
Usually the settings for the scale on the arm are furnished by
the maker for various units of area. It is safer to test this setting
by running the instrument around a known area, such as 4 square
inches and determining the interval passed over by the wheel by
making several tests and by setting the anchor point at different
positions. This interval divided by 4 will be the value of one
square inch of plan area and this is equivalent to a certain number
of square feet of surface, depending upon the scale of the map.
It is important that the sides of the trial square should be laid off
so that they agree with the present scale of the map which, owing
to swelling or shrinking of the paper, is frequently not quite the
same as when it was first drawn (Art. 479, p. 428).*
♦ When areas are desired from U. S. Geological Survey maps on which are
sho^-n parallels of latitude and longitude it is best to refer all planimetered areas
to the areas of a quadrilateral, say, i^ on a side. The area of such quadrilateral
DEFLECTION ANGLES AND CHORDS
341
371. DEFLECnON ANGLES AND CHORDS FOR A CIRCULAR
CURVE. — The computations shown in Fig. 153 refer to the notes
in Fig. 104, p. 237. In the discussion of the simple curve as
6/VEN:-/f'20O, curve to Right, I'Si^-JS'-iO^^PC, ^/e^TZM
Width of Street 70^
T^/rtBfn. 2^^47*40"
iZO0X.^6330«Se.66T
Si* ^.B90H79
SS'^.Oio/8//
.9003360 K2oo^/eo.oaLc
Def/ection Arrg/es,
DefkcHanJL for Soft ^^ x ^S*^7Uo''^^xZS:79^
Log 7zssin:-^tto496
DmfiecHonL for3ao6H:^33SiLyjkH.f6rSaf Log laoM '^^^
eo
Log.60/6 •
3/:
.4*ie'3/'*defl.3a09it,
43:
T09'43''deff.Mff.
/d^Z2MsiZi'Z9'/0
4-ta-3o
RT ia^SlS^^W-^r-^CCheck^
Chords
SOthArc.
Sln7"09'4o"^JZ467
44?0
i/Z47xZ^ s
49868 Cent^Chd.
&7Z7-
S8SB Uft Chd,
41.14 /fight Chd.
30,08 fk Arc
^in4^ie'3o^J07StZ.
3a048(MChd.
\0JS/XZK3Ss> S.2S7^
3S.3/ UftChd,
24.79 Right Chd.
Fig. 153.
can be taken from a publication entitled Geological Tables and Formulas, by
S. S. Gannett, Bulletin No. 232, U. S. Geological Survey, and by simple proper*
tion the desired area found.
34^ COMPUTATIONS [Chap. XIL
applied to city surveying (Art. 259, p. 234) will be found the for-
mulas which have been used in the computations in Fig. 153.
The length of the curve L^ is found by taking from Table VI,
("Lengths of Circular Arcs: Radius — i "), the length of an arc
for 51^, for 35', and for 20^ successively and adding them, which
gives the arc of a curve whose radius is i and whose central angle
is 51^ 35' 20^. This is then multiplied by the radius (200) which
gives the value of £c, which is added to the station of the P.C. to
obtain the station of the P.T.
372. COMPUTATION OF OBSERVATIONS. — The computations
relating to observations for meridian and latitude will be found
in Chapter VII.
COMPUTATION OF VOLUME.
373. BORROW-PITS.* — Fig. 154 is a plan of a portion
of a borrow-pit, at the corners of which the depth of excavation
is marked in feet and tenths. Each of the regular sections of
earthwork is a truncated rectangular prism whose volume is
equal to the average of the four comer heights multiplied by the
area of the cross-section, or expressed as a formula.
Volume Truncated Rectangular Prism = .4 X ^t + h» +^8+^4
4
where A is the area of the cross-section and A„ A„ A,, and h^ are
the comer heights.
For a truncated triangular prism such as dbcy using the same
notation,
h + h •{• h
Volume Truncated Triangular Prism = i4 X ^•
In computing a trapezoidal prism, such ssfdhg, the trapezoid
is subdivided into a rectangle /sA^ and a triangle /de; or for jhds,
into two triangles by diagonal lines, as jhs and kds and their
volumes may be computed by the above formula.
When there are several prisms with the same cross-section, as
shown in Fig. 154, these rectangular prisms can be computed as
one solid by assembling them as follows : — multiply each comer
* For a complete discussion of the computation of Borrow-Pits see Railroad
Curves and Earthwork by Professor C. F. Allen, published by Spon & Chamber-
lain, New York.
BORROW nrs
343
height by the number of rectangular prisms in which it occurs
and then add these results and divide by 4. This is then multi-
plied by the area of the cross-section of one prism. For example,
in Fig. 154, the quantity bounded by amnrsja can be foimd by
m
^
^
•P
Fig. 154. Plan of Portion of a Borrow-Pit.
one computation because it is composed of a series of prisms
having the same cross-section. Li the summation of the heights,
those at a, m, n, f , and s are taken but once, those at such points
as c, g, A, etc. are multiplied by 2, at j the height is multiplied by
3, and at such points as k it is mutliplied by 4.
Where the excavation is completed to a certain level, as in a
cellar, it is a special case of above. The area of the cellar can be
344 COMPUTATIONS [Chap. XIL
divided into rectangles, their comer heights taken, and from these
the volume can be computed.
374. VOLUME OF PRISMOID. — The data obtained from
field notes are usually in the form of cross-sections which are
taken at right angles to some general line of the construction,
thereby dividing the earthwork into prismoidal solids with their
bases parallel and their sides either plane or warped surfaces.
The bases of the solids are the cross-sections which are obtained
by taking sections of trench excavation or of road construction
(Figs. 91 and 92, p. 208).
' 375. End Area Formula. — The simplest method of com-
puting the volume of a prismoidal solid is to average the areas of
the two bases and multiply by the distance between them, which,
expressed as a formula, is
V = di_±A« X / {End Area Formula)
2
in which A^ and A^ are the areas of the two end bases and / is
the distance between them. This method is used to a very great
extent throughout the country, although it does not give suflS-
ciently accurate results for certain classes of work.
376. Prismoidal Formula. — The correct volume of a pris-
moid is expressed by the Prismoidal Formula:
Volume of Prismoid = -(i4, + 4i4n» +-4,)
in which /is the distance between the two bases, A^ and -4,; and
ilm is the '* middle area,^^ i.e., the area half-way between the two
bases, which is obtained by averaging the corresponding dimen-
sions of the two end areas, A^ and -4,; it should not be taken as
the mean of A^ and A^.
377. The end areas can easily be computed by dividing them
into triangles as shown in Fig. 155, the area of which can be
found readily from the dimensions given in the field notes.
VOLUME OF PRISMOID 345
20.0 2I.S
Notes of section: . . + 4-0 ", —
+60 +1.0
Area = 4 X (21.5 + 29) , 20 X (i + 6)
2 2
= 2 X 50.5 + 10 X 7 = 171.
It is also the custom with some surveyors to plot each section
carefully to scale and to obtain its area by use of the planimeter
(Art. 370, p. 339). This is probably the most practical method
when the sections are very irregular since the field work does
not warrant the use of very accurate methods.
There are several other methods employed in computing
earthwork but the above are by far the most common.
Several sets of Earthwork Tables and Diagrams have been
published which reduce the work of computation very materi-
ally.
378. ESTIMATES FOR GRADING.— Estimates for grad-
ing may be conveniently made by means of a topographic map.
On this map will appear the contours of the original surface.
The contours representing the finished surface are also sketched
upon the map, and the smaller the interval between the contours
the more accurate will be the result. In Fig. 156 the full lines
represent the contours of the original surface which is to be al-
tered so that when the necessary cutting and filling has been done
the new surface will have the appearance indicated by the dash
contours. At contour 20 and at contour 25 no grading is to be
done. On the plan, first sketch the lines ABCDEF and
AGHIJB which are lines of "no cut '* and "no fill,'* i.e., lines
which enclose areas that are either to be excavated or filled. The
amount of excavation and embankment must be computed sepa-
rately. In sketching such lines the lines AB^ ED, and Hi, as
will be seen, follow the intersection of the original contours with
the new ones, since at these points there is no cut or fill. There
are no direct data on the plan which define where the earthwork
ends at C but the assumption is here made that the fill will run
out to meet the original surface at about the next contour at C.
In this example the fJl must run out somewhere between the
24-ft. contom: and the 25-ft. contour, for if it ran beyond the 25-ft.
34fi
COMPUTATIONS
[Chap. Xn.
contour there would be another new 25-ft. contour shown on
the plan. Therefore the line BCD has been sketched to repre-
sent the limits of the fill in that vicinity; similarly EFA^ AGH,
and IJB have been sketched.
There are three general methods of computing the earth-
work from the data given on the plan; (i) by computing directly
the amount of cut or fill between successive contours, (2) by
25 -."-'t '^'^
Scale
Fig. 156.
assuming a horizontal plane below the lowest part of the earth-
work and computing the volume of the earth between this plane
and the original surface, then computing the volume between
the same plane and the finished surface; the difference between
these two volumes will be the amount of earthwork, or j) by
drawing on the plan a line of no cut or fill, a line representing,
VOLUMES FROM CONTOURS S47
say, 5 ft. cut or fill, a line representing lo ft. cut or fill and so om
Then compute the volume between these successive 5-ft.
layers.
379. (i ) Referring to Fig. 154 and applying the first method,
the volume of the solid AMPE is that of a solid having two paral-
lel end planes AKEL (a plane at elevation 21) being the lower,
and MNPO (a plane at elevation 22) being the upper pi ne.
The altitude between these two end planes will be the difference
in elevation between 21 and 22, or will be i ft.
The areas of the horizontal planes AKEL, MNPO, QRST,
and BUDV may be obtained by planimeter (Art. 370, p. 339) or
otherwise, and the volume of the solid AKEL-MNPO may be
obta.ned by the End Area Method (Art. 375, p. 344), its altitude
being i ft. If it is desired to obtain the volume by the use of
the Prismoidal Formula the volume of the solid AKEL-QRST
may be found by using AKEL as one base, QRST as the other,
and MNOP as the middle area, the altitude, or length, of the
solid being the difference between 21 and 23, or 2 ft. The solid
AKEL-F may be considered to be a pyramid with a base AKEL
and an altitude equal to the vertical distance between the con-
tour 21 and the point F which is in this case on contour 20, or
a vertical distance here of i ft.
Example.
In Fig. 154 the amount of fiU on the area ABCDEFrs computed below.
Area AELF — 900 sq. ft. 900 x } — 300 cu. ft. (Pyramid)
" MNPO - 1000 _rlr_Z X I - 9SO.
** QRST - 1020 looo -f 1020 ^ J _ j^j^
- BUVD - 680 '^^^-^ ^ X 1 - 850.
2
680 X § » 23a (Pyrandd)
3)3340- cu. ft.
9)1113
124. cu. yds. Total FiU.
380, (2) Referring again to Fig. 156 and applying the sec-
ond method, the area of ABCDEF is found (by planimeter) ; this
is the area of a plane at, say, elevation 20, since none of the fill
348 COMPUTATIONS [Chap. XIL
extends below contour 20. Then the area of A BCD EL is found,
which is the area of the plane cutting the original ground at ele-
vation 21. Similarly the areas of MBCDPO, QBCDST, and
BCDV are found. The volume of the solids between these
planes may be computed by the End Area Method or by use of
the Prismoidal Formula, in which case every other contour
plane is used as a middle area as explained in the preceding
paragraph. The volume of solid whose base is BCDV is a pyr-
amid whose altitude is the vertical distance between the 24-ft.
contour and point C, which in this case is i ft.
By the same general method the areas of ABCDEK,
MBCDPN, etc., which refer to the new surface of the ground,
may be obtained, and the volume of the solids between succes-
sive contour planes computed. The difference between this
quantity and the quantity between a plane at elevation 20 and
the original surface will give the amount of fill.
While in this particular problem the first method is the
shorter, still there are cases where the second method will be
somewhat simpler. It is particularly useful when the actual
amount of cut or fill is not desired but when it is required to know
if the proposed alterations will require more or less earth than
can be easily obtained on the premises and, if so, about how
much the excess will be. In this case the portions of cut and
fill will not have to be computed separately. A line is drawn
around the limits of the entire area where the grading is to be
done, the volume between an assumed plane and the original
surface is found, and then the volume between the same plane
and the proposed surface. The difference between the two
values will give the amount of excess of earthwork.
381. (3) Fig. 157 illustrates a third method of computing
earthwork from the data given on a topographic map. The
original contours are shov n in full lines and the contours of the
proposed surface in dash Unes. Through the intersection of
the new contours with the ori£inal ones is dra^vn the line of "no
cut " (zero line), the line vhere the cut is just 5 ft. (marked 5),
the line of 10 ft. cut (marked 10), etc. These dotted curves
enclose areas which are the horizontal projections of irregular
surfaces which are parallel to the final surface and at 5 ft., 10 ft.,
VOLUMES FROM CONTOURS
349
Ftg. 167.
15 ft., etc., above the final surface. The solids included oe-
tween these 5 ft. irregular surfaces are layers of earth each 5 ft.
thick, and their volumes may be computed by either the End
Area Method or by the Prismoidal Formula as explained in the
preceding methods. The areas of these horizontal projections
are obtained from the map and the vertical dimensions of the
solids are the contour intervals.
382. ROUGH ESTIMATES. — Rough estimates of the quantity
of earthwork are often required for preliminary estimates of the
cost of construction or for monthly estimates of the amount of
work done. For preliminary estimates of road construction, very
3 JO COMPUTATIONS [Chap. XH.
frequently the notes of alignment and the profile of the center line
are the only information at hand. From this profile the center
cuts or fills can be obtained, and the cross-sections can be as-
sumed to be level sections (Art. 232, p. 209) and computed by the
End Area Method. The slight errors resulting will be corrected
in the final estimate.
In obtaining the required data from which to make an ap-
proximate estimate of the quantity of earthwork, the engineer
has an opportunity to exercise his judgment to an unusual degree.
Rough estimates do not, as a rule, call for a large amount of field-
work. It is important that as few measurements as possible
should be taken and that these should also be at the proper
places to give complete data and to allow simple computations.
Too often engineers, as soon as they arrive on the work and
before making a study of their problems, begin to take measure-
ments, consequently they return to the office after hours of
hard work with a mass of figures from which it will take several
more hours to compute the quantities. Whereas, a few mo-
ments' thought given to the choosing of the proper measure-
ments to be taken in the field would give data which could be
computed in a few moments by use of the slide rule, affording
results sufficiently accurate for rough estimates.
PROBLEMS.
1. A series of perpendtcnlar offsets are taken from a straight line to a
carved boundary line. The offsets are 1 5 ft. apart and were taken in the fol-
lowing order: 6.8, 7.2, 4.6, 5.7, 7.1, 6.3, and 6.8.
(a) Find the area between the straight and curved lines by the Trapezoidal
Rule.
(b) Find the same area by Simpson's One-Third Rule.
2. It is desired to substitute for a curved boundary line a straight line which
shall part off the same areas as the curved line. A trial straight line AB has
been run; its bearing is S 10° 15' W, its length is 418.5 ft., and point ^ is on
a boundary line CD which has a bearing S 80^ W. The sum of the areas between
the trial line and the crooked boundary on the easterly side is 2657. ft.; on the
westerly side it is 789 r. ft. It is required to determine the distance BX along
CD such that AX shall be the strsught boundary line desired. Also find the
length of the line AX,
ROUGH ESTIMATES 35 1
3. In the quadrilateral ACBD the distances and angles which were taken
in the field are as follows:
A ^—50.63 A B C— 105** 39' 00"
B C- 163.78 B A Z>-89° zf 30"
CZ>-93.8o
D A^ 160.24
/> ^-167.73
Check the fieldwork by computations, and figure the area of the quadrilateral
by using right triangles entirely.
4. Two street lines intersect at an angle (deflection angle) of 48® \f yf\
The comer lot is rounded off by a drcular curve of 40-ft. radius.
(a) Find the length of this curve to the nearest j^ ft.
(b) Find the area of the land included between the curve and the two tan-
gents to the curve (the two street lines produced).
5. Find the quantity in cubic yards, in the borrow-pit shown in Fig. 154; the
squares are 25 ft. on a side, and the line ast\& straight.
6. At station 6 a rectangular trench was measured and found to be 3 ft
wide and 4 ft. deep. At station 6+70 it was found to be 3.2 ft. wide and 8.6 ft.
deep.
(a) Find by use of the Prismoidal Formula the quantity of eaitliwork be-
tween stations 6 and 6+70. Result in cubic yards.
(b) Find the volume of the same by End Area Method.
7. The following is a set of notes of the earthwork of a road embankment.
« -J7f +^, JM
+ 8.0 +5.6
,, + 60 -3?:? 2^ ^^.o 2^ 2^
+ 10.0 +4.5 +7.5 +6.0
Sta.1, 21± +6.0 2^1
+ 40 +7.2
The base of the road is 30 ft. and the slopes are i) to i.
Tlnd by the End Area Method the quantity of earthwork from Sta. 11 to la. Re-
sult in cubic yaxds.
CHAPTER XIII.
AREA BY DOUBLB MERIDIAN DISTANCES.— COORDINATES.
383. COMPUTATION OF AREA. — The computation of the
area of any piece of property which has been surveyed as a
traverse will in general consist of (i) the computation of the area
enclosed by the traverse and (2), where the traverse does not
follow the property line, the computation of fractional areas to be
added to or subtracted from the area of the traverse as the case
may be.
384. COMPUTATION OF AREA BY DOUBLE MERIDIAN
DISTANCE METHOD. — In the field notes the length and the
bearing of each line of the traverse are recorded. To obtain the
area enclosed the points of the survey are referred to a system
of rectangular coordinates. In Fig. 158 the coordinate axes
chosen are the magnetic meridian through the most westerly
point Fy and a line through F at right angles to the meridian. In
compass surveys it is convenient to use the magnetic meridian
for one of the axes; in transit surveys the true meridian is often
used when its direction is known, but any arbitrary line may be
used as an axis and some convenience results from choosing one
of the lines of the survey as one of the axes.
In computing the area, first find the length of the projection
of each line on each of the coordinate axes, or in other words,
find the northing or southing and the easting or westing of each
line, or course, of the traverse. The projection of any line on the
meridian is called its difference of laiittide or simply its latitude.
The projection of a line on the other axis is called its difference
of departure, or simply its departure,^ In Fig. 158 the latitude
of FA is Fq; the departure of FA is qA. The latitude and de-
parture of each course are computed by solving the right triangle
formed by drawing lines through the extremities of this course
* Some authors use the terms latitude difference and longitude difference.
352
DOUBLE MERIDIAN DISTANCES
353
Fig. 168.
and parallel to the coordinate axes. It is evident from the
figure that
Latitude = Distance x cos Bearing,
and Departure = Distance x sin Bearing.
Latitu'les are called North or South and departures Edst or Westy
depending upon the direction of the course as shown by its letters,
e.g., if the 1 earing is N 30** E thi ; course has a North latitude
and n East departure. North latitudes and East departures are
considered as positive (+), South latitudes and West departures
354
COMPUTATIONS
[Chap. xm.
as negative ( — ). In the figure the courses are assumed to run
from Fto A, from A to 5, etc.
385. After all of the latitudes and departures have been
computed (supposing for the present that the traverse is a closed
Fig. 158
figure) proceed to find the areas of all the trapezoids or trian-
gles, such as DErSf EFr, etc., formed by (i) the courses, (2)
their projections on the meridian, and (3) the perpendiculars
DOUBLE MERIDIAN DISTANCES 355
from the extremities of the courses to the meridian. It is evident
in the figure shown that the area of the field is equal to
{AB wq 4- BCtw 4- CDst) - {DErs + EFr + FAq)y
that is, in this figure the sum of all the areas determined by lines
running northward minus the sum of all the areas determined by
lines running southward. These are known as north areas and
south areas. ^ In computing the areas of these trapezoids it is
convenient as well as customary to find the double areas and
divide the final result by 2 instead of dividing by 2 in figuring
each trapezoid. The area of any trapezoid equals the average
distance of the extremities of the course from the meridian,
multiplied by the length of the projection of the course on the
meridian. This average distance of the ends of the line from
the meridian is known as the meridian distance of the course,
i.e., the meridian distance of the middle point of the course. In
computing the double areas, tvrice this distance, or the double
meridian distance (D.M.D.), is used, which is equal to the sum
of the distances of the ends of the course from the meridian.
In arranging the data for computing the double meridian dis-
tances, the courses must be tabulated in consecutive order around
the traverse, whether they were so taken in the field or not. The
D.M.D. of the course FA is qA which is the departure of the
course FA. The D.M.D. of AB is qA ^ wB == qA -\- qA -¥
zBy i.e., the D.M.D. of course FA + the dep. of FA + the dep.
of AB. The D.M.D. of BC == tC + wB ^ iv + vC + qA +
zB= qA -{-wB + zB+vC^ D.M.D. of AB 4- dep. of AB +
dep. of BC.
Hence the D.M.D. of all of the courses may be computed by
the following rules : —
(i) The D.M.D. of the first course (starting from the
primary meridianf) equals the departure of the course itself.
* If the traverse had been run around the field in the opposite direction these
north areas would become south areas. The result would be the same, however,
ixi either case since it is the algebraic sum of the areas which is obtained.
t Any meridian could have been chosen as the primary meridian, but negative
signs are avoided if the most westerly point is chosen as the starting point.
3S6 COMPUTATIONS [Chap. XIIL
(2) The D.M.D. of any other course equak the D.M.D. of
the preceding course plus the departure of the preceding course
plus the departure of the course itself.
(3) The D.M.D. of the last course should be numerically
equal to its departure, but with opposite sign.
The double areas of all the trapezoids may now be found
by simply multiplying the D.M.D. of each course by the latitude
of the same course, North latitudes being regarded as plus and
South latitudes as miniis. The sum of all the north double areas
minus the sum of all the south double areas equals twice the area
of the field. Be careful to divide by 2 after competing the other
details of the computation.
386. COMPUTATION FOR AREA OF COMPASS SURVEY BY
D.M.D. METHOD. — The details of the above are illustrated
in Fig. 159, which is the computation of the area of the traverse
given in the compass notes in Fig. 50, p. 100. It will be seen
from a study of the notes that there was local attraction of i° at
station 5, and that in the following computations the corrected
bearings are used (Art. 41, p. 30).
In Fig. 159 the hearings, distances, latitudes, departures, and
D.M.D 's, which are recorded on a line with station F are those
corresponding to the course FA; those recorded on a line with
station A refer to the course AB; etc. After the bearings and
distances are entered in the table the places which are to be
blank in the remaining columns are cancelled as shown; this is
a check against putting the results of the computations in the
wrong spaces. In computing the latitudes and departures the
log distance is first entered; the log sin bearing is written below
this and the log cos bearing is recorded above. To obtain the
log latitude add the upper two logarithms; to obtain the log
departure add the lower two logarithms. When the latitude and
departure of a course have been obtained see if the results appear
to be consistent with the given bearing and distance; when the
bearing of a course, for example, is less than 45® its latitude is
greater than its departure and vice versa.
DOUBLE MERIDIAN DISTANCES
357
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Meridian Distance Method.
358 COMPUTATIONS [Chap. XHI.
387. Balancing a Chain and Compass Traverse. — Before
the D.M.D. method can be properly appUed the errors of meas-
urement of the traverse should be so distributed that the figure
becomes a closed polygon. If the field is a closed polygon the
sum of the north latitudes will equal the sum of the south lati-
tudes, and the sum of the east departures will equal the sum of
the west departures. As soon as the latitudes and departures
are computed this test is applied. If the sums differ, the error
is distributed in such a way as to make the sums exactly equal,
and at the same time to give to each latitude and departure its
most probable value. In the case of a compass survey the errors
are fully as likely to be in the bearings, which have been read to the
nearest quarter of a degree, as in the distances; hence if nothing
definite is known in regard to the errors they are assumed to be
proportional to the lengths of the lines and the survey is balanced
by the following rule which alters not only the length of the lines
but also their directions.
388. The correction to be applied ^ ^'*^ j ^ >, / I ^/ ^^J'
course is to the total error ^^ j ^ /, . | ^^ ^^^ length of the course
is to the perimeter of the field.
This rule is based upon purely mathematical considerations
and should be applied only when nothing is known as to where
the errors probably occurred. Usually the surveyor knows where
the error is probably greatest and consequently in balancing the
survey he will place the largest corrections where, in his judg-
ment, they belong. In measuring with the chain, the recorded
distances tend always to be too long, because the sag, poor align-
ment, and poor plumbing, all serve to shorten the chain; conse-
quently the probability is that the recorded measurements are
too long, therefore in balancing it is more logical to subtract
from the latitudes and departures in the columns whose sums
are greater rather than to add anything to the latitudes and de-
partures in the smaller columns. The corrections should of
course be applied in such a way as to decrease the difference
DOUBLE PARALLEL DISTANCES SS9
between the two columns. In the example (Fig. 159) the total
error in latitude is 0.08 and the total error in departure is 0.07.
The perimeter of the traverse is 129.77. Hence the correction
per chain-length is 0.062 links for latitudes, and 0.054 for depart-
ures. The corrected values of the latitudes and departures are
given in the columns headed balanced latitudes and balanced de-
partures.
389. From the balanced departures we then compute the
D.M.D. of each course as shown in the next column. Observe
that the last D.M.D. (point F), as computed from the preceding
one, is exactly equal to the departure of the last course. This
checks the computation of the D.M.D.'s. The D.M.D.'s are
now multipUed by their corresponding latitudes and the prod-
ucts placed in the double a ea columns, those having N latitudes
being placed in the column of north ( + ) double areas and those
having S latitudes in the column of south ( ~ ) double areas. The
sums of these columns differ by 2245.8. One-half of this, or,
1 1 23. is the area of the field in square chains, which equals
1 1 2.3 acres.
By proceeding around the field in the reverse direction the
letters of all of the bearings would be changed, in which case the
column of south double areas would be the larger.
390. Double Parallel Distance. — There is no particular
reason for using the trapezoids formed by projecting the courses
on to the meridian rather than those formed by projecting them
on to the other axis. In the latter case the Double Parallel Dis-
tance {D.P.D,) should be computed, and the result multiplied
by the departure for each course.
In the D.M.D. method the computations have been checked
at every step with the exception of the multiplication of the
D.M.D.'s by the latitudes. A check on this part of the work
can be obtained by figuring the area by use of the D.P.D.'s.
This furnishes an example of a very desirable method of check-
ing, as a different set of figures is used in computing the double
areas, and the opportunity for repeating the same error is thus
avoided. Fig. 160 shows the computation by the D.P.D. method
of the area of the same survey as is calculated by the D.M.D.
method in Fig. 159.
86a
COMPUTATIONS
[Chap. xm.
Sfa
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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 .
where / is the error in latitude and d is the error in departure.
If this distance E is divided by the perimeter of the field the
resulting fraction is called the error of closure, which in this sur-
vey is approximately i^cr^^ (see Art. 132, p. qq).
392. COMPUTATION OF AREA OF A TRANSIT AND TAPE
SURVEY. — The field notes show the lengths of the sides
of the traverse, all of the angles and perhaps the magnetic bear-
DOUBLE MERIDIAN DISTANCES 361
ings of some or all of the courses. If an observation has been
made for determining the direction of the meridian, this affords
the means of computing the true bearings of all of the traverse
lines.
393. The first* step in reducing the notes (provided it has
not already been done in the field) is to see if the difference
between the sum of the right and left deflection angles equals
360®. If interior angles have been measured, their sum should
equal the number of sides of the field times two right angles,
minus four right angles. If there is a small error in the sum
of the angles this is usually adjusted by placing the errcr in the
angles where it probably occurred. If nothing is known as to
where it probably occurred the corrections should be made in
the angles adjacent to the short lines, as any error in sighting or
setting up the transit causes a greater angular error in a short
line than in a long one.
The transit survey is referred to a system of rectangular
coordinates, as in case of the compass survey. If the direction
of the true meridian is known (either from a special observation
or by connection with some other survey referred to the meridian),
it is advisable to use this meridian as one of the coordinate axes.
If the direction of the true meridian is not known the magnetic
meridian may be used. This of course is convenient in some
respects because the bearings taken in the field already refer
to this meridian. If not even the magnetic meridian is known it
will then be advisable to choose some line of the survey (pre-
ferably a long one) as the axis, for using one of the traverse lines
as an axis saves computing the latitude and departure of one
course.
Whatever line is chosen as an axis, the bearings used for
computing the latitudes and departures are to be obtained from
the measured angles (after correction), and not from the observed
bearings. For instance, if some line is selected and its mag-
netic bearing used, then the bearings of all of the other lines
should be computed from this one by^means of the (corrected)
transit angles. In this way the bearings are relatively as accurate
as the transit angles, even though the whole survey may be
referred to an erroneous meridian due to the error of the magnetic
362
COMPUTATIONS
[Chap. xm.
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(The remainder of the computations is in Fig. 161 A.)
DOUBLE MERIDIAN DISTANCES
363
LMCos.Bt(r«
U9 $10. Bear
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(These compatations go with Fig. i6i.)
bearing of the first line. In calculating these bearings the work
should be checked by computing the bearing of each line from
that preceding, the bearing of the last line being followed by the
calculation of a new bearing of the first line of the traverse which
must agree with the magnetic bearing assumed for it, provided
the deflection angles have been adjusted so that their algebraic
sum is 360®. The observed magnetic bearings of the different
courses will serve as a check against large mistakes in this cal-
culation.
3g4. When all of the bearings have been figured the latitudes
and departures are to be computed. In good transit surveys
five places in the trigonometric functions will usually be necessary.
If the angles are measured, by repetition, to a small fraction of
a minute, seven-place logarithmic tables may profitably be em-
364 COMPUTATIONS [Chap. XHI.
ployed, as much interpolation is avoided by their use, but the
logarithm! need not be taken out to more than five or six places.
Seven places, of course, are more than are necessary so far as
precision is concerned (Art. 351, p. 324).
The computation of the latitudes and departures may be
conveniently arranged as shown in Fig. 161 which is the com-
putation of the survey in Fig. 52, p. IC3. After the latitudes and
departures have been calculated they are arranged in tabular
form. The columns of latitudes and the columns of departures
are added and compared just as in a compass survey.
395. Balancing a Transit and Tape Traverse. — In adjusting
(balancing) a transit traverse a different rule is used from the
one given in Art. 388. In the case of a transit survey the error is
chiefly in the measurement of distances, as it is much easier to
secure accurate results in the angular work than in the tape
measurements. Hence the following rule for balancing the
survey is applicable: —
( laiituie )
The correction to he applied ^^^^ ) j j, 4 \^f ^^V course is
,.,,.. . ( latitvde \ . ( latitude ) . .1 .
is to the sum of all of the latitudes (without regard to algebraic
sign).
As in the case of a compass survey, the surveyor's knowledge
of the circumstances should always take precedence over the rule,
and it is probably more nearly correct to shorten the latitudes or
departures in the larger columns than to lengthen them in the
smaller columns. This is because distances are usually recorded
longer than they actually are; the only cases where the distance
is probably too short is when an excessive pull has been given
to the tape or a mistake made in measurement. It will be
observed in the original notes (Fig. 52, p. 103) that the distances
BCf GH, and KA were all questioned, i.e., they were measured
under such conditions that it is probable that there may be one
or two hundredths error in them. In balancing the latitudes and
departures then, this information is used. In Fig. 161 it will
be seen that in balancing the survey the latitudes and departures
DOUBLE MERIDIAN DISTANCES
365
of these questioned measurements have been changed in such a
way as to reduce the length of £C, GjEZ", and KA each one hun-
dredth of a foot.
In balancing the angles^ in which there was an error of 15
seconds, it will be noticed that the correction for this error,
being small, was put into one angle, that at C, one of whose
sides is the shortest Une in the traverse. The area is computed
as explained in Art. 385, p. 354.
396. Fractional Areas. — Fig. 162 is the computation of the
^rea oflandofS/JbsCbkmar. asSpSZ. 3% ^t.^ '^^
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Fig. 162.
Computation of Transit and Tape Survey, including
Fractional Area.
survey shown in Fig. 53, p. 104. The traverse was run with a
transit and tape, the angles being measured to the nearest minute
366 COMPUTATICM^S [Chap. Xm.
and the sides to tenths of a foot. Nothing appears in the fidd
notes to indicate that any of the lines were diflScult to measure,
so it is assumed that any errors in measurement are as likely to
occur in one line as another. Therefore, in balancing the lati-
tudes and departures of this s vey, the rule given in Art. 395 is
applied. In balancing the angles, in which there was an error
of I minute, the entire error was placed in the angle at D where
the side DE is short in comparison with the other sides.
It will be noticed that the distances which appear on the
sketches in the computation are slighdy diflFerent from those
which appear in the field notes (Fig. 53) ; this is due to the fact that
the distances have been corrected for erroneous length of tape
before undertaking to calculate the area. The intermediate
steps in the computation of this traverse do not appear in Fig.
162, but they are the same as in the last traverse. The D.M.D.'s
were computed from Fy the most westerly point. The compu-
tation of the fractional areas is also given.
397. SUPPLYING MISSING DATA, — If any two of the bear-
ings or distances are omitted in the traverse of a field the miss-
ing data can be supplied and the area obtained by computations
based on the measurements taken. As has been shown in
Art. 387, p. 358, the algebraic sum of all the latitudes in a closed
survey must equal zero, and the algebraic sum of all the depart-
ures must equal zero; or, to put it in the form of an equation.
Zi cos A -{- Z2 cos B -h Z^ cos C + etc. =
Zi sin A -f Z2 sin 5 + Z, sin C + etc. =
o
where Zj, Z„ Z,, etc., are the lengths of the corresponding courses.
Therefore from these two equations any two unknown values in
them can be computed. n
The missing data could be any of the following combina-
tions:—
(1) The bearing and length of a line.
(2) The length of a line and the bearing of another line.
(3) The length of two lines.
(4) The bearings of two lines.
SUPPLYING MISSING DATA 367
398* Case (i) where the bearing and length of a line are
missing is by far the most common. Its solution is also more
direct than that of the other cases.
If the latitudes and departures of all of the measured sides
are calculated, the sum of N and S latitudes will be found to
differ, and the amount by which they differ is the latitude of
the omitted side plus or minus the errors of latitudes. Similarly
the amount by which the E and W departures differ is the
departure of the course omitted plus or minus the errors cf
departures. From the latitude and departure of a course its
length and bearing may be readily found.
A practical application of this case is found in the problems
of subdividing a field by a line running from one known point to
another, the direction and length of 'the dividing line not having
been measured. The area of the portion cut off by this line can
readily be computed by the above method. In case the angles
were taken with the transit, the bearing of one line would be
assumed to be correct and all other bearings computed to
correspond.
It is evident from the above that in supplying missing data
the observed measurements must be assumed to be correct, as
there is no way of proving this from the computations. For this
reason it is never advisable, when it can possibly be avoided, to
supply missing data derived from computations on which a field
check has not been obtained.
399. The solutions of the other three cases of missing data
are not so simple, as they involve the use of simultaneous equa-
tions; they will not be discussed here.
400- Besides the four cases mentioned above there are some
special cases which are capable of solution. In Fig. 163 the
lines and angles measured are ^
shown by full lines. The ^rfX^<r^)^^^v^^
bearing of -45 is given. Here ^^^\ ^ — '*^*.^.,^^
one side and two angles are ^v*"^-^ ^^^^^
missing. The solution is as ^s^.,,-^"^"^"^-^ J:J
follows. In the triangle EAB x\ "^--^/^
find £B, £5.4, and i4JE5. In ^ Fig"i63"" *
the triangle EDC find JSC,
DCE, and DEC. Then in the triangle £J5C, in which £C,
36a COMPUTATIONS [Chap. XIH.
£J5, and EBC are known, find ECB, CEB, and BC. All the
angles and sides are then known. Other special cases may be
solved in a similar manner.
401. DETECTING MISTAKES. — Mistakes in fieldwork may
often be detected by means of the calculations. One of the
easiest mistakes to make in surve)ang is to omit a whole tape-
length in counting. If such a mistake were made and the lati-
tudes and departures were computed, the linear error of closure
of the survey would prove to be about a tape-length. In order
to find in which line this mistake probably occurred compute the
bearing of this linear error of closure and examine the traverse to
find a line having a bearing the same or nearly the same. The
error in departure divided by the error in latitude equals the
tangent of the bearing of the line which represents the error of
closure of the traverse. The errors of the survey, of course,
will prevent these bearings from agreeing exactly. If two
mistakes have been made it may be difficult and sometimes im-
possible to determine where they occurred. When an error of
this sort is indicated by the computation the line should be re-
measured. It is bad practice to change an observed measure-
ment because it is found by calculation to disagree with other
measured distances.
It may, and frequently does, happen that there is more than
one line in the traverse which has about the same bearing. In
such a case it is impossible to tell in which of these lines the mis-
take occurred. But if a cut-off line is measured as was sug-
gested in Art. 145, p. 109, and one portion of the survey bal-
ances, the other part will contain the mistake. By proceeding
in this way the number of lines in which the mistake could occur
is reduced so that its location can be determined and checked
by field measurement.
402. The Subdivision of land. — There are a great
many different problems which may arise in the subdivision of
land and which may be solved simply by the application of the
principles of trigonometry. A few of these problems are so
common and so frequently involved in the working out of more
complicated cases that their solution will be given.
SUBDIVISION OF LAND
369
403. To Cut Off from a Traverse a Given Area by a Straight
Line starting from a. Known Point on the Traverse. — In Fig.
164, ABCDE represents the traverse which has been plotted and.
whose area has been computed.
It is desired to cut off a cert in
area by a line running from F
which is at a known distance
fiom A or E. The line FG is
drawn on the plan so as to make
the area FG'DE approximately
equal to the desired area. The
line DC is scaled ofiF and the
scaled distance used as a trial
length. Then the side FG' and
its bearing can be found by the
method explained in Art. 398,
p. 367, and the area FEDG com-
puted in the usual manner. The
difference between the required
area and the area of FEDG is the
amount to be added to or sub-
tracted from FEDG'. If this correction area is a minus area then
the triangle FG'G will represent it. In this triangle the base FG
and its area being known the altitude hG and the distances GG
and FG can be readily computed. In the traverse FGDE^
which is the required area, the length of the missing side FG
and its bearing can be supplied.
Instead of using the trial line FG the line FD might have
been first assumed and the correction triangle would then be
FDG. This method has the advantage of containing one less
side in the first trial area, but the correction triangle is large,
whereas in the method explained above the correction triangJe is
small which may be of advantage in that part of the computa-
tion.
404. To Cut Off from a Traverse a Given Area by a Line
running in a Given Direction. — In Fig. 164, ABCDE repre-
sents a closed traverse from which is to be cut off a given area by a
line running at a given angle (BJK) with AB. On the plot of the
370
COMPUTATIONS
[Chap. xm.
traverse draw the line J'K' in the given direction cutting ofi
J'BCK' which is, as nearly as can be judged, the required area.
Scale the distance BJ' and use this trial distance in the compu-
tations. Then compute the distance J'K' and the area of
J'BCK' either by the method of Art. 398, p. 367, or by dividing
J'BCK' into two oblique triangles. The difference between
this area and the required area
is then found, which is a cor-
rection trapezoid to be added
to or subtracted from J'BCK'.
In this case it will be assumed
that it is to be added to J'BCK',
In this correction trapezoid
the area and one base J'K' are
known; also the base angles, J'
and K'. From these data an
approximate value for the a'ti-
tude of the trapezoid can be
obtained and the length of the
other base K"J" of the trapezoid
computed from this altitude
and the length of J'K', Then
the area of this trapezoid
J'K'K^J" can be accuratdy de-
termined; the difference between this and the required correction
will be small and the dimensions of the second correction trape-
zoid J^K^K'J' can probably be readily Computed from its area
and the length of J^K" which are known. By successive trials,
probably not more than two, the correct line JK can be found.
If lines AB and CD are approximately parallel the trapezoid is
nearly a parallelogram and its correct altitude can then be
quickly determined.
405. To Find the Area Cut Off from a Traverse by a Line
running in a Given Direction from a Given Point in the Traverse.
— This problem may be readily solved by drawing a line from
the given point in the traverse to the comer which lies nearest
the other extremity of the cut-off line. The area of the traverse
thus formed is then computed, and this area corrected by means
of a correction triangle.
TRAVERSES WHICH DO NOT CLOSE
371
Fio. les.
In Fig. 165, ABCDEFG repre-
sents a plot of a field. It is desired
to run the line from £ in a given
direction. EjEZ" and to compute the
area HEFGAB cut off by this line.
The latitude and departure of points
B and E being known the bearing
and length of BE and the area of
ABEFG can be computed. Then
the area and the remaining sides of
the triangle BEH can be obtained
from BE and\he angles at B and E.
It is obvious that the solution
pf such problems as these is greatly facilitated by plotting the
traverse before attempting the computations.
CALCULATIONS RELATHI G TO TRAVERSES WHICH DO NOT CLOSE.
406. To Calculate the
Total distance bjetween
End Points. — Fig. 166 repre-
sents the traverse ABCDEF in
which the distance AF and the
angle BAF are desired. AB can
be assumed as one of a pair of
rectangular coordinate axes and
the coordinates of point F {AH
and HF) computed by the
method explained in Art. 410,
P- 373- ^^ ^^^ the angle BAF
can then be easily found. This
method is of service in check-
ing traverse plots of this type.
407. CUT-OFF LINES. — The calculation of cut-off lines,
like the line CF\n Fig. 166, is the same problem as was explained
in Art. 398, p. 367. The angles DCF and EFC have been
measiu-ed in the field and the traverse CDEF is thus complete
except that the^ength of the line CF is unknown. The length
of CF and the angle it makes with AB can be readily computed
since the coordinates of C and F can be found.
Fig. lee.
37^
COMPUTATIONS
[Chap. xm.
408. COMPUTATION OF AZIMUTHS WHEN CHECKING
ANGLES TO A DISTANT OBJECT. — In this kind of problem
the coordinates of all the points along the traverse can be com-
puted with reference to some coordinate axes. At A and B
(Fig. 167) an^es have been taken to 5, and from these angles
the coordinates of point 5, referred io AB and a line perpen-
dicular to AB as axes, can be computed (Art. 410, p. 373). Co-
ordinates of S referred to the same axes
should have the same value when
figured from BC as a base as when
calculated from the base CD and so on.
If, however, when computed by means
of angles at D and £, the point falls
at 5', and angles E and F give its
location also at 5' there is evidence
of a mistake in the traverse at D. If
the two locations of S and 5' are
such that a line between them is par-
allel to either CD or Z>£, the mis-
take was probably made in the meas-
urement of the line parallel to SS'
and the distance 55' should be ap-
proximately equal to the amoimt of
the mistake in measurement. If,
however, SS' is not parallel to either
CD or DE the mistake probably lies in the angle at D.
Fig. 167.
409. CALCULATION OF TRIANGULATION. — In a triangu-
lation system the base-line is the only line whose length is
known at the start. The sides of any triangle are found from
the law of sines, i.e..
sini4 __ a
sin J3 b
sin -4 __ a
sinC c
tf sing
sin^l
a sin C
sin^l
COORDINATES 373
Assuming a to be the base and the angles A, J3, and C to
have been measured the calculations are arranged as follows:
(i) log a (1400.74) = 3-1463575
(2) Colog Sin^l (57^42' 16'') = 0.0729874
(3) log Sin B (61^ 17' 53^^) = 9-943o639
(4) log Sin C (60^ 59' si'') = 9.9418088
Sum of (i) (2) (3) log* = 3 1624088
Sum of (1) (2) (4) log c = 3.1611537
410. COORDINATES. — In many cities the coordinate sys-
tem of surveying is used (see Chapter IX). In this system the
position of each comer of the different lots is fixed by rectangu-
lar coordinates measured from two lines at right angles to each
other.
Often the origin of coordinates O (Fig. 168) is so chosen that
the whole city is in the first
quadrant YO X. Distances
measured parallel to XX^
are usually called abscissas
and those parallel to YY^
ordinates.
The advantage of this x*
system of surveying lies in
the fact that since all surveys
refer to the same reference
lines, they are therefore tied
to each other; and also in the
fact that a lot can be relo-
cated from the coordinates of
its comers even if all of the corner bounds have been destroyed.
Generally the coordinate lines rim N and S, and E and W,
but when city streets have been laid out at right angles to each
other and not on N and S, and E and W lines, it may be more
convenient to have the system of coordinates parallel to the street
lines.
The coordinates of any unknown point are usually computed
from the coordinates of some other point to which the unknown
V
Fig. 168.
374
COMPUTATIONS
[Chap. Xm.
point is tied by an angle and distance. The difference in coor-
dinates between the known and unknown points will be obtained
as follows: —
Difference in X = distance x sin azimuth angle.
Difference in Y == distance x cos azimuth angle.
Sometimes the unknown point is located by angles from two
other known points, in which case the distance between the two
points whose coJ'rdinates are known can be computed and then
the distance from one of the known points to the unknown point.
The problem is then in the form described in the previous para-
graph.
L..-x..^J
-V
— 1~ X5-r
^
—L.-
I
I
I
I
Fig. 160.
4X1. TO DETERMINE THE AREA OF A FIELD BY REC-
TANGULAR COORDINATES. — The area of the field i, 2, 3, 4
(Fig. 169) is equal to the trapezoids
(a, I, 2, b) + (6, 2, 3, c) - (a, i, 4, d) - (J, 4, 3» <^)-
Expressed as an equation in terms of the coordinates the
area is
AREA BY COORDINATES
375
I, 2, 3, 4 - U -A)^^ +(J, - Js)^y^'
From this equation is derived the following rule for obtaining
the area of a closed field from the coordinates of its corners: —
(i) Number the comers consecutively around the field.
(2) Multiply each | ^^^^cissa | j^^ ^j^^ difference between the
following and the preceding j i^ • { > always subtracting the
preceding from the following, and take one-half of the sum of
the products.
412. Fig. 170 is the computation, by coordinates, of an area
Sia
dearin^
LQftti^iM
Ocfarfiffis
Bakfncm^
X
y
Oot^bkArf^^
/Vf
S'
£i^
w^
U3t
O^p
f
-
0
NSjy^
160
as?
—
/M
*QM7
-/J*
2aS6
2m
-J3fi
//a2
f '
SJ7V
1132
—
/e^
aai
■^23
■%?2
im
???s
*//Jfi
£/6tJ
i
54$'i£
4SJ
—
3Jf
5.28
— ^
-3M
tJZ3
fOQO
mm
^1534
153A
J
N43*t£
a75
3,37
—
B^
—
f^3i3B
'347
aid
6^
'6M7
SKB
4-
NSS^I^
S.O0
4^9
—
—
e.fs
^4^
~tB
il75
J6S?
-M47
3Z9£
f^.33
IS.3S
/i,M
fZ.lf^
^7L1
im
Fig. 170. Computation of Compass Survey by Coordinates.
from the field notes. The origin of coordinates is 10 chains W
and 10 chains S of station 2.
413. Equation (i ) may be developed into the following form:
When this formula is to be used the coordinates may be arranged
in the following simple manner:
(4)
376
DOUBLE MERIDIAN DISTaMCES
(Chap. XHL
From equation (3) it will be seen that the area is equal to the
sum of the products of the ordinates joined by full lines in (4)
minus the sum of the products of the ordinates joined by broken
lines. This formula involves the multiplications of larger num-
bers than in (2), but does not require any intermediate sub-
tractions.
PROBLEMS.
X. The latitude of a line of a traverse is + 106.42 ft; its departure is
— a 73.62. What is its bearing ?
2. From the following notes of a compass survey, compute by the double
meridian distance method the area in acres.
Station.
Bearing.
Distance
(Chains).
A
N46«iW
20.76
B
N5i°iE
13 83
C
East
21.35
D
S56«E
2'. 60
E
S33nW
18.80
F
N74°iW
30. 8
3. In the following notes of a compass survey the length and bearing of one
of the courses were omitted. Substitute the correct values and compute the area
(in acres) by the double meridian distance method.
Station.
Bearing.
Distance
(Chains).
I
S40*>W
17-50
2
N45**W
22.25
3
N 36*^1 E
31-25
4
North
13-50
5
(omitted)
(omitted)
6
ssojw
34.25
7
West
32.50
rkoMXhta
177
4. Rram the notes given in Fig. 52, p. 103, and Ffg. 161, p. 3*2, compute by
the (k>uble meridian distance method the area of the traverse ABCDEK,
5. In the following traverse there are two mistakes. Find where they occur
and determine their amounts.
Station.
Obser\'ed
Bearing.
Deflection
Angle.
Distance
(Feet).
Calculated
Bearings.
Remarks.
A
B
C
D
E
N34^E
S73**iE
Sio'^lW
N26oiW
S52OW
164** 14' R
62«» 16' R
84*» 22' R
I42«49'R
103** 41' L
240.2
163.7
207.6
273-1
147-4
N34*»oo'E
C£=i88.i
BC£= 34** 14'
DEC-^ 81^25'
6. The following b a set of notes of an irregular boundary of a lot of land.
It is desired to straighten this crooked boundary line by substituting a straight line
running from B to the line EF, Find the bearing of the new boundary line and
its length; also the distance along EF from point E to the point where the new line
cuts£F.
Station.
Bearing.
Distance
(Feet).
A
S 89* 14' E
373.62
B
N 13'' 10' E
100.27
C
N o*>i7'W
91.26
D
N270 39'E
112.48
E
N72«>i2'W
346.07
F
S 5**o7'W
272.42
etc.
etc.
7. (a) In the lot of land, A BCD, the lines AB and DC both have a bearing
of N 23** E; the bearing of i4Z> is due East; AD is 600 ft., i4B is 272.7 ft., and DC
is 484.6 ft. Find the length of a line EF parallel toAB which will cut off an area
ABFE equal to half an acre. Also find the length of the lines AE, and BF. (b)
What is the area of EFCD?
8. Given the notes of a traverse, which does not close, as follows: —
Station.
Deflection
Angle.
0
6+40
9+20
14+ 55
17 + 18
20+64
60 17' L
18043'L
12047'R
45^24'L
680o6'R
Find the length
of a straight line
from 0 to 20+ 64
and the angle it
makes with the
line from 0 to 6
+ 40.
37* DOUBLE MERIDIAN DISTANCES [Chap. XHI.
' 9. Compute the area of the foUomng traverse by coordinates.
Station.
Deflection
Angle.
Bearing.
Distance
(Feet).
A
780 10' oo* L
208.64
B
88« 28' oo* L
43<^-79
C
6^'' 02' IS*' L
S6.48
D
33" 39' 15^ L
98.80
E
•go^oo'oo-'R
68.62
F
.220 5I'00*'L
%68o 50' IS-' L
95- 10
G
N36*>i4'oo»W
207.41
H
89* 48' oo*' L
103-75
I
SS" 00' IS- L
96.75
J
530 49'oo«'R
420.77
PART IV.
PLOTTING.
PART IV.
PLOTTING.
CHAPTER XIV.
DRAPTINa IK8TR02ffBNT8 AND MATERIALS.
It is assumed in this section that the student is familiar with
the ordinary drawing instruments such as the T-square, triangles,
dividers, compasses, and scales, as well as with their use.
ENGINEERING DRAFTING INSTRUMENTS.
414. There are several drafting instruments which are used
by engineers and surveyors but which are not so generally em-
ployed in other kinds of drafting work. The most common of
these are briefly described in the following articles.
415. STRAIGHT-EDGE. — Engineering drawings are made
with greater accuracy than much of the drafting work of other
professions. In fact many engineering drawings are limited in
precision only by the eyesight of the draftsman. It is evident,
then, that to use a T-square which is run up and down the more
or less uneven edge of a drawing board will not produce draw-
ings of sufficient accuracy. For this reason in many classes of
engineering work the edge of the drawing board is not relied
upon. Furthermore, in most plots of surveying work the lines
are not parallel or perpendicular to each other except by chance,
but run at any angle which the notes require; and there is there-
fore not so much call for the use of a T-square as there is in archi-
tectural, machine, or structural drawings. All drawings are
usually laid out starting from some straight line drawn on the
paper by means of a straight-edge, which is simply a flat piece
of steel or wood like the blade of a T-square. Steel straight-
edges are more accurate and are more commonly used by engi-
neering draftsmen than the wooden ones, the edges of which are
likely to nick or warp and become imtrue. They can be ob-
38'
382 DRAFTING INSTRUMENTS [Chap. XIV.
tained of almost any length and of any desired weight, the com-
mon length being about 3 feet.
416. ENGINEER'S SCALE. — Practically all engineering
plans are made on a scale of 10, 20, 30, etc. feet to an inch. In
the engineer's scale, therefore, the inch is divided into 10, 20, 30,
etc. parts, instead of into eighths and sixteenths as in the archi-
tect's scale. Engineer's scales are made 3, 6, 12, 18, and 24
inches long. • One form is the flat wooden rule with both edges
beveled and a scale marked on each bevel. Some flat rules are
beveled on both faces and on both edges of each face, thereby
giving four scales on one rule. Still another very common form
is the triangular scale, made of wood or metal, and having six
different scales, one on each edge of the three faces. In such
rules the scales are usually 20, 30, 40, 50, 60, and 80 ft. or 10, 20,
30, 40, 50, and 60 ft. to an inch. Scales are, however, often
made having the inch divided into 100 parts, but in plotting a
map which is on a scale of 100 ft. to an inch the work is probably
more easily done by using a scale of 10, 20, or 50 divisions to an
inch and estimating the fractional part of a division than by try-
ing to plot with a loo-ft. scale which is so finely graduated as
to be very hard to read without the aid of a magnifying glass.
A 20-ft. or 50-ft. scale is more satisfactory for precision than a
lo-ft. scale when it is desired to plot on a scale of 100 ft. to the
inch. A plan on a 206-ft. scale is always plotted by using a 20-
ft. scale, a 300-ft. plan by using a 30-ft. scale, etc.
A map covering considerable area, like the map of a state, for
example, must be plotted to a very small scale, and this is usually
given in the form of a ratio such as i to 500, i to 2500, etc., mean-
ing that one unit on the map is ^ J^, j^j^-^, etc. of the correspond-
ing distance on the ground; this is sometimes called the natural
scale. For plotting such maps specially constructed scales with
decimal subdivisions are used.
4x7. PROTRACTOR, — A protractor is a graduated arc
made of metal, paper, celluloid, or horn, and is used in plotting
angles. There are many varieties of protractor, most of them
being either circular or semicircular.
418. Semicircular Protractor. — Probably the most common
is the semicircular protractor which is usually divided into de-
SCALES AND PROTRACTORS 383
grees, half-degrees, and sometimes into quarter-degrees. Fig.
171 represents a semicircular protractor divided into degrees.
C
Fig. 171. Semicircular Protractor.
In plotting an angle with this protractor the bottom line of the
instrument is made to coincide with the line from which the angle
is to be laid oflF, and the center of the protractor, point C, is
made to coincide with the point on the line. On the outside of
the arc a mark is made on the drawing at the desired reading.
The protractor is then removed from the drawing and the line
drawn on the plan.
Fig. 172. Semicircular Protractor.
Instead of having the 0° and 180^ of the protractor on its
lower edge some instruments are made as shown in Fig. 172.
1
384 DRAFTING INSTRUMENTS [Chap. XIV.
This form is claimed by some draftsmen to be more convenient,
because in handling the protractor by placing the fingers on the
base neither the graduations nor the line on the plan are covered
by the hand.
419. Full-Circle Protractor. — The full-circle protractor is
of use particularly in stadia work or in plotting any notes where
azimuth angles of over 180® have been taken. For such work
as stadia plotting an ordinary paper protractor 8 to 12 inches
in diameter is sufiiciently accurate, and, in fact, paper pro-
tractors of this size will yield more accurate results than .the
smaller metal ones.
420. Some of the metal protractors are provided with an
arm and vernier attachment. These, while giving more precise
results, require more time for manipulation, and a plain metal
protractor with a diameter of, say, 8 inches will give sufficiently
close results for all ordinary work. As a matter of fact a pro-
tractor with a vernier reading to minutes can be set much closer
than the line can be drawn, and it is therefore a waste of time to
attempt to lay oflF the angles on a drawing with any such accuracy.
There is, however, a protractor of this type with a vernier read-
ing to about 5 minutes which may be of use in precise plotting.
421. Three-Armed Protractor. — The three-armed protrac-
tor is used for plotting two angles which have been taken with an
instrument (usually a sextant) between three known points, for
the purpose of locating the position of the observer (the vertex
of the two angles). The protractor has three arms, the beveled
edges of which are radial lines. The middle arm is fixed at the
o® mark and the other two arms, which are movable, can be
laid oflF at any desired angle from the fixed arm by means of the
graduations on the circle, which number each way from the
fixed arm. The two movable arms having been set at the de-
sired angles and clamped, the protractor is laid on the plan and
shifted about until each of the three known points, (which have
already been plotted on the plan), lies on a beveled edge of one
of the three arms of the protractor. When the protractor is in
this position its center locates the point desired which is then
marked by a needle point Only one location of this center
point can be obtained except in the case where the three known
PROTRACTORS AND PANTOGRAPH 385
points lie in the circumference of a circle which passes through
the center.
422. There are several other types of protractor made, but
the principle and use of all of them are much the same as those
of the simple types which have been explained. It is well in
purchasing a protractor to test it to see that the center point lies
on a straight line between the 0° and 180® marks, that the edge
of the protractor is the arc of a true circle, and that the gradua-
tions are uniform.
423. PAMTOGRAPH. — This instrument is composed of several
flat pieces of metal or wood joined in sjich a way as to form
a parallelogram. One of the three points i4, By and C, (Fig.
173) is fixed and the other two movable. The remaining bear*
Fig. 178. The Pantograph.
ing points are not essential except to support and steady the
instrument. The two movable points are so attached to the
instrument that they will trace out exactly similar figures.
The instrument is used for copying a plan either to the same or
to a different scale. There are several different forms of panto-
graph varying considerably in appearance, but they are all based
on the same principle. The essential condition in their design
is that all three points A, B, and C, must lie in a straight line and
each point must be on one of three different sides (or sides pro-
duced) of a jointed parallelogram. Any one of the three points
can be the fixed point. It is evident then that by changing the
relative positions of these points, by moving them up or down the
arms of the parallelogram, but always keeping the points on a
386 DRAFTING INSTRUMENTS [Chap. XIV.
Straight line, the scale of the copy can be made to bear any
desired relation to the scale of the original drawing. These
instruments are usually provided with scales marked on the arms
indicating the proper settings for various reductions or enlarge-
ments. With a pantograph very accurate results cannot as a
rule be obtained because there is lost motion in the several joints
of the instrument. Some of the expensive metal pantographs,
however, will give fairly good results.
424. Parallel ruler. — This is a beveled rule made
of metal and mounted on two rollers of exactly the same diameter.
It is used for drawing parallel lines. This instrument can be
made to do accurate work, but it must be handled with a great
deal of care to prevent the rollers from slipping. It is especially
useful in drafting diagrams of graphical statics in connection
mth structural design, in drawing the parallel sides of buildings,
section lining, blocking out for titles, and in drafting large titles
which require mechanical lettering.
425. BEAM Compass. — This is an instrument used for
drawing the arcs of circles whose radii are longer than can be
set out with the ordinary compass drafting instrument. It is
composed of a strip of wood or metal with two metal attach-
ments which can be fastened to it. One of the§e attachments
carries a needle point and the other, which is usually provided
with a slow-motion screw for exact settings, carries a pencil or
a pen. This instrument is particularly useful in laying out large
rectangles such as are called for when surveys are plotted by
coordinates (Art. 449, p. 401).
426. COHTOUR PEN. — This pen is constructed very much
like an ordinary right-line ruling pen except that it has a metal
shaft, running through the entire length of the holder, to which
the pen is attached. The shaft revolves inside of the holder,
and the pen is so shaped that it drags behind taking a position
ia the direction in which it is being moved. It is used for draw-
ing irregular curved lines such as contours or shore lines. Not
a litde practice is required before one can use a pen of this type
accurately. When skill in its use is once acquired, however, a
plan can be easily made on which the contours all have a uni-
form weight of line giving a very satisfactory appearance. The
BEAM COMPASS. — PROPORTIONAL DIVIDERS 387
purpose of a contour line is to show the facts as to the surface,
and this pen should not be used unless it is found by trial that
it does the work in hand properly. Accuracy is more important
than appearance.
427. PROPORTIONAL DIVIDERS. — Proportional dividers are
substantially an ordinary pair of dividers with both legs pro-
longed through the pivot-point thereby forming another pair of
legs above the pivot. The pivot is movable so that it can be
pushed up and down in a slot in the legs and clamped in any
desired position, thereby altering the relative lengths of the two
pairs of legs. The sliding is accomplished in some dividers by
a rack-and-pinion motion. When the pivot is in the middle
position the legs are equal, and the space between the two points
of one pair of legs is equal to the space between the other pair.
There are marks on the legs showing the proper settings for the
pivot so that the space between one pair of points will bear
any desired ratio to the space between the other pair. The
marks on the legs should not be accepted as correct, but should
be tested by actual trial. One end of the proportional dividers
is used to space off the distances from the original map and the
other end used to plot that distance on the new map. Thus by
means of this instrument a drawing can be enlarged or reduced
to a definite scale without the use of the engineer's scale.
A drawing which is to. be made two-thirds the size of the
original can be readily reduced by scaling the distances from the
original with a 20-ft. scale and plotting them on the new draw-
ing by use of a 30-ft. scale. But when the reduction is some
odd ratio which cannot be readily accomplished by means of the
engineer's scale proportional dividers are very useful.
428. RAILROAD CURVES, FRENCH CURVES, FLEXIBLE
CURVE, AND SPLINE. — For drawing arcs of curves of long
radiij such as occur on railroad plans and on plans of curved
streets, in city work, curves made of wood, hard rubber, celluloid,
or metal are used; these come in sets of about one hundred, with
radii varying from about 2 inches to 300 inches. The metal
curves are the most common and are made with the inside and
outside edges of the same radii both edges being beveled. When
a pencil line is drawn the beveled edges may be used against the
388 DRAFTING INSTRUMENTS [Chap. XIV.
paper, and when ink lines are drawn the curve can be turned
over so that the beveled edges are up, thus preventing the ink
from running in under the curve on the paper. Some curves for
railroad work are made with a short straight edge tangent to
the curve at one end and with the point where the curve begins
marked by a line across it.
429. Irregular curves, called French Curves^ are of a variety
of shapes. They are made of wood, hard rubber, and celluloid,
and are used to guide the pencil or pen in tracing out irregular
curved lines on the map.
430. A Flexible Curve consists of a strip of rubber fastened
to a flexible metal back. This curve can be twisted to conform
to any irregular curved line on the map and can then be used
as a guide against which the pencil or pen is held in tracing out
the curve.
431. A Spline is a long thin flexible piece of wood, hard
rubber, celluloid, or metal which can be bent so as to confcrm
to a curve. It is usually held in position by specially designed
weights with light metal arms which fit into a thin groove in the
top edge of the spline. This instrument is used by naval
architects for drawing long flat irregular curves such as occur
in ship designs. In engineering drafting it is used in drawing
the lines of arches, ^hich frequently are not circxdar.
DRAWING PAPERS.
432. The drawing papers used by surveyors may be divided
into four general classes; (i) those used for plotting plans, (2)
tracing paper or tracing cloth which is used for copying draw-
ings, (3) cross-section and profile papers, and (4) process papers.
433. DRAWING PAPER FOR PLANS. — There are numer-
ous grades of drawing paper ranging from very cheap "detail "
to heavy paper mounted on cloth, called "mounted paper."
For rough plots which are to be copied later or which are for
temporary use only, a manilla detail paper is frequently used;
but where the drawing is to be of a more permanent character
a heavy white or manilla paper is used. Still more permanent
I
DRAWING PAPERS 389
plans, such as the plan of a survey of a city, should be plotted on
heavy mounted paper. There is generally a right and a wrong
side to all papers, which can be distinguished by the "water-
mark "; this will read direct when the right side of the paper is
toward the observer. A paper to be satisfactory for use should
have a surface not too porous to take ink nicely, and of a fiber
such that after scratching with a knife or rubbing with an ink
eraser, the surface will still take ink effectively. No paper,
however, after scratching can be expected to take bottle red ink,
which permeates the fiber with extraordinary ease.
434. TRACING PAPER AKD TRACING CLOTH. — In mak-
ing copies of drawings, a thin transparent paper called tracing
paper is often used. It is not tough enough to withstand rough
handling and is used only for drawings of a temporary charac-
ter. There are, however, certain kinds of transparent bond
paper in use which will withstand considerable hard usage.
435. For more permanent drawings a tracing cloth is used,
made of a very uniform quality of linen coated with a prepara-
tion to render it transparent. Most tracing cloth as it comes
from the manufacturer will not readily take the ink, and it is
necessary to rub powdered chalk or talc powder over the entire
surface of the cloth before inking the drawing. After the sur-
face chalk is brushed off, the tracing cloth is ready for use. Trac-
ing linen generally has one side glazed and the other dull. Pencil
lines can be drawn on the rough side, but the smooth side will not
take even a very' soft pencil; either side may be used for ink draw-
ings. Some draftsmen prefer to use the glazed side but the dull
side is more commonly used. A tracing inked on the glazed side
may be tinted on the dull side either by crayons or by a wash;
the latter will cockle the cloth unless it is put on quite " dry." It
is easier to erase from the glazed than from the dull side, but
the dull side will stand more erasing,* and gives more uniform
lines.
* Erasure of ink lines from a tracing, as well as from any drawing paper, is a
delicate undertaking. Success will result if the followmg suggestions are carefully
observed : — with a smooth sharp knife pick off the ink from the paper ; this can be
done almost without touching the paper. When practically aD of the ink is o&,
rub the line with a pencil eraser. This will take o£f the rest of the line except
390 DRAFTING INSTRUMENTS IChap. XIV.
In making a tracing of another tracing it will be found that
the lines can be more readily seen if a white paper is put under
the lower tracing. It frequently happens that it is necessary
to make a tracing of a blue-print. The white lines of the blue-
print are not easily seen through the tracing linen. An arrange-
ment which will assist greatly in such work is to have a piece of
plate glass set into the top at one end of a drawing table in such
a way that it forms part of the top of the table. The blue-print
is placed over this glass and the light shining through from the
under side of this glass and through the blue-print will make
the white lines easily visible for copying.
It is common practice, after a survey is made and before or
during the computation of it, to plot the field notes accurately
on detail paper and later to copy the plot on tracing cloth, which
is the final drawing of the survey.
From these tracing drawings any number of process prints
can be made (Art. 438), the tracing taking the place of the
negative used in photographic printing.
436. CROSS-SECTION, AlID PROFILE PAPERS. — Paper di-
vided into square inches which, in turn, are divided into small
subdivisions is used to plot cross-sections of earthwork and the
like. The inch squares are usually divided into J'^, ^"^ -j^, or
^''. Cross-section paper can also be obtained divided accord-
ing to the metric system, or with logarithmic divisions. Cross-
section paper usually comes in sheets.
437. Profile Paper which, as the name implies, is used for
plotting profiles comes in rolls of 10 yds. or more. The vertical
divisions are usually much smaller than the horizontal divi-
sions, which makes it easier to plot the elevations accurately.
The horizontal distances to be plotted occur mostly at full sta-
perbaps a few specks of ink which can readily be removed by a sharp knife. This
method of erasing takes more time than the ordinary method of rubbing with an
ink eraser until the line has disappeared, but it leaves the paper in much better
condition to take another line. It is impossible to obtain good results by this
method unless the knife has an edge which is both smooth and sharp Where
the surface of the tracing cloth has been damaged the applicaiion of a thin coat-
ing of coUodioii on the damaged portion will produce a surface which will take
the ink.
PROCESS PAPERS 39I
tion points, which are represented on the profile by the vertical
rulings on the paper.
Both the cross-section and the profile papers come in colors,
(usually red, green, blue, orange, or burnt sienna) so that a black
or a red ink line (the two most commonly used) will show up
distinctly on the paper. These papers can be obtained also of
very thin transparent material or in tracing cloth form, suitable
for use in making process prints. Profile papers usually come
in long rolls 20 inches wide.
438. PROCESS PAPERS. — Blue-Prints. — The most com-
mon process paper used in drafting oflSces is blue-print paper.
It is a white paper coated on one side with a solution which is
sensitive to light. After the solution is applied, the paper is
dried and then rolled and sealed up for the market in light-proof
rolls of 10 yds. or more. Fresh blue- print paper has a greenish-
yellow color. The process of coating the paper and the gen-
eral handling of the blue-print business is so well advanced and
the price of the prepared paper is so low that surveyors now-a-
days seldom coat their own paper. The process is a very simple
one, however, and in emergencies, when commercial blue-print
paper cannot be obtained, it may be very useful to know how to
prepare it. A good formula for the solution is given below.
Make the following two solutions separately (in the light
if desirable) and mix, in subdued light or in a dark room, equal
parts of each of them.
Solution (i)
Citrate of Iron and Ammonia, i part (by weight)
Water, 5 parts C* '' )
Solution (2)
Red Prussiate of Potash (re-
crystalized), i part (by weight)
Water, 5 parts (*' *' )
The mixed solution is applied to the paper by means of a
camel's hair brush or a sponge; this is done in a dark room or
in subdued light. The paper is coated by passing the sponge
lightly over the surface three or four times, first lengthwise of
the paper and then crosswise, giving the paper as dry a coating
39^ DRAFTINO INSTRUMENTS [Chap. XIV.
as possible consistent with having an even coating; it is then
hung up to dry. The above coating will require about 5 minutes
exposure in bright sunlight; for quick printing paper, use a
larger proportion of citrate of iron and ammonia.
The blue-print of a plan is generally made in a printing
frame, which is merely a rectangular frame holding a piece of
heavy glass, with a back to the frame which can be lifted from
the glass. This back is padded so as to fit tight against the glass
when the back is clamped into position. The process of taking
a print is, briefly, to expose the tracing, with the blue-print paper
under it, to the sunlight a proper length of time and then re-
move the blue-print paper and wash it in water.
•439. In detail, the process is as follows. First, turn the
printing-frame over so that the glass is on the bottom, and remove
the back of the frame. Then, after the tracing cloth has been
rolled, if necessary, so that it will lie flat, place it with its face
against the glass. Place the blue-print paper, which has been
cut to the proper size, on top of the tracing with the sensitized
side of the paper next to the tracing. The back of the frame is
then clamped into position and the frame turned over so that
the glass is up. It should then te examined to see that the trac-
ing has been put into the frame with its ink lines against the
glass, that the Hue-print paper is imder the entire tracing, and
that mere are nq wrinkles in the tracing. All of the process to
this stage should be done in subdued light, usually in a rocm
with the shades drawn to keep out most of the sunlight.
The frame is then moved out into the direct sunlight, placed
as nearly as may be at right angles to the rays of sunlight, and
left there a proper length of time, which will depend upon the
sensitiveness of the coating of the paper and the intensity of the
light. Some blue-print papers will print in 20 seconds, others
require 5 or 6 m nutes in direct sunlight. In purchasing, then,
it is necessary to ascertain from the dealer the "speed " of the
paper and govern the exposure accord ngly. Blue-prints can
be made in cloudy weather as well as when the sun is vis ble, the
only difference being that it requires a much longer time for the
exposure. In all cases where the time of exposure is doubtful
the following simple test may be applied. Instead of taking a
BLUE-PRINTING 393
print of the entire tracing the first time, use only a small piece
of the blue- print paper and put it in the frame as explained above
and expose it a given t*me. Take it out and wash it, and from
th s test judge the length of exposure necessary to give the print
of the entire drawing. An under-exposed print, after it has
been washed, will be light blue in color with white lines; an over-
exposed print will be dark blue with bluish-white lines. The
result desired is a dark or meditun blue background with white
lines. It should be borne in mind, in judging the results, that
all prints become a little darker when they are dry.
In washing the print it should be entirely immersed in clear
water at first; care should be taken that no part of the print is
left dry. It should be washed by moving it back and forth in
the water or by pouring water over it until the greenish solution is
entirely washed off its face. The print should be left in the water
for lo to 20 minutes, then it is hung up to dry. It will dry more
quickly if hung so that one corner is lower than the others. It
should not be hung where the sun will shine on it as the sun-
light will fade it.
In taking prints great care must be exercised not to get the
tracing wet. When the prints are being washed the tracing
should always be put in a safe place where the water will not
spatter on it and it should never be handled with moist hands.
It is practically impossible to eradicate the effect of a drop of
water or even the marks made by damp fingers on tracing cloth;
it is sure to show in every subsequent print which is taken from
the tracing.
440. Blue-print cloth is prepared in the same manner as
the blue-print paper. Its advantage over the paper lies solely
in the fact that it does not shrink as badly and is much more
durable. Prints which are to be used on construction work
where they are sure to get rough usage are sometimes made on
cloth.
441. Vandyke Solar Paper. — There has always been a call
for a sensitive paper which will give positive prints, — a black,
a brown, or a blue line on a white background. Such effect was
secured by the old so-called "black print process," but its opera-
tion was not altogether simple and good results were not reason-
394 DRAFTING INSTRUMENTS [Chap. XIV.
ably sure. The Vandyke paper has apparently solved this
difficulty,* and in addition affords other advantages which the
old "black process " paper did not possess.
Vandyke paper is a sensitized paper which is printed in the
same way as a blue-print, except that the tracing is put into the
frame so that the ink lines will be against the Vandyke paper.
The exposure is about 5 minutes in direct sunlight or, more defi-
nitely, until the portion of the Vandyke paper which protrudes
beyond the tracing is a rich dark tan color. Fresh Vandyke paper
is light yellpw in color. The print is washed for about 5 minutes
in clear water (where it grows lighter in color) and then it is put
into a solution consisting of about one-half ounce of fixing salt
(hyposulphite of soda) to one quart of water, where it turns dark
brown. It is left in the fixing bath about 5 minutes, after which
the print is again washed in water for 20 to 30 minutes and then
hung up to dry. The fixing solution may be applied with a
sponge or brush if only a few Vandykes are being made, but it is
better to immerse them in a tank containing the solution.
After the Vandyke print is washed the body is dark brown in
color while the lines are white. This is not the final print to be
sent out; it is simply the negative.
This Vandyke print is then put into the printing-frame in
place of the tracing, the face of the Vandyke being next to the
sensitive side of the process paper, and from it as many prints as
are desired are made on blue-print paper or on any kind of sen-
sitized paper desired. These blue-prints made from Vandykes
have a white background while the lines of the drawing appear
in deep blue lines, for in this case the rays of the sun act only
through the white parts of the Vandyke (the lines), whereas in
making an ordinary blue-print from a tracing the sun's rays act on
the paper through all parts of the tracing cloth except where the
lines appear. Where brown lines on a white background are
desired, the print is made by using a sensitized sheet of Vandyke
paper, in place of the blue- print paper.
One of the advantages of this process is that, as soon as a Van-
dyke has been made from the tracing, the tracing can be filed
away and kept in excellent condition, the Vandyke being used in
making all prints.
VANDYKE PRINTS 395
Another advantage in the use of the blue-prints which hare
been made by this process is that any additions made in pencil or
ink show clearly on the white background of the print which is
not true of the ordinary blue-print, on which corrections must be
made with a bleaching fluid or water-color.
442. Electrical Printing Frames. — The uncertainty of the
sunlight for making prints has brought forward a printing frame
in which an artificial light is used.
One form of electrical printing frame is an apparatus con-
sisting of a hollow glass cylinder, formed of two sections of glass,
and resting on a circular base which is rotated by clock work.
An electric light is suspended in the center line of the cylinder
where it travels up and down by means of a clock work attach-
ment.
The tracing and paper are wrapped around the outer sur-
face of the glass where they are tightly held against the glass by
a canvas which is wound around the cylinder by means of a
vertical roller operated by a handwheel. The cylinder can be
rotated at any desired speed and the light which travels up and
down the axis of the cylinder can be moved through any desired
distance or at any desired speed. These motions are all made
automatically when the apparatus is once adjusted.
In another type of electrical machine several horizontal rollers
are provided, with the light so arranged that as the tracing and
blue-print paper passes from one roller to another the exposure
is made. The speed of the machine is controllable and the length
of the tracing that can be printed is limited only by the length of
the roll of blue-print paper. With this machine, then, long
plans or profiles can be printed without the necessity of frequent
splicing which is required with other types of printing frame;
furthermore the color of the print is also uniform throughout.
The machine is driven by an electric motor. There are several
machines of this general type on the market; some of them are
provided with an apparatus for washing the prints as fast as they
come from the machine.
443. INKS AND WATER-COLORS. — Bottled ink, which is
prepared in various colors, is used extensively on engineering
drawings. The so-called "waterproof" inks differ from other
396 DRAFTINfe INSTRUMENTS [Chap. XIV.
inks in that a water-color wash can be put over the lines without
causing them to "run." Bottled inks are satisfactory for most
drawings, but when very sharp and fine hair-lines are required it
is well to use the stick india ink. This is made by grinding the ink
together with a little water in a saucer made for this purpose,
until the ink is thick and black enough to be used. If the ink
becomes dry it can be restored to as good condition as when first
ground by adding water, a drop or two at a time, and rubbing it
with a piece of cork or a pestle; if the water is added too rapidly
the ink will flake.
While the bottled black inks are fairly well prepared, the red
inks are very unsatisfactory. They will sometimes run on paper
where only very slight erasures have been made; in fact, on some
of the cheaper papers red ink will always run. For tracing pur-
poses red ink is wholly unsatisfactory, as it is impossible to ob-
tain a good reproduction of a red ink line by any of the
process prints. Where red lines are needed the use of scarlet
vermilion water-color will te found to give not only a brilliant
red line on the tracing, but also **body '' enough in the color
so that the lines will print fully as well as the black ink lines.
Scarlet vermilion water-color will give much better lines on any
paper than the bottled red inks. Only enough water should be
used to make the water-color flow well in the pen. Other water-
colors are used in th ^ place of the bottled colored inks, such as
P. ujsian blue i ste d of bottled blue i :k, or ^nt sienna instead
of brow:i ink, and these give much better results.
It is frequently necessary on blue-prints to represent additions
in white, red, or yellow. A white line can easily be put on by
using Chinese white water-color; but sometimes a bleaching fluid
is used which bleaches out the blue leaving the white paper visible.
The best color for a red line on blue-prints is scarlet vermilion
water-color; and for a yellow line none of the ordinary yellow
water-colors gives as brilliant lines as Schoenfeld & Co.'s lighi
chrome yellow.
For tinting drawings water-colors and dilute inks are used.
Effective tinting may be done on tracings by using colored pencils
on the rough side of the linen.
CHAPTER XV.
METBODS OF PLOTTING.
444. LAYDf G OUT A PLAN. — Laying out a plan requires
careful work. If a good-looking plan is to be obtained this part
of the work must be done with not a little judgment. Besides the
plan of the survey or property the drawing must have a title, and
sometimes notes and a needle to show the direction of the merid-
ian. These must all be arranged so that the entire drawing
when completed will have a symmetrical appearance. Often
the plot is of such awkward shape that it is very difficult to lay
out the drawing so that it will look well, and the draftsman's
artistic instincts are taxed to the utmost to produce a satis-
factory result.
445. Scale. — In many cases the scale of the plan as well as
the general arrangement of its parts must be chosen by the
engineer. Surveys of considerable extent which do not contain a
great many details, such, for example, as the preliminary survey
for a railroad, may be drawn to a scale of 400 ft. to an inch. A
plan of a large piece of woodland or a topographical map of a
section of a town may be represented on a scale of from ico ft.
to 400 ft. to an inch. A plan of a city lot for a deed is represented
on a 20- ft. to 80- ft. scale; and city streets, such as sewer plans
and the like, are frequently drawn to a scale of 20 ft. to 40 ft. to
an inch. Sometimes on plans of construction work drawings of
different scale are made on the same sheet. The drawing for a
conduit, for example, may be represented by a general plan on
a scale of 80 ft. to an inch, while on the same sheet the conduit
may be shown in section on a scale of 4 ft. to an inch.
The field maps of the U. S. Coast and Geodetic Survey are
usually plotted on a scale of TTriirTy, but some special maps are
made on scales as large as ^uVtt. The field maps of the U. S.
Geological Survey are mostly plotted to a scale of rahj^-s and
reduced on the lithograph sheets to v-^iis-c or i-^-^vjf-
397
398 . METHODS OF PLOTTING [Chap. XV.
These remarks in regard to scales are not to be con-
sidered in any sense as hard and fast rules to govern all condi-
tions They are suggested simply to give some idea of the exist-
ing practice in this matter.
METHODS OF PLOTTmO TRAVERSES.
446. PLOTTING BY PROTRACTOR AHD SCALE. —The
most common method of plotting angles is by use of the pro-
tractor (Art. 417, p. 382), and of plotting distances, by use of the
engineer's scale. Every traverse consists of a series of straight
lines and angles, which can be plotted by a protractor in the fol-
lowing manner. First, the survey to be mapped should be
sketched out roughly to scale, in order to ascertain its extent and
shape so as to decide the size of paper necessary for any given
scale of drawing and to determine its general position on the
sheet, which will fix the direction of the first line of the traverse,
to be used as a starting line for the entire drawing. This having
been done, the first line is drawn in the proper place on the paper,
its length is scaled off by using the proper scale, and its two ex-
tremities accurately marked by pencil dots or by means of a
needle point, and surrounded by a light penciled circle. The line
should be drawn so that it will extend beyond the next angle
point a distance greater than the radius of the protractor, this
extension of line being of use in the manipulation of the pro-
tractor.
The protractor is placed so that its center is exactly on the
second angle point and so that both the 0° and i8c° marks of the
protractor exactly coincide with the line. The traverse angle taken
from the field notes is plotted, the protractor removed, the line
drawn, and the length of the second course carefully scaled.
Then the protractor is placed along this new line and opposite
the third point, the angle at that point is laid off, the next line
drawn, and the distance scaled. By this process the entire
traverse is plotted.
447. Checks. — On all plotting work, just as on all field-
work and computations, frequent checks should be applied to
insure accuracy.
PROTRACTOR AND SCALE 399
If the traverse is a closed traverse the plot, of course, should
dose on the paper.* If it does not and the error of closure is in a
direction parallel to any one of the lines, there is probably a mis-
take in plotting the length of that line. If there is no indication
of this sort the mistake .may be either in scaling, in laying oflF the
angles, or in both. In such a case the entire plot should be
checked unless there is some reason to think that a certain line
may have been laid off at the wrong angle, in which event that
questionable angle should be replotted. The bearings of all the
lines of the traverse can be computed with reference to the mag-
netic or to any assumed meridian ; any line can be produced to
meet the meridian line, and this angle measured and checked.
Similarly, the bearing of the last line of a traverse which dDes
not close can be computed and the angle the last line makes with
the meridian measured. If it checks the computed angle it is
evident that no error has been made in the angles uiless mis-
takes were made that exactly balance each other, which is not
probable. In this way, by "cutting into" the drawing here
and there, the angular error, if there is one, can be quickly **run
down," without laying out all of the angles again and so possibly
repeating the mistake that was originally made. The angles
measured in applying this check have different values from the
ones first laid out, and the chance of repeating the original mis-
take is thereby eliminated. If no error is found to exist in the
angles, the distances should next be checked. This can be done
in two ways, and in some drawings both of these checks should
be applied.
First, scale each line separately setting down the results in-
dependently upon a sheet of paper. After these are all recorded
(and not before), compare the lengths with the lengths of lines as
taken from the field notes. No error should be allowed to pass
if it is large enough to be readily plotted by the use of the scale.
* Instead of plotting every line of the traverse from its preceding line and
returning, in the case of a closed traverse, to the other end of the starting line, it
may be well to plot half the traverse from one end of the starting line and the
other half from the other end ; the check will then come at a point about half-way
around the traverse. The advantage of this method lies in the fact that accumu-
lative errors are to some extent avoided since they are carried through only half
a8 many courses.
400 METHODS OF PLOTTING [Chap. XV.
Second, take a long straight piece of paper, lay this on the
drawing, and mark off tlie length of the first line on the edge of
the paper; then mark off the length of the second Une starting
from the mark which denotes the end of the first hne, and proceed
in a similar way to the end of the traverse. Apply the scale to
the strip of paper and read the station of each mark; record each
of these independently and afterwards compare them with the
field notes. The entire length of Une should check within a
reasonable amount depending upon the scale; the allowable
error can easily be determined by the principle explained in
Art. 23, p. 14.
By checking angles and distances by the above methods
errors of any consequence can be avoided; in any case a drafts-
man should not allow a drawing to leave his hands which has
not been properly checked and known to be correct.
When the traverse is not closed, such checks as have been
described above must always be applied; otherwise there is no
assurance whatever that the plan is correct. It is especially
necessary to check the bearings of lines frequently, so that the
accumulation of small errors may not become appreciable.
448. Protractor and T-Square. — While the ordinary
T-square is not much used in plotting engineering plans, there
are some occasions where it is convenient to use it. Where a
traverse has been run by bearings or by deflection angles the
T-square with a shifting head can be conveniently used in con-
nection with a protractor for plotting the angles by bearings.
The piper is fastened to a drawing board having a metal edge,
which insures one straight edge to the board. A meridian line
is drawn on the paper, and the shifting head of the T-square
is fastened so that the blade coincides with the meridian line.
Then as the T-square is slid up and down the edge of the draw-
ing board its blade always takes a direction parallel to the meri-
dian. By means of the protractor shown in Fig. 172 the bear-
ing of each line can be readily hid off or checked as illustrated
by Fig. 174 and the distances laid off with the scale. In order
to secure a satisfactory check, the deflection angles should be
laid off directly from the previous line, and the bearings checked
by means of the T-square and protractor.
T-SQUARE AND PROTRACTOR
40t
It is evident that the bearings of the lines may be computed
just as well from any assumed meridian as from the magnetic or
true meridian ; and that the drawing can be fastened to the board
Fig. 174. Laying off Bearings by Use of T-Square and
Protractor.
in such a way that the T-square can be conveniently used.
This method is especially applicable to compass surveys as it ob-
viates the necessity of drawing a new meridian line through each
angle point.
This method can be easily applied also as a means of check-
ing any of the angles of a traverse which have been plotted by
any of the ordinary methods.
449. PLOTTING BY RECTANGULAR COORDINATES. — In plot-
ting by this system all points in the traverse are referred to a
pair of coordinate axes. For convenience thess axes are often
the same as those used in calculating the area enclosed by the
traverse. The advantages of this method are, (i) that all meas-
urements are made by means of the scale only and (2) that the
plotting may be readily checked.
To plot a survey of a field by rectangular coordinates, first
calculate the total latitude and the total departure^ that is, the
ordinate and the abscissa, of eich point in the survey. If the
meridian through the most westerly point and the perpendicular
through the most southerly point are chosen as the axes negative
402
METHODS OF PLOTTING
[Chap. XV.
signs in the coordinates will be avoided. The coordinates of the
transit points are computed by beginning with the most westerly
point, whose total departure is zero, and adding successively the
departure of each of the courses around the traverse. East
departures are called positive and West departures negative.
The total departure of the starting point as computed from
that of the preceding point will be zero if no mistake is made in
the computations. The total latitudes may be computed in a
similar manner beginning, preferably, with the most southerly
point as zero.
450. For plotting the points on the plan, a convenient
method of procedure is to construct a rectangle whose height
equals the difference in latitude of the most northerly and the
most southerly points and whose width equals the difference in
departure of the most westerly and the most easterly points. If
the most westerly and the most southerly points are taken as zero
then the greatest ordinate and the greatest abscissa give the
dimensions of the rectangle. The right angles should be laid
off either by the use of a reliable straight-edge and a triangle or
by the beam compass.
451. The better method, however, is to construct the
perpendiculars by means of a straight-edge and a triangle. It is
Fig. 176. Erecting a Perpendicular with a Straight-edge and
AN Inaccurate Triangle.
RECTANGULAR COORDINATES 403
not at all necessary, although it is always desirable, that the
triangle shall be accurate. It should be used in the following
manner. It is first placed against the straight-edge, as shown by
the full lines in Fig. 175, and a point A, marked on the paper.
Point C is also marked opposite a certain definite part of the
triangle. Then the triangle is reversed to the dotted position
and brought so that its edge coincides with point A, and then
point B is marked opposite point C, as nearly as can be judged.
A point D is plotted midway between B and C and the line AD
is then drawn which is perpendicular to the straight-edge. If the
triangle is accurate point B will fall on point C, so that this
is a method of testing the accuracy of the right angle of any tri-
angle. If it is found to be inaccurate it should be sent to an
instrument maker and be "trued up.** A few cents spent in
keeping drafting instruments in shape will save hours of time
trying to locate small errors, which are often due to the inac-
curacy of the instruments used.
If the compass is used the right angle may be laid off by
geometric construction. On account of the difficulty of judging
the points of intersection of the arcs, very careful work is re-
quired to obtain good results with the compass.
Since the accuracy of all of the subsequent work of a coordi-
nate plot depends upon the accuracy with which the rectangle
is constructed, great care should be taken to check this part of
the work. The opposite sides of the rectangle should be equal
and the two diagonals should be equal, and these conditions
should be tested by scaling or with a beam compass before con-
tinuing with the plot.
452. After the rectangle has been constructed, all points in
the survey can be plotted by use of the scale and straight-edge.
To plot any point, lay off its total latitude on both the easterly
and the westerly of the two meridian lines of the rectangle, be-
ginning at the southerly line of the rectangle. Draw a line
through both of these points by means of a straight-edge.*
* Accurate work, of course, cannot be obtained with a straight-edge that is
not true. A straight-edge can easily be tested by drawing a fine pencil line on the
paper along one edge of the straight-edge; then turn the straight-edge over on
its other side, fit the same edge to the two ends of the pencil line, and see if the
edge coincides with the line.
404
METHODS OF PLOTTING
[Chap. XV.
Then lay off along this line the total departure, beginning at the
westerly side of the rectangle, thus obtaining the desired position
of the point.
The computations of the total latitudes and departures and
the method of plotting a traverse by the coordinate method are
shown in Fig. 176. This is the survey which is shown in the
we.71 >G
< 6IZ32 >
Fig. 176. Computations and Plotting by Rectangular
Coordinates.
calculations in Fig. 161, p. 362, and in the form of notes in Fig.
52, p. 103.
453. Plotting by rectangular coordinates is the most accurate
of all the methods usually employed. It is not very often ap-
plied, however, to traverses which do not close, as there is seldom
any other use for the coordinates of such a traverse, and the
RECTANGULAR COORDINATES 405
labor of computing them for this purpose alone is hardly war-
ranted. For such traverses, therefore, either the protractor and
scale, the Tangent Method, or the Chord Method (which are ex-
plained in the following articles) may be employed. But for
plans of a closed traverse, where the latitudes and departures
have been computed in cpnnection with calculating its area, this
coordinate system of plotting is frequently used.
454. Checks. — When the transit points have been plotted,
the scale distance between consecutive points should equal the
distance measured in the field. It sometimes happens that some
of the transit lines run so nearly parallel to one of the axes that
the distances will scale the right amount even though a mistake
has been made in laying off one of the coordinates. In such a
case any appreciable error can be detected by testing the bear-
ings of the lines by means of a protractor. These two tests,
together with the scaled distances of any cut-off lines which may
have been measured in the field, (Art. 145, p. 109), form a good
check on the accuracy of the plotting. Since all of the points
are plotted independently errors cannot accumulate. If it is
found that any scaled distance fails to check with the measured
distance it is probable that one of the two adjacent lines will also
fail to check and that the point common to the two erroneous
lines is in the wrong position.
It should be remembered that everything depends upon the
accuracy of the rectangle and that nothing should be plotted
until it is certain that the right-angles have been accurately laid
off.
455. PLOTTING BY TANGENTS. — The traverse should first be
plotted approximately on some convenient small scale by use of
the protractor and scale, to ascertain its extent and shape. The
importance of this little plot is often overlooked, with the result
that when the plan is completed it is found to be too close to one
edge of the paper or otherwise awkwardly located on the sheet.
It takes only a few moments to draw such a sketch, and unless
the draftsman is sure of the shape and extent of the plot he'
should always determine it in some such manner before the plan
is started.
The directions of all the lines are referred to some meridian
4o6
METHODS OF PLOTTING
[Chap. XV.
and the bearings determined with an accuracy consistent with the
measured angles. From the auxiliary plot it can be decided
where to start the first course of the traverse on the paper and
in what direction to draw the meridian, so that the lines of the
completed traverse will be well balanced with the edges of the
sheet, and so that the needle will be pointing, in a general way,
toward the top of the drawing rather than toward the bottom.
The bearing of the first line is plotted as follows (Fig. 177).
Fig. 177. Plotting by Tangent Offsets.
Lay off on the meridian line a length Aa of at least 10 inches
and erect a perpendicular at a on the right-hand side of the
meridian if the bearing of the first course is east, and on the
left-hand side if it is west. Look up in the table of natural
functions the tangent of the bearing of the first course and scale
off this distance ai on the perpendicular.* Draw A6 which is
* These distances and also the lo-inch base-lines are all laid off by use of the
engineer*s scale. By using the lo-ft. or loo-ft. scale the tangents can be laid off
without any computation, whereas with the other scales the tangent must be multi-
plied by some number, e.g., by 2 if the 20-ft. scale is used, by 3 if the 30-ft. scale
is used, etc., taking care in the pointing off.
If it is deemed unnecessary to use a base as long as 10 inches, one can be laid
off at the *' 10 ** mark on any engineer's scale and the tangent distances laid off by
using the same scale, e.g., if a 20-ft. scale is used the ^ 10 " mark will give a base-
line 5 inches long.
PLOTTING BY TANGENTS 407
the direction of the first course. On this line scale off ABy the
length of the first course. On this line produced lay off Bg equal
to 10 inches and erect a perpendicular, scaling off on the perpen-
dicular the length gd equal to the tangent of the deflection angle
at B, This determines the direction of ^Cfrom the first course.
The remaining lines of the traverse are plotted in the same manner,
using each time the deflection angle.
456. Checks. — Unless the survey is a closed traverse checks
must be occasionally applied. Every third or fourth course
should be checked by finding the angle between it and the meri-
dian line. This angle should be found by the same method
(tangent offset method) and by using a base of 10 inches as in
plotting the angles. In checking the course £>e, for example, a
meridian is drawn through Z> parallel to Auy De is scaled off 10
inches, and a perpendicular ef erected. The distance ef is scaled
and from the table of tangents the angle /De is obtained. If the
angle that the course makes with the meridian line disagrees
with the calculated bearing of that course by any considerable
amount, say, 10 minutes of angle or more, the previous courses
should be replotted. If the error is less than 10 minutes the
course which is being checked should be drawn in the correct
direction so that even the slight error discovered may not be
carried further along in the plot. Then after the plotting has
proceeded for three or four more courses the check is again
applied.
The bearings of the lines can be checked by use of the pro-
tractor and this will detect errors of any considerable size, but
this method will not disclose any small errors ; moreover, if it is
desired to have the plot when completed as accurate as could be
expected from the precise method employed, it is entirely incon-
sistent to check by use of a method which is far less accurate
than the one used in making the plot. For this reason the
checks on the direction of the lines are applied with the same
care and by the same method as was used in the original layout
of the angles.
Occasionally it is more convenient to plot the complement of
an angle rather than the angle itself, as was done in plotting the
line EF, In this case the right angle erected at E must be laid
4o8
METHODS OF PLOTTING
[Chap. XV.
off with great care, preferably by the method explained in Art.
45 1, p. 402.
It is evident that the direction of each course could have
been plotted by drawing a meridian line through the transit points
and by laying off the bearings by the tangent method. But if
such a method were used there would be no single check applied
that would check all the previous courses, which is an important
feature of the method explained above.
If the traverse is not closed the lengths of the lines of the
traverse should always be checked by the methods explained in
Art. 447, p. 398.
457- PLOTTING BY CHORDS. — This method, which is em-
ployed by many draftsmen in plotting traverse lines, is fairly
good although probably not so accurate as the Coordinate or
as convenient as the Tangent Methods.
Fig. 1 78 represents the traverse ABCDEF which has been
Fig. 178. Plotting by Chords.
plotted by chords. It is the same traverse that is shown in
Fig. 177.
On the meridian line the distance Aa is scaled off equal to
10 inches and the arc ab swung from -4 as a center by use of
the ordinary pencil compass. Then from a table of chords * the
* Tables of chords can be found in Trautwine*s " Civil Engineer's Pocket
Book," published by John Wiley & Sons, New York.
PLOTTING BY CHORDS 409
length of the chord ab is found for the angle aAb, The point b
is sometimes located by setting the dividers at the distance ab
and with ^z as a center intersecting the arc ^ at ^; but the
more accurate method is to scale from point a the chord distance
and mark the point b on the arc. Then the line Ab is drawn and
AB scaled off on it. With 5 as a center the arc gd is drawn
and the chord gd, corresponding to the deflection angle at 5, is
scaled off. Bd is then drawn and BC scaled off on it. In the
same way the entire traverse is plotted.
458. Use of the Sine, — It is evident that the chord
ab^2y. lox sm — •
2'
hence, if a table of chords is not available, a table of sines (always
easily obtainable) can be used. The sine of half the angle can
be taken from the tables and multiplied by 20 mentally. Some
draftsmen use the table of sines and a radius of 5 inches to avoid
the multiplication. This is not recommended because a base of
5 inches is not long enough to insure a very accurate drawing.
The necessity of multiplying by 2 can very easily be done aWay
with by laying off the radius with a 2o-ft. scale and scaling off
the sine of the angle with a lo-ft. scale.
With dividers of the ordinary size it is impossible to lay out
an arc with a lo-inch radius. In such a case either beam com-
passes must be used or the radius employed must be shorter, so
short, in fact, that it will frequently be better to resort to the
Tangent Method.
459, Checks. — Since this method is usually applied to
traverses which do not close it is desirable to check every
fourth or fifth course so that a mistake will not be carried too
far before it is discovered and thereby cause a waste of time.
In Fig. 178 it is desired to check the calculated bearing of De,
The meridian Df is drawn through D parallel to Aa, the arc fe
is swung with Z^ as a center and with a radius of 10 inches, and
the chord ef\^ scaled. From the table of chords (or sines) the
angle fDe (the bearing) can be found. It should agree reason-
ably well with the calculated bearing. The degree of precision
to be expected when plotting by chords is a little less than
4IO METHODS OF PLOTTING [Chap. XV.
that suggested for the Tangent Method in Art. 455, unless the
beam compass is used. The Tangent Method, especially if the
right angles are laid off by reversing the triangle, gives more
accurate results than the Chord Method, for the use of the
ordinary compass in the Chord Method is a fruitful source of
error unless it is handled with the utmost care.
METHOD OF PLOTTING DETAILS.
460- BUILDINGS, FENCES, STREAMS, ETC. — The previous
articles have dealt with the plotting of the traverse lines only,
and these in many cases form merely the skeleton of the final
plan. In the field the details of the survey are located from the
transit line ; and, in a similar manner, the details are located on
the plan from the traverse line which has already been plotted.
Buildings, fences, shore-lines, streams, etc. are all plotted by
means of the scale for distances and the protractor for the
angles. Often a smaller protractor is used for this sort of work
than for the traverse lines. This is permissible, for the lines
which locate the details are usually short in comparison with
the traverse lines and the resulting error is small in any case ;
furthermore any slight error in the location of a detail will not
as a rule affect the rest of the drawing, whereas an error in a
transit line will, of course, have an effect on all of the rest of
the drawing. The plotting of buildings has been taken up in
connection with their location. (See Chapter VI.)
In plotting a set of notes where several angles have been
taken at one point, such as in stadia surveying, it is well to plot
all of the angles first, marking them by number or by their
value, and then to plot the distances with the scale.
461. Contours. — Where contours are located by the cross-
section method (Art. 304, p. 278), this cross-section system is
laid out in soft penciled lines on the drawing. The elevations
which were taken are written at their respective points on the
plan and then the contours desired are sketched. The ground
is assumed to slope uniformly between adjacent elevations, and,
by interpolation between these points, the location of the con-
tours on the plan can be made. When the contours have been
PLOTTING DETAILS 4 1 1
located, the cross-section lines and elevations are erased unless
the plan is intended to be used as a working drawing. As a
rule all useful data, such as construction lines and dimensions,
are left on a working drawing.
When the contours are located by any other means the
principle is the same. The points whose elevations have been
determined are plotted by scale and protractor, and the contours
are interpolated between the elevatibns and sketched on the
plan.
462. CROSS-SECTIONS. — In plotting on cross-section paper,
the rulings of the paper are used as the scale, and all the dimen-
sions of the cross-section, which are to be plotted, are laid off by
counting the number of squares on the cross-section paper.
In highway, railroad, and dam construction it is often neces-
sary to keep a record of the progress made on the earthwork by
plotting the cross-section at each station, and, as the work goes
on, to mark on each section in colored ink the progress of the
work for each month. In this way monthly estimates can be
readily made, and the cross-section sheets will also give a record
of the progress of the work, each month being represented by a
different colored line or by a different style of line.
Where a series of cross-sections like this are to be plotted
the station number and the elevation of the finished grade are
recorded just under or ovier the section. To avoid mistakes in
numbering the sections this should be done at the time of plot-
ting the section.
As these cross-section sheets rarely go outside the office they
are usually considered in the same class with working drawings,
and dimensions, such as the areas of sections or the quantities
of earthwork, are usually recorded on them, together with any
other data which may be of use in calculating the volumes.
463. Profiles. — Profiles are almost always plotted on pro-
file paper, although occasionally they are plotted on the same
sheet with the plan so that the two can be readily compared.
The profile is intended to show (graphically) relative eleva-
tions. In most surveys the differences in elevation are so small
in comparison with the horizontal distances that it is necessary
to exaggerate the vertical scale of the profile so that the eleva-
412 METHODS OF PLOTTING [Chap. XV.
tions can be read from the profile With a reasonable degree of
accuracy. The horizontal scale of the profile should be the same
as the scale of the plan, but the vertical scale should be exagger-
ated, say, 5 to 20 times the horizontal scale, depending upon
how close it is desired to read the elevations from the drawing.
If the horizontal scale of the profile is 80 ft. to an inch its ver-
tical scale should probably be 20, 10, or 8 ft. to an inch.
464. In plotting any 'profile the first step is to lay it out
properly on the paper, i.e., to decide, from an examination of the
range of the elevations, where to start it on the paper so that it
will look well when completed, and so that any additions or
studies which may subsequently be drawn on it will come within
the limits of the paper. Station o of the profile should come on
one of the heavy vertical lines, and the heavy horizontal lines
should represent some even elevation such as 100, 125, 150, etc.
The profile is plotted by using the rulings of the profile
paper as a scale ; it is drawn in pencil first and afterward inked
in. It will be found, if these profile papers are carefully meas-
ured with a scale, that they are not as a rule very accurate.
The rulings may be uniform, but owing to the shrinkage of the
paper the divisions frequently do not scale as long as they should.
In plotting a profile or section on such paper no attempt is made
to use a scale ; the scale of the paper is assumed to be correct
and the intermediate points are plotted by estimation, which can
almost always be accurately done since the rulings of the paper
are quite close together.
The data for a profile of the ground generally consist of
levels taken in the field at such points that the ground may be
assumed to run straight between adjacent elevations. For this
reason, in drawing the profile, the points where the slope of the
ground changes should not be rounded off. On the other hand,
however, the ground probably does not come to an actual angle
at that point. The profile should be plotted therefore as a series
of free-hand straight lines drawn so that the angles are not
emphasized. When a profile is made from a contour map, the
line should be a smooth, rather than an angular line.
465. Profiles of the surface of the ground are generally
made for the purpose of studying some proposed construction
J
PLOTTING PROFILES 413
which is represented on the profile by a grade line, consisting
usually of a series of straight lines. The points where the gra-
dient changes are plotted and connected by straight ruled lines
unless the proposed grade should happen to be a vertical curve
(Art. 268, p. 242). Vertical lines are also drawn from the bot;tom
of the profile to the grade line at these points.
466. When the elevations are such that the profile, if con-
tinued, will run off the top or bottom of the paper the entire
surface line is lowered or raised some even number of feet, such
as 20 or 50 ft., and the plotting continued : the number of feet
represented between two heavy horizontal rulings of the pro-
file paper should determine the drop or rise of the grade line.
This change should be made, when convenient, on one of the
heavy vertical rulings of the piaper or on one of the vertical lines
where the gradient changes.
467. Checks. — After plotting the surface aind grade eleva-
tions in pencil, read off from the profile the station and elevation
of each point as plotted and record both the station and eleva-
tion on a piece of paper. Compare these readings with the data
given and make the necessary corrections. Time can be saved
if one man reads off the station and elevation from the profile
while a second man compares the readings with the note-book.
A quick method of plotting profiles is to have one man read the
notes while the other man plots them, but when the profile is
being checked this method should not be used ; the man, prefer-
ably the one who did not do the plotting, should read from the
profile as plotted and these readings should be compared with the
note-book.
PROBLEMS.
1. Plot the surveys given in Fig. 50, p. 100, and in Fig. 53, p. 104, by Pro-
tractor and Scale, Rectangular Coordinates, Tangents, or Chords.
2. Plot by use of Scale and Protractor the notes given in Fig. 72, p. 168, and
in Fig. 116, p. 266.
Miw)Mt Fork Conmissioit
- s
M
^^"^Z?
CHAPTER XVL
FINISHINa AND nUNG DRAWINGS.
468, WHAT SHOULD APPEAR ON A DRAWING. — Drawings
are made for a great variety of purposes, so that the data which
a plan should contain depend entirely upon the use to which it
is to be put. There are, however, several important things which
should appear on every engineering drawing. In the first place,
it should have a complete title which should be a brief descrip-
tion of the drawing. The title should state whether the drawing
is a plan, cross-section, profile, etc. ; what it represents, — a lot
of land, a sewer, a railroad, etc. ; the name of the owner ; the
place ; the date ; the scale ; and the name of the surveyor. Be-
sides the title, some plans, such as land plans, always require the
names of owners of abutting property, and a meridian. Notes
are frequently added giving such information as is necessary to
interpret the plan. All essential dimensions are lettered in their
proper places.
Besides these it is well to insert in some inconspicuous place
(preferably near the border) the number of the note-book and the
page from which the notes were plotted, and also the initials of
the draftsman who made the drawing and of the man who
checked it.
Fig. 1 79 represents a land plan which contains all of the es-
sentials ; it is a plot of the land shown in the form of notes in
Fig- 52, p. 103; its computations are on p. 362; and its working
plot is illustrated by Fig. 176, p. 404.
469, Traverse lines. — The convenient use of a plan some-
times requires the traverse line to be shown on the completed
drawing. In such a case it is usually shown as a full colored line,
each of the angle points being represented by a very small circle
of the same color, the center of which marks the angle point.
Sometimes the lines of the traverse are drawn to the angle points
415
4i6
FINISHING AND HLING DRAWINGS [Chap. XVL
which are marked by very short lines bisecting the angles. Fig.
1 80 illustrates these two methods of marking transit points.
Fig. 180. Methods of Marking Angle Points on Traverse Lines.
Tnangulation stations are represented by a small equilateral
triangle drawn around the station point. Fig. 115, p. 258, con-
tains several examples of this.
470. PHYSICAL FEATURES. — The boundaries of property
and the physical features which are represented on a plan, such
as streets, buildings, etc., are usually drawn in black ink. Any
additions or proposed changes are frequently drawn in colored
ink, usually in red, although water-color is much better for
the reasons stated in Art. 443, p. 395.
Shore lines and brooks are represented either in black or in
Prussian blue. As a rule the shore line should be one of the
heaviest, if not the heaviest line, on the drawing. Water-lining,
shown in the topographical signs in Fig. 181, adds materially to
the prominence and appearance of a shore line.
471. TOPOGRAPHIC CONVENTIONAL SIGNS. — On topo-
graphic maps certain physical features are shown by conventional
signs which have come to be used so generally that they are
practically standard throughout the country. A few of the
more common of these symbols are shown in Fig. 181. The
one representing " cultivated land " and the horizontal lines of
the "salt marsh" and "fresh marsh" symbols are ruled; the
rest are executed with an ordinary pen, Gillott's No. 303 being
a good one for such work.
It will be noticed that in the symbol for " grass " the indi-
vidual lines of a group all radiate from a center below the group,
and also that they end on a horizontal line at the bottom. This
CONVENTIONAL SIGNS
417
^
fJ!»
.fta.%^
^aiu^gtiA.
Deciduous Trees (Oak).
-•IM -'^ -«^
•Wl*.
.*1X*.
.^1//- «ilM,
*»»/A.
<iU^
^•^«^ *^
*iU<».
M.
.^■».
ai
^Mld.
*»w*.
^//.
4M4»
#/A»
*•/«* *'*'*
.«!//*.
aiiK*.
^MM
-:£&
Grass.
Fresh Marsh.
Waterlining.
-r^X^\t
JU "m OJ •*. • ,/, ^ *
Deciduous Trees (Round Leaf),
v?i.s K s \ \ .s \ \ sX X A
Cultivated Land.
Salt Marsh — Sand.
Ledges — Evergreen Trees.
Fig. 181. Topographic Conventional Signs.
41 8 FINISHING AND FILING DRAWINGS [Chap. XVI.
horizontal line, in the case of "grass" or "marsh" symbols,
should always be parallel to the bottom of the map.
In executing "water-lining " the first line outside the shore
line should be a light full line drawn just as close to the shore
line as possible, and should follow very carefully every irregular-
ity of the shore line. The next water-line should be drawn
parallel to the first but with a little more space between them
than was left between the shore line and the first water-line.
Then the third water-line should be spaced a little farther out,
and so on ; five to ten lines are sufficient to represent this sym-
bol properly. As the succession lines are added farther and
farther from the shore line, the little irregularities of the shore
gradually disappear until the outer water-line shows only a few
irregularities opposite the most prominent ones of the shore.
Wat^r-lining, as well as fresh marsh and salt marsh symbols,
is often represented in Prussian blue. In fact, on some topo-
graphic maps most of the signs are represented by colors, —
the trees by green, the grass by a light green tint, water by a
light blue tint, cultivated land by yellow ochre, and so on.
Contour lines (shown in several of the cuts in Chapter X.)
are almost always drawn in burnt sienna water-color. Every
fifth or tenth contour is usually represented by a line slightly
heavier and also a little darker in color. Gillott's No. 303 i>en
will be found to give good results for this work ; but a contour
pen, if it can be handled well, will give very uniform lines esj)e-
cially where the contours have no sharp turns. In numbering
the contours some prefer to break the lines and place the num-
bers in the spaces, while others prefer to place the numbers
just above or below the contours. Frequently a number is
placed on every contour, but for most plans this is entirely un-
necessary. If the contours are somewhat regular it is only
necessary to number, say, every fifth contour. A good general
rule to follow is to number only those lines which are necessary
in order that the elevation of any contour may be found without
appreciable mental effort. The numbers on the contours should
be small plain figures in burnt sienna.
The shape of the surface of the ground is sometimes repre-
sented by hachure lines, which are illustrated in Fig. 182. The
CONVENTIONAL SIGNS
419
contour lines are first sketched in pencil as a guide to the drafts-
man in drawing the hachure lines, which should be drawn normal
to the contours. The short
lines are drawn from the summit
downward in rows, each row
just touching the next pre-
ceding row. The steepness
of the slope is represented by
the weight and length of the
lines,— the steeper the slope the
heavier and shorter the lines.
The individual lines are equally
spaced, but on the flat slopes
where the lines are lighter they
have the appearance of being
spaced farther apart.
472. Such physical features as railroads, highways, buildings,
Fig. 182. Hachure Lines.
*■ --
1 1 I t I I I I I
I I I I I I I I I
,f- —
□
O
BMxl232
Building. (On large scale maps.)
Bam or Shed. (On large scale maps.)
Buildings. (On small scale maps.)
Fence.
City or Town Boundary.
Stone wall.
Stone Retaining wall.
Single Track Railroad.
Double Track Railroad.
Roads.
Trail.
Bridge.
Triangulation Station.
Stadia Station.
Transit Point. Intersection Point
Bench Mark. (- TnanguUHon.)
Fig. 183.
420
D
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QRST
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421
422
FINISHING AND FILING DRAWINGS [Chap. XVL
and boundaries are usually represented in black ink by the sym-
bols shown in Fig. 183.
473. LETTERUfG.* — The lettering on a drawing probably
has more to do with its appearance than any other feature. To
be able to do good lettering at first is a gift which but few
men possess. It is an art that can be acquired by the most
awkward draftsman, however, if he will study it carefully and
devote a little time to systematic practice.
Several di£ferent styles of lettering are shown in Figs. 184
and 185. The general style to use in 'any given case depends
on the type of drawing and on the use to which it is to be
put. On plans which are to be sent from the office as com-
pleted drawings such letters as the Roman or Gothic may be
appropriate. Stump writing is a style of lettering which is
difficult to execute but whose appearance, when well done, is
very artistic. The ornate lettering in vogue a few years ago
has been superseded by simpler styles which require much less
time to produce. For construction drawings, like a plan of a
bridge or a conduit, for example, the Reinhardt letters are used
Fig. 186. Lettering on Slopes.
* For a complete discussion and illustrations of lettering see any of the fol-
lowing publications : "Plain Lettering," by Professor Henry S. Jacoby, pubHshed
by the Engineering News Publishing Company ; ** Technic of Mechanical Draft-
ing,** by Charles W. Reinhardt, published by the Engineering News Publishing
Company ; " Letter Plates,** by Professor Charles L. Adams, Mass. Inst, of
Technology, published by Professor Adams.
LETTERING 423
to a considerable extent. The title of such a plan looks well
lettered in either erect or inclined Gothic.
All plans should be lettered so as to read from the bottom.
Unless a draftsman exercises considerable care he will find,
when the plan is completed, that some of the lettering is upside
down. Fig. 186 illustrates the proper lettering of lines of vari-
ous slopes.
474. Titles. — The design of the title of a plan gives the drafts-
man an opportunity to exercise good taste. It should be so ar-
ranged and the size of the letters so chosen that the most im-
portant part of the title strikes the eye first. In general, each
line of lettering should be centered, and the spacing between the
lines should be so arranged that no part will either appear crowded
or seem to be floating away from the rest of the title. The gen-
eral outline of the title should be pleasing to the eye. In some
of the larger offices, in order to save the time of the draftsman,
titles are set up in type and printed on the map.
Fig. 187 shows a set of titles which are well balanced and
complete. Fig. 188 shows the style of lettering appropriate for
a profile, a cross-section, or construction details.
424 FINISHING AND FILING DRAWINGS [Chap. XVI.
F^reLtnUnary Slltv^ ftor ct RtLtLnocLcL
/yonv
Crescent Beach, to Wbodlawn Centeteny.
Oct€>ben, /802^
Sccvle '400 rieet to / irtch,
COMMONWEALTH OF MASSACHUSETTS.
METROPOLITAN WATER WORKS.
WACHUSETT DAM
UPPER GATE-CHAMBER.
JULY 9, I900.
UNITED STATES
COAST AND GEODETIC SURVEY
SKETCH OF GENERAL PROGRESS
JUNE 30 1897
Easteni Sheet
Fig. 187. Titles of Plans.
TITLES 4:^5
TRACK ELEVATION.
C. <Sc Vs/. L R. R.
Cross- Section of Brid3e Showing
Floor Construction.
Scale i in.* Ift.
HORIZONTAL SECTIONS
thuouch upper through lower through lower
sluice-gate sluice-gate valve well
01t545«ft
PineliminaiyRvfile
for 9 Railroad from
Bedford Junction to North Liberty
Sta.0 to Sta498*68.7
May/906
Fig. 188. Titles of Profiles
426 FINISHING AND FILING DRAWINGS [Chap. XVI.
475. Notes. — Most drawings require notes of some sort.
These are usually executed with a plain letter like the Reinhardt
alphabet. In Fig. 189 are a few samples the general style of
which is consistent with modern practice.
Note:- Th/s re/nforcemenf is 8-0 "fongf,
ancf comes direcf/y under each track.
Leave amp/e room for brf'of^e-seat
Note:-The datum plane used for con-
tours and soundings on this map is
"Boston City BaseV
Boston City Base is 0.64 ft below
base known as"Mean Low Water at
Navy Yard" which is the datum used
by the U.S. Coast Survey, the U.S.
Engineers Office, and the Mass.
Harbor and Land Commission.
Soundings and Contours confirmed and ex-
tended by data from map (1.-476) on file witti
Massachusetts Hartwr and Land Commission.
Fig. 180. Samples ok Notes.
NOTES — BORDER LINES 427
476. Border Lines. — The border line of a drawing should
consist of a heavy single line or double lines closely spaced. It
should neither be so heavy nor of such fancy design as to be con-
spicuous. Plain clear drawings are the practice of to-day, and the
border line should be in keeping with the rest of the drawing.
For drawings 2 ft. long, the border should be about J" from the
. edge of the sheet : for drawings 4 ft. long, i" to i-J" looks well.
On some, particularly office drawings, the border is unnecessary
and may be undesirable. Fig. 190 gives a few examples of sim-
ple practical border lines.
Fig. 190. Border Lines.
477. Meridians. — On all land plans it is customary to draw
either the true or the magnetic meridian ; often both of them are
represented. To be in keeping with the rest of the drawing this
should be simple in design. Too frequently, however, the drafts-
man attempts to "lay himself out " on the needle with the result
that it is so large and ornate that it is the first thing in the draw-
ing that strikes the eye. The simple meridians shown in Fig.
191 are suggested as suitable for ordinary land plans.
The plan should always be drawn, if possible, so that the
428
FINISHING AND FILING DRAWINGS [Chap. XVI.
meridian will point, in general, toward the top of the drawing
rather than toward the bottom. Sometimes it is drawn with its
upper part above and its tail below the drawing. In such a case
Fig. 191.
5
MERroiANS.
the line of the meridian must never cut any of the lines of the
drawings: it should be interrupted far enough from the drawing
so that it cannot be mistaken for one of the property lines.
478. Scales. — On account of the shrinkage of drawing
paper the scale is sometimes drawn on the plan itself at the time
that the drawing is plotted. It is well to have it sufficiently
long, say, 3 to 10 inches (depending upon the size of the draw-
ing), so that it will be of use in detecting the amount of shrink-
age. This, of course, will determine the shrinkage only in the
direction of the scale. These scales are usually placed directly
under the title or in one of the lower corners. Fig. 192 gives
two examples of scales.
In plotting a coordinate survey, the intersections of the north
and south with the east and west lines should be marked on the
finished drawing, as these are of great assistance in plotting addi-
tions. Moreover the distances between these points give a
reliable measure of the change in scale of the map due to
shrinkage.
479. SHRINKAGE OF DRAWING PAPERS. — All of the papers
in use will shrink and swell more or less with variations of
MERIDIANS — SCALES 429
weather conditions. The heavy mounted papers are affected
the least, but large drawings even on such paper will be found
on examination to change in size perceptibly. The fact that they
do not always shrink the same amount in different directions
T
SCALE ^
?....T «r •¥» *p
_J5.r«r
i * t
"^
SCALe, 1 INCH • ZOO PtlT
B» 0
FiQ. 192. Scales.
•00 ao
makes it difficult to estimate the amount of the change and to
allow for it. This effect can be estimated quite closely, how-
ever, by testing the drawing by measuring accurately a few lines
running in different directions when it is plotted and scaling the
same lines at any other time and making allowance for the
change. Scaled distances on tracing cloth are quite unreliable
if it is not kept in a dry place, and blue-prints generally shrink
in washing so that scale measurements taken from them usually
contain considerable error.
480, MAPS OF LARGE EXTENT. — Some maps, like the
location map of a railroad or the map of a city, are so large that
they must be made in sections. In such cases two slightly differ-
ent methods are employed. One method is to plot the several
sheets so that the drawing on one will extend to but not include
any of the drawing on the adjacent sheet, the limits of the draw-
ings being defined by straight lines. The other method is to
have the drawing on each sheet lap over the drawings on the
adjacent sheets a little. In this case marks are made on all
drawings which make it possible to fit them to the correspond-
ing marks on the adjacent drawings when they are being used
jointly.
In attempting to arrange the sheets of adjacent drawings after
they have been in use for any considerable time, it is often found
that they do not fit well on account of the unequal shrinking and
430 FINISHING AND FILING DRAWINGS [Chap. XVL
swelling of the paper. Moreover in plotting lines on separate
sheets so that they will fit exactly, there are mechanical difficul-
ties which can only be appreciated by the draftsman who has had
experience with them. These objections, together with the fact
that a comprehensive view of the whole situation cannot be taken
in at one time, have led some engineers to prefer large and un-
wieldy drawings to a system of separate sheets, but the latter are
much more convenient when the plans are to be used in the
field.
481. Inking in a profile. — The surface line is usually
shown as a full firm black line and the grade line as a full red
line (Art. 443, p. 395). A horizontal base-line is sometimes drawn
in red a short distance above the bottom of the paper and ver-
tical red lines are drawn from this line to the grade line at every
change of gradient and at both ends of the profile. On these
vertical lines are recorded the grade elevations at these points
and the " plus " if the place where the gradient changes is not
at a full station. On the base-line between these red vertical
lines is recorded the gradient of the grade line above. Under
the base-line is the stationing, which is marked at every heavy
vertical ruling of the profile paper, together with any other notes
of alignment which may be desired.
Information such as the names of streets, brooks, etc., is let-
tered vertically above the profile and at the proper station. A
title and the scale are sometimes placed on the face of the pro-
file ; sometimes these are put on the back of the profile at one
end of it (or both in the case of a long profile), so that the title
can be read when it is rolled up.
482. Cleaning drawings. — Every drawing, during its
construction, collects more or less dirt. Often construction
lines are drawn which must be erased when the plan is
completed. In cleaning a drawing an ordinary soft pencil
eraser is used for the pencil lines while a sponge eraser or stale
bread crumbs will remove the dirt satisfactorily without aflfect-
ing the ink lines.
To take off the pencil lines and dirt from tracing cloth,
wash the drawing with a cloth saturated with gasolene or
benzine. This will remove pencil lines entirely and will clean
FILING PLANS 43 1
the tracing perfectly without any injurious effect on the tracing
cloth.
483. FILING DRAWIHGS. — While the particular method of
filing plans varies considerably in different offices, there are
a few general ideas carried out by all drafting offices in regard
to the preservation as well as the systematic filing of drawings.
There is no doubt that the best method of filing plans is to
keep them flat, but this is not practicable with large plans
which must usually be filed in rolls. In all systems of plan
filing there appears to be a proper use of both flat and rolled
plans.
In large offices plans are, as a rule, made in several standard
sizes prescribed by the rules of the office, and are filed flat in
shallow drawers which are built to fit the different sizes of draw-
ings. In some offices the adherence to standard sizes is very
rigid, and considerable time is often spent to bring drawings
within the limits of one of these sizes. When these sizes are
exceeded the plans are either made in sections of standard size,
as explained in Art. 480, or they are made as large plans which
are rolled and filed away in pasteboard tubes. Sometimes very
large plans are filed flat by hanging them from an overhead
frame.
Plans filed flat are marked each with its proper index
number in one corner, preferably the lower right-hand corner,
so that as the drawer is opened the numbers can be readily exam-
ined. In some offices it is required" that in returning a draw-
ing it shall be placed in its proper order in the drawer as well
as in the proper drawer, while in other offices the plan drawers
are made very shallow, so as to contain only about 15 or 20
drawings, and when a plan is returned no attempt is made to
put it in any particular place in the drawer, there being, at the
most, only a very few drawings to handle to obtain the one
desired.
Rolled drawings are marked on the side of the rolls at each
end so as to be easily read by one standing in front of the shelf
on which the plans are stored. Another style of roll is closed
at one end with a white label on the outside of the closed end.
When the plan has been put into the tube it is so placed on the
43^ FINISHING AND FILING DRAWINGS [Chap. XVI.
shelf that the label on which the plan number is marked is at
the front edge of the shelf where it can be conveniently read.
When the plan is in use the empty tube is left on the shelf
with its open end outward so that its number is in the back part
of the shelf where it cannot be read.
Large plans which are made in sections are often filed in
large folios or books in such a way that they can be readily
taken out and used separately.
484. INDEXING. DRAWINGS. — There are so many systems of
indexing plans that no attempt will be made to explain them other
than to suggest a few of the essentials of any good system.
Every system of numbering the plans should be such that one
can tell from its number whether the drawing is a sketch, a
working drawing, a finished drawing, a tracing, or a process
print. The numbering also should suggest the type of draw-
ing, as a land plan, a construction plan, etc.
For offices where few plans are on file an index book may
suffice for recording the plans, but in large drafting offices the
card catalogue system is used extensively. By a judicious use
of "markers" a card catalogue system can be so devised that
it will be necessary to examine only a very few cards to find
the one corresponding to any plan. Frequently it is necessary
to index a plan by two or three different cards under different
general headings..
485. FILING Note-Books. — Note-books should always be
filed in vaults where they will be protected against fire. Too
frequently through lack of forethought note-books containing
information which it has cost thousands of dollars to collect are
carelessly filed on a shelf in the drafting office. In some offices
the rules require that every note-book and valuable plan shall
be placed in the vault at the end of the day's work, and this
appears to be the proper practice.
Some offices go so far as to require that all notes shall be
copied in ink and the original notes kept permanently filed in
the vault to guard against their loss. Whether a copy is made
or not, the original should be preserved as it has a value, in
a lawsuit for instance, which any copy does not possess. When
copies are made of the original notes they are sometimes made
INDEXING — FILING 43 3
in a loose-leaf book so that if any notes are taken from the office
it is not necessary to take more than a very few leaves of the
copy; the original notes never go from the office except in rare
cases.
486. Indexing Notes. — The notes contained in the field note-
books are often indexed either in a book for this purpose or by
means of a card catalogue. The method of indexing is similar
to that used for plans.
487. Other Records. — Other records, such as borings, sound-
ings, estimates, computations, etc.^ are carefully filed and indexed
so that it will be easy to refer to them.
TABLES.
43S
437
TABLE I.— LOGARITHMS OF NL'MBERS.
N
0123456789
100
00000 00043 00087 OOISO 00173 00217 00260 00303 00346 00389
0432 0475 0518 0561 0604 0647 0689 0732 077i 0817
0860 0903 0946 0988 1030 1072 1115 1157 1199 1242
1284 1326 1368 1410 1452 1494 1536 1578 1620 1662
1703 1745 1787 1828 1870 1912 1953 1995 2036 2078
2119 2160 2202 2243 2284 2325 2366 2407 2449 2490
2531 2572 2612 265i^ 2694 2735 2776 2816 2857 2898
2938 2979 3019 3060 3100 3141 3181 3222 3262 3302
3342 3383 3423 3463 3503 3543 3583 3623 8663 3703
3743 3782 3822 3862 3902 3941 3981 4021 4060 4100
110
04139 04179 04218 04258 04297 04336 04376 04415 04454 04498
4532 4571 4610 4650 4689 4727 4766 4805 4844 4883
4922 4961 4999 5038 5077 5116 5154 5192 5231 5269
5308 5346 5385 5423 5461 5500 5538 5576 6614 6652
5690 6729 5767 5805 5843 5881 6918 6956 5994 6032
6070 6108 6146 6183 6221 6258 6296 6333 6371 6408
6446 6483 6521 6558 6595 6633 6670 6707 6744 6781
6819 6856 6893 6930 6967 7004 7041 7078 7115 7151
7188 7225 7262 7298 7336 7372 7408 7445 7482 7518
7555 7591 7628 7664 7700 7737 7773 7809 7846 7882
120
07918 07954 07990 08027 08063 08099 08135 08171 08207 08248
8279 8314 8350 8386 8422 8458 8493 8529 8565 8600
8636 8672 8707 8743 8778 8814 8849 8884 8920 8956
8991 9026 9061 9096 9132 9167 9202 9237 9272 9307
9342 9377 9412 9447 9482 9617 9552 9587 9621 9666
9691 9726 9760 9795 9830 9864 9899 9934 996810003
10037 10072 10106 10140 10175 10209 10243 10278 10312 0346
0380 0415 0449 0483 0517 0551 0586 0619 0653 0687
0721 0755 0789 0823 0857 0890 0924 0958 0992 1025
1069 1093 1126 1160 1193 1227 1261 1294 1327 1361
180
11394 11428 11461 11494 11528 11661 11694 11628 11661 11694
1727 1760 1793 1826 1860 1893 1926 1959 1992 2024
2057 2090 2123 2156 2189 2222 2254 2287 2320 2352
2385 2418 2460 2483 2516 2548 2681 2613 2646 2678
2710 2743 2775 2808 2840 2872 2905 2937 2969 3001
3033 3066 3098 3130 3162 3194 3226 8258 3290 3322
3354 3386 3418 3450 3481 3513 3545 3577 3609 3640
3672 3704 3735 3767 3799 3830 3862 3893 3925 3956
3988 4019 4051 4082 4114 4145 4176 4208 4239 4270
4301 4333 4364 4396 4426 4457 4489 4520 4561 4582
140
14613 14644 14675 14706 14737 14768 14799 14829 14860 14891
4922 4953 4983 6014 5045 5076 6106 6137 6168 6198
5229 6259 5290 5320 5351 6381 6412 6442 5473 6503
6534 5564 6694 6625 5655 5686 6715 5746 6776 6806
6836 5866 5897 5927 5957 5987 6017 6047 6077 6107
6137 6167 6197 6227 6266 6286 6316 6346 6376 6406
6435 6466 6495 6624 6554 6584 6613 6643 6673 6702
6732 6761 6791 6820 6850 6879 6909 6938 6967 6997
7026 7056 7085 7114 7143 7173 7202 7231 7260 7289
7319 7348 7377 7406 7436 7464 7493 7522 7561 7680
150
17609 17638 17667 17696 17726 17764 17782 17811 17840 17869
438
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456789
150
17600 17688 17667 17606 17725 17764 17782 17811 17840 17869
7898 7926 7956 7984 8013 8041 8070 8099 8127 8166
8184 8218 8241 8270 8298 8827 8865 8884 8412 8441
8469 8498 8626 8564 8688 8611 8689 8667 8696 8724
8762 8780 8808 8887 8865 8893 8921 8949 8977 9006
9088 9061 9089 9117 9146 9178 9201 9229 9267 9285
9812 9840 9868 9896 9424 9461 9479 9607 9685 9562
9590 9618 9646 9678 9700 9728 9766 9788 9811 9888
9866 9898 9921 9948 9976 20008 20080 20068 20086 20112
20140 20167 20194 20222 20249 0276 0808 0880 0868 0385
160
20412 20489 20466 20498 20620 20648 20575 20602 20629 20666
0688 0710 0787 0768 0790 0817 0844 0871 0898 0925
0952 0978 1006 1082 1069 1086 1112 1189 1166 1192
1219 1246 1272 1299 1826 1852 1878 1405 1481 1458
1484 1511 1687 1664 1590 1617 1648 1669 1696 1722
1748 1775 1801 1827 1854 1880 1906 1982 1968 1985
2011 2087 2063 2089 2115 2141 2167 2194 2220 2246
2272 2298 2824 2850 2876 2401 2427 2458 2479 2505
2681 2667 2588 2608 2684 2660 2686 2712 2787 2763
2789 2814 2840 2866 2891 2917 2943 2968 2994 8019
170
23045 23070 23096 28121 28147 28172 23198 23223 23249 23274
3300 3326 3350 3876 3401 8426 3452 3477 3502 3528
3563 3578 3603 8629 8664 3679 3704 3729 3764 8779
3805 3880 3865 3880 3905 3930 3955 8980 4005 4030
4065 4080 4105 4180 4155 4180 4204 4229 4264 4279
4304 4329 4368 4878 4403 4428 4462 4477 4602 4527
4561 4576 4601 4626 4650 4674 4699 4724 4748 4773
4797 4822 4846 4871 4895 4920 4944 4969 4993 6018
6042 5066 6091 6115 5139 6164 6188 6212 5237 5261
6286 5810 5334 6858 5382 5406 6431 5465 6479 5603
180
26627 25661 26676 26600 25624 25648 26672 26696 26720 25744
5768 5792 5816 6840 6864 6888 6912 6985 5959 6983
6007 6081 6065 6079 6102 6126 6160 6174 6198 6221
6246 6269 6293 6816 6340 6864 6887 6411 6435 6468
6482 6606 6629 6653 6576 6600 6628 6647 6670 6694
6717 6741 6764 6788 6811 6884 6858 6881 6905 6928
6961 6975 6998 7021 7045 7068 7091 7114 7138 7161
7184 7207 7281 7264 7277 7300 7323 7346 7870 7393
7416 7439 7462 7486 7608 7631 7664 7677 7600 7623
7646 7669 7692 7716 7788 7761 7784 7807 7830 7862
190
27876 27898 27921 27944 27967 27989 28012 28a% 28058 28081
8108 8126 8149 8171 8194 8217 8240 8262 8285 8307
8830 8363 8375 8308 8421 8448 8466 8488 8511 8533
8666 8578 8601 8628 8646 8668 8691 8713 8736 8758
8780 8808 8825 8847 8870 8892 8914 8987 8969 8981
9008 9026 9048 9070 9092 9115 9137 9159 9181 9203
9226 9248 9270 9292 9314 9330 9368 9380 9403 9425
9447 9469 9491 9513 9535 9567 9679 9601 9623 9645
9667 9688 9710 9732 9754 9776 9798 9820 9842 9863
9886 9907 9929 9961 9978 9994 30016 30038 80060 30081
800
80103 30125 30146 30168 30190 30211 30233 30265 30276 30298
439
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456789
200
30103 3012i 30146 30168 30190 30211 30233 3025^ 30276 30208
0320 0341 0363 0384 0406 0428 0449 0471 0492 0614
0636 0567 0578 0600 0621 0643 0664 0686 0707 0728
0750 0771 0792 0814 0835 0866 0878 0899 0920 0942
0963 0984 1006 1027 1048 1069 1091 1112 1133 1164
1175 1197 1218 1239 1260 1281 1302 1323 1345 1366
1387 1408 1429 1450 1471 1492 1513 1634 1656 1576
1697 1618 1639 1660 1681 1702 1723 1744 1765 1786
1806 1827 1848 1869 1890 1911 1931 1952 1973 1994
2015 2036 2066 2077 2098 2118 2139 2160 2181 2201
210
32222 32243 32263 32284 32305 32326 32346 32366 32387 32408
2428 2449 2469 2490 2510 2531 2562 2672 2693 2613
2634 2654 2675 2695 2716 2736 2766 2777 2797 2818
2838 2858 2879 2899 2919 2940 2960 2980 3001 3021
3041 3062 3082 3102 3122 3143 3163 3183 3203 3224
3244 3264 3284 3304 3325 3345 3365 3386 3406 3426
3446 3465 3486 3506 3526 3646 3566 3686 3606 3626
3646 3666 3686 3706 3726 3746 3766 3786 3806 3826
3846 3866 3885 3906 3925 3946 3966 8985 4005 4025
4044 4064 4084 4104 4124 4143 4163 4183 4208 4223
220
34242 34262 34282 34301 34321 84341 34361 34380 34400 34420
4439 4469 4479 4498 4518 4537 4557 4677 4696 4616
4636 4655 4674 4694 4713 4733 4763 4772 4792 4811
4830 4850 4869 4889 4908 4928 4947 4967 4986 6005
6025 6044 6064 5083 5102 5122 5141 6160 6180 6199
6218 6238 5257 5276 5295 5315 5334 6363 6372 6392
6411 5430 5449 5468 6488 5507 6626 6646 6664 5583
5603 5622 5641 5660 5679 5698 6717 6736 6756 6774
6793 6813 5832 5851 5870 6889 6908 6927 6946 6965
6984 6003 6021 6040 6059 6078 6097 6116 6136 6154
280
36173 86192 36211 36229 36248 36267 36286 36305 36324 36342
6361 6380 6309 6418 6436 6455 6474 6493 6611 6630
6549 6568 6586 6605 6624 6642 6661 6680 6698 6717
6736 6754 6773 6791 6810 6829 6847 6866 6884 6903
6922 6940 6959 6977 6996 7014 7033 7061 7070 7088
7107 7126 7144 7102 7181 7199 7218 7236 7254 7273
7291 7310 7328 7346 7365 738i5 7401 7420 7438 7467
7475 7493 7511 7530 7548 7566 7585 7603 7621 7639
7658 7676 7094 7712 7731 7749 7767 7786 7803 7822
7840 7858 7876 7894 7912 7931 7949 7967 7985 8003
240
38021 38039 38057 38076 38093 38112 38130 38148 38166 38184
8202 8220 82;i8 8256 8274 8292 8310 8328 8346 8364
8382 8399 8417 8435 8453 8471 8489 8507 8525 8543
8561 8578 8596 8014 8632 86^0 8668 8686 8703 8721
8739 8757 8775 8792 8810 8828 8846 8863 8881 8899
8917 8934 8952 8970 8987 9005 9023 9041 9068 9076
9094 9111 9129 9146 9164 9182 9199 9217 9235 9262
9270 9287 9305 9322 9840 9358 9376 9393 9410 9428
9445 9463 9480 9498 9516 9533 9550 9568 9585 9602
9620 9637 9655 9672 9690 9707 9724 9742 9769 9777
250
39794 39811 39829 39846 39863 39881 39898 39916 39933 39950
440
TABLE I.— LOGARITHMS OF NUMBERS.
N
01234567 8 9
250
39794 39811 39829 39846 39863 39881 39898 39915 39933 39950
1
9967 9985 40002 40019 40037 40054 40071 40088 40106 40123
8
40140 40157 0175 0192 0209 0226 0243 0261 0278 0295
8
0312 0329 0346 0364 0381 0398 0415 0432 0449 0466
4
0483 0500 0518 0535 0552 0569 0586 0603 0620 0637
5
0654 0671 0688 0705 0722 0739 0756 0773 0790 0807
6
0824 0841 0858 0875 0892 0909 0926 0943 0960 0976
7
0993 1010 1027 1044 1061 1078 1095 IHI 1128 1145
8
1162 1179 1196 1212 1229 1246 1263 1280 1296 1313
9
1330 1347 1363 1380 1397 1414 1430 1447 1464 1481
260
41497 41514 41531 41547 41564 41581 41597 41614 41631 41647
1
1664 1681 1697 1714 1731 1747 1764 1780 1797 1814
2
1830 1847 1863 1880 1896 1913 1929 1946 1963 1979
8
1996 2012 2029 2045 2062 2078 2095 2111 2127 2144
4
2160 2177 2193 2210 2226 2243 2259 2275 2292 2308
6
2325 2341 2357 2374 2390 2406 2423 2439 2455 2472
6
2488 2504 2521 2537 2553 2570 2586 2602 2619 2635
7
2651 2667 2684 2700 2716 2732 2749 2765 2781 2797
8
2813 2830 2846 2862 2878 2894 2911 2927 2943 2959
9
2975 2991 3008 3024 3040 3056 3072 3088 3104 3120
270
43136 43152 43169 43185 43201 43217 43233 43249 43265 43281
1
3297 3313 3329 3345 3361 3377 3393 3409 34£5 3441
2
3457 3473 3489 3505 3521 3537 3553 3569 3584 3600
8
3616 3632 3648 3664 3680 3696 3712 3727 3743 3759
4
3775 3791 3807 3823 3838 3854 3870 3886 3902 3917
6
3933 3949 3965 3981 3996 4012 4028 4044 4059 4075
6
4091 4107 4122 4138 4154 4170 4185 4201 4217 4232
7
4248 4264 4279 4295 4311 4326 4342 4358 4373 4389
8
4404 4420 4436 4451 4467 4483 4498 4514 4529 4545
9
4560 4576 4592 4607 4623 4638 4054 4669 4685 4700
280
44716 44731 44747 44762 44778 44793 44809 44824 44840 44855
1
4871 4886 4902 4917 4932 4948 4963 4979 4994 5010
2
5025 5040 5056 5071 5086 5102 5117 5133 5148 6163
8
5179 5194 6209 6225 5240 5256 5271 5286 6301 6317
4
6332 6347 6362 6378 5393 6408 6423 6439 6464 6469
5
6484 6500 5515 6530 6646 6561 6576 5691 6606 6621
6
6637 6652 5667 5682 5697 6712 6728 6743 6768 6773
7
6788 6803 5818 6834 6849 6864 6879 6894 6909 6924
8
6939 6954 6969 6984 6000 6015 6030 6045 6060 6075
9
6090 6105 6120 6135 6150 6165 6180 6195 6210 6225
290
46240 46265 46270 46285 46300 46315 46330 46345 46369 46374
1
6389 6404 6419 6434 6449 6464 6479 6494 6509 6523
2
6538 6553 6568 6583 6598 6613 6627 6642 6657 6672
8
6687 6702 6716 6731 6746 6761 6776 6790 6806 6820
4
6835 6850 6864 6879 6894 6909 6923 6938 6963 6967
6
6982 6997 7012 7026 7041 7056 7070 7086 7100 7114
6
7129 7144 7159 7173 7188 7202 7217 7232 7246 7261
7
7276 7290 7305 7319 7334 7349 7363 7378 7392 7407
8
7422 7436 7461 7465 7480 7494 7609 7524 7538 7653
9
7667 7582 7596 7611 7625 7640 7654 7669 7683 7698
800
47712 47727 47741 47756 47770 47784 47799 47813 47828 47842
441
TABLE I.— LOGARITHMS OF NXmBERS.
N
0123456789
800
47712 47727 47741 47756 47770 47784 47799 47813 47828 47842
7867 7871 7885 7900 7914 7929 7943 7958 7972 7986
8001 8015 8029 8044 8068 8073 8087 8101 8116 8130
3
8144 8169 8173 8187 8202 8216 8230 8244 8259 8273
8287 8302 8316 8330 8344 8359 8373 8387 8401 8416
8430 8444 8468 8473 8487 8501 8516 8630 8644 8668
8572 8686 8601 8615 8629 8643 8667 8671 8686 8700
8714 8728 8742 8766 8770 8785 8799 8813 8827 8841
8855 8869 8883 8897 8911 8926 8940 8964 8968 8982
8996 9010 9024 9038 9062 9066 9080 9094 9108 9122
810
49136 49150 49164 49178 49192 49206 49220 49234 49248 49262
9276 9290 9304 9318 9332 9346 9360 9374 9388 9402
9415 9429 9443 9467 9471 9485 9499 9613 9627 9641
9664 9668 9682 9696 9610 9624 9638 9661 9665 9679
9693 9707 9721 9734 9748 9762 9776 9790 9803 9817
9831 984i 9869 9872 9886 9900 9914 9927 9941 9965
9969 9982 9996 50010 50024 60037 6005160065 60079 60092
50106 50120 50183 0147 0161 0174 0188 0202 0215 0229
0243 0256 0270 0284 0297 0811 0325 0338 0352 0366
0379 0393 0406 0420 0433 0447 0461 0474 0488 0601
820
50515 50629 60542 50556 50669 50583 60696 50610 50623 50637
0651 0664 0678 0691 0705 0718 0732 0746 0769 0772
0786 0799 0813 0826 0840 0863 0866 0880 0893 0907
0920 0934 0947 0961 0974 0987 1001 1014 1028 1041
1055 1068 1081 1095 1108 1121 1135 1148 1162 1175
1188 1202 1215 1228 1242 1256 1268 1282 1295 1308
1322 1335 1348 1362 1376 1388 1402 1415 1428 1441
1465 1468 1481 1495 1508 1521 1634 1548 1561 1674
1587 1601 1614 1627 1640 1664 1667 1680 1693 1706
1720 1733 1746 1759 1772 1786 1799 1812 1825 1838
880
51861 51865 51878 61891 51904 61917 51930 51948 51967 61970
1983 1996 2009 2022 2035 2048 2061 2075 2088 2101
2114 2127 2140 2163 2166 2179 2192 2205 2218 2231
2244 2267 2270 2284 2297 2310 2323 2336 2349 2362
2375 2388 2401 2414 2427 2440 2463 2466 2479 2492
2504 2517 2530 2643 2656 2569 2582 2696 2608 2621
2634 2647 2660 2673 2686 2699 2711 2724 2737 2760
2763 2776- 2789 2802 2815 2827 2840 2863 2866 2879
2892 2905 2917 2930 2943 2966 2969 2982 2994 3007
8020 3033 8046 8068 8071 3084 3097 3110 3122 3186
840
53148 58161 53173 53186 53199 63212 63224 53287 63250 63263
8276 8288 8301 8814 3326 8339 3362 3364 8377 3:^90
84a3 3415 3428 8441 8463 8466 8479 8491 8604 3617
3629 3642 3665 8667 8580 8693 8605 8618 3631 3643
3666 8668 8681 3694 8706 3719 3732 3744 3767 3769
8782 3794 3807 3820 3832 8845 8867 3870 3882 3896
8908 3920 3933 8945 8968 8970 3983 3996 4008 4020
4033 4045 4068 4070 4083 4095 4108 4120 4133 4146
4168 4170 4183 4196 4208 4220 4233 4246 4268 4270
4283 4295 4307 4320 4832 4345 4367 4370 4382 4394
360
54407 64419 64432 54444 64460 64469 64481 64494 54506 54618
442
TABLE I.— LOGARITHMS OF NUMBERS.
N
0128456789
850
64407 64419 64432 64444 64466 64469 64481 64494 64606 64618
4631 4643 4656 4668 4680 4593 460^ 4617 4630 4642
4664 4667 4679 4691 4704 4716 4728 4741 4763 4765
4777 4790 4802 4814 4827 4839 4861 4864 4876 4888
4900 4913 492i 4937 4949 4962 4974 4986 4998 6011
6023 6035 6047 6060 6072 6084 5096 6108 6121 6133
5145 5157 5169 6182 5194 6206 5218 5230 5242 6265
5267 5279 5291 6303 5315 5328 5340 5352 5364 6376
5388 5400 6413 6425 6437 6449 6461 6473 6486 6497
6509 6622 6634 6546 5558 5570 6682 6594 6606 6618
860
56630 66642 66664 56666 65678 66691 65703 65715 65727 66739
5751 5763 6775 5787 6799 5811 5823 6835 5847 6859
6871 6883 6895 5907 6919 5931 6943 6955 6967 5979
5991 6003 6015 6027 6038 6050 6062 6074 6086 6098
6110 6122 6134 6146 6158 6170 6182 6194 6205 6217
6229 6241 6253 6265 6277 6289 6301 6312 6324 6336
6348 6360 6372 6384 6396 6407 6419 6431 6443 6455
6467 6478 6490 6502 6514 6526 6538 6549 6561 6673
6685 6697 6608 6620 6632 6644 6656 6667 6679 6691
6703 6714 6726 6738 6750 6761 6773 6785 6797 6808
870
66820 66832 66844 66866 66867 56879 66891 66902 66914 66926
6937 6949 6961 6972 6984 6996 7008 7019 7031 7043
7064 7066 7078 7089 7101 7113 7124 7136 7148 7159
7171 7183 7194 7206 7217 7229 7241 7262 7264 7276
7287 7299 7310 7322 7334 7346 7357 7368 7380 7392
7403 7415 7426 7438 7449 7461 7473 7484 7496 7507
7519 7530 7542 7553 7565 7576 7588 7600 7611 7623
7634 7646 7657 7669 7680 7692 7703 7715 7726 7738
7749 7761 7772 7784 7796 7807 7818 7830 7841 7862
7864 7875 7887 7898 7910 7921 7933 7944 7966 7967
880
57978 67990 68001 68013 58024 68035 58047 68058 68070 68081
8092 8104 8115 8127 8138 8149 8161 8172 8184 8195
8206 8218 8229 8240 8252 8263 8274 8286 8297 8309
8320 8331 8343 8354 8365 8377 8388 8399 8410 8422
8433 8444 8456 8467 8478 8490 8501 8512 8524 8635
8646 8657 8569 8580 8591 8602 8614 8625 8636 8647
8659 8670 8681 8692 8704 8715 8726 8737 8749 8760
8771 8782 8794 8805 8816 8827 883& 8850 8861 8872
8883 8894 8906 8917 8928 8939 8950 8961 8973 8984
8995 9006 9017 9028 9040 9051 9062 9073 9084 9096
890
69106 69118 69129 59140 59151 69162 59173 69184 69195 69207
9218 9229 9240 9251 9262 9273 9284 9296 9306 9318
9329 9340 9351 9362 9373 9384 9396 9406 9417 9428
9439 9450 9461 9472 9483 9494 9506 9517 9528 9539
9650 9561 9572 9583 9594 9605 9616 9627 9638 9649
9660 9671 9682 9693 9704 9715 9'726 9737 9748 9769
9770 9780 9791 9802 9813 9824 9836 9846 9867 9868
9879 9890 9901 9912 9923 9934 9945 9966 9966 9977
9988 9999 60010 60021 60032 60043 60054 60065 60076 60086
60097 60108 0119 0130 0141 0152 0163 0173 0184 0196
400
60206 60217 60228 60239 60249 60260 60271 60282 60293 60304
..J
443
TABLE I.— LOGARITHMS OF NUMBERS.
■isr
O 12^4:5 67 89
400 e0206 60217 60228 60239 60249 60260 60271 60282 60293 60304 |
0314 0325 0336 0347 0358 0369 0379 0390 0401 0412
0423 0433 0444 0455 0466 0477 0487 0498 0509 0520
0531 0541 0552 0563 0574 0584 0595 0606 0617 0627
0638 0649 0660 0670 0681 0692 0703 0713 0724 0735
0746 0756 0767 0778 0788 0799 0810 0821 0831 0842
0853 0863 0874 0885 0896 0906 0917 0927 0938 0949
0959 0970 0981 0991 1002 1013 1023 1034 1045 1055
1066 1077 1087 1098 1109 1119 1130 1140 1151 1162
1172 1183 1194 1204 1215 1225 1236 1247 1257 1268
410
61278 61289 61300 61310 61321 61331 61342 61352 61363 61374
1384 1395 1405 1416 1426 1437 1448 1458 1469 1479
1490 1500 1611 1521 1532 1542 1553 1563 1674 1584
1695 1606 1616 1627 1637 1648 1658 1669 1679 1690
1700 1711 1721 1731 1742 1752 1763 1773 1784 1794
1805 1815 1826 1836 1847 1867 1868 1878 1888 1899
1909 1920 1930 1941 1951 1962 1972 1982 1993 2003
2014 2024 2034 2045 2055 2066 2076 2086 2097 2107
2118 2128 2138 2149 2159 2170 2180 2190 2201 2211
2221 2232 2242 2252 2263 2273 2284 2294 2304 2315
420
62325 62335 62346 62356 62366 62377 62387 62897 62408 62418
2428 2439 2449 2459 2469 2480 2490 2600 2511 2621
2531 2542 2552 2562 2572 2583 2593 2603 2613 2624
2634 2644 2655 2665 2676 2685 2696 2706 2716 2726
2737 2747 2757 2767 2778 2788 2798 2808 2818 2829
2839 2849 2859 2870 2880 2890 2900 2910 2921 2931
2941 2951 2961 2972 2982 2992 3002 3012 3022 3033
3043 3063 3063 3073 3083 3094 3104 8114 3124 3134
3144 3165 3165 3175 3185 3196 3205 3215 3226 3236
3246 3256 3266 3276 3286 3296 3306 3317 3327 3337
480
63347 63367 63367 63377 63387 63397 63407 63417 63428 63438
3448 3458 3408 3478 3488 3498 3508 3518 3528 3638
3548 3558 3568 3579 3689 3599 3609 3619 8629 3639
3649 3659 3669 3679 3689 3699 3709 3719 3729 3739
3749 3769 3769 3779 3789 3799 8809 3819 3829 3839
3849 3859 3809 3879 3889 3899 3909 3919 3929 8939
3949 3969 3969 3979 3988 3998 4008 4018 4028 4a38
4048 4058 4068 4078 4088 4098 4108 4118 4128 4137
4147 4157 4167 4177 4187 4197 4207 4217 4227 4237
4246 4266 4266 4276 4286 4296 4306 4316 4326 4335
440
64345 64366 64365 64375 64385 64395 64404 64414 64424 64434
4444 4454 4464 4473 4483 4493 4503 4513 4523 4532
4542 4552 4562 4572 4582 4591 4601 4611 4621 4631
4640 4650 4660 4670 4680 4089 4699 4709 4719 4729
4738 4748 4758 4768 4777 4787 4797 4807 4816 4826
4836 4846 4856 4865 4876 4885 4895 4904 4914 4924
4933 4943 4953 4963 4972 4982 4992 5002 6011 5021
6031 5040 5050 5060 6070 6079 6089 6099 6108 6118
6128 6137 6147 6157 5167 6176 6186 5196 6206 6215
5225 6234 6244 5264 6263 5273 6283 5292 6302 5312
460
65321 65331 65341 65350 65360 65369 65379 66389 65398 65408
444
TABLE I.— LOGARITHMS OF NUMBERS.
450
0123456789
66821 66331 65341 65360 66360 66369 66379 66389 66398 66408
1
5418 6427 6437 6447 6456 6466 6476 6486 649^ 6604
2
6614 6523 6633 6643 6552 6562 6671 6681 6691 6600
8
6610 6619 6629 6639 6648 6668 5667 6677 6686 6696
4
6706 6715 6725 6734 6744 6763 5763 6772 6782 5792
5
6801 6811 5820 6830 6839 6849 6858 6868 6877 6887
6
6896 6906 6916 5926 6935 6944 6954 6963 6973 6982
7
5992 6001 6011 6020 6030 6039 6049 6068 6068 6077
8
6087 6096 6106 6115 6124 6134 6143 6163 6162 6172
9
6181 6191 6200 6210 6219 6229 6238 6247 6267 6266
460
66276 66285 66295 66304 66314 66323 66332 66342 66361 66361
1
6370 6380 6389 6398 6408 6417 6427 6436 6446 6465
2
6464 6474 6483 6492 6502 6611 6621 6530 6639 6549
8
6658 6567 6677 6586 6596 6605 6614 6624 6633 6642
4
6652 6661 6671 6680 6689 6699 6708 6717 6727 6736
5
6745 6755 6764 6773 6783 6792 6801 6811 6820 6829
6
6839 6848 6857 6867 6876 6886 6894 6904 6918 6922
7
6932 6941 6950 6960 6969 6978 6987 6997 7006 7015
8
7025 7034 7043 7052 7062 7071 7080 7089 7099 7108
9
7117 7127 7136 7146 7164 7164 7173 7182 7191 7201
470
67210 67219 67228 67237 67247 67266 67266 67274 67284 67293
1
7302 7311 7321 7330 7339 7348 7367 7367 7376 7385
9
7394 7403 7413 7422 7431 7440 7449 7469 7468 7477
8
7486 7496 7504 7514 7623 7632 7641 7550 7660 7569
4
7578 7687 7596 7005 7614 7624 7633 7642 7661 7660
5
7669 7679 7688 7697 7706 7715 7724 7733 7742 7752
6
7761 7770 7779 7788 7797 7806 7815 7825 7834 7843
7
7852 7861 7870 7879 7888 7897 7906 7916 7925 7934
8
7943 7962 7961 7970 7979 7988 7997 8006 8016 8024
9
8034 8043 8052 8061 8070 8079 8088 8097 8106 8116
480
68124 68133 68142 68151 68160 68169 68178 68187 68196 68206
1
8215 8224 8233 8242 8251 8260 8269 8278 8287 8296
2
8305 8314 8323 8332 8341 8350 8359 8368 8377 8386
8
8395 8404 8413 8422 8431 8440 8449 8468 8467 8476
4
8485 8494 8502 8511 8520 8529 8538 8547 8566 8566
6
8574 8583 8592 8601 8610 8619 8628 8637 8646 8655
6
8664 8673 8681 8690 8699 8708 8717 8726 8736 8744
7
8753 8762 8771 8780 8789 8797 8806 8816 8824 8833
8
8842 8851 8860 8869 8878 8886 8896 8904 8913 8922
9
8931 8940 8949 8958 8966 8976 8984 8993 9002 9011
490
69020 69028 69037 69046 69056 69064 69073 69082 69090 69099
1
9108 9117 9126 9135 9144 9152 9161 9170 9179 9188
8
9197 9205 9214 9223 9232 9241 9249 9268 9267 9276
8
928S 9294 9302 9311 9320 9329 9338 9346 9366 9364
4
9373 9381 9390 9399 9408 9417 9426 9434 9443 9452
6
9461 9469 9478 9487 9496 9604 9513 9622 9631 9639
6
9548 9557 9666 9574 9583 9592 9601 9609 9618 9627
7
9636 9644 9653 9662 9671 9679 9688 9697 9706 9714
8
9723 9732 9740 9749 9758 9767 9775 9784 9793 9801
9
9810 9819 9827 9836 9845 9854 9862 9871 9880 9888
600
69807 69906 69914 69923 69932 69940 69949 69968 .69966 69976
445
TABLE T.— LOGARITHMS OF NUMBERS.
N
0123456789
500
69897 69906 69914 69023 69932 69940 69949 69958 69966 69975
1
9984 9992 70001 70010 70018 70027 70036 70044 70053 70062
2
70070 70079 0088 0096 0105 0114 0122 0131 0140 0148
8
0157 0165 0174 0183 0191 0200 0209 0217 0226 0234
4
0243 0252 0260 0269 0278 0286 0295 0303 0312 0321
5
0329 0338 0346 0365 0364 0372 0381 0389 0398 0406
6
0415 0424 0432 0441 0449 0458 0467 0475 0484 0492
7
0501 0509 0518 0626 0635 0544 0562 0661 0669 0578
8
0586 0595 0603 0612 0621 0629 0638 0646 0655 0663
9
0672 0680 0689 0697 0706 0714 0723 0731 0740 0749
510
70767 70766 70774 70783 70791 70800 70808 70817 70825 70834
1
0842 0851 0859 0868 0876 0885 0893 0902 0910 0919
2
0927 0935 0944 0962 0961 0969 0978 0986 0995 1003
8
1012 1020 1029 1037 1046 1054 1063 1071 1079 1088
4
1096 1105 1113 1122 1130 1139 1147 1165 1164 1172
5
1181 1189 1198 1206 1214 1223 1231 1240 1248 1257
6
1265 1273 1282 1290 1299 1307 1315 1324 1332 1341
7
1349 1357 1366 1374 1383 1391 1399 1408 1416 1425
8
1433 1441 1450 1458 1466 1475 1483 1492 1500 1508
9
1617 1625 1533 1642 1660 1669 1667 1575 1584 1692
520
71600 71609 71617 71626 71634 71642 71650 71669 71667 71675
1
1684 1692 1700 1709 1717 1725 1734 1742 1750 1759
2
1767 1775 1784 1792 1800 1809 1817 1825 1834 1842
3
1850 1858 1867 1875 1883 1892 1900 1908 1917 1925
4
1933 1941 1950 1968 1966 1975 1983 1991 1999 2008
6
2016 2024 2032 2041 2049 2067 2066 2074 2082 2090
6
2099 2107 2115 2123 2132 2140 2148 2156 2165 2173
7
2181 2189 2198 2206 2214 2222 2230 2239 2247 2265
8
22(53 2272 2280 2288 2296 2304 2313 2321 2329 2337
9
2346 2354 2362 2370 2378 2387 2395 2403 2411 2419
680
72428 72436 72444 72462 72460 72469 72477 72485 72493 72601
1
2509 2518 2526 2534 2542 2550 2668 2667 2675 2583
2
2591 2599 2607 2616 2624 2632 2640 2648 2656 2665
8
2673 2681 2689 2697 2706 2713 2722 2730 2738 2746
4
2754 2762 2770 2779 2787 2795 2803 2811 2819 2827
«
2835 2843 2852 2860 2868 2876 2884 2892 2900 2908
6
2916 2925 2933 2941 2949 2957 2966 2973 2981 2989
7
2997 3006 3014 3022 3030 3038 3046 3064 3062 3070
8
3078 3086 3094 3102 3111 3119 3127 3135 3143 3151
9
3159 3167 3175 3183 3191 3199 3207 3215 3223 3231
540
73239 73247 73255 73263 73272 73280 73288 73296 73304 73312
1
3320 3328 3336 3344 3352 3360 3368 3376 3384 3392
2
3400 3408 3416 3424 3432 3440 3448 3466 3464 3472
8
3480 3488 3496 3504 3612 3520 3628 3536 3644 3652
4
3560 3568 3576 3584 3692 3600 3608 3616 8624 3632
5
3640 3648 3656 3664 3672 3679 3687 3696 3703 3711
6
3719 3727 3735 3743 3751 3759 3707 3775 3783 3791
7
3799 3807 3815 3823 3830 3838 3846 3854.3862 3870
8
3878 3886 3894 3002 3910 3918 3926 3933 3941 3949
9
3957 3965 3073 3981 3989 3997 4005 4013 4020 4028
550
74036 74044 74052 74060 74008 74070 74084 74092 74099 74107
446
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456789
550
74036 74044 74052 74060 74068 74076 74084 74092 74099 74107
1
4115 4123 4131 4139 4147 4165 4162 4170 4178 4186
2
4194 4202 4210 4218 4225 4233 4241 4249 4257 4265
8
4273 4280 4288 4296 4304 4312 4320 4327 4335 4343
4
4351 4359 4367 4374 4382 4390 4398 4406 4414 4421
6
4429 4437 4445 4453 4461 4468 4476 4484 4492 4500
6
4507 4515 4523 4531 4539 4547 4554 4562 4570 4578
7
4586 4593 4001 4009 4617 4624 4632 4640 4648 4666
8
4863 4671 4679 4687 4695 4702 4710 4718 4726 4733
9
4741 4749 4757 4764 4772 4780 4788 4796 4803 4811
560
74819 74827 74834 74842 74850 74858 74866 74873 74881 74889
1
4896 4904 4912 4920 4927 4985 4943 4950 4958 4966
2
4974 4981 4989 4997 5005 5012 6020 5028 5036 6043
8
6051 6069 5066 5074 6082 5089 5097 6105 5113 6120
4
6128 6136 6143 5151 5159 5166 6174 6182 5189 5197
5
6205 5213 6220 6228 5236 5243 6251 6259 5266 5274
6
5282 6289 6297 5305 5312 6320 6328 6335 5343 6351
7
5358 5366 5374 5381 5389 6397 6404 5412 5420 6427
8
6435 6442 5450 6458 6405 6473 5481 6488 6496 6504
9
6611 5619 6526 5534 6542 6549 5667 6565 6672 6580
570
76587 75595 75603 76610 75618 75626 75633 75641 76648 76656
1
6664 5671 5679 5686 6694 5702 6709 5717 5724 5732
2
6740 5747 5755 6762 5770 5778 6786 6793 5800 6808
8
5815 6823 6831 6838 6846 6853 6861 6868 5876 6884
4
5891 6899 6906 6914 6921 5929 6937 6944 6962 5969
5
5967 6974 6982 5989 6997 6005 6012 6020 6027 6035
6
6042 6050 6057 6065 6072 6080 6087 6095 6103 6110
7
6118 6125 6133 6140 6148 6155 6163 6170 6178 6186
8
6193 6200 6208 6216 6223 6230 6238 6246 6253 6260
9
6268 6276 6283 6290 6298 6306 6313 6320 6328 6336
580
76343 76350 76358 76365 76373 76380 76388 76396 76403 76410
1
6418 6425 6433 6440 6448 6455 0402 6470 6477 6485
2
6492 6500 6507 6515 6522 6530 6537 6645 6552 6669
8
6567 6574 6582 6589 6597 6604 6612 6619 6626 6634
4
6641 6649 6656 6664 6671 6678 6686 6693 6701 6708
5
6716 6723 6730 6738 6745 6753 6760 6768 6775 6782
6
6790 6797 6805 6812 6819 6827 6834 6842 6849 6866
7
6864 6871 6879 6886 0893 6901 6908 6916 6923 6930
8
6938 6945 6963 6960 6967 6975 6982 6989 6997 7004
9
7012 7019 7026 7034 7041 7048 7056 7063 7070 7078
590
77085 77093 77100 77107 77115 77122 77129 77137 77144 77161
1
7159 7166 7173 7181 7188 7195 7203 7210 7217 7225
2
7232 7240 7247 7254 7202 7209 7276 7283 7291 7298
8
7305 7313 7320 7327 7335 7342 7349 7357 7364 7371
4
7379 7386 7393 7401 7408 7415 7422 7430 7437 7444
5
7452 7459 7466 7474 7481 7488 7495 7608 7610 7617
6
7525 7532 7539 7546 7554 7561 7568 7576 7683 7690
7
7597 7605 7612 7619 7627 7634 7641 7648 7656 7663
8
7670 7677 7685 7692 7699 7706 7714 7721 7728 7736
9
7743 7750 7767 7764 7772 7779 7786 7793 7801 7808
600
77815 77822 77830 77837 77844 77851 77859 77866 77873 77880
447
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456789
600
77816 77822 77830 77837 77844 77851 77869 77866 77873 77880
7887 789i 7902 7909 7916 7924 7931 7938 7946 7952
7960 7967 7974 7981 7988 7996 8003 8010 8017 802i
8032 8039 8046 8063 8061 8068 8075 8082 8089 8097
8104 8111 8118 8126 8132 8140 8147 8154 8161 8168
8176 8183 8190 8197 8204 8211 8219 8226 8233 8240
8247 8254 8262 8269 8276 8283 8290 8297 8305 8312
8319 8326 8333 8340 8347 8355 8362 8369 8376 8383
aSOO 8398 8405 8412 8419 8426 8433 8440 8447 8465
8462 8469 8476 8483 8490 8497 8604 8612 8519 8626
610
78533 78540 78547 78654 78561 78669 78676 78683 78690 78697
8604 8611 8618 8625 8633 8640 8647 8654 8661 8668
8675 8682 8689 8696 8704 8711 8718 8725 8782 8739
8746 8753 8760 8767 8774 8781 8789 8796 8803 8810
8817 8824 8831 8838 8845 8852 8859 8866 8873 8880
8888 8895 8902 8909 8916 8923 8930 8937 8944 8951
8958 8966 8972 8979 8986 8993 9000 9007 9014 9021
9029 9036 9043 9050 9057 9064 9071 9078 9085 9092
9099 9106 9113 9120 9127 9134 9141 9148 9166 9162
9169 9176 9183 9190 9197 9204 0211 9218 9226 9232
620
79239 79246 79263 79260 79267 79274 79281 79288 79296 79802
9309 9316 9323 9330 9337 9344 9351 9358 9366 9372
9379 9386 9393 9400 9407 9414 9421 9428 9435 9442
9449 9466 9463 9470 9477 9484 9491 9498 9505 9511
9618 9525 9532 9539 9546 9563 9560 9567 9674 9581
9588 9595 9602 9609 9616 9623 9630 9637 9644 9650
9657 9664 9671 9678 9685 9692 9699 9706 9713 9720
9727 9734 9741 9748 9754 9761 9768 9775 9782 9789
9796 9803 9810 9817 9824 9831 9837 9844 9851 9858
9865 9872 9879 9886 9893 9900 9906 9913 9920 9927
680
79934 79941 79948 79955 79962 79969 79975 79982 79989 79996
80003 80010 80017 80024 80030 80037 80044 80051 80058 80065
0072 0079 0085 0092 0099 0106 0113 0120 0127 0134
0140 0147 0154 0161 0168 0175 0182 0188 0196 0202
0209 0216 0223 0229 0236 0243 0250 0257 0264 0271
0277 0284 0291 0298 0305 0312 0318 0325 0332 0339
0346 0363 0359 0366 0373 0380 0387 0393 0400 0407
0414 0421 0428 0434 0441 0448 0455 0462 0468 0476
0482 0489 0496 0502 0509 0516 0523 0530 0536 0543
0560 0657 0564 0570 0577 0584 0691 0598 0604 0611
640
80618 80625 80632 80638 80645 80652 80659 80665 80672 80679
0686 0693 0699 0706 0713 0720 0726 0733 0740 0747
0754 0760 0767 0774 0781 0787 0794 0801 0808 0814
0821 0828 0835 0841 0848 0855 0862 0868 0875 0882
0889 0895 0902 0909 0916 0922 0929 0936 0943 0949
0956 0963 0969 0976 0983 0990 0996 1003 1010 1017
1023 1030 1037 1043 1050 1067 1064 1070 1077 1084
1090 1097 1104 1111 1117 1124 1131 1137 1144 1151
1158 1164 1171 1178 1184 1191 1198 1204 1211 1218
1224 1231 1238 1245 1251 1268 1265 1271 1278 1285
660
81291 81298 81305 81311 81318 81325 81331 81338 81345 81361
448
TABLE I.— LOGARITHMS OF NUMBERS*. \-
N
012 34567 89
650
81291 81298 81305 81311 81318 81325 81331 81338 81345 81361
1358 1365 1371 1378 1385 1391 1398 1405 1411 1418
1425 1431 1438 1445 1461 1468 14(J5 1471 1478 1485
1491 1498 1605 1611 1618 1525 1531 1538 1644 1551
1668 1664 1571 1578 1584 1591 1698 1604 1611 1617
1624 1631 1637 1644 1661 1657 1664 1671 1677 1684
1690 1697 1704 1710 1717 1723 1730 1737 1743 1750
1757 1763 1770 1776 1783 1790 1796 1803 1809 1816
1823 1829 1836 1842 1849 1856 1862 1869 1875 1882
1889 1895 1902 1908 1915 1921 1928 1935 1941 1948
660
81964 81961 81968 81974 81981 81987 81994 82000 82007 82014
2020 2027 2033 2040 2046 2053 2060 2066 2073 2079
2086 2092 2099 2106 2112 2119 2126 2132 2138 2145
2161 2168 2164 2171 2178 2184 2191 2197 2204 2210
2217 2223 2230 2236 2243 2249 2256 2263 2269 2276
2282 2289 2296 2302 2308 2315 2321 2328 2334 2341
2347 2354 2360 2367 2373 2380 2387 2393 2400 2406
2413 2419 2426 2432 2439 2445 2452 2468 2465 2471
2478 2484 2491 2497 2504 2610 2617 2523 2630 2636
2643 2649 2666 2562 2669 2575 2582 2688 2695 2601
670
82607 82614 82620 82627 82633 82640 82646 82653 82669 82666
2672 2679 2686 2692 2698 2705 2711 2718 2724 2730
2737 2743 2750 2756 2763 2769 2776 2782 2789 2796
2802 2808 2814 2821 2827 2834 2840 2847 2853 2860
2866 2872 2879 2885 2892 2898 2905 2911 2918 2924
2930 2937 2943 2950 2956 2963 2969 2975 2982 2988
2995 3001 3008 3014 3020 3027 3033 3040 3046 3052
3059 3065 3072 3078 3085 3091 3097 3104 3110 3117
3123 3129 3136 3142 3149 3165 3161 3168 3174 3181
3187 3193 3200 8206 3213 3219 8226 3232 3238 3245
680
83261 83267 83264 83270 83276 83283 83289 83296 83302 83308
3315 3321 3327 3334 3340 3347 3353 8369 3366 3372
3378 3385 3391 3398 3404 3410 3417 3423 3429 3436
3442 3448 3465 3461 3467 3474 3480 3487 3493 3499
3606 3612 3618 3525 3531 3637 8644 3660 3656 3563
3669 3575 3682 3588 3594 3601 3607 3613 3620 3626
3632 3639 3645 3651 3658 3664 3670 3677 3683 3689
8696 3702 3708 3715 3721 3727 3734 3740 3746 3763
8759 3766 3771 3778 3784 3790 3797 3803 3809 3816
3822 3828 3835 3841 3847 3853 3860 3866 3872 3879
690
83885 83891 83897 83904 83910 83916 83923 83929 83935 83942
3948 3964 3960 3967 3973 3979 3985 3992 8998 4004
4011 4017 4023 4029 4036 4042 4048 4065 4061 4067
4073 4080 4086 4092 4098 4105 4111 4117 4123 4130
4136 4142 4148 4165 4161 4167 4173 4180 4186 4192
4198 4205 4211 4217 4223 4230 4236 4242 4248 4255
4261 4267 4273 4280 4286 4292 4298 4305 4311 4317
4323 4330 4336 4342 4348 4354 4361 4367 4373 4379
4386 4392 4398 4404 4410 4417 4423 4429 4435 4442
4448 4464 4460 4466 4473 4479 4485 4491 4497 4604
700
84610 84516 84522 84528 84535 84541 84547 84553 84559 84666
449
TABLE I.— LOGARITHMS OF NUMBERS.
N
0128456789
700
84510 84516 84522 84528 8453i 84541 84547 84553 84559 84566
4572 4578 4584 4590 4597 4603 4609 4615 4621 4628
4634 4640 4646 4652 4658 466^ 4671 4677 4688 4689
4696 4702 4708 4714 4720 4726 4783 4789 4745 4751
4757 4763 4770 4776 4782 4788 4794 4800 4807 4813
4819 4825 4831 4837 4844 48^0 4856 4862 4868 4874
4880 4887 4893 4890 4905 4911 4917 4924 4930 4936
4942 4948 4954 4960 4967 4973 4979 4985 4991 4997
5003 5009 5016 5022 5028 6034 5040 5046 5052 5058
6065 5071 6077 6083 5089 6095 5101 5107 5114 5120
710
85126 85132 85138 85144 85150 85156 85163 85169 85175 85181
5187 5193 5199 5205 5211 5217 5224 5230 5236 5242
5248 5254 5260 5266 5272 5278 5285 5291 5297 5303
6309 5315 5321 5327 5333 5339 5345 6352 5358 5364
5370 5376 5382 5388 5394 5400 5406 6412 5418 5425
6431 6437 6443 5449 5455 6461 6467 6473 6479 5485
6491 5497 5503 6509 6516 6522 5528 5534 6540 5546
6552 6558 5664 6570 5576 5582 6688 6594 5600 6606
6612 5618 5625 5631 6637 5643 5649 6655 6661 6667
6673 6679 6685 6691 5697 5703 5709 5715 5721 5727
720
85788 85739 85746 85751 85757 85763 85769 85775 85781 85788
6794 6800 5806 6812 5818 5824 6830 5836 5842 5848
5854 5860 5866 6872 5878 5884 5890 5896 5902 6908
6914 6920 5926 5932 5938 5944 5950 6956 5962 6968
5974 6980 5986 5992 5998 6004 6010 6016 6022 6028
6034 6040 6046 6052 6058 6064 6070 6076 6082 6088
6094 6100 6106 6112 6118 6124 6130 6136 6141 6147
6153 6159 6165 6171 6177 6183 6189 6196 6201 6207
6213 6219 6225 6231 6237 6243 6249 6255 6261 6267
6273 6279 6285 6291 6297 6303 6308 6314 6320 6326
780
86332 86338 86344 86350 86356 86362 86368 86374 86380 86386
6392 6398 6404 6410 6415 6421 6427 6433 6439 6445
6451 6457 6463 6469 6475 6481 6487 6493 6499 6504
6610 6616 6522 6528 6534 6540 6546 6552 6558 6564
6670 6676 6581 6587 6593 6599 6605 6611 6617 6623
66"?^ 6635 6641 6646 6652 6658 6664 6670 6676 6682
6688 6694 6700 6705 6711 6717 6723 6729 6735 6741
6747 6753 6759 6764 6770 6776 6782 6788 6794 6800
6806 6812 6817 6823 6829 6836 6841 6847 6853 6859
6864 6870 6876 6882 6888 6894 6900 6906 6911 6917
740
86923 86929 86935 86941 86947 86953 86958 86964 86970 86976
6982 6988 6994 6999 7005 7011 7017 7023 7029 7035
7040 7046 7052 7058 7064 7070 7075 7081 7087 7093
7099 7105 7111 7116 7122 7128 7134 7140 7146 7151
7157 7163 7169 7175 7181 7186 7192 7198 7204 7210
7216 7221 7227- 7233 7239 7245 7251 7256 7262 7268
7274 7280 7286 7291 7297 7303 7309 7315 7320 7326
7332 7338 7344 7349 7355 7361 7367 7373 7379 7384
7890 7896 7402 7408 7413 7419 7425 7431 7437 7442
7448 7454 7460 7466 7471 7477 7483 7489 7495 7500
750
87606 87512 87518 87523 87529 87535 87541 87547 87662 87568
450
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456789
750
87S06 87612 87618 87623 87529 87635 87641 87647 87662 87668
7564 7670 7576 7681 7687 7693 7699 7604 7610 7616
7622 7628 7633 7639 7645 7651 7666 7662 7668 7674
7679 7686 7691 7697 7703 7708 7714 7720 7726 7731
7737 7743 7749 7764 7760 7766 7772 7777 7783 7789
7795 7800 7806 7812 7818 7823 7829 7835 7841 7846
7852 7858 7864 7869 7875 7881 7887 7892 7898 7904
7910 7915 7921 7927 79;^ 7938 7944 7950 7966 7961
7967 7973 7978 7984 7990 7996 8001 8007 8013 8018
8024 8030 8036 8041 8047 8063 8068 8064 8070 8076
760
88081 88087 88093 88098 88104 88110 88116 88121 88127 88133
8138 8144 8150 8156 8161 8167 8173 8178 8184 8190
8195 8201 8207 8213 8218 8224 8230 8236 8241 8247
8262 8258 8264 8270 8276 8281 8287 8292 8298 8304
8309 8315 8321 8326 8332 8338 8343 8349 8365 8360
8366 8372 8377 8383 8389 8395 8400 8406 8412 8417
8423 8429 8434 8440 8446 8461 8467 8463 8468 8474
8480 8486 8491 8497 8502 8508 8513 8519 8525 8630
8536 8542 8547 8563 8559 8564 8570 8676 8581 8587
8693 8698 8604 8610 8616 8621 8627 8632 8638 8643
770
88649 88655 88660 88666 88672 88677 88683 88689 88694 88700
8706 8711 8717 8722 8728 8734 8739 8745 8760 8766
8762 8767 8773 8779 8784 8790 8795 8801 8807 8812
8818 8824 8829 8835 8840 8846 8852 8867 8863 8868
8874 8880 8885 8891 8897 8902 8908 8913 8919 8925
8930 8936 8941 8947 8963 8968 8964 8969 8975 8981
8986 8992 8997 9003 9009 9014 9020 9026 9031 9037
9042 9048 9053 9059 9064 9070 9076 9081 9087 9092
9098 9104 9109 9115 9120 9126 9131 9187 9143 9148
9164 9159 9165 9170 9176 9182 9187 9193 9198 9204
780
89209 89215 89221 89226 89232 89237 89243 89248 89264 89260
9265 9271 9276 9282 9287 9293 9298 9304 9310 9316
9321 9326 9332 9337 9343 9348 9354 9360 9366 9371
9376 9382 9387 9393 9398 9404 9409 9415 9421 9426
9432 9437 9443 9448 9454 9459 9465 9470 9476 9481
9487 9492 9498 9504 9509 9515 9620 9626 9631 9637
9542 9548 9553 9559 9564 9570 9576 9681 9686 9592
9697 9603 9609 9614 9620 9626 9631 9636 9642 9647
9653 9658 9664 9669 9675 9680 9686 9691 9697 9702
9708 9713 9719 9724 9730 9735 9741 9746 9762 9757
790
89763 89768 89774 89779 89785 89790 89796 89801 89807 89812
9818 9823 9829 9834 9840 9845 9861 9866 9862 9867
9873 9878 9883 9889 9894 9900 9906 9911 9916 9922
9927 9933 9938 9944 9949 9955 9960 9966 9971 9977
9982 9988 9993 9998 90004 90009 90015 90020 90026 90031
90037 90042 90048 90053 0059 0064 0069 0075 0080 0086
0091 0097 0102 0108 0113 0119 0124 0129 0135 0140
0146 0151 0167 0162 0168 0173 0179 0184 0189 0195
0200 0206 0211 0217 0222 0227 0233 0238 0244 0249
0255 0260 0266 0271 0276 0282 0287 0293 0296 0804
800
90309 90314 90820 90326 90331 90336 90342 90347 90362 90368
451
TABLE I.— LOGARITHMS OF NUMBERS.
K
0123456780
800
90300 00314 90320 90326 90331 90336 90342 90347 90352 90358
0363 0369 0374 0380 0385 0390 0396 0401 0407 0412
0417 0423 0428 0434 0439 0445 04^0 0455 0461 0466
0472 0477 0482 0488 0493 0499 0504 0509 0515 0520
0526 0531 0536 0542 0547 0553 0558 0563 0569 0574
0580 0585 0590 0596 0601 0607 0612 0617 0623 0628
0634 0639 0644 0650 0655 0660 0666 0671 0677 0682
0687 0693 0698 0703 0709 0714 0720 0725 0730 0736
0741 0747 0752 0757 0768 0768 0773 0779 0784 0789
0795 0800 0806 0811 0816 0822 0827 0832 0838 0843
810
90849 90864 90859 90865 90870 90875 90881 90886 90891 90897
0902 0907 0913 0918 0924 0929 0934 0940 0945 0950
0956 0961 0966 0972 0977 0982 0988 0993 0998 1004
1009 1014 1020 1026 1030 1036 1041 1046 1062 1057
1062 1068 1073 1078 1084 1089 1094 1100 1105 1110
1116 1121 1126 1132 1137 1142 1148 1153 1158 1164
1169 1174 1180 1185 1190 1196 1201 1206 1212 1217
1222 1228 1233 1238 1243 1249 1254 1259 1265 1270
1275 1281 1286 1291 1297 1302 1307 1312 1318 1323
1328 1334 1339 1344 1350 1355 1360 1365 1371 1376
820
91381 91387 91392 91397 91403 91408 91413 91418 91424 91429
1434 1440 1445 1450 1455 1461 1466 1471 1477 1482
1487 1492 1498 1503 1508 1514 1519 1524 1529 1535
1540 1546 1561 1566 1661 1666 1572 1577 1582 1587
1693 1598 1603 1609 1614 1619 1624 1630 1635 1640
1645 1651 1656 1661 1666 1672 1677 1682 1687 1693
1698 1703 1709 1714 1719 1724 1730 1735 1740 1746
1751 1766 1761 1766 1772 1777 1782 1787 1793 1798
1803 1808 1814 1819 1824 1829 1834 1840 184^ 1850
1866 1861 1866 1871 1876 1882 1887 1892 1897 1903
880
91908 91913 91918 91924 91929 91934 91939 91944 91950 91955
1960 1966 1971 1976 1981 1986 1991 1997 2002 2007
2012 2018 2023 2028 2033 2038 2044 2049 2064 2059
2065 2070 2075 2080 2085 2091 2096 2101 2106 2111
2117 2122 2127 2132 2137 2143 2148 2153 2158 2163
2169 2174 2179 2184 2189 2195 2200 2205 2210 2215
2221 2226 2231 2236 2241 2247 2252 2267 2262 2267
2273 2278 2283 2288 2293 2298 2304 2309 2314 2319
2324 2330 2335 2340 2346 2360 2365 2361 2366 2371
2376 2381 2387 2392 2397 2402 2407 2412 2418 2423
^TcV
92428 92433 92438 92443 92449 92464 92469 92464 92469 92474
2480 2485 2490 2496 2500 2606 2611 2616 2621 2526
2631 2536 2642 2647 2562 2567 2662 2667 2572 2578
2683 2588 2693 2698 2603 2609 2614 2619 2624 2629
2634 2639 2645 2650 2665 2660 2666 2670 2675 2681
2686 2691 2696 2701 2706 2711 2716 2722 2727 2732
2737 2742 2747 2762 2768 2763 2768 2773 2778 2783
2788 2793 2799 2804 2809 2814 2819 2824 2829 2834
2840 2845 2850 2865 2860 2866 2870 2875 2881 2886
2891 2896 2901 2906 2911 2916 2921 2927 2932 2937
850
92942 92947 92952 92957 92962 92967 92973 92978 92983 92988
45^
TABLE L— LOGARITHMS OF NUMBERS.
N
0123456780
850
92042 02947 92952 92957 92962 92967 92978 92978 92983 92988
2993 2998 3003 3008 3013 3018 3024 3029 3034 3039
3044 3049 3054 3059 3064 3069 3075 3080 3085 3090
3095 8100 310$ 3110 3115 3120 8125 3181 3136 3141
3146 3151 3156 3161 3166 3171 3176 3181 3186 8192
3197 3202 3207 3212 3217 3222 3227 3232 8237 8242
3247 3252 3258 8263 3268 3273 3278 3283 3288 3293
3298 3303 3308 3313 3318 3328 3328 3334 3339 3344
3349 3354 8359 3364 3369 3374 3379 3384 3389 3394
3399 3404 3409 3414 3420 3425 3430 3435 8440 3445
860
93450 93455 93460 93465 93470 93475 93480 93485 93490 93495
3500 3505 3510 3516 8520 3526 3531 3536 3541 3546
3551 3556 3561 3566 3571 3576 3581 3586 3591 3596
3601 3606 3611 3616 3621 3626 8631 3636 3641 3646
3651 3656 3661 3666 3671 3676 3682 3687 3692 3697
3702 3707 3712 3717 3722 3727 3732 3737 3742 3747
3752 3757 3762 8767 3772 8777 3782 3787 3792 3797
3802 3807 3812 3817 3822 3827 3832 3837 3842 3847
3852 3857 3862 3867 3872 3877 3882 3887 3892 3897
3902 3907 3912 3917 3922 3927 3932 3937 3942 3947
870
93952 93957 93962 93967 93972 93977 93982 93987 93992 93997
4002 4007 4012 4017 4022 4027 4032 4037 4042 4047
4052 4057 4062 4067 4072 4077 4082 4086 4091 4096
4101 4106 4111 4116 4121 4126 4131 4136 4141 4146
4151 4156 4161 4166 4171 4176 4181 4186 4191 4196
4201 4206 4211 4216 4221 4226 4231 4236 4240 4245
4250 4255 4260 4265 4270 4275 4280 4285 4290 4295
4800 4305 4310 4315 4320 4325 4330 4335 4340 4345
4349 4354 4359 4364 4369 4374 4379 4384 4389 4394
4399 4404 4409 4414 4419 4424 4429 4433 4438 4443
880
94448 94453 94458 94463 94468 94473 94478 94483 94488 94493
4498 4603 4507 4512 4517 4522 4527 4532 4537 4542
4547 4552 4557 4562 4567 4571 4576 4581 4586 4591
4596 4601 4606 4611 4616 4621 4626 4630 4635 4640
4645 4650 4655 4660 4665 4670 4675 4680 4685 4689
4694 4699 4704 4709 4714 4719 4724 4729 4734 4738
4743 4748 4753 4768 4763 4768 4773 4778 4783 4787
4792 4797 4802 4807 4812 4817 4822 4827 4832 4836
4841 4846 4851 4856 4861 4866 4871 4876 4880 4885
4890 4895 4900 4905 4910 4915 4919 4924 4929 4934
890
94939 94944 94949 94954 94969 94963 94968 94973 94978 94983
4988 4993 4998 5002 5007 5012 5017 5022 5027 5032
5036 6041 5046 5051 5056 5061 5066 5071 6075 5080
5085 5090 5095 5100 6105 5109 5114 5119 5124 5129
5134 6139 5143 5148 5153 5168 5163 5168 6173 5177
5182 5187 6192 5197 5202 5207 5211 5216 5221 5226
5231 5236 5240 5245 6250 6255 5260 5265 5270 5274
6279 5284 5289 5294 6299 5303 5308 5313 5318 5323
5328 5332 5337 5842 5347 5352 5357 5361 5366 5371
5376 5381 5386 5390 5395 5400 5405 6410 5415 5419
900
95424 95429 95434 95439 95444 95448 95458 95458 95463 95468
453
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456780
900
95424 95429 954:M 95439 96444 95448 96453 95458 95463 95468
5472 5477 5482 5487 5492 5497 5501 5606 5511 5616
5521 6525 65.S0 5535 5540 654^ 65^0 6664 6569 5664
6669 6574 5578 5583 5688 5593 6598 5602 5607 6612
6617 5022 5626 5631 5630 5641 5646 5650 5665 5660
6665 6670 6674 5679 5684 6689 5694 6698 6703 5708
5713 6718 6722 5727 6732 6737 6742 6746 6751 5756
6761 6766 5770 5775 6780 5785 6789 5794 5799 5804
5809 6813 5818 5823 5828 6832 6837 6842 5847 6862
5856 6861 5866 5871 6876 6880 6885 6890 6895 6899
910
95904 95909 95914 96918 95923 96928 95933 95938 95942 96947
5952 6957 5061 5966 6971 6976 6980 5985 5990 6995
5999 6004 6009 6014 6019 6023 6028 6033 6038 6042
6047 6052 6057 6061 6066 6071 6076 6080 6085 6090
6005 6099 6104 6109 6114 6118 6123 6128 6133 6137
6142 6147 6152 6156 6161 6166 6171 6176 6180 6185
6190 6194 6199 6204 6209 6213 6218 6223 6227 6232
6237 6242 6246 6251 6256 6261 6265 6270 6275 6280
6284 6289 6294 6298 6303 6308 6313 6317 6322 6327
6332 6336 6341 6346 6350 6355 6360 6365 6369 6874
920
96379 96384 96388 96393 96398 96402 96407 96412 96417 96421
6426 6431 6435 6440 6445 6450 6454 6459 6464 6468
6473 6478 6483 6487 6402 6497 6501 6506 6611 6616
6620 6525 6530 6534 6539 6544 6548 6653 6668 6562
4
6567 6572 6577 6581 6586 6591 6696 6600 6605 6609
6614 6619 6624 6628 6633 6638 6642 6647 6652 6656
6661 6066 6670 6676 6680 6685 6689 6694 6699 6703
6708 6713 6717 6722 6727 6731 6736 6741 6746 6760
6755 6759 6764 6769 6774 6778 6783 6788 6792 6797
6802 6806 6811 6816 6820 6825 6830 6834 6839 6844
980
96848 96853 96858 96802 96867 96872 96876 96881 96886 96890
6895 6900 6904 6909 6914 6918 6923 6928 6932 6937
6942 6946 6951 6956 6960 6965 6970 6974 6979 6984
6988 6993 6997 7002 7007 7011 7016 7021 7025 7030
7035 7039 7044 7049 7053 7058 7063 7067 7072 7077
7081 7086 7090 7095 7100 7104 7109 7114 7118 7123
7128 7132 7137 7142 7146 7151 7166 7160 7165 7169
7174 7179 7183 7188 7192 7197 7202 7206 7211 7216
7220 7225 7230 7234 7239 7243 7248 7253 7257 7262
7267 7271 7276 7280 7285 7290 7294 7299 7304 7308
940
97313 97317 97322 97327 97331 97336 97340 97345 97350 97364
7359 7304 7368 7373 7377 7382 7387 7391 7396 7400
7405 7410 7414 7419 7424 7428 7433 7437 7442 7447
7451 7456 7460 7465 7470 7474 7479 7483 7488 7493
7497 7502 7506 7511 7516 7520 7525 7629 7534 7639
7543 7548 7552 7667 7562 7566 7571 7576 7580 7685
7589 7594 7598 7603 7607 7612 7617 7621 7626 7630
7635 7640 7644 7(549 7653 7658 7663 7667 7672 7676
7681 7685 7690 7695 7699 7704 7708 7713 7717 7722
7727 7731 7736 7740 7745 7749 7754 7759 7763 7768
960
97772 97777 97782 97786 97791 97795 97800 97804 97809 978U
454
TABLE I.— LOGARITHMS OF NUMBERS.
N
0123456 7'80
960
97772 97777 97782 97786 97791 97796 97800 97804 97809 97813
7818 7823 7827 7832 7836 7841 7846 7860 7865 7869
7864 7868 7873 7877 7882 7886 7891 7896 7000 7905
7909 7914 7918 7923 7928 7932 7937 7941 7946 7980
7965.7969 7964 7968 7973 7978 7982 7987 7991 7996
8000 8005 8009 8014 8019 8023 8028 8032 8037 8041
8046 8060 8055 8069 8064 8068 8073 8078 8082 8087
8091 8096 8100 8105 8109 8114 8118 8123 8127 8132
8137 8141 8146 8150 8155 8169 8164 8168 8173 8177
8182 8186 8191 8196 8200 8204 8209 8214 8218 8223
960
98227 98232 98236 98241 98246 98250 98264 98269 98263 98268
8272 8277 8281 8286 8290 8295 8299 8304 8308 8313
8318 8322 8327 8331 8336 8340 8345 8349 8364 8:^68
8363 8367 ^372 8376 8381 8386 8300 8394 8399 8403
8408 8412 8417 8421 8426 8430 8435 8439 8444 8448
8463 8467 8462 8466 8471 8476 8480 8484 8489 8493
8498 8602 8607 8511 8516 8620 8625 8629 8634 8638
8543 8547 8552 8556 8561 8566 8670 8674 8579 8683
8588 8592 8597 8601 8006 8610 8614 8619 8623 8628
8632 8637 8641 8646 8660 8665 8669 8664 8668 8673
970
98677 98682 98686 98691 98696 98700 98704 98709 98713 98717
8722 8726 8731 8736 8740 8744 8749 8763 8768 8762
8767 8771 8776 8780 8784 8789 8793 8798 8802 8807
8811 8816 8820 8825 8829 8834 8838 8843 8847 8861
8866 8860 8865 8869 8874 8878 8883 8887 8892 8896
8900 8905 8909 8914 8918 8923 8927 8932 8936 8941
8945 8949 8964 8958 8963 8967 8972 8976 8981 8986
8989 8994 8998 9003 9007 9012 9016 9021 9026 9029
9034 90;38 9043 9047 9062 9056 9061 9065 9069 9074
9078 9083 9087 9092 9096 9100 9105 9109 9114 9118
980
99123 99127 99131 99136 99140 99145 99149 99154 99168 99162
9167 9171 9176 9180 9185 9189 9193 9198 9202 9207
9211 9216 9220 9224 9229 9233 9238 9242 9247 9261
9266 9260 9264 9269 9273 9277 9282 9286 9291 9296
9300 9304 9308 9313 9317 9322 9326 9330 9335 9339
9344 9348 9352 9357 9301 9366 9370 9374 9879 9383
9388 9392 9396 9401 9406 9410 9414 9419 9423 9427
9432 9436 9441 9445 9449 9464 9468 9463 9467 9471
9476 9480 9484 9489 9493 9498 9602 9606 9611 9516
9620 9624 9628 9633 9537 9542 9646 9660 9665 9669
990
99664 99568 99572 99677 99581 99586 99690 99694 99599 99603
9607 9612 9616 9621 9625 9629 9634 9638 9642 9647
9651 9656 9660 9664 9669 9673 9677 9682 9686 9691
9695 9699 9704 9708 9712 9717 9721 9726 9730 9734
9739 9743 9747 9752 9756 9760 9765 9769 9774 9778
9782 9787 9791 9795 9800 9804 9808 9813 9817 9822
9826 9830 9835 9839 9843 9848 9862 9866 9861 9865
9870 9874 9878 9883 9887 9891 9896 9900 9904 9909
9913 9917 9922 9926 9930 9935 9939 9944 9948 9962
9957 9961 9966 9970 9974 9978 9983 9987 9991 9996
1000
00000 00004 00009 00013 00017 00022 00026 00030 00085 00a39
455
TABLE II.— LOGARITHMIC SINES AND COSINES.
t
©•
1
•
«•
/
SiDe
CkMdne
Sine
Oosine
Sine
Cosine
0
—00
10.00000
8.84186
9.99993
6.54268
9.99974
60
1
6.46878
00000
84908
:>9998
54648
99978
60
9
78476
00000
85609
54999
99978
58
8
04066
00000
86304
99993
55354
99978
57
4
7.00579
00000
26968
99992
56706
99978
56
5
16870
00000
27661
99992
56054
69971
65
0
84188
00000
28324
QOOOO
56400
99971
54
7
80888
00000
28975
99992
56748
99970
58
8
88688
00000
29621
99992
57064
99970
58
9
41797
00000
80256
99991
57421
99969
61
10
7.46878
10.00000
8.30679
9.99991
8.57757
9.99969
60
11
50518
00000
81495
99991
58069
99986
49
18
54891
00000
82106
99990
56419
99966
48
18
57767
00000
82702
99990
58747
99967
47
14
60966
00000
88208
99990
58078
99967
46
16
68968
00000
88875
99990
50895
99967
46
18
66784
00000
81450
99989
59715
99960
44
17
69417
9.99999
S.^18
99989
60088
99966
48
18
71900
OOQQQ
WtfW
85678
99969
60849
99965
48
19
74848
86131
99969
60668
VWwV
41
90
7.76475
9.99999
8.36676
9.99986
8.60978
9.99964
40
81
78694
OQOQO
VWW
37217
99988
61288
99963
89
88
80615
87750
99968
61589
99968
88
88
88545
99999
88276
99987
61894
99062
87
84
84898
99999
88796
99967
68196
99903
86
85
86166
00099
89310
99987
(B497
99961
85
26
87870
99999
89618
99986
62796
99961
84
37
80509
99999
40890
99986
68091
99980
88
88
91088
99999
40816
99966
68886
99980
82
89
98618
OMQR
41807
99085
68876
99959
81
80
7.94064
9.99996
8.41798
9.99966
8.68968
9.99959
80
81
95506
90998
42272
99983
64256
99958
29
88
96697
99998
42746
99964
64548
99968
26
88
96283
99908
48216
99984
64827
99957
87
84
99520
QQAQO
48680
99964
65110
99956
26
85
8.00779
09006
44139
99983
65891
99956
86
86
02008
QQQQO
44594
99988
65670
99955
84
87
08198
99997
45044
99963
65947
99965
23
88
04350
99997
45489
99982
66228
99954
22
89
05478
99997
45930
99962
66497
99954
21
40
8.0657B
9.99997
8.46866
9.99962
8.66709
9.99958
20
41
07050
99997
46799
99961
67089
19
48
06696
99997
47226
99981
67306
99952
18
48
09718
99997
47650
99061
6757B
99951
17
44
10717
OOOCM
WMID
46069
99960
67841
99951
16
45
11698
99996
48486
99980
68104
99960
15
46
12647
OQQOA
48896
99979
68367
99949
14
47
18581
99996
49304
99979
68627
99949
13
48
14496
OQQOA
49706
99979
68866
99948
12
49
15891
99996
50106
99976
60144
99918
11
60
8.16868
9.99996
8.50504
9.99978
6.60400
9.99947
10
51
17126
99996
50807
99977
69654
99946
0
58
17971
99996
51287
99977
609O7
99946
8
68
18796
90995
51678
99977
70159
99045
7
54
19610
99995
58055
99976
70409
99944
6
66
20407
QQQOA
52484
99976
70658
90944
5
56
81189
99994
52810
99975
70905
99943
4
57
819.58
QQQQti
53188
99975
' 71151
99942
8
58
22718
90994
53552
991)74
71396
99942
2
50
28156
99994
53019
99974
71636
99941
1
60
24186
90003
54282
99974
71880
99940
0
$
Cosine
Sine
Cosine
Sine
Cosine
Sine
t
89°
88
«
87
o
4S6
TABLE II.— LOGARITHMIC SINES AND COSINES.
9
go
4*
«•
/
Sine
Cosine
Sine
Cosine
Sine
Cofdne
0
8.71880
9.99940
8.84858
9.99894
8.94080
9.99884
60
1
72120
99940
84589
0S8Q8
94174
99888
59
9
Tum
99988
84n8
99692
94817
99682
58
8
78597
99988
84807
99691
94461
99681
57
4
78884
99988
85076
99691
94608
99880
66
6
78069
99987
85252
99690
94746
99889
55
6
78308
99986
86429
99889
94887
99628
54
7
78685
99936
85606
99888
96029
99827
53
8
78T67
99985
85780
99887
95170
99625
62
9
78997
99934
85966
99886
06810
99624
61
10
8.74826
9.99984
8.86128
9.99686
8.06460
9.99688
50
11
74454
99988
86301
99684
95569
99622
49
18
74680
99982
86474
99888
9.5?^
99821
48
18
74906
99982
86646
99682
93867
99620
47
14
75130
99981
86816
99681
96005
99819
46
15
75868
99980
80967
99880
06148
99817
45
16
75575
99929
87156
99679
96280
99616
44
17
75795
99929
87825
99679
96417
99815
48
18
76015
99928
87494
99678
96568
99814
48
19
76234
99927
87661
90877
96689
99618
•41
80
8.76451
9.99926
8.87829
9.99876
8.96825
9.99618
40
81
76667
99926
87996
99675
96960
99610
89
23
76888
99925
88101
99674
97095
99609
88
88
77097
99924
88326
99678
97829
99606
87
84
77810
99928
88490
998<2
97868
99807
86
85
77522
999-28
88G54
99671
97496
99606
35
26
77738
99922
88817
99870
97629
99804
84
27
77948
99921
88080
99869
97762
99808
88
88
78162
99920
89142
99868
97894
99F02
82
89
78360
99920
89304
90667
98026
99801
31
80
8.78668
9.99919
8.89464
9.99666
8.98157
9.99600
80
81
78T74
99918
89683
99665
96288
99798
29
38
78979
99917
89784
99864
98419
99797
28
88
79188
99917
89948
99668
98549
99796
27
84
79386
99916
90102
99662
96679
99795
28
86
79588
99916
90260
99661
98806
99798
25
86
79789
99914
90417
99860
98937
99792
24
87
79990
99918
90574
99650
99066
99791
88
88
80189
99918
90780
99658
99194
99790
82
89
80388
99912
90886
99657
99822
99788
21
40
8.80585
9.99911
8.91040
9.99656
8.99450
9.99787
80
41
80782
99910
91195
99855
99577
99786
19
42
80978
99909
91849
99854
99704
99785
18
48
81178
99909
01502
99853
99630
99783
17
44
81367
99906
91655
99868
99056
99782
16
45
81560
99907
91807
99651
9.00082
99781
16
46
81752
99906
91959
99850
00207
99780
14
47
81944
99905
92110
99648
00:^32
99r78
18
48
82134
99904
92261
99647
00456
99777
18
49
82324
99904
92411
99646
00681
99776
11
50
8.82518
9.99903
8.98561
9.99645
9.00704
9.99776
10
51
82701
99902
92710
99644
00628
99778
9
58
R2888
09901
92859
99848
00951
99772
8
58
83075
99900
93007
99842
01074
99771
7
54
88261
90699
9:3154
99841
01196
99769
6
55
88446
9'.«98
93301
99840
01818
99768
6
56
8'i630
99S98
03448
99839
01440
99767
4
57
83818
99897
93594
99838
01561
8
58
83996
99896
98740
99887
01682
99764
S
59
84177
99895
08885
99836
01803
1
60
84868
99894
94030
99884
01923
99761
0
f
Cosine
Sine
Cosine
Sine
Cosine
Sine
9
86«
86«
84-
457
TABLE n.— LOGARITHMIC SINES AND COSINES.
#
6
e
7»
%•
/
Bine
Cosine
Sine
Cosine
Sine
Cosine
0
9.01988
9.99761
9.08589
9.99675
9.14856
9.99575
60
1
0^48
99760
06692
99674
14445
99574
59
2
02168
99759
08795
99672
14585
9957%
58
8
02388
99W7
06897
99670
14624
99570
67
4
0^402
09756
06999
99669
14714
99568
56
6
Oisao
99766
00101
99667
14808
99566
66
6
ami
99758
Qoao-i
99660
14891
99666
54
7
02757
9am
09S04
99664
14980
99508
58
8
0tf8r4
09406
99663
16069
99561
52
9
0299;3
90749
09506
99661
15167
99559
51
10
9.03109
9.99748
0.09606
0.99659
9.15245
0.99657
50
11
08326
99747
09707
99658
15388
99656
49
19
03342
99745
09807
99656
15421
99564
48
18
03458
99744
0^907
99055
15608
99662
47
14
08574
99748
10006
99653
16696
90650
40
15
aMS90
99741
10106
99G51
15688
99548
45
16
03S03
99740
10205
99650
16770
99546
44
17
mio
99788
10304
99648
15657
99545
48
18
01034
99787
10402
99647
15944
99548
42
19
01149
99786
10501
99645
16060
99541
41
20
9.04262
9.99784
9.10609
9.99648
0.16116
9.99689
40
21
04876
99783
10697
99642
16203
99537
88
23
04490
99781
10795
99640
16269
09585
88
23
04608
99780
10603
99688
16874
09633
87
24
04715
99^
10990
99637
164GO
99582
86
25
04828
99727
11087
99635
16545
99580
35
26
04940
99726
11184
99638
16631
99628
84
27
06052
99724
11281
99682
16716
99526
88
28
05164
99723
iiarr
99630
16801
99524
82
29
05276
99721
11474
99629
16886
99522
31
80
9.05886
9.99720
0.11670
0.99627
0.16970
9.99520
80
81
06497
99718
11G66
99625
17055
99518
29
88
05607
99717
11761
99624
17139
99517
28
88
05717
99716
lia57
99622
17228
99515
27
84
05827
99714
11952
99620
17807
99518
26
85
05037
90718
12047
99618
17391
99511
25
86
06046
99711
12142
99617
17474
99509
24
87
06155
99710
12286
99615
17568
99507
28
88
06.264
99706
1233i
99618
17641
99605
22
89
0687:2
99707
12425
9U612
17724
99508
21
40
9.06481
9.99706
9.12619
&.99610
9.17807
9.99501
20
41
06580
99704
12612
99608
17890
99499
19
42
06696
99702
12706
99607
17978
99497
18
43
06804
99701
12799
99605
18055
99496
17
44
06911
QQAQQ
12892
99608
18187
99494
10
45
07018
99608
12985
99601
18220
99492
15
40
07124
QQMIA
18078
99600
18902
99490
14
47
07231
99605
18171
99596
18388
99488
18
48
07837
99693
13268
99596
18465
99486
12
49
07442
99692
18365
99696
18547
99484
11
60
9.07548
9.99690
9.18447
0.99693
9.16628
9.99482
10
51
07653
99689
18589
99591
18709
99480
0
62
07768
99687
13030
90589
18790
99478
8
68
07868
99686
18722
99588
18871
99476
7
64
07968
90684
18818
99586
18952
99474
6
66
08072
99683
18904
99584
19088
99472
5
66
06176
99681
i:i994
99582
19118
99470
4
67
08^80
99680
14085
99581
19193
99468
8
68
06888
99678
14175
99579
19273
99J66
8
80
06480
99677
14266
99577
19358
99464
1
60
08589
99675
14366
99575
19483
99462
• 0
#
' Cosine
Sine
Cosirift
Sine
Cosine
Sine
/
8r»
82»
81»
4S8
TABLE 11.— LOGARITHMIC SINES AND COSINES.
t
BO
10«
IP 1
/
Sine
Cosine
Sine
Coeine
Sine
Cosine
9.19488
9.99482
9.28967
9.99886
9.28060
9.99195
60
19618
09460
24089
99888
28185
99192
69
19602
99468
24110
99831
28190
99190
66
19672
90456
24181
99888
28854
99187
67
19751
99454
24258
99396
28819
99186
66
19880
99462
24824
99684
28884
99182
65
19909
99450
24805
09822
28448
99180
64
19088
09448
24466
99319
28512
99177
68
20067
99446
24536
99317
28677
99175
69
20146
99444
24607
99315
28641
99172
61
0.20828
9.99442
9.24677
9.99818
9.88706
9.99170
60
80802
09440
24748
99810
28769
99167
49
90880
09438
24818
99308
28888
99165
48
90468
99486
24888
99806
88896
99162
47
90685
99434
24968
99804
28960
99160
46
90618
99482
2SQ86
99801
29024
99157
46
90601
99429
26098
99899
29087
99156
44
<0768
99427
25168
99897
29150
99162
48
90646
99426
95287
99894
29814
99160
42
90022
99428
25807
99898
29877
99147
41
90
9.90999
9.99421
0.26876
0.99890
9.89340
9.99146
40
21
91076
99419
25445
99888
29408
99148
89
23
91168
99417
25514
99885
29466
99140
88
28
8!«»
99416
25688
99888
29529
99187
87
24
91806
99418
25652
99881
29591
99185
86
25
21862
90411
25721
99278
29654
99188
35
23
21466
99400
25700
99876
29716
99180
84
27
21984
09407
85858
99874
29779
99127
38
28
21010
99404
26927
99871
29841
99124
88
29
21686
99402
26995
99269
29908
99122
31
80
0.21761
9.99400
9.26068
9.99867
9.29966
9.99119
80
81
99396
26181
99864
80088
99117
29
82
219B7
99896
26199
99862
80090
99114
26
88
99894
26267
99860
80161
99112
27
84
29062
99392
26336
99857
80818
99109
26
86
22187
99390
96408
99856
80875
09106
25
80
22211
99388
26470
99252
80886
90104
24
37
22S86
99.385
26688
OfiStfO
80898
90001
28
88
22861
99388
26605
99848
80450
99009
22
89
22485
99881
26672
99845
80621
99096
21
40
9.29609
9.99379
9.86789
9.99848
9.80688
0.99096
20
41
22588
99377
26806
99241
80648
99091
19
42
22667
99375
26878
99888
807t)4
99088
18
48
22781
99378
26040
99886
80766
99086
17
44
28806
99870
27007
99288
80826
99068
16
45
22878
99866
27078
90881
80887
99060
16
46
22962
99366
27140
99829
80947
99078
14
47
28025
99864
97206
99226
81006
99075
18
48
28096
99362
27878
99284
81068
99072
12
49
28171
99859
27889
99281
81129
99070
11
50
9.28244
9.99867
0.27405
9.99819
9.81180
0.99067
10
51
28317
99256
27471
99217
81250
99064
9
62
83890
99868
27fS87
99814
81810
99062
8
68
23462
99361
27602
99812
81870
99060
7
64
28585
99348
27666
99809
81480
09066
6
66
28607
99346
27734
99807
81490
99054
6
50
23879
90844
27799
99804
81540
99061
4
67
28763
9934?
27864
99808
81609
99048
8
66
28823
99340
27980
99800
81669
99046
2
SO
23895
993:J7
£7995
91>197
317S8
90043
1
m
wm
B9335
280ft0
HOI 95
3178ft
990*0
0
#
COBlJie
Sine
Cosine
Sine
Cosine
Sine
*
W
7t*
yg-
459
TABLE n.— LOGARITHMIC SINES AND COSINES.
/
!«•
IS*
14-
/
Sine
Cosine
Sine
Cosine
Sine
Cosine
0
9.81788
9.99040
9.85200
9.06872
9.88868
9.96690
60
1
81847
99088
85263
06869
88418
96687
50
2
31907
99066
85318
96867
38469
96684
58
8
81966
99082
85878
06864
88519
96681
57
4
880:25
99080
35427
98861
88570
96678
56
5
82084
99027
85481
08868
88620
98675
56
6
88143
99024
86536
96865
88670
98671
54
7
82202
99028
85590
96862
88721
96668
58
8
82261
99019
85644
98649
88771
96665
52
9
88819
99016
86696
08846
88821
98662
51
10
9.82878
9.99018
9.85752
9.96848
9.88871
0.98659
60
11
38487
99011
85606
96840
38921
96666
49
18
32495
99006
85860
08837
38971
96652
48
18
82553
99005
85914
98684
89021
98649
47
14
38612
99002
85968
86881
80071
96646
46
15
88670
99000
86022
96828
89121
06648
45
16
82728
96997
86075
96625
89170
96640
44
17
32786
96004
86129
98822
89220
98636
48
18
82844
98991
86182
98819
39270
96688
42
19
88908
96969
36286
06816
89819
96(380
41
80
9.82960
9.96986
9.86280
0.06818
9.89660
0.086^
40
81
88018
96088
86842
98810
88418
96623
89
88
88075
96060
86895
06807
39467
96620
38
88
83183
96978
86449
86804
89517
06617
.97
84
33190
96975
86502
98601
89566
98614
86
85
83248
96972
86655
96708
89615
98610
85
86
83305
98969
86608
96796
89664
98607
84
87
83362
98967
86660
98798
89718
98604
S3
88
88420
98964
86718
96789
89762
96601
82
80
83477
98961
86766
96786
89611
96597
31
80
0.33584
9.98958
9.86819
9.98788
9.89660
0.98594
30
81
38591
96955
86871
96780
89909
98591
29
88
38647
98853
86924
96777
39958
98588
26
S3
33704
96950
36976
98774
40006
98584
27
84
88761
96947
87028
98771
40055
98581
26
85
88818
98944
87081
96768
40103
98578
25
86
88874
98941
87138
98765
40152
96574
24
87
83931
96038
37185
96708
40200
96571
88
88
88987
98936
87287
96750
40249
98568
22
89
84043
98988
37288
96756
40297
96565
81
40
9.84100
9.98900
9.87841
0.96758
9.40846
9.98561
20
41
84156
98927
87398
98750
40894
96558
19
48
84212
96924
87445
98746
40442
96566
18
43
84268
96921
87497
98743
40490
98561
17
44
84324
98919
37549
96740
40538
98548
16
45
84380
98916
87600
98787
40586
98546
15
46
34436
98918
87652
98734
40684
98541
14
47
34491
98910
37708
98781
40682
98538
18
48
34547
98907
87755
96728
40780
98535
12
49
84602
96904
87806
98725
40778
96531
11
60
9.34658
9.98901
9.87858
9.96722
0.40625
9.98528
10
51
34713
98898
87909
96ri9
40878
96525
9
68
84769
98896
37960
96715
40921
98521
8.
53
84824
98893
38011
96712
40968
98518
7
54
34879
98890
38062
96700
41016
96515
6
65
84984
98887
88118
08706
41068
9a-)ii
5
56
34989
98884
88164
96708
41111
96508
4
57
85044
98881
38215
98700
41158
98505
8
68
35099
9W78
mm
9^97
41305
98501
2
59
SB154
Qf^Tr.
S6:i37
U*i*i
41i^J
9S49H
1
m
mm9
&S8::J
3im)^
98COU
41300
HfUM
0
t
Cofrtno
sine
CkrislDe
Sine
Cofltne
Sfne
f
77-
7«*
76'
460
TABLE II.— LOGARITHMIC SINES AND COSINES.
9
16«
16«»
II*
0
Sine
CkNiine
Sine
Cosine
Sine
Cosine
0
9.41800
9.96494
9.44084
9.98884
9.46504
9.98060
60
1
41847
98191
44078
98281
46635
98066
60
2
41804
96488
44122
98277
46676
96068
66
8
41441
98484
44166
98878
46717
98048
67
4
41488
96481
44210
98270
46758
08044
66
5
41585
98477
44258
98266
46800
98040
66
0
41588
98474
44297
96262
46841
98086
54
7
410;®
98471
44341
98259
46888
98088
68
8
41675
98467
44385
98255
46023
9800
58
9
41722
98464
44428
98251
46964
98ffii5
51
10
9.41768
9.98460
9.44472
9.98248
9.47006
9.96081
60
11
41815
98457
44516
98244
47045
98017
49
12
41861
98458
44559
96240
47066
08018
48
18
41908
98450
44G02
98237
47187
98009
47
H
41954
98447
44646
98283
47168
96006
46
15
42001
96448
446K9
98229
47209
98001
45
16
42047
9B440
447^
96226
47^249
97997
44
17
42093
98436
44776
98222
47290
97998
48
18
42140
98483
44810
96218
47380
97969
42
19
42186
98429
44862
98215
47371
97986
41
20
9.42232
9.96426
9.44906
9.96211
9.47411
9.97988
40
21
42278
98422
44948
96207
47452
97978
80
82
42324
98419
44998
98204
47498
97974
88
23
42370
9W15
4.'W)3»
98200
47583
97970
87
24
42416
98412
45077
98196
47573
97966
86
25
42461
98409
46120
98192
47618
97962
85
26
42507
96406
45168
96189
47654
97958
84
27
42553
98402
45206
96185
47694
97954
83
28
42599
98396
45249
96181
47734
97960
8<
29
42644
06895
45292
98m
47774
97946
81
80
9.42690
9.98891
9.45834
9.98174
9.47814
9.97948
80
81
42735
98.388
45377
98170
47854
97988
29
82
42781
98384
45419
98166
47894
97984
28
83
42826
96381
45463
96168
47934
97980
27
84
42872
96377
45504
98159
47974
97926
26
85
42917
96878
45547
96155
4.-014
97928
85
38
42968
96370
45589
96151
48054
97918
84
87
43008
98366
4.^632
98147
48094
97914
88
88
48058
9a363
4.'>674
98144
48188
97910
88
89
43098
96359
45716
96140
48178
97906
21
40
0.48148
9.96856
9.45758
9.96186
9.48818
9.97908
80
41
43188
98352
45801
96188
^ 48252
97898
19
42
48283
98^9
45848
96129
48298
97894
18
43
48278
98345
45885
98125
48382
97890
17
44
48328
9a342
45927
98181
48371
97686
16
45
48367
98338
45969
98117
48411
97882
15
40
43412
98334
46011
98118
48450
97878
14
47
48157
98331
46058
96110
48490
97874
18
48
43502
98327
46095
98106
48629
97870
12
49
48546
98324
46186
96108
48568
07866
11
50
9.43591
9.98320
9.46178
9.98098
9.48007
9.97861
10
51
98317
462-iO
98094
48647
97857
9
52
43680
0^813
46J68
98090
48686
97858
8
58
43724
98:309
46.308
9S087
48?25
97849
7
54
43769
98:W6
46.345
98083
48784
97845
6
55
43813
98302
46:^
98079
48808
97841
5
56
43857
96299
46428
98075
4aS42
97887
4
57
43901
98295
46469
9S071
48881
97838
8
58
43946
98291
46511
98067
48920
97829
8
59
48090
98388
46558
98063
48959
97885
1
60
44034
96284
46594
98060
48996
97821
0
#
CoRlne
Sine
Cosine
Sine
Cosine
Sine
f
74«
78<»
720
46i
TABLE 11.— LOGARITHMIC SINES AND COSINES.
/
IS'*
!»•
20O
/
Sine
Cosine
Bine
Coeine
Bine
Cosine
0
9.4899R
9.97821
9.61864
9.97507
9.58406
9.97299
60
1
49037
97817
51801
97563
68440
97294
69
2
49076
97812
51838
97658
68475
97889
66
8
49113
97808
61374
97554
68609
97886
67
4
49153
97804
61411
97550
68644
97280
66
6
49192
97800
61447
97545
68678
66«
6
49231
97796
61484
97541
68618
97871
64
7
49269
97792
51520
97586
68647
97866
68
8
49308
97788
61557
97582
68682
97268
68
9
49847
97784
51598
97528
68716
9Rtt7
51
10
9.49885
9.97779
9.61689
9.97K8
9.58751
9.97868
60
11
49424
97776
61666
97519
58786
97248
49
n
49462
97771
61702
97516
63819
97848
48
18
49500
97767
51788
97510
63854
97888
47
14
49539
97763
51774
97506
53888
vnoi
46
15
49577
97759
51811
97601
53922
97829
46
16
49615
97754
51847
97497
63967
97884
44
17
49654
97750
61883
97498
68991
97220
48
18
49692
97746
61019
97488
54085
97216
48
19
49730
97742
61966
97484
f059
9?210
41
20
9.49768
9.97738
9.51991
9.97479
9.64098
9.97806
40
21
49806
97784
62027
97476
54127
9?201
89
22
49844
9r729
52063
97470
64161
97196
88
23
49882
97725
62099
97466
54196
97192
87
24
499*^0
97721
52135
97461
64229
97187
86
25
49958
97717
52171
97457
54288
97188
85
26
40996
97718
52207
97458
54297
97178
84
27
50084
9T708
62242
97448
64881
9n78
88
28
60072
97704
62278
97444
54365
97168
88
29
50110
97700
52314
97489
54399
97168
81
80
9.50148
9.97696
9.52350
9.97435
9.54488
9.97169
80
81
50185
97691
52385
97480
54466
97154
89
82
60228
97687
62421
97426
54500
97149
28
83
50261
97688
524.56
97421
64584
97145
87
34
50298
97679
52498
97417
54567
97140
86
85
50836
97674
52527
97412
54601
97186
85
86
50^74
97670
525G3
97408
64685
97180
84
87
50411
97666
52598
97403
54668
97126
88
88
50449
97662
52634
97899
54702
97121
88
89
50486
97657
52669
97394
54736
97116
81
40
9.50523
9.97658
9.52706
9.97890
9.54769
9.97111
80
41
60561
97649
5t.'740
97:«5
54802
97107
42
50599
97645
527T5
97381
64836
97108
48
50C35
97640
52811
97876
54869
97097
44
50673
97636
52846
97372
54908
97098
45
60710
97632
5'J881
97867
54936
97087
46
50747
97628
52916
97368
64960
97088
47
50784
97623
62951
97858
65008
97078
48
508J1
97619
52986
97358
65036
97078
49
50a58
97615
53021
97349
66060
97068
SO
9.50896
9.97610
9.68056
9.97344
9.55108
9.97068
51
509-3;3
97606
53092
97340
65186
97059
52
60970
97602
63126
97835
55169
97054
58
51007
97597
53161
97881
65208
97049
54
510^
97598
58196
97826
65285
97044
56
51080
97589
63231
97828
65.?68
97039
56
51117
97584
53266
97317
55301
97086
57
51154
97580
53301
97312
5.5384
97080
58
61191
97576
53.336
97808
55367
97025
69
51227
97571
63370
97803
55400
97020
60
51264
97567
53405
97299
56433
9J016
f
Ck>8lne
Sine
Cosine
Sine
Cosine
Sine
i
710
70*
«9»
462
TABLE n.— LOGARITHMIC SINES AND COSINES.
/
«!•
220
w 1
§
Sine
CkMine
Sine
Cosine
Sine
Cosine
0
9.56488
9.97016
9.57368
9.96717
9.59168
9.96408
00
1
OOvOO
97010
67889
96711
59218
96897
60
s
66499
97005
67490
96706
69247
96898
68
8
666ftS
97001
67451
96701
59877
968^7
57
4
66564
96906
67488
96696
69807
96881
66
6
65697
96991
67514
96691
59386
96876
65
6
56680
96986
67546
96686
5986«
96870
54
7
56668
96961
57576
96681
59896
96865
53
8
56695
96976
96676
59485
96860
62
9
66788
96971
67688
96670
69456
96854
61
10
9.65781
9.96966
9.57660
9.96666
9.59484
9.96849
60
11
56798
9G008
67700
96660
59514
96848
49
18
6o8<!6
96957
67731
96656
50548
96336
48
18
66858
96958
67763
96650
69578
96883
47
14
66691
96947
57798
96646
69608
96827
46
16
55988
96948
57884
96640
69682
96882
45
16
65956
96987
67856
96684
69661
96316
44
17
55968
96988
67885
96629
69690
96311
48
18
56081
96987
67916
96684
69780
96806
48
19
66068
96928
67947
96619
60749
96800
41
90
9.56065
9.96917
9.67978
9.96614
9.69778
9.96294
40
81
66118
96918
68008
96608
69808
96269
89
83
66160
9(3907
58089
96608
50687
96284
38
88
66188
96908
68070
96698
59666
96278
37
84
56215
96898
68101
96596
59896
96273
86
85
66•^47
96898
58181
96588
50924
96267
86
86
66879
96888
68168
96588
69954
96262
84
87
56311
96888
58198
96577
69088
96256
83
88
96878
68288
96578
60018
96251
83
89
56376
96878
58258
96567
60041
96246
81
80
9.60408
9.96868
9.58884
9.96568
9.60070
9.06840
80
81
56440
96863
58814
96666
60099
06234
29
38
66478
96858
58845
96561
60126
96829
88
83
66504
96S53
68376
90546
60157
9e8;'8
27
84
56686
96848
58406
96541
60186
9«UI8
S6
85
66566
96848
68486
96585
60815
96818
85
86
KitKOQ
96838
58467
96530
60844
96207
84
87
56631
96838
68497
96525
60278
96801
83
88
66668
96828
58527
96580
60808
96196
98
89
66695
96828
58557
96514
60831
96190
81
40
9.56727
9.96818
9.58588
9.96509
9.60359
9.96186
80
41
56759
96818
68618
96504.
ws^
96179
43
66790
96808
58648
96496
60417
96174
48
56882
96803
68678
96498
60446
96168
44
56854
96708
68709
96486
60474
96188
45
56886
96798
68739
96488
60508
96157
46
56917
96788
68769
96477
60638
96151
47
56949
96788
58799
96478
60561
96146
48
56980
96778
58889
96467
60589
96140
49
67013
96778
68859
96461
60618
96185
60
9.57044
9.96767
9.58889
9.96456
9.60646
9.96189
61
57075
96768
58019
96451
60676
96188
68
57107
96757
68949
96445
60704
96118
58
67188
96758
58979
96440
60788
96118
54
67169
96747
69009
96435
60761"
96107
66
57201
96748
59OT9
96489
60r89
96101
66
67-i82
i»787
59069
96484
60618
96095
57
57264
9G738
59098
96419
60646
96090
58
67295
96727
69188
96418
60875
960H4
50
67826
96728
59156
96406
60908
96079
60
67858
96717
59188
96403
60981
96078
i
Cosine
Sine
Cosine
Sine
Cosine
Sine
»
68*
«?•
W
463
TALBE n.— LOGARITHMIC SINES AND COSINES.
/
24*
26*
«6»
/
Sine
Cosine
Sine
Cosine
Sine
Coffine
9.60981
9.96078
9.62695
9.96788
9.64184
9.96866
60
60900
96067
62022
95722
64210
95360
69
00988
96068
62649
95716
64286
95854
58
01010
96056
02676
95710
64262
95348
67
01045
96060
62708
95704
64288
96841
66
OlOTB
96045
62730
95696
64813
95886
65
01101
96039
62757
05692
64889
95829
54
01129
90034
62784
95686
64866
95828
68
01158
96028
62811
95680
64891
95817
52
01180
96022
62888
95674
64417
95810
51
9.01214
9.90017
9.62805
9.96668
9.64442
9.95804
60
01212
96011
02892
95663
64468
96286
49
01270
90005
62918
95657
64494
95292
48
01:298
9600O
62915
95661
64510
95286
47
01328
96091
02972
95646
64545
95279
46
01354
95988
02990
95689
64571
96278
45
01882
95962
95038
64696
95267
44
01411
95977
03052
95627
64622
95261
48
18
01488
95971
08079
95621
64647
96254
42
19
61460
95905
03100
95615
64678
96248
41
SO
9.01494
9.96900
9.63133
9.95609
9.64608
9.96242
40
21
01522
95954
08159
95608
64724
95286
89
33
01560
95948
03186
95597
64749
95229
88
28
01578
96942
68218
95591
64775
95228
87
24
01600
959.S7
63230
95585
64800
95217
86
25
01681
95981
03260
93579
64826
95211
85
20
01062
95925
68292
95578
64851
96201
34
27
61689
95920
63319
95567
64877
95196
88
28
61717
95914
68345
95561
64902
96192
82
89
61746
96008
68372
95655
64927
95185
31
80
9.61778
9.95902
9.68398
9.95549
9.64953
9.95179
80
81
61800
95897
63425
95548
64978
96178
29
32
618-J8
95891
63451
95587
65003
96167
28
83
61856
968S5
03478
95531
65029
95160
27
34
61883
95879
C3504
95525
65054
95154
26
85
61911
95873
03581
96519
65070
95148
25
86
61989
95868
08567
95518
65104
95141
24
87
61966
95862
0.3583
95507
65180
95135
28
88
61994
95830
08610
95500
65155
95189
28
89
62021
95850
63686
95494
65180
95122
21
40
9.63049
9.96844
9.68662
9.95488
9.65305
9.95116
SO
41
62076
95839
63689
95482
65280
• 95110
19
42
62104
95838
63715
95476
66255
95108
18
48
62131
95827
63741
95170
65281
95097
17
41
62159
96821
63767
95464
65806
95000
16
45
6S186
95815
63794
95458
65881
95081
15
46
62214
95810
63820
95452
65356
95078
14
47
62241
95804
63846
95446
65881
95071
18
48
62268
95706
63R72
05440
65406
95065
18
49
62296
95793
63S98
95484
65431
95059
11
50
9.62328
9.95780
9.03924
9.95427
9.65456
9.95058
10
51
62850
95780
68950
95421
65481
95046
9
52
62877
95775
63976
95416
65506
05039
8
58
62405
96709
04002
95409
ftWtt
95033
7
54
62432
95757
64028
95408
65566
95027
6
55
62459
04054
95897
65580
95020
5
60
62486
95751
64060
95391
06605
96014
4
67
62513
95745
04100
95384
06680
95007
8
68
62541
95739
04182
95378
06056
95001
2
59
012568
95738
04158
95872
66G80
94995
1
00
62596
04181
95866
65705
94968
0
r
Cosine
Sine
Cosine
Sine
Cosine
Sine
/
66*
«4*
68*
464
TABLE n.— LOGARITHMIC SINES AND COSINES.
/
«7«
28«»
290
i
Sine
Cosine
Sine
Cosine
Sine
Cosine
0
9.66705
9.94968
9.6n61
9.94598
9.68557
9.94188
60
1
65789
94988
67186
94687
68580
94176
60
i
66764
94975
67208
94580
68608
94168
58
8
66779
94969
67888
94573
68625
94161
57
4
66804
94968
678S6
M567
68648
94164
66
5
65888
94956
67280
94560
68671
94147
66
6
65858
94949
67808
94568
68C94
94140
64
7
66878
91948
67887
94546
68716
94133
68
8
65008
94936
67850
94540
68739
94186
68
9
65987
94980
67874
94588
68768
94119
51
10,
9.66068
9.94988
9.67398
9.94526
9.68784
9.94118
60
11
66976
94917
67421
94519
68807
94106
40
12
66001
94911
67445
94513
68829
94098
48
;8
66085
94904
67468
U4506
68852
iM090
47
14
660S0
94898
67492
94499
66875
94063
46
16
66075
94891
67516
94493
68897
94076
45
16
66099
94885
67539
94485
68920
94069
44
17
66184
94878
67562
94479
68948
94068
48
18
66148
94871
67586
94472
68965
94055
48
19
66173
04865
67609
94465
68967
94046
41
80
9.66197
9.94868
9.67688
9.94458
9.69010
9.94041
40
21
66881
94858
67656
94451
94034
89
82
66846
94845
67680
94445
600r>5
94027
38
88
66270
94889
67708
94438
69077
94080
87
84
66895
94882
67726
94431
69100
94018
86
85
66819
94826
67750
91484
69128
94006
85
86
66348
94819
67778
94417
69144
98996
84
87
66868
94818
67796
94410
60167
93991
83
88
66898
94806
67KJ0
94404
69189
93984
82
89
66416
94799
67848
94397
69818
98977
81
80
9.66441
9.94798
9.67866
9.94890
9.69284
9.98970
80
81
66405
94786
67890
94883
69866
98968
89
88
66489
94780
67918
94376
60879
98956
26
83
66518
94778
67986
94369
69801
98948
27
84
66587
94767
67959
94868
69823
93941
86
85
6G568
94760
67982
94355
69346
93984
85
86
66586
94768
68006
94349
69368
03987
84
87
66610
94747
68029
94343
69390
98980
88
88
66684
94740
68052
94385
69418
98918
88
89
66668
94734
68075
94828
69434
93906
81
40
9.66688
9.94787
9.68096
9.94821
9.69456
9.93898
80
41
66706
• 94720
68121
94314
69479
98891
19
48
66781
94714
68144
94307
60ri01
93884
18
43
66765
94707
68167
94300
69628
98876
17
44
66T79
94700
68190
94293
69546
<93869
16
45
66803
94694
68218
94286
69567
9S862
15
46
66887
94687
68237
94279
60589
93856
14
47
66851
94680
68260
94278
69611
93847
18
48
66875
94674
68283
94266
69638
98840
18
49
66899
94667
68805
94269
69055
98838
11
60
9.66988
9.94660
9.68828
9.94258
9.69677
9.93886
10
61
66946
94654
68S51
94245
, 69699
93819
9
63
66970
94647
68874
94238
69781
98811
6
63
66994
94640
68397
94231
69748
93804
7
54
67018
94684
66420
94284
69766
98797
6
55
67048
94627
68448
94217
69787
93789
6
66
67066
94620
68466
94210
69809
93788
4
67
67090
94614
68489
94208
69881
98776
8
68
67113
94607
68518
94196
69858
98768
8
69
67187
94600
68534
94189
69876
93760
1
60
67161
94598
68567
94188
69897
98758
0
#
Cosine
Sine
Cosine
Sine
Cosine
Sine
/
«ao
6P
60»
465
TABLE n.— LOGARITHMIC SINES AND COSINES.
f
$0<»
81»
82«»
/
Sine
Cosine
Sine
Cosine
Sine
Cosine
0
9.69697
9.98758
9.71184
9.98807
9.79421
9.92842
60
1
69919
9S746
71205
86299
72441
92884
69
8
69941
98788
71226
98991
72481
92826
68
8
69968
98731
71247
93284
72482
92818
67
4
69964
03794
71268
98276
72502
92810
56
5
70006
93717
71289
98269
72529
92803
65
6
98700
71810
98261
72542
92795
54
7
70090
93709
71881
98258
72fi62
9-2787
68
8
70072
98696
71852
98946
72588
92779
69
9
70008
93687
71878
93238
72609
92771
61
10
9.70115
9.93680
9.71398
9.98280
9.72622
9.92768
60
11
70187
93673
TI414
93228
72648
92765
49
18
70159
98665
71486
98216
72668
92747
48
18
70180
93658
71456
93207
72883
92739
47
14
70902
98650
71477
98-200
T/m
92781
46
16
70224
93648
71498
98192
72798
92728
46
18
70245
96686
71619
98184
72748
92715
44
17
70267
93698
71589
98177
72768
92707
48
18
70288
93621
71660
98169
72788
92G99
49
19
70810
93614
71581
98161
72808
92691
41
90
9.70382
9.98606
9.71608
9.98154
9.72828
9.92683
40
21
70858
08599
71629
98146
72848
92676
39
82
70875
98591
71648
93138
72868
92667
88
28
70896
98584
71664
93181
72888
92659
87
24
70418
98577
71686
98128
79902
9-2651
36
25
70489
98569
71706
93U5
72922
99648
.35
26
70461
93.569
71726
98108
72942
92685
84
97
70489
98554
71747
96100
72962
92697
88
28
70604
98547
71767
98099
72988
92619
82
29
70525
96589
71788
98084
78002
99611
81
80
9.70547
9.93588
9.71809
9.98077
9.78088
9.92608
80
81
70508
98595
71899
98069
78041
92595
99
82
70690
98517
71850
98061
78061
92587
26
88
70611
93510
71870
93058
78081
92579
97
84
70638
03509
71891
98046
78101
92571
26
85
70654
98496
71911
1»8088
78121
92568
85
86
70675
98487
71939
98080
78140
92555
94
87
70697
98480
71952
93029
78160
92646
98
88
70718
98479
71978
93014
73180
22
89
70739
93465
71994
98007
7820O
99580
91
40
9.70761
9.93457
9.72014
9.92999
9.7S819
9.99529
80
41
70782
93450
72034
92991
78289
92514
19
42
70808
98442
72055
92088
78269
92506
18
48
70624
98436
78076
92976
78278
92496
17
44
70846
98427
72096
92968
78298
92490
16
45
70867
93420
72116
92960
78318
09488
16
46
70888
98412
721.37
92959
78337
99478
14
47
70909
93405
72157
92944
78a'i7
92465
18
48
70931
93397
72m
92986
78877
99457
19
49
70939
96390
72198
92929
73896
99449
11
60
9.70978
9.93882
9.72918
9.92921
9.78416
9.99441
10
61
70994
93375
72288
92918
78485
92438
«
52
71015
98367
72259
92906
78455
92425
8
58
71036
93360
72279
92897
78474
. 92416
7
64
71068
93352
72299
92889
78494
92408
6
56
71079
0^344
72.320
92881
78518
92400
6
56
71100
M387
72340
92874
73588
92399
4
67
71121
98329
72860
92866
78562
92884
8
66
71149
98392
72381
9-2858
78579
92876
9
69
71168
93314
72401
92850
78691
92367
1
60
71184
93807
72421
92842
78611
9-2869
0
/
Cosine
Sine
Cosine
Sine
Cosine
Sine
/
69«
68«
67»
466
TABLE II.— LOGARl'l'HMIC SINES AND COSINES.
/
88»
S4*
S6- 1
/
Sine
Cosine
Sine
Cosine
Sine
Cosine
0
9.78611
9.98869
9.74766
9.91857
9.76859
9.91886
60
1
78680
98851
74776
91849
75877
91828
69
3
78650
93848
74794
91840
76806
91319
68
8
78669
98885
74813
91882
75918
91810
67
4
78689
98888
74881
91828
75931
91801
56
6
92818
74860
91815
75949
91898
66
6
78727
92810
74868
91806
75967
91288
54
7
78747
98808
74887
91798
75985
91874
58
8
78766
92298
74906
91789
76008
91266
62
9
78T85
92285
74924
91781
76021
91867
61
10
9.78805
9.92277
9.74948
9.91772
9.76039
9.91248
60
11
78824
92269
74961
91763
76057
91289
49
18
73848
92260
74960
91756
76076
91280
48
18
78868
92268
74999
91746
76098
91881
47
14
^T«88
98844
76017
91788
76111
91818
46
16
78901
92286
7S086
91729
76129
91208
45
16
78921
92227
75054
91780
76146
91194
44
17
78940
92219
75078
91712
76164
91186
48
18
78959
92811
75091
91703
76188
91176
48
19
78978
98802
75110
91695
76200
91167
41
20
9.78997
9.92194
9.75188
9.9ir86
9.76218
9.91158
40
81
74017
92186
75147
91677
76286
91149
89
28
74086
92177
75166
91669
76268
91141
88
8S
74065
98169
75184
91660
76271
91188
87
24
74074
92161
76208
91651
76289
91128
86
25
74098
92162
75221
91648
76307
91114
35
80
74118
92144
76289
91684
76324
91105
84
27
74182
92186
76258
91625
76342
91096
38
28
74151
92127
75276
91617
76860
91087
82
29
74170
92119
76294
91608
76878
91078
31
90
9.74189
9.92111
9.76818
9.91509
9.76895
9.91069
80
81
74208
92102
76381
91591
76418
91060
89
88
74227
92094
76860
91582
76481
91061
88
88
74246
92086
75368
91578
76448
91048
27
84
'i4885
92077
75886
91566
76466
91088
86
85
74284
92069
75405
91556
76484
91028
85
86
74808
92060
75423
91547
78501
91014
84
8/
74828
98062
75441
915S8
76619
91006
28
88
74841
92044
75459
91680
76537
90996
28
89
.74360
98086
76478
91521
76554
90987
21
40
9.74879
9.98087
9.75496
9.91512
9.76572
9.90978
90
41
74898
92018
75514
91504
76590
00969
19
42
74417
92010
76583
91495
76607
90960
48
74486
92002
75561
91486
76625
90961
44
74455
91998
75569
91477
76642
90949
45
74474
91985
75587
91469
76660
90988
46
74496
91976
76605
91460
76677
90924
47
74518
91968
76624
91451
76696
90916
48
74581
91959
76642
91442
76718
90906
49
74549
91961
76660
91488
V6780
90696
60
9.74!M»
9.91943
9.75678
9.91425
9.76747
9.90887
61
74587
91964
75696
91416
76766
90878
68
74606
91926
75714
91407
767B8
90809
58
74625
91917
91898
76800
90860
64
74644
91908
75751
91389
76817
00861
66
74668
91900
75769
91881
76836
90849
66
74681
91891
75787
91872
768.'$8
90888
67
74700
91888
75805
91368
76870
90828
68
74719
91874
75828
91354
76887
90814
69
74787
91866
75841
91346
76904
00806
60
74756
91867
75860
91836
76028
90796
9
Ck)8iiie
Sine
Cosine
Sine
Cosine
Sine
/
66*
fiS'
ft4*
467
TABLE II.— LOGARITHMIC SINES AND COSINES.
#
se* '
S?'
88«
/
Gttne
Cosine
Sine
Cosine
Sine
Cosine
0
9.76099
9.90796
9.77946
9.90985
9.780S4
9.69658
60
1
76989
90787
77968
90996
78950
89648
59
»
76957
90777
7;t«o
90916
78967
89638
58
8
76974
90768
77997
90206
78983
89624
57
4
76091
90758
78018
90197
78999
89614
56
5
77009
90750
78000
90187
79015
89604
55
8
770«
90741
78047
90178
7O081
80594
54
7
77048
00731
78068
90168
79047
89584
58
8
77061
90799
78080
• 90159
79068
89574
59
9
77078
90718
78097
90149
79079
80564
61
10
9.77096
9.90704
9.78118
9.90130
9.79095
9.89564
50
11
77119
90694
78180
90180
79111
89544
49
18
77130
90685
78147
90190
79198
89684
48
18
77147
90676
78183
90111
79144
89524
47
14
77164
90667
78180
90101
79160
89514
46
15
77181
90657
78197
90091
79176
89604
46
16
77199
90648
78218
90089
79199
89495
44
17
77216
90689
78230
90079
79906
89486
48
18
77988
90630
78246
90068
79224
89475
49
19
77250
90620
7B263
90058
79240
89465
41
90
9.77968
9.90611
9.7S280
9.90048
90084
9.79956
9.89456
40
91
77285
90609
78296
79279
89445
80
99
77302
90592
78813
90024
79988
89485
88
28
77319
90583
78829
90014
79804
89425
87
94
77386
90574
78340
90005
70819
89415
86
95
77858
90565
78869
89995
79385
89405
85
98
77870
90555
78379
89965
79851
80896
84
97
77387
90546
78396
89976
79867
89386
88
98
77406
90587
78419
89966
79388
89375
89
99
77429
90627
78428
80950
79899
89364
81
80
9.77489
9.90618
9.78445
9.89947
9.79415
9.89854
80
81
77456
90509
78461
89937
79481
89844
99
89
77478
90499
78478
89927
79447
89334
28
88
77490
90490
78494
89918
79468
89824
97
81
77507
90480
78510
89906
79478
89314
96
85
77524
90471
78527
89808
79494
89304
95
86
77541
90469
78548
ovooo
79510
89294
94
87
77558
90152
78560
80879
79596
89284
93
88
77575
90448
78576
80869
79549
89274
29
89
77598
90434
78599
89859
79558
89964
91
40
9.77609
9.90424
9.78609
9.80849
9.79678
9.89254
90
41
77696
90415
78626
89840
79580
89944
19
49
77648
90406
78649
80830
89288
18
48
T?«60
90896
78658
898,»0
78621
89228
17
41
77677
90386
78674
89810
79686
89918
16
45
77694
90377
78691
89801
79659
80908
16
46
77711
9030S
78707
89791
79668
89198
14
47
77728
90358
78728
89781
79684
89188
18
48
77744
90349
78789
89771
79699
89178
19
49
77761
90389
78756
89761
79715
89169
11
SO
9.77778
9.90830
9.78772
9.89759
9.79731
9.80159
10
51
77795
90820
78788
89749
79746
89149
9
59
77819
90311
78805
89739
79769
89139
8
58
7?»J9
90301
78821
80729
79778
89129
7
54
77846
90-.^2
78837
89719
79798
89119
6
65
77889
90282
78853
89709
79809
89101
6
56
77879
90278
7S8C9
89693
79825
89091
4
57
77896
90963
78886
89688
79640
80061
8
58
77918
90254
78002
89078
79856
89071
9
50
r79:»
90244
78918
89663
79872
89060
1
(SO
T7946
90235
78984
89653
79887
89050
0
/
Ckwlne
Sine
Cosine
Sine
Cosine
Sine
/
680
62«
51- 1
468
TABLE 11.— LOGARITHMIC SINES AND COSINES.
/
89*
40«
41»
/
Sine
Cosine
Sine
Oodne
Sine
Cosine
0
9.79887
9.89060
9.80007
9.88426
9.81694
••2X32
87767
60
1
70908
89040
80882
88415
81709
69
2
79918
89060
80887
88404
81728
87756
68
8
79934
89020
80862
86894
81788
87745
67
4
79950
89009
80667
88888
81782
87784
66
5
79966
88999
80882
88872
81767
87788
66
6
79981
88989
80897
88808
81781
87718
54
7
79996
88078
80912
88851
81796
erroi
68
8
80012
88968
80927
88840
81810
87090
68
9
800(27
QQQeO
OOWUO
80942
86880
81826
87679
51
10
9.80048
9.88948
9.80967
9.88819
9.81889
9.87668
60
11
80068
88067
80972
88808
81664
87667
49
12
80074
88927
80987
88208
81868
87646
48
18
80089
88917
81002
8B287
81882
87686
47
14
80106
88906
81017
88276
81897
87624
46
15
80120
88806
81082
88266
81911
87818
45
16
80186
88886
81047
88266
81926
87601
44
17
80161
88875
81061
88244
81940
87690
48
18
80166
88865
81076
88284
81956
87879
42
19
80182
88855
81091
88S28
61909
87688
41
80
9.80197
9.88844
9.81106
9.88212
9.81968
9.8;i»7
40
21
80218
88884
81121
88201
81998
87646
80
22
80228
88824
81186
88191
88019
87586
88
28
80244
88818
81161
88180
8S026
87584
87
24
80250
8S808
81166
88169
82041
87518
86
25
80274
88798
81180
88168
82066
87501
86
26
80290
88782
81195
88148
82069
87490
84
27
80805
81210
88187
880S4
87479
88
28
80820
88761
81225
88126
88006
87468
88
29
80386
88751
81240
88115
82112
87457
81
80
9.80851
9.88741
9.81254
9.88105
9.82186
9.87446
80
81
80866
88780
81269
88094
82141
87484
89
82
80882
88720
81284
88088
82166
87488
88
88
80897
88709
81299
88072
82169
87412
27
84
80412
88699
81314
88061
82184
87401
86
85
80428
88688
81888
88051
82198
87880
86
86
80448
88678
81848
88040
82212
24
87
80458
88668
81858
68089
82226
87867
28
88
80478
88657
81872
88018
82240
87866
22
89
80489
88647
81887
88007
82256
87846
21
40
9.80604
9.88686
9.81402
9.87996
9.88269
9.87884
90
41
80619
88626
81417
87986
82288
8788S
19
42
80534
88616
81481
87976
88297
87811
18
48
80660
88605
81446
87964
88811
87800
17
44
80565
88594
81461
87958
88820
87888
16
45
80580
88584
81475
87942
82840
87877
15
46
80595
8S578
81490
87931
82854
87866
14
47
80610
88663
81606
87920
82868
87855
18
48
60625
88562
81519
»2882
67248
12
49
80641
88542
81584
87898
88896
87282
11
60
9.80656
9.88581
9.81549
9.87887
9.82410
0.87S21
10
61
80671
88521
81663
87877
82424
9
52
80686
88510
81578
87866
82489
87196
8
58
80701
88499
81592
87866
82468
87187.
7
64
80716
88489
81607
87844
82467
87176'
6
55
80781
88478
81632
87838
88481
87164
6
66
80746
88468
81686
87882
82496
67168
4
57
80762
88457
81651
87811
88609
67141
8
58
80777
88447
81666
67800
68523
87180
2
50
80792
88486
81680
87789
88587
87119
1
60
80807
88425
81694
82551
87107
0
/
Cosine
Sine
Cosine
Sine
Cosine
Sine
/
i
W
490
48*
469
TABLE II.— LOGARITHMIC SINES AND COSINES.
/
4t*
48-
44«»
/
Sine
Coeiue
Sine
Coaine
Sine
Cowne
9.82561
9.87107
9.88878
9.86418
9.84177
9.85693
60
82566
87096
83392
86401
84190
85681
59
62679
870S5
88405
8G880
64208
65669
58
8^596
8T073
88419
86877
84216
a5657
67
82607
87062
83482
86866
84229
86645
66
82621
87050
83446
86354
64242
66682
55
82685
87039
83469
86842
64255
85020
54
82649
87028
83478
8G330
84260
65606
58
82668
87016
83486
80318
84282
65506
52
82677
87005
88500
86806
84206
65588
51
9.82691
9.86993
9.88518
9.86295
9.84306
9.65571
60
82T05
86982
88527
86283
84381
85569
49
82719
86970
88540
86271
84384
85647
46
82788
86059
83554
86250
84347
65584
47
82747
86947
88567
86247
64860
65622
46
82r61
86936
88381
86235
64878
85610
46
82775
86924
8S594
86228
84885
85497
44
82788
86918
83608
86211
84396
65485
48
18
82802
86902
83621
86200
64411
86478
48
19
82816
86890
83684
86186
64424
85460
41
90
9.82880
9.86879
9.88648
9.86176
9.84487
0.86448
40
21
8-2814
86867
88661
86164
84450
85436
89
S2
828S6
86855
83674
86152
84468
65428
86
fa
82872
86844
83688
86140
84476
85411
87
S4
82885
86832
88701
86128
64489
65899
86
25
82899
86821
83715
86116
84508
65886
.35
28
8;>918
86809
83728
86104
64515
85874
84
27
82927
86798
88741
84526
85361
83
28
82941
86786
83755
86080
84540
65349
82
29
82956
86775
88768
86008
64553
86837
31
80
9.»^y68
9.86768
9.88781
9.86056
9.84566
9.86384
30
81
8J982
867S2
83795
86044
64579
86312
29
88
82996
86740
83806
860ffi
64592
65299
88
88
sw\o
86728
88821
86020
64605
65287
27
84
83028
86717
83884
86008
84618
65274
26
«i
88037
86705
83848
85996
84680
65262
85
86
83051
86694
88861
86964
64643
65250
24
87
8:^063
86682
88874
85972
64656
85237
28
88
R3078
8C670
83887
86060
64660
85225
22
89
88092
86659
83901
85948
84682
85212
81
40
9.83106
9.86647
9.83914
9.85966
9.84694
9.a5200
80
41
88120
86635
88927
85924
64707
65187
19
42
83133
86624
88940
85912
64?20
85175
16
48
83147
86612
83954
85900
64788
86162
17
44
83161
86600
83967
a5888
84745
85150
16
45
83174
86589
83960
85876
84758
85187
16
46
83i88
80577
a3TO3
85881
84771
85125
14
47
88202
86566
84006
85861
84784
65112
18
48
88il5
86564
g405JO
85830
64796
65100
12
49
832;!9
86542
84a»
86827
64809
85087
11
80
9.83242
9.80530
9.84046
9.85815
9.R4R22
9.86074
10
61
83256
86518
84059
85H03
84835
85002
9
62
83v>70
86507
84072
85791
84847
A5(»49
8
58
83-283
86495
840«S
85779
84860
86037
7
54
83297
86488
84098
86766
64873
86024
6
55
83:^10
86472
84112
857.54
84885
85012
5
56
83324
86460
84125
86742
64S98
64999
4
57
88338
8644S
84188
85730
84011
64986
8
58
88ail
86436
84151
85718
84928
64974
8
59
83865
86425
84164
85706
81936
84961
1
60
83378
86413
84177
85698
64949
64949
0
/
Ooeine
Sine
C!o8iiie
Slne_
Cosine
Sine
/
470
46«
46*
470
TABLE m
.—LOG.
TANGENTS AND COTANGENTS.
/
©•
!•
9
•
/
Tta
CJotan
Tan
Ootan
Tan
Cotan
— 00
00
8.94199
11.75808
8.54808
11.45699
60
6.46878
18.58637
94910
76090
54669
45881
59
76476
28584
95616
74884
66097
44973
68
94066
05916
96819
73668
66882
44618
57
7.06579
12.93421
96996
78004
66784
44966
66
16270
88780
97669
72331
66063
48917
65
84188
75819
98839
71668
66429
48571
54
80688
69118
28986
71014
66778
48927
53
86688
68818
99629
70871
6ni4
49886
68
41T97
58208
80263
69737
67469
49548
61
7.46878
19.58687
8.80688
11.69119
8.67788
11.49919
60
60618
49488
81505
68496
68121
418W
49
64S91
45709
S2119
67888
66461
41649
48
67767
42283
82711
67289
58779
41991
47
60986
89014
88302
6669R
59106
40696
46
68988
86018
83886
66114
60428
40679
45
66786
88215
84461
65639
60749
40251
44
09418
80582
85029
64971
60068
89989
48
71900
28100
86590
64410
60884
89616
49
74248
25758
86148
68857
60696
89809
41
90
7.76476
18.23584
8.86689
11.68311
8.61000
11.88991
40
81
78595
21405
87220
62771
61819
88681
89
8S
60616
19885
87762
estm
61696
88874
88
88
88546
17454
88288
61711
61931
88069
87
84
84394
15606
3b809
61191
62884
87766
86
85
86167
13838
89328
60677
62535
87466
85
86
»7871
12129
89832
60168
62834
87166
84
87
80510
10490
40884
69666
63131
86809
88
88
01089
08911
40880
69170
63426
86674
88
89
98618
07887
41321
58679
68718
86889
81
80
7.94066
19.05914
8.41807
11.68198
8.64009
11.86991
80
81
95510
04490
42287
67718
64998
85708
99
38
96889
08111
42768
57288
64586
85415
98
83
08225
01775
48239
66768
64870
85180
97
84
99588
00478
48696
66304
65154
84846
96
85
8.00781
11.99219
44166
66844
65485
84665
95
88
0^004
97996
44611
65389
66715
84285
94
87
08194
96S06
45061
54939
65098
84007
28
88
04353
96647
46607
54493
66869
83781
89
89
05481
94510
46948
54053
66548
88457
91
40
8.06581
11.93419
8.46885
11.63615
8.66816
11.88184
90
41
07658
92847
46817
53188
67087
88918
19
4S
08700
91300
47246
52756
67856
82644
18
48
09728
90278
47669
52331
67624
82376
17
44
10720
80280
48089
51911
67890
82110
16
45
11696
88304
48505
51495
68154
81846
15
46
18651
87340
48917
51C83
68417
81583
14
47
13586
86415
49335
60675
68678
81322
18
48
14500
85500
49729
60271
68038
81062
19
49
16896
84605
60180
49870
69196
80804
11
50
e.ioifl^
11.88787
8.80627
11.49473
8.69468
11.30547
10
61
17188
82867
50920
49080
69708
80299
9
58
17976
82024
61310
48690
69969
80088
8
68
18804
81196
61696
48304
70214
99766
7
54
19616
80384
62079
47921
70465
29685
6
66
80418
7».')87
62459
47541
70714
99286
5
66
21195
78805
628:»
47165
70968
99088
4
57
21964
78036
58208
46792
71208
98799
8
68
22720
77280
53578
46422
71458
2f547
9
60
28462
76538
63945
46055
71697
28808
1
60
24192
76808
64308
45692
71940
28060
0
9
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
89«
88»
S7-
471
TABLE III.— LOG. TANGENTS AND COTANGENTS.
/
8-
!•
6»
/
Tan
Ootan
Tan
Cotan
Tan
Cotan
0
8.71940
11.88060
8.84464
11.15586
8.94195
11.06805
60
1
78181
87819
84646
15854
94840
05660
59
a
78480
87580
84826
15174
94485
06515
58
8
78659
87841
86006
14994
94680
05870
57
4
78896
87104
85185
14815
94778
05327
66
5
73133
86668
85868
14637
94917
05063
55
8
78866
86634
85540
14460
95060
04940
54
7
73600
26400
85717
14283
95808
04798
68
8
78832
26168
85898
14107
96844
04656
63
9
74068
85987
86060
139S1
95486
04514
51
10
8.74898
11.85708
8.86843
11.18757
8.95697
11.04378
60
11
74521
85479
86417
13588
95767
04838
49
12
74748
85853
86591
18409
95906
04098
48
18
74974
26026
86763
18237
96047
08958
47
14
75199
84801
86935
13066
96187
08818
46
16
76428
24577
87106
12894
96385
03075
45
16
75645
84365
87877
12728
96464
03536
44
17
76867
84183
87447
12568
96608
03896
48
18
76067
83913
87616
12384
96739
08261
48
19
76806
28694
87785
12215
96877
08128
41
20
8.76525
11.83475
8.87968
11.12047
6.97018
11.08967
40
31
76748
83258
88120
11880
97160
02860
88
88
76968
83043
88287
11718
97285
02715
88
88
77178
82827
88458
11547
97421
02579
87
84
77887
22613
88618
11383
97556
02444
86
85
77600
38100
88783
11217
97691
08309
85
80
77811
88189
88948
11053
97825
08175
84
87
78088
21978
89111
10689
97969
02041
88
88
78238
81768
89274
107^6
96092
01906
82
29
78441
81559
89437
10668
98225
01775
81
SO
8.78649
11.81351
8.89698
11.10403
8.96358
11.01648
80
81
78866
21145
89700
10240
96490
01610
89
88
79061
20939
89980
10060
96628
01878
88
83
79866
80784
90080
09920
98753
01847
37
84
79470
80530
90240
09760
98884
01116
86
85
79678
80327
90399
09601
99015
00965
85
86
79875
80125
90657
09443
99145
00855
84
87
80076
19924
90716
09^85
99".'75
00725
88
88
808n
19728
90872
09128
99405
00595
88
89
80476
19584
910-29
08971
99634
00466
81
40
8.80674
11.19386
8.91185
11.08815
6.99663
11.00888
90
41
80878
19128
91340
08660
99791
00209
49
81068
18938
91495
06505
99919
00081
48
81864
18736
91650
06350
9.00040
10.99954
44
81459
18641
91803
08197
001T4
99826
45
81658
18347
91957
08043
00301
99099
46
81846
18154
92110
07890
00427
99573
47
88088
17962
92262
07788
00553
99447
48
82830
17770
92414
07686
00679
99821
49
88490
17580
92565
07485
00805
99195
60
8.88610
11.17890
8.92716
11.07884
9.00980
10.90070
51
82799
1T201
92866
07134
010)5
96945
58
82987
17018
93016
06984
01179
96881
58
83175
16825
93165
06835
01808
98697
54
83361
16638
98313
06o87
01427
98578
55
83547
16453
93468
06538
01660
98450
56
88738
16268
93609
06391
01678
98327
67
88916
16084
93756
06244
01796
96204
58
84100
15900
93903
06097
01918
98082
59
84283
15718
94049
05951
02040
07960
60
84464
15536
94195
05805
02162
97888
»
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
se*
85-
84*
472
TABLE III.— LOG. TANGENTS AND COTANGENTS.
/
«
e
7*
a*
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.02162
10.97888
9.08914
10.91086
9.14780
10.66220
60
1
021288
97717
09019
90961
148ni
85128
69
2
02404
97596
09123
90877
14963
85087
58
8
02.Vi6
97475
09227
90778
15054
84946
67
4
09645
97365
09830
90670
15146
84855
56
6
02766
97284
09484
90566
15286
64764
55
6
02886
97115
09537
90463
15827
84678
64
7
08006
96995
09640
90360
15417
84583
58
8
08124
96876
09742
90258
15506
84493
58
9
08242
96758
09845
90155
15698
84402
61
10
9.08361
10.96689
9.09W7
10.90058
9.16688
10.84812
50
11
03479
96521
J0049
89951
15777
84228
49
12
08597
96408
10150
69850
16867
64138
48
18
03714
96286
10252
89748
15966
64044
47
14
06882
96168
10663
89647
16046
63964
46
IS
08948
96052
10454
89546
16)35
63865
45
16
04065
96983
10665
80145
16284
83776
44
17
04181
95819
10656
69344
16812
83688
48
18
wa97
95703
107D6
89244
16401
88599
42
19
04418
95587
10656
69144
16488
88611
41
20
9.04528
10.g54?3
9.10956
10.89044
9.16577
10.88428
40
21
04048
95357
11056
88944
16665
63885
89
22
04758
95242
11155
88846
16758
88847
38
28
04878
95127
11254
88746
16841
88169
87
24
04987
95018
11868
88647
16928
88078
86
25
05101
94899
11452
88548
17016
82964
36
26
06214
947%
11651
88449
17108
62897
34
27
05328
94672
11649
66361
17190
82810
38
28
06441
94559
11747
882C8
17277
82723
88
29
05558
94447
11845
68155
17368
82687
31
80
9.05666
10.94384
9.11943
10.88057
9.17460
10.82560
30
81
05778
94222
12040
879G0
17536
82464
29
32
05890
94110
12188
87862
17622
82378
88
88
06003
93998
12235
87/65
17708
82298
27
84
06118
93887
12382
87668
17794
82S06
86
85
06224
93776
12428
87572
17880
82120
26
86
06335
98665
12525
67475
17965
82085
24
87
06445
93556
12621
87379
16051
81949
28
88
06556
93444
12717
87283
18186
81864
22
89
06666
93334
12818
87187
18221
61779
21
40
9.06775
10.9322S
9.12909
10.87091
9.18306
10.81694
20
41
06885
93115
13004
8C096
18891
61609
42
06994
93006
13009
8(J901
18475
61525
48
07103
92897
13194
86806
18560
81440
44
07.211
92789
13289
6G7I1
18644
81856
45
07820
92C80
13884
8G616
16728
81272
46
07428
92572
13478
86522
18812
81188
47
07536
92464
13573
86427
18806
81104
48
07648
92857
13667
863S8
18979
81021
49
07751
92249
13761
66239
19068
80987
60
9.07858
10.92142
9.13854
10.86146
9.19146
10.80K54
61
07964
92036
13948
86052
19229
80771
52
08071
91929
14041
85069
19312
80688
58
08177
91828
14134
85866
19395
80606
64
08288
91717
14227
85778
19478
80522
56
08389
91(311
14320
85680
10561
60489
56
08495
91506
14412
85568
19643
80367
57
08t>00
91400
14504
65496
J9725
80875
68
08705
91295
14597
65408
19807
80198
59
08810
91190
14688
85312
10889
80111
60
0H914
91086
14780
85220
19971
60029
§
Cotan
Tan
Cotan
Tan
Cotan
Tan
#
88«»
82*
81»
473
TABLE III.— LOG.
TANG]
ENTS AND COTANGENTS.
$
»•
10"
II*
t
Tan
Cotan
Tan
Ootan
Tan
Cotan
0
9.19971
10.80039
9.94689
10.75368
9.38866
10.71186
60
1
30068
79947
. 84706
75294
98988
71067
59
9
80184
79866
94T79
75S91
99000
71000
58
8
90916
79784
94858
751^
90067
70883
57
A
30897
79708
94936
75074
99184
70666
56
6
80878
79688
95000
75000
99801
70799
56
6
20459
79541
95078
74997
99966
70788
54
7
80540
79460
95146
74854
99335
70665
58
8
80621
79379
35919
74781
99408
70598
63
9
80701
79399
96299
74706
29466
70539
51
10
9.80788
10.79918
9.35865
10.74635
9.39585
10.70465
SO
11
80868
79188
95487
74668
39601
70899
49
18
80948
79a'W
95510
74490
90668
70889
48
18
81088
78978
95588
74418
99784
70286
47
14
31108
78806
95655
74845
90600
70200
46
15
81188
78818
96737
74278
99666
70134
45
18
91861
78789
95709
74901
99889
70068
44
17
81341
78669
95871
74199
99096
70009
48
18
31480
78580
95948
74057
30064
69936
49
19
81499
78501
96015
78965
30180
69670
41
80
9.81578
10.78499
9.96066
10.73014
9.80106
10.69606
40
81
21657
78348
96158
73649
80361
60789
38
88
81786
78964
90230
78771
80896
69674
36
88
81814
78186
96801
7S699
80891
69609
37
84
81808
78107
96873
78098
80457
69548
36
85
81971
78039
96448
73567
80539
69478
35
86
88049
77951
96614
78486
80567
69413
34
87
38187
77878
8G585
78416
30659
69348
83
88
82903
77795
96655
78845
30717
68268
83
89
83883
77717
88796
78374
80789
60218
31
80
9.89361
10.77689
9.96797
10.78906
9.80S46
10.69154
80
81
38488
77563
96867
78188
30911
69069
99
88
98516
77484
96967
78068
80975
69035
28
88
98598
77407
97008
7'2992
81040
68960
97
84
88670
77830
97078
79939
81104
66896
36
fti
98747
77858
97148
73859
31166
68839
95
86
9;»34
77176
87318
72789
81988
68767
34
87
99901
77099
97288
73719
81297
68703
33
88
33977
77038
97357
73648
31361
68689
99
88
»M)54
76946
97427
73578
81495
68575
91
40
9.38130
10.76870
9.97496
10.73504
9.31489
10.68511
30
41
88M6
76794
97566
79484
31559
68448
19
4'^
93«3
76717
97635
73366
81616
66384
18
48
88869
78641
97704
72296
81679
68321
17
44
93435
76565
97778
73387
81743
68257
16
45
98510
76490
97849
73158
81806
6S194
15
46
98586
76414
97911
79069
31870
68180
14
47
98661
76839
97960
73030
81938
68067
18
48
93787
76368
98049
71951
81906
68004
19
49
83813
76188
98117
71888
83059
67941
11
60
9.33887
10.78118
9.3R186
10.71814
9.39199
10.67H78
10
51
93963
76088
36S54
71746
83186
67815
9
53
94037
75963
88338
71677
83348
6T763
8
58
94113
75888
98891
71609
8-1811
67689
7
54
91186
75814
38459
71541
89878
67637
6
65
94361
75789
38537
71473
83430
67564
6
56
94335
75665
38595
71405
33498
67603
4
87
94410
75690
98669
71886
8SS61
67489
8
58
84484
75516
98730
71370
83623
67377
9
69
94558
75443
98796.
71909
82685
67315
1
60
34633
75368
98865
71135
32747
67253
0
/
Cot&n
Tan
Cotan
Tan
Cotan
Tan
80»
79*
780
474
TABLE in.— LOG. TANGENTS AND COTANGENTS.
f
19»
18«
140
/
Tan
Cotan
Tan
Otan
Tan
Cotan
0
9.82747
10.67258
9.36336
10.63664
9.89677
10.60828
60
1
82810
67190
86394
63C06
89781
60269
69
2
82872
67128
86459
63548
89785
60215
68
8
82983
67067
86509
63491
89838
60162
67
4
82995
67005
86566
68484
88693
60108
66
6
83057
66048
86624
68376
89945
60056
65
6
38119
66661
86681
63319
80999
60001
54
7
83160
66830
86738
63263
40062
69948
63
8
83242
66758
86795
63205
40106
69894
63
9
83303
66697
86852
68148
40159
69811
61
10
9.83865
10.66636
9.36909
10.63091
9.40212
10.59788
60
11
a^26
66574
86966
63034
40366
60784
49
18
88487
66618
87028
62077
40319
69681
48
18
88548
66459
37060
62920
40372
60628
47
14
83609
66891
87137
628C3
40425
60575
46
16
88670
66830
87193
62807
40478
69523
45
i6
83731
66269
8?250
62750
40581
69469
44
17
88792
66908
87306
62G94
40584
69416
43
18
83858
66147
87363
62637
40636
69364
42
19
83918
66067
27419
62581
40689
69811
41
90
9.88974
10.66026
9.87476
10.62524
9.40742
10.69268
40
81
84034
65966
87532
624C8
40795
69806
89
23
84095
66006
87588
62412
40647
69158
88
88
84165
66845
87644
62356
40900
69100
87
84
84216
65765
87700
62300
40062
69048
86
S5
84276
66724
87T66
62244
41005
68996
86
96
84836
65664
87812
62188
41057
68943
84
97
84806
65604
37868
62132
41109
58891
88
98
84456
65544
87924
62076
41161
58839
88
99
84516
65484
37960
62020
41814
68786
81
80
9.34576
10.65424
9.38036
10.61965
9.41266
10.58784
80
81
84636
65365
38091
61909
41818
58683
29
33
84695
65306
88147
61853
41370
68630
28
88
84756
66245
38202
61798
41422
68578
27
84
84814
65166
38257
61743
41474
68526
86
85
84874
65126
38313
61687
41636
68474
26
86
84833
66067
88368
61632
41578
68432
84
87
84999
65008
38423
61577
41629
68371
88
88
85051
64949
38479
61521
41661
68319
28
80
86111
64889
88534
61466
41733
68367
21
40
9.85170
10.64830
9.88589
10.61411
9.41784
10.68316
90
41
85229
64771
38644
61356
41836
68164
19
42
85288
64712
38G99
61301
41887
68118
18
48
85847
64663
88754
61246
419:»
68061
17
44
85406
64595
88808
61192
41990
68010
16
45
85464
64536
38868
61137
42041
67959
16
46
86523
64477
8S918
61082
42093
67907
14
47
35581
64419
88972
61028
42144
67856
18
48
85640
64360
39037
60973
42195
67805
49
85698
64303
89062
60918
42246
67754
60
9.86757
10.64248
9.89136
10.60864
9.42297
10.57708
51
85815
64185
89190
60810
42348
67652
63
85878
64127
89246
60755
42399
67601
68
85931
64069
39299
60701
43460
67550
54
35969
64011
39863
60647
42601
67499
55
36047
63953
30407
60593
42652
67448
66
36105
68895
89161
60539
42603
67397
57
36163
63837
39515
60485
42653
67347
68
86221
63779
89569
60431
42704
67296
SO
36279
€87*31
39^23
60377
427S5
5T24B
60
363-16
ekdc^fl^
3t«77
flO?t2S
42805
571flfi
'
CoUu
Tan
CoUn
T&n
(JOULD
Tan
*
37*
«•
76-
47S
TABLE ra.— LOG. TANGENTS AND COTANGENTS.
*
16»
!«•
17»
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.49805
10.57196
9.4B750
10.64860
9.48684
10.61466
60
1
42856
67144
45797
64208
48579
61421
69
2
42000
67094
46846
64156
48624
61876
68
8
42967
67048
45898
64106
48669
51831
^7
4
48007
66998
46040
64060
48714
61286
66
6
48057
56943
46987
64018
48760
61241
56
e
48108
56898
46086
63965
48804
61196
54
7
48158
66842
46088
68918
48849
61151
63
8
48206
66792
46180
68870
48894
61106
68
9
48258
66742
46177
68628
48069
61061
61
10
9.48808
10.66608
9.46884
10.68770
9.48964
10.51016
60
11
48858
66642
46271
68729
49029
60971
49
12
66598
46819
63681
49078
60927
48
18
48468
66548
46860
63634
49118
60662
47
14
48506
56498
66442
46418
68567
49168
60887
46
16
4.S558
46460
68540
49207
60798
45
10
48607
66898
46507
68493
49858
60748
44
17
48657
66843
46664
68146
49296
50704
43
18
43707
66298
46601
68889
49341
60659
42
19
48766
66244
46648
68352
49885
60616
41
20
9.48806
10.56104
9.46604
10.68306
9.49480
10.60670
40
21
48866
66145
46741
63269
49474
60586
89
e
489a5
66095
46788
63212
49519
S0481
88
28
48954
66046
46885
63166
49608
60437
87
24
44004
65996
46881
63119
49607
60893
80
25
44058
65947
46928
68072
49668
60848
86
M
44102
65896
46975
6S0S6
49690
60804
34
27
44161
&'^9
47021
62979
49740
60260
88
28
44201
66799
47068
62082
49784
60216
82
29
442Q0
65750
47114
62886
49628
60178
81
80
9.44299
10.66701
9.47160
10.62840
9.49678
10.60128
80
81
44848
66668
47207
62798
49910
60064
29
82
44397
65608
47268
62747
49960
60040
88
88
44446
65554
47299
62701
60004
49996
27
84
44495
66506
47346
62654
60048
49952
86
88
44544
56466
47892
62606
60098
49906
85
88
4459-^
65408
47488
62562
60130
49864
84
87
44641
65359
47484
62516
60180
40620
88
88
44690
65310
47680
68470
60228
49777
28
89
44788
65262
47670
62424
60267
49738
81
40
9.44787
10.66213
9.47628
10.68878.
9.50311
10.49689
80
41
44886
65164
47668
62332
60366
49646
42
44884
66116
47714
62286
60396
49602
48
44933
55067
47780
68240
60442
49558
44
44961
6S019
47806
62194
60486
49515
46
46029
64971
47a52
62146
60629
49471
46
45078
64922
47897
62106
60678
49428
47
45126
64874
47943
68057
60616
49884
48
45174
64826
47989
52011
60660
49841
49
45223
54778
48036
61066
60708
49297
60
9.45271
10.64789
9.48080
10.51920
9.60746
10.49254
51
45819
64681
4H126
61874
60780
49211
52
45367
64683
48171
61829
60633
49167
58
45416
64566
48217
61783
60876
49124
64
45463
64537
48262
51786
60919
49081
65
46511
64480
48307
61698
60962
49088
66
455.'>9
54441
48]i53
61647
51005
48995
57
45606
64^J94
48:^96
61608
61046
48952
68
46654
64:M6
4H443
61567
61092
48908
69
46702
64298
48489
61611
51185
4»«5
60
45750
64250
48534
61466
61178
48822
0
Co tan
Tan
Cotan
Tan
Cotan
Tan
.
74»
78«
72»
476
TABLE m.— LOG. TANGENTS AND COTANGENTS.
/
18*
19*
eo«»
i
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.61178
10.48882
9.53697
10.46303
9.66107
10.48883
60
1
61281
48779
58788
46268
56146
43854
59
8
61264
48736
53779
46821
66186
48816
58
8
51306
48694
53820
46180
66284
48776
57
4
51349
48651
58861
46139
66864
43786
56
6
51398
48608
53902
46098
66308
48697
55
6
51435
48565
53948
46057
56342
43658
54
7
61478
48528
53984
46016
56381
48619
58
8
51580
48480
54025
45976
56480
48580
52
9
61663
48437
54065
45935
uOw>V
48541
51
10
9.61606
10.48394
9.54106
10.45894
9.56498
10.43508
50
11
51648
48858
54147
45853
56587
43463
49
12
51691
48309
54187
46818
66576
48484
48
13
51784
48266
54828
45772
66616
48885
47
14
61776
48224
54269
45781
56654
4:1846
46
15
51819
48181
54309
45691
56693
43807
45
16
51861
48139
54350
45650
56738
43868
44
17
51903
48097
54390
45610
56771
48829
48
18
51946
48054
54431
45569
66810
48190
48
19
51988
48012
64471
45529
66849
48151
41
90
9.5S031
10.47969
9.54518
10.45488
9.56887
10.48118
40
81
52073
47927
54552
45448
66986
43074
89
88
58115
47885
54593
45407
56965
43085
38
83
62157
47843
54633
45367
67004
42996
37
84
58200
47800
54673
45827
67048
42958
86
85
52842
47768
54714
45286
57081
48919
35
86
52284
47716
54754
45246
57180
42880
84
87
52328
47674
54794
45206
57168
48848
88
88
52368
47638
54835
45165
57197
42803
88
89
58410
47590
54875
45125
67235
48765
31
80
9.52458
10.47548
9.54915
10.45085
9.57274
10.48726
30
31
62494
47506
54955
45045
57318
48688
89
38
52r>3A
47464
54995
45006
57851
48649
88
83
52578
47422
55035
44965
57889
48611
27
34
52620
47880
55075
44925
67428
48572
86
35
52661
47339
55116
44885
57466
42534
85
86
62703
47,'97
55155
44845
57504
42496
84
37
62715
47S55
55195
44805
67543
42467
88
88
52787
47:218
55235
44765
57581
48419
28
89
52889
47171
55275
44'ra5
57619
42381
21
40
9.52870
10.47130
9.55815
10.44685
9.57658
10.42848
80
41
52918
47088
55355
44645
57696
42304
19
42
52953
47047
55895
44605
67734
42266
18
48
58995
47006
65434
44566
57778
42288
44
53037
46968
55474
445-26
57810
48190
45
53078
4G922
65514
44486
57849
48151
4&
m-M
4^880
55554
4UiG
h7W*7
42113
47
531(11
im'^
firsr.03
4m7
670-:5
4ii075
48
Mt}02
4fi:;is
S-T/^
AWil
670f4
4m^
4»
63^14
4(5750
DGOTS
44.1ii7
5^101
410S9
GO
9.r>?^P5
io.4onjv
9.,')!i7ia
t0.4l^
9.58039
30.4ia6l
51
53327
46S73
55752
442 J8
58077
41923
W
5330S
4fiG3^2
5&701
44^09
6SII5
41885
53
&1509
46r,oi
B-VflUl
44100
58153
41847
54
53450
4fir>so
5.W0
44130
68191
41809
56
GSin
4ft508
5,-1010
44000
58223
417T1
66
^m^
4filfj7
fir/HS
44051
.-ie-s;
41733
57
6-^'iT4
4(hJSfl
^'>o^^
44011
58804
41696
fiS
53fil5
mi^
50(KS
490?i
5834i
41558
59
5.%M
4fMI44
5eOG7
43033
5>^B0
41030
60
63607
40303
56107
43S)3
ES418
4158a
/
Cot&n
Tan
Cotan
Tan
Cotan
Ttan
t
71*
70-
6»'
477
TABLE in.— LOG. TANGENTS AND COTANGENTS.
$
21.
22«
28-
1
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.68418
10.41588
9.60641
10.80850
9.62785
10.37815
60
1
68166
41545
60677
89383
62820
87180
59
8
58498
41507
60714
89288
62855
87145
58
8
68581
41460
60750
89250
62890
87110
57
4
6R569
41481
60788
89214
62926
37074
56
5
68606
41894
80828
89177
62961
87039
55
8
68641
41856
60859
89141
62998
87004
54
7
58681
41819
60695
39106
68031
86909
58
8
56719
41281
60981
89069
68068
86984
58
9
58757
41248
60067
89068
83101
86899
51
10
9.58794
10.41908
9.61004
10.38996
9.63135
10.86866
50
11
58888
41168
81040
88960
68170
86880
49
13
58869
41181
61076
88924
63206
36795
48
18
5S907
41098
81118
88888
68240
86760
47
14
58044
41056
61148
88858
63275
86725
46
16
58961
41019
81184
88816
63310
86690
46
16
59019
40981
81220
88780
6a345
86655
44
17
50066
40944
81256
88744
6S879
86621
48
18
50094
40008
81292
38706
63414
86586
48
19
59181
40860
81326
86672
68449
86551
41
90
9.59168
10.40688
9.61864
10.88688
9.63484
10.86516
40
81
59306
40795
61400
88600
68519
36481
39
28
59848
40767
61486
88564
68558
86447
88
£8
59280
40720
61473
88528
63588
86412
87
24
58617
40688
61506
8H498
68623
86377
36
85
59854
40646
81544
88458
68657
36318
85
86
50891
40609
81579
88421
68692
86808
84
87
59429
40571
61615
88385
68726
86274
33
28
50466
40534
•'61651
38.349
68761
362'J9
38
29
59508
40497
81687
88818
68796
86204
31
80
9.50640
10.40460
9.61722
10.88278
9.63830
10.36170
SO
81
59577
40428
61758
38242
68865
36135
29
88
59614
40886
61794
38206
68899
86101
26
88
59651
40849
61830
38170
68934
86066
87
84
59688
40818
61865
38135
63968
86032
26
85
59725
40275
61901
88099
64003
85997
25
86
59768
40238
81936
38064
64037
35963
24
87
40201
61972
38028
64072
85928
23
88
50885
40165
62006
87992
64106
35894
22
89
50678
40128
62043
87057
64140
35860
21
40
9.59909
10.40091
9.82079
10.37981
9.64175
10.35825
20
41
59946
40054
83114
87886
64209
85791
19
48
69068
40017
62150
87850
64243
85757
18
48
60019
89961
82185
87815
64278
85722
17
44
60056
89944
68221
87779
64312
85688
16
45
60093
89907
62256
87744
64346
85654
15
46
60180
89870
62292
87706
64381
35619
14
47
60166
89634
62327
87678
64415
35585
18
48
60206
89797
62862
87688
84449
36551
12
49
60240
62398
37602
64483
85517
11
60
9.60876
10.89784
9.62438
10.87567
9.64517
10.85488
10
51
60318
89687
68468
87588
64562
85448
9
58
60849
89661
62504
87496
64586
35414
8
68
60386
89014
62539
87461
64620
35880
7
54
60428
89578
62574
87426
64654
85346
6
65
60459
89541
62609
87391
64688
85312
6
66
60495
89505
62645
37855
647^
85278
4
67
60588
89468
62680
87320
64756
35244
8
68
60608
89438
62715
37285
64790
35210
8
50
60606
89895
68750
37250
64824
35176
1
60
60641
89859
62785
87215
64858
85142
0
/
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
68«
87«
W
478
f ABLK ITT
.—LOG.
TANGENTS AND COTANGENTS.
/
«4«
«5«»
28-
/
Tan
Cotan
Tan
Coran
Tnn
Cotan
0
9.64868
10.85142
9.66867
10.33138
9.68818
10.81182
00
1
64892
85108
06900
88)00
68850
81150
60
2
64926
85074
66933
88007
68882
81118
68
8
64960
85040
66966
83034
68914
81066
67
4
64994
85006
66999
83001
68946
81064
66
6
65028
84972
67032
82968
68078
81022
65
6
65062
84938
67066
82936
69010
t0990
54
7
66096
84904
67098
82902
69042
80968
63
8
66180
84870
67131
32869
69074
80926
62
9
65164
84886
67163
82887
69106
80604
61
10
9.65197
10.84803
9.67196
10.82804
9.60188
10.30662
60
11
65281
84769 •
67229
82771
69170
80630
49
12
65265
34785
6?262
82788
09202
80798
48
13
65299
84701
67295
82705
60234
30766
47
14
65883
84667
67327
82678
69266
80734
46
15
65S66
84634
67860
32640
69296
80702
45
18
65400
84600
67398
82607
69829
80671
44
17
65484
34566
67426
82574
69361
80689
48
18
65467
84533
67458
82542
69393
80607
42
19
65501
84499
67491
82509
694;e5
80676
41
90
0.65585
10.84465
9.67624
10.82476
9.69457
10.80643
40
21
65668
8448d
67556
82444
69488
80512
89
22
65602
84396
67589
&2411
69620
80480
88
28
66686
84364
67622
82378
60552
80448
87
24
65669
84881
67654
82846
69584
80416
86
25
65708
84297
67687
82313
69615
80885
85
26
65786
84264
67719
82281
69647
80658
84
27
65770
84230
67752
32248
89679
80821
88
28
66803
84197
67785
82215
69710
80290
88
29
65837
84163
67817
82188
09742
80258
81
80
9.65870
10.84180
9.678IJ0
10.82150
9.69774
10.80220
80
81
65904
84096
67882
32118
69805
80196
29
32
66937
34063
67915
82085
69S87
80168
28
88
65971
84020
67947
82053
69868
80182
87
84
66004
83996
67980
82080
69900
80100
26
85
66038
83962
68012
81988
69932
80068
86
W
66071
83929
68044
81956
69963
80037
84
87
66104
83896
68077
81923
69995
80006
88
88
66188
83S62
68109
31691
70026
29974
88
89
66171
838:.>9
68142
31858
70058
29942
21
40
9.66204
10.88796
9.68174
10.31886
9.70089
10.29911
80
41
66288
83762
68:208
31794
70121
29870
42
66271
837^
68239
81761
70152
29848
48
66804
88696
68371
81729
70184
29616
44
66887
88668
68303
81697
70216
29785
45
66371
83629
68386
81664
70247
29758
40
66404
a36n6
68308
81682
70278
89722
47
66487
S&563
68400
31600
70809
29G91
48
66470
aS580
68432
81668
70841
29669
49
66603
83497
66465
81585
70372
29628
60
9.66637
10.33463
9.68497
10.31.'i03
9.70404
10.29596
51
66670
384.30
68539
31471
70485
29665
52
66608
83.397
68?)61
81489
70466
29584
58
66636
88864
es^9Z
81407
70498
89602
54
66660
83881
68626
81374
70529
204n
55
66702
83298
68658
81342
70560
29440
56
667&5
88265
6H690
81810
70592
29406
57
66768
8323>
68722
81278
70623
99877
58
66801
83199
68754
81246
70654
89346
59
66834
38166
68786
81214
70685
29816
60
66667
83133
68818
81182
70717
29888
/
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
•
86»
64«
W
479
TABLE ni.
—LOG.
TANGENTS AND COTANGENTS.
#
87«
28*
290 1
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.T0ri7
10.89288
9.72667
10.27438
9.74875
10.95625
60
1
70748
292S8
72598
27402
74405
25596
59
2
29S21
78628
27872
74435
25666
58
8
70610
29190
72659
27841
74465
85586
57
4
70841
29159
72689
27811
74494
26506
66
6
70878
89127
7^^720
87280
74524
26476
55
8
70904
29096
72750
87260
74554
25446
54
7
70985
29065
72780
74588
85417
68
8
70966
29034
72811
27189
74618
85887
62
9
70997
29008
T2S41
87159
74643
26857
61
10
9.71028
10.88978
9.78878
10.2n28
9.74678
10.25327
60
11
71059
:28941
72902
27098
74702
85296
49
1«
71090
28910
72968
27068
74788
26266
48
18
71181
28879
72063
27087
74762
25836
47
14
71158
28847
72998
27007
74791
85209
46
15
71184
88816
78028
26977
74821
95179
45
18
71215
28785
78054
26946
74851
25149
44
17
71246
28754
78084
26016
74880
26120
48
18
71277
28728
73114
26886
74910
25090
48
19
71808
28692
T8144
26856
74030
25061
41
80
9.71389
10.28661
9.78175
10.26825
9.74969
10.25061
40
81
71870
28680
78:205
26706
74906
86002
89
88
71401
28599
78285
26705
75028
24072
88
£8
71481
28569
73266
86786
75058
84942
87
24
71468
28588
78295
26705
75087
84918
86
86
71408
28607
78328
26674
75117
24888
86
88
71524
28476
78856
96644
75146
84854
84
87
71555
28445
78886
26614
75176
24624
88
28
71686
28414
78416
26584
76206
84795
88
89
71617
28888
78446
96664
75285
84765
81
80
9.71648
10.28868
9.78476
10.26524
9.76264
10.84786
80
81
71679
28821
73507
26498
75294
84706
29
88
71709
28291
78687
26468
75828
84677
28
88
71740
28260
78567
26433
75368
24647
27
34
71771
28229
78697
26406
75388
24618
26
85
71802
28196
78627
263T3
75411
24589
26
86
71833
28167
78867
26343
75441
24559
24
87
71863
28187
73667
26818
75470
24580
28
88
71894
28106
73717
26283
75500
24500
22
88
71085
28075
78747
26258
75529
24471
21
40
9.71956
10.28045
9.787T7
10.26228
9.75558
10.24442
80
41
71986
28014
73807
26103
75588
24412
48
72017
27068
78887
26163
75617
84383
48
72048
27952
73867
26188
76647
2435.)
44
72078
27922
73897
26108
75676
24824
45
72109
27991
78927
26073
75706
84295
46
72140
27860
73957
26048
75735
84266
47
72170
27830
78987
26018
76764
24286
48
72201
27799
74017
26883
75798
24207
49
72231
27769
74047
25958
76622
84178
50
9.72262
10.27738
9.74077
10.25928
9.76652
10.24146
51
72298
27707
74107
26806
75881
24119
58
72823
27677
74137
S6868
75910
84090
58
72854
27646
74166
2.5884
76989
24061
54
72384
27616
74196
25804
75969
24081
66
72415
27586
74226
25774
75996
24002
56
72445
27555
74256
25744
78027
28973
57
72476
27524
74286
25714
76056
28944
58
72606
27494
74316
26684
76086
28914
50
72587
27468
74345
26655
76115
28885
60
72667
27483
74875
25625
76144
88856
/
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
6f
61*
60»
48o
TABLE III.— LOG. TANGENTS AND COTANGENTS.
/
80«
81-
82»
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.76144
10.23866
9.77m
10.22128 '
9.79579
10.80421
00
1
76178
23827
77906
82004
79607
20898
60
2
76202
28798
77985
22066
79685
80866
58
.8
78281
77963
2S087
79668
80887
67
4
76261
28789
77992
88006
79691
20809
66
5
76290
28710
78020
81980
79719
80861
56
6
76819
23681
78049
21961
79747
80858
64
7
76348
23G52
78077
81928
79776
80294
58
8
76377
28628
78106
81894
79604
80196
59
9
76406
28694
78185
21865
79688
80168
61
10
9.76435
10.23666
9.78168
10.21887
9.79660
10.20140
50
11
764G4
23536
78192
21808
79688
20118
40
18
76198
23507
78820
21780
79916
80064
46
18
76622
28478
78249
817V1
79944
20056
47
14
76551
23449
78277
21728
79978
S0OS6
46
15
76580
23420
78806
21694
80000
80000
45
16
76609
28391
78334
81666
80028
19078
44
17
76639
28861
78868
81687
80056
19944
48
18
76668
28882
78391
81609
80064
19916
48
19
76697
23308
78419
81681
80118
19668
41
SO
9.76725
10.23276
9.7B448
10.81558
9.80140
10.19600
40
21
76754
28246
78476
81524
80168
19832
80
22
76788
28217
78506
21495
80195
19605
88
28
76812
28188
78538
21467
80288
19777
87
24
76841
281.59
78562
81438
60%1
19749
86
25
76870
28180
78590
81410
80279
19781
85
26
76899
28101
78618
81882
80807
19698
84
27
76928
28072
78647
81858
60885
19666
88
28
76967
28043
78675
81326
60868
19687
88
29
76986
23014
78704
81296
80891
19609
31
80
9.rroi6
10.22985
9.78788
10.21268
9.80419
10.19681
30
81
77044
22956
78760
21240
60447
19668
80
82
77073
22987
78789
81211
80474
10686
88
83
77101
22899
78817
81188
80502
19496
97
84
77180
22870
78846
21165
60580
19470
86
85
77159
22841
78874
81126
60666
19448
85
86
77188
22812
78902
81098
60686
10414
94
37
77217
22783
78980
81070
60614
10886
88
88
77246
22754
78959
81041
60648
10868
88
89
77274
22726
78987
81018
60669
10881
91
40
9.77303
10.22697
9.79016
10.80986
' 9.60607
10.10808
90
41
77882
2-J668
79043
20957
60785
1087S
42
77361
22639
79072
20928
80768
10247
43
77890
22610
79100
20900
60781
10810
44
77418
22582
79128
20878
80806
10108
45
77447
22553
79156
80844
80886
10164
46
77476
22524
79185
80815
80864
10186
47
77605
22495
79218
80787
80808
10106
48
77533
22467
79241
80759
80919
10061
49
77562
22488
79269
20781
80947
10068
50
9.77591
10.32409
9.79297
10.80708
9.800ni
10.10085
51
77619
22381
79826
20674
61008
16097
52
7764S
22352
79354
20646
61080
18970
58
77677
22328
79888
80618
61056
18948
54
77706
22294
79410
80590
81086
18914
66
77784
22266
79488
20668
81118
16867
66
77763
22237
79466
20584
61141
18659
57
77791
22209
79496
80506
81169
18881
58
77820
22180
79528
20477
61196
16804
59
77849
22151
79551
80449
81224
18778
60
77877
22128
79679
80421
61252
16748
/
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
6»»
58*
67«
48i
TABLE III.— LOG. TANGENTS AND COTANGENTS.
/
SS*
S4'
•S-
/
Tan
Ootan
Tun
Cotan
Tan
Cotan
9.81858
10.18748
9.88809
10.17101
9.84588
10.15477
60
81-279
18781
82986
17074
81560
15460
60
81807
18608
88958
17047
84678
15484
58
81886
18665
88980
17080
84606
16397
57
81868
18688
88008
16008
84680
15870
66
81890
18610
88085
16965
84657
16348
66
81418
18588
83068
16888
84684
16816
64
81446
18565
88089
16911
84711
15889
68
81478
18587
88117
16888
84788
15868
58
81600
18500
88144
16856
84764
15886
61
10
9.81688
10.18478
9.83171
10.16889
9.84701
10.16809
60
11
13
81566
18444
83198
16808
84818
16188
49
81588
18417
83835
16775
84845
16165
48
18
14
16
16
81611
18880
88258
16748
84873
16188
47
81638
18868
83280
16780
84899
16101
46
81666
18881
88307
16698
84925
15075
45
81098
18807
88334
16666
84958
15048
44
17
817^
18879
83861
16639
84970
16081
48
18
81748
18858
838S8
16618
86006
14994
48
19
81776
18884
88415
16586
85068
14967
41
SO
9.81806
10.18197
9.88148
10.16668
9.86060
10.14941
40
81
S8
81681
18169
83470
16580
85088
14914
89
81868
18148
88497
16508
86118
14887
88
81886
18114
88584
16476
86140
14860
87
M
81918
18087
88551
16449
85166
14884
86
S6
81941
18060
88578
16488
86198
14807
86
26
81968
18088
83605
16895
85880
14780
84
87
81906
18004
88688
16868
86847
14758
88
88
88088
1T977
83650
16841
85878
14787
88
SO
82051
17949
88686
16814
86300
14700
81
80
9.88078
10.17988
9.88718
10.16887
9.86387
10.14678
80
81
88
88106
17894
88740
16860
85364
14646
80
88188
17897
83768
16288
85880
14620
88
88
84
88161
17830
88795
16805
85407
14508
87
88188
17818
88888
16178
85484
14666
86
86
88815
17786
83849
16151
85460
14540
85
80
«2848
17757
88876
16184
85487
14618
84
87
88-^ro
17780
83008
16007
86514
14486
88
88
82898
17708
83930
16070
85540
14460
88
89
88385
17676
83957
16048
85567
14488
81
40
9.83858
10.17048
9.83984
10.16016
9.85594
10.14406
80
41
8a2880
17680
84011
15969
86680
14880
48
88407
17598
84038
15968
86647
14368
48
88435
17566
84066
15935
85674
14886
44
82468
17588
84002
15908
86700
14300
45
88489
17511
84119
16881
85T27
14878
46
88517
17483
84146
15864
86754
14846
47
88544
17456
84178
16887
85780
14890
48
88671
17489
84800
15800
85807
14198
49
88509
17401
84837
15778
85834
14166
60
9.88086
/O. 17874
9.84854
10.15746
9.85860
10.14140
51
8S658
17847
84880
15780
86887
14118
68
88681
17819
84807
15698
86913
14087
68
88706
17898
84884
15666
85940
14060
64
82786
17866
84361
15689
86967
14033
66
88768
17888
84888
15618
85993
14007
66
88790
mio
84415
16685
86080
18980
57
88817
17188
84448
15558
86046
18964
68
82844
37:56
84460
16631
86078
18987
60
60
88871
17189
81406
16504
86100
13900
88899
17101
84583
16477
86186
13874
0
/
Ck>tan
Tan
Cotan
Tan
Ck>tan
Tan
/
M*
66»
64»
482
TABLE m.— LOG
. TANGENTS AND CO'
FANGEN
TS.
/
86«»
' $7»
88«
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.86186
10.18874
9.87711
10.18289
9.89881
10.10719
60
1
86168
18847
87788
18868
89807
10693
60
a
86179
18881
87764
18236
89888
10667
68
8
86806
13794
87790
18810
89859
10641
57
4
86888
13768
87817
18188
89885
10616
66
0
86859
18741
.87848
18157
89411
10580
55
6
86886
18716
•87869
12131
89487
10568
54
7
86818
18688
87896
18106
89463
10587
63
8
66388
18668
87928
18078
89489
10511
68
9
86865
13635
87948
18068
89516
10485
61
10
9.86888
10.13608
9.87974
10.18026
9.89541
10.10459
60
11
86418
18568
88000
12000
89567
10483
49
18
86446
18666
88087
11978
89598
10407
48
18
86471
18629
88068
11947
89619
10881
47
14
86496
13608
88079
11981
89646
10866
46
16
86584
18476
88105
11895
89671
10889
46
16
86561
13449
88181
11869
89697
10806
44
17
86677
18488
88158
11848
10877
48
18
86608
13397
88184
11816
89749
10251
48
19
86680
18370
88810
11790
89776
10886
41
90
9.86666
10.18844
9.88886
10.11764
9.89601
10.10199
40
81
86688
18817
88868
11738
88887
10178
89
23
86709
18891
88289
11711
89658
10147
88
88
86786
18864
88316
11685
89879
10181
87
84
86768
13888
88341
11659
89906
10095
86
86
86789
18811
88867
11633
89931
10069
85
88
86816
18185
88393
11607
89957
10043
34
87
86849
18168
88420
11560
89983
10017
38
88
86868
18188
88446
11554
90009
09991
88
89
86894
18106
88478
11528
90085
09966
31
80
9.86981
10.18079
9.88498
10.11508
9.90061
10.09989
80
81
86947
18058
88524
11476
90086
09914
89
38
86974
• 13086
88550
11450
90118
09888
88
88
87000
18000
88577
11488
90188
09868
97
84
87027
18978
88608
11397
90164
00686
86
85
87068
18947
88629
11871
90190
09610
85
86
87079
18981
88655
11845
90216
09:84
84
87
87106
12894
88681
11819
90248
09758
88
88
87188
18868
88707
11293
90268
09782
88
89
87168
18848
88783
11867
90894
09706
81
40
9.87186
10.18815
9.88759
10.11841
9.90380
10.09680
80
41
87811
18789
88786
11814
90346
09654
48
87888
18768
88818
11188
90S71
09689
48
87264
18786
88838
11168
90897
09608
44
87290
18710
88864
11186
90488
09577
45
87817
18688
88800
11110
90449
09561
46
87343
12657
88916
11084
90475
09585
47
87369
12631
88942
11058
90501
09499
48
87396
18604
88968
11088
90587
09478
49
87488
18578
88994
11006
90558
09447
50
9.87448
10.18858
9.89020
10.10980
9.90578
10.09498
61
87475
18525
89046
10964
90604
08396
68
87501
18499
89078
10927
• 90630
09370
68
87537
18478
89099
10901
90656
09844
64
87664
18446
89185
10875
90688
09818
55
87580
12420
89151
10849
90708
09298
66
87606
18894
89177
10823
90734
09266
67
87638
12367
89203
10797
90769
09841
68
8T659
12841
89229
10771
90785
09815
59
87685
12315
89256
10746
90611
00189
60
87711
12289
89281
10719
90837
09168
/
Cotan
Tan
Cotan
Tan
Cotan
Tan
/
58-
««•
61«
483
TABLE III.— LOG. TANGENTS AND COTANGENTS.
• /
sr
40-
41'
/
Tau
Cotan
Tan
Ctotan
Tan
Cotan
0
9.90687
10.09163
9.92381
10.07619
9.93916
10.06084
60
]
90668
09187
92407
07593
93942
06058
69
2
00889
09111
92438
07567
93907
06038
68
8
90914
09086
92458
07642
98906
06007
67
4
00940
09060
92484
07516
94018
05982
56
6
90966
00034
92510
07490
94044
06956
65
6
90992
09008
92535
07465
94069
06931
64
7
91018
08982
92661
0:439
94095
06906
68
8
91048
08S)S7
02587
07418
94120
05880
62
9
91000
08081
92612
07888
94146
06854
61
10
9.91095
10.08905
9.92688
10.07368
9.94171
10.05829
60
11
91121
08879
92663
07887
94197
05803
49
12
91147
08853
92689
07811
94222
06778
48
18
91172
088-^8
92715
07286
91248
06762
47
14
91198
08802
92740
07260
94278
06787
46
16
91224
08776
92766
07234
94299
05701
46
16
91250
08750
92792
07208
94824
06676
44
17
91278
08724
92817
on83
94a'30
05660
48
18
91801
08699
92813
07167
94376
05626
42
19
91827
08673
92868
07182
94401
06599
41
20
9.91338
10.08647
9.92894
lo.onoo
9.94426
10.06574
40
21
91379
08621
92920
07080
94452
05548
89
22
91404
08596
02946
07066
94477
05528
86
128
91480
08570
92971
07029
94603
05497
87
24
91466
oa>i4
92006
07004
94628
06472
86
25
9148^
08518
93022
06978
94564
06446
86
26
91507
08493
93048
06052
94579
. 05421
84
27
91538
08467
98078
06927
94604
05896
88
28
91559
08441
06001
94680
06370
82
29
91585
08416
93124
06876
04656
05345
81
80
9.91610
10.08890
9.0615O
10.06850
9.94681
10.05819
80
81
91036
08364
98176
06825
94706
a^294
29
82
91662
0»i88
93201
06799
94782
05268
28
88
9168S
08312
93227
06778
04757
06248
27
34
91713
08J87
93252
06748
94788
05217
26
85
917B9
08261
93278
06722
94806
06192
25
86
91765
08285
98308
06697
94834
05166
24
87
91791
08209
06329
06671
94859
06141
28
88
91816
08184
93354
06646
94884
06116
22
89
91842
08158
93380
06620
94910
06090
21
40
9.01868
10.08132
9.93406
10.06594
9.94985
10.06066
20
41
91893
08107
93481
06569
94961
05089
19
4;!
91919
0S081
93457
06548
94986
05014
18
4.3
91945
08055
93482
06518
95012
04968
17
44
91971
08029
93508
06492
96037
04968
16
45
91996
08004
06583
06467
06062
04988
16
46
92022
07978
93559
06441
95088
04912
14
47
92048
07962
06584
06416
95118
04887
18
48
93073
07927
fr^eio
06390
96139
04861
12
49
92099
07901
98686
063&4
95164
04886
11
50
9.92125
10.07875
9.08661
10.06389
9.06190
10.04810
10
51
92160
07850
93687
06318
95216
04786
0
52
92176
07H24
93712
06288
95240
04760
8
f>3
92»2
07798
93738
06262
95266
04784
7
54
92227
07V/8
93763
06237
95v>91
04709
6
55
92258
07747
93789
06211
95817
04688
6
56
92279
07721
93814
06186
95842
04658
4
57
9-.'304
07696
93840
06160
96368
04C32
8
58
92380
07670
93865
06185
95898
04607
2
59
92366
07644
98891
06109
95418
04682
\
60
92881
07619
93916
06064
96444
04656
0
»
Ck>taii
Tan
Cotan
Tan
Cotan
Tan
/
M*
490
48»
TABLE ni.— LOG. TANGENTS AND COTANGENTS.
'
42*
48*
44»
/
Tan
Cotan
Tan
Cotan
Tan
Cotan
0
9.95444
10.04666
9.90966
10.08084
9.96484
10.01616
60
1
9&4(i0
0453]
96991
06009
96600
01491
69
8
95495
04606
97016
08984
96684
01466
68
8
95620
04480
97048
08968
96660
01440
67
4
96545
04465
97067
02988
96686
01416
60
6
96571
04429
97098
08908
96610
01890
66
6
96596
04404
9ni6
09888
96686
01865
64
7
96628
04878
97143
02867
98661
01839
68
8
96647
04868
97166
02838
96686
01814
68
9
96678
04888
97196
08607
98711
01889
61
10
9.96698
10.04308
9.97819
10.0S781
9.96787
10.01868
60
11
96728
04277
Vm4
02766
96768
01886
49
18
96748
04288
97209
02781
96787
01818
48
18
95774
04226
97296
08705
96818
01188
47
14
96799
04^1
97880
08680
96888
01168
46
16
95826
04176
97845
02656
96868
01187
46
16
95660
04160
97871
08089
98886
01118
44
17
95876
04126
97896
08604
96918
01067
48
18
95901
04099
97421
08679
96989
01061
48
19
96926
04074
97447
08668
98964
01066
41
90
9.96068
10;04048
9.97478
10.08626
9.98969
10.01011
40
21
96977
04088
97497
97588
08603
99016
00966
89
28
96002
08996
08477
99040
00960
88
23
96028
08978
97546
02468
99066
00965
87
24
9tK)5S
08947
97678
08427
99090
00910
88
25
96078
03922
97696
02403
99116
006B4
36
86
96104
08896
97624
08876
99141
00669
84
27
96129
08871
97649
02851
99166
00684
88
88
96156
08846
97674
08326
99191
00609
88
89
96180
08880
97700
02800
99817
00788
31
80
9.96206
10.08796
9.977S5
10.08276
9.99848
10.00766
ao
81
962JJ1
08769
97760
08260
99867
89
88
96866
08744
97778
09884
99898
OOiW
88
88
96281
06719
97801
08199
90816
00662
87
34
96307
08698
978M
02174
99848
00667
86
8R
96338
08668
97851
08149
99368
00688
86
86
96857
08648
97877
02128
00606
84
37
96888
08617
97908
08006
99419
0C681
88
88
96406
08692
97927
02073
O0666
28
89
96438
08567
97B68
02047
99460
00681
81
40
9.964!)9
10.08641
9.97978
10.08028
9.99496
10.00606
80
41
96484
08516
96008
01997
99620
00480
19
48
96610
03490
98029
01971
99645
00466
18
48
96586
08466
96054
01946
9067U
00480
17
44
96560
08440
98079
01981
99606
00404
16
45
9C688
08414
96104
01896
99081
00879
16
46
96611
08389
96180
01870
mow
00864
14
47
96686
03364
96166
01845
99678
O0BS6
18
48
96668
03338
96180
01890
99697
00808
18
49
96667
03318
98806
01794
99788
00878
11
60
9.90718
10.03288
9.96231
10.01769
9.99747
10.00258
10
61
96738
08268
96256
01744
99773
00827
9
68
96763
03237
96261
01719
99798
00008
8
68
96788
03212
98807
01693
99688
00177
7
64
96814
08186
983S8
01668
99848
00168
6
66
96889
08101
96367
01648
99674
00180
6
66
96864
06186
96868
01617
99889
00101
4
67
96690
08110
96406
01698
99994
00076
8
68
96915
08066
96488
01667
99949
00061
8
69
96940
03060
96466
01548
99976
00085
1
60
96966
00084
98484
01516
10X10000
00000
0
/
CotAD
Tan
Cotan
Tan
Cotan
Tan
/
4T
46»
♦«•
485
TABLE IV.— NATURAL SINES AND COSINES.
/
0»
V 1
2- 1
8- 1
4- 1
Sine ;CosIn
TOOOOO One.
Sine
.01745
Cosin
.99985
Sine
:03490
Cosln
.9i««i9
Sine
.06284
Cosin
799863
Sine
Cosin
.06976
.99756 60
1
.000291 One.
.01774
.99984
.03519
.99ff«
.05263
.99861
.07006
.99754169
2
.00068 One.
.01803
.99984
.03548
.99937
.05292
.99660
.07084
.99752168
8
.00087 One.
.01833
.99963
.08577
.99936
.05321
.99858
.07063
.99750! 57
4
.001161 One. , .018621
.99988
.08606
.90935
.05350
.99657
.07092
.99748166
5
.00145 One.
.01891
.99962
.06635
.99934
.05379
.99855
.07121
.99746! 65
6
.00175, One.
.01920
.99962
.06664
.99933
.05406
.99854
.07150
.997441 54
7
.00204 One.
.01949
.99981
.08693
.99932
.05487
.99652
.07179
.997421 53
8
.00283 One.
.01978
.99960
.03723
.99931
.06466
.99661
.07208
.99740 52
0
.00262 One.
.02007
.99980
.08752
.99930
.05495
.99649
.07237
.99V38I 51
10
.00891 One.
.02086
.99979
.08781
.99929
.06624
.99647
.07266
.00786. 60
11
.00620 .99909
.02065
.99979
.06610
.90927
.98846
.07296
.99784' 49
12
.00849 .99909
.02004
.99978
.03839
.99020l
.05582
99644
.07824
.99731,48
18
.006781.99999
.02123
.999'i7
.08866
.99925,
.06611
!99642
.07868
.99729 47
14
.00407 .99999
.02152
.99977
,08897
.999241
.05640
.99641
.07382
.99727 46
15
.0(M36,. 99909'
.02181
.99976
.08926
.99923
.06609
.99639
.07411
.99725 45
10
.00465 .999991
.02211
.99976
.03955
.99922
.06696
.99688
.07440
.99733 44
17
.0M96 .90999'
.02240
.99975
.08084
.99921
.05727
.99686,
.07469
.99721148
18
.00604 .99999,
.02269
.99974
.04013
.999191
.05756
.998841
.07498
.99719 42
19
.00658 .999981
.02296
.99974
.04042
.99918
.05786
.99833;
.07527
.99716
41
20
.00682 .99996;
.02827
.99973
.04071
.99917
.06814
.998811
.07556
.99714
40
21
.00611 .99998*
.08366
.99972
.04100
.99916'
.05H44
.99629
.07585
.09712
89
22
.00610 .990981
.02385
.99972
.04129
.99915, .05873
.99627
.07614
.09710
88
28
.00669. 999981
.02414 .99971
, .04159
.09913 .06902
.99826
.<7643
.99706
87
24
.00698 .09998'
.02443
.99970 1.04186
.99912, .06931
.99824
.07672
.99705
36
26
.00727' 99997
.02472
.9996911.04217
.99911! .06960
.996:^
.07701
.99708
85
26
.00756 .99997
.02501
.99969
.04246
.99910 .06989
.99821
.07780
.09701
84
27
.00785 .99997
.02530
.99968
.04275
.9990911.06018
.99819
.07759
00699
88
28
.008141.99997
.02560
.99967
1.04304
.99907; 1.06047
.99817
.07788
.99696
82
29
.00844 .99996
.02589
.99966
.04883
.99906,,. 06076
.99815
.07817
.99694
81
ao
.00673 .99996
.02618
.99966
.04862
.99905 !|. 06105
.99813
.07846
.99602
80
81
.00902 .99996
.02647
.99966
.04891
.099041,. 06184
.99612
.07875
.09689
29
82
.00981 .99996
.02678
.99964
.04420
.99903 .06163
.90810
.07904
.99687
28
83
.00960 .99995
.02706
.9996b
.04449
.999011!. 06192
.99608
.07938
.99685
27
84
.00969*. 99995
.08784
.99968
.04478
.99900 ' .06221
.99806
.07962
.99683126
86
.01018 .99995
.02763
.99962
.04507
.99698; .06250
.99804
.07991
.99680 25
86
.010471.99995
.02792
.99961
.04586
.99697
.06279
.99803
.08Q!»
.99678! 24
87
.01076 .99991
.02821
.99960
.04565
.99896
.06908
.99801
.08049
.99676! 23
88
.01105 .99994
.02850
.99959
.04504
.99894
.06387
.99799
.08078
.99673' 22
80
.01184 .99994
.02879
.99959
.04823
.9989311.06366
.99797
.08107
.99671121
40
.01164 .99998
.02906
.99958
.04658
.99692, '06396
.99890 '1.06424
.99795
.06186
.99668
20
41
.01196'. 99993
.02988
.99957
.04682
.99793
.06165
.99666
19
42
.01^ .99998
.02967
.99956 .04711
. 99889 1I.OO453
.99792
.08194
.99664
18
48
.01251 .99992
.02996
.999551 .04740
.99888; .06482
.99790
.08223
.99661
17
44
.01280 .999921
.08025
.999541 .04769,. 99886 1.06511
.99788
.08252
.99659
16
45
.01809 .99991 >
.03054
.999531 .04796
.99885 .06&40
.99786
.06281
.99657
15
46
.01388 .99991 1
.09063
.99952 ,'.04827
.998831 .06569
.99^
.08310
.99654
14
47
.01367. 99991
.08112
.999521!. 04866
.09883 .06596
.99782
.08339
.99652! 13 1
48
.01396 .99990
.08141
.99951 1 .04885
.99881 1.06627
.99780
.06368
.99649
12
49
.01425 .999901
.08170
.99060, .04914
.99879 1.06656
.99778
.08397
.99647
11
50
.014M|.99989
.08199
.999491,. 04948
.99678 i .06686
.99776
.08426
.99644
10
61
.014881.99960
.08828
.09048 1.04972
.09876
.06714
.99774'
.06466
.99642
9
02
.016181.999691
.09257
.99947 1 1.05001
.99675
'. 06743 i.OOTTS
.08484 '.99689
8
58
.01542'. 99968
.03286
.99946 .05080
.99673
.067731.99770
.06518!. 99687
7
54
.015711.99966
.08316
.99945 ,.05050
.99872
.06802 .99768
.Oa'>42,. 99635
6
66
.01600 .90967
.08845
.99944. .06068
.99670
.06831 .99766
.08571
.99632
6
66
.01629 .99967
.06374
.99943,1.06117
.99669;!. 06860!. 99764
.06600
.99630
4
67
.01668 .99966
.06403
.99042 , .05146
.99867 1.068801.99762
.99866". 06918'. 99760
.08629
.99627
8
68
.01667 .99966
.03482
.99041
1.06175
.06658
.99625
2
60
.017161.99966
.08461
.99940
.06206
.09664! 1.06947 1.99758
.08687
.99622
1
60
/
.01746 .99985
Cosln fSlne
.08490
CkwiD
.99989
Sine 1
.05284
Cosln
.99863 .06976 .99756
.06716
.99619
_0
/
Sine
Coain|Sine |
Cosin
Sine
88*
81
!• 1
87- 1
1 86- i
>9-
486
TABLE IV.— NATURAL SINES AND COSINES.
"o
6-
6o ,
70 ,
8»
9*
/
60
Sine
.08716
Oodn
.99619
Sine
Cosin
.99452
Sine
.12187
Cosin
.9ft255|
Sine
:i3917
Cosin
.99027
Sine
, .15643
Cosin
.10453
.96769
1
.08746
.99617
.10482
.99449
.12816
.99251'
.13946
.99023
1.15672
.98764
60
2
.08774
.99614
.10511
.99446
.122451. 99248,
.13975
.99019
.15701
.96760
58
8
.08803
.99612
.10540
.99443
.12274 .99844
.140W
.99015
1.15780
.96755
67
4
.06831
.99609
.10569
.99440
.12308 .99240'
.14033
.99011
.15758
.96751
66
6
.08860
.99607
.10597
.994371
.12331 .998371
.14061
.99006
1 .15787
.98746. 56 1
6
.06889
.99804
.10626
.99134'
.12360 .992331
.14090
.99008
.15816
.98741
54
7
.08918
.99602
.10655
.99131;
.123891.99230,
.14119
.98998
1.15845 .96737
53
8
.08947
.99599
.10684
.994281
.12418
.992361
.14148
.98994
.15878
.98782
58
9
.08976
.99596
.10713
.99421'
.12447
.99823'
.14177
.96990
.15002
.98728
61
10
.09006
.99594
.10742
.99421
.12476
.992191
.14205
.96966
.15981
60
11
.09084
.99691
.10771
.99418
.12504
.99215
.14234
.96962
1.16959
.98718
49
12
.09063
.99588
.10800
.99415 {
.12533
.99211
.14263
.98978
.15968
.96714
48
18
.09092
.99586
.10829
.99412
.12562
.99208:
.14292
.98973
.16017
.98709
47
14
.09121
.99583
.10858
.99409'
.12591
.99204
.14320
.96969
'.16046
.96704
46
15
.09150
.99580
.10887
.99406
.12620
.99200'
.14349
.98965
.16074
.98700
46
16
.09179
.99578
.10916
.99408
.12649
.99197
.14878
.98961
.16103
.98605
44
17
.09206
.99573
.10945
.99399
.12678
.99193
.14407
.98957
,.16132
.98690
48
18
.09237
.995?2
.10973
.99396
.12706
.99189
.14436
.98953
1.16160
QfiUfiA
48
19
.09266
.99570
.11008
.99393
.12735
.99186
.14464
.98948
I .16189
.96681
41
20
.09295
.99667
.11081
.99390
.12764
.99183
.14493
.96944
.16218
.98676
40
21
.09324
.99564
.11060
.99386
.12798
.99178
.14582
.98940
1.16946
.96871
80
22
.09353
.99562
.11089
.99383
.18822
.99175
.14651
.98936
,.16875
.98667
88
28
.09382
.99559
.11118
.993801
.12851
.99171 1
.14580
.98931
M6804
.98662
87
24
.09411
.99556
.11147
.993771
.12880
.991671
.14608
.98927
.16838
.96657
86
25
.09440
.99553
.11176
99374
18<¥V^
99163
14687
98983
1 16361
96652
85
28
.09469
.99551
.11205
.99370
.12937
!99160
!l4666
.'98919
:!l6S90
!96648
84
27
.09498
.99548
.11231
.99367
.12966
.99156'
.14695
.98914
1 .16419
.96643
83
28
.09527
.99545
.11263
.99364
.12995
.99158
.14';^
.98910
.16447
.96688
38
29
.09556
.99M2
.11291
.99360
.18024
.99148
.14758
.98906
.16476
.98638
81
SO
.09585
.99540
.11820
.99337
.13053
.99144
.14781
.98902
1.16505
.98689
80
81
.09614
.99637
.11849
.99354
.18081
.99141
.14810
.98897
.16688
.98684
89
32
.09642
.99534
.11878
.993311
.13110
.99137
.14838
.98893
1.16662
.98619
88
88
.09671
.99531
.11407
.99347
.13139
.99133
.14867
.98889
.16691
.98614
87
84
.09700
.99528
.11436
.99344
.13168
.99129
.14896
.96884
.16680
.96609
86
85
.09729
.99526
.11465
.99341'
.13197
.99125
.14925
.98880
.16648
.98604
85
86
.09758
.99523
.11494
.99337,
.18226
.99128
.14954
.98876
.16677
.96600
84
87
.09787
.99520
.11523
.99334 1
.13854
.99118
.14982
.96871
.16706
.96595
83
88
.09816
99517
.11552
.993311
.13883
.991141
.15011
.98867
.10784
.96690
88
80
.09845
.99514
.11580
.993271
.13312
.991101
.15040
.98863
.16768
.98585
21
40
.09874
.99511
.11609
.99384
.13311
.99106,
.15069
.98868
.16798
.96680
80
41
.00908
.99508
.11638
.99380'
.18370
.99108
.15097
.98854
.16820
.98576
19
42
.09932
.99506
.11667
.993171
.13399
.990981
.15126
.98849
.16849
.96570
18
48
.09961
.99503
.11696
.99314'
.13427
.99094'
.15155
.98845
.16878
.96565
17
44
.09990
.99500
.11725
.99310 1
.13456
.99091,
.15184
.98841
.16906
.96661
16
46
.10019
.99497
.11754
•99307
.13485
.99087'
.15218
.98836
.16935
.98556
16
46
.10048
.99494
.11783
.993031
.13514
.99083
.15241
.98832
.16964
.96551
14
47
.10077
.99491
.11812
.99800
.13543
.99079
.15270
.98827
.16992
.96546
18
48
.10106
.99488
.11840
.992971
.13572
.99075
.15299
.98883
.17021
.98541
18
40
.10135
.99485
.11869
.992931
.18600
.99071
.15327
.96818
.17060
.98536
11
60
.10164
.99482
.11898
.99290'
.13629
.99067
.15866
.96814
.17078
.98531
10
61
.10192
.99479
.11927
.998861
.18858
.99063
.15886
.96809
.17107
.98686
9
62
.10221
.99476
.11956
.99883]
.18687
.99a59
.15414
.98805
.17186
.96621
8
68
.10250
.99473
.11985
.99279
.13716
.99055
.15448,. 98800
.17164
.96616
7
64
.10879
.99470
.18014
.99276
.13744
.99051
.15471 .98796
.17193
.98611
6
65
.10808
.99467
.12043
.99878!
.13773
.90017
.15500 .98791
.17282
.98606
5
66
.10337
.99464
.12071
.992691
.13802
.99043
.15529!. 98787
.17250
.96601
4
67
10366
.99461
.12100
.99885
.18831
.99039
.155571.98782
.17879
.98496
8
68
.10895
.99158
.12189
.998621
.13860
.99035
.15586 .98778
.17806 .96491
8
69
.10424
.99455
.12158
.99258'
.13889
.99a31
.15615 .98r?8
.17386
.96488
1
!2
/
.10453
.99152
Sine
.12187
Cosin
.99255
Sine
.13917
Cosin
.99027
Sine
.15643 .98769
Cosin Sine
.17865
.98481
_0
t
Cosin
CoBln
Sine
84- 1
88- 1
82- 1
81*
80* 1
487
^' TABLE IV.— NATURAL SINES AND COSINES. ^
Sine ,Co8in
[7865 .96481
[7888 .96476
[742S .96471
17461 .96466
17479 .96461
17506 .96455
17587 .96460
17565 .96445
17594 .96440
[7623 .96485
[76511.96480
17660
17T06
17787
17766
17794
I78S3
17852
17880
17909
17987
16424
18452
18481
18509
18861
18010
18988
18967
18995
19024
19052
19061
Cosin
98425
96420
96414
96409
98404
96899
98894
98357
96825
II'
96250
96245
,98218 I
.20861
.20635
.20563
Cofllnl
.98163
.98157
.96152
.96146
.96140
.98135
.96129
.96124
.^118
.96112
.98107
.98101
.96096,
.98090
.960641
.960791
98073,
96067'
.980611
96066|
.96060
96OI4I
96089
98033
96027,
960211
.98016
98010,
960O4I
97996,
97992
979671
97961 1
97975I
97969
97963 I
97958
97952 I
97946 I
97940 I
97931 I
120
Sine
.20791
.20648
.20877
.20905
.20962
.20990
.21019
.21017
.21076
.21104
.21188
.21161
.21189
.21218
.2t»16
.21275
.21808
.21381
.21860
.21886
.21417
.21445
.21474
.81502
.21580
.21550
.21587
.21616
.21644
.21672
.21701
.21729
.21758
.21786
.21814
.21843
.21871
.21899
.21928
97928 '.21956
97922,1.21965
97916 ' .22013
97910 .22041
97905 '.22070
978991
97893;
978871
978811
97875
22096
.22126
.22155
.221831
.22212,
OosinI
Sine
97869' .28240
9786311.22268
97857 1.22297 1
97851:1.22325
978451 .22353
97839 '.22382
97833 ,. 22110 :
97827 1.22138
97821 .22467;
,97815 .22495,
.97815,
.97809
.97808
.97797
.97791
.977841
.97778
.97772
.97766
.97760,
.97754'
.97748'
.97742
.977861
.97789
.977831
.977171
.97ni|
.97705
.97696
.97692
.97688*
.97680
97673;
976671
97661:
.97655,
.97648'
97612 1
.97638
97630 1
97823|
97617
976111
976011
97596
.97592
97585
.97579
97573
.97566;
97S60
97553!
97547,
975411
97584'
97528
975211
97515 1
,975061
,97602 1
,97496
.97489
.97483
.97476
.97470
.97468
.97467 J
.97450
.97444
.97437
18«
Sine
.82496
.82722
.22760
.22778
.82807
.22977
.83005
.88090
.88118
.83146
.28175
,88316
,83845
.88878
83401
,83429
.83458
.83486
.23514
,28542
.23571
.88687
.88656
.83712
.23740
.23797
Cosin I Sine Cosin | Sine
78* il 77»
.28995
.24023
.24051
.24079
.24106
.24136
.84164
.81102
Cosin
Cosin
.97487
.97480
.97484
.97417
.97411
.97404
4)7896
.97891
.97364
.97878
.97871
.97866
.97868
.97851
.97846
.97886
.97881
.97885
.97818
.97811
.97804
.97896
.97891
.97884
.97878
.97871
.978U
.97857
.97851
.97844
.97887
.97880
.97!!^
.97217
.97810
.97208
.97190
.97189
.971821
.97176
.97169
.97162,
.97156
.97148
.971411
.97184,
.97127 1
.971201
.971181
.971081
.97100
.97093'
.97066
.97079
.970781
.97065
.97058'
.97061,
.97044'
.97087 1
■97030 I
Sine !
14*
Sine Cosin
.24198
.24849
.24877
.84805
.84390
.84418
.84446
.84474
.84608
.84581
.84560
.84587
.84616
.84644
.84672
.84700
.84786
.84756
.84784
.84841
.84897
.84954
.85010
.85086
.86066
.85091
.25122
.25151
.25179
.26807
.25380
.85848
.85876
.8&104
.85460
.85616
.85645
.85573
.85601
.97030
.97023
.97016
.97006
.97001
.96091
.96967
.96960
.96978
.96966
.96960
.96946
.96987
.96030
.96016
.96008
.96894
.96880
.96878
.96866
,96858
.96851
,96644
.96887
,96829
,96815
,96807
96800
96793
96786
96778
96771
96764
96756
96749
96742
96734
96787
96719
96712
96705
96697
966901 18
96662 18
96675 11
96667 10
.85657
.96660
.96658
.96645
.85713
.85741
.85769
.85796 .966151
96606)
966001
25682 .96593
966301
.25864
.25682
Cosin
Sine
76-
76»
488
TABLE IV.— NATURAL SINES AND COSINES.
/
"o
16"
ie«
17* 1
18«
1 19«
/
60
Sine ICosin
Sine iCosin
Sine
Cosin
Sine
Cosin
.95106
1 Sine
.82567
Cosin
.258821.96598,
.275641.96128
.29237
.95630 :80902
.04558
1
.25910 .965851, .27592 .96118' .29265
.95622. .80929
.95097
.32584
.94542
60
?.
.25988|.96578|l .27820,. 96110|; .29293
.96618; .80967
.96088
.82612
.94583
58
8
.25966.96570' .27&18'. 96102,1 .29821
.96606! .80985
.95079
.a»89
.94523 57
4
.260941.966621
.27676 1.960941;. 29348
.96696, .81012
.96070
.82667
.94514 56
5
.26022;. 96555
. 27704,. 960861 1.29376
.95588; .81040
.95061
.82694
.04504
65
6
.26060 .96&17I
.27781 .96078 1.29404
.96579! 1.81068
.960^2
.38722
.04495
54
7
.28079
.96540
.27769|-96070 .2W32
.955n .81096
.96048
.82749
.04485
58
8
.26107
.96632
.27787 .96062 1.29460
.278161.96054 1.29487
.95662 ; .81128
.96088
.82777
.04476
52
9
.26185
.96624
.96664 1 .81161
.96024
1.82804
.04466
51
10
.26168
.96617
.27848 .96046 1,. 29515
.96646 , .81178
.06016
.82882
.04457
60
11
.26191
.96609
.27871
.96087 .29648
.96686 ' .81206
.96006
'.82860
.04447
40
12
.26219 .966021
.27899
.96029 .29671
.9562811.31233
.94997
1.82887
.04488
48
13
.262471.96494,
.27927
.96021 1 .29599
.95519, .81261
.94968
.82914
.04488
47
14
.26275 .96486'
.27955
.96013 .29626
.95511 II. 81289. 94979
.82942
.04418
46
16
.268081.96479!
.27988; .960051 1 .29654; .96502 1 .31316, .94970
1 .82060
.04409
45
16
.26331
.96471 .280111.959971,. 29682I.9&493 .81344 .94961
.82097
.94899
44
17
.26359
.96463
.28039!. 96989 .297101 .95486 ; .31372
.94962
.88024
.94890
48
18
.26887
.96466)
.28067 .95981 .29737 '.96476, .81899
.94943
.88051
.94880
48
19
.26415
.96448
.28096.95972 .297651.95467 .81427
.94983
! .88070 .94870, 41
20
.26443
.96440
.281281.96964
.29793 .95459 .81464
.949»4
' .88106 .04861
40
21
.26471
.96488!
.28160 1.96966
1.29821 .96450!;. 81482
.94915
.88184 .04351
80
22 .26600
.96425
.28178. 96948
.29849 .95441, .81510
.94906
.881611.04842! 88
28 1.26528
.964171
.282061.95940
1.29876 .95483'!. 81537
.94897
1 .88189. 94832 87
24
.26566
.96410,
.28234. 96031
1.29904'. 95424 11.31565
.94888
.382161.94822 88
25
.26684
.96402
.28262 .9»928
, .29932 .95415 .81503
.94878
1.38244,. 04818 85
26
.26612
.96394,
.282901.96915
1.29960 .95407, .81620
.94869
.83271
.94803
84
27
.26640
.96386,
.28818 .96907
.29987 .95898l|. 81648
.94860
.83208
.04293
38
28
.26668
.963791
.28346 .96808
.800151.95389, .81675
.94861
.88826
.94284
82
29
.26696
.96371
.28374 .95890
.800431.95380 .81708
.94842
.88858
.04274
81
80
.26721
.96863
.28402 .96682
.80071 .96872 1 .81780
.94882
.88881
.04264
80
81
.26752
.96865
.28429 .96874
.80098 .95863'i. 81758
.04828
.88408
.04254
20
82
.26780
.963471
.28457 .95865
.80126 .95354,1.81786
.94814
.88486
.04245,28
38
.26808
.96340,
.28185 .95857
.801541.95345 !. 81813
.94806
;. 83468
.04285 27
84
.26836
.96382'
.28513 .95*49
.80182
.953371 1.81841
.947%
.88490 .04225126
85
.26864
.96S34
.28541 .95841
.80209
.95828 I .81808
.94786
1.88518 .04215 25
86
.26892
.96316
.28569 .95882
.80237
.95319 ' .81896
.94777
.88645 .04206124
87
.26920
.90308
.28597 .95824
.30265
.95310 1 .81923
.04788
1.88573 .04106,28
38
.26948
.96301
.28625 '.95816
!. 30292
.95301 .31951
.04758
1. 88600 1.04180 122
39
.26976
.96293
.28652 .aVW
1. 30320;. 95293 1.81979
.94749
.88627 '.04176 21
40
.27004
.96285
.28680 .95799
.80848 .95284 1.82006
.04740
.88655 .94167:20
41
.27T)82
.9627?'
.28708 '.95791
.80876 .95275 '1.82034
.04780
'.88682 .04157119
42
.27060
.96269,
.28736 .95;^
.30403 .95266 i .82061
.04721
1.88710 1.04147 18
43
.27088
.96261
.28764 .95774
.30431 . 95257 1{. 32089
.04n2
'.88787,. 04137
1 .8»r64 .04127
17
44
.27116
.962531
.28792 .95766
.30459 .95248 .82116
.04702
16
45
.27144
.96246
.28820
.95757
.80486 .95240 .82144
.04693
1.33702 '.941 18
15
46
.2n72
.962381
.28847
.96749
.30614
.952311
.82171
.046*4
1.38819 .04108
14
47
.27200
.96280
.28875
.95740
1.30542
.95222
.82199
.04674
.88846 .04098
18
48
.27228
.962221
.28903' 96732
1.80570
.962131
.82227
.04665
.88874 .94088
12
49
.27256
.96214
.28931 .95724
.80597
.952041
.82254
.04656
1 83901 .94078
11
50
.27284
.962061
.28959 .95716
.30625
.96195,
.82282
.0464d
.88929 .04068
10
51
.27812
.96198
.289871.96707
'.80658
.961861
.82809
.04637
.889561.94068
0
52
.27340
.96190
.29015'. 96698 .30680
.95177
.82387
.04627
' .38983 .04049
8
58
.27368 .96182
.29042,. 95690
.30708
.95168'
.82364 .94618
.84011 1.94089
7
54
.27396'. 96174
.290^/0 .0)681
.30736
.95159
.82392
.04609
.84038 .94029
6
65
.274241.96166
.29098 .95673
.30763:. 96150 1.32419
.04699
1 .84065 .04010
5
56
.27452 .96158
.29126 .95664 .307911.95142 ' .82447
.94690
.34093!. 04009
4
57
.27480 1.96150
.29154. 95656
.80819 .95133 ; .82474
.94680
, .84120 .98999
8
58
.27508. 96142
.29182 .95647
.80846 .96124 .82502
.04571
' .84147 .98969
8
69
.27536 .96134
.29209 .95639
.30874 .95115 1 .82529
.94561
.84175 .08979
1
60
.27564 {.96126
.292:^ .95680
.30902 95106 | .32557
.94.V>2
,.84202 .98969
_0
/
Cosin, Sine
Cosin Sine
Cosin I Sine Ck>8in
Sine
Cosin {Sine
9
740
78-
72» 71« 1
70*
489
>>-.
TABLE IV.— NATURAL SINES AND COSINES.
20»
Sine Cosiii
.84208
.84229
.84257
84811
84839
,84888
.84421
.84448
.84475
.84608
.84680
84667
.84584
.84612
.84689
.84666
.84694
.84721
.84748
.84775
.84808
.84880
.84867
.84884
.84912
.84966
.84996
.86021
.85048
.86075
.85102
.85180
.85157
.85184
.85211
.85847
.86875
.85402
.85429
85456
.85484
.86511
.86565
.85608
.85619
.85047
.85674
.85701
.85728
.86756
85782
.85810
98969
98949
98989
.98919
98879
.98849
.98819
.98809
.98799
.98789
.98779
.98769
.98759
.96748
.98788
.98728
.98718
.98708
.986Tr
.98667
.98657
.98M7
.996371
.98616
.93606
93596
.98575
.93565
.98555
.93544
.93581
.98514
.98508
.98493
.93472
.93462
.98462
.98441
.98431
.93420
.98410
.98400
.98879
Cosin I Sine
69^
^Ine Coain __
.40674 .91355 60
.40700 .9iai3 59
.40727 1. 91831 58
.40753,. 91319, 57
.40780 .91307 156
.406061.91296 56
.40833 '.91283 54
.40860 .91272 58
.40686,. 91 260 52
.409181.91248 51
.40989
.91236 50
.91224)49
.91212 48
.91200 47
.91188 46
.91176, 45
.91164 1 44
.91152 48
.91140 42
.91128 41
.911161 40
.91104 39
.91092 38
.91080 87
.91068 36
.91056,36
.91044 34
.91082 38
.91020, 82
.91006' 81
.9099^130
.36948 .92924
.86975 .929131
.87002 '.92002
.87029
.92892
.87066
.92881
.37088
.92870
.37110
.92859
.371371.928491
.37164
.92888
.8n91
.92827
.87218
.98816
.87245
.98805
.87272
.92794
.87899
.92784
.87826
.92778
.87863
.92782
.87380
.92751
.87407
.98740
.87434
.92729
.37461
.92718
Cosin
Sine
61
!•
490
TABLE rV.— NATURAL SINES AND COSINES.
25* 1
26<' 1
27» 1
28* 1
29*
/
60
Sine
Coslii
.90631
Sine
.48887
Cosin
789879
Sine
.45399
Codn
.89101
Sine
.46947
Codn
Sine ICosIn
.42262
.88295
.48481
.87462
1
.42288
90618
.48863
.89867
.45425
.89067
.46973
.8^81
.48506
.87448
59
2
.42815
90606
43889
.80664
.46451
.89074
.46999
.88267
.48582
.87484
58
8
.42841
.90594
48916
.89841
.46477
.89061
.47024
.88264
.48567
.87420
57
4
.42387
90582
.43942
.89888
.45503
.89048
.47050
.88240
.48588
.87406
66
6
.42394
.90569
.48968
.89816
.45529
.89035
.47076
.88226
.48606
.87891
65
6
.42420
.90557
43994
.89803
.45554
.89021
.47101
.88213
.48684
.87877
64
7
.42446
.90545
.44020
.89790
.45580
.89008
.47127
.88199
.48659
.87863
58
8
.42478
90532
.44046
.89777
.45600
.88995
.47153
.88185
.48684 .87849
62
9
.42499
.90520
.44072
.89764
.46632
.88981
.47178
.88172
.48710
.87886
61
10
.42626
.90607
.44098
.89752
.45658
.88968
.47904
.88168
.48786
.87821
60
11
.42652
90495
.44124
.89789
.45684
.88965
.47220
.88144
.48761
.87806
49
12
.42678
90483
,44151
.89726
.45710
.88942
.47255
.88180
.48786
.87802
48
18
.42604
90470
.44177 .89713
.45786
.88928
.47281
.88117
.48811
.87278
47
14
.42681
90458
.44203 .89700
.45762
.88915
.47306
.88108
.48887
.87264
46
16
.42667
.90446
.44229
.89687
.45787
.88902
.47332
.88089
.48862
.87250
45
16
42683
.90483
.44255
.89674
.46813
.88888
.47858
.88075
48888
.87235
44
17
.42709
.90421
.44281
.89662
.45839
.88875
.47883
.88062
.48918
.87281
48
18
.42786
9040S
.44307
.89649
.45865
.88862
.47409
.88048
.48988
.87207
42
19
.42762
.90896
.44333
.89686
.45891
.88848
.47434
.88034
.48964
.87198
41
20
.42788
.90883
.44859
.89623
.45917
.88835
.47460
.88020
.48060
.87178
40
21
.42816
.90371
.44885
.89610
.45942
.88822
.47488
.88006
.49014
.87164
89
22
.42841
90858
.44411
.89597
.45968
.88808
.47511
.87998
.40040
.87150
88
28
.42867
90346
.44437
.89584
.45994
.88795
.47537
.87979
.49066
.8n86
87
24
.42894
90834
.44464
.89571
.46020
.88782
.47562
.87965
.49090
.87121
96
25
.42920
90321
.44490
.89558
.46046
.88768
.47588
.87961
.49116
.87107
86
26
.42946
.90809
.44516
.89545
.46072
.88755
.47614
.87937
.49141
.87098
84
27
42972
90296
.44%42
.89532
.46097
.88741
.47039
.87923
.49166
.87X)7»
88
28
.42999
.90284
.44568
.89519
.46123
.88728
.47665
.87909
.49192
.87064
82
20
.48025
.90271
.44694
.8950G
.46149
.88715
.47090
.87896
.49217
.87060
81
80
.48061
.90259
.44620
.89493
.46175
.88701
.47716
.87882
.49242
.87086
30
81
.48077
.90246
.44646
.89480
.46201
.88688
.47741
.87868
.40268
.87081
29
82
.48104
.90233!
.44672
.89467
.46226
.88674'
.47767
.87854
.40298
.87007
28
88
.48180
90221;
.44698
.89454
.46252
.88661
.47793
.87840
.49318
.80098
27
84
.48156
90208!
.44724
.89441
.46278
.88647
.47818
.87826
.40844
.86078
26
86
.48182
90196,
.44750
.89428
.46304
.88634
.47844
.87812
.49860
.86064
25
86
.48209
.90183
.44776
.89415
.46330
.88620
.47869
.87798
.49894
.86948
24
87
.48285
.90171
.44802
.89402
.46355
.88607
.47895
.87784
.49419
.86086
28
88
.43261
: 90158
.44828
.89389
.46381
.88593
.47920 i.srr/o
.49446
.86981
22
89
.43287
.90146
.44854
.89376
.40407 '.885801
.47W6
.87756
.49470
.86906' 21 1
40
.48818
.90133
.41880
.89363
.46483
.88566
.47971
.87743
.48496
.86802
20
41
.48840
.90120
.44906
.80350
.46458
.88553
.47997
.87TO9
.40621
.88878
10
42
.48866
.90108,
.44932
.89337
.40484
.88589
.48022 '.87715
.49546
.86868
18
48
.43892
.900951
.44958
.89324
.46510
.88526
.48048 .87701
.49571
.86840
17
44
.48418
90082
.44984
.89311
I .46536 .885121
.480731.87687
.49596
.86884
16
45
.43445
.900'.^
.45010
.89298
1 .46561 1 .88499
.48099 .87678
.49622
.86820
16
46
.48471
.90057
.45036
.89285
.465871.88485
.48124 .87659
.49647
.86806
14
47
.43497
.90045
.45062
.89272
1.406131.88472
.48150 .87645
.49678
.86791
18
48
.43528
.90032
.45088
.89259
.46689;. 88458
.48175 .87681
.49697
.86777
12
40
.43549
.90019
.45114
.89245
1.46664
.88445
.482011.87017
.49788
.86762
11
60
.48575
.90007
.45140
.80232
'.46600
.88481
.48226 .87608
.48748
.86748
10
51
.43602
.89994
.46166
.89219
.46716
.88417
.48252 .87589
.49778
.86788
9
6S
.48628
.45192
.892061 .46742
.8^404
.48277 .87575
.49798
.86n9
8
68
.48654
!89968
.45218
.89193
.46767
.SK^IK)'
.48308 .87561
.40624
.86704
7
64
.43680
.89956,
.46243
.89180
,.46793
.88:^77
.48828
.87546
.49849
.86600
6
66
.48706
.899431
.45269
.89167
1.46819
.88363
.48354
.87582"
.49674
.88676
5
56
.48783
.89930
.45295
.891.53 .46844
.88*19 ; .48379
.87518,:. 49809
.86661
4
57
.43759
.89918
.45321
.89140, .46870
.rnVVS .48405
.87504
.49924
.86646
8
68
.43785
.89905
.4&^7
.89127 .46896
,88:^22 .48430
.87490
.49950
.86682
2
69
.438111.89892
.45373
.89114
, .46921
.saws , .48456
.87476
.49976
.86617
1
60
9
.438371.89879
.45399
Cosin
.89101
"Sine
' .46947
(Josin
.88205
Sine
.484811.87462
Cosin 1 Sine
.50000
Cosin
.86608
Sine
_0
/
Cosin 1 Sine
64-
e3»
i 62- "1
ei'
1 0O<»
491
TABLE IV.—
NATURAL SINES AND COSINSSl
"o
80* 1
t 81«» 1
V 82-
88«
84-
i
00
Sine
.50000
CoBin
.86608
Sine
:51'504
Cosin
Sine I Cosin
.52992 .84805
SinelCosin
Sine
Cosin
.85717
.54464 '.83867
.56019
.82904
1
.50025
.86588
.61529
.85702
.63017 .84789
.64488 .88851
.55948
.82887; 6©
2
.50050
.86578
.61554
.85687
.68041 .84774
.64518 .88835
.55968
.828n|68
8
.50076
.86659
.51579
.85672
.68066 .84759
.64587 .83819
.55992
.a»65!67
4
.50101
.86544
.51604
.85657
.530911.84748
.64661 .83804
.56016
.82889 66
5
.50126
.86530
.61628
.85642
.531151.84728
.54586 .88788
.56040
.82822
65
6
.50151
.86515
.61653
.85627
.68140
.84712,
.64610 .8877%
.56064
.82806
64
7
.50178
.86501
.6167«
.85612
.68164
.84697
.54685 .83756
.50088
.82790
68
8
.50201
.8ft486
.61703
.86597
.58189
.846811
.54659 .88740 i .56112
.82778
62
0
.50227
.86471
.61728
.a'5582
.53214
.84666 1. 54688. 83724; .56186
.82757
61
10
.60252
.86467
.61763
.85667
.68838
.84660
.647061.83708 .56160
.82741
60
11
.60277
.86442
.61778
.85561
.68263
.84686*
.54782!. 88092 1.66184
.8S724
49
12
.50802
.86427
.51803 '.855361
.63288
.84619
.54756 .83676 1.50208
.82706
48
18
.50827
.86413
.518281.85521 1
.53312
.84604
.64781 .83660
.66282
.82692
47
14
.50852
.86398
.51853
.85506
.63337
.84588
.64805 .88645
.66256
.82875
46
15
.50377
.86384
.61877
.85491
.53361
.84573
.64829 .83629
.66280
.82659
45'
16
.50108
.86:369
, .61902
.8&476
.58386
.84557
.64854 .88613
.66805
.82648:44
17
.50428
.86354
.51927
.85461
.6.3411 .84542
.54878 •.88597
.66329
.82626 48)
18
.50458
.86810
.51952
.85446
.68435
.84526,
.64902 .88581
.56858
.82610 42
19
.50478
.86325
.61977
.85431
.63460
.84511
.54927 .88565
.66877
.82503 41
SO
.50608
.86310
.62002
.85416
.5»m
.844951
.64961
.88649
.66401
.82677,40
21
.60528
.86295
.52026
.85401
.68509
.84480
.64975
.88688'
.66425
.82561189
22
.50553
.86281
.5aaji
.85885
.63584
.84464 ! .64999
.83517,
.66449
.82544 86
28
.50578
.86266
'52076
.85370
.5^58
.84448
.550241.88501
.66473
.82528; 87
»t
.50603
.86251
1 .52101
.85355
.53383
.84433
.55048 .88485
.66497
.82511 86
25
.50628
.86237
.52126
.85840
.53607
.SU17
.65072 .88469
.66521
.82495, 85
26
.50654
.86222
.62151
.85325
.53632 .84402
.55097 .88453
.66645
.82478 84
27
.50679
.86207
.52175
.83310
.53666 .84886
.65121 .88487
.56569
.82462 88
28
.50704
.86192
.52200
.85294'
.53681
.84370
.55145 .834211
.66593
.82446 82
29
.50729
.86178
.52225
.85279
.63705
.ai;J55
.55169 .8ai05'
.66617
.82429 81
SO
.60754
.86168
.62250
.85264
.63730
.84889
.65194 .88389 j
.66641
.88418, 80
81
.50779
.86148
.52275
.85249
.5875-4
.84324
.65218 .888781
.66665
.82896
28
82
.506&1
.86183
.52299
.85234
.63779
.84308
.55242. 88856'
.66689
.82380
28
88
.50829
.86119
.52324
.85218
.63804
.84292
.55286,. 88340
.66713
.82368
27
Si
.50854
.86104
.52349
.852031
.53828
.84277
.55201 .83324
.66736
.82847
26
85
.50879
.86089
.52374
.85188'
.53853
.84261
.558151.83308,
.66760
.82380
25
86
.50904
.86074
.52399
.851731
.538771.84245
.55389 .83292
.66784
.82814' 24
87
.50929
.86059
.52423
.85157
.5.3902,. 84230
.5536:3 .83276
.568081.82297 28
88
.50954
.86045
.52448
.85142
.53926
.84214
.55388 .88260,
.66832
.82281 22
39
.50979
.80030
.5247:}
.85127
.53951
.ail98
.554121.83244
.66856
.82264
21
40
.61004
.86015,
.52498
.85112
.58975
.84182,
.55436 83228, |. 56880
.82248
20
41
.51029
.860001
.52522
.86096 1
.54000
.84167
.65460 1.88212' .66904
.82281
19
42
.510M
.859851
.52547
.85081 1
.M024
.84151
.554*4 '.83193 .66928
.82214
18
48
.510791.85970
.525?2
.85006:
.54049
.84185
.55509,. 83179 j .66962
.82198) 17
44
..')1104.. 85956,
.52397
.85031
.54073
.84120
.5f>333'. 88163 1 1.56976
.821811 16
45
.51129
.85941
.52621
.85035
.54097
.84104
.655571.88147 i .57000
.82165 15
46
.51154
.85926
.52646
.86020
.54122
.W088
.555811.88181 1.67024
.82148 14
47
.51179
.85911
.52671
.8500)
.54146
.84072
.65605 .88115 1 .57047
.82182, 13
48
.51204
.85896
.52696
.84989
.54171
.84057!
.65680
.83098 .67071
.82115 12
49
.51229 .a'>881 •
.52720
.84974
. 54195 ;.&4041'
.65654
.83082 .67095
.820981 11
60
.61254
.85866
.52745
.84959 1
.54220
.84025
.65678
.83066 .57119
.62082^ 10
61
.61279
.86861 1
.62770
.849481
.64244
.8i009
.56702
.88050 .67143
.82066
9
62
.51304
.85836
.52794
.84928
.64269
.8.3994
. 55726
.88084 ; .67167
.82048
8
63
.51829
.85821
.62819
.&1913
.54293
.88978
.66750
.83017;
.6n91
.82032
7
64
.51854
.85806
.52844
.848{>7
.54317
.83962
.56775
.830011
.67215
.82015
6
65
.61879
.85792
.52869
.WH82
.54.S12
.&3946
.66799
.82985
.67238
.81999 6
66
.61404
.85777
.52893
.^4^^
.54.366
.83930
.55823
.82969'
.57262
.81962 4
67
.51429
.86762
.62918
.ai851
.54391
.83916,
.65847
.82953
.67286
.81965 8
68
.51464
.85747
.62948
.ai836
.544 15. 88899!
.55871
.829361
.67310
.81949 2
69
.61479
.85792
.62967
.84820
.54440 .83883!
.5^895
.82920!;. 67834
.81932 1
60
.51504 .85717
.62992
.84805
.544W .838071
.55919
.82904 .67358
.81915; 0 I
/
Cosln Sine
Cosin
Sine ,
Cosin rsine"
Cosin
Sine l| Cosin
Sine
^ ,
69«
58- 1
67«
56' II SS'
_
493
TABLE IV.— NATURAL SINES AND COSINES.
"o
860 I
86*
ar* 1
88* 1
89-
/
60
Sine
Ooain
.81915
Sine Cosin
.587791.80902
Sine
.60188
Cosin
.79864
Sine
.61566
Cotdnl
:'re80i
Sine
Cosin
.57S58
.62982
.77715
1 .67881
.81899
.58808
.806851
.60805
.79646
.61589
.78783
.62965
.77696
59
2
.57405
.81882
.58886
.80867'
.60228
.79829
.61618
.78766
.62977
.77678
58
8
.57429
.81865
.68S49
.80650
.60251
.79811
.61685
.78747
.63000
.77660
57
4
.57458
.81848
.58878
.808881
.60874
.79798
.61658
.787^
.63022
.77641
66
6
.57477
.81832
.58896
.808161
.60296
.79';76
.61681
.78711.
.68045
.77828
55
6
.57501
.81815
.58920
.807991
.60821
.79758
.61704
.78694
.68068
.77606
54
7
.57524
.81798
.58948
.807881
.60344
.79741
.61726
.786761
.68090
77586
58
8
.67548
.81788
.58967
.80765
.60367
.79728
.61749
.78658
.68118
.77568
62
0
.67572
.81766
.58990
.80748
.60890
.79706
.61772
.78640
.68185
.77860
51
10
.57596
.81748
.59014
.80780
.60414
.79688
.61796
.78622
.68168
.77681
60<
11
.57619
.81781
.59037
.80718
.60487
.79671
.61818
.78604
.68180
.77518
49
IS
.57648
.81714
.59061
.80696
.60460
.79658
.61841
.78686
.68806
?74fM
48
18
.57667
.81698
.69064
.80679
.60488
.79685
.01864
.78668
.68225 .T747«
47
14
.57891
.81681
.59108
.80668
.60506
.79618
.61887
.78660
.68248 .77458
46
15
.57n5
.81664
.50181
.80644
.60529
.79600
.61909
.78582
.68871,. T7489
46
16
.57788
.81647
.59154
.80687
.60658
.79588
.61982
.78514
.68293 !.774S1
44
17
.67762
.81681
.59178
.80610
.60676
.79565
.61955 .7B496
.68816,. 77402
48
18
.57788
.81614
.59801
.80503'
.60599
.79547
.61»re
.78478
.68888 1.77884
42
19
.57810
.81597
.59885
.805761
.60622
.79580
.62001
.78460
.68361 .77866
41
20
.57888
.81680
.69248
.80558
.60645
.79512
.62024
.78442
.68888 .77847
40
21
.57867
.81668
.69872
.80541
.60668
.79494
.62046
.78424
.68406 .77889
89
22
57881
81546
69295
80524
60091
79477
62069
TRJnK
68428 77810
88
28
.57904
.'81580
!59818
!80507
!60714
.'79450
.62092 .78887
.68451 .77282
87
24
.57988
.81518
.59842
.80489
.60788
.79441
.62115 .78869
.68473 .77278
86
25
.57952
.81496
.59365
.80472
.60761
.79424
.62188
.78851,
.68496 .77866
85
26
.57976
.81479
.59389
.80455
.60784
.79406
.62160
.78838,
.68618. T7886
84
27
.57999
.81468
.59418
.80488
.60607
.79888 .62188
.783151
.68540 .77218
88
28
.58028
.81445
.80480
.60830
.79371
.62806
.78297
.68668
.77199
82
20
.58047
.81488
.59459. 804031
60653
.79358
.62829
.78879
.68585
.77181
81
SO
.58070
.81412
.59482,. 80386;
.60676
.79885
.62251
.78861
.68606
.77168
80
81
.58094
.81395
.59506
.80368
.60899
.79318
.62274
.78248
.68680
.77144
29
82
.58118
.81878
.59529
.80351,
.60928
.79300
.62297
.78225
.68858
.7n26
28
88 .58141
.81361
.59552
.803341
.60M5
.79888
.62820
.78206
.68676
.77107
27
84
58165
81844
59570
80316
60968
79264
.68842
78188
68698
.77068
26
85
!58189
.81887
.59599
!80899;
.60991
! 79247
62865
i-ffilTO
!68720
.77070
25
86
.58212
.81810
.59622
.80882:
.61015
.79229
.68888
.78152
.68742
.77051
24
87
.56286
.81298
.59646
.808641
.61088
.79211
.62411
.78184
.68765
.75X188
28
88
.58260
.81276
.59669
.802471
.61061
.79198
.62488
.78116
.68787
.77014
28
89
.58888
.81250
.59693
.808801
.61084
.79176
.62456
.78098
.63810
.76996
21
40
.58807
.81242
.59716
.80812
.61107
.79158
.62479
.V80'<»|
.68882
.78977
80
41
.58380
.81225
.59789
.80195
.61180
.79140
.62602
.TWWll
.68854
.76960
19
42
.68354]. 812081
.59763
.80178;
.61153
.79128
.62524
.78048
.68877
.76940
18
48
.58878 .81191'
.59786 .80160:
.61176
.79105
.62547
.78025
.68899
.76921
17
44
.58401 .81174
.59809
.80148'
.61199
.79087
.62570
.78007
.63982
.76908
16
45
.58425 '.'81157
.69632
.801251
.61228
.79069
.62598
.77988
.639441.76884
15
46
.58449,. 81140
.59856
.80108
.61245
.79051
.62615
.77970
.68966
.76866
14
47
.58472
.81128
.59879
.80091
.612681.79088
.62688
.77952
.68989
.76W7
13
48
.58496
.81106,
.59902
.80078
.61291
.79016
.68660
.77984
.64011
.76888
12
49
.58519
.81089
.59986
.80066
.61814
.78998
.68688
.77916
.64083
.76810
11
60
.58548
.81078
.59949
.80038
.61887
.78980
.62706
.77897
.64056
.76791
10
61
.58667
.81065
.59972
.80021
.61360
.78962
.68788
.77879
.64078
.76772
9
52
.58590
.81038,
.59995
.80003
.61883 .789441
.68751
.77861
.64100
.7B7H
8
58
.58614
.81081
.60019
.79986
.61406
.78986
.62774
.77843
.64128
.78736
7
54
.58687
.81004,
.60048!. 79968
.61429
.78008
.62796
.77884;
.64145
.76717
6
66
.58661
.80987
.60065
.79951!
.61461
.78891
.68819
.77806-
.64167 .76696
5
56
.58684
.80970 ; .60089
.79934
.61474
.78878
.68842
.77788
.641901.76679
4
67
.58708
.80953
.60118
.79916
.61497
.78865
.62864
.77769,
.64212 .76661
8
68
.58781
.809381
.60135
.79899
.61680
.78887 1.68887
.77751
.64284 .78642
8
59,. 58786
.80919
.60158
.79881
.61548
.78819
.62909
.Tr?83,
.64256 .76688
1
60 .58779
.80908
.60188
.79864
.61566
.78801
.62938
.77715
.64279 .76604
J9
/
Cofiin
Sine
Ck>ain
Sine
Cosin
Sine
Cosin
"StaVi
Cosin, Sine
54- 1
6J
J»
52- 1
61- 1
W i
493
TABLE IV.— NATURAL SINES AND COSINES.
40*
Sine
Codn
0
Mm
.76604
1
.64801
.70586
8
.6438S
.76567
8
.648M
.70548
4
.64368
.76530
6
.64390
.76511
6
.6441S
.76493
7
.64485
.75473
8
.64457
.76455
9
.61479
.76436
10
.64501
.78417
11
.64584
.76896
12
.64546
.78880
18
.64568
.76361
U
.64590
76343
15
.64618
.76388
16
.64685
.78804
17
.64057
.78886
18
.64679
.76867
19
.64701
.76848
SO
.64788
.70889
81
.64746
.70810
88
.64788
.78198
88
.64790
.76178
84
.64818
.78154
85
.64884
.76185
86
.64866
.76116
87
.64878
.780971
88
.64901
.78078
89
.64983 .760691
80
.64945
.76041
31
.64967
.76083
88
.64969
.70003
88
.6S011
.75984
84
.65088
.75065
85
.65065
.759461
86
.66077
.75087
87
.65100
.75906
88
.65183
.75889
89
.65144
.75870
40
.65166
.75851
41
.66188
.75888'
48
.65310
.75813
48
.65883
.75794
44
.651^
.75775
46
.658r6;. 75756!
46
.66806
.757381
47
.65880
.76719
48
.66848
.75700
49
.66864
.75680
50
.65886
.75661
51
.06408
.75648
68
.65480
.76083
58
.65458
.75604
54
.66474
.75585
55
.65496 .75566
66
.66518 .75547
57
.66540
.75528
m
.75509
69
tMKHA
.00004
.75490
60
.66606
.75471
/
Oosin
Sine
41
^
41»
Sine
.66606
.66050
.05678
.05004
.05710
.06738
.65759
.65781
,65808
.65847
.65913
.66956
.66078
.60000
,00044
66000
.00068
.00109
,60181
,00153
60175
86197
60818
00384
00300
00887
00340
00371
06414
66436
66158
.66480
66501
6651o
66566
66610
Cosin
.7W71
.75152
.75133
.75414
.75395
.75375
.75356
.75387
75318
.75299
.75380
.75861
.75841
66653
.66675
.66697
.66718
.66740
.60708
.00783
.06806
75803
.75184
75106
.75146
.75186
,75107
7B088
75069
.75050
75090
75011
.74932
•4978
.749531
74984
749151
74800
74878
748571
74888
748181
74799
74780
74760
74741 i
74722
747031
746831
74664
74644'
74625
74006
74586
74567
745481
745281
74509 1
74489
74470
,744511
.74431'
.74412
.66870
.66018
Cbsin
.74373
.74358
.74884
.74314
Sine
48»
42«
Sine
,66918
.66956
.67021
.67048
.67064
.67086
.67107
.67189
.07151
.67178
.67194
.67315
.07887
.07858
.07880
.07301
.07823
.07844
.67866
.67887
.67409
.67490
.67458
.67473
.67495
.67516
.67538
.67559
.67580
.67608
Cosin
.74314
.74295
.74276
.74256
.67615
,67866
,67709
,67730
67752
,67773
,67795
.67816
,67837
.67859
.67880
.67901
67944
,67965
67987
.08006
.74217
.74198
.74178
.74159
.74139
.74180
.74100
.74080
.74061
.74041
.74082
.74008
.73963
.73044
.73084
.78904
.73865
.78816
.78806
.78787
.73767
.73747
78788
73706
.78688
78669
78649
.78629
78610
.73590
73670
.73551
78531
78511
73101
73172
73462
73432
73413
73398
73373
78853
73333
,78814
78894
,73274
.73254
Cosinj
.73815
.73195
.73175
.73155
,78135
Sine
47*
48*
Sine
.68864
.68306
.68370
.08891
.08418
.08484
.08455
.08476
.68497
.68518
.68561
.68645
.68709
.68730
.68751
.68773
.68793
.68814
.68857
.68878
.68920
.68941
,69001
,69016
,69067
,69088
,69109
,69130
,69151
,69172
,69214
.69377
,69340
.09361
.69103
.69424
.69445
.69166
Cosin
Cosin
.73135
.73116
.73090
.78070
.73050
.73030
.73016
.72996
.72976
.72957
.78987
.72917
.72897
.72877
.78857
.72837
.72817
.72797
.72Trr
.72757
.72787
.78717
.72697
.786771
. 78667 ■
.78637,
.786171
.72597
.78577,
.72557
.78537
.78517
.78497
.784rr'
.724571
.78437
.7TM17
.72397
.72377
.72357
.78387
.78817
72277
72257
78210
72196
78176,
72156
78136
72116
72095
72075
73056
72035
72015
71995
,71974
.71964
.71934
440
Sine
46»
Sine
.69487
.69508
.69529
.60549
.69570
.69591
.69618
.69688
.69654
.09075
.09090
.09717
.09787
.09758
.09779
.09800
09821
C06in{ _
.71934,00
.71914 69
.71894 58
.71873 57
.71853 50
.71838 55
.71813
.71792
.71772
.71752
.71788
.71711
.71091
.71071
.71050
.71080
.71610
.71590
.09904
.09925
.00946
.69642 .71569
.69862 .71549
.71589
.71508
.71488
.71468
.71447
.71427
.71407
.71886
.71366
.71345
.71885
.69987
,70008
.70049
.70070
.70001
.70118
70132
.70153
.70174
.70195
.70215
.70236
.70257
.70277
.70298
.70319
.70339
.70360
.70381
.70101
.70422
.70143
.70468
.7W^
.70505
.70525
.70546
.70567
.70687
.70608
.70828
.70649
.70670
.70690
.70711
.71805
.71284
.71264
.71243
.71203
.711821 28
.71162 28
71141 81
71181 80
.71100 19
71080 18
.71059 17
.71089
.71019
.70998
.70978
.7095:
.70937
.70916
70896
,70875
,70855
,70634
,70813
,70793
.70772
.70752
.70731
.70711
Cosin I Sine
46^*
494
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
O* 1
1- 1
1 2« 1
8<»
/
60
Tang
.00000
Cotangr
Tang
.01746
Ck>tang
Tang
Cotang
Tang
.05241
Cotang
Infinite.
67.2900
.03492
28.6868
19.0811
1
3487.75
.01776
66.8506
.08521
28.8994
.06270
18.9755 60
2
.00058
1718.87
.01804
66.4416
.08560
28.1664
.06299
18.8711 68
8
.00087
1145.92
.01888
64.5618
.08579
27.9872
.05828
18.7678 67
4
.00116
859.486
.01862
68.7066
.08609
27.7117
.05857
18.6656 66
5
.00146
687.649
.01891
62.8821
.06688
27.4899
.06387
18.5646 I55
6
.00175
672.957
.01920
62.0807
.08667
27.2715
.05416
18.4645 ,64
7
.00204
491.106
.01949
61.8032
.03696
27.0566
.06445
18.8655
68
8
429.718
.01978
60.5485
.03725
26.8450
.05474
18.2677
52
9
.00262
381.971
.03007
49.8167
.03754
26.6367
.05508
18.1708
61
10
.00291
848.774
.02036
49.1039
.08788
26.4316
.06588
18.0760
60
11
.00820
812.521
.02066
48.4121
.08812
26.2296
.05662
17.9602
49
12
.00349
286.478
.02095
47.7396
.03842
26.0307
.05591
17.8888
48
13
.00878
264.441
.02124
47.0853
.03871
25.8348
.05620
17.7934
47
14
.00407
245.552
.02158
46.4489
.03900
25.6418
.OoG49
17.7016
46
15
.00436
829.182
.02182
46.8294
.03929
25.4517
.05678
17.6106
45
16
.00465
214.838
.022U
45.2261
.03958
25.2644
.05708
17.5805
44
17
.00495
2U2.219
.02240
44.6386
.03987
25.0798
.05787
17.4814
48
18
.00321
190.984
.02269
44.0661
.04016
24.8978
.05766
17.8482
42
19
.0a">.53
180.932
.02298
43.5081
.04046
24.n85
.05796
17.2558
41
20
.00682
171.886
.02328
42.9641
.04075
24.6418
.06824
17.1008
40
21
.00611
168.700
.02857
42.4835
.04104
24.8675
.05854
17.0887
89
22
.00640
156.258
.02386
41.9158
.04138
24.1957
.05888
16.9990 '881
28
.00669
149.466
.02416
41.4106
.04163
24.0268
.05912
16.9150
87
24
.00098
148.287
.02144
40.9174
.04191
23.8598
.05941
16.8310
86
25
.00727
187.607
.02478
40.4358
.04220
23.6946
.06970
16.7496
85
26
.00756
182.219
.02508
89.9655
.04250
23.6321
.06999
16.6681
84
27
.00785
127.821
.02531
89.5059
.04279
28.8n8
.06029
16.5874
88
28
.00816
122.774
.02560
89.0568
.04308
28.2187
.06058
16.5075
82
29
.00844
118.540
.02589
88.6177
.04387
28.0577
.00087
16.4288
31
80
.00878
114.589
.02619
88.1885
.04366
22.9038
.06116
16.8490
80
81
.00902
110.892
.02648
87.7686
.04896
22.7519
.06145
16.2722 ^29
82
.00931
107.426
.02677
87.8579
.04424
22 6020
.06175
16.1952 {S8
88
.00960
104.171
.02706
86.9560
.04454
22.4541
.06204
16.1190 27
84
.00989
101.107
.02735
86.5627
.04483
22.3081
.06238
16.0485
28
85
.01018
98.2179
.02764
86.1776
.04512
22.1640
.06262
15.9687
85
86
.01047
95.4895
.02798
85.8006
.04541
22.0217
.06291
15.8946
24
87
.01076
92.9085
.02822
85.4818
.04570
21.8813
.06821
15.8211
23
88
.01105
90.4688
.02851
85.0695
.04599
21.7426
.06350
15.7488
22
89
.01185
88.1436
.02881
84.7151
.04628
21.6056
.06379
15.6782
21
40
.01164
85.9898
.02910
84.8678
.04658
21.4704
.06406
15.6048 20
41
.01198
83.8435
.02989
84.0273
.04687
21.8869
.06487
15.6840 19
42
.01222
81.8470
.02968
88.6935
.04716
21.2049
.06467
15.4688 18
43
.01251
79.9434
.02997
83.3662
.04746
21.0747
.0^96
16.3948 17
44
.01280
78.1268
.03026
88.0452
.04774
20.9460
.06525
15.8254 16
45
.01309
76.3900
.03055
32.7303
.04803
.20.8188
.06554
15.2671 15
46
.01388
74.7292
.08084
32.4218
.04838
20.6982
.06584
15.1893 14
47
.01867
78.1390
.08114
32.1181
.04862
20.6691
.06618
15.1222
13
48
.01396
71.6151
.03148
31.8206
.04891
20.4465
.06642
15.0557
12
49
.01426
70.1583
.03172
81.5284
.04920
20.3253
.06671
14.9898
11
50
.01465
68.7501
.08301
81.2416
.04949
20.2066
.06700
14.9244
10
61
.01484
67.4019
.03280
80.9609
.04978
20.0872
.06780
14.8696
0
62
.01518
66.1055
.03259
80.6838
.0o<T07
19.9708
.06759
14.7954
8
53
.01542
64.8580
.03288
80.4116
.05037
19.8546
.06788
14.7317
7
54
.01571
63.6567
.03317
80.1446
.06066
19.7403
.06817
146685
6
65
.01600
62.4992
.03346
29.8828
.06095
19.62'/8
.06847
14.6059
6
66
.01629
61.3829
.08376
29.6246
.06124
19.5156
.06876
14.5438
4
57
.01638
60.3a'58
.03406
29.3711
.05153
19.4051
.06905
14.4823
8
58
.01687
59.2659
.08434
29.1220
.06182
19.2969
.96934
14.4212
2
59
.01716
68.2612
.03463
28.8771
.05212
19.1879
.06968
14.3607
1
60
/
.01746
57.2900
.03492
Cotang
28.6:363
.06241
Cotang
19.0811
.06993
Ctotang
14 8007
_0
Cotang
Tang
Tang 1
Tang
Tang
\ S9-
8«» 1
87- 1
1 86« 1
495
TABLE V.~NATURAL TANGENTS AND COTANGENTS.
/
"0
4*
6'> 1
1 e» 1
1 7«
/
60
Tang
Cotang '
Tang
.06749
Cotang
Tang 1 Cotang 1
Tang
Cotang
.06998
14.3007 1
11.4801
.10610
9.51488
.12278
8.14485
1
.07028
14.8411
.08778
11.8919
.10540
9.48781
.12308
8.18481
59
2
.07051
14.1881
.08807
11.8540
.10569
9.46141
.12338
8.1068G
58
8
.07080
14.1835
.06837
11.8168
.10599
9.48515
.12367
8.06600
57
4
.07110
14.0655
.08866
11.2789
.10688
9.40904
.12397
8.08674
66
5
.07189
14.0079
.08895
11.2417
.10057
9.38807
.12426
8.04756
55
6
.07168
13.9607
.06925
11.2048
.10087
9.85784
.12456
8.02848
54
7
.07197
13.8940
.08954
11.1681
.10716
9.83155
.12485
8.00948
68
8
.07227
18.8378
.08988
11.1316
.10746
9.80599
.12515
7.99068
52
9
.07256
13.7821
.09013
11.0954
.10775
9.28058
.12544
7.97176
51
10
.07286
18.7887
.09042
11.0594
.10606
9.1^580
.12574
7.96808
50
11
.07814
13.6719
.09071
11.0237
.10834
9.28016
.18808
7.96488
49
IS
.07844
13.6174
.00101
10.9682
.106G3
9.20516
.12638
7.91588
48
18
.07878
13.5634
.09130
10.9529 1
.10693
9.18028
.12668
7.89784
47
14
.07402
18.5098
> .09159
10.9178
.10922
9.15554
.18098
7.87895
46
15
.07431
18.4566
.09189
10.8829
.10952
9.18098
.12722
7.86064
46
16
.07461
18.4089
1 .09218
10.8483
.10961
9.10&46
.12751
7.84242
44
17
.07490
13.8515
1 .09847
10.8189
.11011
9.08211
.12781
7.88428
48
18
.07519
18.2996
.00277
lo.vvyy
.11040
9.05789
.12810
7.80688
42
10
.07548
18.2180
.09306
10.7457
.11070
9.03379
.12840
7.78825
41
80
.07578
18.1969
.09385
10.7119
.11099
9.00988
.12869
7.77085
40
81
.07fi07
18.1461
.09865
10.6788
.11128
8.98598
.12809
7.75854
89
28
.07686
18.0058
.09394
10.0450
1 .11158
8.90227
.12929
7.73480
88
88
.07665
18.0458
.09423
10.6118
.11187
8.93867
.12958
7.71715
87
24
.07695
18.9968
.09453
10.5789
.11217
8.91520
.12988
7.69967
86
25
.07724
12.9469
.(ms2
10.5462
.11246
8.89185
.18017
7.68206
85
26
.07753
12.8981
.09511
10.5136
.11270
8.86862
.13047
7.66466
84
27
.07782
12.8196
.09541
10.4813
.11806
8.84551
.13076
7.64738
38
28
.07812
12.8014
.09570
10.4491
.11885
8.82252
.13100
7.68005
38
29
.07841
12.7536
.00600
10.4172
.113ft4
8.79964
.13186
7.61887
81
ao
.07870
12.7062
.09689
10.8854
.11894
8.77689
.18165
7.69575
80
81
.07899
12.6591
.09658
10.8538
.11423
8.76425
.18195
7.57872
29
82
.07989
12.6121
.00688
10.8224
.11452
8.7^172
.18224
7.56176
88
83
.07958
12.5660
.09717
10.2913
.11188
8.70931
.13254
7.54487
87
84
.07987
12.5199
.09746
10.2602
.11511
8.68701
.13284
7.52806
86
85
.08017
12.4742
.09776
10.2894
.11541
8.66482
.18313
7.51188
86
86
.06046
12.4288
.09806
10.1988
.11570
8.64276
.18343
7.49465 '24
87
.08075
12.8838
.09834
10.1683
.11600
8.62078
.18372
7.47B06 83
88
.08104
i2.a^
.09864
10.1381
.11629
8.59893
.13402
7.46154 182
39
.08184
12.2916
.09893
10.1080
.iie>^
8.57718
.13432
7.44609 '21
40
.08163
12.2505
.09923
10.0780
.11688
8.56555
.18461
7.42871 laO
41
.08192
12.2067
.09952
10.0488
.11718
&.6W08
.18491
7.41840 '19
42
.08221
12.1632
.09981
10.0187
.11747
8.51250
.13521
7.89616
18
48
.08251
12.1201
.10011
9.98931
.11777
8.49128
.13550
7.87999
17
44
.06280
12.077^
.10040
9.96007
.11806
8.47007
.13580
7.86389
16
45
.06809
12.0346
.10069
9.98101
.11836
8.44896
.13609
7.84786
15
46
.06339
11.9983
.10099
9.90211
.11865
8.42795
.13689
7.83190
14
47
.06868
11.9504
.10128
9.87838
.11895
8.40705
.18669
7.31600
18
48
.08397
11.9087
.10158
9.84488
.11924
8.88685
.13698
7.80018 'l8|
49
.06427
11.8673
.10187
9.81641
.11954
8.86555
1 .18728
7.88448
11
50
.06456
11.8262
.10816
9.78817
.11963
8.84496
.18758
7.26873
10
51
.06485
11.7868
.10846
9.76009
.12018
8.82446
.18787
7.25310
68
.06514
11.7448
.10275
9.73217
.12042
8.3O406
.18817
7.23754
68
.06644
11.7045
.10305
9.70441
.12072
8.28376
.13846
7.22804
64
.08578
11.6645
.10334
9.67680
.12101
8.26355
.13876
7.20661
66
.08602
11.6248
.10363
9.64935
.12131
8.24345
.18906
7.19125
66
.06638
11.5853
.10393
9.62206
.12160
8.22344
.18035
7.17594
57
.08661
11.5461
.10428
9.59490
.12190
8.20352
.13965
7.16071
68
.08690
11.5078
.10458
9.56791
.12819
8.18870
.18996
7.14653
69
.06780
11.4685
.10481
9.54106
.12249
8.16398
.14084
7.13042
60
/
.08749
11.4301
; .10510
Cotang
9.51486
.12278
Cotang
8.14436
.14054
7.11587
/
Ck>tangi Tang
Tang
Tang
Cotang
Tang
860
1 840 1
>s° 1
8S-
49*
TABLE V.—NATURAL TANGENTS AND COTANGENTS.
/
8»
90
10-
!!•
/
Taofl^ Cotang
Tang
Cotang
Tifcng
Cotang
Tang
Cotang
"o
.14054 7.11587
.15838
6.31875
.17688
5.07128
.19488
6.14466
60
1
.140^ 7.10088
.15868
6.80189
.17668
6.66165
.19488
6.18668
60
2
.14118
7.08M6
.15898
6.29007
.17698
6.66205
.19496
6.18688
68
8
.14148
7.07069
.15928
6.27829
.17728
6.M248
.19529
6.18060
67
4
.14178
7.05579
.15958
6.26665
.17758
6.68296
.19KW
6.11879
56
6
.14202
7.04106
.15988
6.25486
.17788
6.62844
.19589
6.10400
65
8
.14232
7.08687
.16017
6.24821
.17818
6.61897
.19619
6.00704
54
7
.14262
7.01174
.16047
6.23160
.1W48
6.60452
.19649
6.06021
58
8
.14291
6.99718
.16077
6.22008
.17878
6.59511
.19680
6.06180
62
9
.14821
6.96268
.16107
6.20651
.17903
6.58578
.19710
6.07360
51
10
.14851
6.96828
.16187
6.19708
.17988
6.67686
.19740
6.06684
60
11
.14881
6.95886
.16187
6.18550
.17968
6.66706
.19770
6.06800
40
12
.14410
6.93852
.16196
6.17419
.17998
6.65777
.19801
6.06087
48
18
.14440
6.92525
,WSA
6.16288
.18028
6.64851
.19681
6.(Vt867
47
14
.14470
6.91104
.16256
6.15151
.18058
6.68927
.10681
6.08480
46
15
.14499
6.89688
.16286
6.14028
.18068
.19801
6.02784
45
16
.14529
6.88278
.16316
6.12899
.18118
6.62090
.10021
6.oion
44
17
.14559
6.86874
.16846
6.11779
.18148
6.61178
.19068
6.01810
48
18
.U'iSS
6.8547B
.16376
6.10864
.18178
6.60264
.10068
6.00451
42
19
.14618
6.&4082
.16405
6.09562
.18206
5.49856
.20018
4.0060ff
41
20
.14648
6.88694
.16485
6.06444
.18288
6.48451
.80048
4.06040
40
21
.14678
6.81312
.16465
6.07840
.18288
6.47548
.80078
4.06188
80
2d
.14707
6.79936
.16405
6.06240
.18298
6.46&48
.80108
4.97488
88
28
.14737
6.7^564
.16525
6.06148
.18828
6.45761
.80188
4.06800
87
24
.14767
6.77199
.16556
6.04061
.18858
6.44857
.80164
4.06045
86
26
.14796
6.75838
.16686
6.02962
.18884
6.48966
.20104
4.06801
85
26
.14826
6.74483
.16615
6.01878
.18414
6.48ar7
.80224
4.04480
84
27
.14856
6.73138
.16(M6
6.00797
.18444
6.42192
.20254
4.08m
88
28
.14886
6.71789
.16674
6.99720
.1W74
6.41809
.80886
4.08064
82
29
.14915
6.70450
.16704
5.98646
.18604
6.40420
.80816
4.02240
81
80
.14945
6.69116
.16784
6.97576
.18584
6.89662
.80846
4.01616
80
81
.14975
6.67787
.16764
6.96510
.18664
6.88877
.80878
4.00765
89
32
.16005
6.66468
.16794
6.95448
.18504
6.87806
.80406
4.00066
88
83
.16034
6.65144
.16824
6.94890
.18624
6.86986
.80436
4.80880
87
84
.15064
6.63831
.16854
6.98385
.18654
6.86070
.80486
4.88805
86
86
.15094
6.62528
.16884
6.92288
.18684
6.86206
.80407
4.878ffl
85
86
.15124
6.61219
.18914
6.91236
.18714
6.84845
.80687
4.87188
84
87
.15158
6.69921
.16944
6.90191
.18746
6.88487
.80667
4.86444
88
88
.15188
6.58627
.16974
6.89151
.18775
6.82881
.80588
4.86737
88
80
.15218
6.57339
.17004
6.88114
.18805
6.81778
.20818
4.86018
81
40
.16248
6.56066
.17088
6.87060
.18886
6.80928
.80648
4.84800
80
41
.15272
6.64777
.17068
5.88061
.18885
6.80080
.80870
4.88800
10
42
.15302
6.53508
.17098
5.85024
.18896
6.29286
.80700
4.88888
18
48
.15332
6.52234
.17128
6.84001
.18926
6.28898
.80730
4.82175
17
44
.15862
6.50970
.17158
5.82962
.18955
6.27658
.80770
4.81471
16
45
.15391
6.49710
.17188
6.81966
.18966
6.28716
.80600
4.80700
15
46
.15421
6.48466
.17218
6.80958
.19016
6.25860
.80680
4.80068
14
47
.15451
6.47206
.17848
6.79944
.19046
6.26048
.80661
4.70870
18
48
.15481
6.46961
.17278
6.7B988
.19076
6.24218
.80601
4.78878
18
49
.15611
6.44720
.17808
6.77986
.19106
6.28891
.80081
4.77078
11
50
.15640
6.48484
.17888
6.76987
.19186
6.22666
.80068
4.77866
10
61
.15670
6.42858
.17868
6.75941
.19186
6.21744
.800Bi
4.7B886
0
52
.15600
6.41026
.17898
5.74949
.19197
6.20925
.81018
.81048
4.76006
8
58
.15630
6.89804
.17423
6.78960
.19227
6.20107
4.76210
7
54
.15660
6.88587
.17453
6.?2974
.19257
6.19298
.81078
4.74584
6
66
.15689
6.37374
.17483
6.71992
.19287
6.18480
.81104
4.78851
6
56
.15719
6.36165
.17513
5.71018
.19817
6.17671
.21184
4.78170
4
67
.15749
6.34961
.17548
S.TIXW/
.19847
6.16888
.81164
4.78490
8
58
.15779
6.83761
.17578
6.69064
.19878
6.18068
.81106
4.71818
8
59
.15809
.17608
5.68094
.19406
6.16286
.81285
4.71187
1
60
9
.15888
6.31375
.17638
Cotang
5.67128
.19438
6.14456
.21256
4.70468
Cotang
Tang
Tang
Cotang
Tang
Cotang
Tang
81- 1
1 80- 1
70- 1
78« 1
497
TABLE v.— NATXmAL TANGENTS AND COTANGENTS.
18-
W I
W I
W
9
60
Tang
.21256
Cotong
Tang
.28087
Ck>tang
Tang
.24088
Cotang
Tang
.26795
Cotang
4.70468
4.88148
4.01078
8.78905
1
.21286
4.69791
.28117
4.88578
.24964
4.00682
.26826
8.78771
60
%
.21816
4.69121
.28148
4.82001
.24995
4.00066
.26857
8.78888
66
8
.21847
4.68462
.28179
4.31430
.25026 ! 8.99692 I
.26888
8.71907
67
4
.81877
4.67786
.28209
4.80660
.25056
8.99009
.26920
8.71476
66
6
.21408
4.67121
.23240
4.80291
.25087
8.98607
.26051
8.71046
66
6
.21488
4.66468
.23271
4.29724
.25118
8.98117
.26968
8.70616
64
7
.21460
4.65797
.23301
4.29150
.25149
8.97827
.27018
8.70188
68
8
.21490
4.65188
.23332
4.28595
.25180
8.97189
.27044
8.69761
68
9
.215£9
4.64480
.23863
4.280S2
.25211
8.96661
.27076
8.69885
61
10
.21560
4.68825
.28388
4.27471
.25248
8.96166
.27107
8.66909
60
11
.21600
4.63171
.28424
4.80911
.25878
3.96680
.27188
8.66485
49
IS
.21631
4.62518
.23456
4.26852
.25904
8.95196
.2n69
8.68061
48
18
.21651
4.61868
.23485
4.25796
.25335
8.94718
.27201
8.67638
47
14
.21682
4.61219
.23516
4.25289
.85366
8.94388
.87882
8.67817
46
15
.21712
4.60572
.23547
4.84685
.85897
8.93751
.27268
8.66796
45
16
.21748
4.59927
.28578
4.24132
.86428
8.98271
.27294
8.66376
44
17
.21778
4.59288
.28606
4.28580
.25459
8.93798
.27826
8.66957
48
18
.21804
4.58641
.28639
4.23080
.25490
8.93316
.27357
8.66538
42
19
.21884
4.58001
.28670
4.22481
.25531
8.91889
.27388
3.65121
41
ao
.21804
4.57868
.28700
4.21988
.25658
8.91864
.87419
8.64705
40
21
.21895
4.56726
.23781
4.21887
.26588
8.90690
.27461
8.64289
89
23
.21985
4.56091
.28762
4.30642
.25614
8.90417
.27482
8.63874
88
23
.21956
4.65458
.28798
4.20396
.25646
8.89945
.27518
8.68461
87
24
.21986
4.54826
.23828
4.19756
.25676
8.89474
.27546
8.68048
86
25
.22017
4.54196
.28854
4.19215
.25707
8.89004
.27578
8.62636
85
26
.22047
4.53568
.23885
4.18675
.25788
8 88586
.27607
8.62224
84
27
.22078
4.52941
.23916
4.18187
.25769
8.88068
.27688
8.61814 '8S
28
.22108
4.52316
.23946
4.17600
.25800
8.87601
.27670
8.61405 82
29
.22189
4.51608
.28977
4.17064
.25831
8.87186
.27701
8.60990 81
80
.22160
4.51071
.24006
4.16680
.26862
8.86671
.27782
8 60588 |80
81
.22200
4.60451
.24069
4.16997
.25898
8.86206
.27764
3.60181
89
82
.22231
4.49682
.84069
4.15465
.25924
8.85745
.87795
8.50775
86
88
.22261
4.49216
.24100
4.14984
.25955
8.85384
.87826
8.59870
27
84
.22292
4.48600
.24181
4.14406
.26986
8.84824
.27858
8.58966
86
85
.22822
4.47986
.24168
4.18877
.26017
8.84364
.27889
8.68562
85
86
.22858
4.47874
.24108
4.13360
.26048
8.83906
.27921
8.58160
24
87
.22388
4.46764
.84228
4.12825
26079
8.88449
.27952
8.57758
28
88
.22414
4.46156
.24254
4.12301
.26110
8.83992
.27988
8.57857
22
89
.22444
4.45548
.24385
4.1ir?8
.26141
8.83537
.28015
8.56967
21
40
.22475
4.44942
.84316
4.11266
.26172
8.82068
.28046
8.66667
80
41
.22606
4.44888
.84847
4.10786
26308
3.81680
.28077
8.56169
19
42
.2»86
4.48785
.84877
4.10216
!36285
8.81177
.28109
8.55761
18
48
.22567
4.48184
.24408
4.09699
.38366
8.80728
.28140
8.55864
17
44
.22507
4.42584
.24439
4.09182
.30397
3.80376
.28172
8.&4968
16
45
.22628
4.41986
.84470
4.06666
.26336
8.79827
.28208
8.54573
15
46
.22658
4.41840
.84501
4.06162
.26359
8.79378
.28334
8.54179.
14
47
.22689
4.40745
.84588
4.07689
.26890
8.78981
.28286
8.58785
18
48
.22719
4.40152
.84562
4.07127
.26421
8.78465
.28397
8.53398
12
49
.22750
4.39560
.24598
4.06616
.aM62
8.78040
.28329
8.53001
11
60
.22781
4.88960
.24684
4.06107
.26483
8.77596
.28860
8.58609
10
61
.22811
4.88881
.24665
4.06S90
.86515
8.T7188
.28891
8.62819
9
52
.22842
4.87798
.24686
4.05098
.86646
8.76709
.28438
8.51829
8
68
.22872
4.87307
.24717
4.04586
.86577
8.76368
.28454
8.61441
7
54
.22008
4.36628
.24747
4.04061
.86606
8.75826
.28486
8.51058
6
65
.22984
4.86010
.24778
4.08578
.86639 8.75388 I
.28517
8.50666
5
56
.22964
4.8M59
.24809
4.08076
.86670
8.74950
.28549
8.60279
4
57
.22996
4.84879
.24840
4.02574
.86701
8.74518
.28680
8.49894
8
68
.28026
4.84800
.24871
4.03074
.26738
8.74073
.28612
8.49609
2
69
.28056
4.83738
.1M908
4.01576
.26764
8.73640
.28648
8.49125
1
60
/
.28087
4.38148
.24938
4.01078
^26795^
,Ck)tang
3.73305
Tang
.28675
8.48741
/
Cotang
Tang
Cotang
Tang
Cotang
Tang
77- 1
1 7e- 1
1 76- 1
74*
498
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
"o
!«• I
If- 1
ISO
190
/
60
Tang
.28675
Cotang
Tang
Cotang
Tang 1 Cotang
Tang
.34483
Cotang
8.48741
.30578
8.27085
' .82492
3.07768
2.90421
1
.28706
8.48859
.80605
8.26746
.82524
8.07464
.84465
2.90147
60
2
.28738
8.47977
.30637
8.26406
: .32656
8.07160
.84498
2.80678
68
8
.28769
8.47596
.30669
8.28067
' .32588
8.06857
.84530
2.89600
67
4
.28800
8.47216
.ao-zoo
8.26729
.82621
8.06554
.84563
2.89887
56
6
.28882
8.46887
.30732
8.25892
1 .82658
8.00252
.^596
2.89066
56
6
.28864
8.46458
.30764
8.25065
.32685
8.05950
.84628
2.88788
64
7
.28895
8.46080
.80796
8.24719
1 .82717
8.05649
.84661
2.88611
68
8
.28927
8.45703
.80828
8.24388
.32749
8.05349
.84608
2.88240
62
9
.28958
8.45327
.30860
8.24049
, .82782
8.06049
.34728
2.8'iVrO
51
10
.28090
8.44961
.80891
8.28714
1 .82814
8.04749
.84758
287700
60
11
.29021
8.44576
.80928
8.28881
.82846
8.04460
.84791
2.87430
49
12
.29058
8.44202
.80055
8.28048
.82878
8.04152
.84824
2.87161
48
18
.29084
8.48829
.80987
8.22716
.82911
8.08854
.84856
2.86882 ;47l
14
.29116
8.48456
.81019
8.22884
.82948
8.0^56
.84889
2.86624
46
16
.29147
8.48084
.31051
8.22058
.82976
8.03260
.84922
2.86856
45
16
.29179
8.42713
.31088
8.21722
.88007
8.02963
.84954
2.86089
44
17
.29210
8.42818
.81115
8.21392
.83040
8.02667
.84987
2.83»22
48
IS
.29242
8.41978
.81147
8.21068
.33072
8.02372
.85020
2.85555
42
19
.29274
3.41604
.81178
8.20784
.33104
8.02077
.86052
2.85289
41
20
.29805
8.41236
.81210
8.20406
.83136
8.01783
.85086
2.85028
40
21
.29337
8.40869
.81242
8.20079
.88169
8.01480
.86118
8.847S8
80
22
.29868
8.40502
.81274
8.19752
.33201
8.01196
.85150
2.84494
88
23
.29400
8.40136
.31806
8.19426
.83233
3.00906
.85183
2.84229
87
24
.29482
8.89771
.81338
8.19100
.88266
8.00611
.86216
2.8396ri
86
25
.29168
8.39406
.81370
8.18775
.88298
8.00319
.85248
2.83702
85
26
.29495
8.39042
.31402
8.18451
.88880
8.00028
.85281
2.83489
84
27
.29526
8.88679
.81484
8.18127
.83368
2.99788
.86814
2.88176
83
28
.29558
8.38317
.31466
8.17804
.33895
2.99447
.85846
2.82914
82
29
.29590
8.87955
.81498
8.17481
.88427
2.99158
.85879
2.82658
81
80
.29621
8.87594
.81530
8.m59
.83460
2.98868
.86412
2.82801
80
31
.29658
8.87284
.81562
8.16888
.88492
2.98580
.86445
2.82180
20
32
.29685
8.86875
.81594
8.16517
.83624
2.98292
.86477
2.81870
28
83
.29716
8.86516
.81626
8.16197
.38557
2.98004
.85610
2.81610
27
84
.29748
8.86158
.31658
8.15877
.38589
2.97717
.85548
2.81850
26
35
.29780
8.35800
.81690
8.15558
.83621
2.97430
.85576 : 2.81091
26
86
.29811
8.35448
.81722
8.15240
.33654
2.97144
.85608
2.80888
24
87
.29848
8.36087
.31754
8.14922
.33686
2.96858
.86641
2.80674
28
88
.29875
8.84782
.81786
8.14605
.83718
2.96578
.86674
2.80816
22
89
.29906
8.34377
.81818
8.14288
.88751
2.96288
.85707
2.80050
21
40
.29988
8.84028
.81860
8.18072
.83788
2.96004
.85740
2.79802
20
41
.29970
8.38670
.81882
8.18656
.83816
2.95721
.85772
2.79545
10
42
.80001
8.33817
.81914
3.13841
.83848
2.95437
.85805
2.79280
18
48
.80038
8.82965
.81946
3.13027
.a3881
«. 95155
.85838
2.79038
17
44
.80065
8.32614
.81978
8.12718
.88913
2.94872
.85871
2.78778
16
45
.80097
8.32264
.32010
8.12i00
.83945
2.94691
.85904
2.78523
15
46
.30128
3.81914
.82042
8.12087
.88978
2.94309
.85937
9.76269
14
47
.80160
8.31566
.32074
8.11775
.84010
2.94028
.85969
2.78014
18
48
.80192
8.31216
.82106
8.11464
.84043
2.93748
.86002
2.77781
12
49
.30224
3.80868
.82139
8.11158
.34075
2.93468
.86086
2.77807
11
60
.80265
8.30521
.82171
8.10842
.84106
2.98189
.86068
2.77254
10
61
.80S87
8.80174
.82208
8.10582
.84140
2.92910
.86101
2.T?002
9
62
.80819
3.29829
.32236
8.10228
;84178
2.92682
.86184
2.76750
8
68
.30351
3.29483
.82267
8.09914
.84205
2.92354
.86167
2.76498
7
54
.30382
3.29139
.82299
8.09606
.34238
2.92076
.86199
2.76247
6
55
.36414
3.28795
.82331
8.09298 1
.81270
2.91799
.86232
8.7S006
6
56
.80446
8.28452
.82368
8.08991
.84303
2.01523
.86266
2.75746
4
57
.30478
8.28109
.82396
8.08685 '
.843:55
2.91246
.86298
2.76496
8
58
.80509
8.27767
.82428
8.08379
.843()8
2.90071
.86881
2.75246
8
59
.80541
8.27426
.83460
8.08073
.84400
2.90696
.86364
2.74097
1
60
/
.80573
Cotang
3.27085
.32492
8.07768
.84433
Cotang
2.90421
.86397
8.74748
_0
Tang
Cotang
Tang 1
Tang
Cotang
Tang
78- 1
7S« 1
71» 1
70»
499
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
20-
SI"* 1
82- 1
S8-
/
60
Ton^
Cotazig
Tang
.88886
Cotang
Tang
.40103
Cotang
Tang
Cotang
.86397
2.74748
2.60509
2.47509
.42447
8.86686
1
.86180
2.74400
.88420
2.60288
.40486
8.47808
.42188
8.85896
60
2
.86468
2.74261
.88458
2.60067
.40470
8.47096
.42516
8.80806
68
8
.86196
2.74004
.88187
2.59881
.40504
8.46888
.42551
8.86016
67
4
.86520
2.78756
.88690
2.59606
.40588
8.46688
.42585
8.84896
66
6
.86562
2.78500
.88558
2.60381
.40672
8.46476
.42619
8.84686
66
6
.86695
2.73968
.88687
8.50166
.40606
8.46870
.42654
8.84447
64
7
.86628
2.78017
.88620
8.58988
.40640
8.46066
.42888
8.84968
68
8
.86661
2.72771
.88654
8.58708
.40674
8.46860
.42722
8.84060
68
0
.86694
2.72586
.88687
8.58484
.40707
8.46666
.42757
8.88881
61
10
.86727
2.72281
.88721
8.68961
.40741
8.46461
.42791
8.88698
60
11
.88780
2.78086
.88754
8.66088
.40775
8.48846
.48836
8.88806
48
18
2.71798
.88787
8.67816
.40800
8.46048
.49860
8.88817
48
18
.86896
2.71648
.88881
8.67698
.40848
8.44889
.49894
8.88180
47
14
.86860
2.71806
.88864
8.57871
.4fl8rr
8.44686
48089
8.88948
46
16
.86802
9.71068
8.67160
.40911
8.44488
.'48968
8.88766
46
16
.88996
8.70810
!88991
8.56928
.40046
8.44880
.48996
8.88670
44
17
.86068
8.70W7
.88066
2.56707
.40079
8.44087
.48088
8.88888
48
18
.86901
8.70886
.88988
8.56487
.41018
8.48885
.48007
8.88197
48
10
.87024
8.70004
.89082
8.56866
.41047
8.48628
.48101
8.88018
41
90
.87067
8.60868
8.56046
.41061
8.48488
.48186
8.81886
40
21
.87000
8.69618
.88080
8.65897
.41116
8.48280
.48170
8.81641
88
29
.87128
8.69871
.88122
8.55606
.41149
8.48019
.43206
8.81466
88
98
.8n67
8.60181
.89156
8.55889
.41188
8.42810
.48880
8.812n
87
94
.87190
8.68899
.88190
8.56170
.41217
2.42618
.48874
8.81086
86
96
.87228
8.68658
.89298
8.61958
.41261
2.42418
.48806
8.80908
85
96
.87256
8.68414
.89957
8.64784
.41286
2.42218
.48848
8.80718
84
97
.87280
8.68176
.89990
2.M516
.41819
2.42019
.48878
8.80684
88
98
.87822
8 67087
.89394
2.61200
.41858
2.41819
.48418
8.80861
88
90
.87356
2.67700
.89857
2.64062
.41387
2.41620
.48447
8.80167
81
80
.87888
2.67468
.88881
2.58866
.41481
2.41481
.48461
8.88984
80
81
.87422
8.67885
.80495
2.68648
.41456
2.41228
.48616
8.8d801
89
88
.87466
8.66080
.80158
2.58132
.41490
2.41026
.48560
8.89610
88
88
.87488
2.66758
.80192
2.58217
.41524
2.40S27
.48586
8.80487
87
84
.87sn
2.66616
.89696
8.53001
.41558
2.40629
.48620
8.89864
86
86
.87564
2.66981
.89559
8.62786
.41598
2.40432
.48664
8.890?8
86
86
.87588
2.66046
.89598
8.52571
.41686
2.40236
.48689
8.98891
84
87
.87821
2.66811
.88626
2.68867
.41660
2.40088
.48724
9.98710
88
88
.87661
2.66676
.89660
2.62142
.41694
8.39641
.48756
8.88688
88
88
.87687
2.65842
.89604
2.51959
.41728
8.39646
.48798
8.98848
81
40
.87720
8.65100
.80727
2.51716
.41768
8.39448
.48898
8.88167
80
41
.87764
8.64875
.89761
2.51502
.41797
8.89858
.48868
8.87987
10
49
.87787
2.64642
.80795
8.61889
.41881
8.39058
.48897
8.87806
18
48
.87890
2.61410
.89820
8.51076
.41866
8.88868
.48988
8.87686
17
44
.87868
2.64177
.88662
8.50661
41809
8.88668
.48966
8.87447
16
46
.87887
8.63946
.89696
8.50658
.41983
8.38478
.41001
8.87967
16
46
.87920
8.63714
.89930
2.50440
.41968
8.88279
.44086
8.87088
14
47
.87058
8.68488
.80068
2.50889
.42002
8.38084
.44071
8.86000
18
48
.87986
8.68252
.89997
2.50018
.42036
8.87801
.44106
8.86780
18
40
.88020
8.68091
.40031
2.49807
.42070
8.87697
.44140
8.80668
11
60
.88068
8.69701
.40066
2.49597
.42106
8.87504
.44175
8.86874
10
61
.88086*
8.62661
.40098
2.40686
.42189
8.87811
.44210
8.86196
0
68
.88120
2.62882
.40182
2.49177
.42178
8.37118
.44244
8.86018
8
68
.88168
2.62108
.40166
2.48967
.42907
8.36926
.44970
8.86840
7
64
.88186
2.61874
.40200
8.48758
.42219
8.36788
.44814
8.86668
6
66
.88920
2.61646
.40284
8.48549
42276
8.86541
.44849
SSS
6
66
.88968
8.61418
.40267
248840
.42310
2.86349
.44884
8.86809
4
67
.88986
8.61190
.40301
2.48182
.42815
2.36158
.44418
8.86188
8
68
.88320
8.60968
.40886
2.47924
.42879
2.35067
.44468
8.94066
8
60
.88868
8.60T86
.40369
2.47716
.48418
2.35776
44488
8.84780
1
60
/
.88886
2.60600
.40408
2.47509
.42447
2.35585
.44528^
Cotang
8.84604
Tang
JO
/
CJotang
Tang
Ootang
Tang
Cotang
Tang
- 68- 1
es-
67- 1
66<»
Soo
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
"o
24- I
25*
26- 1
27°
60
Tang
Ck>tang
Tang
.46681
Cotaxkg
Tang
Cotang
Tang
.50958
Cotang
.44523
2.24604'
2.14461
.48773
2.06080
1.90261
1
.44558
2.24428
.46666
2.14288
.48809
8.04870
.60069
1.96120
60
2
.44598
2.24252
.46702
2.14126
.48846
2.04728
.61026
1.96079
58
8
.44627
2.24077
.46787
2.18968
.48881
2.04577
.51068
1.95888
67
4
.44663
2.28902
.46772
S.18801
.48917
2.04426
.61099
1.96098
66
6
.44697
2.23727
.46806
2.18688
.48958
8.04270
.61186
1.966S7
56
6
.44732
2.23568
.46848
2.184T7
.48989
8.04126
.61173
1.96417
64
7
.44767
2.23378
.46879
2.13816
.49026
8.08975
.51209
1.90277
68
8
.44803
2.28204
.46914
2.18164
.49062
8.06825
.51246
1.96187
58
9
.44837
2.28080
.46060
2.12998
.49096
af.08675
.61288
1.94897
61
10
.44872
2.22857
.46985
2.12888
.49184
8.06686
.51819
1.94866
50
11
.44907
2.29688
.47021
2.12671
.49170
2.08870
.61866
1.94718
40
12
.44942
2.22510
.47066
2.12511
.49206
8.08227
.61898
1.94570
48
18
.44977
2.22887
.47092
2.12350
.49242
2.00078
.61480
1.94440
47
14
.45012
2.22164
.47128
2.12190
.49878
8.02029
.61407
1.04801
40
15
.46047
2.21992
.47168
2.12080
.48815
8.02780
.61506
1.04102
46
10
.46062
2.21819
.47199
2.11871
.49861
2.02681
.61540
1.94028
44
17
.46117
2.21647
.47284
2.11711
.49887
2.02488
.61577
1.98886
48
18
.45152
2.21475
.47270
2.11562
.49428
8.02835
.51614
1.98740
42
19
.46187
2.21804
.47805
2.11892
.49459
8.02187
.61051
1.98008
41
20
.45222
2.21182
.47341
2.11288
.40485
8.02089
.51088
1.96470
40
21
.45257
2.20961
.47377
2.1107B
.49682
2.01881
.51784
1.93838
89
22
.45292
2.20790
.47412
2.10916
.49566
2.01748
.61701
1.98196
88
28
.45827
2.20619
.47448
2.10758
.49604
2.01696
.61796
1.98067
87
24
.45882
2.20449
.47488
2.10600
.49640
2.01449
.61885
1.98920
80
26
.45897
2.20278
.47519
2.10142
.49677
2.01802
.51872
1.98788
85
26
.45482
2.20108
.47555
2.10284
.49718
2.01165
.61909
1.99645
84
27
.45467
2.19988
.47590
2.10126
.49749
2.01008
.61940
1.92606
88
28
.45502
2.19769
.47626
2.09969
.49786
2.00602
.51968
1.92871
82
29
.45588
2.19599
.47662
2.00811
.49822
2.00715
1.98286
81
80
.45578
2.19480
.47698
2.09654
.49666
2.00660
.58067
1.98006
80
81
.45(A)6
2.19261
.47788
2.00496
.49694
2.00428
.62094
1.91902
29
82
.45648
2.19092
.47769
2.09841
.49981
2.00277
.02181
1.91826
28
88
.45678
2.18928
.47806
2.09184
.49967
8.00181
.62166
1.91090
27
84
.46718
2.18755
.47840
2.09028
.50004
1.99980
.62205
1.91664
26
86
.45748
2.18587
.47876
2.08872
.50040
1.99841
.62248
1.91418
26
86
.46784
2.18419
.47912
2.0ffn6
.60078
1.99096
.58279
1.91282
24
87
.46819
2.18251
.47948
2.06560
.60118
1.99650
.62816
1.91147
28
88
.46854
2.18064
.47964
2.08405
.60149
1.09406
.62868
1.91012
22
89
.45889
2.17916
.48019
2.06250
.60185
i.onn
.62890
1.90670
21
40
.45924
2.17749
.48056
2.06094
.60222
1.99116
.62487
1.90741
20
41
.45960
2.17582
.48091
2.07989
.60268
1.98978
.52404
1.900O7
19
42
.45996
2.17416
.48127
2.07786
.60295
1.968S8
.88601
1.90478
18
48
.46080
2.17349
.48168
2.07680
.50881
1.96684
.58586
1.90887
17
44
.46065
2.17088
.48198
2.07476
.60868
1.96540
.62576
1.90808
10
46
.46101
2.16917
.48234
2.07821
.60404
1.98896
.62618
1.90009
15
46
.46186
2.16751
.48270
2.0n67
.50441
1.98258
.62050
1.89085
14
47
.46171
2.16585
.48806
2.07014
.50477
1.96110
.68667
1.86801
18
48
.46206
2.16420
.48842
2.06860
.50514
l.OTWMS
.52784
1.80007
12
49
.46242
2.16255
.48878
2.O6706
.50550
1.97828
.52761
1.89638
11
60
.46277
2.16090
.48414
2.06568
.60687
1.97081
.68796
1.89400.
10
61
.46812
2.15925
.48450
2.06400
.50628
1.97588
.68886
1S%
9
62
.46848
2.15760
.48486
2.06247
.60660
1.97896
.68878
1.89188
8
68
.46888
2.15596
.48521
2.06094
.60696
1.97258
.52910
1.89000
7
64
.46418
2.15482
.48657
2.05042
.80738
1.97111
.68947
1.68807
0
66
.46454
2.16268
.48593
2.06790
.60709
1.96909
.58966
1.88734
6
66
.46480
2.15104
.48629
2.06687
.60806
1.90827
.68088
*§?S
4
67
.46625
2.14940
.48665
2.06486
.60848
1.96686
.68050
1.88409
a
68
.46560
2.14777
.48701
2.06338
.60879
1.90644
.68096
1.86887
2
69
.46695
2.14614
.48787
2.06182
.50916
1.96402
.68184
1.88806
1
60
/
.46681
Cotang
2.14451
.48778
2.05080
.50958
1.90961
.68171
1.88078
_0
Tang
Tang
Ck)tang
Tang
Cotang
Tang
6S- 1
64° 1
68- 1
82*
50I
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
28<»
W 1
1 SO-
81»
/ •
60
Tang
.63171
Cotang
Tang
.5M81
Cotang
_Tang
.67735
Cotang
Tang
Cotang
1.88078
1.80405
1.78805
.60086
1.66428
1
.53206
1.87941
.56469
1.80881
.5rr74
1.78069
.60186
1.66818
69
8
.63246
1.87809
.65507
1.80158
.57818
1.72978
.60165
1.66209
58
8
.58288
1.87677
.55545
1.80084
.57851
1.72867
.60805
1.66099
57
4
.53320
1.87546
.55588
1.79911
.57890
1.72741
.60846
1.66990
56
5
.63858
1.87415
.56681
i.Tvm
.57989
1.78625
.60284
1.66881
55
6
.53396
1.87288
.55659
1.79665
.57968
1.78509
.60824
1.65778
54
7
.53482
1.87158
.55697
1.79548
.58007
1.72896
.60864
1.65668
58
8
.53470
1.87081
.56786
1.79419
.58046
1.72278
.60408
1.65664
68
0
.58507
1.86891
.65774
1.79896
.58085
1.72168
.60448
1.66445
51
10
.58545
1.86780
.55812
1.79174
.58124
1.72047
.60488
1.66887
50
11
.68582
1.86680
.55850
1.79051
.58162
1.719&3
.60628
1.66888
49
12
.58630
1.86499
.56888
1.78929
.58801
1.71817
.60562
1.65120
48
18
.58657
1.86869
.56026
1.78807
.58840
1.71708
.60608
1.66011
47
14
.58694
1.86889
.56964
1.78685
.58279
1.71588
.60642
1.64908
46
15
.58782
1.86109
.56008
1.78568
.58818
1.71478
.60681
1.64795
45
16
.58769
1.85979
.56041
1.78441
.58857
1.71858
.60721
1.64687
44
17
.58807
l.a5850
.56079
1.78819
.58896
1.71844
.60761
1.64579
48
18
.58844
1.85720
.56117
1.78198
.58485
1.71189
.60801
1.64471
42
19
.58882
1.85591
.56156
1.78077
.58474
1.71015
.60641
1.64868
41
SO
.58980
1.85468
.56194
1.77956
.58618
1.70901
.60861
1.64256
40
21
.58067
1.86888
.56888
1.77884
68558
1.70787
.60921
1.64148
89
22
.58995
1.85804
.56870
1.77718
.'58591
1.70678
.60960
1.64041
88
28
.54062
1.86075
.56309
1.77608
.68681
1.70560
.61000
1.68964
87
24
.54070
1.84946
.56347
1.77471
.58670
1.70446
.61040
1.68826
86
25
.54107
1.84818
.56386
1.77851
.58709
1.70888
.61060
1.68719
86
26
.54145
1.84689
.66424
1.T7230
.58748
1.70819
.61120
1.68612
84
27
.54188
1.84661
.66468
i.mio
.58787
1.70106
.61160
1.68S06
88
28
.54280
1.84488
.56601
1.76990
.58826
1.69992
.61200
1.68898
82
29
.54258
1.84806
.56639
1.76869
.58865
1.69879
.61840
1.68898
81
80
.54296
1.84177
.56577
1.76749
.58906
1.69766
.61880
1.68185
80
81
.54883
1.84049
.56616
1.76689
.58944
1.69653
.61820
1.68079
29
82
.54871
1.83988
.56654
1.76610
.58068
1.69541
.61860
1.62972
28
88
.54409
1.88794
.56693
1.76890
.69028
1.69428
.61400
1.62866
27
84
.54446
1.83667
.56731
1.76271
.59061
1.69316
.61440
1.62760
86
86
.64484
1.88540
.56769
1.76151
.59101
1.69808
.61480
1.62654
85
86
.54522
1.83418
.56806
1.70082
.59140
1.69091
.61620
1.62548
24
87
.54560
1.83886
.56846
1.75918
.59179
1.68979
.61561
1.68442
83
88
.54697
1.83159
.56885
.56^
1.75794
.59218
1.68866
.61601
1.62336
88
89
.M685
1.83038
1.75675
.59858
1.68754
.61641
1.62290
81
40
.54678
1.88906
.56968
1.75666
.59897
1.68648
.61681
1.62185
80
41
.54711
1.88780
.57000
1.75487
.50886
1.68581
.61721
1.68019
19
42
.54748
1.88654
.57069
1.75819
.69376
1.68419
.61761
1.61914
18
48
.54786
1.88588
.57078
1.75800
.59415
1.68806
.61801
1.61806
17
44
.54884
1.88408
.57116
1.75082
.59454
1.68196
.61848
1.61708
16
45
.54862
1.82876
.57156
1.74964
.59494
1.68085
.61888
1.61698
15
46
JS4900
1.88150
.57198
1.74846
1 .59583
1.67974
.61928
1.61493
14
47
.54988
1.88085
.57838
1.74788
.50578
1.67868
.61968
1.61368
18
48
.54976
1.81899
.57271
1.74610
.59618
1.67758
.62003
1.61883
18
49
.56018
1.81774
.57809
1.74498
.59651
1.67641
.68048
1.61179
11
50
.56051
1.81649
.57848
1.74875
.59691
1.67530
1.61074
10
51
.56060
1.81584
.57886
1.748W
.59780
1.67419
.68184
1.60070
9
58
.66127
1.81399
.57425
1.74140
.59rro
1.67800
.62164
1.60865
8
68
.55165
1.81874
.57464
1.74088
.59600
1.67198
.62204
1.60761
7
54
.56206
1.81160
.57508
1.78905
.50649
1.67088
.68245
1.60667
6
55
.65241
1.81025
.67541
1.78788
.59888
1.66978
68885
1.60558
5
56
.56279
1.80001
.57580
1.78671
.69928
1.66887
.68885
1.60449
4
57
.56817
1.80777
.57619
1.73556
.59967
1.66757
.62366
1.60845
8
58
.55855
1.80658
.67657
1.78488
1 .60007
1.66647
.68406
1.60841
8
59
.55393
1.80589
.57696
1.73321
1 .60046
1.66538
.62446
1.60187
1
60
1
.5M:n
l.M(VI05
.57735
Cotang
1. 73305
.60086^
Cotang
1.66428
Tang
.62487
1.60088
0
Tang
Tang
Cotang [ Tang
6
!•
60° 1
59<» 1
58-
502
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
820 1
88- 1
a4» 1
85<»
/
60
.62487
Ck>taiier
Tang
.64941
Cotang
Tang
Cotang'
Tang
.70021
Cotang
1.60088
1.58986
.67451
1.48266 1
1.42816
1
.62527
1.60980
.64982
1.63888
.67493
1.48168 I
.70064
1.42786
60
2
.62568
1.69826
.66024
1.68791
.67536
1.48070 1
.70107
1.49688
68
8
.62608
1.69728
.66065
1.68608
.67578
1.47977
.70151
1.42660
67
4
.62649
1.59620
.66106
1.68595
.67620
1.47885
.70194
1.42462
66
5
.62689
1.60517
.66148
1.58497
.67663
1.47792
.70238
1.42874
56
6
1.69414
.65189
1.68400
.67705
1.47699
.70281
1.42S86
64
I
.62770
1.69311
.66281
1.68302
.67748
1.47807
.70325
1.42196
68
8
.68811
1.69208
.66272
1.63205
.67790
1.47514
.70388
1.42110
68
9
.62862
1.59105
.65314
1.58107
.67832
1.47422 1
.70412
1.42022
61
10
.(SW92
1.69002
.65355
1.63010
.67876
1.47880
.70455
1.41984
60
11
.62988
1.68900
.66897
1.62918 1
.67917
1.47238
.70499
1.41847
40
12
.62978
1.68797
.66488
1.62816 ,
.67960
1.47146
.70642
1.41760
48
18
.63014
1.58696
.66480
1.62719
.68002
1.47063
.70586
1.41078
47
14
.68065
1.58593
.66521
1.52622
.68045
1.46962
.70629
1.41584
46
16
.63005
1.68490
1.52525 1
.68068
1.46870
.70673
1.41497
45
16
.68136
1.58388
.65604
1.62429
.68130
1.46778
.70717
1.41409
44
17
.63177
1.68286
.00040
1.52882
.68173
1.46686
.70760
1.41822
48
18
.68217
1.68184
.65688
1.52286
.68215
1.46605
.70804
1.41285
42
19
.68258
1.68068
.66729
1.62189
.68258
1.46503
.70848
1.41148
41
20
.63299
1.67981
.65771
1.62043
.68801
1.46411
.70891
1.41061
40
21
.68840
.65818
1.61946
.68848
1.46320
.70036
1.40974
89
22
.68880
1.57778
.65854
1.51850
.68886
1.46229
.70979
1.40887
88
28
.68421
1.67676
.66896
1.51754
.68429
1.46137
.71023
1.40800
87
24
.63462
1.67675
.65938
1.51658
.68471
1.46046
.71066
1.40714
86
25
.68606
1.67474
.65980
1.51662
.68514
1.45955
.71110
1.40627
85
26
.68644
1.57872
.max
1.61466
.68657
1.45864
.71154
1.40540
84
27
.68684
1.67271
.66068
1.51370
.68600
1.46778
.71198
1.40464
88
28
.63625
1.67170
.66105
1.61275
.68642
1.45682
.71242
1.40867
88
29
.68666
1.67069
.66147
1.51179
.68686
1.45592
.71286
1.40S81
81
80
.68'/U'/
1.66969
.66189
1.51084
.687S8
1.46501
.71829
1.40195
80
81
.68748
1.66868
.66280
1.60988
.68771
1.46410
.71878
1.40109
29
82
.63'/89
1.56767
.66272
1.50893
.68814
1.46320
.71417
1.40028
28
83
.68880
1.66667
.66814
1.50797
.68857
1.45229
.71461
1.89986
87
84
.68871
1.66566
.66356
1.60702
.68900
1.46139
.71605
1.89650
86
85
.68912
1.66466
.66898
1.50607
.68942
1.46049
.71649
1.89764
26
86
.68058
1.66866
.66140
1.50612
.68985
1.44958
.71698
1.89670
24
87
.68994
1.66266
.66482
1.50417
.69028
1.44868
.71687
1.89608
28
88
.64086
1.56165
.66524
1.50322
.69071
1.44778
.71681
1.89607
82
89
.64076
1.66065
.66566
1.50228
.69114
1.44688
.71725
1.89421
81
40
.64117
1.66966
.66608
1.60188
.69157
1.44608
.71769
1.39886
80
41
.64168
1.66866
.66650
1.60088
.69200
1.44508
.71818
1.89850
19
42
.64109
1.65766
.66092
1.49944
.69248
1.44418
.71867
1.89165
18
48
.64240
1.65666
.ee*;^
1.49649
.69286
1.44829
.71901
1.89079
17
44
.64281
1.55667
.66776
1.49755
.69329
1.44239
.71946
1.88094
16
46
.64322
1.65467
.66818
1.49661
.698V2
1.44149
.71990
1.88909
15
46
.64863
1.66868
.66860
1.49566
.69416
1.44060
.72084
1.88884
14
47
.64404
1.65269
.66902
1.49472 1
.69459
1.43970
.72078
1.38788
18
48
1.66170
.66944
1.49378 1
.69502
1.43881
.72122
1.38668
18
49
.64487
1.55071
.66086
1.49284 I
.69545
1.48792
.72167
1.88566
11
60
.64528
1.64972
.67028
1.49190
.69588
1.48708
.72211
1.88484
10
61
.64669
1.64878
.67071
1.49097
.60681
1.48614
.7S266
1.88890
0
52
.64610
1.54774
.67118
1.49008
.69675
1.43525
.T^eeoo
1.88814
8
68
.64658
1.64675
.67155
1.48909
.69718
1.43486
.78844
1.88229
7
54
.64693
1.54576
1 .67197
1.48816
.69761
1.48347
.72388
1.88146
6
66
.64784
1.64478
.67239
1.48722
.69804
1.43258
.72482
1.88000
6
66
.64776
1.54879
.6728J
1.48629
.69847
1.48169
.72477
1.87976
4
67
.64817
1.54281
.67324
1.48536
.69891
1.48080
.72521
1.87801
8
68
.64868
l.&il83
1 .67886
1.48442
.69934
1.42992
.72565
1.87807
8
59
.64899
1.64086
.67409
1.48349
.69977
1.42908
.72610
1.87722
1
60
t
.64941
1.68986
; .67451
1.48256
.V0021
1.42815
.72664
1.87688
#
Ck>tang
Tang
Cotang
Tang
Cotang
Tang
Cotang
Tang
57*
1 se-
55» 1
1 64*
503
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
"0
86» I
S?' 1
88- 1
89»
/
60
Tang
Cotang
Tang
.75355
Cotang
Tang
.78129
Cotang
Tang i Cotang
.72654
1.87688
1.82704
1.27994
.80978
1.28400
1
.72609
1.87554
.75401
1.8S624
.78175
1.27917
.81027
1.28416
60
a
.72748
1.87470
.75447
1.82544
.78222
1.27841
.81075
1.28818
68
8
1.87886
.75492
1.82464
.78269
1.27764
.81123
1.28270
67
4
.72882
1.87802
.75588
1.82384
.78316
1.27688
.81171
1.28196
66
6
.78877
1.87218
.75584
1.82804
.78863
1.27611
.81220
1.28128
66
6
.72921
1.87184
.75029
1.82224
.78410
1.27686
.81268
1.28060
64
7
.72966
1.37050
.75675
1.82144
.78457
1.27468
.81316
1.22977
63
8
.78010
1.86967
.75721
1.82064
.78604
1.27382
.81864
1.22904
52
0
.78066
1.36888
.75767
1.81984
.78551
1.27806
.81418
1.8S831
61
10
.78100
1.86800
.76812
1.81904
.78598
1.27230
.81461
1.22768
60
11
.78144
1.86716
.75858
1.81825
.78646
1.27158
.81510
1.28666
40
19
.78189
1.86688
.75904
1.81745
.78692
1.27077
.81568
1.22612
48
18
.78284
1.86549
.75950
1.81666
.78;«)
1.27001
.81606
1.22680
47
14
.78278
1.86466
.76096
1.81586
.78786
1.26026
.81656
1.22467
46
16
.78828
1.86%8
.76012
1.81507
.78834
1.26849
.81708
1.22894
45
16
.78868
1.36800
.78088
1.31427
.78881
1.28774
.81752
1.22821
44
17
.78418
1.86217
.76134
1.31348
.78928
1.26696
.81800
1.22249
48
18
.78467
1.86134
.76180
1.81269
.78975
1.26622
.81849
1.22178
42
19
.78603
1.86061
.76226
1.31190
.79022
1.26546
.81898
1.22104
41
ao
.78547
1.85068
.76272
1.81110
.79070
1.26471
.81946
1.22081
40
21
.78602
1.36RR5
.76318
1.81081
.79117
1.26806
.81996
1.21099
80
22
.78687
1.85802
.76864
1.80952
.79164
1.26319
.82044
1.21886
88
28
.78681
1.85719
.76410
1.80873
.79212
1.26244
.82092
1.21814
87
24
.78726
1.85687
.76456
1.80796
.79269
1.26160
.82141
1.21742
86
25
.78771
1.85654
.76502
1.80n6
.79306
1.26098
.82190
1.21670
85
26
.78816
1.35472
.76548
1.80687
.79354
1.26018
.82238
1.21696
84
27
.78861
1.85889
.76594
1.30558
.79401
1.25043
.82287
1.21626
88
28
.78906
1.35307
.76640
1.30480
.7W49
1.25867
.82886
1.21454
82
29
.78961
1.85224
.76686
1.30401
.79496
1.25792
.82385
1.21882
81
80
.78996
1.85142
.76768
1.80328
.79544
1.25717
.82484
1.21810
80
81
.74041
1.85080
1.80244
.79591
1.25642
.82488
1.21288
29
82
.74086
1.84978
.76825
1.80166
.79689
1.25567
.82531
1.21166
28
88
.74181
1.84896
.76871
1.30087
.796H6
1.25492
.88580
1.21094
27
84
.74176
1.84814
.76918
1.30009
.79784
1.25417
.82629
1.21028
26
86
.74221
1.34732
.76064
1.29931
.79781
1.25348
.82678
1.20951
26
86
.74267
1.84660
.77010
1.29853
.79829
1.25268
.82727
1.20879
24
87
.74812
1.84568
.77057
1.297/5
.79877
1.25193
82776
1.20608
28
88
.74857
1.84487
.77103
1.29696
.79924
1.25118
.82825
1.20786
22
89
.74402
1.84406
.77149
1.29618
.79972
1.25044
.82874
1.20665
21
40
.74447
1.84823
.77196
1.29541
.80020
1.24969
.82923
1.20608
20
41
.74492
1.34212
.77242
1.29463
.80067
1.24896
.82972
1.20622
19
42
.74688
1.84160
.7TO89
1.29885
.80115
1.24820
.83022
1.20451
18
48
.74688
1.84070
.77335
1.29307
.80163
1.24746
.88071
1.20679
17
44
.74628
1.83998
.77882
1.29229
.80211
1.24672
.83120
1.20608
16
45
.74674
1.88916
.77428
1.29152
.80868
1.24697
.83169
1.20237
16
46
.74719
1.88835
.77473
1.29074
.80806
1.24528
.88218
1.20166
14
47
.74764
1.83754
.77521
1.28997
.80854
1.24449
.88268
1.20096
18
48
.74810
1.83673
.77568
1.28919
.80402
1.24375
.88317
1.20024
12
49
.74865
1.8&592
.77615
1.28M2
.80450
1.24301
.88366
1.19958
11
60
.74900
1.38511
.77661
1.28764
.80498
1.24227
.88415
1.19682
10
51
.74946
1.88430
.77708
1.28687
.80546
1.24168
.88466
1.19611
0
62
.74991
1.83349
.77754
1.28610
.80604
1.24079
.83514
1.19740
8
68
.76067
1.88268
.77801
1.28533
.80642
1.24006
.83564
1.19669
7
64
.76082
1.83187
.7?848
1.28456
.80690
1.28981
.83618
1.19699
6
66
.76128
1.83107
.77895
1.28379
.80738
1.28858
.88662
1.19628
6
66
.75178
1.33026
.77941
1.28302
.80786
1.23784
.88712
1.19467
4
67
.75219
1.82946
.77988
1.28225
.80834
1.28710
.83761
1.19687
8
68
.75264
1.32B65
.78035
1.28148
.80882
1.23637
.83811
1.19316
a
60
.76810
1.82785
.78082
1.28071
.80930
1.23668
.88860
1.19246
1
60
/
.758B5
Ootang
1.82704
.78129
1.27994
.80978
1.23490
.83910
1.19176
J
/
Tang
Cotang
Tang
Cotang
1 Tang
Cotang 1 Tang
58*
1 620
1 61-»
i w
S04
TABLE v.— NATURAL TANGENTS AND COTANGENTS.
40»
4P 1
42«
48»
/
60
Tangr lOotan^
Tang
.86929
Cotang
Tang 1 Cotang
Tang 1 Cotang
.83910
1.19176
1.15087
.90040
1.11061
.98252
i.ora87
1
.83960
1.19105
.86980
1.14960
.90093
1.10996
.93806
1.07174
59
2
.84009
1.19085
.87081
1.14802
.90146
1.10981
.98880
1.07112
68
8
.a«)69
1.18964
.87082
1.14884
.90199
1.10867
.08416
1.07049
57
4
.81108
1.18804
.87188
1.14787
.90281
1.10602
.98469
1.06087
56
6
.a»58
1.18824
.87184
1.14699
.90304
1.10787
.98624
1.06925
56
6
.81:^
1.18754
.87286
1.14682
.90357
1.10672
.98578
1.06662
M
7
.84258
1.18684
.87287
1.14566
.90410
1.10607
.98688
1.06800
58
8
.84307
1.18614
.87388
1.14496
.90468
1.10548
.08688
1.06736
62
9
.84357
1.18544
.87889
1.14430
.90516
1.10478
.93742
1.06676
61
10
.84407
1.18474
.87441
1.14363
.90669
1.10414
.98797
1,06618
60
11
.84457
1.18404
.87492
1.14296
.90621
1.10849
.98862
1.06661
49
12
.84507
1.18884
.87548
1.14229
.90674
1.10285
.08806
1.06488
48
18
.04000
1.18264
.87505
1.14162
.90727
1.10220
.98961
1.06427
47
14
.84606
1.18194
.87646
1.14095
.90781
1.10156
.94016
1.06865
46
15
.84666
1.18125
.87698
1.14028
.90634
1.10091
.94071
1.06808
45
16
.84706
1.18065
.87749
1.18961
.90887
1.10027
.94125
1.06241
44
17
.84756
1.17988
.87801
1.18894
.90940
1.09968
.94180
1.06179
48
18
.84806
1.17916
.87852
1.18888
.90998
1.09609
.04286
1.06117
42
19
.84866
1.17846
.87904
1.18761
.91046
1.09834
.94290
1.06056
41
80
.84906
1.17777
.87955
1.18604
.91099
1.097TO
.94845
1.05094
40
21
.84956
1.17708
.88007
1.13627
.91158
1.00706
.94400
1.06882
89
22
.86006
1.17688
.88059
1.13561
.91206
1.00642
.94466
1.06670
86
88
.86067
1.17669
.88110
1.13494
.91250
1.00678
.94510
1.06809
87
24
.85107
1.17500
.88162
1.13428
.91818
1.09614
.94665
1.05747
86
25
.85157
1.17480
.88214
1.18361
.91866
1.09460
.94680
1.06685
86
86
.86207
1.17861
.88265
1.13296
.91419
1.00886
.94676
1.06624
84
27
.86257
1.17892
.88317
1.18226
.91478
1.09822
.94781
1.06662
88
88
.86806
1.17228
.88369
1.13162
.91586
1.09256
.94786
1.05601
82
20
.85858
1.17154
.88421
1.13096
.91580
1.09196
.94841
1.06480
81
80
.86406
1.17066
.88478
1.13020
.01688
1.09181
.94806
1.05878
80
81
.86458
1.17016
.88524
1.12968
.91687
1.09067
.V49DV
1.G6817
29
82
.85609
1.16947
.88576
1.18807
.91740
1.09008
.95007
1.06856
28
88
.85550
1.16878
.68628
1.18881
.91704
1.06940
.96062
1.05194
27
84
.86609
1.16809
.88680
1.18766
.91847
1.06676
.96118
1.06188
26
86
.86660
1.16741
.88782
1.12699
.91901
1.06818
.96178
1.06078
25
86
.86710
1.16672
.88784
1.12688
.91965
1.06749
.96889
1.06010
24
87
.85761
1.16606
.88836
1.12567
.92008
1.06686
.95264
1.04049
28
88
.85811
1.16585
88888
1.18501
.92062
1.06622
.95840
1.04888
22
89
.86862
1.16466
.88940
1.12435
.92116
1.06559
.06895
1.04887
21
40
.86012
1.16896
.88992
1.12869
.92170
1.06406
.95461
1.04766
20
41
.86968
1.16829
.89045
1.12308
.92224
1.06482
.95606
1.O4705
19
42
.86014
1.16861
.89097
1.12288
.92277
1.06360
.95562
1.04644
18
48
.86064
1.16192
.89149
1.12172
.98381
1.06806
.96618
1.04588
17
44
.86115
1.16124
.89201
1.12106
.92385
1.06248
.96678
1.04588
16
45
.86166
1.16066
.89258
1.12041
.92489
1.08179
.95729
1.04461
15
46
.86216
1.16987
.89606
1.11975
.92498
1.06116
.957B5
1.04401
14
47
.86267
1.16919
.89858
1.11909
.92547
1.06058
.95641
1.04840
18
48
.86818
1.16851
.89410
1.11844
.92601
1'07990
.95897
1.04279
12
40
.86368
1.15788
.89468
1 11778
.92656
1.07927
.96052
1.04218
11
60
.86419
1.15715
.80515
1.11718
.92709
1.07864
.96008
1.04158
10
51
.86470
1.15647
.89567
1.11648
.92768
1.07801
.96064
1.04097
9
62
.86621
1.15579
.89620
1.11562
.98817
1.07788
.96120
1.04006
8
58
.86572
1.15511
.89672
1.11517
.98872
1.07678
.96176
1.08976
7
64
.86628
1.15448
.80725
1.11452
.92926
1.07618
.96282
1.08015
6
66
.86674
1.16375
.69777
1.11387
.92980
1.07660
.96886
1.08866
5
66
.86725
1.15806
.89680
1.11821
.98034
1.07487
.96844
1.08794
4
57
.86778
1.15240
.89883
1.11256
.98068
1.07425
.06400
1.08734
8
66
.86827
1.15172
.89935
1.11191
.98148
1.07862
.96457
1.08674
2
60
.86878
1.15104
.89988
1.11126
.98197
1.07299
.96618
1.08618
1
60
^929
Cotang
1.15087
.90040
1.11061
.93252
Cotang
1.07287
.96669
Cotang
1.08668
0
Tang
Cotang 1 Tang
Tang
Tang
49-
480
1 47« i
1 46«
505
TABLE v.— NATXJRAL TANGENTS AND COTANGENTS.
/
44*
/
/
44<»
/
/
440
/
Tang
Ootaog
Tang
Cotang
Tang
Cotang
0
1
2
8
4
5
6
7
8
9
10
tl
12
18
14
15
16
17
18
19
90
.96569
.96685
.96681
.96738
,967»1
.96850
.96907
.96963
.97030
.97076
.97138
.97189
.97846
.97808
.97869
.97416
.97478
.97589
.97586
.97648
.97700
1.08563
1.08498
1.08488
1.08878
1.08818
1.06858
1.06198
1.08188
1.08078
1.08018
1.08958
1.08898
1.08888
1.08r?8
1.08718
1.08668
1.08598
1.08588
1.08474
1.08414
1.08856
60
50
56
67
56
56
54
68
58
61
50
49
48
47
46
46
44
48
48
41
40
80
81
88
88
84
85
26
87
88
29
80
81
88
88
84
85
86
87
88
89
40
/
.97700
.97756
.97813
.97870
.97987
.»7984
.98041
.98098
.98155
.96818
.96870
.98887
.96384
.96441
.98499
.96566
.96618
.98671
.96786
.96786
.96848
1.08855
1.08896
1.08836
1.08176
1.08117
1.08057
1.01996
1.01938
1.01879
1.01880
1.01761
1.01708
1.01648
1.01583
1.01584
1.01466
1.01406
1.01847
1.01886
1.01889
1.01170
40
89
88
87
86
35
84
83
88
81
80
89
88
27
86
85
81
83
88
81
80
40
41
48
48
a
45
46
47
48
49
50
51
58
58
64
65
56
57
58
59
60
.98848
.98001
.08958
.99016
.99078
.99181
.99189
.99247
.99804
.99368
.99480
.99478
.99586
.99694
.99668
.99710
.99766
.99686
.99884
.99948
1.00000
1.01170
1.01118
1.01058
1.00994
1.00985
1.00876
1.U0618
1.00759
1.00701
1.00648
1.00588
1.00585
1.00467
1.00408
1.00850
1.00891
1.00888
1.00175
1.00116
1.00058
1.00000
80
19
18
17
16
16
14
18
18
11
10
9
8
7
6
6
4
8
8
1
0
/
Cofang
Tang
/
Cotang
Tang
/
/
Cotang
Tang
/
460
46- 1
46* 1
5o<5
TABLE VI.— LENGTHS OF CIRCULAR ARCS: RADIUS = i.
Sec.
Length.
1
.0000048
S
.0000007
8
.0000146
4
.0000194
5
.0000248
6
.0000891
7
.0000688
8
.0000688
0
.0000486
10
.0000486
11
.0000688
18
.0000688
18
14
.0000680
.0000679
16
16
17
18
19
80
81
88
88
84
86
86
87
88
89
80
81
88
88
84
86
86
87
88
89
40
41
48
48
44
46
46
47
48
49
60
61
68
68
64
65
66
57
68
69
60
.0000787
.0000776
.0000684
.0000873
.0000081
.0000970
.0001018
.0001067
.0001116
.0001164
.0001218
.0001961
.0001809
.0001867
.0001406
.0001464
.0001608
.0001661
.0001600
.0001648
.0001697
.0001745
.0001794
.0001848
.0001891
.0001989
.0001968
.0008086
.0008066
.0008138
.0002188
.0002280
.0002279
.0002827
.0002876
.0008424
.0008473
.OOQ2S7X)
.0002618
.0002666
.0008716
.0002768
.0008818
.0002860
.0002909
Min.
1
8
8
4
6
6
7
8
9
10
11
12
18
14
16
16
17
18
19
90
21
22
83
84
26
26
27
28
89
80
81
88
88
84
36
86
87
88
89
40
41
42
48
44
46
46
47
48
49
60
61
68
68
64
66
66
67
68
69
60
Length.
.0008909
.0006818
.0008727
.0011636
.0014644
.0017463
.0020868
.0023271
.0026180
.0029069
.0061906
.0064807
.0087816
.0010784
.0043688
.0046648
.0049461
.00G286O
.0066269
.0068178
.0061087
.0068996
.0066904
.0009613
.0072728
.0076681
.0078540
.0061449
.0064868
.0067866
.0090176
.0098064
.0096993
.0096902
.0101811
.oiwrao
.0107629
.0110538
.0118446
.0116865
.0119264
.0122173
.0126068
.0127991
.0180900
.0183809
.0186717
.0188626
.0142536
.0146444
.0148868
.0161868
.0164171
.0167060
.0159969
.0162887
.0165806
.0168715
.0171624
.0174683
I>eg.
1
8
3
4
6
6
7
8
9
10
11
18
18
14
16
16
17
18
19
20
21
84
26
26
2r
28
29
30
81
32
88
84
86
86
1 37
I 88
89
I 40
> 41
42
43
44
46
46
47
48
49
60
61
62
63
64
66
66
67
68
Length. Deg.
.0174688
.0849066
.0678666
.1047196
.1221780
.1670796
.1746829
.1919662
.2094396
.2448461
.2617994
.2792527
.2967060
.8141693
.8816126
.8490669
.8666191
.a889r84
.4014267
.4188790
.4637866
.4712889
.6061466
.6410621
.6686064
.5769687
.6984119
.6106662
.6457718
.6682261
.6806784
.6961317
.7165860
.7604916
.7679440
60
.8028615
.8208047
.8877580
.8662113
.8786646
.8901179
.9075712
.9260246
.9424778
.9599311
.9778644
.9948877
1.0122910
1.0297448
1 .0471976
I 61
62
63
64
66
66
67
68.
69
70
71
78
78
74
76
76
77
78
79
80
84
86
86
87
88
89
90
91
98
98
94
96
96
97
96
99
100
101
' 108
103
I 104
106
I 106
' 107
106
I 109
! 110
{I 111
I I 118
118
! 114
; 115
1 1 116
I, 118
,1 119
18J
Length.
1.0646508
1.0681041
1.0996674
1.1170107
1.1844640
1.1619178
1.1608706
1.1868839
1.2042772
1.2817306
1.2666871
1.8740904
1.2916486
1.8069969
1.8264608
1.8480066
1.8618668
1.8788101
lA
1.4187167
1.4811700
1.4488888
1.4660!66
1.4886899
1.6009682
1.6184864
1.6868807
1.6638480
1.6707968
1.6
1.6067089
1.6281568
1.6406006
1.6680688
1.6766161
1.6820694
1.71042S7
1.7278760
1.7468898
1.7808868
1.7976891
1.8161484
1.6826967
1.8600490
1.8676088
1.8849666
1.9024089
1.9196688
1.9678166
1.9647688
1.9788221
1.9696768
2.00n286
2.0846819
2.0420868
2.0694S86
2.0769418
8.0948051
REFRACTIONS IN DECLINATION
507
TABLE VII.
Mban Refractions in Declination.*
to be used with the solar attachment.
(Compated by Edward W. Anns, C. E., for W. & L. E. Garley, Troy, N. Y.)
i
1
DECLINATIONS.
For Latttudb 2" 30'.
+»<»
+ir
+ir
+5»
©•
-«•
—10«
— W
— ao»
Ob.
—18*
—12'
—07'
—02'
-1-02'
07'
12*
18'
28'
—18
—12
—07
—02
--02
07
12
18
23
—17
—11
—06
—01
--03
08
13
19
26
—16
—10
—06
0
--06
10
16
21
27
S 1
—10
—06
0
+06
10
16
20
26
32
For Latttudb 6"*. |
Ob.
—16'
—10'
—06'
0'
+06'
--06
ity
16'
20'
27'
—15
—10
—06
0
10
16
20
27
—13
—08
—03
--02
--06
07
12
17
23
29
—10
—06
0
10
16
20
27
32
—06
0
+ 06
10
16
20
27
32
40
For Latttudb 7* 30'. ' 1
Ob.
—13'
—08'
—02'
+02*
08'
13'
18'
24'
29*
—12
—07
—01
-03
09
14
19
26
31
—10
-f-06
0
--06
10
16
20
26
32
—06
0
+ 06
10
16
20
26
32
39
+07
12
17
23
29
36
43
61
I'Ol'
For Latitude 10". 1
Ob.
-10*
-06'
0'
+06'
10'
16'
20'
26'
32*
—07
-03
--02
--03
07
12
17
22
28
34
—06
0
08
13
19
26
31
38
0
06
/lO
16
20
26
32
39
46
+ 15
20
26
32
39
46
66
1'06'
I'lO'
For Latttudb 12^30'. 1
Ob.
—08'
—02'
+02'
--06
8'
13'
18'
24'
30'
86'
—06
00
10
16
20
26
32
39
+02
07
12
17
23
29
36
43
61
04
09
14
20
26
31
40
48
66
21
27
33
40
48
67
1'08'
1'23'
1'41'
For Latttudb 16*. 1
Ob.
-06'
0'
+06'
W
16'
21'
27'
33'
40'
—03
+ 02
07
12
18
23
29
86
43
+01
06
11
16
. 22
28
34
41
49
08
12
10
24
30
37
44
63
1'04
29
34
41
49
69
no
1*24
r43
208
• Printed by permission of W. & L. E. Gurley.
5o8
REFRACTIONS IN DECUNATION
1
0t
DECLINATIONS.
For Latitude 17° 30'.
+*o°
+ 15°
+ 10°
+»'
0°
— »°
—10°
—15°
— *0°
Oh.
—02'
+02'
08'
13'
18'
24'
30'
36'
44'
0
05
10
15
21
27
33
40
48
+02
10
15
21
27
33
40
48
57
13
18
23
29
36
43
51
I'Ol
1'13
34
41
49
58
I'lO
1'23
1'41
206
2 42
For Latitude 20**.
Oh.
0'
05'
10'
15'
21'
27'
33'
40'
48'
03
07
13
18
24
30
36
44
52
06
13
18
24
30
36
44
52
1'02
17
22
28
35
42
50
VOO
I'll
126
30
47
57
1'07
1'20
1'37
200
2 32
325
For Latitudb 22° 30'.
Oh.
02'
08'
13'
18'
24'
30'
36'
44'
52'
06
11
15
21
27
33
40
48
57
11
15
21
27
33
40
48
57
1'08
20
26
32
39
46
56
1'07
1'19
137
45
53
1'03
1'16
1'31
1'52
2 21
3 07
428
For Latitude 25°.
Oh.
06'
10'
15'
21'
27'
33'
40'
48'
67'
08
14
19
25
31
38
46
54
1'05
12
18
24
30
37
44
53
1'04
118
23
29
35
45
53
1'03
1'16
131
152
49
59
I'lO
1'24
1'52
2 07
244
346
5 43
For Latitude 27° 30'.
Oh.
08'
13'
18'
24'
30'
36'
44'
52*
1'02'
11
16
22
28
34
41
49
I'OO
1 10
17
22
28
35
42
50
I'OO
1 11
126
28
35
42
50
I'OO
I'll
126
143
209
54
1'05
1'18
1'34
154
2 24
3 11
438
8 15
For Latitude 30°. 1
Oh.
10*
15'
21'
27'
33'
40'
48'
67'
1'08'
14
19
25
31
38
46
64
1'05
118
20
26
32
39
47
55
1'06
119
136
32
39
46
62
1'06
1'19
135
157
229
I'OO
I'lO
1'24
1'52
2 07
244
346
543
13' 06
For Latitude 32° 30'.
Oh.
13'
18'
24'
30'
36'
44'
52*
1'02'
1'14'
17
22
28
35
42
50
I'OO
1 11
126
23
29
35
43
51
I'Ol
1 13
128
147
35
43
51
I'Ol
1'13
127
146
2 13
2 54
1'03
1'16
1'31
153
220
3 06
425
7 36
fiSFftAcnous ja declination
509
1
p
0
DECLINATIONS.
For Latitude 36°.
+*0°
+ 16°
+ 10°
+»°
0°
KO
—10°
—16°
— to°
Oh.
16"
21'
27'
33'
40'
48'
57'
1'08'
1'21'
20
26
32
38
46
56
1'06
118
136
26
33
30
47
66
1'07
121
138
200
30
47
66
1'07
1'20
136
169
2 32
325
1'07
1'20
1'38
200
2 34
329
6 14
10 16
For Latitude 37"* 30'.
Oh.
18"
24'
30*
36'
44'
52'
1'02'
1'14'
1'29
22
28
36
42
50
I'OO
112
126
146
20
36
43
62
1'02
114
129
149
2 16
43
61
I'Ol
1'13
127
149
2 14
2 54
406
I'll
1'26
164
2 10
2 40
356
6 16
14 68
For Latttudb 40®.
Oh.
21'
27'
33'
40'
48'
67'
1'08'
1'21'
1'39'
25
32
39
46
62
1'06
1 19
136
157
33
40
48
67
ro8
121
138
202
236
47
66
1'06
1'19
136
158
230
3 21
469
1'16
1'31
161
220
306
426
734
26 18
For Latitude 42° 30'. 1
Oh.
24'
30'
36'
44'
62'
1'02'
1'14'
1'29'
1'49'
28
36
39
50
I'OO
1 12
126
145
2 11
36
43
52
ro2
113
129
149
2 17
2 69
60
I'OO
I'll
126
144
2 10
2 49
356
6 16
1'19
136
158
230
3 22
500
9 24
For Latitude 46°.
Oh.
27'
33'
40*
48'
67'
1'08'
1'21'
1'39'
2'02'
32
39
46
62
1'06
1 19
135
157
229
40
47
56
1'07
121
138
200
2 34
329
64
1'04
1'16
133
154
2 24
3 11
4 38
8 16
1'23
141
206
2 41
340
540
12 02
For Latttudr 47° 30'.
Oh.
30*
36'
44'
62'
1'02'
1'14'
1'29'
1'49'
2'18'
35
42
60
I'OO
112
126
146
2 01
2 51
43
51
I'Ol
113
128
147
2 15
2 56
408
66
1'09
123
140
205
240
339
6 37
11 18
i'27
146
2 12
2 62
4 01
630
16 19
For Latitudb 60°. 1
Oh.
88'
40*
48'
67'
1'08'
1'21'
1'39'
2'02'
2'36'
88
46
66
1'06
1 18
135
167
228
3 19
47
56
1'06
119
136
229
2 31
323
5 02
1'02
1'14
129
148
2 16
258
4 18
6 59
19 47
130
161
2 19
3 04
4 22
728
24 10
5IO
REFRACTIONS IN DECLINATION
9
0
DECLINATIONS.
4
PS
p
For Latitude 52° 30'.
+to°
+ 15°
+ 10°
+ 5°
0°
—6°
—10°
—15°
— *0°
eh.
86'
44'
52'
1'02'
1'14'
1'29'
1'49^
2'18'
3'05'
1
43
50
59
111
126
142
223
249
3 65
t
50
I'OO
I'll
126
146
2 11
2 51
258
622
4
1'05
118
135
2 10
228
3 19
453
842
6
134
156
227
3 16
4 47
8 62
For Latitude 65°. |
Ob.
40'
48'
67'
1'08'
1'21'
1'39'
2'02'
2'86'
3'38'
1
46
55
1'06
1 18
134
156
230
3 15
4 47
S
56
1'06
1 19
135
168
230
3 21
458
9 19
4
I'lO
123
142
206
243
344
5 49
12 41
6
137
2 01
2 34
328
6 15
10 18
Fob Latitudb 67° 30'.
Oh.
44'
52*
1'02'
I'M'
1'29'
1'49'
2'18'
3'05'
4*37'
1
60
50
1 11
125
143
209
2 47
3 51
604
S
68
I'lO
124
142
2 07
243
346
650
12 47
4
I'll
125
143
2 10
260
356
614
14 49
S
141
206
242
342
646
12 26
For Latitude 60°. |
Oh.
48'
57'
1'08'
1'21'
1'39'
2'02'
2'36'
3'33'
5'28'
1
64
1'04
1 17
133
154
2 24
3 12
438
8 16
S
1'08
116
130
151
220
3 04
4 24
7 31
24 44
4
1 18
134
156
228
3 18
460
853
S
145
2 11
260
357
6 21
16 32
For Latitude 62° 30'.
Oh.
52'
1'02'
1'14'
1'29'
I'SO*
2'18'
S'OO'
4'17'
7'13'
1
58
1'09
123
141
206
243
344
550
12 44
t
1'07'
123
138
2 01
236
330
6 16
10 24
4
123
140
205
240
340
5 37
1150
S
148
2 17
259
4 14
7 03
For Latitudb 65°. |
Oh.
57'
1'08'
1'21'
1'39'
2'02'
2'36'
3'33'
5'23'
10*51'
1
I'OS'
1 16
131
162
2 21
307
428
744
t
112
127
146
2 12
2 52
402
633
4
127
147
2 13
2 54
405
640
S
162
222
308
430
7 52
For Latitude 67° 30'. |
Oh.
1'02'
1'14'
1'29'
1'50'
2'18'
3'00'
4'17'
7'13'
1
108
122
140
2 03
239
3 37
5 32
1128
t
1 17
134
155
2 26
3 14
444
8 34
4
132
163
223
3 14
435
805
S
156
228
3 17
440
8 61
For Latitude 70°. |
Oh.
1'08'
1'21'
1'39'
2'02'
2'36'
3'33'
5'23'
10'51'
114
120
160
2 18
300
4 17
7 13
123
143
206
2 41
3 41
5 59
12 15
137
200
2 34
328
520
10 12
2 02
233
327
6 11
10 06
TRIGONOMETRIC FORMULAS S'l
TABLE VIII. TRIGONOMETRIC AND MISCELLANEOUS FORMULAS.
RiOHT Trxangxji
Obuqub Trianglb
SOLUTION OF RIGHT TRIANGLES.
c
o
sec yf — - a> cosec ^
o
^ c^b d
c c
c
cot ^»--tan^
a
coaocA^-^wecB
a
- e
ezsec^—-
a^c ^n A ^btSLn A m,c COB B^b cot B^V{c+d)(c—i)
d'^ccos A t^a cot A '^c^B^a talk B — V(i'+tf) (<•— tf)—^'— ^ vers^
d^cyersA e^cexaecA
d
cos ^ sin ^ sin ^ cos A vers ^ exsec A
^b-\-htxi/K,A
SOLUTION OF OBLIQUE TRIANGLES.
Given.
A,B,a
A,a,b
C,atd
A, By C,a
Cya,b
Sought.
byc
B,c
A^B
A
area
area
area
Formulas.
b^
sinyf
• sin^,
sin^« by
a
sin
a
sinyf
-sin(^+i5)
sin C,
tanJ(-<-^)-^tanJ(-<+^)
If ..J(«+*+0. Bin M-V^^^^^
cos
sin^-
vers^-«
g\/j(j-a)(j-^)(j-0
be
be
area— \// {s—a) (s—b) (s—c)
a* sin i9 . sin C
area— : — -z
2smA
area— }a^ sin C.
512 TRIGONOMETRIC FORMULAS
TABLE VIII. TRIGONOMETRIC AND MISCELLANEOUS FORMULAS.
GENERAL TRIGONOMETRIC FORMULAS,
sin A =«2 sin \ A cos } A «*\/i — cos' A «« tan A cos ^ = \/J(i— cos 2 A)
cos yl => 2 cos' Ji4— I « I — 2 sin' } ^^ =»cos' 1^4 —sin* \A^i —vers A
' sin A Vi— cos'i^ sin 2 i4
tan^« —
COtytfa
COS^ cos >l I + cos 2 i4
cos A sin 2 yf sin 2 yl
sin ^ I— cos 2^ vers 2 ^
vers A = i —cos ^ =8in A tan \A^2 sin' ^ yf
exsec .<4 —sec A—i «.tan ^^ tan i ^ » 7-
' COSi4
sin 2 ^ «■ 2 sin yf cos ^
cos 2 ^ » 2 cos' A— I -aCos*.<4 —sin' /f -■ i — 2 sin' ^
2 tan^^
tan 2 A mm
cot 2 yf s
I - ton' A
cot'^-i
2 cot ^
vers 2^ = 2 sin'^ «- 2 sin ^ cos A tan ^
2 tan' i<
exsec 2 A^-
I — tan* -^
sin' A + cos' .<4 «B I
sin {A ^ ^) —sin ^^ cos ^ ^ sin ^9 cos A
cos {A J- ^) =cos -<4 cos ^ ^ sin i4 sin B
sin ^ + sin ^»2 sin } (^ + B) cos ^ (^ —B)
sin ^ —sin -^ = 2 cos J (-4 +-^) sin J {^4 — ^)
cos A +COS ^=" 2 cos 1 (^ + ^) cos J (^ -^)
cos B — cos yf = 2 sin J (^ H-^) sin ^ (i4 — -^)
sin' A - sin' ^ -= cos' B - cos' -4 - sin (i4 + -^) sin {A - -<P)
cos' -4- sin' ^=cos (A+B) cos (A-B)
sin{A-^B)
t^nA-\-i9JiB-
tan i4 — tan B
cos ^ cos B
sin (A-B)
cos -<4 cos B
CIRCULAR CURVES — GEOMETRIC FORMULAS
513
TABLE IX. CIRCULAR CURVE FORMULAS
^ » Radius
/ = Central Angle
T -* Tangent Distance
£ =^ External Distance
T' « ^ tan J /
^ = ^ exsec i I
C =» 2 ^ sin i /
L,='Rx Circular Measure / .
M » Middle Ordinate
L, a> Length of Arc
C -Chord
/ » Tangent Offset .
L,-C «^T^ (Approximate)
M^-^— (Approximate)
/«^
zR
TABLE X. GEOMETRIC FORMULAS.
Required.
Area of
Circle
Sector of Circle
Segment of Circle
Ellipse
Surface of
Cone
Cylinder
Sphere
Zone
Volume of
Prism or Cylinder
Pyramid or Cone
Frustum of Pyramid
or Cone
Sphere
Given.
Radius =r
Radius a r, Arc=Z«
Chordae, Middle Ordinate
Semi-axes «a and^
Radius of Bases r; Slant
Height »» J
Radius sr, Height ssi
Radius — r
Radius of Sphere =r, Height
of Zone— ^
Area of Base = b ; Height = h
A rea of Base = b ; Height » h
Area of bases -'^ and b' \
Height — h
Radius = r
Formulas.
2
\ CAf (Approximate)
icab
icrs
2Trh
2 irrh
bh
bh^
3
514 LINEAR AND SQUARE MEASURE
TABLE XI. LINEAR MEASURE.
I foot M 12 inches
I yard =■ 3 feet
I rod — 5i yards — 16} feet
I mile ■■ 330 rods a- 1 760 yards —5280 feet
TABLE XII. SQUARE MEASURE.
I sq. foot — 144 sq. inches
I sq. yard— 9 sq. feet— 1296 sq. inches
I sq. rod » 30^ sq. yards— 272^ sq. feet
I acre — 160 sq. rods— 4840 sq. yards —43,560 sq. feet
I sq. mUe— 640 acres —102^00 sq. rods— 27,878400 sq. feet
TABLE XIII. LINEAR MEASURE — METRIC SYSTEM.
I myriameter— 10 kilometers
I kilometer — 10 hectometers
I hectometer— 10 decameters
I decameter — 10 meters
I meter — 10 decimeters
I decimeter — 10 centimeters
I centimeter — 10 millimeters
TABLE XIV. SQUARE MEASURE — METRIC SYSTEM.
I centare— i sq. meter
I are — 100 sq. meters
I hectare— 100 ares— 10,000 sq. meters
CONSTANTS
S15
TABLE XV. CONSTANTS.
1
Number.
Logarithm.
Ratio of circumference to diameter
3M159
0.49715
Base of hyperbolic logarithms
2.71828
0.43429
Modulus of common system of logs
0.43429
9 .63778-10
Length of seconds pendulum at N. Y. (inches)
39.1017
I .59220
Acceleration due to gravity at N. Y.
32.15949
2 .36361
Cubic inches in i U. S. gallon
231
Cubic feet in i U. S. gallon
0.1337
9.12613-10
U. S. gallons in i cubic foot
7.4805
0.87393
1.79588
Poun(ls of water in i cubic foot
62.5
Pounds of water in i U.S. gallon
8-355
0.92195
Pounds per square inch due to i atmosphere
14.7
I. 16732
Pounds per square inch due to i foot head of
water
0.434
9.63749-10
Feet of head for pressure of i pound per square
inch
2.304
0.36248
Inches in i centimeter
0.3937
9. 59517-10
Centimeters in i inch
2.5400
0.40483
0.5159^
Feet in i meter
3.2808
Meters in i foot
0.3048
9.48402-10
Miles in i kilometer
0.62137
9.79335-10
Kilometers in i mile
Square inches in i square centimeter
0.1550
9. 19033-10
Square centimeters in i square inch
6.4520
0.80969
Square feet in i square meter
10.764
I .03197
Square meters in i square foot
0.09290
8.96802-10
Cubic feet in i cubic meter
35.3156
I .54797
Pounds (av.) in i kilogram
2 .2046
0.34333
Kilograms in i pound (av.)
0.4536
9.65667-10
Ft.-lbs. in I kilogram-meter
7 .23308
0.85932
APPROXIMATE VALUES OF SINES.
Natural sine of i*^ = ^ e "^ ZZ (roughly)
Natural sine of o<» i'« °-°3 Jt.
100 ft.
Natural sine of 0° 00^ 01" = ^il^
I mile
Sl6 . GREEK ALPHABET
Greek
Alphabet.
LBTTKRS
MAHB
A, a,
Alpha
B,i8.
Beta
r,y.
Gamma
A, 8,
Delta
E,c,
Epsilon
z.^
Zeta
H, ij,
Eta
©.O,
Theta
I.t,
Iota
K, K,
Kappa
A,X,
Lambda
M,/t,
Mu
N.v,
Nu
H,^,
Xi
0,0,
Omicron
n,ir,
Pi
p,p,
Rho
S, «r, s,
Sigma
T.T,
Tau
Y,v,
Upsilon
*,<^,
Phi
X, X.
Chi
♦,1^,
Psi
0,*),
Omega
INDEX.
Abbrevkdons in notes, 1x3.
Abnej7 Ic^d and clinometer, 293.
Abscissa, 356
Accidental errors, 14.
Accumulative errors, 14.
Accuracy,
in measuring lines, xa.
of measurements, 14.
of traverses, xxx.
Acre, 3-
Additions, to blue prints, 396.
Adit, 385.
Adjusting screws, 95.
Adjustment,
of chain, 4.
of compass, 35.
of dumpy levd, 91.
of level, 88.
bubble. 89.
cross-hairs, 88.
direct method, 91 .
indirect method, 89.
peg method, 91 •
wyes, 90.
of Locke level, 93.
of scdar attachment, 70.
of transit, 56.
bubble, 56.
coUimation, 58.
cross-hairs, 57.
loDjg bubble on telescope, 59.
objective sGde, 60.
standards, 59.
of traverses, 358, 364.
an^es. 333.
Agomc line, 35.
AUgimient, 8, 11, 13.
Allen, C. F., 335, 343.
Almanac NautioU, 68, 184, 188.
Altitude, of sun, 69, 100, 193.
American Ei^iemeris (see almaiutc).
Angles,
brass, 83.
calculated from bearings, 30.
doubling, 108.
deflection, xo8.
horizontal, 47.
interior, 108.
laying off, 50.
measured by repetition, 48.
parallel lines, 103.
vertical, 54.
Angular,
convergence of meridians, 139, 154.
value dt one division of level, 73-
Aimual variation, ao.
Apparent,
motion of heavenly bodies, 180.
decUnation of sun, X95.
Ap«,
of vein, 385.
atathJTfg out, 316.
Approximate times of culmination and elonga-
tion of Polaris, x86.
Areas,
averaging end, 344,
computation of,
by double meridian distances, 35a.
by douUe parallel distances, 359.
by offsets, 333.
by Simpson's one-third rule, 334.
by trapezoidal rule, 334.
by triangles, 336.
coordinates, 374.
airved comer lot, 338.
quadrilateral, 337.
Arrangement of computations, 330.
Astronomy,
observation for,
azimuth, 180.
latitude, 196.
Atmosphere, refraction, *' boiling of," 87.
Attachment,
for {dumbing rods, 83.
solar, 65.
Attraction, local magnetic 39.
Aurora Borealis, 35.
Automatic tide gauge, 313.
Auxiliary level, 59-
Averaipng end areas, 344.
Axe, 19.
Axis,
pohu-j 65.
adjustment of, 70.
Aamutn,
angles, x?, xo8.
assumed, 109.
at any hour, 188.
in dty triangulation, 36a.
needle checks on, no.
of line, 17.
of Polaris, 183.
to distant object, 37a.
Back,
bearing, 19.
ground, 53.
sight, 51, 86.
Balancing surveys, 358, 364.
Ball and socket joint, 16, 3a.
Baltimore triangulation, 358.
Base, 309'
board, 106.
line, 133, 134, 3^1 •
Batterboards, 353.
Bauer, L. A., 30.
Beam compass, 386.
Bearings, 16.
method of taking, x8.
reverse i9-
Bed of stream, ownership, 118.
Bench, 303.
marks, 303.
establishing, 198.
in dties, 336.
Bent.
needle, 36.
pivot, 37.
517
Si8
INDEX
BiaectioDj
methoa, ija,
target, 80.
Black prints, 393.
Blazing trees, 107, 130.
Blue prints, 391 .
Blocks.
irregular, 247.
size of, 933.
"Boiling " of atmosphere, 87.
Border unes, 437.
Bore holes, 307, 315.
Borrow pits, sio, 343*
Boston rod, 78.
Boundaries,
of land
conventional sign for, 4x9.
disputed, 117.
identification, 106, xi6.
irregular, 105.
relocating, xi8.
Bounds,
control, X16.
stone, 330.
Breast, 30X.
Bubbles,
levels, construction of vial, 7a-
angular value, one division. 73-
Buildings, location of. 159, 168.
Bulletin of University of Wisconsin. 69.
Bureau of Equipment, Navy, X84.
Bureau of Standards, 3x7.
Burnt sienna, 396.
Calculated bearings, xio.
Calculating angles from bearings, 39.
Calculation (see computation).
Caroel's-hair brush, 61, 391.
Capstan-head screw, 36.
Card, for sighting, 53.
Care of instruments, 61 .
Carrying the transit, 63.
Cassiopeia, 183.
Catenary, X3.
Celestial sphere, 65, x8o.
Cellar, excavation, 343.
Centering pivot point, 37.
Chaining,
xnethod of, 7.
on slopes, 9.
Chains,
Gunter's, 3, 99.
engineer's, 3.
testing, 4.
wearing, ia.
Chain surveymg, xo6.
Chainmen, 7, ix.
Change, magnetic, 90, X19.
Chart, isogonic 33.
Cheddng,
allies, no.
azimuths, X09.
bench marks, 199.
buildings, x6o.
compuution, 330, 337, 339.
distances, 8, xi.
field work, X09.
lots, dty, 949-
target reading, 84.
traverses, 100, 103, X05, xxi, 156.
traverses not dosed, 156. ^
Chinese white, 396.
Chords, plotting by, 408.
Circular curve, 933.
Qty surveys, 3x6.
City surveys (Continued)
blocks, 333, 347.
contour map, 3x9.
datum, 336.
laying out town site, 319, 337, 239.
location of streets, 330.
marking comers, 930.
monuments, 338.
permanent boich works, 336.
plans, 355.
provision for growth, 2x9.
sewer system, 337.
staking out, 351.
standard, 3x8.
steel tapes, 3x6.
streets, 319, 351, 359.
cross-section, 343, 346.
curb lines, 349.
grades, 334, 34X, 359.
gutters, 344.
location, 330.
width, 333.
Claims, mining, 313.
Clamps, 33.
Cleaning drawings, 430.
Clinometer, 393, 3x6.
Clips, 89.
Closure, error of, 99.
limits, public lands, X41
Cloth tape, 5.
Coast and Geodetic Survey, 30, 39, 35, 77, 89,
XI9, 184, 303.
(Coefficient of expansion, steel, 13.
Co-latitude, 67.
O>llimators, 58.
(Compass, beam, 386.
box, 16.
(Compass,
surveyor's, x6.
adjustments, 35.
reading bearings, x8.
remagnetizing, 38.
running traverse with, 99.
pocket (compass), x8.
prismatic (compass), x8.
(Compartment, 385.
Compensating errcrs, X4.
Comstock's refraction fcnnula, 69.
(Computations,
area, 334.
arrangement of, 330.
book, 330.
borrow pits, 343.
checks, 330.
coordinates, 373.
deflection anglra, 341.
estimates, 345, 349.
logarithmic 337.
preparing notes for, 333.
prismoids, 344*
short cuts, 338.
rignifirant figures, 333.
slide rule, 330.
triangulation, 373.
volumes, 343.
Concrete, 331.
Connecting surface with undetgroond, 998.
Cmstellations, x8x.
(Contour, 365, 4x0.
found by cross-sections, 376.
incorrectly sketched, 376.
interpolated, 376.
locating, 378.
locating by hand levd, 379.
maps, 370.
INDEX
519
Gntour (Coatiiuied)
pea, 586.
probfem* 381.
properties of, 368.
Omtncts, X3I, xsS.
Coatrol by triangulatioD, 356, 364.
Conventional signs, 417, 4x9.
Convergence of meridians, isi, 139, 154.
~ ■ , Chief Justice, 1x8.
area by, 374-
calcolation, 373.
plotting by, 40X.
system for dties, 355.
Copying notes, 433.
Copying plans, 38$, 391.
Corners,
locating aixd marking. 106, 147 •
monuments, 338, 147.
section, X33.
selecting, xo6.
Correction, »
line, 133.
triangle, 370.
Corrections,
for refraction,
in altitude, 193.
in dedinatioa, 69.
on slopes, 3. x3> ax 7, 330.
Counting, chain lengths, 8.
Counterweight, on needle, X9.
Courts, XX7.
Cross-cuts, 385.
Cross^hairs, 34t 36.
illuminating, x83.
Croasrsection,
areas of, 344.
contours determined by, 376.
for earthwork, 307.
paper, 390.
xtMui, 307.
Crow foot, 108, XT3.
Crown, glass, 36.
of pavements, 3^4.
Cidminatioa of Polaris, x8o.
Cultivated lands, conventional sign, 4x7.
Cumulative errors, X4.
Curb lines, 349-
Currents, electric 19, 311 •
Curvature and refraction, 3, 87.
Curves,
laying out, 334*
prinaples, 333.
street comers, 939.
vertical, 343.
Cut, 3o8.
Cut-oflf line, xxi, 157, 37X.
Daily, variation of ikeedle, 30.
Dam, 373*
Dark glass, 190.
Data,
to be noted (public lands). X39.
missing, 366.
Datum plane, 198, 336.
establishing, six.
Day, dvil and astronomical, X87.
Decimals of foot in inches, 6.
Dedmal division of vernier, 44.
Declination,
oi needle. 19, XX9.
variation. 30, XX9.
daily, 30.
secular, 30.
irregular. 30.
Declination (Continued)
determination by observation, 35.
of sun, 68, X94.
correction for refraction, X93.
Deed,
description, XX5.
interpretation, xx7.
looking up. X30.
recordmg. x3o.
survey for, xx4.
Deflection angles, xxo, a^, 407.
Degree of longitude, length of, 155.
Deucacy of level, 73.
Departure,
■ computation of, 353-
total, 40X .
Depth, cut or fill, 308.
Detecting local attraction, 39.
Dial, miner's, 387.
Difference in elevation, 85.
Differential leveling, 85.
Dip df needle, 19.
of vein, 385.
Direct, adjustment, 9x .
position of instrument, 34.
Direction,
measurement of, x6.
Discharge of streams, 3x0.
measurement, 3x0.
miner's inch, 3x0.
Distances,
with chain, 7.
with stadia, 6.
with tape. xo.
between end points. 371.
on slope. 13, ai7. 333-
Distant object, checking angles, X58.
Disturbed comers, 333.
Dividers,
proportional. 387.
Division of land, 368.
Division of level tube, 73*
Double area, 355.
meridian distance, 355.
rodded lines, 30x .
vemiers, 39.
Doubling angles, 50, X56.
Drafts in shaft, 396.
Drafting instruments, 381.
Drainage,
area, 319, 373-
provision for, 230, 337.
Drawing paper, 388.
Drift, a86.
DriU-hole. 339, 300.
Drying prints, 393, 393.
Dumpy level, 74.
adjustment of, 9X .
Duty of surveyor, xx6.
Earth,
curvature, 3, 87.
magnetism, X9.
Earthwork,
computation, 34a.
end area method, 344.
estimate of from, contour map, 345, 340*
prismoidal formula. 344-
Easting. 3^3.
Eccentric bearing telescope, 390.
Eccentridty,
of drcles, ^4-
Economic prindples, mining, 3x6.
Effect of sa^, 13.
Elaborate, titles, etc., 427.
520.
INDEX
Electric aixrent, 19.
Electric prmting frame, 395.
Electrification, compass cover, 18.
Elevation,
difl^ftnce of, 85.
FJimination of errors, 61, 87, 183, 300.
Elongation,
aamuth at, x8i .
of Polaris, times of, x86.
meridian obeervatioQ at, x8a.
Embankment, ao8.
End area method, 344.
Engineer's,
chain, 3, 4.
scale, 38a.
transit. 34.
Enlarging pi^n«, 385.
Ephemens, 68, 184, 188.
Equal backsights and foresights, 87.
Equation <A tmie, 196.
Equator, 66.
Erecting eyepiece, 37-
ErrocB.
accidental, 14.
amount of, xa.
cumulative, 14.
compensating, 14.
in compan work, a8.
in measuring lines, xi.
in tape, 1 1 .
in use of level, 94.
in use of transit, 6a.
of adjustment, 61 .
elimination of, 61, 87.
of dofiure, 99.
of graduation, 49, 6x.
Establishing comers, 130.
Estimates, earthwork, 345.
Etched, marks on tape, 5.
European surveys, 8a.
Excavation, 308.
Expansion,
coefficient of, steel, X3.
of ipaper, 4a8.
Extenor, township, xa7.
Extenu&l secant. 339.
Eztralateral rights, 314-
Eyepiece, erecting and inverting, 37.
Farm surveys, 99.
Famham, I. T., 339.
Fedend laws, mining, 313.
Fences,
measurement of, xxo.
conventional signs, 4x9.
Field,
book (see notes)
checks, X09.
methods, loi.
notes. Pub. Lands, 143, 146.
Land Surveying. 100, X03.
Leveling, aoo, ao4.
Topographical, a66.
. work, 99.
Figures, significant, 333.
Fifing,
drawings, 415-
note-books, 431-
Fill, ao8.
Finding, lost comers, xi6.
Finishing nail, 300.
Floats, s^i 310.
Flow,
of streams, 3x0.
miner's inch, 310.
Focus, principal, 35*
Focusing, ^5.
" F(dding ' vernier (Fig. 36), 43.
Foresights, 51, sa. 53. 06, 87.
Formulas,
area bv coordinates, 374.
area of tiiangle, 336, 337-
area curved comer lot, 338.
azimuth by Polaris, X83, X87.
azimuth by sun, 190, 194.
convergence of meridians, 154.
cross-section of street, 344.
curves, 333.
prismoid, ^44-
sag and alignment, 13.
Simpson's rule, 334.
side and top telocope. %oS.
tangent o£fset and middle ordinate* 339.
trapezoid, 334.
Fractional areas, 335.
French curves, 387.
Frost, 50, 339.
Functions, judicial, of the surveyor, xx6.
Gallery, 394, 305, 306.
Gauge,
seU registering, 313.
tide staff, 31a.
Geodetic surveying, 3.
Geographical meridian, x6.
Geological Survey, U. S.
bench marks, aoa.
Geology, relation to topography, 373.
(Geometric principles, 159.
Giving line, xii.
Glass, magnifying, 47.
Grade <rf street, 334.
line, 307, 319, 251 •
levding for, 3x0.
Graduation,
errors of, 49, 61.
compass-bcnc, x6.
numoering, 33.
tapes, 5.
Grantee, X17.
Grantor, 1x7.
Gravity, 73.
Great dipper, x83.
Greenwich time, 68, 187.
Group, of daims, 314.
Growth of dties, 319.
Guide meridians, X36.
Gunter's chain, 3.
Hachure, 367.
Handkerchief, signals, 54.
Hand level, 77.
adjustments, 93*
topography wim. 378.
Hayford, J. F^ 188.
Head,
chainman, 7.
shifting, 31.
Heading, 386.
Height of instrument, 198.
Hiffh water mark, 135.
Holman, S. W^ 333-
Horizon, 66.
Horizontal,
angles, 47-
measurements, 3.
line, defined, 73.
Hourly change m declination of sun, 68, 191.
Houses,
conventiona] agns, 4x0.
INDEX
521
Houses (Continued)
locating, 159.
Hypotenuse, 13-
Identification of bounds, xo6.
Illuminatioa,
cross-hairs, x8a.
plumb line, 187 •
Image, 3^ 37 -.
Inaccessible distances. 174-x 78-
Inch, not used in surveymg, 3.
miner's, 310.
Inclination,
correction for, 3, i3t 2i7i 333*
measured With clinometer, 393.
Index correction, 54.
Indexing, 43a.
India ink, ^96.
Indirect, adjustment, 89.
Information in notes, ix4*
Initial point, ia4.
Ink, 395-
Instructions, Manual of, 134.
Instrument,
care of, 61.
height of, 198.
Instrumental errors,
elimination of, 61, 87, 183, aoo.
Instruments, drafting, 381 .
Instruments, surveying and plotting,
Abney level, 393.
chain, 3.
compass, 16.
pocket, x8.
insmatic 18.
level, 79. 73.
rod, 78.
Locke, 77.
lining pole, 7*
measuring rod, 6.
pins, 7.
planimeter, 339-
protractor, 38a, 384* 396.
slide rule, 330.
stadia, 6.
targets, level rod, 80.
lamp, a99.
telescope, 34-
Thatcher r\Ue, 333-
transit, 31*
solar attachment, 65.
Interdiangeable side and top telescope, 390.
Interior angles, 108.
Interpolating contours, 376.
Interval, contour, a67.
Inverting eyepiece, 37-
Iron, effect on needle, 19.
Irregular,
areas, 344.
variation, 35.
Isotonic
chart, as, as
lines, 35.
Judicial functions of the surveyor, 116.
Keys, 19.
Lamp targets, 999.
Land Office, 134, xs6.
Land surveying, ,
by compass and chain, 99.
hy transit and tape, xox.
error of closure, 99*
Land stirveying (Continued)
computiiu; area
by DVM. D.'s, 356.
by coordinates, 374*
additional areas, 334*
dividing land, 368.
obstacles, 170.
resurveys, xi8.
supplying omissions, 1x7, 366.
survey for deed, ix4-
U. S. Public Lands, xax.
Lateral,
adjustment, 89.
movement of transit hesd, 3a >
Latitude,
difference, 35a-
error in, 358, .^64.
observation, 196.
of a line, 35a-
parallel, X48.
total, 401.
Laws, mining, 3x3*
Laying out curves, 337-
Laying out land, aaa.
Laying out town site, axo.
Least squares, method of, 14, a6a.
Legal lines, 106.
Length,
absolute of, tape, a 17.
standard of, 318.
Lenses, 36, 78.
Lettering on maps, 4ao, 4ax, 4aa.
Level,
bubble, 73-
dumpy, 74-
hand, 77-
line, 7-
Locke, 77.
precise, 73-
rod, 78.
section, axo.
surface, 7a.
(term in mining), a86.
use oi, 83.
vial, 7a.
wye, 73-
Leveling, x^.
across a river, 3x3.
for bench marks, 198.
for grade line, axo.
for profile, 303.
instruments used in, 7a.
rods, 78.
screws, 33.
signals used in, 85.
Lcv«^ ....
divergence m duphcate hues, aox.
double rodded lines, aox.
Limb,
of sun, 69, 19a.
of transit (graduated), 3X.
Lines,
clearing out, X50.
horizontal, j.
legal, X06.
of coUimation, 34-
stretchiiMS, 330.
Lining in, 8, X3.
Lining-poles, x8.
Link, 3.
Local,
attraction, 39.
hiws, 3M. „ .
mean time, i87« X90.
522
INDEX
Locadng,
buildinss, 159.
points from transit line, 265.
property lines, 363.
Locke hand levd, 77.
Lode,
claim, 3x3.
cdoring en maps, 306.
Logarithms,
computing by, 327.
Long rod, 80.
Lonmtude,
difference, 3Sa-
table of lengths, 155.
Lost moauments, xo6, 116.
Lots, dty, aaa.
Magnet, earth a, zg.
bar, used, 38.
Magnetic,
attraction, 39.
bearing, z6.
dedinatifui, xp.
variation of. ao.
observations for, zSa.
meridian, 16.
needle, x6, 3a.
ore, 394<
storms, 35.
Magnetian, weakened, 37.
of eardi, 19.
Magnifying glass, 47.
Magnifying power, 37.
Bian-hole (mining), a86.
** Manual of Instructions," 134, z88, 3x5.
Map,
Baltimore, 358.
Washington, 333.
Maps,
aty, 355.
large, 429.
lettering on, 430.
mine, 305.
topo^aphical, 370.
requirements, 1x4.
Marking points in tape measurement, xo.
Marsh, conventional sign, 4x7.
Massachusetts,
observed declinations, ao.
Mean,
tide, Z98, 3X3.
time, X87, X96.
Meandering, X34.
Meander lines, X35.
Meandering a stream, X36.
Measurements,
accuracy of, 14.
chain, 7.
mistakes in recording, ix.
odometer, 6.
on slope-reducing, 338.
pacing, 9.
stadia, 6.
tape, xo.
Measuring,
angles by repetition, 48.
base line, 361.
differences in elevation, 73.
distances, 3.
Mental estimates, xa, 330.
Meridians,
angular convergence of, X3x, X39i X54.
determination of, 180.
bv sdar attachment, 66.
guiae, X36.
Meridians (Gmtinued)
magnetic and true, x6, 17,
on maps, 437.
principal, X3a, X36.
Metallic tane, 5.
Method of least 89uares, X4, 363.
Method of surveymg field, xox.
Metric chain, 5.
Metropolitan Water and Sewerage Boazd, 8z.
Microscope, 35.
Mill-hole, 386.
Mine,
bore-holes, 3x5.
boundaries, 3x3.
claims, 3x3.
carrying meridian into, 394.
connectmg surface and undeigroond, 298.
laying out work, 307.
ma-pping, 305.
patent surveying, 31a.
placer claims, 3x4.
surface surveys, 3x1.
surveys, 285.
underground leveling;, 306.
underground surveying, 394.
Miner's dial, 387.
Mining, claims, 313.
terms, 385.
Missing data, 366.
Missouri River Commission, aoa.
Mistakes,
in compass work, a8.
in level work, ^4.
in measuring hues, ix.
in transit work, 6a.
in using solar attachment, 70.
Moisture, effect on level rod, 83.
Monuments,
at section comer, X47.
in dty work, 338.
lost, 1x6.
significance and authority of, xx6.
Motion of fingers in leveling, 46.
Multiplication by slide rule, 333.
Nails, in marking points, xo, 300.
Natural, functions, 337.
scale of maps, 38a.
Nautical Almanac, 68. X84, x88.
Navigators, ao.
Needle, compass,
bent, a6.
remagnetiang, a8.
New York rod, 80.
Night, observations at, x8a.
Noon observation, X96.
Normal position of telescope, 34.
North star, x8o.
Northing, 353.
Note book, 84, xoo, xxa.
Notekeepcr, ix.
Notekeeping, xia.
bench levels, aoo.
cross-section, ao6.
double rodded lines, aoz.
mine, 301.
profile, ao4.
road, cross-section, 309.
topographical survey, 366.
Notes, 1X3.
comi>ass and chain, xoo.
transit and tape, 103.
traverse, xs7-
INDEX
523
Numb
grtduatKms, 3a.
townships, X3».
. "3.
Objective, ^5-
slide, adjustment, 60.
Obstacles, X701 173-
Observations,
for dedinatioa of needle, 35.
for meridian by solar, 66.
on Pdlaris, x8a.
on Polaris, with compass, 187.
Odometer, 6.
Office,
plans, 431-
work, X07.
Offsetting transit Une, x 70, 173.
Offsets, 105, X59.
Omissions, supplied, 366.
Optical, azist 35. .
center, 36.
Ordinates, ac6.
Ore, testing by electric currents, 3x1.
Organization of transit party, xxx.
Origin of coordinates, 956.
Original notes, xoo, xia.
Outcrop, a86.
Owners of property, xx7.
Ownership of surveys, xx4.
Pacing, 9.
Pantograph, 385.
Paper, 388.
expansion of, 4a8.
mounted, 388.
protractor, 384-
Parabolic curve, a4a.
Paraffin, 83.
Parallax, X9, 34» 48.
Parallel,
of latitude, X48.
ruler, 386.
standard, xaa, ia6.
Party, transit, xxx.
Patent, mining claims, 3x3.
Peg adjustment, 91.
aidjustment for hand level, 93.
Pen, contour, 386.
Pencil, 53, 9x, 1x3.
Permanence, bench marks, 303.
Philadelphia rod, 8x.
Photograph of landscape, 37X.
Pins, 7.
Pitch, 386.
Pivot, x6, 77-
bent, 37.
cap, 59-
Placer, 3x4, 3x5.
Plain transit, 34.
Plan, 1x4. ass-
Plane intersecting surface, 380.
Plane surveying, 3.
Planimeter, 340.
Plates, of transit, 31.
Plotting. 397'
by coorcfinates, 401.
by chords, 409.
by protractor, 398.
by tangents, 405-
cross-sections, 4">
details, 4x0.
land surveys, 397*
Plumb-bob, 3a, 397*
Plumb line, 5a, 396.
in chaining, 9.
Plumbing,
attachment. 8a.
down a shajft, 396.
rod. 83.
Plummet lamp, 300.
Plus, sution, X56.
Pocket compass, x8.
Polar axis, 65.
Pole, S3, x8o.
Polaris,
elongation of, 186.
observation on, 35, x8o.
polar distance, xSo, 184.
Pole star, x8, 180.
Posidon, zero point, ix.
Precautions,
in using level, 88.
in using transit, 55.
Pr«dae level, 73-
rod. 83.
Predsian, 334.
Preliminary railroad surveys, 6.
Primary txiangulation, 356, 364.
PrindpeJ focus, 35.
Principal meridian, X33, X36.
Printing frames, 395.
Prints, olue, 39X.
Vandyke, 393-
Prismadc compass, x8.
eyepiece, 393.
Prismoidal formula, 344-
Prismoids, 344*
Process papers, 391.
Profile,
leveling, 303.
ProfiC 430.
Projection, 371.
Proportion (see balancing surveys).
Proportional dividers, 387.
Protractors, 38a, 398.
Prussian blue, 396.
Public land surveys, xai.
PuU, ,
chain, 9.
tape, X3.
Purposes, of surveys, 3.
Rack and pinion, 34. . ,
Radius, of curve, level vial, 7a.
Raihtiads,
curves, 387.
surveys, 6.
tracks, 35X.
Raise, 386.
Random Une, si> 130« x<^>
Range,
line, X33, X59f x6o.
pole, 7, 59t 53f x8o.
ips, IM.
Raymond, W. G^ 1x4.
Reading glass, 47-
Rear chainman, 7.
Records, ass- ,, ,,
Rectangular coordinates, 9SS-
Reducing slope measuxements, 3a8, 333*
524
INDEX
Reference measurements, 107 .
RefracticMi,
aurvature and, 89.
in altitude, 193.
in declination. 507.
Registry of deeds, lao.
Remagnetizina needle, a8.
Repeating angles, 48.
tape measurements, xo.
Report (see U. S. Coast and Geodetic Survey).
Results, 393.
Resurveys, xi8.
Reverse bearings, 99.
Reverse position of telescope, 34.
Reversing £. and W. points, 18.
Reversion, 56.
Revolution of wheel, 6.
Right ascension, 188.
Ring, cross-hair, 58.
Rivers, xi8.
Roads,
measuring on, xo.
Rod, 3.
Rod levels, 8a.
Rodman, 78.
Rod readings, 84.
for grade, 307.
Rods,
leveling, 78.
Boston, 78.
measuring, 6.
New York, 80.
Philadelphia, 8x.
precise level, 8a.
special, 81 .
tape, 81.
Roof, transit on, 17a.
Rule,
parallel, 3^.
slide, 330.
two foot, 6.
Runner, on slide rule, 331.
Sag, 9. 13.
Sand, 417.
Scales,
engineer's, 389.
logarithmic, 331.
of maps, 397, 439.
Screen, for sun's image, 65.
Screw, leveling, 3a.
Sea-level (see datum).
Secant method, 148.
Secondary triangulation, 363.
Section, of level, 74.
of transit, ^i.
Section, pubuc lands, 133.
Secular, variation, ao.
Self-reading rods, 81.
Setting up transit, 45.
Sewer, aao, aa7, 351.
Shading (see hachures).
Shafts, a86, 994, apS.
Sherman, C. W., ai8.
Shifting head of transit, 31, 46.
Shirley, Maj.T.W.,a58.
** ShootinjK in '^ grades, an.
Shop adjustments, 6x .
Shore lines, aSo.
Short rod, 80.
Shrinkage of paper, 438.
Side telescope, 387.
Sidereal time, 188.
Siffhts, compass, x6, 18.
length of, 87.
Sights (Continued)
diowing, 5a.
Sighting rod, 7.
Signals,
level work, 85.
transit work, 53.
triangulation, 356.
Signs, conventioQai, 4x7, 4x9.
Simpson's <nie-third rule, 334*
Sines, 485.
Sketch, X99-
Sketching contours, 976.
in minmg, 31a.
Slide rule, 33X.
Thatcher rule, 333.
Slopes,
rnsining on, 9.
reducing slope measurements, 338, 333.
slope stakes, 308.
Slow motion, 33.
Solar.
attachment, 65, xss.
observations, X90, 193.
Sources of error,
compass work, 38.
level work, 94.
measuring lines, xi.
transit work, 6a.
Southing, 35a.
Soundings, 310.
Speaking rods (see self-reading).
Sf^iexical triangle, 67.
Specimen notes, 146.
Speed of floats, 3x0.
Spider lines, 36.
Spindle, 31 .
Spirit level, x6,'3a, 7a.
Spline, 387.
Spring baunce, ax6.
Stadia, 6.
Staff gauge, 3x3.
Staking out lots, 237.
Standard,
length, ^ X3, 3x8.
of transit, 33.
parallel, 136.
time, 68.
State surveys, 158.
Staticms, xo, X56.
marking, 300.
Steel tape, 5, 10, x 3.
measurements, xo.
Stopes, 386.
Stop screw. 88.
Straight-edge, 38X.
Straight lines, 51, 5a.
Straightening crooked boundary, 336.
Streams, a8o.
flow of, ^xo.
meandexmg, 136.
Streets, 3x9, 351.
Stretch, tapes, 5.
Striding level, 387.
Strike. 386.
StuU, 386.
Sub-grade, 307.
Subdivision <rf land, 368.
Supidying missing data, 366.
Surveying.
plain and geodetic, 3.
lor area. 99.
for deed, 114.
public lands, lax.
Siirveyor general, 131, 314.
INDEX
525
Surveyor's,
chain, 3*
compass, x6.
transit, 32
Surveys, accuracy of, 14.
Swamp, conventional tia
Swing o£fset, 150.
System, Public Lands,
.417.
lax.
T-aquare, 381, 400.
Tack, X08.
Tables,
convergence of meridians, xag*
dedxDJUS of foot in inches. 6.
hourly chuige in sun's declinati(m, 191.
length of degree of longitude, 155.
mean polar distances of Polaris, 184.
observed declinations in Massachusetts, ao.
refraction in altitude, 193 •
tangent method, parallel, 15a, 15.3 •
times of elongation and culmination, x86.
secant method, parallel, X49.
Tag, on chain, 4.
Tangent, method, X50.
plotting by, 405.
screws, 3a.
Tape-rod, 8x, ao6.
Tapes, dty work, ai6.
metallic 5.
steel, s.
Target,
bisection, 80.
rod. 80.
Telescope, 34t 287.
Temperature, 13.
Tension handle, 3x6.
Tests, on transit, 55.
Thacher slide rule, 333-
Thermometer, on levding rod, 8a.
Three arm protractor, 384.
Tide.
gauge, axa.
observation of, axx.
Ties, lox, X07, XS9.
Tiers, townships, 133.
Time (see mean time).
Titles, 433*
Top telescope, 387-
Topographical surveying, 364.
Topographical symbols, 4x7, 4x9-
Topography,
contour mao, 370.
field methods, 364-
methods of representing, 4i7> 4X9>
Total latitude and departure, 401.
Township, X3x.
comers, X47.
exteriors, 137.
numbering, X33.
subdivision, ia8
Tradng,
doth, X90, 389*
paper, 389-
Transit, 31-
adjustment of, 56.
carrying, 6a.
engineer's, 34.
measuring angles with, 47i 4Dt 54*
plain, 34'
running straight line, 51*
setting up, 45-
idar attachment, 65.
Transit (Continued)
traversing, xox.
used in land surveys, xas*
used in mining, 387.
Traverse, definition, 100.
adjusting, 364*
Traversing,
with compass, 99*
with transit, xox.
Traverses,
in dty work, 363.
inking in, 4x5.
locating from, 159-
underground, 399.
Trench excavation, 3x0.
Triangles, inaccurate, 40a.
Triangulation,
calculation, 373.
scheme, 358.
sutions, 356.
system, 356, 393.
Tripod, ^3, 45. 61.
Tube reflector, 300.
Tunnel, 386.
Ttiming points, 86, X98, 303.
double (see double rodded lines).
True,
bearing, X7.
meridian (sec observations).
Two-foot rule, 6, 53.
Underground surveys, 394-
United States,
Coast and Geodetic Survey, 30, 35. 77i xi9t
184. 3oa.
Deputy surveyors (see deputy).
Geological Survey, aoa.
Isogonic Chart, 33-33.
Public Lands, Surveys, lai.
ang'^^flf convergence of meridians, X39.
baseline, 134.
comers, X47.
guide meridians, 136.
kgal requirements, xa3.
meandering, 134.
notes, XA3.
prindpal meridian, X36.
scheme of subdivision, i3X.
standard parallels, 126.
subdivision of township, 138.
township, X37.
standard of length, X3.
surveyor general, X3x, 3x4.
Units, J, 4*
Ursa Major, x8a.
Value of one division, level, 73-
Valley, a so, 370.
Vandyke paper, 393- ... . ^
Variation of compass (see dechnauon).
of dedination, 30.
Veins, 3x6.
Vdodt^ of stream, 3x0.
VermihoQ, 396.
Vernier, 38.
Vertical,
angle, 54. 308.
drdc, 34.
Volumes, 343.
Wan, a86.
conventional sign, 4x9.
Washington, aa^.
Water, boundaries, xx8.
526
Water (Continued)
color, 395.
lininff, 4x7.
supply, 227.
Waterproof ink, 395<
Watershed, a6o, 273*
Waving the rod. 84.
Wearing of chains. z»
Westing, 35a.
INDEX
Wheel, mroltttiaa oC, 6.
Wind, e£feGt of , xx.
Win2e,986.
Wires, illuminating, i8a.
Wooden rods, 6*
•* Working into line,* 51-
T^t*', 66.
Zero point, ii«
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