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Geodetic surveying and the ad ustment of 




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GEODETIC SURVEYING 



Published by the 

McGrav/ - Hill Book. Company 

Ne-w Yoric 

iSucce&sors to theBooKDepartments of the 

McGraw Publishing Company Hill Publishing' Company 

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GEODETIC SURVEYING 



AND 



THE ADJUSTMENT OF OBSERVATIONS 

(METHOD OP LEAST SQUARES) 



BY 



EDWARD L. INGRAM, C.E. 

Professor of Railroad Engineering and Geodesy, University of Pennsylvania 



McGRAW-HILL BOOK COMPANY 

239 WEST 39TH STREET, NEW YORK 

6 BOUVERIE STREET, LONDON, E.G. 

1911 



COPTKIQHT, 1911 
BY 

McGRAW-HILL BOOK COMPANY 



PREFACE 



After a careful examination of existing books, the University 
of Pennsylvania has failed to find a satisfactory text from which to 
teach its civil engineering students the fundamental principles of 
geodetic surveying and the adjustment of observations as it 
feels they should be taught to this class of men. A canvass of 
the leading colleges of the country has shown that the same lack 
of a suitable book has been felt by many other institutions. The 
present volume has been prepared to meet this apparent need. 
No attempt has been made, therefore, to treat the subject 
exhaustively for the benefit of the professional geodesist, but 
rather to build up a book colitaining everything that can be con- 
sidered desirable for the student or useful to the practicing civil 
engineer. In order to make the book complete for such engineers 
it has been necessary to include a large amount of matter not 
desirable or suitable for class-room work, the arrangement of 
the college course being left to the judgment of the instructor. 

In writing the book in two parts the aim has been to make each 
part complete in itself, so that either part may be read intelligently 
without having read the other part. Those who wish to make 
a study of geodetic work without entering into involved mathe- 
matical discussions, will fimd a complete treatment of geodetic 
methods and the rules for making the necessary adjustments 
in the first part of the book. Those who wish to become familiar 
with the fimdamental principles of least squares, or those famihar 
with geodetic work who wish to understand the mathematical 
theory on which the rules for adjusting observations are based, 
may read the second part of the book alone. The book has 
been written with the intention, however, that engineering students 
shall take the two parts in succession. 



VI PEEFAGE 

In the first part of the book the initial chapter takes up the 
principles of triangulation work as the best introduction to geodetic 
work in general. Nothing new of any special importance is 
available in the general scheme of triangulation, and the chapter 
is written as briefly and logically as possible. 

The second chapter treats of the subject of base-line measure- 
ment, including measurements with base-bars, steel tapes, invar 
tapes, and steel and brass wires. Special care 'has been taken 
to have such constants as the temperature coefficient, the modulus 
of elasticity, and the specific weight correct and complete for 
the different materials involved. The mathematical treatment 
of the corrections required in base-line work has been made as 
simple as possible, avoiding needless transformations of mathemat- 
ical formulas to cover unusual methods of work. 

The third chapter takes up the subject of angle measurement, 
and is intended to make clear the most approved methods of using 
the instruments and performing the actual work ia the field. 
The repeating method is given in much detail on accoimt of the 
excellent results obtainable by this method with the ordinary 
engineer's transit. 

The fourth chapter includes the computations and adjust- 
ments required in triangulation work, and is intended to cover 
all points of interest to the civil engineer. 

The fifth chapter takes up the subject of computing the geodetic 
positions from the results of the triangulation work. The mathe- 
matical treatment of this subject is so difficult that the formulas 
to be used are given without demonstration, but all the rules and 
constants are given that the engineer will ever require. 

The sixth chapter is devoted to geodetic leveling, and contains 
the familiar knowledge on this subject arranged as briefly as is 
consistent with clearness and completeness. 

The seventh chapter is devoted to astronomical determina- 
tions, giving in detail such work as falls within the province of the 
engineer, and in outline such general information as the educated 
engineer should possess, but which is seldom found in engiueering 
text-books. The number of methods for making astronomical 
determinations is almost without limit, but the older and well- 
tried methods are here retained as best adapted to the needs of 
the engineer. 

The eighth chapter considers the principal methods of map 



PEEFAOE vii 

projection, and differs from the treatment found in other books 
chiefly by including the formulas which alone make it possible 
to use the different methods. 

Chapters IX to XVI form the second part of the book, devoted 
to the development of the Method of Least Squares and its apphca- 
tion to the adjustment of observations. 

Chapter IX includes the necessary classification of values, 
quantities and errors, and also the laws of chance on which the 
theory of errors is founded. This is followed in Chapter X by 
the development of the mathematical theory of errors, which is 
the fundamental basis from which all the rules for adjustments 
are derived. 

Chapter XI develops the mathematical methods for obtaining 
the most probable values of independent quantities in general 
from their observed values, and Chapter XII extends the methods 
so as to include conditioned and computed quantities. 

Chapter XIII explains the meaning of and methods of obtaining 
the probable error for both observed and computed quantities. 
The derivation of the necessary formulas is considered too 
abstruse for the average student, and these formulas are given 
without demonstration. 

Chapters XIV, XV, and XVI, deal respectively with the 
application of the theory of least squares to the various condi- 
tions met with and adjustments required in angle work, base- 
line work, and level work, covering all cases likely to be of interest 
to the civil engineer. 

In the preparation of the text the following points have been 
kept constantly in view: to bring the book up to date; to make the 
treatment of each subject as clear and concise as possible; to 
use the same symbols throughout the book for the same meaning, 
adopting the symbols having the most general acceptance; to 
define each symbol in a formula where the formula is developed, 
so that the user of the formula is never required to hunt for the 
meaniag of its terms; to give for every formula the unit in which 
each symbol is to be taken; to clear up any doubt as to what 
algebraic sign is to be given to a symbol in a formula, as the sign 
required in a geodetic formula is not infrequently the opposite 
of what would naturally be supposed; to make perfectly rigid 
such demonstrations as are given; where demonstrations are 
not given to state where they may be found; to give the best 



viii PEEFAOE 

obtainable values for all constants required in geodetic work; 
and to state the accuracy attaiuable with different instruments 
and methods, so that a proper choice may be made. Attention 
is called to the very large number of illustrative examples that 
are given, and which are worked out in detail so that every 
process may be thoroughly understood. 

E. L. I. 
Philadelphia, Pa., December, 1911. 



TABLE OF CONTENTS 



INTRODUCTION 

ART. PAGE 

1. Geodesy Defined 1 

2. Importance of Geodetic Work 1 

3. Geodetic Work in the United States . . ■ 1 

4. Historical Notes 1 

5. Scope of Geodesy 2 

6. Geodetic Surveying 2 

7. Adjustment op Observations 3 



PART I 
GEODETIC SURVEYING 



CHAPTER I 

PRINCIPLES OF TRIANGULATION 

8. General Scheme 4 

9. Geometrical Conditions 6 

10. Special Cases 8 

11. Classification of Triangulation Systems 9 

12. Selection of Stations 10 

13. Rbconnoissance 10 

14. Curvature and Refraction 12 

15. Intervisibility of Stations 14 

Example 15 

16. Height of Stations 17 

Example 17 

17. Station Marks 17 

18. Observing Stations and Towers 18 

ix 



X TABLE OF CONTENTS 

ART. PAGE 

19. Station Signals or Targets 18 

Board Signals 20 

Pole Signals 20 

Heliotropes 21 

Night Signals 23 

CHAPTER II 

BASE-LINE MEASUREMENT 

20. General Scheme 24 

21. Base-bars and their Use 24 

22. Steel Tapes and their Use 30 

23. Invar Tapes 32 

24. Measurements with Steel and Brass Wires 32 

25. Standardizing Bars and Tapes 33 

26. Corrections Required in Base-line Work 33 

27. Correction for Absolute Length 36 

28. Correction for Temperature 36 

29. Correction for Pull 38 

30. Correction for Sag 39 

31. Correction for Horizontal Alignment 40 

32. Correction for Vertical Alignment 42 

33. Reduction to Mean Sea Level 43 

34. Computing Gaps in Base Lines 44 

35. Accuracy of Base-line Measurements 45 

Example 46 

CHAPTER III 

MEASUREMENT OF ANGLES 

36. General Conditions 47 

37. Instruments for Angular Measurements 47 

38. The Repeating Instrument and its Use 52 

39. First Method with Repeating Instrument 52 

40. Second Method with Repeating Instrument 53 

40a. Reducing the Notes 56 

406. Illustrative Example 57 

40c. Additional Instructions 58 

41. Adjustments of the Repeating Instrument 59 

42. The Direction Instrument and its Use 60 

43. First Method with Direction Instrument 61 

44. Second Method with Direction Instrument 64 

45. The Micrometer Microscopes 65 

46. Reading the Micrometers 67 

46a. Run of the Micrometer 68 

Example 72 



TABLE OF CONTENTS xi 

AHT. PAGE 

47. Adjustments of the Direction Instbttmbnt 71 

48. Reduction to Center 75 

49. Eccentricity of Signal 78 

50. Accuracy of Angle Measurements 78 

Example 79 



CHAPTER IV 

TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 

51. Adjustments 81 

52. Theory of Weights 81 

53. Laws op Weights 82 

Examples 82 

54. Station Adjustment 84 

Examples 84 

55. Figure Adjustment 87 

56. Spherical Excess 88 

57. Triangle Adjustment 89 

58. The Geodetic Quadrilateral 90 

59. Approximate Adjustment of a Quadrilateral 92 

Example 95 

60. Rigorous Adjustment of a Quadrilateral 96 

Example 101 

61. Weighted Adjustments and Larger Systems 100 

62. Computing the Lines op the System 102 

63. Accuracy op Triangulation Work 102 



CHAPTER V 

COMPUTING THE GEODETIC POSITIONS 

64. The Problem 103 

65. The Figure op the Earth 104 

66. The Precise Figure 105 

67. The Practical Figure 106 

68. Geometrical Considerations 106 

69. Analytical Considerations 109 

70. Convergence op the Meridians Ill 

71. The Puissant Solution 113 

72. The Clarke Solution 116 

73. The Inverse Problem 118 

74. Locating a Parallel op Latitude 120 

75. Deviation op the Plumb Line 124 



TABLE OF CONTENTS 



CHAPTER VI 

GEODETIC LEVELING 

ART. PAGE 

76. Principles and Methods 125 

77. Determination of Mean Sea Level 125 

A. Bakometeic Leveling 

78. Instruments and Methods 126 

79. The Computations 128 

Example 128 

80. Accuracy op Barometric Work 129 

B. Trigonometric Leveling 

81. Instruments and Methods 130 

82. By the Sea Horizon Method 131 

83. By an Observation at One Station 133 

84. By Reciprocal Observations 136 

85. Coefficient op Refraction 138 

86. Accuracy of Trigonometric Leveling 139 

C. Precise Spirit Leveling 

87. Instrumental Features 139 

88. General Field Methods 143 

89. The European Level 145 

89o. Constants of European Level 146 

Exam-pies 147 

89b. Adjustments of European Level 150 

Example , . . 152 

89c. Use of European Level 152 

Example 154 

90. The Coast Survey Level 153 

90a. Constants of Coast Survey Level 155 

906. Adjustments of Coast Survey Level 155 

90c. Use of Coast Survey Level 156 

Exaip,ple 157 

91. Rods and Turning Points 158 

92. Adjustment op Level Work 160 

Example 160 

93. Accuracy op Precise Spirit Leveling 161 



TABLE OF CONTENTS xui 



CHAPTER VII 

ASTRONOMICAL DETERMINATIONS 

ART, PAGE 

94. General Considerations 163 



Time 

95. General Principles 164 

96. Mean Solar Time 164 

96a. Standard Time 165 

966. To Change Standard Time to Local Mean Time and vice 

versa 165 

Examples 168 

97. Sidereal Time 168 

98. To Change a Sidereal to a Mean Time Interval and vice 

versa 169 

99. To Change Local Mean Time or Standard Time to Sidereal . . 169 

Example 169 

100. To Change Sidereal to Local Mean Time or Standard Time . . 170 

Example 170 

101. Time bt Single Altitudes 171 

101a. Making the Observation 172 

1016. The Computation 173 

Example 175 

102. Time bt Equal Altitudes 176 

102a. Making the Observation 176 

1026. The Computation 177 

Example 180 

103. Time by Sun and Star Transits 181 

103a. Sun Transits with Engineering Instruments 181 

1036. Star Transits with Engineering Instruments 182 

103c. Star Transits with Astronomical Instruments 183 

104. Choice of Methods 184 

105. Time Determinations at Sea 184 



Latitude 

106. General Principles 186 

107. Latitude prom Observations on the Sun at Apparent Noon .... 188 

108. Latitude by Culmination of Circumpolar Stars 190 

109. Latitude by Prime-vertical Transits 192 

110. Latitude with the Zenith Telescope 193 

111. Latitude Determinations at Sea 196 

112. Periodic Changes in Latitude 196 



xiv TABLE OF CONTENTS 



Longitude 

ABT, PAGE 

113. Genbhal Peinciples 197 

114. Difference op Longitude by Special Methods 198 

By Special Phenomena 198 

By Flash Signals 198 

115. Longitude by Lunak Observations 198 

By Lunar Distances 199 

By Lunar Culminations 199 

By Lunar Occultations 199 

116. Difference OF Longitude BY Transportation op Chronometers. 199 

117. Difference of Longitude by Telegraph 200 

By Standard Time Signals 201 

By Star Signals 201 

By Arbitrary SigruUs 202 

118. Longitude Determinations at Sea 203 

119. Periodic Changes in Longitude 203 

Azimuth 

120. General Principles 203 

121. The Azimuth Mark 204 

122. Azimuth by Sun or Star Altitudes 205 

122a. Making the Observation 205 

1226. The Computation 206 

123. Azimuth from Observations on Circumpolar Stars 207 

123o. Fundamental Formulas 208 

1236. Approximate Determinations 213 

123c. The Direction Method 215 

Example 216 

123d. The Repeating Method 218 

Example 219 

123e. The Micrometric Method 221 

Example 223 

124. Azimuth Determinations at Sea 225 

125. Periodic Changes in Azimuth 226 



CHAPTER VIII 

GEODETIC MAP DRAWING 

126. General Considerations 227 

127. Cylindrical Projections 229 

127a. Simple Cylindrical Projection 229 

1276. Rectangular Cyhndrical Projection 231 

127c. Mercator's Cyhndrical Projection 231 

128. Trapezoidal Projection 234 



TABLE OF CONTENTS xv 

ART. PAGE 

129. Conical Phojbctions 234 

129a. Simple Conic Projection 235 

1296. Mercator's Conic Projection 236 

129c. Bonne's Conic Projection 238 

129d. Polyconic Projection 239 



PART II 

ADJUSTMENT OF OBSERVATIONS BY THE. 
METHOD OF LEAST SQUARES 



CHAPTER IX 
DEFINITIONS AND PRINCIPLES 

130. General Considebations 241 

131. CLASSinCATION OF QUANTITIES 241 

132. Classification of Values 242 

133. Observed Values and Weights 243 

134. Most Probable Values and Weights 243 

135. True and Residual Errors 245 

136. Sources of Error 247 

137. Nature of Accidental Errors 247 

138. The Laws of Chance 248 

139. Simple Events 248 

140. Compound Events 249 

141. Concurrent Events 249 

142. Misapplication of the Laws of Chance 250 

CHAPTER X 
THE THEORY OF ERRORS 

143. The Laws of Accidental Error 252 

144. Graphical Representation of the Laws of Error 253 

145. The Two Types of Error 254 

146. The Facility of Error 255 

147. The Probability of Error 256 

148. The Law of the Facility op Error 257 

149. Form of the Probability Equation . j, 257 

150. General Equation op the Probability Curve , 260 

151. The Value of the Precision Factor 262 

152. Comparison of Theory and Experience 264 



xvi TABLE OF CONTENTS 



CHAPTER XI 
MOST PROBABLE VALUES OF INDEPENDENT QUANTITIES 

AKT. PAGE 

153. General Considerations >. 266 

154. Fundamental Principle op Least Squares 266 

155. Direct Observations op Equal Weight 267 

Example 268 

156. General Principle op Least Squares 268 

157. Direct Observations op Unequal Weight 270 

Example 271 

158. Indirect Observations 271 

159. Indirect Observations op Equal Weight on Independent 

Quantities 273 

Examples 274 

160. Indirect Observations op Unequal Weight on Independent 

Quantities 276 

Examples 277 

161. Reduction op Weighted Observations to Equivalent Obser- 

vations OP Unit Weight 278 

Examples 279 

162. Law op the Coeppicients in Normal Equations 280 

Example 281 

163. Reduced Observation Equations 281 

Examples 282 

CHAPTER XII 

MOST PROBABLE VALUES OF CONDITIONED AND COMPUTED 

QUANTITIES 

164. Conditional Equations 284 

165. Avoidance OP Conditional Equations 285 

Examples 286 

166. Elimination op Conditional Equations 288 

Example 289 

167. Method op Correlatives .- 290 

Example 295 

168. Most Probable Values op Computed Quantities 296 

CHAPTER XIII 

PROBABLE ERRORS OF OBSERVED AND COMPUTED 
QUANTITIES 

A. Op Observed Quantities 

169. General Considerations 297 

170. Fundamental Meaning op the Probable Error 297 



TABLE OF CONTENTS xvii 

ABT. PAGE 

171. Graphical Representation op the Probable Error 298 

172. General Value op the Probable Error 299 

173. Direct Observations op Equal Weight 300 

Example 300 

174. Direct Observations op Unequal Weight 301 

Example 302 

175. Indirect Observations on Independent Quantities 302 

Example 303 

176. Indirect Observations Involving Conditional Equations 304 

177. Other Measures op Precision 304 

B. Op Computed Quantities 

178. Typical Cases 306 

179. The Computed Quantity is the Sum or Difference op an 

Observed Quantity and a Constant 306 

Example 307 

180. The Computed Quantity is' Obtained prom an Observed 

Quantity by the Use op a Constant Factor 307 

Example 308 

181. The Computed Quantity is any Function of a Single Ob- 

served Quantity 308 

Example 308 

182. The Computed Quantity is the Algebraic Sum op Two or 

More Independently Observed Quantities 308 

Examples 309 

183. The Computed Quantity is any Function op Two or More 

Independently Observed Quantities 310 

Examples 310 

CHAPTER XIV 
APPLICATION TO ANGULAR MEASUREMENTS 

184. General Considerations 312 

Single Angle Adjustment 

185. The Case op Equal Weights 312 

Example 312 

186. The Case of Unequal Weights 313 

Example 313 

Station Adjustment 

187. General Considerations 313 

188. Closing the Horizon with Angles op Equal Weight 313 

Example 315 



xviii TABLE OF CONTENTS 

AHT. PAGE 

189. Closing the Horizon with Angles of Unequal Weight. . . 315 

Example 317 

190. Simple Summation Adjustments 317 

Examples .' 318 

191. The General Case 319 

Examples 320 

Figure Adjustment 

192. General Considerations 321 

193. Triangle Adjustment with Angles of Equal Weight 322 

Example 323 

194. Triangle Adjustment with Angles op Unequal Weight 323 

Example 324 

195. Two Connected Triangles 325 

Example 325 

196. Quadrilateral Adjustment 326 

Example 330 

197. Other Cases of Figure Adjustment 329 

Examples 331 



CHAPTER XV 

APPLICATION TO BASE-LINE WORK 

198. Unweighted Measurements 333 

Example 333 

199. Weighted Measurements 333 

Example 334 

200. Duplicate Lines 334 

Example 335 

201. Sectional Lines 335 

Examples 336 

202. General Law of the Probable Errors 336 

Example 337 

203. The Law op Relative Weight 337 

204. Probable Error op a Line op Unit Length 338 

205. Determination of the Numerical Value op the Probable 

Error of a Line op Unit Length 339 

Example 341 

206. The Uncertainty op a Base Line 342 

Examples 343 



TABLE OF CONTENTS xix 



CHAPTER XVI 
APPLICATION TO LEVEL WORK 

ABT. PAGE 

207. Unweighted Measurements 344 

Example 344 

208. Weighted Measurements 345 

Example 345 

209. Duplicate Lines 346 

Example 346 

210. Sectional Lines 347 

Example 347 

211. General Law of the Probable Errors 347 

Example 348 

212. The Law of Relative Weight , 348 

213. Probable Error op a Line of Unit Length 348 

214. Determination of the Numerical Value of the Probable 

Error op a Line of Unit Length 349 

Example 350 

215. Multiple Lines 350 

Example 351 

216. Level Nets 352 

Example 353 

217. Intermediate Points 355 

Example 356 

218. Closed Circuits 357 

Example 358 

219. Branch Lines, Circuits, and Nets 359 



FULL-PAGE PLATES 

Example op a Triangulation System 6 

Example of a Tower Station 19 

EiMBECK Duplex Base-bar 28 

Contact Slides, Eimbeck Duplex Base-bar 29 

Repeating Instrument 49 

D;rection Instrument 50 

Altazimuth Instrument 51 

Reduction to Center 77 

Location op a Boundary Line 123 

European Type op Precise Level 141 

Coast Survey Precise Level 142 

Molitoh's Precise Level Rod and Johnson's Foot-pin 159 

Celestial Sphere 166 

Portable Astronomical Transit 185 

Map op Circumpolar Stars 191 

Zenith Telescope 195 



XX TABLE OF CONTENTS 

TABLES 

PAGE 

I. Curvature and Refraction in Elevation 363 

II. Logarithms of the Puissant Factors 364 

III. Barometric Elevations 366 

IV. Correction Coefficients to Barometric Elevations for 

Temperature (Fahrenheit) and Humidity ' 368 

V. Logarithms of Radius of Curvature (Metric) 369 

VI. Logarithms of Radius of Curvature (Feet) 370 

VII. Corrections for Curvature and Refraction in Precise 

Spirit Leveling 370 

VIII. Mean Angular Refraction 371 

IX. Elements of Map Projections 372 

X. Constants and Their Logarithms 373 

BIBLIOGRAPHY 

References on Geodetic Surveying 374 

References on Method op Least Squares 375 



GEODETIC SURVEYING 

AND 

THE ADJUSTMENT OF OBSERVATIONS 

(METHOD OF LEAST SQUARES) 

INTRODUCTION 

1. Geodesy is that branch of science which treats of making 
extended measurements on the surface of the earth, and of 
related problems. Primarily the object of such work is to furnish 
precise locations for the controlling points of extensive surveys. 
The determination of the figure and dimensions of the earth, 
however, is also a fundamental object. 

2. The Importance of Geodetic Work is recognized by all 
civilized nations, each of which maintains an extensive organi- 
zation for this purpose. The knowledge thus gained of the earth 
and its surface has been of great benefit to humanity. In further- 
ance of this object an International Geodetic Association has been 
formed (1886), and includes the United States (1889) in its mem- 
bership. 

3. Geodetic Work in the United States is carried on by the 
United States Coast and Geodetic Survey, a branch of the Depart- 
ment of Commerce and Labor. The valuable papers on geodetic 
work published by this department may be obtained free of 
charge by addressing the " Superintendent United States Coast 
and Geodetic Survey, Washington, D. C." 

4. History. Plane surveying dates from about the year 
2000 B.C. Geodesy literally began'about 230 B.C., in the time of 
Erastosthenes and the famous school of Alexandria, at which 
time very fair results were secured in the effort to determine the 



2 GEODETIC SURVEYING 

shape and size of the earth. Modem geodesy practically began 
in the seventeenth century in the time of Newton, owing to 
disputes concerning the shape of the earth and the flattening of 
the poles. (See Chapter III for further treatment of this subject.) 

5. The Scope of Geodesy originally involved only the shape 
of the earth and its dimensions. Modern geodesy, covers many 
topics, the principal ones being about as follows : 

Leveling (on land) ; 

Soundiags (oceans, lakes, rivers); 

Mean Sea Level; 

Triangulation; 

Time; 

Latitude (by observation) ; 

Longitude (by observation) ; 

Azimuth (by observation) ; 

Computation of Geodetic Positions (latitude, longitude, and 

azimuth by computation) ; 
Problems of Location; 
Figure and Dimensions of the Earth; 
Configuration of the Earth; 
Map Projection; 
Gravity ; 

Terrestrial Magnetism; 
Deviation of the Plumb Line; 
Tides and Tidal Phenomena; 
Ocean Currents; 
Meteorology. 

6. Geodetic Surveying. This class of surveying is distin- 
guished from plane surveying by the fact that it takes account 
of the curvature of the earth, usually necessitated by the large 
distances or areas covered. Work of this character requires the 
utmost refinement of methods and instruments, 

1st, Because allowing for the curvature of the earth is in 

itself a refinement; 
2nd, Because small measurements have to be greatly 

expanded; 
3rd, Because the magnitude of the work involves an accumu- 
lation of errors. 
The fundamental operations of geodetic survejang are Triangu- 
lation and Precise Leveling. These in turn require the deter- 



INTRODUCTION 3 

mination of time, latitude, longitude, and azimuth; the deter- 
mination of mean sea level; and a knowledge of the figure and 
dimensions of the earth. The first part of this book covers 
such points on these subjects as are likely to interest the civil 
engineer. 

7. The Adjustment of Observations. vUl measurements are 
subject to more or less unlmown and unavoidable sources of 
error. Repeated measurements of the same quantity can not 
be made to agree precisely by any refinement of methods or 
instruments. Measurements made on different parts of the same 
figure do not give resiilts that are absolutely consistent with the 
rigid geometrical requirements of the case. Some method of 
adjustment is therefore necessary in order that these discrepan- 
cies may be removed. Obviously that method of adjustment 
will be the most satisfactory which assigns the most probable 
values to the unknown quantities in view of all the measurements 
that have been taken and the conditions which must l)e satisfied. 
Such adjustments are now imiversally made by the Method of 
Least Squares. The application of this method to the elementary 
problems of geodetic work forms the subject-matter of the second 
part of this book. 



PART I 

GEODETIC SURVEYING 



CHAPTER 1 

PRINCIPLES OF TRIANGULATION 

8. General Scheme. The word triangulaiion, as used in 
geodetic surveying, includes all those operations required to 
determine either the relative or the absolute positions of different 
points on the surface of the earth, when such operations are 
based on the properties of plane and spherical triangles. By the 
relative position of a point is meant its location with reference 
to one or more other points in terms of angles or distance as may 
be necessary. In geodetic work distances are usually expressed 
in meters, and are always reduced to mean sea level, as explained 
later on. By the absolute position of a point is meant its loca- 
tion by latitude and longitude. Strictly speaking the absolute 
position of a point also' includes its elevation above mean sea 
level, but if this is desired it forms a special piece of work, and 
comes imder the head of leveling. Directions are either relative 
or absolute. The relative directions of the lines of a survey are 
shown by the measured or computed angles. The absolute 
direction of a line is given by its azimuth, which is the angle it 
makes with a meridian through either of its ends, counting clock- 
wise from the south point and continuously up to 360°. For 
reasons which will appear later the azimuth of a line must always 
be stated in a way that clearly shows which end it refers to. 

In the actual field work of the triangulation suitable points, 
called stations, are selected and definitely marked throughout 
the area to be covered, the selection of these stations depending 
on the character of the co'untry and the object of the survey. 

4 



PRINCIPLES OF TRIANGULATION 5 

The stations thus established are regarded as forming the vertices 
of a set of mutually connected triangles (overlapping or not, as 
the case may be), the complete figure being called a triangula- 
tion system. At least one side and all the angles in the triangula- 
tion system are directly measured, using the utmost care. All 
the remaining sides are obtained by computation of the successive 
triangles, which (corrected for spherical excess, if necessary) 
are treated as plane triangles. The line which is actually measured 
is called the base line. It is common to measure an additional 
line near the close of the work, this line being connected with 
the triangulation system so that its length may also be obtained 
by calculation. Such a line is called a check base, forming an 
excellent check on both the field work and the computations of the 
, whole survey. In work of large extent intermediate bases or check 
bases are often introduced. Lines which are actually measured on 
the ground are always reduced to mean sea level before any further 
use is made of them. It is evident that all computed lengths will 
therefore refer to mean sea level without further reduction. 

The stations forming a triangulation system are called triangu- 
lation stations. Those stations (usually triangulation stations) 
at which special work is done are commonly given corresponding 
names, such as base-line stations, astronomical stations, latitude 
stations, longitude stations, azimuth stations, etc. 

An example of a small triangulation system (United States 
and Mexico Boundary Survey, 1891-1896) is shown in Fig. 1, 
page 6, the object being to connect the " Boundary Post " on 
the azimuth line to the westward with " Monument 204 " on the 
azimuth line to the eastward. The air-line distance between 
these points is about 23 miles. The system is made up of the 
quadrilateral West Base, Azimuth Station, East Base, Station 
No. 9; the quadrilateral Pilot Knob, Azimuth Station, Station 
No. 10, Station No. 9; the quadrilateral Pilot Knob, Azimuth 
Station, Station No. 10, Monument 204; and the triangle Pilot 
Knob, Boundary Post, Azimuth Station. The base line (West 
Base to East Base) has a length of 2,205 meters (1. 37 -f- miles), 
and the successive expansions are evident from the figure. 

9. Geometrical Conditions. The triangles and combinations 
thereof which make up a triangulation system form a figure involv- 
ing rigid geometrical relations among the various lines and angles. 
The measured values seldom or never exactly satisfy these con- 



GEODETIC SURVEYING 



U4 



45' 

.Pilot Knob 



114 40 



Boundary Post 



^East Base* 




Iriangulation 

in vicinity of 

Yuma, Arizona; 

International Boundary Survey 

United States and Mexico, 

1891-1896. 

Scale=l:180,000. 



Fig. 1.— Example of a Triangulation System. 
From Report of U. S. Section of International Boundary Commission. 



PEINCIPLES OF TRIANaULATION 7 

ditions, and must therefore be adjusted until they do. In the 
nature of things the true values of the lines and angles can never 
be Icnown, but the greater the number of independent conditions 
on which an adjustment is based the greater the probability that 
the adjusted values lie nearer to the truth than the measured 
values.- It is for this reason that work of an extended character 
is arranged so that some or aU of the measured values wiU be 
involved in more than one triangle, thus greatly increasing the 
number of conditions which must be satisfied by the adjustment. 

The simplest system of triangulation is that in which the work 
is expanded or carried forward through a succession of independent 
triangles, each of which is separately adjusted and computed; 
and where the work is of moderate extent this is usually all that is 
necessary. The best triangulation system, under ordinary circum- 
stances, when the survey is of a more extended character, or 
great accuracy is desired, is that in which the work is so arranged 
as to form a succession of independent quadrilaterals, each of 
which is separately adjusted and computed. (In work of great 
magnitude the entire system would be adjusted as a whole.) 
A geodetic quadrilateral is the figure formed by connecting any 
foxir stations in every possible way, the result being the ordinary 
quadrilateral with both its diagonals included ; there is no station 
where the diagonals intersect. The eight comer angles of the 
quadrilateral are always measured independently, and then 
adjusted (as explained later) so as to satisfy all the geometric 
requirements of such a figure. Other arrangements of triangles 
are sometimes used for special work. More complicated systems 
of triangles or adjustment are seldom necessary or desirable, 
except in the very largest class of work. Since triangiilation 
systems are usually treated as a succession of independent figures 
it evidently makes no difference whether the figures overlap or 
extend into new territory. . 

Every triangulation system is fundamentally made up of 
triangles, and in order that small errors of measurement shall not 
produce large errors in the computed values, it is necessary that 
only well shaped triangles should be permitted. The best shaped 
triangle is evidently equilateral, while the best shaped quadri- 
lateral is a perfect square,- and these are the figures which it is 
desirable to approximate as far as possible. A well shaped 
triangle is one which contains no angle smaller than 30° (involving 



GEODETIC SUEVEYING 



the requirement that no angle must exceed 120°). In a quadri- 
lateral, however, angles much less than 30° are often necessary 
and justifiable in the component triangles. 

10. Special Cases. It is often desirable and feasible (espe- 
cially on reconnoissance) to connect two distant stations with a 
narrow and approximately straight triangulation system, as shown 
diagrammatically by the several plans in Fig. 2. In these diagrams 
the heavy dots represent the stations occupied, all the angles at 
each station being directly measured. The maximum length 

I II III 




Fig. 2. 

of sight is approximately the same in each case. The stations 
to be connected are marked A and B. In an actual survey, of 
course, the location of the stations^ could only approximate the 
perfect regularity of the sketches. 

In System I the terminal stations are connected by a simple 
chain of triangles. This plan is the cheapest and most rapid, 
but also the least accurate. 

System II is given in two forms, which are substantially alike 
in cost and resvilts, the hexagonal idea being the basis of each 
construction. This system not only covers the largest area, 
but greatly increases the accuracy attainable. The large num- 



PRINCIPLES OF TEIANGULATION 9 

ber of stations in this system necessarily increases both the labor 
and the cost. 

System III is formed by a continuous succession of quadri- 
laterals, and is the one to use where the highest degree of accuracy 
is desired. The area covered is less than in System I, but the cost 
and labor approximate System II. 

11. Classification of Triangulation Systems. It has been 
foimd convenient to classify triangulation systems (and the 
triangles involved) as primary, secondary and tertiary, based on 
the magnitude and accuracy of the work. 

Primary triangulation is that which is of the greatest magnitude 
and importance, sometimes extending over an entire continent. 
In work of this character the highest attainable degree of accuracy 
(1 in 500,000 or better) is sought, using long base lines, large and 
well shaped triangles, the highest grade of instruments, and the 
best known methods of observation and computation. Primary 
base lines may measure from three to ten or more miles in length, 
with successive base lines occurring at intervals of one hundred 
to several hundreds of miles (about 30 to 100 times the length 
of base), depending on the character of the country traversed and 
the instrument used in making the measurement. In primary 
triangulation the sides of the triangles may vary from 20 to 100 
miles or more in length. 

Secondary triangulation covers work of great importance, 
often including many hundred miles of territory, but where the 
base lines and triangles are smaller than in primary systems, and 
where the same extreme refinement of instruments and methods 
is not necessarily required. An accuracy of 1 in 50,000 is good 
work. Base lines in secondary work may measure from one to 
three miles in length, and occur at intervals of about twenty to 
fifty times the length of base. The triangle sides may vary from 
about five to forty miles in length. 

Tertiary triangulation includes all those smaller systems 
which are not of sufficient size or importance to be ranked as 
primary or secondary. The accuracy of such work ranges upwards 
from about 1 in 5,000. The base lines measure from about a 
half to one and a half miles long, occurring at intervals of about 
ten to twenty -five times the length of base. The triangle sides may 
measure from a fraction of a mile up to about six miles in length. 

In an extended survey the primary triangulation furnishes 



10 GEODETIC SUEVEYING 

the great main skeleton on which the accuracy of the whole survey 
depends; the secondary systems (branching from the primary) 
furnish a great many well located intermediate points; ^nd the 
tertiary systems (branching from the secondary) furnish the 
mioltitude of closely connected points which serve as the reference 
points for the final detailed work of the survey. 

12. Selection of Stations. This part of the work calls for the 
greatest care and judgment, as it practically controls both the 
accuracy and the cost of the survey. Every effort, therefore, 
should be made to secure the best arrangement of stations con- 
sistent with the object of the survey, the grade of work desired, 
and the allowable cost. The base line is usually much smaller 
than the principal lines of the triangulation system, and there- 
fore requires an especially favorable location, in order that its 
length may be accurately determined. Approximately level 
or gently sloping ground (not over about 4°) is demanded for 
good base-line work. It is also necessary that the base line be 
connected as directly as possible with one of the main lines of the 
system, using a minimum number of well shaped triangles. The 
base-line stations and the connecting triangulation stations are 
consequently dependent on each other, in order that both objects 
may be served. In flat country the greatest freedom of choice 
would probably lie with the base-line stations, while in rough 
country the triangulation stations would probably be largely 
controlled by a necessary base-line location. 

The various stations in a triangulation system must be selected 
not only with regard to the territory to be covered and the for- 
mation of well shaped triangles, but so as to secure at a minimvim 
expense the necessary intervisibility between stations for the 
angles to be measured. Clearing out lines of sight is expensive 
in itself, and may also result in damages to private interests. 
Building high stations in order to see over obstructions is like- 
wise expensive. A judicious selection of stations may materially 
reduce the cost of such work without prejudicing the other 
interests of the survey. It is important that lines of sight shoiild 
not pass over factories or other sources of atmospheric disturb- 
ance. These and similar points familiar to surveyors must all 
receive the most careful consideration. 

13. Reconnoissance. The preliminary work of examining 
the country to be surveyed, selecting and marking the various 



PEINCIPLES OF TEIANGULATION 11 

base-line and angle stations, determining the required height 
for tower stations, etc., is called reconnaissance. As much infor- 
mation as possible is obtained from existing maps, such as the 
height and relative location of probable station points and desir- 
able arrangement of triangles. The reconnoissance party then 
selects in the field the best location of stations consistent with the 
grade and object of the survey and in accordance with the prin- 
ciples laid down in the preceding article. The reconnoissance 
is often carried forward as a survey itself, so that fairly good 
values are obtained of all the quantities which will finally be 
determined with greater accuracy by the main survey. When a 
point is thought to be suitable for a station a high signal is erected, 
such as a flag on a pole fastened on top of a tree or building, 
and the surrounding comitry is scanned in all directions to pick 
up previously located signals and to select favorable points for 
advance stations. 

The instrumental outfit of the reconnoissance party is selected 
in accordance with the character of the information which it 
proposes to obtain. In any event it must be provided with 
convenient means for measuring angles, directions, and eleva- 
tions. A minimum outfit would probably contain a sextant for 
measuring angles, a prismatic compass for measuring directions, 
an aneroid barometer for measuring elevations, a good field glass, 
and creepers for climbing poles and trees. 

A common problem for the reconnoissance party is to estab- 
lish the direction .between two stations which can not be seen 
from each other until the forest growth is cleared out along 
the connecting line. Any kind of a traverse rim from one station 
to the other woidd furnish the means for 
computing this direction, but the follow- 
ing simple plan can of ten, be used: 

Let AB, Fig. 3, be the direction it is 
desired to establish. Find two inter- 
visible points C and D from each of 
which both A and B can be seen. 
Measure each of the two angles at C 
and D and assume any value (one is Fig. 3. 

the simplest) for the length CD. From 

the triangle ACD compute the relative value of AD. Sim- 
ilarly from BCD get the relative value of BD. Then from the 




12 



GEODETIC SDEVEYING 



triangle ABD compute the angles at A and B, which will give 
the direction of AB from either end with reference to the point 
D. All computed lengths are necessarily only relative because 
CD was assumed, but the computed angles are of course correct. 

The required intervisibUity of any two stations must be finally 
determined on the ground by the reconnoissance party, but a 
knowledge of the theoretical considerations governing this ques- 
tion is of the greatest importance and usefulness. 

14. Curvature and Refraction. Before discussing the inter- 
visibUity of stations it is necessary to consider the effect of curva- 
ture and refraction on a line of sight. In geodetic work curvature 




Fig. 4. 

is understood to mean the apparent reduction of elevation of 
an observed station, due to the rotundity of the earth and 
consequent falling away of a level line (see Art. 76) from a 
horizontal line of sight. Refraction is understood to mean the 
apparent increase of elevation of an observed station, due to 
the refraction of light and consequent curving of the line of sight 
as it passes through air of differing densities. The net result 
is an apparent loss of elevation, causing an angle of depression 
in sighting between two stations of equal altitude. In Fig. 4 
the circle ADE represents a level line through the observing 
point A, necessarily following the curvature of the earth. Assum- 



PEINCIPLES OF TRIANGULATION 



13 



ing the line of sight to be truly level or horizontal at the point 
A, the observer apparently sees in the straight line direction 
AB (tangent to the circle at A), but owing to the refraction of 
light actually looks along the curved line AC (also tangent at 
A). The observer therefore regards C as having the same eleva- 
tion as A, whereas the point D is the one which really has the 
same elevation as A. There is hence an apparent loss of eleva- 
tion at C equal to CD, as the net result of the loss BD due to 
curvature and the gain BC due to refraction. Just as C appears 
to lie at B, so any point F appears to lie at a corresponding point 
G. The apparent difference of elevation of the points A and 
F is measured by the line BG, the true difference being DF. 
As DF = BG + BD - FG, the apparent loss equals BD - FG, 
which does not ordinarily differ much from CD. 




Fig. 5. 

So far as the intervisibility of two stations is concerned it is 
only necessary to know the effect of curvature and refraction 
with reference to a straight line tangent to the earth at mean 
sea level. Referring to Fig. 5, BD represents the effect of 
curvature, and BC the effect of refraction, as in the previous 
figure. By geometry we have 

A52 =BD X BE. 



The earth is so large as compared with any actual case in 
practice that we may substitute AD (=distance, called K) 



14 GEODETIC SUEVEYING 

for AB, and DE ( = 2R) for BE, without any practical error, 
and write 

„_ , Distance^ K^ 

BD = curvature = 7 t= 7 — rrr = ?rn, 

Aver. diam. of earth 2R ' 

in which all values are to be taken in the same units. (For 
mean value of R see Table X at end of book.) As the result of 
proper investigations we may also write 

, ,. Distance^ ^2 K^ 

BC = retraction = m-. ^ — 7 -;- = »w-b- = 2m— j^ , 

x\ver. rad. of earth it 2R 

in which m is a coefficient having a mean value of .070, and K and 
R are the same as before. (For additional values of m see Art. 85.) 
We thus have 

BD-BC=CD= curv. and refract. = (1 - 2m)-^ . 

Table I (at end of book) shows the effect of curvature and 
refraction, computed by the above formula, for distances from 
1 to 66 miles. 

15. Intervisibility of Stations The elevation (or altitude) 
of a station is the elevation of the observing instrument above 
mean sea level. This is not to be confused with the height of a 
station, which is the elevation of the instrument above the natural 
ground. In order that two stations may be visible from each 
other the line of sight must clear all intermediate points. The 
necessary (or minimum) elevation of each station will therefore 
be governed by the following considerations: 

1. The elevation of the other station. Obviously a line of sight 
which is required to clear a given point by a certain amount can 
not be lowered at one end without being raised at the other. 

2. The profile of the intervening country. It is evidently not 
only the height of an intermediate point but also its location 
between the two stations that will determine its influence on their 
intervisibility. An elevation great enough to obstruct the line 
of sight if located near the lower station might be readily seen 
over if located near the higher station. 

3. The distance between the stations. Owing to the curvature 
of the earth it is necessary in looking from one point to another 
to see over the intervening rotundity, the extent of which depends 



PRINCIPLES OF TRIANGULATION 15 

on the distance between the stations. Since lines of sight are 
nearly straight this can not be accomplished imless at least one 
of the stations has a greater elevation than any intermediate 
point. Owing to the refraction of light the line of sight is not 
really a straight line, but in any actual case is practically the arc 
of a circle, with the concavity downwards, and a radius about 
seven times that of the earth. This fact slightly lessens the 
elevation necessary to see over the rotundity, but otherwise 
does not change the conditions to be met. Thus in Fig. 5 the 
points F and C are just barely intervisible, though F and C both 
have greater elevations than A. 

In view of the above facts it is usually necessary to place 
stations on the highest available ground, such as ridge lines, 
summits, or mountain peaks, increasing the height, if necessary, 
by suitably built towers. 

The simplest question of intervisibility is illustrated in Fig. 5, 
where all points between station F and station C lie at the eleva- 
tion of mean sea level. If the elevation of F is given or assumed 
the corresponding distance HA to the point of tangency is taken 
out directly from Table I (interpolating if necessary) . The 
value CD corresponding to the remaining distance AD is then 
taken out from the same table, and gives the minimum elevation 
of C which will make it visible from F. Thus if HD = 30.0 
mUes, and elevation oi F = 97.0 ft., we have HA = 13.0 mUes, 
and the remaming distance AD = 17.0 miles, calling for a min- 
imum elevation of 165.8 ft. for station C. 

In general the profile between two stations is more or less 
irregular, and the question can not be handled in the above 
simple maimer. It is usually necessary to compute the elevations 
of the line of sight at a number of different points and compare 
the resxilts with the ground elevation at such points. The critical 
points are usually evident from an inspection of the profile. 
Owing to the imcertainties of refraction accurate methods of 
computation are not worth while; different methods of approx- 
imation give slightly different results, but all sufficiently near 
the truth for the desired purpose. 

The following example will show a satisfactory method of pro- 
cedure in any case that may arise in practice. The line AEJP, 
Fig. 6, page 16, is the natural profile of the ground, and it is desired 
if possible to estabhsh stations at A and P. The critical points 



16 



GEODETIC SURVEYING 



that might obstruct the line of sight are evidently at E and J. 
Assume the following data to be known: 

Distances (at mean sea level) . Elevations (above M. S. L.) . 
BH =30.0 miles A =1140.6 ft. =AB 

HN = 10.1 " E= 1322.7 " =EH 

NR = 10.7 " 7 = 1689.0 '•• =JN 

P =2098.3 " =PE 

For an imaginary line of sight BQ, horizontal at B we have 
from Table I (by interpolating) : 

C G = 516.4 ft. =GH. 
Elevation of M = 922.8 " =MiV. Hence PQ = 617.4. ft. 
[ Q = 1480.9 " =QR. 




Fig. 6. 



Assuming the lines of sight BP, AP, and AO to have the same 
radius of curvature as BQ, we may write approximately 



FG 
PQ 



BG 
BQ 



BH 
BR 



and 



LM BM 



BN 
BR' 



PQ BQ 

giving, by substitution, FG = 364.6 ft. and LM = 498.3 ft. 

Hence we have elevation of 



By the similar approximations 
DF FP HR 



|P = 881.0 ft. 



1421.1 ft. 



AB BP BR 



and 



KL 
AB 



LP 
BP 



NR 
BR ' 



PRINCIPLES OF TRIANGULATION 17 

we find DF = 467.0 ft. and KL = 240.2 ft. 

Ti V, 1 +• f [-D =1848.0 ft. 

Hence we nave elevation oi -^ „ ,^o, o jj 

yK = 1661.3 ft. 

Hence the line of sight AP clears E by 25.3 ft., but fails to 
clear J by 27\7 ft. 

16. Height of Stations. Referring to the previous article, 
suppose it is desired to erect a tower OP, so that the line of sight 
OA shall clear the obstruction J. It was found that the line 
PA failed to clear .7 by 27.7 ft., and it is not desirable to have a 
line of sight less than 6 ft. from the ground, hence IK should be 
about 34 ft. Using the approximation 

0P_-^=^ OP^ _ 50^ 

IK " AK~ BN °^ 34.0 ~ 40.1 ' 

we find OP =43.1 ft. 

Hence a suitable tower at P should not be less than 43 ft. 
high. If it were desired to build a smaller tower at P, the instru- 
ment at A would also have to be elevated, the amount being 
determined by a similar plan of approximation. It is evident 
that the least total height of towers is obtained by building a 
single tower at the station nearest to the obstruction. If the 
obstruction is practically midway between the stations the com- 
bined height of any two corresponding towers would of course 
come the same as that of a suitable single tower. If more than 
one obstruction is to be seen over, the most economical arrange- 
ment of towers is readily found by a few trial computations. 

In heavily wooded country tower stations extending above the 
tree tops are frequently more economical than clearing out long 
lines of sight, and their construction is therefore justified even 
though the intervening country would not otherwise demand 
their use. In general it is not wise to have a line of sight near 
the ground for any large portion of its length, on account of the 
unsteadiness of the atmosphere and the risk of sidewise refraction. 

17. Station Marks. Any kind' of a survey requires the station 
marks to remain luichanged at least during the period of the 
survey. When work is of sufficient magnitude or importance 
to justify geodetic methods and instruments, permanent station 
marks are usually desirable. The best plan seems to be to place 
the principal mark below the ground, as least likely to suffer 



18 GEODETIC SURVEYING 

disturbance by frost, accident, or malicious interference. Though 
many plans have been tried, the common imderground mark 
consists of a stone about 6"X6"X24" placed vertically with 
its top about 30" below the surface of the groimd, the center 
point being marked by a small hole or copper bolt. The imder- 
ground mark is of course only used in case there is reason to 
think the surface mark has been moved. The siirface mark 
usually consists of a similar stone, reaching nearly down to the 
bottom stone and extending a few inches above the surface, 
with the station point similarly marked. Three witness stones 
are commonly set near the station (where least likely to be dis- 
turbed, ordinarily 200 or more feet from the station, and forming 
approximately an equilateral triangle), with their azimuths 
and distances recorded, so that the station might be restored 
if entirely destroyed. Stones about 36" long and projecting 
about 12" above the surface have proven satisfactory. Other 
means of establishing permanent stations wiU suggest themselves 
to the surveyor when the surrounding conditions are known. 

18. Observing Stations and Towers. In addition to the station 
mark a suitable support is required to carry the observing instru- 
ment. Unless the tripod is very heavy and stiff it will not prove 
satisfactory. In such a case a rigid support must be provided. 
Heavy posts well set in the ground may serve as the basis for 
such a construction for a low height, bracing as may prove neces- 
sary for rigidity. If an observing platform is built it must not 
be connected in any way with the structure that carries the instru- 
ment. A low masonry pier makes an excellent station. Under 
15 ft. in height a tripod can be built at the station heavy enough 
to be satisfactory as an instrument support. For greater heights 
a regular tower should be built to carry the instrument, so braced 
and guyed as to be absolutely immovable and free from, vibra- 
tion. The observer's platform must be carried by an entirely 
independent structure surrounding the instrument tower with- 
out being in any way connected with it, or in any way possible 
to come in contact with it. A light awning on a framework 
attached to the observer's platform should shelter the instrument 
from the sun. Fig. 7 shows a common form of tower station. 

19. Station Signals or Targets. These terms (used more or 
less interchangeably) refer to that object at a station which is 
sighted at by observers at other stations. A satisfactory target 



PEINCIPLES OF TRIANGULATION 19 




Fig. 7. — Tower Station. 
From Appendix No. 9, Report for 1882, U. S. C. and G. S. 



20 GEODETIC SUEVEYING 

must be distinctly visible against any background and of suit- 
able width for accurate bisection, and preferably free irom phase. 
When the face of a target is partially illuminated and partially 
in shadow, the observer usually sees only the illuminated portion 
and thus makes an erroneous' bisection, the apparent displace- 
ment of the center of the target being called phase. Targets of 
this kind have been used and rules for correction for phase devised, 
but targets free from phase are much to be preferred. The target 
may be a permanent part of the station (such as a flagpole carried 
by an overhead construction so as to clear the instrument), or 
only brought into service when the station is not occupied (such 
as flagpoles, heliotropes or night signals). In any case a signal 
must of course be accurately centered over the station. Eccentric 
signals are sometimes used, involving a corresponding reduction 
of results, but where the instrument and signal can not occupy 
the same position it is more common to regard the signal as the 
true station and the instrument as eccentric. 

Board Signals. Approximately square boards, three or more 
feet wide, painted in black and white vertical stripes or other 
designs, have been tried as targets and found usually unsatis- 
factory, except for distances of a few miles only. The painted 
designs are hard to see unless in direct sunlight and not easy to 
bisect even then. Thej' present their full width in only one 
direction. If two such boards are placed at right angles (whether 
as a cross or one above the other) so as to give a good apparent 
width in any direction, the shadow of one board on the other 
produces the very phase difficidty that board targets were designed 
to prevent. 

Pole Signals. Roimd (sometimes square) poles, painted black 
and white in alternate lengths, are frequently used for signals. 
Against a sky background they give good results, but against 
a dark background they may give the usual trouble from phase. 
Their diameter should be about 1^ inches for the first mile, 
increasing roughly as the square root of the distance. Their size 
becomes prohibitory for distances of over 15 or 20 miles. The 
equivalent of a pole signal, made out of wire and canvas and free 
from phase, was found very satisfactory on the Mississippi River 
Survey. The general construction consisted of four vertical 
wires forming a square, held in place by wire rings (all con- 
nections soldered), black and white canvas being stretched 



PRINCIPLES OF TRIANGULATION 



21 



across the diagonal wires between the successive rings, so as to 
form a vertical series of black and white planes at right angles 
to each other and showing both colors in both directions. The 
distance between the rings was made several times the diameter 
of the rings, so that any shadow or phase effect would affect only 
a very small part of the length of each canvas. In addition to 
being accurately centered any pole or equivalent signal must of 
course be set truly vertical. 

Heliotropes. When the distance between stations exceeds 
about 15 or 20 miles resort is had to reflected sunlight as a signal. 
If the reflecting surface is of proper size such a signal is entirely 
satisfactory for any distance from the smallest to the largest, 
on account of the certainty with which it is seen. Any device 






Fig. 8. — Heliotrope. 



by which the rays of the sun may be reflected in a given direction 
is called a heliotrope, the essential features being a plane mirror 
and a line of sight. A simple form of such an instrument is 
shown in Fig. 8. An additional mirror (called the back mirror) 
is also required, in order to reflect the sunlight onto the main 
mirror when it can not be directly received. The heliotrope is 
generally mounted on a tripod, with a horizontal motion for 
lining in with the distant station, and is centered over its own 
station with a plumb bob. 

In more elaborate forms a telescope with imiversal motion 
furnishes the line of sight, the mirror and vanes being mounted 
on top of it. 

In using the instrument it is pointed towards the observing 



22 GEODETIC SURVEYING 

station by means of the sight vanes or telescope, and the mirror 
is turned so as to throw the shadow of the near vane centrally 
on the farther vane, an attendant moving the mirror slightly 
every few minutes as required. The cone of rays reflected by 
the mirror subtends an angle of about 32 minutes (the angular 
diameter of the sun as seen from the earth), or about 50 feet 
in width per mile. The light will therefore be seen at the observ- 
ing station if the error of pointing is less than 16 minutes or about 
25 feet per mile. The topographical features of the country 
generally enable the heliotroper to locate a station with this degree 
of approximation without any other aid, though it is well to be 
provided with a good pair of field glasses if the heliotrope has no 
telescope. The observing station usually has a heliotrope also, 
so that the two stations may be in communication by agreed 
signals or by using the telegraphic alphabet of dots and dashes 
(long flashes for dashes and short ones for dots, swinging a hat or 
other handy object in front of the mirror to obscure the light as 
desired) . When each station has a heliotrope they soon find each ■ 
other by swinging the light around slowly until either one catches 
the other's light, when the two heliotropes are quickly and 
accurately centered on each other. 

The best size of mirror to use depends on the character of the 
observing instrument, the state of the atmosphere, and the dis- 
tance between stations. In order to have a signal capable of 
accurate bisection it must be neither dangerously indistinct nor 
dazzlingly bright. Between these limits there is a wide range 
of light which is satisfactory. If the light is too bright it is 
readily reduced by covering the mirror with a cardboard disc 
containing a suitable sized hole. A miri'or whose diameter is 
proportioned at the rate of 0.2 inch per mile of distance will 
answer well for average conditions of climate and instruments. 
In the dry climate of our western states one-half this rate will 
prove sufBcient. In the southern part of California the writer 
has seen a six-inch mirror for 80 miles across the Yuma desert with 
the naked eye, but this required exceptionally favorable conditions. 

The apparent size of the heliotrope light varies remarkably 
with the time of day and the condition of the atmosphere, this 
phenomenon being an actual measurable fact and not an optical 
illusion. At simrise and sunset the light appears as srnall as a 
star, almost covered by the vertical hair, and giving a perfect 



PRINCIPLES OF TEIANGULATION 23 

pointing. Anywhere within about two hours of sunrise and sunset 
the image is circular, clean cut, aryi readily bisected, the size 
of the image increasing rapidly with the distance of the sim above 
the horizon. After the sun has risen a couple of hours above 
the horizon \uatil noon the image gradually gets more and more 
irregular in outline and gains in size at an enormous rate, some- 
times filling 25 per cent of the field of view of the telescope at 
noon. The image then decreases in size and becomes gradually 
more regular in outline, becoming fit to observe again about two 
hours before sunset. When the wind blows strongly the image 
elongates like an ellipse, and appears to wave and flutter like 
a flag. If the attendant neglects his work, so that either the 
back mirror or main mirror is poorly pointed, the image loses 
rapidly in brilliancy. On the United States Boundary Survey, 
however, it was found by the most careful micrometric experi- 
ments that the center of the apparent image always corresponded 
with the true center of station. 

Only one objection has been urged against the heliotrope, 
namely, that it can only be used when the sun is shining, while 
angles are best measured on cloudy days. Nevertheless, the 
heliotrope furnishes the best solution for long distance signals 
in the daj'time, and good resijlts can be obtained by making the 
measurements close , to sunrise and sunset. For the best class 
of work the afternoon period is much the best, as great risk of 
sidewise (lateral) refraction always endangers the work of the 
morning period. 

Night signals. A great deal of geodetic work has been done 
at night, using an artificial light as a signal, aided by a lens or 
parabolic reflector. Up to about forty miles a kerosene light with 
an Argand burner is entirely satisfactory. Over forty miles a 
magnesium ribbon burned in a special lamp meets every require- 
ment. Other kinds of lights have been successfully used, but 
those above given have the advantage that only unskilled labor 
is required to operate them, such as can operate heliotropes in 
the daytime. Up to midnight fully as good work can be done 
as in the daytime, but the remainder of the night does not pro- 
vide favorable atmospheric conditions for close work. The chief 
advantage of night work is, of course, the fact that it pi-actically 
doubles the number of hours per day available for good work. 



CHAPTER II 

BASE-LINE MEASUREMENT 

20. General Scheme. The accurate measurement of base 
hnes required for geodetic work may be accomplished with rigid 
base-bars placed successively end to end, or with flexible wires 
or tapes stretched successively from point to point. Base-bars 
were formerly used exclusively for the highest grade of work, 
but tape or wire measurements are rapidly growing in favor. 
The Corps of Engineers, U.S.A., uses steel tapes for its base- 
line work, while the U. S. Coast and Geodetic Survey uses both 
base-bars and steel tapes. The convenience of the steel tape is 
apparent, and the ease and rapidity with which it can be used 
are strong points in its favor. 

No form of measuring apparatus maintains a constant length 
at all temperatures, nor is it often possible to measure along a 
mathematically straight line. Base lines can seldom be located at 
sea level. The adtual length of a bar or tape under standard 
conditions (called its absolute length) is seldom found to be 
exactly the same as its designated length. Tapes and wires are 
elastic, and their length varies with the tension (pull) under 
which they are used. The weight of tapes or wires (when unsup- 
ported) causes them to sag and thus draw the ends closer together. 
In base-bar work corrections may hence be required for absolute 
length, temperature, horizontal and vertical alignment, and reduc- 
tion to mean sea level. With tape or wire measurements correc- 
tions may be required for absolute length, temperature, pull, 
sag, horizontal and vertical alignment, and reduction to mean sea 
level. These corrections will be considered in turn after describ- 
ing the types and use of bars and tapes. 

21. Base-bars and Their Use. The fundamental idea of a 
base-bar is a rigid measuring unit, such as a metallic rod. The 
general scheme of measuring a base requires the use of two 
such bars. The first bar is placed in approximate position, 

24 



BASE-LINE MEASUREMENT 25 

supported at the quarter points by two tripods or trestles, care- 
fully aligned both horizontally and vertically, and moved longi- 
tudinally forward or backward until its rear end is vertically 
over one end of the base line. The second bar, similarly supported 
and aligned, is then drawn longitudinally backward untU its rear 
end is just in contact with the forward end of the first bar. The 
first bar and its supports are then carried forward, alignment and 
contact made as before, and the measurement so continued to 
the end of the base. In the simple form outlined above the 
method would not produce results of sufiicient accuracy for 
geodetic work, but with the perfected methods and appa- 
ratus in actual use measurements of extreme precision may be 
made. 

Several features are more or less common to all types of base- 
bar. The actual measuring unit is generally made of metal and. 
protected by an outer casing of wood or metal. Mercurial 
thermometers are located inside the casing for temperature 
measurements. Means are provided for aligning the bars hori- 
zontally, usually a telescope suitably mounted at the forward 
end of the bar. Vertical alignment is provided for, usually 
by a graduated sector carrying a level bubble, moimted on the 
side of the bar near its central point, so that the bar may be made 
truly horizontal or its inclination determined. A slow motion 
is' provided for making the contact with the previous bar; the 
slow motion is produced by turning a milled head at the rear of 
the bar, which moves the measuring unit only, the casing remain- 
ing stationary in its approximate position on the tripods on 
account of the friction due to the weight of the bar. The rod 
(or tube) constituting the measuring unit terminates at its for- 
ward end with a small vertical abutting plane; the rear end 
of the rod carries a sliding sleeve pressed outward by a light 
spring and ending in a small straight knife edge for making the 
contact with the abutting plane of the previous bar; the length 
of the bar is the distance between the knife edge and the abutting 
plane of its measuring unit when the sliding sleeve is in its proper 
place, indicated by a mark on the sleeve coinciding with a mark 
on the rod; the forward bar is therefore brought into proper 
position without disturbing the rear bar, the only pressure on 
the rear bar being that due to the light spring controlling the 
contact sleeve while the forward measuring unit is slowly drawn 



26 GEODETIC SUEVEYING 

backward luitil the coincidence of the indicating lines shows that 
the bar is in its proper place. 

One of the earlier forms of bar used by the U. S. Coast and 
Geodetic Survey is described in Appendix No. 17, Report for 1880, 
and called a perfected form of a contact-slide base apparatus. 
This bar was an improvement on similar bars in previous use, 
and besides the features enumerated above contained a new 
device for determining its own temperature. The actual measuring 
imit was a steel rod 8 mm. in diameter. A zinc tube 9.5 mm. 
in diameter was placed on each side of the steel rod (not quite 
reaching either end) . The rear end of one zinc tube was soldered 
to the rear end of the steel rod, and the forward end of the other 
zinc tube was soldered to the forward end of the steel rod. By 
suitable scales on the steel rod and the free ends of the zinc tubes 
the apparatus was thus converted into a metallic thermometer 



Fig. 9. — Thermometric Base-bar. 

(zinc having a coefficient of expansion about 2| times that of 
steel), so that the temperature of the bar became very accurately 
measured. In Fig. 9 the arrangement is shown in outline, 
the light line indicating steel and the heavy lines zinc. This bar 
was 4 meters long. 

In Appendix No. 7, Report for 1882, a compensating bar is 
described. This bar is made of a central zinc rod and two side 
steel rods, as shown in Fig. 10. The ends of this bar remain 



Fig. 10. — Compensating Base-bar. 

nearly the same dista"Qce apart at all temperatures. The com- 
pensation is not absolutely perfect, however, and the scales at 
each end indicate the temperature so that the final small correc- 
tion may be made for this cause. This bar was 5 meters long. 
In Appendix No. 11, Report for 1S97, the Eimbeck duplex 
base-bar is described, this bar having almost entirely superseded 
those previously discussed. This bar is a bi-metallic contact- 



BASE-LINE MEASUEEMENT 27 

slide apparatus consisting of two measuring units of precisely 
similar construction, one of steel and one of brass, each 5 meters 
in length, and weighs complete 118 pounds. The measuring 
imits are made of tubing | inch in diameter, each having a 
thiclmess of wall corresponding to the conductivity and specific 
heat of the material of which it is made, so that under changing 
conditions each tube shall keep the same temperature as the 
other one, which is an essential requirement. The two measuring 
tubes are carried in a brass protecting casing, which turns on 
its longitudinal axis in an outer brass protecting casing which 
remains stationary. The inner casing is rotated 180° from time 
to time to equalize temperature distribution. This bar is illus- 
trated LQ Figs. 11 and 12. The two measuring miits are entirely 
disconnected, and contact is always made brass to brass and steel 
to steel, so that two independent measures of the base are 
obtained, one by the brass unit and one by the steel imit. The 
difference in the length of these two measurements furnishes 
the key to the average temperature of the bars during the 
measuring, so that the correction for temperature can be very 
closely determined. Since the coefficient of expansion for brass 
is about li times that for steel, the two measuring units are 
seldom of the same length, and the shorter one continually 
gains on the longer one. To overcome this difficulty the meas- 
uring imits are provided with vernier scales, and the brass 
bar is occasionally shifted a small amount which is read from 
the scales and recorded for an evident purpose. The duplex bar 
is superior to the bars previously described both in speed 
and accuracy. A speed of forty bars per hour is readily main- 
tained. 

The tripods used to support base-bars must be absolutely 
rigid. Special heads are provided so that both quick and slow 
motion are available for raising the bar support. The rear tripod 
usually has a knife-edge support and the front one a roller sup- 
port. By easing the weight on the edge support the bar may 
be readily moved on the roller support and quickly brought into 
proper position. 

Satisfactory work is accomplished with base-bars at all hours 
of the day. In order to protect the bars from the extreme heat 
of the sun, however, a portable awning is often placed over them, 
which is dragged steadily forward as the work advances. 



28 



GEODETIC SUEVEYING 




■9 § 



•I s 

I bo 

1—1 ^ 

T-4 O 

. ^ 

f^ a 

o 

(14 



BASE-LINE MEASUREMENT 



29 




pq CJ 
X •§ 

3 O 



a 



13 
CO 






^ O 

. ^ 

o 



30 GEODETIC 8UEVEYING 

22. Steel Tapes and Their Use. Steel tapes for base-line work 
do not differ materially from ordinary tapes except in length. 
Surveyors generally use tapes 50 or 100 feet long, and with 
proper precautions a high grade of work can be done. Better 
or quicker work, however, can probably be done with longer 
tapes, such tapes usually being also somewhat smaller in cross- 
section. Experience shows that tapes 300 to 500 feet in length 
and with about 0.0025 square inch cross-section are entirely 
satisfactory. 

It is seldom desirable to use the tape directly on the ground, 
on account of the uneven surface and the \.mcertainties of fric- 
tion. The usual way is to support the tape at a number of equi- 
distant points (20 to 100 feet), letting it hang suspended between 
these points and computing the corresponding correction for 
sag. In order to avoid any friction the supports are usually 
wire loops swinging from nails driven in carefully aligned stakes. 
Unless the points of support are on an even and determined 
grade it is necessary to measure the elevation of each such point, 
in order to make the necessary reduction for vertical alignment, 
that is, reduction to the horizontal. The points of support 
must have such elevations that the pull on the tape will not 
lift it free of any of the supports. No change of horizontal 
alignment is allowable within a single tape length. It is evident 
that good work can not be done with a suspended tape if an 
appreciable wind is blowing. 

The pull on the tape must be exerted through the medium 
of a spring balance or other device attached to the forward end. 
The pull adopted may be from 12 to 20 pounds, depending on the 
weight of the tape and the distance between supports, so as to 
prevent excessive sagging and to hold the tape in line. For an 
accuracy of 1 in 50,000 the pull may be made with a good spring 
balance, properly steadied by connection with a good stake. 
For extreme accuracy the pull must be known within a question 
of ounces, and special stretching devices attached to firmly driven 
stakes are required. The desired amount of pull can be very 
accurately made through the simple device of a weight acting 
through a right-angled lever turning on a knife-edge fulcrum; 
the device must be so mounted that the lever arms can be brought 
into a truly vertical and horizontal position when the strain is 
on the tape. ■ 



BASE-LINE MEASUEEMENT 31 

The length of a steel tape is materially modified by a moderate 
change of temperature, so that the greatest care is required in 
making the corresponding correction. It is foimd iii practice 
that a high grade of work can not be done in direct sunlight, 
owing to the difficulty of ascertaining the temperature of the tape, 
a mercurial thermometer held near the tape or in contact with it 
failing to give the true value by many degrees. An accuracy 
of 1 in 50,000 requires the mean temperature of the tape to be 
known within a degree, and an accuracy of 1 in 500,000 to within 
one-fifth of a degree. The highest grade of work can therefore 
be done only on densely cloudy days or at night. 

In the common method of using steel tapes the tape is stretched 
(suspended) between two tripods (or posts driven or braced imtil 
immovable), the rear one being carried forward in turn for each 
new tape length. Intermediate supports are provided as previ- 
ously described, if necessary. The rear end of the tape is con- 
nected with a straining stake a few feet back of the rear tripod; 
the front 'end is connected with the spring balance or other device 
for giving the desired pull, the strain at this end also being 
resisted by a suitable stake or stakes beyond the forward tripod; 
in this way no strain is allowed to come on either tripod. A 
small strip of zinc is secured to the top of each tripod, and each 
tripod is set with sufficient care so that the end mark on the 
tape will come somewhere on the zinc strip, the exact point being 
marked by making a fine scratch on the zinc with any suitable 
instrument. In regard to temperature measurements tapes 100 
feet or less in length ought to have two thermometers tied to 
them, one at each quarter point; longer tapes, up to about 300 
feet, ought to be equipped with three thermometers, one at the 
center, and one about one-sixth the length from each end. 

Professor Edward Jaderin of Stockholm has obtained the very 
best results in a method slightly differing from the above. Profes- 
sor Jaderin prefers a tape 25 meters long, 5 centimeters each 
side of the 25-meter mark being graduated to millimeters and 
read by estimation to the nearest tenth of a millimeter. Each 
tripod carries a single fixed graduation, and the distance between 
the marks on two successive tripods must not vary more than 5 
centimeters either way from 25 meters. By means of the end 
scale on the tape the exact distance from tripod to tripod is 
determined and the whole base found by the sum of the results. 



32 GEODETIC SURVEYING 

The best work can only be done on densely cloudy days or at 
night. 

23. Invar Tapes. By alloying steel with about 35 per cent 
of nickel a material is produced possessing an exceedingly small 
coefficient of expansion, this discovery being due to C. E. Guil- 
laume (of the International Bureau of Weights and Measures, 
near Paris). For this reason the name "invar" (from "inva- 
riable ") has been applied to this material. Tapes made of invar 
have proven extremely satisfactory for the accurate measure- 
ment of base lines, errors in determining the temperature of the 
tape being of so much less importance than with steel tapes, which 
makes it possible to do first class work at all hours of the day. 

The coefficient of expansion of invar is about 1 : 28 that of 
steel, or about 0.00000022 per degree Fahrenheit. The modulus 
of elasticity is about 8 : 10 that of steel, or about 23,000,000 pounds 
per square inch. The tensile strength is about 100,000 pounds 
per square inch, or about half that of the ordinary steel tape, but 
amply sufficient for the purpose. The yield point is about 70 
per cent of the tensile strength. 

■ In 1905 the Coast Survey purchased six invar tapes from 
J. H. Agar Baugh, London, Eng., for the purpose of subjecting 
them to the actual test of field work and comparing them with 
steel tapes tinder similar conditions. (See Appendix No. 4, 
1907.) These tapes averaged about 0".02X0".25 in cross- 
section, about 53 meters in length, looked more like nickel than 
steel, and were full of innumerable small kinks which, however, 
did not cause any inaccuracy in actual service. They were very 
soft and easily bent, being much less elastic than steel, and requir- 
ing reels 16 inches in diameter to prevent permanent bending. 
Steady loads up to 60 pounds caused no permanent set. While 
rusting more slowly than steel tapes oiling and care were found 
to be necessary. 

The experience of the Coast Survey with invar tapes indicates 
that they possess no properties derogatory to their use for base- 
line work, and that under similar conditions both better and 
cheaper work can be done than with steel tapes. They are used 
in all respects like steel tapes, using special care to avoid injury 
from bending. 

24. Measurements with Steel and Brass Wires. Professor 
Edward Jaderin of Stocldiolm has found it possible to do excellent 



BASE-LINE MEASUEEMENT 33 

base-line work throughout the entire day by using steel and brass 
wires instead of steel tapes. (See U. S. C. and G. S. Appendix 
No. 5, Report for 1893.) The object of using the metal in wire 
form instead of tape form is to minimize the effect of the wind, 
since the circular cross-section (for the same area) exposes much 
less surface to the action of the wind than the flat surface of 
the tape form. The method used is the same as described in the 
last paragraph of Art. 22, except that two values are obtained 
for the distance between each pair of tripods, one with the steel 
and one with the brass wire. Two measurements of the whole 
base line are thus obtained, and from their difference the average 
temperature of the wires is deduced and hence the corresponding 
correction. The assumption is made that the wires are always of 
equal ,temperature, both being given the same surface (nickel 
plate, for example), the same cross-section, and the same hand- 
ling. The principle is identical with that of the Eimbeck 
duplex base-bar described in Art. 21. 

25. Standardizing Bars and Tapes. The nominal length of a 
bar or tape is its ordinary designated length, as, for example, a 
fifty-foot tape or a five-meter bar. The actual length seldom 
equals the nominal length, but varies with changing conditions. 
The absolute length is the actual length under specified conditions. 
If the absolute length is known, the laws governing the change of 
length with changing conditions, and the partictdar conditions 
at the time of measuring, then the actual length of the measuring 
imit becomes known, and consequently the actual length of the 
line measured. By standardizing a bar or tape is meant deter- 
mining its absolute length. Such an expression as the " tem- 
perature at which a bar or tape is standard " means the tempera- 
ture at which the actual and designated lengths agree. 

The absolute length of a bar or tape may be determined in a 
number of ways, but the essential principle in each case is the 
same, namely, the comparing of the unknown length with some 
known standard length at an accurately known temperature. 
If the comparison is made in-doors, the room must be one (such 
as in the basement of a building) where the temperature remains 
practically constant for long periods, so that the temperature of 
the measuring units will be the same as that of the surroimding 
air. If the comparison is made in the open air the work must be 
done on a densely cloudy day or at night, for the same reason. 



34 GEODETIC SUEVEYING 

Tapes are generally standardized supported horizontally 
throughout their length, at any convenient pull and temperature, 
the Coast Survey reducing the results by computation to a stand- 
ard pull of 10 pounds and temperature of 62° F. The absolute 
length of a tape may be found by measuring it with a shorter 
unit (such as a standard yard or meter bar); by comparing it 
with a similar tape whose absolute length is known; by comparing 
it with fixed points whose distance apart is accurately known; 
or by measuring with it a base line whose length is already accur- 
ately known. For a nominal fee the Coast Survey at Washington 
will determine the absolute length of any tape up to 100 feet in 
length. 

Any device or apparatus which permits a measuring unit to 
be compared with a standard length is called a comparator. It 
is quite common at the commencement of a survey to fix two 
points at a permanent and well determined distance apart, and 
compare all tapes used with these points from time to time; the 
standard or reference distance thus established would be called 
a comparator. In the laboratory the comparator may be a very 
elaborate piece of apparatus with micrometer microscopes, by 
which the most accurate comparisons may be made, or with 
which a measuring unit may be most accurately measured by a 
shorter standard. 

Base-bars are probably most readily and accurately standard- 
ized by measuring a base line of known length with them. The 
actual length of the bar thus becomes known, by computation, 
for the temperature at which the measurement was made; and 
by means of its coefficient of expansion its length becomes known 
at any temperature. 

Measuring the same base with the same bar or tape, at widely 
different temperatures, furnishes a good means of determining 
the coefficient of expansion if it is not otherwise Icnown. With 
the compensating bar the coefficient of the residual expansion 
(since the compensation is never perfect) may be thus obtained. 

If a base line of known length is measured with a duplex 
base-bar at a certain average temperature, the average actual 
length of each component bar (steel and brass) becomes Imown 
for that temperature, and the difference in these average lengths 
indicates that particular temperature and that particular length of 
each bar. The absolute length of each component is thus known 



BASE-LINE MEASUREMENT 35 

for that particular temperature. If the same thing is done at a 
widely different temperature the same information is obtained 
at the new temperature. Since the average length of each com- 
ponent is obtained at the two different temperatures the coefficient 
of expansion of each component becomes known. Since the differ- 
ence in the lengths of the components is loiown at two widely 
separated temperatures, and since this difference changes uniformly 
from the lower to the higher temperature, the temperature corre- 
sponding to any particular difference in the length of the bars also 
becomes known. In measuring an unknown base with a duplex 
bar (provisionally using the absolute length of each component 
at the standard temperature on which the coefficient of expansion 
is based) the total difference by the two component bars becomes 
known, hence the average difference per bar length, hence the 
average temperature, hence by combination with the coefficient 
of expansion the actual length of each component at the time 
of measurement, hence the actual length of the base line. The 
result must, of course, be the same whether fiiiaUy deduced from 
the steel or from the brass component, thus furnishing a good 
check on the computations. When base lines are measured" 
with steel and brass wires these wires are standardized and used 
in the same manner as the duplex base-bar. 

A base line of known length, to be used for standardizing 
bars or tapes, may be one that is measured with apparatus already 
standardized, or one measured with a base-bar packed in melting 
ice so as to ensure a constant and known temperature. 

26. Corrections Required in Base-line Work. As explained 
in Art. 20, if a base line is measured with base -bars corrections 
may be required for absolute length, temperature, horizontal and 
vertical alignment, and reduction to mean sea level. If the base 
line is measured with supported tapes or wires an additional 
correction may be required for puU. If unsupported tapes or 
wires are used additional corrections may be required for both 
pull and sag. With a simple or a compensating base-bar, there- 
fore, it is necessary to know its absolute length and coefficient 
of expansion before it can be used for base-line work. With a 
duplex base-bar (and correspondingly with double wire measure- 
ments) it is necessary to know the absolute length and coefficient 
of expansion of each of the component \mits. With tapes and 
wires it is necessary to loiow the absolute length, coefficient of 



36 GEODETIC SURVEYING 

expansion, modulus of elasticity, area of cross-section, and weight. 
Except in work of great accuracy average values may be assumed 
for the weight, coefficient of expansion, and modulus of elasticity 
for the material of which the wire or tape is made. 

The above corrections are relatively so small that they may be 
computed individually from the uncorrected length of base line, 
and their algebraic sum taken as the total correction required. A 
plus correction means that the uncorrected length is to be increased 
to obtain the true length, and a minus correction the reverse. 

27. Correction for Absolute Length. The absolute length of 
a measuring unit is generally stated as its designated length plus 
or minus a correction. The total correction will have the same 
sign, and be equal to the given correction multiplied by the num- 
ber of tape or bar lengths in the base (including fractional lengths 
expressed in decimals); or what amounts to the same thing, 
multiply the given correction by the length of the base and divide 
by the length of the measuring unit. 

If Ga =correction for absolute length; 

c= correction to measiiring unit; 

il =imcorrected length of measuring unit; 

L =uncorrected length of base; 

then r 

^ _Lc 

In duplex measurements the absolute lengths are used directly 
in the computations in order to determine the average temperature. 

The quantities L and I must be expressed in the same unit 
(feet or meters, for instance), and Ca will be in the same unit as c 
(which need not be the same as used for L and I) . 

28. Correction for Temperature. In measuring a base line 
the temperature usually varies more or less during the progress 
of the work, but it is found entirely satisfactory to apply a 
correction due to their average temperature to the sum of all 
the even bar or tape lengths, and add a final correction for any 
fractional lengths and corresponding temperatures. 

If Ct = correction for temperature ; 
a =. coefficient of expansion; 
Tm — mean temperature for length L; 
Tg= temperature of standardization; 
L = length to be corrected; 



BASE-LINE MEASUREMENT 37 

then practically, since the measuring unit changes length uni- 
formly with the temperature, 

Ct =a{T„-T,)L. 

Ci will be in the same unit as L and must be applied with its 
algebraic sign. 

The coefficient of expansion for steel wires and tapes may vary 
from 0.0000055 to 0.0000070 per degree F., and if its value is 
not known for any particular case may be assumed as 0.0000063 
(Coast Survey value). For the most accurate work the coeffi- 
cient of expansion for the particular tape or wire ought to be 
carefully determined, either in the laboratory or by measuring a 
known base at widely different temperatures. 

The coefficient of expansion for brass wires was foimd by 
Professor Jaderin to average 0.0000096 per degree F. 

The coefficient of expansion of invar may be 0.00000022 per 
degree F., or less. 

In the case of duplex measurements the average temperature 
and corresponding corrections may be deduced as follows: 

Let Ls = provisional length of base, using absolute length of 
steel component at the standard temperature 
(usually 32° F. or 0° C.) to which coefficient of ex- 
pansion refers; 
Lj, = same for brass; 
A, = coefficient of expansion of steel; 
Ab = same for brass; 

T = average number of degrees temperature above 
standard ; 

then the true length of base in terms of steel component 

= Ls + LgAgT, 

and in terms of brass component 

= Li + LiAbT. 

Equating and reducing, we have 

LbAb — LgAg 



38 GEODETIC SUEVEYING 

and correction for steel-component measurement 

n —TAT— ^s(Le— Li,)Ag 

or practically 

Cta = correction to measurement by steel component 

and similarly 

Cth = correction to measurement by brass component 

= (L3-L6)- 



Ai 



'Ai-A/ 

These corrections will be in the same tmit as L^ and Lj and are to 
be used with their algebraic signs. 

29. Correction for Pull. This correction only occurs with 
tapes and wires; if the pull used is not the same as that to which 
the absolute length is referred a corresponding correction must 
be made. 

Let Cj) = correction for pull; 

Pm = pull while measuring base line; 
Pa = pull corresponding to absolute length; 
S = area of cross-section of tape; 
E = modulus of elasticity of tape; 
L = uncorrected length of line; 



then practically 



^"^ SE 



If E is taken in pounds per square inch, then P^ and Pa must 
be in pounds, L in inches, and S in squares inches, whence Cp 
will be in inches, and is to be applied with its algebraic sign. 

If the cross-section is unknown it may readily be foimd by 
weighing the tape or wire (without the box or reel), and finding 
its volume by comparison with the specific weight of the same 
material. The cross-section then equals the volume divided by 
the length. The weight of a cubic foot may be assumed as 490 
pounds for steel tapes, 500 poimds for steel wires, 520 pounds 
for brass wires, and 510 pounds for invar tapes. 



BASE-LINE MEASUREMENT 39 

If the modulus of elasticity is unknown it may be found as 
follows: Support the tape horizontally throughout its length, 
and apply two widely different pulls, noting how much the tape 
changes in length due to the change in the amount of pull. 
Let Pj = smaller pull; 
Pi = larger pull; 
I = length of tape; 

Ic = change in length caused by change in pull; 
S = cross-section of tape; 
E = modulus of elasticity; 



then 



(Pi-P,)l 



If Pi and Ps are taken in pounds, I and Z„ in inches, and S in 
square inches, then E will be in poimds per square inch. 

Except for the most accurate work E may be assumed as 
follows: 

for steel, E = 28,000,000 lbs. per sq. in. 
for brass, E = 14,000,000 " 

for invar, E = 23,000,000 

30. Correction for Sag. This correction only occurs in the case 
of unsupported tapes and wires. In any actual case in practice 
the catenary curve thus formed will not differ sensibly in length 
from a pafabola. The correction required is the difference in 
length between the curve and its chord. 

Let Ca = correction for sag for one tape length; 

c = correction for sag for the interval between one 

pair of supports; 
I = length of tape ; 

d = horizontal distance between supports (for which the 
uncorrected distance given by the tape is used 
in practice without sensible error) ; 
V = the amoimt of sag; 
P = the pull; 
w = weight of a unit length of tape. 

The difference in length between the arc and chord of a very flat 
parabola (such as occurs in tape measurements) is found by the 



40 GEODETIC SUEVEYING 

calculus to be very nearly — — , but the formula is never used in 

3a 

this form since it is inconvenient and unnecessary to measure 

V in actual work. Passing a vertical section midway between 

supports, and taking moments around one support, we have 







^-TX4- = 


ivd^ 
■ 8 ' 


from which 




wd^ 




whence 










8^2 
3d 


d{wdf 
= 24P2 or - 


d{wdf 
24P2 



and if there are n intervals per tape 

nd(wd)" l(iL'd)^ 

The correction to the whole base line is found by multiplying 
the correction per tape length by the number of whole tape 
lengths, and adding thereto the corrections for any fractional 
tape lengths (which must be computed separately) . 

If w is taken as pounds per inch,, then P must be taken in 
pounds and d and I in inches, whenpe Cj will be in inches. 

The normal tension of a tape is such a tension as will cause 
the effects of pull and sag to neutralize each other, so- that no 
correction need be made for these effects. Since the effects of 
pull and sag are opposite in character (pull increasing and sag 
decreasing distance between ends of tape) such a value can always 
be found by equating the formulas (for a tape length) for sag 
and for pull, and solving for P„ or pull to be used during measure- 
ment of line. 

31. Correction for Horizontal Alignment. Ordinarily base 
lines are made straight horizontally, but sometimes slight devi- 
ations have to be introduced, forming what is called a broken 
base. Fig. 13 shows a common case of a broken base, a, b, and 
d being measured, and c found by computation, some unavoidable 



BASE-LINE MEASUREMENT 41 

condition preventing the direct measurement of c. From trig- 
onometry we have 

a^ + b^ + 2ab cos d = c^, 

so that c can always be found. If, however, is very small 
(say not over 3°) we may proceed as follows : 

Let Cfcb = correction for broken base; then 

Ctt = -[{a + b) ~ c]; 



but 



a2 + 62 + 2ab cos = c^; 
a2 _|_ 52 _ g2 = _ 2ab cos 6. 



Adding 2ab to both members 

a2 + 2ab +b^ - c2 = 2ab - 2ab cos d; 

(a + by - c2 = 2ab (1 - cos 6). 

Substituting (1 — cos d) = 2 sin^ ^6, 

[{a + b) - c] X [(a + 6) + c] = 4a& sin^ ^d. 
Hence 

Aab sin2 \d 



Cbb = — 



{a + b) + c ■ 



If d is very small (which is practically always the case) Cbb will 
be very small, and we may substitute 

sin iO = id sin 1' and (a + 6) + c = 2(a + 6), 
whence 

„ _ a&g2 sin2 1^ 
^""^ a + b^ 2 ' 

in which 6 must be expressed in minutes, and Cbb will be in the 
same unit as a and b. 

^^ = 0.00000004231. 



42 GEODETIC SURVEYING 

32. Correction for Vertical Alignment. When measurements 
are taken with wires or tapes the elevations of the different 
points of support will usually be different, though frequently 
a number of successive points may be made to fall on the same 
grade. 

Let h, I2, etc., be the successive lengths of uniform grades; 
hi, h2, etc., be the differences of elevation between the 

successive ends of these grades; 
Ci, C2, etc., be the numerical corrections for the single 

grades; 
Cg = total correction for grade; 
then for any one grade 

c =1 - \/P - h^, 



c -I = - Vp - h^, 

c2 - 2lc + 12 =12 - h?, 

c2 - 2Zc = - h?, 

2lc - c2 = h?, 

h2 



c = 



21 -c' 



but since c is very small in comparison with I we may write with 
sufficient precision 



h^ 

c = 

whence 



'=21' 



C, = - f ^ + ^ . + ^ 

\2li 2I2 2ln 

If the grade lengths are all equal, as, for instance, when h 
is taken at every tape length, 

Cg= - |(Al2 +h2^...+ hn^) = - ^. 

Fractional tape lengths must be reduced separately. 

When base-bars are used the angles of inclination are measured, 
and the correction is the same for the same angle whether the 
angle is one of elevation or depression. 



BASE-LINE MEASUREMENT 



43 



Let Cg== grade correction for one bar length; 
I = length of bar; 

= angle of inclination from the horizontal; 
then 

C,= - Z (1 - cos 5) = - 21 sin2 ^6. 

If d is less than 6° we may write without material error 



whence 



or 



sin i^ = \d sin 1', 






Cg = - 0.00000004231 dH, 



with the understanding that 6 is to be expressed in minutes, 
and Cg will be in the same unit as I. The grade correction for 
the entire line will be the sum of the individual corrections for 
the several bar lengths. 

33. Reduction to Mean Sea Level. In geodetic work all 
horizontal distances are referred to mean sea level, that is, the 
stations are all supposed to be 
projected radially (more strictly, 
normally) on to a mean-sea-level 
surface, and all distances are 
reckoned on this surface. All the 
angles of a triangulation system 
are measured as horizontal angles, 
and are not practically affected by 
the different elevations which the 
various stations may have. If the 
lines which are actually measured 
(bases and check bases) are re- 
duced to mean sea level, all com- 
puted lines will correspond to this 
level without further reduction. 
It is necessary, therefore, to con- 
nect the ends of base lines with 
the nearest bench marks whose 

elevations are known with reference to mean sea level. (See 
Art. 77 for determination of mean sea level.) 




44 GEODETIC SURVEYING 

Let C„jj = reduction to mean sea level; 
r = mean radius of earth; 
a = average elevation of base line; 
B = length of base as measured; 
6 = length of base at mean sea level; 

then, from Fig. 14, page 43, 

r + o _ r 

or since a is always very small as compared with r, we may write 

r - _ :?^ 

in which a and r must be in the same unit, and in which C,„s; 
will be in the same unit as B (need not be in the same unit as 
for a and r). 

r (in meters) = 6,367,465 log. = 6.8039665. 

r (in feet) =20,890,592 log. = 7.3199507. 

34. Computing Gaps in Base Lines. Sometimes an obstacle 
occurs which prevents the direct measurement of a portion of 
a straight base line, as, for instance, between B and C in Fig. 15. 
In such a case if two auxiliary points A and D (on the base) 
are taken, x can be computed if the distances a and h and the 
angles a, p, and 6 are measured. Draw BE and CF perpendicular 
to AO, and CG and BH perpendicular to DO. Then 

BE BA BO sin a a 

or 



whence 



Also 



CF CA CO sin (a +/?) x + a' 



BO _ a sin (a + p) 
CO (x + a) sin a 



(1) 



BH_ _BD BO sin {^ + d) _ x + h 

CG ~ CD ^"^ CO am ^ b ' 



BASE-LINE MEASUREMENT 45 

whence 

BO _ ix + b)sha.d 

CO bsin(fi + e) ^^ 

Comparing (1) and (2) 

g sin (c : + /?) _ (x + b) sin d _ 
(x -)- a) sin a 6 sin (/?+&)' 



or 



(X + a) (X + 6) = «&sin(a+^)sin(/?+g) 

sm a sm & ' 



which gives 



^ _ ^ U sin (a+^) sin (/?+e) ^ /, 
\ sin asm 6 \ 



ab sin (a+^) sin jj^+d) , /a - b\^ a + 6 



, 2 / 2 

A a B 




It is evident that good results can not be obtained unless the 
points A, D, and are selected so as to make a well shaped 
figure. 

35. Accuracy of Base-line Measurements. The accuracy 
possible in the determination of the length of a base line depends 
on the precision with which the various constants of the meas- 
uring apparatus have been obtained and the precision with which 
the field work is done. The instrumental constants can be 
determined with a degree of precision commensurate with the 
highest grade of field work. The precision attainable in the field 
is judged by making repeated measurements of the same base 
with the same apparatus and comparing the results. From the 
discrepancies in these measurements the probable error (Chapter 
XIII) of the average (arithmetic mean) of the determinations 



46 



GEODETIC SURVEYING 



is found and compared with the total length of the line as a 
measure of the precision attained. This measure of precision 
is called the uncertainty. 

An exact comparison of the merits of different base-line 
apparatus is manifestly impossible, but under similar conditions 
the following results have been obtained : 

Uncertainty of Mean Length of Base. Steel tapes in cloudy 
weather or at night, 1 in 1,000,000 or better. Invar tapes at all 
hours, 1 in 1,000,000 or better. Steel and brass wires at all 
hours, 1 in 1,000,000 or better. Ordinary base-bars, 1 in 2,000,000 
or better. Duplex base-bars, 1 in 5,000,000 or better. 

The probable error of a base line is obtained as follows : 

Let r„ = probable error of mean length; 

Ml, M2, etc. = value of each determination; 
z = mean length of line; 

residuals ; 



Ml ~z 
Ah -2 



then 



etc., 

Hv- = sum of squares of residuals; 
n = number of measurements; 





ra= ± 0.6745 


/ Iv^ 






\n{n - !)• 




Example. Five measurements of a base line were made : 




Observed Values. 


Arithmetic Mean. 


V. 


vl 


6871.26 ft. 
6871.31 " 

6871.27 " 
6871.30 " 

6871.28 " 


6871.284 ft. 
6871.284 " 
6871.284 " 
6871.284 " 
6871.284 " 


- 0.024 
-1- 0.026 

- 0.014 
-1- 0.016 

- 0.004 


0.000576 
0.000676 
0.000196 
0.000256 
0.000016 


5)1.42 


0.000 


0.001720 



.284 

The algebraic sum of the residuals is zero, as it always should be. Then 
for ra, the probable error of the mean length, we have 



ra = ziz 0.6745 



4 



001720 



= ± 0.0093 ft.; 



' 5(5 - 1) 

and for Ua, the uncertainty of the mean length, we have 

0.0093 1 



[/. = 



6871.284 738848" 



CHAPTER III 
MEASUREMENT OF ANGLES 

36. General Conditions. Assuming that the stations and 
signals have been arranged to the best advantage, as described 
in Chapter I, the finest grade of instruments and especially 
favorable atmospheric conditions are required for the highest 
grade of work. In clear weather only fairly good work can be 
done during a large part of the day except under special con- 
ditions. From dawn to sunrise (and within about an hour 
after sunrise if heliotropes are used), and from about four o'clock 
in the afternoon until dark, represent the only hours available 
for the highest grade of work; even the early morning period 
frequently proves unsatisfactory. In densely cloudy weather 
work may be carried on all day. If night signals are used (see 
Art. 19), good work can be done up till about midnight. Accu- 
rate results can not be expected if the instrument is exposed 
to the direct rays of the sun immediately before or during the 
measurement of an angle. The effect of the sun's rays is to 
cause heat radiation, producing an apparent unsteadiness of all 
objects seen through the telescope, due to the irregular refraction 
caused by the currents of air of different temperatures; an 
uncertain amount of sidewise refraction, even if the unsteadiness 
is not sufficient to prevent a good bisection of the signal; a 
disturbance of the adjustments of the instrument and bubbles, 
and an actual twisting of the instrument on a vertical axis, both 
caused by unequal expansion and contraction; and a twisting 
of the station itself on a vertical axis, if it have any particular 
height (the twisting being generally toward the sun's movement, 
and amounting to as much as a second of arc per minute on- a 
75-foot tower) . 

37. Instruments for Angular Measurements. Two types of 
instrument are in use for fine angle work, the Repeating Instru- 
ment, and the Direction Instrument, the latter being considered 

47 



48 GEODETIC SURVEYING 

the best in the hands of well-trained observers. If either instru- 
ment is provided with a vertical arc or circle it is called an 
Altazimuth Instrument. The term Theodolite is frequently applied 
to any large instrument of high grade, though more correctly 
limited to istruments in which the telescope can not be reversed 
without being lifted out of its supports (on account of the low- 
ness of the standards). When an instrument has to be reversed 
in this manner the telescope must be turned end for end without 
reversing the pivots in the wyes. The illustrations are all of 
high grade instruments, Fig. 16 being a repeating instrument. 
Fig. 17 a direction instrument, and Fig. 18 an altazimuth instru- 
ment (in this case also a repeating instrument). In general, 
geodetic instruments are larger than surveyors' instruments, 
though experience has shown that horizontal circles greater than 
10 or 12 inches in diameter offer no further advantage in the 
accuracy of the work that can be done with them. Such instru- 
ments are made of the best available material and with the greatest 
care, the utmost care being taken with the graduations and 
the making and fitting of the centers. Lifting rings are often 
provided to avoid strain in handling. The instruments are 
supported on three leveling screws (instead of four as ordinarily 
found on surveyors' transits), and in addition a delicate striding 
level is provided for direct application to the horizontal axis 
of the telescope. All the levels are more delicate than on a 
common transit, the plate levels running from about 10 to 20 
seconds per division, and the striding level from 1 to 5 seconds per 
division. Repeating instruments are usually read by verniers, 
an 8-inch instrument reading to 10 seconds and a 10- or 12-inch 
instrument even down to 5 seconds, attached reading glasses 
of high power taking the place of the ordinary vernier glass. 
Direction instruments generally read to single seconds, as described 
in detail later on. The leveling screws (which support the 
instruments) are pointed at the lower ends and rest in ^'-shaped 
grooves, so that they are not constrained in any waj-. If tri- 
pods are used the grooves are usually cut in round foot plates 
(about H inches in diameter) properly placed on the tripod 
head by the maker. Extra foot plates are often provided which 
can be screwed to piers or station heads as desired. A trivet 
is a device often used for the same purpose, consisting of a frame 
containing three equally-spaced radial "\^-shaped grooves cut in 



MEASUREMENT OF ANGLES 



49 




<U O 

I. d 

1-4 o 



50 



GEODETIC SUEVEYING 




Fig. 17. — Direction Instrument. 
From a photograph loaned by the U. S. C. and G.[S. 



MEASUEBMENT OF ANGLES 



51 




Fig. 18. — Altazimuth Instrument. 
From a photograph loaned by the U. S. C. and G. S. 



52 GEODETIC SURVEYING 

suitable arms. A three-screw instrument is leveled by setting 
a bubble parallel to a pair of leveling screws and bringing it 
to the center by turning that pair of screws equally in opposite 
directions; the crosswise bubble is then leveled by using only 
the single screw that is left. 

38. The Repeating Instrument and its Use. Besides the 
features common to all first-class instruments, as described in 
the previous article, the repeating instrument must contain the 
special feature of a double vertical axis (as is always the case 
in the surveyor's transit), thus permitting angles to be measured 
by the method of repetition. The fundamental idea of measuring 
an angle by repetition is to measure the angle a number of times 
without resetting the plates to zero between the successive 
measurements,, and dividing the accumulated result by the 
number of repetitions. It was at first thought that any desired 
degree of accuracy could be obtained by this method by simply 
increasing the number of repetitions, but it is now known that 
increasing the number of repetitions beyond a certain limit does 
not improve the result, on account of systematic errors introduced 
by the instrument itself, chiefly due to the clamping attach- 
ments. The method is nevertheless very meritorious, and excel- 
lent work can be done. The object of the repetition is twofold: 
First, the errors ia the pointings tend to compensate each other, 
and the remaining error is largely reduced by the division; 
Second, the accumulated reading is theoretically correct to the 
least count of the vernier, and the division by the number of 
repetitions tends to make the reduced value as close as if the 
least count were just that much finer. There are two ways of 
measuring an angle by the method of repetition, each designed 
to eliminate as far as possible the various instrumental errors, 
but based on somewhat different arguments. 

39. First Method with Repeating Instrument. The common, 
but not the best, method consists in repeating the angle any 
desired number of times, measuring from the left-hand to the 
right-hand station, with telescope direct, and dividing by the 
number of repetitions to obtain one value of the angle; then 
measuring the same angle in the reverse direction (right-hand 
to left-hand station), using the same number of repetitions, but 
with telescope reversed, and dividing as before to obtain a second 
value of the angle; the average of the two determinations is then 



MEASUREMENT OE ANGLES 53 

taken as the value of the angle (as given by that set, and of 
course as many sets as desired may be averaged together). The 
number of repetitions in each set is commonly so taken as to 
make each of the accumulated readings approximately equal 
to one or more times 360°, in order to eliminate errors of gradu- 
ation. If this plan would require an unreasonable number of 
repetitions, a number of smaller sets may be taken from sym- 
metrical points around the graduated limb, and the results 
averaged. Thus four independent sets might be taken, the start- 
ing point for vernier A for each set being respectively 0°, 45°, 
90°, and 135°. The reversal of the telescope is designed to elimi- 
nate errors caused by imperfect adjustment of the collimation 
and the horizontal axis of the telescope. Measuring in opposite 
directions between stations is designed to eliminate errors caused 
by the clamping apparatus. The reading of the instrument at 
any time is understood to be the mean of the readings of the 
two verniers, as the eccentricity of the verniers and of the centers 
is thus eliminated. The argument advanced in favor of this 
method is that reversing all the processes for the second half 
of a set ought to reverse the signs of the various errors, so that 
theoretically they ought to largely vanish from the mean value. 
As this method is not recommended it is not given in any further 
detail. 

40. Second Method with Repeating Instrument. In this 
method, considered the best, the instrument is always revolved 
about its vertical axis in the same direction 
(almost Toniversally clockwise), no matter which 
clamp is loosened nor how great the angle 
through which it must be turned to point to 
the desired station. The fundamental scheme 
of this method is to measure (see Fig. 19) the 
desired angle from A to B (called the interior 
angle) , and also to measure the other angle (called 
the exterior angle) from the B the rest of the way 
around to A, measuring this remaining angle being 
called closing the horizon. The interior angle A to Fig. 19. 

B is repeated as many times as desired with the 
telescope direct (often called normal) and an equal number of 
times with the telescope reversed, and the accumulated reading 
divided by the total number of repetitions for the provisional 




54 GEODETIC SUEVEYINQ 

value of this angle. The exterior angle 5 to A is measured in 
exactly the same way with the same number of repetitions, 
etc. The values thus obtained for the interior and exterior 
angles are added together, and if the resiilt is not exactly 360° 
the discrepancy is equally divided between the two angles. 
The entire operation makes one set. The argument in favor 
of this method is that since the exterior angle is measured in 
identically the same way as the interior angle it ought to be 
subject to exactl}" the same error; adding the two angles together, 
therefore, should double the error; and the value of this double 
error be made apparent by the failure of the sum to equal 360°. 
The assumption is evidently made that the errors which it is 
sought to eliminate by this method are independent of the size 
of the angle, and this is generally believed to be true. In practice 
the verniers are not reset to zero after completing the measure- 
ment of the interior angle, but become the starting point for the 
measurement of the exterior angle just as they stand; the 
instrument is thus made to automatically add the interior and 
exterior angles on its own graduations, and the verniers should 
therefore read zero (360°) at the completion of the set Lf no errors 
were involved. It is more common for the combined angles to run 
under than over 360°, about 10" per repetition not being an 
unusual amoimt. It is found by experience with this method that 
six repetitions (3' direct and 3 reversed) of the interior angle, and 
the same for the exterior angle, make a very satisfactory set; 
and the average of two such sets (if in close agreement) gives 
a very good determination of the desired angle. The plates are 
not reset to zero between the two sets, but left undisturbed as 
a starting point for the second set, so that the vernier readings 
become slightly different each time and the mind is free from 
bias. The complete program fbr a double set would be as follows : 

PROGRAM 
First Set. 

1. Level up, set vernier d to zero, read vernier B. 
Set telescope direct and 

2. Undamped below, turn clockwise and set on left station. 

3. " above, " " right " 

4. Unclamp below, and read vernier ^1. 



MEASUEEMENT OF ANGLES 55 

Leaving verniers unchanged, 

6. Undamped below, turn clockwise and set on left station. 

6. " above, " " right " 

7. " below, " " left 

8. " above, " " right 

Reverse telescope and 

9. Undamped below, turn clockwise and set on left station. 

10. " above, " " right 

11. " bdow, " " left 

12. " above, " " right 

13. " bdow, " " left 

14. " above, " " right 

15. Unclamp below and read both verniers. 

Leaving telescope reversed and verniers unchanged, 

16. Undamped below, turn clockwise and set on right station. 

" " left " 

right 
left 

right " 
left 

Set telescope direct and 

22. Undamped below, turn clockwise and set on right station. 

23. " above, 

24. " below, 

25. " above, 

26. " bdow, 

27. " above, " 

28. Unclamp below and read both verniers. 

Second Set. 

1. Leaving verniers unchanged from previous set, relevel 
with lower motion undamped. 

Set telescope direct and 

2. Undamped below, turn clockwise and set on left station. 

3. " above, " " right " 
■ etc. etc. 



17. 




above, 


18. 




below, 


19. 




above, 


20. 




below. 


21. 




above 



left 


right 


left 


" right ' 


left 



66 GEODETIC SUEVEYING 

40a. Reducing the Notes. The following points are taken 
advantage of to save labor in reducing the notes : 

First. In finding the average value of the six repetitions 
by dividing by six, it will be noted that the remainder from the 
degrees gives the first figure of the minutes, and the remainder 
from the minutes gives the first figure of the seconds, so it 
becomes unnecessary to reduce these remainders to the next 
lower unit, as would be required with any other number of 
repetitions. For example, let the accumulated reading be 
250° 57' 15", 

6)250° 57' 15" 
41 49 32.5' 

6 into 250 goes 41 times and 4 over, and 4 is the first figure of 
the minutes; 6 into 57 goes 9 times and 3 over, and 3 is the first 
figure of the seconds. 

Second. The same numerical result can be reached without 
carrying out the reduction exactly as described in the explanation 
of the method. 

Let a = mean of verniers at beginning of a set ; 

b = mean of verniers after six repetitions on interior 

angle ; 
c = mean of verniers after six repetitions on exterior 

angle; 
n = number of times vernier passes initial point in the 

six repetitions of the interior angle ; 
/ = interior angle as measured; 
E = exterior angle as measured; 

V = adjustment to be added to either angle as measured; 
A = adjusted value of interior angle. 

Since the interior and exterior angles together make 360°, 
and each has been repeated six times, the total angle turned 
through must be 360° X 6, or what amounts to the same thing, 
5 complete circuits plus the indications of the verniers and the 
correction for the accumulated errors; so that if n equals the 
number of complete circuits involved in the six repetitions of 
the interior angle, then (5 — n) must represent the number of 
complete circuits involved in the six repetitions of the exterior 
angle. Hence 



MEASUREMENT OF ANGLES 



57 



I = 



E = 



I + E = 



360n + b — a 
6 ' 

360(5 -n) + c -b 
6 

360 X 5 + c - a 



. =1(360 - 

A =1 + V, 
360n 4 



6 

360 X 5 + c 



1/ 360 -c + a 
''2\ 6 



)■ 



- a 1/ 360 - c + a \ 
u '^2\ 6 J 

^l/ 360n + b -g 360n + (360 -I 
~2\ 6 "^ 



b) - cN 



=K(' 



60n + 



b — a 



QOn + 



6 
(360 + b) -c 



)]■ 



In actual work no attempt is made to observe the value of n, 
as its value is always evident from the approximate value of 
the angle as given by the first reading. The remainder of the 
formula involves very simple operations on the three mean vernier 
readings. 

40b. Illustrative Example. A complete example of notes and reduc- 
tions for a double set of angle measurements is here given to illustrate the 
above method. 



Station occupied = A. 
Date = Aug. 28, 1911. 
Time = 4.30 p.m. 


• Angle = Sta. B to Sta. C. 
Observer = J. H. Smith. 
Instrument = Brandis No. 17. 


Telescope. 


Ver. A. 


Ver. B. 


Mean. 


Angle. 


Average. 


— 


0° 00' 00" 


180° 00' 10" 


0° 00' 05" 






1. D 


75 12 30 










6. D&R 


01 14 50 


271 14 50 


91 14 50 


75° 12' 27.5" 




6. R&D 


359 58 50 


179 59 00 


359 58 55 


75 12 39.2 


75° 12' 33.4" 


6. D&B 


91 13 50 


271 13 50 


91 13 50 


75 12 29.2 




6. R&D 


359 57 50 


179 57 50 


359 57 50 


75 12 40.0 


75 12 34.6 








Mean angle = 


75° 12' 34.0" 



58 GEODETIC SUEVEYING 

It will be noted that vernier A was set to zero to begin with, and vernier 
B read 180° 00' 10" This setting to zero is, of coiirse, not essential, but 
convenient, as the next reading at once gives a close value of the desired 
angle without computation. There is no object in reading vernier B for 
this approximate determination. The remaining readings are taken at the 
proper time just as the instrument reads, paying no attention to the number 
of times the 360° point has been passed. 1. D means one measurement of 
the angle with the telescope direct. 6. D & R means six repetitions, usmg 
the telescope equally both direct and reversed (hence 6. D & R means the 
result after 3 direct and 3 reversed measurements). It will also be noted 
that no resetting of verniers has taken place at any time throughout the 
complete double set. Vernier B is only read in order to average out instru- 
mental errors (which are always very small), and therefore in filling in this 
column the degrees are recorded the same as given by vernier A, that 
is, the constant difference of 180° between vernier A and B is not allowed 
to affect the mean. In filling out the column marked angle the first and 
the final reading of each set are subtracted from the middle reading (adding 
360° if necessary to make the subtraction possible), dividing the remainder 
by 6, and adding as many times 60° as may be needed to make the result 
correspond to the 1. D reading. 



91° 


14' 


50" 


91° 


13' 


50" 





00 


05 


360 






6)91 


14 


45 


451 


13 


50 


15 


12 


27.5 


359 


58 


55 


60 






6)91 
15 


14 
12 


55 


75 


12 


27.5 


29.2 








60 








75 


12 


29.2 


91" 


14' 


50" 


91° 


13' 


50" 


360 






360 






451 


14 


50 


451 


13 


50 


359 


58 


55 


359 


57 


50 


6)91 


15 


65 


6)91 


16 


00 


15 


12 


39.2 


15 


12 


40.0 


60 






60 






75 


12 


39.2 


75 


12 


40.0 


75 


12 


27.5 


75 


12 


29.2 


75 


12 


39.2 

2)66.7 

33.4 


75 


12 


40.0 

2)69.2 

34.6 



In actual practice the 360° and the 60° would have been added mentally as 
needed. 

40c. Additional Instructions. If it is desired to attempt to 
eliminate errors of graduation, several double sets may be taken at 
different parts of the circle, symmetrically disposed. Modern 




MEASUEEMENT OF ANGLES 59 

instruments are so well graduated, however, that it is doubtful 
if any increased accuracy is gained by this refinement when 
measuring angles by any method of repetition. 

If it is desired to measure more than one angle at the same 
station, as for instance AOB and BOC, Fig. 20, we may take six 
repetitions on each of these angles and close the horizon by six 
repetitions on the angle from C clockwise around to A, and divide 
the failure to total 360° equally among the three angles; or we 
may measure AOB and its exterior angle without regard to station 
C, and then measure BOC and its exterior 
angle without regard to station A. 

In using the above or any other methods 
of measuring an angle by repetition it is 
presumed the surveyor will use every pre- 
caution possible in the handling of the instru- 
ment. . Avoid walking around the instrument, 
if supported on a tripod; unclamp the lower 
motion and revolve the instrument if it is 
desired to read the verniers. Do not relevel 
during the progress of measuring an angle j^iq. 20. 

except at such times as the upper motion is 
clamped and the lower motion free. Revolve the instrument 
very carefully on its vertical axis to avoid slipping the plates. 
Read each vernier independently, without regard to what the 
other one may have read. 

41. Adjustments of the Repeating Instrument. For the 
measurement of horizontal angles the required adjustments 
include : 

The plate-bubble adjustment; 
The striding-level adjustment; 
The collimation adjustment; 
The horizontal-axis adjustment. 

These adjustments may be made as here described, but there 
is usually more than one way of making the same adjustment. 

The Plate-bubble Adjustment. This is made in the same 
manner as with a surveyor's transit. Place one bubble parallel 
to two of the leveling screws, and bring both bubbles to the center. 
Turn the instrument 180° on the vertical axis, and adjust each 
bubble for one-half of its movement. Level up and test again, 



60 GEODETIC SUEVEYING 

and so continue until revolution on the vertical axis causes no 
movement of the bubbles. 

The Striding-level Adjustment. Level up the instrument by 
the plate bubbles (not absolutely necessary but convenient). 
Place striding level in position with telescope parallel to one pair 
of screws. Bring striding-level bubble to center with remaining 
screw. Lift striding level off, and replace in reversed position. 
Adjust it for one-half the bubble movement. Again bring bubble 
to middle as before with the leveling screw, test again, and repeat 
until reversal of the striding level causes no movement of its 
bubble. 

The Collimation Adjustment. This is the same as with a 
surveyor's transit. Set up on nearly level ground, level up 
with the plate bubbles, and then perfect the leveling with the 
striding level, so that revolution on the vertical axis of the 
instrument causes no movement of the striding-level bubble. 
Unless the horizontal axis is in adjustment this stationarj'- posi- 
tion of the bubble will not be in the middle. With the instrument 
clamped set a point about 200 feet away, plunge and set a second 
point about the same distance in the opposite direction, with 
the telescope reversed. Un clamp, revolve on vertical axis, set 
on first point with telescope reversed. Plunge and set a third 
point near the second point. Adjust by bringing the vertical 
hair back one quarter of the disagreement. Repeat the whole 
process until no discrepancy can be detected. 

The Horizontal-axis Adjustment. This is the same as with the 
surveyor's transit. Level up perfectly with the striding level 
near an approximately vertical wall or equivalent. Set on a 
high point, with instrument clamped. Drop the telescope and 
mark a low point about level with the telescope. Unclamp, 
revolve on vertical axis, and set on high point with the telescope 
reversed. Drop the telescope and set a low point abreast of the 
first low point. Adjust the horizontal axis so that the line of 
sight will pass through the high point and bisect the space between 
the low points. If the striding level and the horizontal axis are 
both in adjustment and the instrument level, the striding-level 
bubble should stay unmoved in its middle position while the 
instrument is turned completely around on its vertical axis. 

42. The Direction Instrument and its Use. Besides the 
features common to all first-class instruments, as described in 



MEASUREMENT OF ANGLES 61 

Art. 37, the direction instrument has two distinguishing features: 
First, it has only "one vertical axis, so that angles can not be meas- 
ured by repetition (means often provided for shifting the limb 
between sets of readings must not be used for angle repetition) ; 
Second, it is provided with two or more micrometer microscopes 
for reading the angles measured. The single center and clamp, 
instead of the two centers and clamps of the repeating instru- 
ment, imdoubtedly add to the stability of the instrument and 
the trueness of its motion. The limb of a 10-inch or 12-inch 
direction instrument is commonly graduated into 5-minute spaces, 
and the micrometer microscopes enable an angle to be read at 
once to the nearest second, as described later on. 

In using the direction instrument each angle is read a number 
of times, and the results averaged, to eliminate errors of pointing ; 
all the microscopes are read at each pointing, to eliminate eccen- 
tricity of vertical axis or microscopes; half of the readings are 
taken with the telescope direct and half with it reversed, to 
eliminate errors of coUimation and horizontal axis; half of the 
readings are taken to the right and half to the left, to eliminate 
errors due to twisting of the instrument and station. In the 
highest grade of work the limb of the instrument is shifted between 
each set of readings an amount equal to 180° divided by the num- 
ber of sets, ia order to eliminate errors of graduation. Modern 
direction instruments are so well graduated that in ordinary 
work this last refinement may be omitted. 

43. First Method with Direction Instrument. The instru- 
ment having been set up and leveled with the telescope in its 
normal position is directed to the first station, and all of the 
micrometers read, and so on to the right (clockwise) to each 
station in order, the values of the different angles being obtained 
by taking the differences of the successive readings, as will be 
illustrated by an example when the method of using the microm- 
eters is explained. When the last station to the right has been 
reached the instrument may be turned still further in the same 
direction until it reaches the initial station, called closing the 
horizon, and any difference between the initial and final readings 
equally divided among all the angles, but experience does not 
appear to show any advantage in thus closing the horizon, and 
it is commonly not done. When the last pointing to the right 
has been made, the instrument is brought back station by station 



62 



GEODETIC SURVEYING 



to the initial point, thus making a new series of values for the 
angles., The right and left pointings are agam repeated, this 
time with, the telescope reversed. The four series of values thus 
obtained constitute one set, and as many sets as desired may be 
averaged together. When for any cause a set is incomplete or 
inconsistent the entire set is rejected. When there are several 
angles to be measured at one station they are sometimes measured 
in various combinations as well as singly, the method of adjust- 





ment appearing later. The program in measuring a single angle. 
Fig. 21, is as follows: 

PROGRAM 

First Set. 

1. Level the instrument. 
Set telescope direct and 

2. Set on A and read micrometers. 



3. 



B 



4. " A " " 
Reverse telescope and 

5. Set on A and read micrometers. 

6. " B 

y HA CI CI 

Second Set. 

1. Shift limb. Relevel. 
Leave telescope reversed and 

2. Set on A and read micrometers. 

3. " B " " 

4. " A " " 



MEASUREMENT OF ANGLES 63 

Set telescope direct and 

5. Set on A and read micronaeters. 

6. " B " " 

17 it A IC CC 

If there were t"^o angles to be measured at a station, as illus- 
trated in Fig. 22, the program would be as follows : 

PROGRAM 
First Set. 

1. Level the instrument. 
Set telescope direct and 

2. Set on A and read micrometers. 

3. " B 

4. " C 

5. " B 

6. " A 

Reverse telescope and 

7. Set on A and read micrometers. 



8. 


" B 


9. 


" C 


10. 


" B 


11. 


" A 


Second Set. 


1. 


Shift limb. 



Relevel. 
Leave telescope reversed and 

2. Set on A and read micrometers. 



3. 


Ct 


B 


ii 


(i 


4. 


•ii 


C 


it 


ii 


5. 


li 


B 


a 


li 


6. 


a 


A 


it 


it 


Set telescope 


direct 


and 




7. 


Set on A and read micrometers, 


8. 


tc 


B 


ii 


it 


9. 


ii 


C 


C£ 


ii 


10. 


ti 


B 


11 


ii 


11. 


it 


A 


" 


ii 



64 GEODETIC SUEVEYING 

and similarly for any number of angles at one station. It will be 
noted that in the above method the telescope is reversed in posi- 
tion only at the initial station. 

44. Second Method with Direction Instrument. If it is not 
desired to make so many pointings (in order to reduce the labor and 
time) the telescope may be reversed at both the initial and final 
stations and the number of pointings be greatly reduced. 
The determination of the different angles, however, by this second 
method would not be considered as good on account of the decreased 
number of pointings. If a sufficient number of sets were taken 
to equalize the number of pointings the two methods would, of 
course, be equivalent. Referring to Fig. 21, page 62, the program 
for a single angle by the second method would be as follows : 

PROGRAM 

First Set. 

1. Level the instrument. 

Set telescope direct and 

2. Set on A and read micrometers. 

3. " B " " 

Reverse telescope and 

4. Set on B and read micrometers. 

5. " A 

Second Set. 

1. Shift limb. Relevel. 

Leave telescope reversed and 

2. Set on A and read micrometers. 

3. " B " " 

Set telescope direct and 

4. Set on B and read micrometers. 

5. " A " 

Referring to Fig. 22, page 62, the program by the second 
method for two angles at a station would be as follows : 



MEASUREMENT OF ANGLES 65 

PROGRAM 
First Set. 

1. Level up instrument. 
Set telescope direct and 

2. Set on A and read micrometers. 

3. " B 

4. " C 

Reverse telescope and 

5. Set on C and read micrometers. 

6. " B " " 

Y "A " " 

Second Set. 

1. Shift limb. Relevel. 
Leave telescope reversed and 

2. Set on A and read micrometers. 

3. " B " " 

4. " C " " 

Set telescope direct and 

5. Set on C and read micrometers. 

6. " B " " 

n It A II It 

and similarly for any number of angles at a station. 

45. The Micrometer Microscopes. When the direction instru- 
ment is set up and leveled at a station the graduated plate occupies 
a fixed position for the time being. The framework which 
supports the telescope carries two or more microscopes symmetric- 
ally disposed around the center of the instrument and focussed 
directly on the graduated ring. As the telescope is swung around 
from station to station the zero point of each microscope passes 
over the graduations an equal angular amount. If the exact 
position (on the graduated ring) of the zero point of any one of the 
microscopes is noted for two different stations, then the difference 
of these readings gives the angle through which the instrument 
has been turned, and consequently the angle between these 



66 



GEODETIC SURVEYING 



stations. Combined with each microscope is an instrument called 
a filar micrometer, by means of which the exact position of the 
zero point of the microscope on the scale may be determined. 

Fig. 23 represents diagrammatically a sectional view of a filar 
micrometer. A is the micrometer box, attached to the microscope 




Fig, 23. — Filar Micrometer, 



as seen in Fig. 17. The micrometer is made up of the following 
parts: 

A, micrometer box; 

h, b, fixed guide rods; 

c, ^movable frame carrying comb scale d; 

d, comb scale attached to movable frame c; 

e, movable frame carrying cross-hairs /; 

/, cross-hairs attached to movable frame e; 

ff} fft 9) spiral springs to take up lost motion of movable 

frames c and e; 
h, fixed screw whose revolution adjusts movable frame c; 
m, micrometer screw attached to movable frame e; 
n, fixed nut whose revolution moves cross-hairs across 

field of view; 
p, milled head for revolving nut n; 
s, graduated head for indicating fractional revolutions of 

nut n; 
t, fixed index for reading scale on graduated head s; 
V, dust cap to protect micrometer screw m. 

The central notch of the comb scale is marked by a small hole 
drilled behind it (or greater depth to that notch, or other equiv- 
alent), and is intended to be practically at the center of the field 



MEASUREMENT OF ANGLES 67 

of view. Every fifth notch is indicated usually by its greater 
depth and square bottom. All coiinting is done with the notches 
and not with the points of the teeth. Each revolution of the 
micrometer screw moves the cross-hairs over a space equal to the 
distance between the bottoms of two adjacent notches. When 
the microscope is properly adjusted the image of the graduated 
ring is formed in the plane of the micrometer cross-hairs, so that 
both image and cross-hairs are seen sharply defined on looldng 
into the eyepiece, the microscope ordinarily having a magnifying 
power of 30 to 50 diameters. The comb scale is placed as close as 
possible to the cross-hairs without touching them, and hence is 
seen at the same time and in sufficiently good focus. As ordinarily 
arranged the limb of the instrument is graduated into five-minute 
spaces, and the micrometer head into sixty spaces, and five 
revolutions of the micrometer screw carry the cross-hairs across 
the image of the limb from one five-minute division to the next 
five-minute division; so that one notch on the comb scale or 
one revolution of the micrometer screw indicates one minute of 
angle, and each division on the head indicates one second. 

46. Reading the Micrometers. The cross-hairs /, Fig. 23, 
consist of two parallel spider threads, placed just a little further 
apart than the width of the graduation lines on the instrument, 
so that when a graduation line comes central between the hairs 
a narrow illuminated line appears to lie on each side of the gradu- 
ation. It is found in practice that the hairs can be centered over 
a graduation in this way better than by any other plan (such 
as a single thread or intersecting threads). Everythiug being 
in good adjustment the zero point of the microscopes is the 
center of the space between the cross-hairs when they are opposite 
the central notch of the comb scale and the zero of the head is 
opposite the index line. It is important to note that the comb 
scale is not an essential part of a micrometer, but simply a con- 
venience, enabling the observer to see at any moment how many 
complete revolutions of the micrometer screw have taken place 
at any time without keeping track of the matter while the screw 
is being turned; no attempt must be made to get the value of a 
reading by the comb scale beyond its intended purpose of indicat- 
ing whole revolutions or single minutes, the seconds being read 
entirely from the micrometer head; as long as the comb scale 
serves its intended purpose, therefore, of coimting whole revolu- 



68 



GEODETIC SUEVEYING 



tions, it does not matter whether its position in the field of view 
is microscopically exact or not. Referring to Fig. 24, a greatly 
exaggerated view is given of what is seen through one of the 
microscopes for a certain pointing of the telescope. As seen in 
the microscope (which reverses the actual fact) the scale reads 
from left to right. Assuming the cross-hairs set to their index 
or zero point the reading is seen to be 65° 10' plus the value 
between the 10-minute division and the center between the 
two hairs. Running the micrometer screw backwards until 
the hairs exactly center over the 10-minute division it is found 




Fig. 24. 

that one notch has been passed over by the hairs, but that they 
have not gone far enough to center over the second notch from 
the middle one. The screw has therefore been turned through 
more than one but less than two revolutions. The numbers 
on the micrometer head increase as the hairs run toward the left, 
and assuming the index to stand opposite 25 the complete microm- 
eter reading is 1' 25.0", making the complete reading for the 
pointing 

65° 10' + 1' 25.0" = 65° 11' 25.0", 



if no corrections were required. The head reading is usually 
estimated to the nearest tenth of a second. 

46a. Run of the Micrometer. It is not found practicable 
in actual work to adjust the microscopes so perfectly that the 



MEASUREMENT OF ANGLES 69 

screw will be turned through exactly five revolutions (or indicate 
exactly 300 seconds) in drawing the hairs from one five-minute 
division to the next one, but the excess or deficiency should not 
exceed about 2". The closer the microscope is brought to the 
graduated ring the larger becomes the image in the plane of the 
cross-hairs. It is almost impossible to get the image the exact 
size that corresponds to precisely five revolutions of the screw; 
even if this result were accomplished at one part of the graduated 
ring the instrument is seldom made so true that it would hold 
good all around the ring, either on account of slight errors in the 
graduations or a lack of perfect trueness of the ring itself, and 
many other reasons; owing to temperature changes and other 
reasons it will not remain true or the same at the same part of 
the ring. In running from one scale division to another the amount 
by which the micrometer measurement varies from 300 seconds 
is called the run of the micrometer between those divisions, and 
must be determined at the time the pointing is made. Whatever 
the micrometer head may read when the hairs are set over one five- 
minute division, it must necessarily read the same when the hairs 
are advanced to the next five-minute division, provided there is 
no run of the micrometer, that is, provided that the screw turns 
through precisely five revolutions or 300". If the two head read- 
ings are not the same the difference gives the value of the run of 
the micrometer between these two divisions. In drawing the 
hairs from left to right the head readings decrease, so that the 
micrometer overruns when the forward head reading is less 
than the backward head reading, and vice versa. The run of the 
micrometer for the 300" space, therefore, equals the backward 
head reading minus the forward head reading, and the micrometer 
measurement of the 300" space equals 300" plus the run of the 
micrometer for this space. Since the micrometer does not 
measure the five-minute (300") space correctly, it follows that a 
proportionate error exists for intermediate points; or for any 
intermediate point we have 

Correction for run 



Run of micrometer for 300" space 



Micrometer measurement for intermediate point 
Micrometer measurement of 300" space 



70 GEODETIC SUEVEYING 

Let n = number of full turns to back division; 
= back head reading; 
p = forward head reading; 
6 = backward reading in seconds = 60 n + o; 
/ = forward reading (so called) in seconds = 60 m + p; 
&+/ 

d = run of micrometer for 300" space =o — p = h — f; 
c = correction for run to value 6; 
D = 300"; 
A = micrometer measurement of 300" space = 300 + d = 

D +d; 
M = adjusted micrometer reading to add to scale read- 
ing = b — c; 



then 



c _ 6 ^ b 
d~ /S ~ D + d' 

db 
"' D + d' 

71*- 7 db 

M =b ~ 



D + d 



substituting 



b +f , b -f d 

d I md 



M = m + 



2 \D + d 



D +dj' 



but since d is always very small in comparison with D we may 
write instead the extremely close approximation. 

i,r , d md 

in which care must be taken to use d algebraically with its correct 
sign. Since the adjusted reading is based entirely on b and / it 
is evidently unnecessary to set the micrometer to its zero point 
before reading either b or /. When a pointing is made in actual 
work b is taken as the mean value of all the back readings of the 



MEASUREMENT OF ANGLES 71 

different microscopes, and / as the corresponding mean of the 
forward readings, and only one reduction is made for a pointing. 
The scale reading is taken for one micrometer only. The eyepiece 
of each microscope must be very carefully focussed by the observer, 
as any perceptible parallax renders good work impossible. 

A complete example of notes and reduction is given on pages 
72 and 73. 

47. Adjustments of the Direction Instrument. For the 
measurement of horizontal angles the required adjustments 
include : 

The plate-bubble adjustment; 

The striding-level adjustment; 

The coUimation adjustment; 

The horizontal -axis adjustment; 

The microscope and micrometer adjustment. 

These may be made as here described, but there is usually 
more than one way of making the same adjustment. 

The Plate-bubble Adjustment. This is made in the same 
manner as with a surveyor's transit. Place one bubble parallel 
to two of the leveling screws, and bring both bubbles to the 
center. Turn the instrument 180° on the vertical axis, and 
adjust each bubble for one-half its movement. Level up and 
test again, and so continue until revolution on the vertical axis 
causes no movement of the bubbles. 

The Striding-level Adjustment. Level up the instrument by 
the plate bubbles (not absolutely necessary but convenient). 
Place striding level in position with telescope parallel to one pair 
of screws. Bring striding-level bubble to center with remaining 
screw. Lift striding level off, and replace in reversed position. 
Adjust it for one-half the bubble movement. Again bring bubble 
to middle as before with the leveling screw, test again, and repeat 
until reversal of striding level causes no movement of its bubble. 

The CoUimation Adjustment. This is the same as with a 
surveyor's transit. Set up on nearly level ground, level up with 
the plate bubbles, and then perfect the leveling with the strid- 
ing level, so that revolution on the vertical axis of the instrument 
causes no movement of the striding-level bubble. Unless the 
horizontal axis is in adjustment this stationary position of the 
bubble will not be in the middle. With the instrument clamped, 



72 



GEODETIC SURVEYING 



ANGLE MEASUREMENT WITH 



Station occupied = Sta. A. 
; Date = May 15, 1910. 
Time = 5.00 p.m. | 


Station 


Instru- 
ment. 


Microm- 
eter. 


Scale. 


b 


/ 


m 


B 


D 


A 


65° 10' 


85".0 


82".7 








B 




83 .4 


87 .6 










Mean 


84 .2 


85 .2 


84".70 


C 


D 


A 


75° 12' 


126 .4 


124 .0 








B 




124 .2 


125 .2 










Mean 


125 .3 


124 .6 


124 .95 


C 


R 


A 




125 .2 


123 .1 








B 




123 .0 


126 .3 










Mean 


124 .1 


124 .7 


124 .40 


B 


R 


A 




82 .5 


80 .3 








B 




81 .0 


84 .6 










Mean 


81 .8 


82 .5 


82 .15 






Limb Shifted about 90°. 






B 


R 


A 




72".6 


69".4 








B 




69 .8 


71 .8 










Mean 


71 .4 


70 .6 


71".00 


C 


R 


A 




112 .8 


110 .0 








B 




109 .8 


111 .6 










Mean 


111 .3 


110 .8 


111 .10 


C 


D 


A 




111 .5 


110 .1 








B 




109 .1 


111 .2 










Mean 


110 .3 


110 .7 


110 .50 


B 


D 


A 




70 .1 


69 .0 








B 




68 .0 


71 .2 










Mean 


69 .1 


70 .1 


69. 60 

















MEASUREMENT OF ANGLES 



73 



THE DIRECTION INSTRUMENT 



Angle = Sta. B to Sta. 0. 

Observer = Wm. S. Browa. 

Instrument = Brandis No. 20. 


d md 
2 ~D 


M 


Pointing. 


Angle. 


Average. 






















- 0".22 


84".48 


65° 11' 24".48 


























+ .06 


125 .01 


75 14 05 .01 


10° 02' 40".53 
























- .05 


124 .35 


75 14 04 .35 




10° 02' 41 ".45 






















- .16 


81 .99 


65 11 21 .99 


10 02 42 .36 


































+ .21 


71 .21 


65 11 11 .21 


























+ .06 


111 .16 


75 13 51 .16 


10 02 39 .95 
























- .05 


110 .45 


75 13 50 .45 




10 02 40 .54 






















- .27 


69 .33 


65 11 09 .33 


10 02 41 .12 








Mean angle = 


10° 02' 41 ".00 



74 GEODETIC SURVEYING 

set a point about 200 feet away, plunge and set a second point 
in the opposite direction with telescope reversed. Unclamp, 
revolve on vertical axis, set on first point with telescope reversed. 
Plunge and set a third point near the second point. Adjust by- 
bringing the vertical hair back one quarter of the disagreement. 
Repeat the whole process until no discrepancy can be detected. 

The Horizontal-axis Adjustment. This is the same as with the 
surveyor's transit. Level up perfectly with the striding level near 
an approximately vertical wall or equivalent. Set on a high 
point, with instrument clamped. Drop the telescope and mark 
a low point about level with the telescope. Unclamp, revolve 
on vertical axis, and set on high point with the telescope reversed. 
Drop the telescope and set a low point abreast of the first low 
point. Adjust the horizontal axis so that the line of sight will 
pass through the high point and bisect the space between the 
low points. If the striding level and the horizontal axis are both 
in adjustment and the instrument level, the striding-level bubble 
should stay xmmoved in its middle position while the instrument 
is turned completely around on its vertical axis. 

The Microscope and Micrometer Adjustment. It is necessary 
to have the graduated arc pass practically across the center of 
the field of view, and the supporting frame is generally provided 
with self-evident means of making this adjustment. Sometimes 
all but one of the microscopes may be moved circumferentially 
so as to space them equally around the circle, but frequently 
they are permanently mounted by the makers in their proper 
places. The microscope tube may \be rotated on its own axis 
until the cross-hairs are exactly parallel to the graduation lines. 
The microscope can be adjusted so as to change the distance 
between the objective and the cross-hairs, and the whole micro- 
scope can be moved so as to change the distance between the 
objective and the graduated plate; if the micrometer overruns, 
the image of the graduations is too large, and must be made 
smaller, by decreasing the distance between the objective and 
cross-hairs slightly, and then carefully moving the whole micro- 
scope away from the graduations imtil a perfect focus is again 
obtained exactly in the plane of the cross-hairs, as shown by the 
fact that properly focussing the eyepiece shows both the hairs 
and the graduations sharply defined and without parallax; if 
the micrometer tmderruns the image is too small, the objective 



MEASUREMENT OF ANGLES 75 

must be moved away from the cross-hairs, and the whole micro- 
scope moved toward the graduations; this adjustment should be 
perfected until the error does not exceed 2". The zero point 
of each microscope can be changed by shifting the comb scale 
and revolving the graduated head on the micrometer screw; 
this adjustment enables two microscopes to be set exactly 180° 
apart, three microscopes 120° apart, etc. Great care and skill 
are necessary to properly adjust the microscopes and micrometers. 
48. Reduction to Center. It is sometimes impossible to set 
up an instrument exactly over a given station, a flag pole or steeple, 
for instance. In such a case an eccentric station is taken as near 
the true station as possible, and the eccentric angle is measured 
with the same precision as would have been used for the real 
angle. From the location of the true station with reference to 
the eccentric station a correction is computed which wiU reduce 
the eccentric angle to what it would have been if measured at 
the true station, this operation being known as reduction to center. 
The true station is generally referred to the eccentric station 
by an angle and a distance, a single measurement of the angle 
being sufficiently accurate for the pur- 
pose. Referring to Fig. 25, C is the 
true station, E the eccentric station, 
BCA the desired angle, BEA the 
angle actually measured, and a and r 
the angle and distance connecting the 
true station with the eccentric station. 
In the triangle ABC the angles at A 
and B are known by actual measure- 
ment, and one of the sides of the 
triangle must be known by measure- Fig. 25. 

ment or by computation from its con- 
nection with the triangulation system. Having one side and two 
angles given we may regard all the parts of the triangle ABC as 
known with sufficient accuracy for the present reduction, on 
accoimt of the desired correction always being very small. Oppo- 
site angles at D being equal, we have 

C + y =E+x^ 
or 

C =E + {x - y), 




76 
but 

hence 



GEODETIC SURVEYING 



sin a; r , sm y r 

= - and -. — - = — ; 

sm a a 



sm X = 



sin {E + a) b 
r sin {E + a) 



and sin y = 



r sm a 



b "a 

Since x and y are very small angles, we may write 
sin X = X sin 1" and siny = y sin 1", 
r \sin (E + a) 



or 



X = 



sin 1' 



r \ sm a 



and w = \~. — 77-. I , 



whence 



C = E + ^^r?HL(^JliO _ sing;"! 
sinl"L b o- i 



in which the correction to be applied to E will be in seconds, 
and may be essentially positive or negative, since the true angle 

may be either larger or smaller than 
the eccentric angle. If care is taken 
to use the proper value of a, to re- 
member that angles between 180° and 
360° have negative sines, and to work 
out the formula for C algebraically, the 
correct value of C will be obtained 
whether it be larger or smaller than E, 
and without knowing what the plotted 
figure would look like. If measured 
from r the angle a must be taken 
counter-clockwise all the way around to 
the line EB no matter how large it may come; if measured 
from EB it must be taken clockwise around to r; thus in 
Fig. 26 the angle a is the one so marked and not the insi'.e 
angle BEC. 

The correction to be applied to E to obtain C depends entirely 
on the values of x and y, and these may be computed directly 
if preferred, and combined in the proper way by inspection of 
the figure, since the observer can scarcely be ignorant of how the 
different stations are related to each other and hence can quickly 




Fig. 26. 



"MEASUREMENT OF ANGLES 



77 







o 





3 








78 GEODETIC SURVEYING 

draw a sketch of the actual conditions. All the possible cases 
are shown in Fig. 27, page 77, for any angle less than 180°. 

49. Eccentricity of Signal. It sometimes becomes necessary 
in measuring an angle to sight on an eccentric signal ; for instance, 

as in Fig. 28, it may be necessary to 
sight to B' instead of the true station 
B. The measured angle ACB' must 
therefore be corrected by the small 
angle BCB' to obtain the desired angle 
ACB. In the triangle ABC the angles 
at A and B are measured, and one side 
is always known through connection 
with the rest of the system, so that 
the side BC can be computed with 
sufficient closeness for the present 
purpose. The distance BD, perpen- 
dicular to CB' and called the eccen- 
tricity, is either directly measured or computed from a measure- 
ment of the distance B'B and the angle at B'. Then 

BCB' (in seconds) = 




BC sin 1' 



50. Accuracy of Angle Measurements. When the same 
instrument is used by a skilled observer imder the same condi- 
tions results are obtained which differ but slightly from each other. 
In measuring an angle with an ordinary 30-second transit of 
good make two sets taken by the method of repetition, in accord- 
ance with the example given on page 57, should not differ by 
more than 5". A 10-inch repeating instrument used in the same 
way, or a 10-inch direction instrument used in accordance with the 
example on pages 72 and 73, shoxild give sets differing by less than 
2". A great many sets may be taken at the same time and agree 
with each other within these limits, but it does not follow that the 
value of the angle is obtained with this degree of precision. If the 
same observer measures the same angle with the same instrument 
imder different conditions a new series of values may be obtained 
closely agreeing with each other, but the mean of the values 
belonging to the first series may differ several seconds from the 
mean of the second series; in fact, the two means may differ more 
from each other than the result of any one set differs from the 



MEASUREMENT OF ANGLES 79 

maen of its own series. Morning measurements often differ 
from afternoon measurements, even when the atmospheric 
conditions appear to be the same. In the finest work an angle 
is measured on many different days (sometimes with an equal 
number of a.m. and p.m. measurements), under as different 
conditions as possible, and a general average taken of all the values 
obtained, called the arithmetic mean. 

In the Coast Survey work the probable error (Chapter XIII) 
of a primary angle must not exceed 0".3, and primary triangles 
must close within 3". In secondary work the probable error 
of an angle must not exceed 0".7, and triangles must close 
within 6". In work of less importance a greater probable error 
is allowable, but, the triangles are expected to close within about 
12". A sufficient number of measurements must be taken to 
bring about these results, but in primary work in any event 
at least five double sets like those given in the examples ought 
to be taken. 

The probable error of an angle is obtained as follows: 

Let r„ = probable error of mean angle (in seconds) ; 

Ml, M2, etc. = value given by each set; 
z = mean value of angle; 



Ml — z = vi 
M2 — z = V2 



etc. =residuals (in seconds); 

HtP = sum of squares of residuals; 
n = number of sets. 



then 



\n{n — 



±0.6745^,..,.. "^ 



Example. Six measurements of an angle were taken : 
Observed Values. Arithmetic Mean. v 



7° 


16' 9". 2 


7" 


16' 9" 


.7 


- 0' 


.5 


0.25 


7 


16 '12 .1 


7 


16 9 


.7 


+ 2 


.4 


5.76 


7 


16 8 .4 


7 


16 9 


.7 


- 1 


.3 


1.69 


7 


16 6 .7 


7 


16 9 


.7 


- 3 





9.00 


7 


16 10 .3 


7 


16 9 


.7 


+ 


.6 


0.36 


7 


16 11 .5 


7 


16 9 


.7 


+ 1 


.8 
.0 


3.24 




6) 58". 2 


0" 


20.30 



9 .7 



80 GEODETIC SUEVEYING 

The algebraic sum of the residuals is zero, as it always should be. 



r. = ±0.6745Vgf:?^ = ±0".55. 

If the several determinations of the angle are not considered 
equally good (on account of a difference in the number of repe- 
titions or in the atmospheric conditions, etc.), and the values are 
correspondingly weighted, each value is multiplied by its weight 
and the sum of the products divided by the sum of the weights 
giving the weighted arithmetic mean, the probable error of which is 



r^ = ± 0.6745 J. '^^^ 



Sp(n - 1)' 



in which llpt^ equals the sum of the results obtained by multiplying 
each squared residual by the corresponding weight; and 2p equals 
the sum of the weights. 



CHAPTER IV 

TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 

51. Adjustments. After the field work of angle measurement 
has been completed there still remains the office adjustment 
of the angles necessary to satisfy the rigid geometrical conditions 
involved; thus all the angles around a point must add up to 
360°, the three angles of a triangle must add up to 180°, etc. 
All such geometrical conditions must be satisfied before .the 
lengths of the various lines of the system are computed. The 
adjustment of the angles at any station withoXit regard to meas- 
urements taken at other stations (such as making the angles 
around a point add up to 360°), is called station adjustment. The 
mutual adjustment of the several angles of a given figure (such 
as making the angles of a triangle add up to 180°), is called figure 
adjustment. Easily applied rules for simple cases of adjustment 
can be derived by the method of least squares or the theory of 
weights; more complicated cases are better adjusted directly 
by the method of least squares, as explained in Part II of this 
book. The object in any case of adjustment is, of course, to 
find from the measured values the most probable values con- 
sistent with the geometrical conditions involved. 

52. Theory of Weights. The weight of a quantity is defined 
as its relative worth. The term weight, therefore, is purely relative 
and must never be understood in an absolute sense. A distance of 
3 feet or 3 miles is an absolute and definite distance; a weight of 3 
does not represent any definite degree of precision, but is simply a 
comparison with that which is assigned a weight of 1. The basis of 
comparison is fundamentally the number of observations of imit 
weight from which the given value is derived; thus if 5 measure- 
ments of an angle were regarded as equally reliable, expressed 
mathematically by assigning to each a weight of 1, the mean value 
of the angle (by definition) would have a weight of 5. Weights are 
often arbitrarily assigned as a matter of judgment, however, 

81 



82 GEODETIC SURVEYING 

where the corresponding number of observations does not exist; 
thus a measurement obtained under unusually favorable condi- 
tions might be considered as good as the mean of two measure- 
ments taken xmder less favorable conditions, and hence a weight 
of 2 assigned to this single favorable measurement. Since, 
therefore, the numbers representing weight are purely relative, 
and do not necessarily represent a corresponding number of 
observations, any number, whole or fractional, may be so used; 
thus two quantities may be said to have the weights respectively 
of 1 and 2, or ^ and 1, or 0.12 and 0.24, and their relative worth 
is the same in either case. The mean value as understood above 
is the arithmetic mean, and is only used when the quantities are 
of equal weight. When the different values are of unequal weight 
each value is multiplied by its weight, and the sum of the products 
is divided lay the sum of the weights, the result obtained being 
called the weighted arithmetic mean. 

53. Laws of Weights. The following principles (established 
by the method of least squares) govern the use of weights : 

1. The weight of the arithmetic mean (with measurements 
of unit weight) equals the number of observations. 

Example. Angle A by different mensurements equals 

29° 21' 59". 1, weight 1 

29 22 06 .4, " 1 

29 21 58 .1, " 1 

3 )88° 06' 03". 6 

Arithmetic mean = 29° 22' 01". 2, weights. 

2. The weight of the weighted arithmetic mean equals the sum 
of the individual weights. 

Example. Base hne AB by different measurements equals 

4863.241ft., weight 2 
4863.182 ft., " 1 

whence 

4863.241 X2 = 9726.482 
4863 , 182 X 1 = 4863.182 
3 )14589.664 
Weighted arithmetic mean= 4863.221, weight 3. 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 83 

3. The weight of the algebraic sum of two or more numbers is 
equal to the reciprocal of the sum of the reciprocals of the indi- 
vidual weights. 

Example. Angle A = 46° 14' 11" .2, weight 2 
Angle -B = 11 21 19 .6, " 3 

1 6 



A+B = 56° 35' 30". 8, weight = 
A- B =33° 52' 51". 6, weight = 



i+i 5' 
1 6 



4+i 



4. Multiplying a quantity by a factor divides its weight by 
the square of that factor. 

Exampk. Angle A = 67° 10' 12". 5, weights, 

3 3 



2 A = 134° 20' 25". 0, weight = 



2X2 4" 



5. Dividing a quantity by a factor multiplies its weight by 
the square of that factor. 

Example. Base AB = 2716. 124 ft., weight 3, 

AB 

-^= 1368.062 ft., weight = 3 X 4 = 12. 

■ 

6. Multiplying an equation by its own weight (or dividing it 
by the reciprocal of its weight), inverts its weight. 

Example. %{x + ?/) = 400, weight }; multiplying by J (or dividing by 
J), we have 

4 
1{x-\-y) = 300, weight g. 

7. Changing all the signs of an equation, or combining the 
equation with a constant by addition or subtraction, leaves the 
weight unchanged. 

Example, a; + 2/ = 11° 10' 14". 6, weight 2.3, 
and 

360° - (a; + 2/) = 348° 49' 45". 4, weight 2.3. 



84 



GEODETIC SURVEYING 



54. Station Adjustment. This consists, as explained in 

Art. 51, of making the angles at a station geometrically consistent, 

such as making all the angles around 

a point add up to 360°. The following 

cases are worked out as shown : 

Case 1. The angles at a point have 
been measured with equal care (giving 
them equal or unit weight). In this 
case any discrepancy is equally dis- 
tributed among the three angles. Thus 
in Fig. 29, if the angles x, y, z, as 
measured, added up to 360° 00' 06", 
then each measured value would be re- 
duced by 2". 

As an application of the theory of weights, let us suppose 
we have by measurement 

X = a, weight 1 

y-\ "1 

2 = f, "1 

From third observation 360° — z = 360° — c, weight 1 

or X + y = 360° - c, "1 

By second observation y = h, " 1 




Fig. 29. 



By subtraction 

By first observation 



X =360° - 6 -c, weight i 
X = a, "1 



Taking the weighted arithmetic mean of these values of x, 



By addition 
whence 



\x = ^(360° - 6 - c) 
X = a 



ix =a + i(360° -b -c) 
z =ia + i(360° -b -c) 
= a + K360° -a -b -c) 
= a + I [360° - (a -I- 6 -f- c)]. 



which indicates that the most probable value of x is found by 
correcting the measured value a by one-third the discrepancy ; and 
the same result would be reached for y and z. In combining the 
observations as above it is to be noted that each observation can 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 85 

be used but once, as otherwise additional observations would be 
implied that in fact have not been taken. The above rule for 
the distribution of the station error is, of course, the same as 
would be obtained by the method of least squares. 

Case 2. The aligles as measured around a point. Fig. 29, 
have been assigned different weights. In this case any discrep- 
ancy is distributed inversely as the weights. Thus if the weights 
are 

for X, 1, for y, 2, for z, 3, 

the distribution of error would be as 

i-l- i 
1 • 2 ■ 3' 

which is the same as 

6.3.2 

6 ■ 6 " 6' 
which is equivalent to 

6:3:2; 

and since 6 + 3 + 2 = 11, we have 

correction for x = — of discrepancy ; 

3 

correction for y = — of discrepancy; 

2 
correction for 2 = — - of discrepancy. 

Case 3. Several angles at a point, and also their sum, have 
been measured with equal care. In 
this case any discrepancy is to be 
equally distributed among all the meas- 
ured values (including the measured 
sum). When the measured sum of 
several angles is greater than the sum 
of the individual measurements, the 
correction is positive for the single 
measurements and negative for the 
measured sum, and vice versa. Thus -piQ. 30. 

in Fig. 30, if the entire angle measured 
8" more than the sum of the single measurements, then the x, y, 




86 



GEODETIC SURVEYING 



and z measurements would each be increased by 2" , and the 
measured sum would be reduced by 2". 

Ca&e 4. Several angles at a point, and also their sum, have 
been measured, and different weights have been assigned to the 
measured values. In this case any discrepancy is distributed 
among aU the measured values inversely as their weights. Thus 
in Fig. 30, page 85, suppose 

X measured with weight 2; 

y " " 1; 

z " " 3; 

(x +y +z) " " 1, 



the division of error would be as 



which is the same as 



which equals 



i 1 i 1 

2 ■ 1 '3 ■ 1' 

3 6 2 6 
6 "6 ■ 6 ■ 6' 



3:6:2:6; 
and since 3 + 6 + 2 + 6 = 17, we have 



correction for 



x=^of 


discrepancy; 


y 17 


tc 


17 


11 


ix + y + z) =^" 


It 



If the measured values of x, y, and z add up to less than the 
measured sum (x + y + z), then the corrections for x, y, and z, 
are to be added, and the correction for {x + y + z) subtracted, 
and vice versa. 

General Rule. Any case of station adjustment in which the 
coefficients in the equations are all unity and the signs are all 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 87 

positive (as is usually the case), and in which the horizon has not 
been closed or the closing has been evaded in the equations by 
subtracting one or more angles from 360°, and in which the weights 
of the final results are not desired, may be solved as follows: 
Multiply each equation by its own weight ; add together separately 
all the new equations containing x, y, z, etc., as shown in the 
following example, and solve the resulting equations as simulta- 
neous. 



Example. Observed values. Fig. 31, 

X = 14° 11' 17".l, weight 1 

y = 19 07 21 .3, " 

x+y = 33 18 43 .4, " 

z = 326 41 18 .2, " 

2/ + Z = 345 48 39 .2, " 

Subtracting the angles involving z from 360°, 



weight 1 
'■' 2 




Fig. 31. 



X = 14° 11' 17".l, 

2/ = 19 07 21 .3, 

X + 2/ = 33 18 43 .4, 

360° - z = a; + 2/ = 33 18 41 .8, 

360° - (2/ + z) = a; = 14 11 20 .8, 

Multiplying each equation by its weight, 

X ' . =14° 11' 17".l' 

+ 22/ = 38 14 42 .6 

a; + 2/ = 33 18 43 .4 

2a; + 22/ = 66 37 23 .6 

3a; = 42 34 02 .4 

Combining separately all equations containing x, and all equations contain- 
ing y, we have 

7x-\-Zy = 156° 41' 26".5 
3x + 5y = 138 10 49 .6 

which, solved as simultaneous equations, give 

x = 14° 11' 20".14 
y = 19 07 21 .83 

the sum of which subtracted from 360° gives 

z = 326° 41' 18".03. 



55. Figure Adjustment. Having foimd by measurement and 
station adjustment the best attainable values of the different 
angles of a system, the next step is to make the figure adjustment. 
(If the work is very important and the angles so involved that 



88 GEODETIC SURVEYING 

making the figure adjustment would disturb the station adjust- 
ment, all the adjustments would have to be made in one operation 
by the method of least squares.) The figure adjustment, as 
explained in Art. 51, consists in making such slight changes in 
the various measured angles as will make the figure geometrically 
consistent, such as making the angles of a triangle add up to 180°, 
the angles of a quadrilateral add up to 360°, etc. The adjust- 
ment required in any case could be made in an infinite number 
of ways, but the adjustment that is sought is the one that assigns 
the most probable values to the various angles in view of their 
actually measured values. Since all the angles measured are 
spherical angles, it is necessary to compute the spherical excess 
in work of any magnitude before it can be determined to what 
extent the measured values are geometrically inconsistent. 

If all the triangulation stations (referred to mean sea level) 
were connected by chords instead of arcs, we would have a net- 
work of plane triangles perfectly locating all the stations, and 
through which the computations could be carried with perfect 
accuracy, provided the plane angles were known and used. 
These plane angles become as well known as the actually 
measured spherical angles by a proper reduction for spherical 
excess. On account of the simplicity and saving of labor the 
computations in practice are always made on the basis of plane 
triangles. In carrying forward the azimuths of the various lines, 
however, the reduction for spherical excess must be restored to 
the adjusted plane angles, and a further allowance made for 
convergence of meridians, as explained in Chapter V. 

56. Spherical Excess. The sum of the angles of a spherical 
triangle is always greater than 180° by an amount directly pro- 
portional to the area of the triangle and inversely proportional 
to the surface of the sphere, the value of the increase being called 
the spherical excess. It follows that the rule must also hold good 
for any spherical polygon, since such a figure can always be divided 
up into spherical triangles. Owing to, the shape of the earth, 
which is not a perfect sphere, the spherical excess for the same 
area decreases slightly as we advance from the equator toward 
the poles; except for very large areas it may be taken as 1" 
for every 76 square miles, the true value for this area being 
1".0035 + in latitude 18° and 0".9925 + in latitude 72° It may 
ordinarily be disregarded entirely where the area is less than 10 



TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 89 

square miles. The precise formula for any triangle may be 
written, 

area X (1 — ^ sin^ ^)2 

^= c ' 

in which 

£ = spherical excess in seconds of arc ; 
^ = latitude at center of triangle; 
loge2 = 7.8305026 - 10; 

[■ 1.8787228 (for area in square miles) 
log C = 9.3239906 ( " square feet) 

[ 8.2920224 ( " square meters). 

For logarithms of (1 — e2 sin2 ^) see Table IX. 

It is evident that neither the area nor the latitude need be 
known with extreme precision for the present purpose, and may 
be estimated before any adjustments have been made. 

57. Triangle Adjustment. The failure of the measured 
values of the angles of a triangle to add up to 180° is due to the 
spherical excess and the errors of measurement. If the spherical 
excess be computed, as explained in the previous article, the 
balance of the discrepancy represents the errors of measurement; 
or in order words, 180° + spherical excess — sum of angles = 
errors of measurement. The recognized adjustment for spherical 
excess is a deduction of one-third of the total excess from each 
angle, which is not mathematically correct unless the angles are 
all equal, but which may be so considered in any case that arises 
in practice; the reason for this is found in the fact that the excess 
is always a small quantity (rarely reaching 60"), and also that 
the triangles are always well shaped in this class of work. The 
theoretical adjustment for errors of measurement is to divide the 
amotint among the three angles inversely as their weights; if 
the angles are of equal weight this results in correcting each angle 
by one-third of the error. In view of the above considerations 
the failure of the angles of a triangle, as measured, to add up to 
180° is adjusted as follows: 

1. If all the angles as measured are considered equally reliable 
(of equal weight) the discrepancy is divided equally among the 
three angles. The spherical excess need not be computed in 
this case, imless it is desired for other purposes. 



90 GEODETIC SUEVEYING 

2. In important work where the angle measurements have 
different weights, each angle is first reduced by one-third of the 
spherical excess, and then corrected for the errors of measure- 
ment inversely as its weight. 

3. In small triangles or work of minor importance, where the 
angle measurements are of unequal weight, the total discrepancy 
is divided among the angles inversely as their weights. 

58. The Geodetic Quadrilateral. A geodetic quadrilateral is 
formed when the four stations. A, B, C, D, are connected as 
shown in Fig. 32. The size of the largest quadrilateral occurring 




Fig. 32. — The Geodetic Quadrilateral. 

in practice is relatively so small as compared with the size of the 
earth that we may always assume without material error that 
the four stations lie in a plane. In such a quadrilateral one side, 
as AD, must be known, either by direct measurement or connec- 
tion with the system; and the eight angles a, b, c, d, e, f, g, h, 
must be measured. If the quadrilateral is of sufficient size to 
require it the measured angles must be reduced for the spherical 
excess; in minor work this may be distributed equally among 
the eight angles; in more important work each of the four triangles 
formed by the intersection of the diagonals would be treated 
separately — ^that is, each angle would be reduced by one-third of 
the excess appropriate to its own triangle. In the plane quadri- 
lateral ABCD there are seven angle conditions and three side 



TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 91 

conditions that must be satisfied to make such a figure geometric- 
ally possible, and these ten conditions can all be covered by three 
angle equations and one side equation. 

The seven angle conditions are as follows : 

1. The sum of the eight comer angles must be exactly 360°. 
This furnishes one angle condition. 

2. The opposite angles where the diagonals cross must be 
equal. This furnishes two angle conditions. 

3. In each of the four triangles formed among the stations, 
such a.s ABC, the sum of the three angles must be exactly 180°. 
This furnishes four angle conditions. 

These seven conditions are so involved, however, that if any 
three independent ones are satisfied the other four are also satis- 
fied. As the first three conditions are independent all the angle 
conditions will be satisfied if we have 

a + h-^c + d + e+f + g+h=- 360°; 
a + 6 = e -f- /; 
c + d = g + h. 

The three side conditions arise from the fact that each unknown 
side is contained in two different triangles, so that each side may 
be found by two independent computations which must give 
identical results; thus the tmknown side BC may be computed 
from the known side AD through the triangles ACD and BCD, 
or through the triangles ABD and ABC, and the two values thus 
obtained must be the same. These three conditions are not 
independent, however, for if any one of them is satisfied the other 
two are also satisfied. It is well to note that all the seven angle 
conditions may be satisfied without satisfying any of the side 
conditions. From the figure we have 



also 

whence 

or 



sm a sm o sm a' 

bc = cd'^ = ad'^'^, 

, sm c sm e sm c ' 

BC _ sin g sin g _ sin / sin h 
AD sin b sin d sin c sin e ' 

sin a sin c sin e sin g 



sin b sin d sin / sin h 



= 1, 



92 GEODETIC SURVEYING 

which is called the side equation. When this equation is true 
the side conditions will all be satisfied. Writing the side equation 
in logarithmic form, which is the most convenient form for use, 
we have 

(log sin a + log sin c + log sin e + log sin g) 

— (log sin b + log sin d + log sin/ + log sin h) =0. 

59. Approximate Adjustment of a Quadrilateral. Assuming 
the angles to have been measured with equal care (and reduced 
for spherical excess, if necessary), a quadrilateral of moderate 
size or minor importance can be adjusted with sufficient approx- 
imation and with comparatively little labor by the method here 
given. 

Referring to Art. 58, the equations of condition which must 
be satisfied are as follows : 

Angle equations, 

a+b + c+d + e+f + g+h= 360°; 
a + b = e + f; 
c + d = g + h. 

Side equation, 

(log sin a + log sin c + log sin e + log sin g) 

— (log sin 6 + log sin d + log sin / + log sin K) = 0. 

The adjustments for the three angle equations are made first; 
since these three equations are independent the adjustments 
required to satisfy them may be made in any order, and will not 
disturb each other. Since the angles are supposed to be equally 
well determined the adjustments made to satisfy an}' one of the 
angle equations ought to have the same value for each angle 
affected. Therefore, if the eight angles fail to add up to 360°, 
each angle is corrected by one-eighth of the discrepancy; thus 
if the sum of the eight angles were 360° 00' 08", each angle would 
be reduced 1". If a + & fails to equal e + f each angle is cor- 
rected by one-fourth the discrepancy, reducing the larger side 
of the equation and increasing the smaller one; thus if a ^- 6 
exceed e + / by 8", a and b must each be reduced by 2" and e 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 93 

and / must each be increased by 2". Similarly, ii c + d fails 
to equal g + h, then each of these angles must be corrected for 
one-quarter of this discrepancy. 

The adjustment for the side equation is then made as follows : 
Let A, B, etc., represent the measured angles as thus far 
adjusted; 
I, represent the value of the first member of the side 

equation when A, B, etc., are substituted for a, b, etc.; 
I', represent the numerical value of I; 
'"a, ^6 , etc., represent the numerical change in seconds 
required in A,B,etc., in order to satisfy the side equation; 
da, db , etc., represent the tabular differences for 1" for log 
sin A, log sin B, etc. Then 

(log sin A + log sin C + log sin £^ + log sin G) 

— (log sin B + log sin D + log sin F + log sin H) = I. 

Since the adjustment of the angles must reduce I to zero (with a 
minimum change in each angle), it is seen from this equation 
that when I is positive the first four terms must be reduced and 
the last four increased, and vice versa when I is negative. This 
is equivalent to saying that if I is positive, the angles A, C, E, 
and G must be reduced if less than 90°, and increased if greater 
than 90°, and the angles B, D, F, and H increased if less than 
90°, and decreased if greater than 90°; and that if I is negative, 
the angles A, C, E, and G must be increased if less than 90°, and 
decreased if greater than 90°, and the angles B, D, F, and H 
decreased if less than 90°, and increased if greater than 90°- 
It therefore only remains necessary to find the numerical values 
of the corrections. In either case, in order that I may vanish, 
the numerical sum of the logarithmic changes must equal the 
numerical value of I. Since changing the angle A by Va changes 
log sin A by Vada , etc., we have 

Vada + Vcdc + Me + Vgdg + V^db + Vddd + V^f + Vjjdh = I', 

in which all the terms are to be made positive. Since this equation 
contains eight imknown quantities, Va,Vc, etc., it can not be solved 
imless some additional relationship among the unknowns is 
assumed. This relationship is found in the fact that the values 



94 GEODETIC SURVEYING 

Va, Vc, etc., are to be the most probable ones; and it is generally 
admitted that the most probable values are those that are pro- 
portional to their influence in building up the quantity V. Thus 
if da is twice dc , then, second by second, Va is twice as effective 
as Vc in building up the total, V ; and this effectiveness should be 
recognized by allotting twice as many seconds to v^ as are allotted 
to Vc , and so on. We thus have 

Va-.Vc :Ve, etc. = da : dc : de, etc. 
But if ^" = §^, ^ = f, etc., 

Vc dc Ve de 

th n " = — — = — etc 

Vcdc d^ ' Vedc d^ ' ■' 



or Vada : Vc^c '■ Vede , etc. = dip : dp : dp, etc. 

Referring to the equation to be solved, therefore, we see that 
V is to be divided into 8 pieces which shall be in the ratio of the 
numbers d,p, dp, dp, etc., giving for the successive terms of the 
equation the values 

dgH' dpi' dPl/_ 
2d2' Sd2' 2^2' • 



Hence 



d 2/' f] 27' 

V d = " vd = " pto 

'^a^'a V/-72 ' "^c^c VW2 ' ^i^^. 



and we have the numerical values 

Va = da (^2^ j , Vc=dc (^-^^j , etc, 

the signs of these corrections having been determined as pre- 
viously explained. 

The side-equation adjustment (having been derived without 
regard to the angle-equation requirements) will probably disturb 
the angle-equation adjustment slightly, but seldom seriously. 
If necessary, the two adjustments may be repeated in turn \mtil 
both are satisfied with sufficient approximation. 



TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 95 



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96 



GEODETIC SURVEYING 



A complete example of adjustment by this method is 
worked out in the table on page 95. In this particular 
case the side-equation adjustment has disturbed the angle-equa- 
tion adjustment to a maximum extent of 1".49. If this approxi- 
mation is not as close as desired the adjusted values may be 
treated like original values, and be readjusted by the same 
method. A second adjustment gives the following values: 



a = 46° 18' 38" .47 

6 =53 26 11 .92 

99 44 50 .39 

c = 42 11 27 .26 

d = 38 03 42 .35 

80 15 09 .61 

e = 58 19 10 .54 



/ = 41 



39 .90 



99 44 50 .44 

g =3A 33 47 .38 

h =A5 41 22 .18 

80 15 09 .56 

360 00 00 .00 



log sin 


= 9.S591959 




log sin 


= 


9.9048230 


log sin 


= 9.8271126 




log sin 


= 


9.7899405 


log sin 


= 9.9299248 




log sin 


= 


9.8206448 


log sin 


= 9.7538238 




log sin 


= 


9.8546489 




39.3700571 


39.3700572 



An examination of these values shows an almost perfect adjust- 
ment. It is interesting to compare the results of both the first 
and the second adjustment with the results of the rigorous adjust- 
ment of the same example as given in Art. 60. 

60. Rigorous Adjustment of a Quadrilateral. Assuming the 
angles to have been measured with equal care (and reduced for 
spherical excess, if necessary), and that the work is of too much 
importance for only approximate adjustment (or that a little 
extra labor on the computations is not objectionable), the follow- 
ing method may be used, the results being the same as would be 
obtained by the method of least squares. 

Referring to Art. 58, the equations of condition to be satisfied 
are as follows: 



TRIANGULATION ADJUSTMENTS AND COMPUTATIONS 97 
Angle equations, 

a + b+c+d + e+f + g+h=^ 360°; 
a + b = e +f; 
c + d = g + h. 
Side equation, 
(log sin a + log sin c + log sin e + log sin g) 

— (log sin b + log sin d + log sin / + log sin h) = 0. 

As in the case of the approximate method, a provisional adjust- 
ment is first made that will satisfy the angle equations, being 
made in the same way as there explained because it recognizes 
as far as possible the fact that all the angles have been measured 
with equal care. This adjustment is made as follows :> 

If a + b + c +, etc., fails to equal 360°, correct each angle 
by y of the discrepancy. 

li a + b fails to equal e + f, increase each member of the 
smaller sum and decrease each member of the larger sum by ^ 
of the discrepancy. 

li c + d fails to equal g + h, increase each member of the 
smaller sum and decrease each member of the larger sum by | 
of the discrepancy. 

The side-equation adjustment is then made, but made in 
such a way as will not disturb the angle-equation adjustments. 
Let A, B, etc., represent the angles as thus far adjusted; 

I, represent the value of the first member of the sid.e 

equation when A, B, etc., are substituted for a, b, etc.; 

Va, vt, etc., represent the total corrections in seconds to 

A, B, etc., to satisfy the side equation; 
X, Xi, X2, Xs, X4, represent the partial corrections of which 

Va, Vt, , etc., are composed; 
da, db, etc., represent the tabular differences for 1" for log 
sin A, log sin B, etc., taken as positive for angles less 
than 90° and negative for angles greater than 90°; 
then 

(log sin A + log sin C + log sin E + log sin G) 

— (log sin B + log sin D + log sin F -f- log sin H) = l\ 



98 GEODETIC SUEVEYING 

and in order that the logarithmic corrections shall cause I to 
vanish we must have 

(Vada + V4c + 1>ede + Vgdg) — (Vbdi + Vddd + Vjdf + v^dh) = — I, 

in which such values must be assigned to v^, Vt , etc., as will not 
disturb the angle-equation adjustments already made. These 
adjustments have given us 

(A + B) + {C + D) + {E + F) + (G + H) =0; 

{A +5) = {E +.F); 

iC +D) = (G + H). 

It is evident from these three equations of condition that there 
are only two possible ways in which the adjusted angles A, B, 
etc., can be modified without disturbing the angle-equation 
adjustments. First, any correction can be made to the sum of 
A and B, provided the same correction is made to the sum of 
E and F, and at the same time an equal and opposite correction 
is made to each of the other two sums; since the two angles 
of any sum are equally reliable the same numerical change 
must be made to each angle and wUl be denoted by x. Second, 
any group, such as {A + B), may have any correction applied to 
one of its members, provided an equal and opposite correction 
is made to its other member; these corrections are independent 
of the first correction and of each other, and will be represented 
by Xi, X2, X3, and x^. In accordance with the above considera- 
tions the side-equation adjustments must have the following 
relative values: 

Va = + X + Xi Ve = + X + X-s 

Vb = + X — Xi Vf = + X — Xz 

Vc = — X + X2 Vg ■= — X + Xi 

Vd = — X — X2 Vh = — X — X4 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 99 
Substituting these values in our conditional side equation 

{Vada + Mo + Vede + Vgdg) — {Vbdj, + Vddd + Vfdf + Vftdft) ^—l, 

and rearranging the terms, we have 

[{da + dd + de -\- dh) — (4 + do + df + dg)]x + (da + 4) Xi 

+ {dc + dd)x2 + (de + d^)x3 + {dg+ dh)Xi = -I, 

which for convenience we write 

Cx + CiXi + C2X2 + C3X3 + C4X4 = — I. 

Since this equation contains five unknown quantities it can not 
be solved unless some additional relationship among the unknowns 
■ ■ is assumed. The most probable relationship is therefore taken, 
namely, that the unknowns are proportional to their average 
effectiveness per angle in building up the quantity ( — . Hence, 
since x affects 8 angles and the other imlmowns only 2 each, we 
write 



C Ci C2 C3 G4 ^ ri /^ ri n 

X : xi : X2 : xz : Xi = -^ : -^ : — : -^ : — = -r : C-i_ : d : C3 : C. 



But if 



then 



8 ■ 2 ■ 2 ■ 2 ■ 2 4 



C 
x_ ^ 4_ ^ ^Ci X2 ^Ch 
XX ~ Cx' X2 ~ C2' X3 ~ C3' ''^''■' 



91 
Cx _ 4 Cixi _ (7i2 C2X2 _ C^ , 

— 2 1 etc.. 



^4. 



Cixx Ci^' C2X2 C^' C3X3 X32 



or 



n2 
Cx : Cixi : C2X2 : C3X3 : C4X4 = ^-Ci^: Cz^ : Cz^ : G^. 

Referring to the equation to be solved, therefore, we see 
that ( — Z) is to be divided into five pieces which shall be in the 



100 GEODETIC SURVEYING 

ratio of the numbers — , Ci^, C-^, C^, C^, giving for the succes- 
sive terms of the equation the values 



, etc. 



Hence, writing S = ™ , we have 

^ + Ci2 + C22 + (73^ + C42 
Q2 (J 

Cx = -rS, whence x = --rS; 
4 4 ' 

Cixi = Ci^S, " xi = Ci;S; 

C2X2 = C2^S, " X2 = C2S; 

C3X3 = Cs^S, " X3 = C3S; 

dxi = Ci^S, " Xi = CiS. 

Combining these values of x, xi, X2, etc., to form Va, v^, etc., and 
applying these corrections to A, B, etc., we obtain the most 
probable values of the angles a, b, etc., consistent with the geo- 
metrical necessities of the figure and with the fact that all the 
angles were measured with equal care. 

A complete example of adjustment by this method is worked 
out in the table on page 101, using the same quadrilateral 
that was adjusted by the approximate method (pages 95 and 96) in 
order to compare results. It wiU be noted that the first approxi- 
mate adjustment has a maximum variation of only 0".42 from 
the rigorous adjustment, and that the second approximation 
comes within 0".02 of the rigorous values. 

61. Weighted Adjustments and Larger Systems. If the 
measured angles of a triangle have different weights, the adjust- 
ment is made as already explained. If the measured angles of 
a quadrilateral or other figure are not of equal weight, the adjust- 
ment is best made by the method of least squares. 



TEIANGULATION ADJUSTMENTS AND COMPUTATIONS 101 



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102 GEODETIC SURVEYING 

In work of moderate extent or importance a system composed 
of a series of triangles or quadrilaterals woiild have each triangle 
or quadrilateral independently adjusted. In work of the highest 
importance, such as primary triangulation, the entire system 
would be adjusted simultaneously by the method of least squares. 

62. Computing the Lines of the System. After a figure or 
system is satisfactorily adjusted the distances between the various 
stations are computed, solving each triangle in order (as a plane 
triangle) by the ordinary sine ratio. In the case of the quadri- 
lateral the two diagonals and the sides adjacent to the known 
side (called the base) are computed from the triangles involving 
the base; the side opposite the base is then computed from both 
the triangles in which it occurs, and the mean of the two results 
taken as its value. These two values would of course be exactly 
alike if the angle adjustments were perfect, but these adjust- 
ments are only correct as far as they are carried out decimally; 
a material disagreement in the two values would indicate errors 
in the computations. 

63. Accuracy of Triangulation Work. The accuracy of this 
class of work is judged by measuring a check base at the end of 
the system, if the work is of moderate extent, with intermediate 
check bases if the work covers a large territor5^ The length of 
the check base as computed through the triangulation system 
should agree closely with its measured length. In triangulation 
work by the U. S. Coast and Geodetic Survc}-, extending over 
several states in one system, extremely close results are reached. 
In systems 600 to 800 miles in length the computed and measured 
values of check bases may agree within fractions of an inch. 



CHAPTER V 

COMPUTING THE GEODETIC POSITIONS 

64. The Problem. After the triangulation system has been 
computed as described in the last chapter the relative positions' 
of the various stations are known. By computing the geodetic 
positions is meant computing the absolute positions (latitudes 
and longitudes) of the triangulation stations from their relative 
positions; this computation can be made if we have the latitude 
and longitude of one of the stations and the azimuth of one of 
the lines through that station, provided we know the shape and 
dimensions of the earth. The jDroblem, then, may be stated 
as follows: Given the latitude and longitude of a station and the 
azimuth and distance to another station, to find the latitude and 
longitude and the back azimuth at the second station. This 
problem is often called the L. M. Z. problem, the letters meaning 
latitude, longitude (meridian), and azimuth. The back azimuth 
at the second station will seldom be the same as the forward 
azimuth at the first station, on account of the convergence of 
the meridians. Having found the latitude, longitude, and back 
azimuth at the second station, the azimuths of the other lines 
at that station become known through the adjusted angles at 
that station, remembering that azimuths are counted clockwise 
from the south point continuously up to 360°, and that if the 
spherical excess has been- removed from any angle it must be 
restored for the present purpose. By proceeding with the com- 
putations in the same manner from station to station we obtain 
the latitudes, longitudes, and azimuths for the whole system. 
There are many methods of solving the given problem, depending 
on the distance involved and the precision required; all methods 
are somewhat complicated on account of the shape of the earth. 
Two of the best solutions will be considered after discussing the 
figure of the earth. 

103 



104 GEODETIC SURVEYING 

65. The Figure of the Earth. It is doubtful when it was first 
realized that the surface of the earth is not a plane. Early 
Greek philosophers believed in solid figures of various shapes. 
Aristotle (340 B.C.) gives reasons for believing the shape to be 
spherical, geometers estimating the circumference at 300,000 
stadia. The famous School of Alexandria appears to have made 
the first actual measurements of the curvature of the earth, 
and hence its radius, the earliest measurement being made by 
Erastosthenes, about the year 230 B.C., and a second one a little 
later. Erastosthenes concluded that the circumference of the 
earth was about 250,000 stadia in length, but the exact length of 
the stadium is now unlmown. The knowledge which the Greeks 
obtained of the size and shape of the earth was lost during the 
declining civilization that followed, and no further measurements 
were made for upwards of a thousand years. About the year 825 
the Arabs made a very good determination of the radius of the 
earth by measuring the arc of a meridian on the plains of Mesopo- 
tamia. This was followed bj- another lapse of about 700 years 
before any further measurements were undertaken. During 
the middle ages Europeans generally believed the earth to be 
flat until about the 15th century, when a few men, such as Colum- 
bus, declared it to be globular. In the 16th century general 
belief in the spherical shape of the earth was again established. 

From the earliest measurements to the present time the 
principle emploj-ed has been essentially the same, but a very 
much higher degree of accuracy is now reached on account of the 
great refinement in detail. The fundamental idea is to obtain 
both the linear and the angiolar measure of the arc of a meridian, 
whence the distance divided by the number of degrees gives the 
length of one degree, and this multiplied by 360 gives the length 
of the entire circumference. In early times the meridian arc 
was actually staked out and its length obtained by direct meas- 
urement, but modern methods of measuring and computing are 
so improved that distances measured in any direction may be 
utilized. The angular measure of the arc is the angle between 
its two end radii (which meet near the center of the earth), and 
its value is obtained by finding the latitude at each end and 
taking their difference. 

When Newton discovered the law of gravitation late in the 
17th century he proved that the earth as a revolving plastic body 



COMPUTING THE aEODETIC POSITIONS 105 

subject to its own attraction should have taken the form of a 
slightly flattened sphere, while an arc measured in France between 
1683 and 1716 indicated an elongated sphere. To settle the 
question an arc was measured in the equatorial regions of Peru 
(1735-1741) and another in the polar regions of Lapland (1736- 
1737), which showed that a degree of latitude was longer near 
the pole than near the equator and that Newton's theory was 
correct. Since these dates a large amount of geodetic work has 
been done, in which France, Great Britain, Germany, Russia, 
and the United States have taken a leading part. Among the 
more recent arcs measured may be mentioned the Anglo-French 
arc, extending from the northern part of the British Isles south- 
ward into Africa; the great Russian arc, extending from the 
Arctic Ocean to the northern boundary of Turkey; the great Indian 
arc, extending from the southern point of India to the Himalayas; 
the European arc of a parallel, extending from southern Ireland 
eastward to central Russia; and in the United States, the trans- 
continental arc, extending along the 39th parallel from the 
Atlantic Ocean to the Pacific Ocean, and the eastern oblique arc, 
extending parallel to the Atlantic coast from Maine to Louisiana. 
These six arcs joined end to end would reach about two-fifths 
of the way around the earth. 

66. The Precise Figure. Various names have been applied 
to the earth from time to time in the attempt to describe its 
shape more exactly as our knowledge has advanced. Roughly 
it may be called a sphere, since the flattening at the poles is rela- 
tively very small; a model with an equatorial diameter of fifty 
feet would only be flattened one inch at each pole. As the result 
of many precise measurements the shape has been found to be 
such that with considerable exactness any section parallel to the 
equator is a circle, and any section through the poles is an ellipse; 
the flgure is such as may be generated by revolving an ellipse 
about its minor axis and is called an oblate spheroid. To be 
still more exact, the equatorial section is not exactly circular 
but very slightly elliptical, so that a section in any direction 
through the center would be an ellipse; such a figure is called 
an ellipsoid. Still further exactness indicates that the southern 
hemisphere is a trifle larger than the northern, and that all polar 
sections are therefore slightly oval, leading to the name ovaloid. 
As a matter of absolute precision no geometrical solid exactly 



106 GEODETIC SURVEYING 

represents the shape of the earth, and this has been recognized 
by applying the special name geoid. 

67. The Practical Figure. AH the computations in geodetic 
work are based on the assumption that the figure of the earth 
is an oblate spheroid; this is found to be amply precise, since the 
variations from this figure are relatively very small. The most 
important determinations of the elements of the spheroid, 
founded on the best available data, are those made by Bessel in 
1841 and Clarke in 1866. Bessel's spheroid is still largelj' used 
in Europe, but all computations in the United States are made on 
the basis of Clarke's spheroid, which conforms better to the actual 
surface of this country. According to Clarke's comparison of 
standards a meter contained 3.2808693 feet, a result which is now 
laiown to be too large. In the legal units of the United States 
the meter contains exactly 39.37 inches, which equals 3.2808333 
feet, a value which is believed to be very close to the exact truth. 
The elements of Clarke's spheroid in U. S. legal units are as 
follows : 



Semi-major axis = a = 



Semi-minor axis = b = ■ 



6,378,276.5 meters, log = 6.8047033 
20,926,062 feet, log = 7.3206875 

6,356,653.7 meters, log = 6.8032285 
20,855,121 feet, log = 7.3192127 

EUipticity = ^— ^ = e = 0.00339007, log = 7.5302093 - 10 



Eccentricity =J ^— = e =0.08227184, log = 8.9152513- 



10 



a2 — 7,2 
Eccentricity2 = ~ — = e2 = 0.0067686580, log = 7.8305026- 10 

Ratio of axes =~ = |||^, log = 9.9985252 - 10 

68. Geometrical Considerations. In Fig. 33 the" ellipse 
WNES represents a polar section of the earth, in which WNES 
is the meridian; NS, the polar axis, or minor axis of the ellipse; 
WE, the equatorial diameter, or major axis of the ellipse; n, 
any point on the meridian; nt, the tangent at n; nlpm, the normal 
at n, or the direction of the plumb line if there is no local deflection ■ 



COMPUTING THE GEODETIC POSITIONS 



107 



np, the radius of curvature at n; no, the radius of the small circle 
or parallel of latitude at n; f, f, the foci of the ellipse; (p, the 
latitude of the point n. It is to be noted that the normal nm 
from the point n does not pass through the center c (except when 
n is at the poles or on the equator), and that the radius of curva- 
ture (and hence the length of a degree of latitude) increases from 
the equator to the poles; that the radii of curvature for different 




Fig. 33. 



latitudes on a meridian do not intersect unless produced; and 
that for different latitudes not on the same meridian the normals 
(which include the radii of curvature) do not intersect at all. 

Since the normals for two points of different latitudes and 
longitudes do not intersect, they do not lie in a plane; hence, 
Fig. 34, page 108, the vertical plane at A(AaB) which includes 
B and the line of sight from A to B, is not the same as the vertical 



108 



GEODETIC SURVEYING 



plane at B {BbA) which includes A and the line of sight from BtoA. 
The lines which these planes cut at the surface of the spheroid 
are called elliptic arcs. In setting points from A to 5 an observer 
at A would mark out the line AaB, while an observer at B would 
mark out the line BbA; the greatest discrepancy between the 
lines would be practically at the center, and under extreme con- 
ditions might amount to about an inch for 50 mile lines and 10 
feet for 500 mile lines; the angles bAa and bBa might approx- 
imate 0".l for 50 mile lines and 2".0 for 500 mile lines. For 
lines 100 miles or so long, therefore, it is evident that the two 
elliptic arcs may usually be regarded as identical, but that for 
greater distances the question may often be of considerable 




Fig. 34. 



importance. If an observer should set up his instrument at any 
intermediate point on either elliptic arc he would not find himself 
in line with A and B; if he sighted on A, for instance, he could not 
sight on B by simply transiting his telescope, as the angle between 
A and B would not measure 180°. An alignment curve (as repre- 
sented by the dotted line CD, Fig. 34) is such a line that at any 
intermediate point a vertical plane can be established that will 
pass through both end stations; as seen from any intermediate 
point the two end stations are always 180° apart; such a line is 
a line of double curvature, slightly less in length than the elliptic 
arcs between which it lies, and tangent to the line of sight at each 



COMPUTING THE GEODETIC POSITIONS 109 

end. A geodesic line is the shortest line that can be drawn between 
two points on a spheroid, and is a line of double curvature resem- 
bling the alignment curve, but the reverse curvature is not so 
pronounced. Between any two points on the earth that are 
actually intervisible all the lines described may be regarded as of 
equal length. 

In geodetic work the term latitude always refers to the angle 
(j) (Fig. 33, page 107) or geodetic latitude, and not to the angle ncd 
or geocentric latitude. The astronomical latitude, or angular 
distance from the equator to the zenith, is the same as the geodetic 
latitude except where there is local deflection of the plumb line. 
By longitude is meant the angular distance from some fixed meridian 
(usually Greenwich) to the given meridian, positive when coimted 
westward. By the azimuth of a line (or a direction) from a given 
point is meant its angular divergence from the meridian at that 
point, counted clockwise from the south continuously up to 360°. 
Thus in Fig. 34, the angle DAa is the azimuth at A towards B 
(AaB being the line of sight from A), and the angle SBb (clock- 
wise as marked) is the azimuth from B towards A. The azimuth 
(or forward azimuth) of a line means taken forward along the 
line, and back azimuth means in the reverse direction; the 
azimuth and back azimuth at the same point differ by 180°. 
The angles NAa and NBb, inside the two polar triangles NAB, 
are called azimuthal angles, the angle at each station being taken 
to the line of sight from that station; the relation of these angles 
to the azimuth above described is self evident. In solving either 
of the triangles NAB the angles at both A and B must be taken 
in the same triangle, the necessary reduction being made by 
means of the auxiliary angles bBa and hAa. 

69. Analytical Considerations. The most important section 
of the spheroid is the meridian section, Fig. 33, page 107, of which 
A^^ and R are the principal functions. 

Let N = the normal nm; 

R = radius of curvature np; 
r = radius no of parallel of latitude; 
T = tangent nt; 
(f> = latitude (geodetic) ; 
/? = geocentric latitude; 
p = radius vector nxi; 



no 



GEODETIC SURVEYING 



then from analytical geometry 
a 



N = 



(1 — e^sm^(j))^' 

&2 



R = 



R at equator 

r = N cos ^, 

nl = Nil - e2), 

&2 
tan /? = -2 tan 0, 



a(l -e2) 
(l-e2sin2 0)8' 

2 



i? at poles = 






r = iV cot (}), 
nd = A'^Cl — e2) sin ^, 

jO = a(l — e2 sin2 /?)*, 



VRN = radius of osculatins; sphere at n = :; „ . „ , , 

1 — e^ sin^ 0' 

in which the logarithms of the constants are as follows: 



Quantity. 


Metric. 




Feet. 




a 


6.8047033 




7.3206875 




b 


6.8032285 




7.3192127 




e2 


7.8305026 - 


-10 


7.8305026- 


-10 


(1 - e2) 


9.9970504- 


- 10 


9.9970504- 


- 10 


a(l - e2) 


6.8017537 




7.3177379 




aVl - e2 = 6 


6.8032285 




7.3192127 




- = o(l - e2) 
a 


6.8017537 




7.3177379 




a2 a 


6.8061781 




7.3221623 




b Vl - e2 




? = .-^ 


9.9970504- 


-10 


9.9970504 - 


-10 



The section of next importance at any point, after the merid- 
ian section, is that which is cut from the spheroid by the 
prime vertical, which is the vertical plane at the given point 
that is perpendicular to the meridian through that point. The 
ellipse that is thus cut from the spheroid is tangent to the 
parallel of latitude through the given point, and hence a straight 
line run east or west from any point is commonly called a tangent. 
The radius of curvature, Rp, of a prime-vertical section at the 
point where it originates has the same length as the normal 
N at that point, that is. 



Rr 



:N = 



(1 -e2sin2</))i" 



COMPUTING THE GEODETIC POSITIONS 111 

A vertical plane at a given point that is neither a meridional 
plane nor a prime-vertical plane, is called an azimuth plane; such 
a plane cuts an azimuth section from the spheroid and traces 
an azimuth line on its surface, that is, a straight line whose initial 
direction is not at right angles to the meridian. All the prop- 
erties of an azimuth section may be deduced from those of the 
prime-vertical and meridional 
sections. Thus, for instance. 

Let « = azimuth of azimuth 
line at initial point; 
N = normal at same point ; 
R = radius of curvature 
of meridian section 
at same point; 
i2„ = radius of curvature 
of azimuth section 
at same point; 
then 



R(x~ 



R 



l + ^^tan^a 




70. Convergence of the 
Meridians. On account of the 
convergence of the meridians 
the azimuth of a line varies 
from point to point, unless 
the given line be the equator 
or a meridian. By the con- 
vergence of the meridia7is is 
meant their angular drawing 
towards each other in passing pj^ 35 

from the equator to the poles. 

Any two meridians are parallel at the equator or have a zero 
convergence (meaning no inclination towards each other); 
in moving towards the poles the meridians incline more and 
more towards each other, until at the ■ poles the convergence 
is just equal to the difference of longitude. Referring to Fig. 35, 
the convergence at any two points, n, n', which are in the 
same latitude ^1, is found by drawing tangents from n and n' 



112 GEODETIC SURVEYING 

to their intersection t on the polar axis, in which case the angle 
6 is the convergence for those two meridians for the common 
latitude <f>i. When the two points P and P' are not in the same 
latitude the convergence for the middle (average) latitude is 
understood; so that if ^ and (j)' represent the latitudes of the 
two points we may write in any case ^i = i{f + 4-'), and n 
and n' represent points on the middle parallel of latitude. 

Let 4>\ = the common latitude for the points n and n' (or the 
average latitude for any two latitudes 4> and ^') ; 

^A = difference of longitude for the two meridians; 

no = r = radius of parallel of latitude at n; 

nt = T = tangent at n. 

From the figure 

Chord nn' = 2Tsmid--= 2rsini(iA). 

Substituting r = T sm4>i, 

2T sin id = 2T sin (f>i sin 4(JA), 
or 

sin id = sin i(^A) sin </-'i, 

which in terms of the latitudes ^ and <pf, becomes 

sin id = sin i(iA) sin i{4> + ^'). 

When the difference of longitude, ^^, is small, 6 will also be small, 
and we may write with great closeness 

= (JA) sin i(<l> + <//), 

in which 6 will be in the same unit as ^^ (usually taken in minutes 
or seconds). Thus in an average latitude of 40° and a difference 
of longitude of one degree, or about 60 miles, the error of the 
approximation would be less than the one thousandth part of 
a second. 

Referring to Fig. 36, let rr' be a straight line in the plane 
stv, and passing as close as possible to the points P and P'. In 
any case occurring in practice the angle rpv will differ but very 
little from the forward azimuth at P of a true geodetic line from 
P through P', and the angle rp's will closely represent the corre- 



COMPUTING THE GEODETIC POSITIONS 



113 



spending forward azimuth at P'. We may therefore write with 
great closeness 

Change of azimutfi = rp's — rpv. 

But from the figure 

6 = rpv — rp's, 



or 



Change of azimuth = — = — {AX) sin h{<t' + ^')- 



Hence, in passing from one station to another, the change of 
azimuth is very closely the same in numerical value as the corre- 
sponding convergence of the 
meridians. The error in the 
approximation in running 60 
miles in any direction in the 
neighborhood pf 40° latitude 
would not exceed one tenth of 
a second. In the northern 
hemisphere the azimuth of a 
line decreases in running west- 
ward, and increases in running 
eastward, and vice versa in 
the southern hemisphere. • The 
minus signs in the last formula 
must therefore be changed to 
plus in the southern hemi- 
sphere. In running approxi- 
mately east and west in about 
40° latitude the change of azi- 
muth will be over half a minute per mile. The back azimuth 
of a line is equal to the forward azimuth at the same point 
plus 180° (less 360° if this number is exceeded). 

71. The Puissant Solution. Given the latitude and longitude 
of a station and the azimuth and distance to a second station, 
the problem (Art. 64) is to find the latitude, longitude, and 
back azimuth at the second station. The Puissant solution 
(as modified by the U. S. C. & G. S.) is found amply precise 
if the distance between the stations does not exceed about 1° 
of arc or about 69 miles (in which case the errors of the com- 
puted values might run from 0.001 to 0.003 seconds). For a 




Fig. 36. 



114 GEODETIC SURVEYING 

less degree of accuracy the method may be used up to about 
iOO miles. The Puissant method has the advantage that only 
seven place logarithms are required. With the help of special 
tables for certain factors in the formulas the actual work of 
computation is not very great. For a derivation of the for- 
mulas, examples of their use, and a complete set of tables, see 
Appendix No .9, Report for 1894, U .S. Coast and Geodetic Survey. 
These formulas (in slightly different form) are as follows: 
Let ^ = the known latitude at the first station; 
A = the known longitude at the first station; 
a = the known azimuth at the first station; 
4>' = the unknown latitude at the second station; 
A' = the unknown longitude at the second station; 
oi' = the unknown back azimuth at the second 
station; 
s = the known distance between the stations; 
A, 5, etc., = certain factors required in the formulas; 

then by successive steps we have 

' - h = s cos a. . B, 

- d<j>= ^ {h + s^sin^a.C f- /i.s^sin a-E), 

or with ample precision 

S4> (for 15 miles or less) = — {h + s^ sin^ a-C). 

In either case 



and 



and 



^' - (f> + Jcj) = latitude of second station; 
V cos / 



i^' = ^ + J /< = longitude of second station ; 

or with ample precision 

'ia (for 15 miles or less) = — (-/O sin i{<p + (/>'), 



COMPUTING- THE GEODETIC POSITIONS 115 

which agrees with the result of Art. 70. The sign of ^a is for 
the northern hemisphere, and is to be reversed in the southern 
hemisphere. Then 

«' = c: + Jn: + 180° = back azimuth at second station. 

In the above formulas the values of ^i^, '^^, and ^a are obtained 
in seconds. In using the formulas both north and south latitude 
are to be taken as positive, west longitude as positive and east 
longitude as negative, and the trigonometric functions are to be 
given their proper signs. The lettered factors of the formulas 
have the following values : 

A = A'il - e^ sin2 ^')i, D = D'(f^^), 

^ \1— e^ sm^ (f>/ 

B =£'(l-e2sin2 96)?, E =E'{1 + 3 ta-n^ 4>) (1 - e^ sin^ <p) , 

C = C'(l - e2 sin2 ^)2 tan <f>, F = F' (sin cos^ 4>), 

G = value determined by second part of Table II, 

in which the logarithms of the constants are as follows: 

Metric. Feet. 

8.5097218 - 10 7.9937376 - 10 

8.5126714 - 10 7.9966872 - 10 

1.4069381 - 10 0.3749697 - 10 

2.6921687 - 20 2.6921687 - 20 

5.6124421 - 20 4.5804737 - 20 

8.2919684 - 20 8.2919684 - 20 

With the help of these constants it is not difficult to find 
the values of the factors A to F for any latitude. If the distance 
s is given in meters these factors may be taken from Table II, 
at the end of the book, this table being an abridgment of the 





Constant. 


A' 


1 


a arc 1" 


B' 


1 


a(l - e2) arc 1" 


C 


1 


2a2(l-e2)arcl" 


D' 


= |e2arcl" 


E'- 


1 
6a2 


F' 


= ±arc2 1" 



116 GEODETIC SURVEYING 

Coast Survey tables referred to (and corrected to agree with 
the U. S. legal meter of 39.37 inches). 

72. The Clarke Solution. This solution of the problem (Art. 
64) is adapted to greater distances than the previous one, being 
sufficiently precise for the longest lines (say about 300 miles) that 
could ever be directly observed. It has the advantage of being 
reasonably convenient in use, even without specially prepared 
tables, but requires not less than nine place logarithms for close 
work, on account of the size of the numbers involved. In this 
method the azimuthal angles are used in the computations 
instead of the azimuths themselves. The azimuthal angles 
(shown in Fig. 34, page 108, and explained at end of Art. 68), 
are the angles (at the stations) inside the polar triangles which 
are formed by the nearest pole and the two stations, the relation 
to the corresponding azimuths being always self-evident. The 
formulas used in this solution (taken from Appendix No. 9, 
Report for 1885, U. S. Coast and Geodetic Survey, but modi- 
fied in form) are as follows: 

Let 4> = the known latitude at the first station; 

X = the known longitude at the first station; 
a = the known azimuthal angle at the first station; 
j)' = the unknown latitude at the second station ; 
X' = the unknown longitude at the second station; 
a' = the imknown azimuthal angle at the second station; 

s = the known distance between the stations; 

6 = the angle between terminal normals; 

X, = auxiliary azimuthal angle at second station; 
AX = X' — k = difference of longitude; 
J<p = (j)' — ^ = difference of latitude; 

Y = 90° — <p = colatitude at first station; 
N = normal (to minor axis) at first station ; 
R = radius of curvature of meridian at middle latitude; 

i{4> + (j)') = middle latitude. 

From Art. 69, 

iV= " R - "(1 -«^) 

(1 - e2 sin2 <P)i' [1 - e2 sin2 i(<l> + <j>')]r 

Then 

^ = M ^ Ml + ( Lf"^ 2\ r^ ^'^^^ 9^ ^=082 a. 
N sm 1 \ 6(1 — e'') / 



COMPUTING THE GEODETIC POSITIONS 117 

But if s is not over about 100 miles we may write with ample 
precision 



N sin 1"" 



In either case s and N must be in the same unit, and 6 is obtained 
in seconds. If the second term is used in finding d the approx- 
imate value of 6 is used in that term. The value of this second 
term is always extremely small. Then 

p Sin 1 

' 6^ 008^ (p sin 2a, 



4(1 -e2) 

in which ^^ is obtained in seconds and is always a very small 
quantity; 

tan P=^-^^! cot J 

, „ cosKr-^) .a 

tan Q = . / , „, cot -^, 

cos^ (7- + d) 2' 

from which values 

a' = P + Q — ^ = azimuthal angle at second station; 

Ak=Q - P; 

X' = X + AX = longitude at second station. 

The difference of latitude is found from the formula 

f'sin i{a' + ^— a)' 



J^ = 



i?sinl"Vsini(a:'+ ^ + a) 



/sinM: 
^\ 12 



-)^2cos2J(a'-a)l 



in which ^^ is obtained in seconds, and in which s and R must 
be in the same unit. Then 

<j)' = (p + J<f) = latitude at second station. 

It must be noted, however, that the Acj) formula requires the 
use of R for the middle latitude, which is not known until Acj) 
is found. A(f> must therefore be found by successive approximation 
— that is, an approximate value of R must first be used to obtain 
an approximate value of ^0, a greatly improved value of R thus 
becoming available to find a much closer value of 4^, and so on. 



118 



GEODETIC SURVEYING 



A few trials will soon give a value of R which is consistent with the 
value of <p' to which it leads. As with the Puissant formulas, 
both north and south latitude are to be taken as positive, west 
longitude as positive and east longitude as negative, and trig- 
onometric functions used with their proper signs. The constants 
which enter into the above formulas have the following values: 



Quantity. 
a (metric) 
a (feet) 
e2 
e^ sin2 1" 
6(1 -e2) 
e^ sin 1" 

4(1-62) 



Log. 
6.8047033 
7.3206875 
7.8305026 - 10 

6.4264506 - 20 
1.9169671-10 



Quantity. 
a(l — e^) (metric) 
a(l - e2) (feet) 
(1 - e2) 

sin 1" 

sin2 1" 



12 



Log. 
6.8017537 
7.3177379 
9.9970504-10 

4.6855749-10 
8.2919684-20 



When the distance is so great jthat the Clarke solution is not 
satisfactory, resort must be had to more direct solutions, requiring 
at least ten place logarithms. The solutions by Bessel (1826) 
and Helmert (1880) are of this character. 

73. The Inverse Problem. In this case the latitude and 
longitude are known at each of two stations, and the problem 
is to find the coimecting distance and the mutual azimuths. 
The solution may be effected with either the Puissant or the 
Clarke formulas. 

By the Puissant Formulas. There are several ways of 
securing the desired result; the one here given is chosen on 
account of its directness and simplicity. By transforming and 
combining the formulas in Art. 71, omitting terms which are 
too small to be appreciable, and writing x and y for the resulting 
values, we have 



s sm a= y = 



(J/<) cos 0' 



scosa = x=-^[J<p + C-y^ + D{A4>)^ + E{A<}>)y^ + E-C-y% 



from which we obtain 



tan a = — and 

X 



y 



X 



sm a cos a 



COMPUTING THE GEODETIC POSITIONS 



119 



The closest value of s is obtained from the fraction whose numer- 
ator is the smallest. Then, from Art. 71, 

1 



za = - r 



{JX)smi(<l>+4>') 



{AX)^-F 



cosi(J^) 

Aa (for 15 miles or less) = —{AX) sin i(^+^'); 

and in either casi^ 

a' = a + Aa + 180°- 

Either station may be called the first station, so that the problem 
may be worked both ways as a check, if desired, in which case 
Aa need not be computed at all. As in Art. 71, the values 
Aa^ J(j)^ and AX are expressed in seconds, and s will be in the 
same xmit as that on which the factors A, B, etc., are based. 

By the Clarke Formulas. In this method the desired values 
are found by successive approximation. The Puissant method 




Fig. 37. 

is applied first, therefore, to obtain as close an approximation 
as possible to begin with. The approximate values of s and a 
(changed to the azimuthal angle) are then substituted in the 
Clarke formulas, calling either station the first station, and com- 
puting the latitude and longitude- for the second station. The 
computed values will usually disagree a small amount with the 
known latitude and longitude of the second station, and a new 
trial has to be made with s and a slightly changed, and so on 
until the assimied values of s and a satisfy the known con- 
ditions. The disagreement to be adjusted is always very small, 
and when all the circumstances are known it is not difficult to 



120 GEODETIC SUEVEYINQ 

judge which way and how much to modify s and a to remove 
the difficulty. Referring to Fig. 37, let the lines NS represent 
meridians, the line CB a parallel of latitude, and A and B the 
points whose latitude and longitude are known. With the assumed 
distance s and the assumed azimuthal angle a suppose, for 
instance, that the computation gives us the point B' instead 
of the desired point B. We then have 

BC = error in longitude in seconds of arc; 
B'C = error in latitude in seconds of arc ; 



BB' (in seconds) = V BC^ + B'C^; 

b = BB' in distance = {BB')R sin 1" (approximately) ; 

tanC5'5 = |g; 

BB'D = 180° - a' - CB'B; 

b cos BB'D = B'D = approximate error in the assumed value 
for distance s; 

: — -71— = BAD (nearly) in seconds = approximate error in 

s sm 1" 

assumed value of angle a. 

74. Locating a Parallel of Latitude. For marking bound- 
aries, or other purposes, it often becomes desirable to stake out 
a parallel of latitude directly on the ground. Points on the 
parallel are most conveniently found by offsets from a tangent 
(Art. 69). Thus in Fig. 38, ABD is a tangent from the point 
A, and ACF is the corresponding parallel; the point C on the 
parallel, for instance, is determined by the offset BC and the 
back-azimuth angle SB A. It is seldom desirable to run a tangent 
over 50 miles on account of the long offsets required ; if the parallel 
is of greater length it is better to start new tangents occasionally. 
The computations may be made by either the Puissant (Art. 
71), or tlie Clarke (Art. 72) formulas, which are much simplified 
by the east and west azimuths. Using the Puissant formulas, 
substituting 90° (westward) or 270° (eastward) for a, and omitting 



COMPUTING THE GEODETIC POSITIONS 121 

inappreciable terms, we have with great precision for a hundred 
miles or more 

J4> (in seconds) = — s'^-C, 

n (in seconds) = f '^''''^^ ^ + 1 -^77 ; 
[ running E, — J cos <j)' 

whence 

J4> (in linear units) = — (s^-C) R sin 1" = om > 




in which either formula may be used as preferred, and in 'which 
all linear quantities must be taken in the same unit. The 
expressions for N and R are given in Art. 69. For the change 
of azimuth we have 



Ja (in seconds) = 



■^;'^"'!f'^";;}[(i^)sini(^+^')+(i^)^-i^]; 

or for the field work (within one-tenth of a second), 

J r JN f N. hemisphere, -|.,,^ . ,, 

Aa (m seconcjs) = \a u 1 | ('^'^) ^^"^ ^9- 

It is seen from the above formulas that the offsets (in seconds 
or linear units) may be taken to vary directly as the square of 
the distance, and the change of azimuth directly as the change 
of longitude. 

In actual practice the point A may have to be located, or 
may be given by description or monument; in either case the 
latitude and meridian at A are determined by astronomical 



122 GEODETIC SURVEYING 

observations, and the tangent AB (or a line parallel thereto) 
run out by the ordinary method of double centering. At the 
end of the tangent the computed value of the back azimuth 
should be compared with an astronomical determination; in 
the writer's experience on the Mexican Boundary Survey with 
an 8-inch repeating instrument (with striding level), and heliotrope 
sights ranging in length from 6 to 80 miles, the back-azimuth 
error was readily kept below one-tenth of a second per mile, 
regardless of the number of prolongations in the line. The 
conditions met with in the survey referred to are illustrated 
in Fig. 39, which shows also the adjustment made for back- 
azimuth error. The boundary line was intended to be the parallel 
of 31° 47', but according to treaty all existing monuments had 
to be accepted as marking the true line. The astronomical 
station was conveniently located, and proved to be slightly south 
of the desired parallel, which in turn passed south of the old 
monument L. When the last point on the tangent was reached 
the back azimuth measured less than the theoretical value, 
indicating that the tangent as staked out swerved slightly to 
the south from its original direction. Assuming all corresponding 
distances on tangents and parallels to be equal and the azimuth 
error to accumulate uniformly from A to d, 

Let E = azimuth error at d; 
Eh — azimuth error at &; 



then 



Eb = ^,E; dD = ^E sinl"; bB = i^Esml 



If . 



Ad ' 2 ' 2Ad 

DF = A(j> (linear) for AD; BC = J<j) (linear) for AB; 
dM and AL are known by measurement ; 
FG =CH = AL; 
GM = dM - dD - DF - AL; 

HP==GM^~=GM^^. 
LG Ad 

Hence for any point P, no the adjusted boundary, we have 

bP =- bB + BC + CH + HP. 



OOMPTJTINa THE GEODETIC POSITIONS 



123 




124 GEODETIC SURVEYING 

75. Deviation of the Plumb Line. There is always more or 
less uncertainty at any station as to the plumb line hanging 
truly vertical, or normal to the surface of the spheroid; it is 
not uncommon for the deviation to amount to 10 or more seconds 
of arc, with occasional values of 15 to 30 seconds. This fact 
is forced on our notice in a number of ways; if, for instance, 
the computed latitudes and longitudes of the stations in a triangu- 
lation system are tested by astronomical observations, the dis- 
crepancies are often greater than can be charged to either 
determination; if a parallel of latitude is staked out and tested 
astronomically at different points, the same discrepancies appear. 
By a proper combination of geodetic and astronomical measure- 
ments involving a num'ber of stations, the probable deviation 
at each station and the probable errors in the latitude and long- 
itude determinations can be computed. Astronomical and 
computed azimuths disagree for the same reason, and require 
similar adjustment. In moderate sized trig-ngulation systems, 
such as are likely to engage the attention of the civil engineer, 
adjustments of this kind are rarely called for; but in extended 
systems astronomical latitudes, longitudes, and azimuths are 
taken at many stations, in order that such adjustments may be 
made. 



CHAPTER VI 

GEODETIC LEVELING 

76. Principles and Methods. Leveling is the operation of 
determining the relative elevations of different points on the 
surface of the earth. By relative elevation is meant the difference 
of elevation between any two points compared. The absolute 
elevation of a point is its elevation above some particular point 
or surface of reference, mean low water, for instance; in geodetic 
work elevations are commonly referred to mean sea level. A 
level line is a line having the same absolute elevation at every point. 
By geodetic leveling is meant that class of leveling in which extra 
precision is sought by refinement of instruments and methods. 

Three principal methods are available for determining dif- 
ferences of elevation, (A) Barometric Leveling, (B) Trigonometric 
Leveling, (C) Precise Spirit Leveling. Barometric leveling, based on 
determinations of atmospheric pressure, is briefly treated below 
on account of its usefulness in reconnaissance work. Geodetic 
leveHng is generally understood to mean either trigonometric 
leveling, based on vertical angles (corrected for curvature and 
refraction), or precise spirit leveling, which differs from ordinary 
spirit leveling only in the refinement of its details. 

77. Determination of Mean Sea Level. By mean sea level is 
meant the average elevation of the surface of the sea due to 
its continuous change of level; and not, as might be supposed, 
the mean elevation of its high and low waters. In order to 
average out the irregularities due to winds and other causes 
the observations at any point should extend over a period of 
several years. Further, since tidal variations are relatively large 
during a lunar month, only complete lunations can be allowed 
in the reductions; if any storm period, for instance, is rejected 
on account of its excessive irregularities, that entire lunation 
must be rejected. 

Observations of the varying elevation of the surface of the 
sea are best made by means of automatic tide gauges. An 

125 



126 GEODETIC SURVEYING 

automatic or self-registering tide gauge consists essentially of 
a well made clock and attached mechanism, by which a sheet 
of paper is drawn continuously past a pencil point which is moved 
crosswise of the paper by connection with a float; a rising and 
falling curve is thus traced on the paper, in which the ordinate 
of any point shows the elevation of the water at the time indi- 
cated by the corresponding abscissa. The float moves up and 
down in a vertical box admitting water only through a small 
opening in the bottom, which practically prevents oscillation 
of the float by wave action. A catgut cord or fine wire connects 
the float with the pencil through a suitable reducing mechanism. 
Pin points are often arranged to prick the even hours on the 
paper. The clock is often designed to run a week without 
rewinding, and the paper to last a month without changing. 
A scale of one inch per foot and f of an inch per hour makes 
a very good record. 

A staff tide gauge is always placed as near as possible to the 
automatic gauge, and its zero point connected by accurate 
leveling with a permanent bench mark near by. At least once 
a week the attendant carefully raises and lowers the float so that 
the pencil of the automatic gauge will mark the true direction 
of the ordinates at that time; and near the ordinate thus made 
he records the date, the staff reading, and the clock reading and 
error. The attendant's visits should be so timed that his staff 
readings will be alternately near high and low water, thus fur- 
nishing scales for different parts of the sheet that will practically 
neutralize errors due to stretching or shrinking of the paper or 
float connections. Hourly ordinates are drawn on all the records 
obtained at a station, and the average value of these ordinates 
is taken as the staff reading of mean sea level at that station. 
The relation of the permanent bench mark to the zero of the 
staff having been determined, as previously described, the ele- 
vation of the bench mark with reference to mean sea level becomes 
known, and furnishes the basis of the precise level lines that 
are extended to inland points. 

A. Barometric Leveling 

78. Instruments and Methods. The instruments available 
are the familiar types of aneroid and mercurial barometers. 



GEODETIC LEVELING 127 

The mercurial barometer is the standard instrument for indicating 
atmospheric pressure, but lacks the aneroid's advantage of 
convenience in portability. The aneroid barometer is decidedly 
inferior to .the mercurial barometer as a pressure indicator, 
but is sufficiently accurate for many purposes, such as recon- 
naissance work. Pocket aneroids (about 3 inches in diameter) 
are found to be as reliable as the larger sizes. Aneroids are 
intended to read the same as mercurial barometers under the 
same conditions, being compensated for the effect of tempera- 
ture on their own construction; they are not compensated for 
the effect of temperature on atmospheric pressures. The aneroid 
requires careful handling, should be kept in its case at all times 
and away from the heat of the body, should be read in the open 
air and in a horizontal position, and' should be gently tapped 
when reading to overcome any friction among its moving parts. 

If all the conditions were the same at two different stations, 
the difference in atmospheric pressure would correspond to the 
difference in altitude; for points not over about 100 miles apart 
the conditions may be assumed to be nearly the same at the same 
time in ordinary calm weather. Two barometers are necessary 
for good work, the office barometer which is kept at the reference 
station, and the field barometer, which is carried from point 
to point. If the office barometer is an aneroid it must be stand- 
ardized, that is, adjusted by the small screw at the back imtil it 
reads the same as a mercurial barometer. During the period of ob- 
servations the office barometer and attached thermometer are read 
at regular intervals (about 15 or 30 minutes), so that by inter- 
polation the readings are assumed to be known for any instant. 
The time and temperature are recorded whenever a field reading 
is taken, so that comparison may be made with the office 
readings for the same time. If the field barometer is an aneroid 
its readings will need correction for . initial error and inertia. 
Before starting out to take readings with the field barometer 
it is compared with the office barometer and any difference is 
its initial error, which will affect all its readings to the same 
extent. On returning to the office after one or more observations 
the field barometer is again compared with the office one, and 
the amount by which the initial error has changed is called 
the inertia error; this error is distributed among the different 
readings in proportion to the elapsed time. 



128 



GEODETIC SURVEYING 



79. The Computations. The complete barometric formula 
(for which see Appendix No. 10, Report for 1881, U. S. Coast 
and Geodetic Survey) is very complicated and the smaller terms 
are generally omitted in ordinary work. Assuming all readings 
reduced to the standard of the office barometer, 

Let H = elevation in feet of the office barometer above a 
plane corresponding to a barometric pressure of 
30 inches for dry air at a temperature of 50° F.; 

h = the same for the field barometer; 

B = reading of office barometer in inches; 

6 = corrected reading of field barometer in inches; 

t = Fahrenheit temperature at office barometer; 

t' = Fahrenheit temperature at field barometer; 

C = correction coefficient for mean temperature —^ for 

average conditions of humidity; 
z= difference of elevation of the two barometers in feet; 

then we have, nearly, 

H= 62737 log ^, A = 62737 log y, 



and 



z = {h~H){l + G), 



in which H and -h may be obtained from Table III, and C from 
Table IV opposite {t + t'). 

Example. In the following table the field observations were taken with 
an aneroid and require the corrections described above. 

Field Notes and Reductions, May 17, 1910. 



Station. 


Time. 


Barom. 


Temp. 


Initial 
Corr. 


Inertia 
Corr. 


Thermom. 
Corr. 


A 


8.00 a.m. 


29.124 


73° F. 


+0.040 




+ 1° 


B 


11.10 a.m. 


28.247 


70° F. 


+ 0.040 


-0.006 


+ 1° 





1.30 P.M. 


29.216 


79° F. 


+ 0.040 


-O.OU 


+ 1° 


A 


4.00P-M. 


29.182 


79° F. 


+ 0.040 


-0.016 


+ 1° 



GEODETIC LEVELING 129 

OFnoE Notes aud Reductions, May 17, 1910. 



Station, 


Office. 


Field (reduced). 


Diff. Elev. 


Elevation. 


Barom. 


Temp. 


Barom. 


Temp. 


A 


29.164 


74° F. 


29.164 


74° F. 




1867 ft. 


B 


29.179 


76° F. 


28.281 


71° F. 


897 


2764 ft. 


G 


29.189 


79° F. 


29.245 


80° F. 


-55 


1812 ft. 


A 


29.206 


80° F. 


29.206 


80° F. 




1867 ft. 



From Table III (by interpolation) 



6 = 28.281, 
B = 29.179, 



h = 1607 
H = 756 

h- H = 851 



6 = 29.245, 
B = 29.189, 



h = 
H = 



694 
746 



h- H = -52 



From Table IV (by interpolation) 



159°, C = 


+ 0.0667 




- 52 X 0.0667 = - 


3 + 


- 52 + ( 


-3) = - 


55 



147°, C = + 0.0544 
851 X 0.0544 = 46 + 
851 + 46 = 897 

In this example the elevation of station A was known from previous 
determinations. 

80. Accuracy of Barometric Work. For exploration, recon- 
naissance, and other classes of work where close results are not 
required, the barometer serves a very useful purpose- For 
stations only a few miles apart, or not differing much in altitude, 
the errors in the determinations may not exceed a few feet. 
For long distances or large differences in altitude the results 
are very disappointing, notwithstanding that the utmost refine- 
ments of theory and practice are employed, and daily readings 
averaged for a number of years. In general the values obtained 
in the heat of the day are too great, and in the morning and evening 
too small; and similarly too great in summer and too small in 
winter. Professor Whitney, in his Barometric Hj^sometry, gives 
the results of three years' observations at Sacramento and Summit, 
California, from October, 1870, to October, 1873, in which the 
monthly average determinations of the difference of elevation 



130 GEODETIC SUfeVEYING 

varied from 6900 to 7021 feet, the average for the three years 
being 6965 feet; according to railroad levelings the true dif- 
ference is 6989 feet. Summit is about 77 miles from Sacramento 
in an air line; the altitude of Sacramento is about 30 feet above 
mean sea level. The example given is a fair illustration of 
the general experience in this class of work, with plus and minus 
errors about equal. The chief source of error in barometric 
work seems to be due to the lack of knowledge of the true 
average, temperature of the air column between the levels of 
any two given stations, the mean of the temperatures themselves 
being only a fair approximation. 

B. Trigonometric Leveling 

81. Instruments and Methods. Trigonometric leveling can 
be done with any instrument capable of measuring angles 
of elevation and depression, but good work can be done 
only when the angles can be measured with precision. While 
the ordinary surveyor's transit may read vertical angles only 
to the nearest minute, a fine altazimuth instrument may be 
provided with micrometer microscopes reading such angles to 
single seconds. In round numbers a minute of arc corresponds 
to a foot and a half per mile, and a second to three-tenths of an 
inch; with moderate sized vertical angles, such as would usually 
occur in trigonometric leveling, the resulting effect in altitude 
is practically the same. It is presumed that the observer under- 
stands how to adjust and use his particular instrument to the 
best advantage. 

The elevation of a station from which the open sea is visible 
can be determined by measuring the angle of depression to the 
sea horizon. The difference of elevation of two stations whose 
distance apart is known can be determined by measuring the 
angular elevation of one of them as seen from the other, con- 
stituting an "observation at one station," or by measuring the 
angular elevation of each station as seen from the other, con- 
stituting "reciprocal observations." From the nature of the 
case the effects of curvature and refraction are necessarily involved 
in any form of trigonometric leveling. The best results are 
obtained between 9.00 a.m. and 3.30 p.m., during which time the 
refraction has its least value and is comparatively stationary. 



GEODETIC LEVELING 



131 



82. By the Sea Horizon Method. If a station is so situated 
as to command a view of the open sea its elevation above the 
surface of the water may be determined by measuring the angle 
of depression to the sea horizon. The advantage of this method 
lies in the fact that no distance is required to be known. Fig. 40 
represents the vertical plane of the measured angle, in which 
A is the station whose elevation is 
desired; SS is an elliptic arc at the 
level of the sea horizon, but it is here 
assumed to be the arc of a ciroic; 
AE is & straight line from A tangent 
to the arc SS at the point E or true 
sea horizon; BC (on the vertical line 
AC) is the radius of the arc SS, and 
is assumed to be equal to the mean- 
sea-level radius of the section for the 
point A, the point C being in general 
not at the center of the earth. E' 
is the false horizon caused by the 
refraction of light; i3 is the apparent 
and C the true angle of depression 
to the sea horizon; and BD is a 
tangent at B. From well known Fi°- 40. 

geometrical principles the angles 

GAD, ADB, and BCE are equal, and the line DC bisects the 
angle at C. 

Let R = BC = the mean-sea-level radius of the section for 
the point A ; 

C = the true angle of depression = angle at center; 

d = the apparent angle of depression; 

Z = 90° + d = apparent zenith distance of sea horizon; 

m = coefficient of refraction; 

h = AB = elevation of station A above surface of sea; 

then from the figure we have 

h = BD tan C, 

C 




BD =R tan 



2' 

(J 
■■ R tan -^ tan C; 



132 GEODETIC SURVEYING 

or since C is always a small angle (rarely 60'), 
h =^C2tan2 1". 

On account of refraction (Arts. 14 and 14a) the observer does 
not sight along the true line AE, but in the direction of the 
dotted line from A, which is tangent to a curved line of sight 
from A to the false horizon E' . The practical result of the 
refraction is to make the measured angle 5 too small by the 
amount mC, so that 

5 = C - mC, 



1 — r?i' 
and 

2\1— m/ 

whence by transposition 

in which In, and r must be taken in the same imit, and b must 
be taken in seconds. By many experiments the mean value of 
m on the New England coast has been found to be 0.078* if we 
use this value we may write 

1°<2(I^^2) = 9-1406579 -20. 

In order to secure the best results it is necessary to measure 
the azimuth of the plane in which the angle of depression is 
taken, and use the mean-sea-level value of R for this azimuth 
and the latitude of the station. This value may be taken from 
Tables V and VI, or computed as explained in Art. 69. If errors 
which may range up to say about 1 in 300 are not objectionable, 
we may use a mean value of R and write 



log 



/ tan2 1" \ ^l r metric, 5.9446244- 10 1 , 
U(l-m)2J^ I = I feet, 6.4606086 - 10 j (^PP^o^i^iate), 



GEODETIC LEVELING 133 

or 

, f metric, 0.000088^2] 

'^^(feet, 0.000289 52 j (approximate), 

in which d must be taken in seconds of arc. 

83. By an Observation at One Station. When the distance 
between two stations is known their difference of elevation can 
be computed if the vertical angle of either as seen from the 
other is measured. The advantage of this method over the 
reciprocal method (Art. 84) lies in the economies due to occupying 
only one station, but the results are not likely to be so good 
on account of the uncertainty in the assumed value for the 




coefficient of refraction. Fig. 41 represents a plane through 
the two stations A and B, taken vertical at their middle lat- 
itude, and assumed to be vertical at both stations; SS is the 
elliptic arc cut from the spheroid, but it is here assumed to be the 
arc of a circle; the radius of the arc SS is taken as the mean- 
sea-level radius of the section at the middle latitude, the center 
C being in general not at the center of the earth; AC and BC 
are drawn to the center C and assumed to be vertical; Z is the 
apparent zenith distance of A as seen from B, and is in error 
by the small angle rnC due to refraction. 



134 GEODETIC SURVEYING 

Let h = AM = elevation of station A above mean sea level; 
h' = BN = elevation of station B above mean sea level; 
K = MN = mean-sea-level distance between stations 

A and B; 
R = MC = mean-sea-level radius of section at middle 

latitude between A and B • 
C = central angle ACB; 

Z = apparent zenith distance of A as seen from B; 
a = 90° — Z = apparent elevation of A as seen from B; 
mC = elevation of line of sight due to refraction; 

then 

AG -\-BC _ 2R +h + h' _ t&n^jABC + BAG) 
AC-BC" h-h' tan HABC - BACy 

ABC + BAC = 180° - C, 
tan i{ABC + BAC) = tan/^90° - g) = cot -■ 

ABC = 180° -Z ~'mC 

BAC = Z + mC -C 

ABC - BAC = 180° -2Z -2mC + C 

iiABC - BAC) = 90° -(z + mC - |V 
taniiABC - BAC) = cot(z + mC - ^V 

2R + h + h' °°^2 ■ 1 



^ ^' cot(z + mC-^\ i&n^Goiiz + mC -^\ 

h-h' = {2R + h + h') tan ^ cot(z + mC - - j. 



C . 
Expanding tan — in series, we have 



. c c .c^ 

*^^2 = 2+24+- 



GEODETIC LEVELING 135 

But C (in arc) = -j^ , 

li 

whence 

Hence by substitution and reduction and the omission of an inap- 
preciable factor, we, have 

C (in seconds) 



Also 

K 



R sin 1" ' 
whence 

h-h' =Kcot [z+(m-i)^^](l+'^ + ^) 

= if tan [a + H - m)^^,] (l + ^-^ + ^g,), 
or approximately (error seldom over 1 in 3000) 
h — h' = K cot\Z + (m — i)n—- — tt, (approximate), 

= K tan a + (-^ — m)^ — -. — -y, (approximate). 

The value of (h — h') is always found first by the approximate • 
formula, after which a closer value may be obtained from the 
complete formula if so desired. In these formulas h, h', K, and 
R must all be in the same unit. The coefficient of refraction 
m will average about 0.070 inland, and about 0.078 on the coast. 
The radius R is to be taken for the middle latitude of A and B 
and the approximate azimuth of the line joining them; this 
value may be taken from Tables V and VI, or computed as 
explained in Art. fi9. If errors which may reach or possibly 
exceed about 1 in 500 are permissible we may use a mean value 
of R and write 

metric, 6.80396651 
■ feet, 7.3199507 , 



\ogR 



, i^mean value. 



136 GEODETIC SURVEYING 

84. By Reciprocal Observations. When the distance between 
two stations is known their difference of elevation can be com- 
puted without assuming any particular value for the coefficient 
of refraction if the vertical angle of each station as seen from 
the other is measured. This result is brought about by assuming 
that the refraction is the same at each station, which is probably 
very nearly true if the observations are made at the same time 
on a calm day, although this is not always done. The advantage 
of this method over the single observation method (Art. 83) 
lies in the increased accuracy of the results. Fig. 42 (as in Fig. 41, 
Art. 83) represents a plane through the two stations A and B, 
taken vertical at their middle latitude and assumed to be 




vertical at both stations; SS is the elliptic arc cut from the 
spheroid, but it is here assumed to be the arc of a circle; the 
radius of the arc 8S is taken as the mean-sea-level radius of the 
section at the middle latitude, the center C being in general 
not at the center of the earth; AC and BC are drawn to the 
center C and assumed to be vertical; Z and Z' are the apparent 
zenith distances of the stations as seen from each other, each 
angle being assumed equally in error by the small angle mC 
due to refraction. 

Let h = AM = elevation of station A above mean sea level; 
h' = BN = elevation of station B above mean sea level; 



then, 



GEODETIC LEVELING 137 

K = MN = mean-sea-level distance between stations 

A and B; 
R = MC = mean-sea-level radius of section at middle 

latitude between A and B; 
C = central angle ACB; 

Z = apparent zenith distance of A as seen from B; 
Z' = apparent zenith distance of B as seen from A ; 
a = 90° — Z = apparent elevation of A as seen from B; 
a' = 90° — Z' = apparent elevation of B as seen from A; 
mC = elevation of lines of sight due to refraction; 

AC + BC _ 2R +h +h' ^ tan jjABC + BAC) 
AC -BC ~ h -h' ~ tan ^{ABC - BAC)' 

ABC + BAC = 180° - C, 

tan i{ABC + BAC) = tan^gO" - ^) = cot ^, 

ABC = 180° - Z -mC 
BAC = 180° -Z' -mC 





ABC 


-BAC = 


Z' -Z 




tan i {ABC 


-BAC) 


= tan iiZ' 


-Z), 


2R + h + h' 




,C 
cot 2 




1 



h-h' tani(^'-Z) tan§tanK^'-Z)' 

h-h' = (2R +h + h') tan ^ tan ^{Z' - Z). 

C 
Expanding tan ^ in series, we have 

, C C _,C^ , 
*^^ 2 = 2 + 24 +• ■ • 
But 

C (in arc) = -^ , 



whence 



^^'^ 2 " 2R ^24723 +' 



138 GEODETIC SURVEYING 

Hence by substitution and reduction and the omission of an 
inappreciable factor, we have, 

h-h'=KUnh iZ' - Z) (l+ ^- + ^) 
= if tan i (a - a') \\ + -^ + -~^, 

or approximately (error seldom over 1 in 3000) 

/i —In! = K tan \ {Z' — Z) (approximate), 
= K tan 'J (a — a') (approximate) . 

The value of Qi — h') is always found first by the approximate 
formula, after which a closer value may be obtained from the 
complete formula if so desired. In these formulas h, h! , K, 
and R must all be in the same unit. Except for very important 
work the mean value of R as given in Art. 83 is sufficiently precise. 
For very exact results the radius R is to be taken for the middle 
latitude of A and B and the approximate azimuth of the line 
joining them; this value may be taken from Tables V and VI, 
or computed as explained in Art. 69. 

85. Coefficient of Refraction. If the distance between two 
stations is known, the coefficient of refraction m, may be obtained 
as follows: 

1st. If the angular elevation of either station as seen from 
the other is measured, and the difference of elevation is obtained 
by spirit leveling, we have from Art. 83, 



h -h'= K cot 
= K tan 



Z + {m -i) ^ 



R sin 1' 

a + (J - m)-5— 7 — -7 
R sm 1 



^ 2R ^ 12R2J 
2E ^ 12i?2/' 



in either of which expressions it is only neccessary to substitute 
the known values and solve for m. The exact value of R is to 
be used, as explained in Art. 83. 

2nd. If the angular elevation of each station as seen from 
the other is measued, we halve" from Fig. 42, page 136, 

Z + mC - C = 180° - Z' - mC; 

whence 2mC = 180° ~ Z - Z' + C, 

180° - Z - Z' + C a + a' + C 
and m = ^, = ^ , 



QEODETIO LEVELING 139 

in which all the angular values must be expressed in the same 
unit (degrees, minutes, or seconds). From Art. 83 we have 

77' 

C (in seconds) = ^r-- — tt, , 
it sm 1" 

in which K and R must be in the same unit. 

The average value of the coefficient of refraction from many 
Coast Survey observations (Appendix No. 9, Report for 1882), 
is as follows: 

Across parts of the sea near the coast 0.078 

Between primary stations 0.071 

In the interior of the country 0.065 

86. Accuracy of Trigonometric Leveling. The U. S. Coast 
and Geodetic Survey has done a large amount of leveling of this 
class in connection with its triangulation work, with sights 
sometimes exceeding a hundred miles in length in mountainous 
regions. The best results are obtained by reciprocal observations, 
taken on a niunber of different days so as to average up the 
atmospheric conditions. When the work is conducted in this 
manner on lines not over about 20 miles in length the probable 
error may be kept down to about one inch per mile. When the 
lines exceed about 20 miles in length it is necessary to take a 
great many observations under especially favorable conditions to 
secure good results. In order to prevent an accumulation of 
errors in the elevations determined by trigonometric leveling, 
connection is made at various points with precise-level bench 
marks, and the trigonometric leveling Ls adjusted to fit the precise 
leveling between these points. 



C. Pbecise Spirit Leveling. 

87. Instrumental Features. The instruments used for precise 
leveling are the same in principle as the various types of engineers' 
levels, the essential feature being a telescopic line of sight and 
a spirit level (detachable or fixed) to determine its horizontality. 
Engineers' levels are designed to be as rapid and convenient 
in use as possible, consistent with the requirements of engineering 
work. Precise levels are designed to attain the highest possible 



140 GEODETIC SURVEYING 

degree of precision in the work which is done with them. Such 
instruments are made in various forms, two of which are shown 
in Figs. 43 and 44 and described in Arts. 89 and 90. Certain 
features are more or less common to all types of precise level. 
A rigid construction and the highest grade of material and 
workmanship are demanded. Especial care is taken to make 
the line of collimation true for all distances. The telescope is 
made inverting (the increased illumination permitting a higher 
magnifying power), and has three horizontal hairs (as equally 
spaced as possible) whose mean position determines the line of 
sight. The convenience of having the line of sight at right 
angles to the vertical axis of the instrument is abandoned in 
order to place a delicate control of the position of the bubble 
in the hands of the observer; this is accomplished by pivoting 
the telescope near the object-glass end, and providing a fine 
screw motion near the eyepiece end, so that the inclination of 
the telescope can be changed as desired. Such a screw is commonly 
called a micrometer screw because it was originally provided 
with a graduated head for measuring the value of small changes 
of inclination. The level vial is placed above the telescope, 
and a mirror or other means provided to enable the observer 
to see the bubble at the moment of taking an observation. A 
sensitive bubble is used, one division corresponding to about 
1 to 3 seconds of arc (against about 20 seconds in the ordinary 
wye or dumpy level). The level vial is chambered, permitting 
the observer to adjust the bubble to its most efBcient length, 
and is so mounted that it is free to expand and contract. The 
instrument is supported on three pointed leveling screws resting 
freely in V-shaped metal grooves on the tripod head. Such 
an instrument is leveled by setting the bubble parallel to a pair 
of leveling screws and bringing it to the center by turning that 
pair of screws equally in opposite directions, then turning the 
bubble in line with the remaining leveling screw and bringing 
it to the center with that screw alone; then turn the instrument 
180° on its vertical axis, and if the bubble moves from the center 
bring it half way back by the micrometer screw of the telescope 
and relevel both ways as before; when the bubble will stay within 
a few divisions of the center all the way around the leveling 
is satisfactory, as the precise leveling of the line of sight is accom- 
plished with the micrometer screw while taking the observation. 



GEODETIC LEVELING 



141 




Ph 



3 

E-1 



(3 



142 



GEODETIC SURVEYING 




^ s 

03 CO 

gp 

02 § 
+= o 
tn ^ 
03 .a 
o a 
O g 



f^ I 



GEODETIC LEVELING 143 

The tripods used with these instruments must be strong and rigid. 
Rods of special pattern and metallic turning points, as described 
in Art. 91, are used in this class of work. 

88. General Field Methods. In order to secure a high degree 
of precision in leveling the greatest care is required in the field 
work and methods. Five sources of error have to be guarded 
against, namely, errors of observation, instrumental errors, 
curvature and refraction errors, atmospheric errors, and errors 
from unstable supports. 

Errors of observation are kept as small as possible by care 
on the part of the observer; by keeping the rods plumb; by 
using a proper length of sight, 100 meters or about 300 feet 
being suitable for average conditions; by comparing at every 
sight the two intervals furnished by the readings of the three 
wires, any material disagreement (more than 2 millimeters) 
denoting an erroneous reading; by the fact that each pointing 
is taken as the mean of the three wire readings; and by the 
further fact that every line is run in duplicate in the reverse 
direction and a limit set on the allowable discrepancies. 

Instrumental errors are kept as small as possible by keeping 
the instrument in good adjustment; by determining the instru- 
mental constants with care and applying the corresponding 
corrections when necessary; by using a program of observations 
adapted to the type of instrument used, so as to eliminate the 
instrumental errors as far as possible; by making the length 
of each foresight nearly equal, if possible, to that of the corre- 
sponding backsight; by balancing any extra long or short fore- 
sight by a similar long or short backsight elsewhere, and vice 
versa; and by keeping the sum of the lengths of the foresights 
as nearly equal as possible to the sum of the lengths of the back- 
sights, with suitable corrections for the net difference. If the 
foresights and backsights were all exactly equal no correction 
would be required for instrumental errors. The effect of the 
various instrumental errors is to give the line of sight an inclina- 
tion with the horizontal. The value of the inclination becomes 
known through the instrumental constants, as explained later. 
The required correction in elevation is found by multiplying 
the net difference in length of sights by the sine of this incli- 
nation. 
• Curvature and refraction errors exist in every line of sight, 



144 GEODETIC SURVEYING 

as explained in Art. 14, but are obviously eliminated if the 
foresights and backsights are kept equal. If these sights are 
kept nearly balanced, as explained in the previous paragraph, 
and a suitable correction made for the net difference, the effects 
of curvature and ordinary refraction are practically reduced to 
zero. The correction which is made is the value of the curva- 
ture and refraction for the net difference in the lengths of the 
foresights and backsights. The net difference should be kept 
so small that no such correction may be necessary, but if required 
it can be taken from Table VII or computed as explained in 
Art. 14. 

Atmospheric errors are those due to an actual unsteadiness 
of the rod or instrument, caused by the wind; an apparent 
unsteadiness of the rod, caused by heated air currents, commonly 
called heat radiation; an irregular vertical displacement of the 
line of sight, caused by variable refraction; and the disturbance 
of the relation between the line of sight and the axis of the bubble, 
caused by unequal expansion and contraction of the different 
parts of the instrument. Moderate winds do not prevent good 
work, especially if wind shields are used around the instrument; 
but when the wind reaches about eight miles an hour it becomes 
impracticable to do first class work. When the rod becomes un- 
steady through heat radiation it becomes necessary to decrease 
the length of the sights in order to read the rod satisfactorily, but 
the increased number of sights increases the probable error of 
the result; if it becomes necessary to decrease the length of sight 
below 50 meters, or about 150 feet, it is not advisable to continue 
the work. Refraction is nearly stationary and has its least 
value between about 9.00 a.m. and 3.30 p.m., but during this 
period heat radiation is apt to be very troublesome; outside 
of these hours the refraction may be very variable. The result 
is that in perfectly clear weather the best class of work is only 
possible during a few hours of the day. In order to guard against 
unequal expansion and contraction the instrument is protected 
with a large sunshade (umbrella), and never exposed to the direct 
rays of the sun either while in use or while being carried to a new 
set-up. 

By the errors from unstable supports are meant the errors 
caused by the instrument or turning points changing their eleva- 
tions slightly between readings. It is shown by experience that 



GEODETIC LEVELING 145 

either rising or settling may take place, though settling is the 
most common. If the instrument settles between the backward 
reading and the forward reading the final elevation will be too 
high; the same result will occur if the rod settles between the 
forward reading and the backward reading on it. Errors of this 
class are kept as small as possible by planting the instrument 
firmly; by using well driven metallic turning points; by taking 
both readings from each set-up with as little intermediate delay 
as possible, using two rodmen for this reason as well as the saving 
of time; by reading the back rod first for every other set-up, 
and the fore rod first for the intermediate set-ups; and by duplicat- 
ing each line in the opposite direction, and correcting for half 
of the discrepancy. 

Certain field methods have been discarded, after years of 
extensive use, because the results have not proven as satisfactory 
as by other methods. Among these may be mentioned methods 
involving computations based on readings of the micrometer 
screw. The best results are obtained when all the observations are 
taken with the bubble in the center, the micrometer screw being 
used simply as the means of keeping it there. Another unsatis- 
factory method is the running of so-called simultaneous lines, 
in which readings are taken at each set-up to the turning points 
of two separate lines, as a substitute for running duplicate lines 
in opposite directions. 

89. The European LeveL An instrument of this form, but 
of American manufacture, is illustrated in Fig. 43 (page 141). 
The European type of instrument is essentially a wye level, 
in which different makers have followed the same general 
design, but with modified details. The telescope may be rotated 
in the wyes or lifted from the wyes and reversed. The level 
is separate from the instrument, being an ordinary striding 
level with the addition of a movable mirror over the bubble; 
by holding the eyes in a vertical line the image of the bubble 
may be seen with one eye while the rod is seen through the tele- 
scope with the other eye, the bubble being kept in the center 
with the micrometer screw while the observation is being made. 
The magnifying power is about forty-five diameters. Besides 
the above special features the instrument has all the general features 
. of a good instrument, as described in Art. 87. With this type of 
level there are three so-called constants and two adjustments. 



146 GEODETIC SURVEYING 

89a. Constants of European Level. The three constants of 
this instrument, which should be examined at least once a year, 
are as follows: 

1. The angular value of one division of the bubble, meaning the 
change in inclination which causes the bubble to shift its position 
by one division on the bubble scale. Modern level vials are 
ground so nearly uniform in curvature that it is customary to 
measure the change of inclination for the whole run of the bubble, 
dividing by the number of divisions through which the bubble 
moves to obtain the average value of one division. By the posi- 
tion of the bubble, or the movement of the bubble, is meant the 
position or the movement of its central point; the ends of the 
bubble are constantly changing their position on account of the 
changing length of the bubble, but the center remains stationary 
as long as there is no change of inclination. Bubble tubes are 
sometimes graduated from one end, but more frequently both 
ways from the center, in which case the divisions one way from 
the center are called positive and the other way negative. The 
reading of the center of the bubble is the algebraic mean of its 
two end readings. The movement of the bubble between any 
two positions is the algebraic difference of its two center readings. 
The practical operation of finding the value of one division is as 
follows: Level up the instrimient with the striding level in place, 
and have a leveling rod held at a fixed point at a known distance 
of about 200 feet. Turn the micrometer screw until the bubble 
comes near one end of its run, note each wire reading on the rod 
as closely as possible, and each end reading of the bubble to the 
nearest tenth of a division. Rim the bubble to the other end 
of the tube and note the rod and bubble readings for this posi- 
tion. Take a number of readings in this way at both ends, with 
the bubble in slightly different positions so as to obtain unbiassed 
values. Compute the position of the center of the bubble for each 
reading, then the average of the center readings for each end of 
the run, and then the movement corresponding to these average 
centers, which will be the average movement of the bubble. 
Subtract the mean of the lower readings from the mean of the 
upper readings on the rod for the average movement of the line 
of sight, which divided by the distance times the sine of 1" will 
give the average change of inclination in seconds of arc. The 
angular value of one division of the bubble in seconds will be this 



GEODETIC LEVELING 



147 



average change of inclination divided by the average movement 
of the bubble. In this process the rod readings and the dis- 
tances must be expressed in the same unit. In the following 
example illustrating the above principles the bubble tube is 
graduated each way from the center and a metric rod is held 70 
meters from the instrument. Each recorded rod reading is the 
average of the three wire readings. 

Example. — Asgvias, Valitb op One Division of Bubble Titbb 



Looking Up. 


Looking Down . 


Rod. 


Bubble. 


Rod. 


Bubble. 


Left. 


Bight. 


Center. 


Left. 


Right. 


Center. 


1.5245 


-36.9 


+ 3.1 


-16.9 


1.5000 


-1.5 


+ 37.8 


+ 18.2 


1.5240 


-36.1 


+2.8 


-16.7 


1.5005 


-1.7 


+ 36.7 


+ 17.5 


1.5245 


-36.0 


+2.0 


-17.0 


1.5010 


-2.0 


+35.4 


+ 16.7 


1.5250 


-35.6 


+ 1.1 


-17.3 


1.5005 


-1.4 


+ 35.9 


+ 17.2 


1.5245 


-35.8 


+ 1.8 


-17.0 


1.5005 


-1.5 


+ 35.7 


+ 17.1 


7.6225 


Slims 


-84.9 


7.5025 


Sums 


+86.7 


1.5245 


Means 


-16.98 


1.5005 


Means 


+ 17.34 


sin 1"=0. 0000048 
70Xsinl"=0. 0003393 


48 
6 

".72 
".72 


1-.6245 


Algebraic 
differences 


-16.98 


0.0240 


34.32 


0.204 
Chang 


Oh-0.000 
e of inclii 


33936= 7C 

ation= 7C 


Angular 


70". 72-=- 34. 32=2". 06 
value of one division =2". 1 



2. The inequality of the pivot rings, meaning the angle between 
the line joining the tops of the pivot rings (the telescope collars 
that rest in the wyes) and the center line of these rings. This 
angle would of course be zero, if there were no inequality in the 
size of the rings; but a small angle generally exists, due usually 
to unequal wear. It follows that when the tops of the rings are 
in a level plane, as indicated by the striding level, the line of 
sight or center line of the rings must be inclined to the horizontal 
to the extent of this angle. In order to determine this value 



148 GEODETIC SUEVEYING 

the instrument is approximately leveled, and clamped on its 
vertical axis. Bubble readings are then taken with the telescope 
direct and also when reversed end for end in the wyes. If the 
striding level and telescope were reversed together (as one piece) 
the movement of the bubble would measure twice the angle 
between the axis of the bubble and the bottom line of the pivot 
rings. If the striding level were in perfect adjustment (axis of 
bubble parallel to line of feet) this would mean the same thing 
as twice the angle between the top line and bottom line of the 
rings, or four times the pivot inequality (angle between center 
line and tops of rings). The striding level is seldom in perfect 
adjustment, but its error is eliminated by taking its average 
reading for its direct and reversed positions for each position 
of the telescope. The telescope is generally reversed a number 
of times and the average result taken. It is found in practice 
that the inclination of the telescope is liable to be changing 
during the progress of the observations, and thus lead to erroneous 
conclusions. Readings are therefore not only taken for alternate 
positions of the telescope, but the last position is made the same 
as the first position; the assumption is then made that the mean 
of the direct sets and the mean of the reverse sets correspond 
to the same instant of time. When the pivot inequality is 
obtained in bubble divisions its angular value is found by multiply- 
ing this result by the angular value of one division of the bubble. 
In the following example illustrating the above principles the 
level tube is graduated both ways from the center, and is called 
direct with the marked end towards the eyepiece. 

It will be noted in this example that the average effect 
of reversing the telescope (from eye-end left to eye-end right), 
is to cause the bubble to move to the right or towards the eye- 
end, showing the eye-end ring to be larger than the other ring 
which it replaces; when the tops of the rings are in a level plane, 
therefore, as indicated by the striding level, it follows that the 
line of sight (center line of the rings) must look up. If the tele- 
scope looks up it will cause the final elevation to be too low for 
an excess in the foresights and too high for an excess in the back- 
sights, and vice versa when the telescope looks down. The 
amount of the correction required will be equal to the excess 
distance multiplied by the angular inequaHty of the pivots and 
by the sine of 1". 



GEODETIC LEVELING 
Example, — Inequality of Pivot Rings 



149 



Telescope. 


Level. 


Bubble Readings. 


Left. 


Right. 


Left. 


Right. 


Eye-end left 


Direct 

Reversed 


- 26.6 

- 28.0 


+ 23.7 
+ 22.4 






right .... 


Direct 

Reversed 






-24.0 
-25.4 


+ 26.5 
+ 25.0 


" left 


Direct 

Reversed 


- 26.9 

- 28.4 


+ 23.8 
+ 22.4 






right 


Direct 

Reversed 






-24.2 
-25.8 


+ 26.8 
+ 25.2 


" left 


Direct 

Reversed 


- 27.2 

- 28.6 


+ 24.1 
-1- 22.8 






Sums 


-165.7 


+ 139.2 


-99.4 


+ 103.5 


Means 


- 27.62 


+ 23.20 


-24.85 


+ 25.88 


Center of buhhle 


-2.21 


+ 0.52 


Eye-end ring large 


Bubble moves to right = 2 . 73 div. 


Telescope looks up 


Inequahty of pivots =—0.68 " 


0.68X2". 1=1". 428 


Ang. inequality of pivots = — r'.4 



3. The angular value of the wire interval, meaning the ratio 
between the solar focus or principal focal length of the objective 
and the distance between the outer cross-hairs. The telescope 
may be regarded as set for a solar focus when it is f ocussed on any 
distant object. 

Let D = unknown distance between level rod and vertical 

axis of level ; 

(S = corresponding rod intercept between outer cross-hairs; 

(i = a known distance from axis of level; 

s = corresponding intercept; 

/ = distance from cross-hairs to objective for solar focus; 

c = distance from vertical axis to objective for solar focus; 

i = distance between outer cross-hairs; 

f 
A = -= angular value of wire interval; 



150 GEODETIC SURVEYING 

then, from the theory of stadia measurements, 

^ = / = ^ - (/ + ^) 

i s ' 

D =A-S+ if+c), 

in which formulas D, S, d, s, f, and c must all be taken in the same 
unit. The field work of finding A consists in focussing on a 
distant point and measuring on the telescope the values of / 
and c; then measure a distance of about 100 meters or about 300 
feet from the vertical axis of the instrument, and take the rod 
readings at this point (with the instrument leveled) for the upper 
and lower hairs; the intercept s of the formula is the difference 
of these readings; then substitute the values d, s, f, and c in the 
formula for A. The value of A may run from about 100 to about 
300, the instrument maker usually setting the hairs as near as 
possible for an even hundred. With the value of A known and 
the recorded rod readings a simple substitution in the formula 
for D at once gives the distance between the instrimient and cor- 
responding turning point. Since the corrections for instrumental 
errors are only applied to the excess distance between foresights 
and backsights, a running total is kept of the corresponding 
wire intervals, and the formula for D applied to this excess interval 
only, omitting the small constant (/+c). 

89b. Adjustments of European Level. The two adjustments 
of this instrument, which should be examined daily, are as follows : 

1. The collimation adjustment, meaning the adjustment of 
the position of the ring that carries the cross-hairs so that the 
actual line of sight (as indicated by the mean position of the hairs) 
shall coincide with the true line of sight or center line of the rings. 
This adjustment is made by leveling up the instrument and sight- 
ing at a rod (about 100 meters distant) with the telescope both 
direct and inverted. If the mean of the three wire readings is 
not the same in each case the reticule is moved in the apparent 
direction needed to correct the error and an amount equal to 
half the discrepancy. It is essential that the instrument be 
perfectly leveled for each reading. When the discrepancy is 
brought down to about two millimeters it may be considered 
satisfactory, as it is easy to apply a correction for the residual 



GEODETIC LEVELING 151 

error, or the error may be eliminated by the method of observing. 
The collimation error is the angular amount by which the 
actual line of sight (determined by mean position of cross-hairs) 
deviates from the center line of the rings. The collimation error 
only affects the excess distance, hke all the other instrumental 
errors. 

Let C = collimation correction for excess distance D; 

D = excess distance between backsights and foresights; 
c = collimation error; 
rf = a known distance; 
Ri = mean rod reading for d with telescope normal; 
R,i = mean rod reading for d with telescope inverted; 

then evidently, 

c = ^'~^' and C=cD, 

in which all values must be taken in the same unit. 

2. The bubble adjustment, meaning the adjustment by which 
the axis of the bubble is made parallel to the line joining the feet 
of the striding level. This adjustment is made by leveling up 
the instrument, clamping the vertical axis, bringing the bubble 
exactly central with the micrometer screw, and then reversing 
the striding level without disturbing the telescope. If the bubble 
is not central after reversal it is to be adjusted for one-half of 
its movement. Relevel with the micrometer screw, reverse 
again, and so on until the adjustment is satisfactory (within 
about one division of the scale). The bubble error or inclination 
of the bubble is the angle between the axis of the bubble and the 
line joining the feet of the striding level; this angle would be 
zero if the bubble were in perfect adjustment. To determine 
the bubble error level up the instrument approximately, clamp the 
vertical axis, bring the bubble near the center with the micrometer 
screw, and then read the bubble a number of times in direct and 
reversed positions, making the last position the same as the first 
position. The bubble error in bubble divisions is half the average 
movement of the bubble; the inclination of the bubble is the error 
in bubble divisions multiplied by the angular value of one 
division. In the following example illustrating the above principles 
the level tube is graduated both ways from the center, and is 
called direct with the marked end towards the eyepiece. 



152 



GEODETIC SURVEYING 
Example. — Inclination of Bubble 



■3 

o 

i 


striding Level. 


Bubble. 


Left. 


Bight. 


Left. 


Right. 




-'2li.6 


+ 23.7 














-28.0 


+ 22.4 






-2(i.9 


+23.8 














-28.4 


+ 22.4 






-21.-2 


+ 24.1 








Sums 


-80.7 


+ 71.6 


-56.4 


+ 44.8 


Means 


-26.90 


+ 23. S7 


-28.20 


+ 22.40 


Center of bubble 


-1.52 


-2.90 


Level direct— Telescope looks ilown. 


Bubble error= +0.69 division. 


0.69X2". 1 = 1". 441) 


Inclination of bubble= + r'.4 



It will hv noted in the above example that the average effect 
of reversing the striding level (putting marked end towards 
object glass) is to cause the bubble to move away from the 
marked end, showing that the marked end has the shortest leg; 
when the bubble is in the center, therefore, if the marked end of 
the striding level is nearest the eyepiece the telescope looks down. 
If a line of levels were run with the striding level in a fixed posi- 
tion a correction would be required for the excess distance, the 
value of which would equal the inclination of the bubble multiplied 
by the excess distance and the sine of 1". The sign of the cor- 
rection for excess of foresights would be positive for telescope 
looking up and negative looking down, and vice versa for excess 
of baclcsights. 

89c. Use of European Level. The best results are obtained 
when all the rod readings are taken with the bubble precisely 
centered, and the observations so arranged as to eliminate as 
far as possible the effects of the instrumental errors. All the 
precautions of Art. 88 are to be carefully observed. Among 
these may be again mentioned the necessity of keeping the 
instrument sheltered by the imibrella from the sun and wind at 
all times; making each foresight approximately equal to the 



GEODETIC LEVELING 153 

previous backsight (pacing is satisfactory); keeping the sum of 
the foresights nearly equal to the sum of the backsights, as 
indicated by the corresponding sums of the wire intervals; plant- 
ing the instrument firmly and making the turning points solid; 
keeping the rod plumb; watching the wire intervals at every 
sight, and taking a new reading of each of the three wires when- 
ever the half intervals disagree by more than two millimeters; 
and running a duplicate line in the opposite direction as a check, 
and in order to eliminate errors from unstable supports (by using 
the mean difference of elevation as the true value). 

Program of observations for each set-up. Level up the instru- 
ment; sight at the back rod; take each of the three wire readings 
with the bubble kept centered with the micrometer screw; sight 
on the forward rod and read with bubble central as before; remove 
striding level, invert telescope in wyes, replace striding level 
reversed end for end; read forward rod with bubble central; 
sight on back rod and read with bubble central. This method 
of observing eliminates both the bubble error and the coUimation 
error, even with the foresights and backsights unbalanced. The 
correction for inequality of pivots, however, must be applied to 
any excess distance, as also the correction for curvature and 
refraction if the excess distance makes the amount appreciable. 
An example of notes and reductions is given on the next page. 
In this case the backsights are in excess, but not enough to require 
appreciable corrections. 

90. The Coast Survey Level. Previous to 1900 the precise 
leveling of the U.S. Coast and Geodetic Survey was done with 
the European type of instrimaent. Commencing with the summer 
of 1900 this work has been done with a type of instrument designed 
by the Department and known as the Coast Survey level. A 
view of this level is shown in Fig. 44, page 142. The instrument 
is essentially a dumpy level, as the telescope does not rest in wyes, 
can not be removed from its supports, and can neither be inverted 
nor reversed. The base of the instrument is of the usual three 
leveling screw type, except that the center socket is xmusually 
long and extends downwards through the tripod head. An 
outer protecting tube through which the telescope passes is 
rigidly attached to the vertical axis; the telescope is pivoted at 
one end of this outer tube, and has its inchnation controlled by a 
micrometer screw at the other end. The collimation adjustment 



154 



GEODETIC SURVEYING 



FORM OF NOTES— EUROPEAN LEVEL 
(Left-hand page.) (Right-hand page.) 



Forward Line. 
Backsights. 


B. M. 4 to B. M. S. 
Date, June 18, 1911. 


Point. 


Rod. 

and 

Temp. 


Thread Readings. 


Mean. 


Intervals. 


Remarks. 


1 


2 


3 


Each. 


Sums. 


B.M.4 

r. P. 1 


2 
76 


2.518 
2.522 


2.616 
2.618 


2.714 
2.716 


2.6170 

2.3967 
+ 5.0137 


0.1950 
0.1625 


0.1950 
0.3575 


Elevation of 
B.M. 4=117.617 

/Desoription\ 
\of B. M.4.J 


Means 

5 

78 


2.5200 

2.313 
2.318 


2.6170 

2.395 
2.398 


2.7150 

2.476 
2.480 


Means 


2.3155 


2.3965 


2.4780 


(Left-hand page,) (Right-hand page,) 


Forward Line. 
Foresights. 


B. M. 4 to B. M. 5. 
Date, June 18, 1911. 


Point. 


Rod. 

and 

Temp. 


Thread Readings. 


Mean. 


Intervals. 


Remarks. 


1 


2 


3 


Each. 


Sums. 


r. P. 1 

B.M.5 


5 

77 


1.167 
1.173 


1.260 
1.266 


1.354 
1.361 


1.2635 

0.8008 

-2.0643 
+ 5.0137 


0.1875 
0.1585 


0.1875 

0.3460 

0.3460 
0.3575 


/Desoription\ 
\ of B. M. 5./ 

Elevation of 
B.M. 4=117.617 
+ 2.949 


Means 

2 

78 


1.1700 

0.720 
0.723 


1.2630 

0.800 
0.802 


1.3575 

0.880 
0.880 


Means 


0.7215 


0.8010 


0.8800 




+ 2.9494 




0.0115 


B.M. 5=120.566 



is permanently fixed by the maker. The level tube is attached 
to the telescope, but has provision for adjustment. A strong 
point of the instrument is the closeness of the bubble to the line 
of sight, the level tube being let part way into a slot cut in the 
top of the telescope tube, the top of the level tube coming about 
flush with a slot in the top of the outer tube. The level vial is 
chambered for adjusting the length of the bubble. Attached to 
the left side of the instrument is a light auxiliary tube through 



GEODETIC LEVELING 155 

which the left eye may see an image of the bubble while the right 
eye is observing the rod, the head being held in its natural posi- 
tion, and the tube being adjustable sideways to suit the eyes 
of different observers. Besides the lens in its eyepiece the tube 
contains two prisms, adjustable for length of bubble, and placed 
opposite a slot running abreast of the level vial. The bubble is 
brought within the view of the left eye through the eye lens, the 
two prisms, and a mirror attached to the telescope. The telescope 
tube and outer casing are made of a nickel-iron alloy that has 
a coefi&cient of expansion which is only one-fourth that of brass, 
while the micrometer screw and other important screws are made 
of nickel-steel having a coefficient of expansion as low as 0.000001 
per degree centigrade. A detailed description of this instrument 
(from which the above notes have been gathered) is given in 
Appendix No. 3, Report for 1903, U. S. Coast and Geodetic 
Survey. Work with this level has been extremely satisfactory, 
better results being secured with greater rapidity and a much 
reduced cost. The Coast Survey level has two constants and 
one adjustment. 

90a. Constants of Coast Survey Level. The two constants of 
this instrument, which should be examined at least once a year, 
are as follows: 

1. The angular value of one division of the bubble. This is 
found by the optical method, as described in Art. 89a. 

2. The angular value of the wire interval. This is also 
foimd as described in Art. 89a. 

90b. Adjustments of Coast Survey Level. The only adjust- 
ment of this instrument, which should be examined daily, is as 
follows: 

To make the axis of the bubble parallel to the line of sight. 
This adjustment is made by the ordinary peg method (as adapted 
to this type of instrument), the bubble tube being raised or lowered 
at the adjusting end as may be required. The cross-hairs must 
never be disturbed as these have been permanently adjusted for 
collimation by the instrument maker. In testing the adjustment 
the rod reading is taken as the mean of the three wire readings, 
and the rod interval as the difference between the outside wire 
readings, the bubble being kept exactly centered while reading 
each of the three wires. Two pegs or turning-point pins are 
firmly driven about 100 meters apart, each rod being kept 



156 GEODETIC SURVEYING 

on its own point if two rods are used, or one rod being shifted as 
required. The instrument is set up approximately in line with 
the two points, first about ten meters beyond one point, and then 
about the same distance beyond the other point. The rod read- 
ing is taken for each point in each position of the instnunent, 
the ternas near rod and distant rod being used to indicate the 
relative position of the rods for each set-up. Having taken 
the four readings we have 

^ _ (sum of near -rod readings) — (sum of distant-rod readings) 
(sum of distant-rod intervals) — (sum of near-rod intervals) ' 

in which C is called the bubble error or constant for the day's 
work. If C does not exceed 0.010 (numerically) it is not advisable 
to change the adjustment. The telescope looks down when C 
is positive and up when C is negative, so that if an adjustment 
is found to be necessary the line of sight (here taken as the middle 
wire) is raised or lowered on the distant rod by C times that 
distance, and the bubble tube adjusted to bring the bubble 
central. A new determination of C is always made after each 
adjustment, and in very precise work the distant-rod readings 
are corrected for curvature and refraction (Table VII) before 
using in the formula, as these errors double up instead of canceling 
out in this method of adjustment. A correction equal to C times 
the excess interval between the foresights and backsights is 
applied to the final elevation; if the backsights are in excess the 
correction has the same sign as C, and the opposite sign when the 
foresights are in excess. 

90c. Use of Coast Survey Level. In order to obtain the best 
results with this instrument all the precautions given in Art. 88, 
and briefly summarized in Art. 89c, must be observed. The 
program of observations is much simpler than with the European 
level, there being nothing to do at each set-up except to obtain 
the three wire readings on each rod, with the bubble kept exactly 
centered while reading each wire. It is considered advisable 
to read the fore rod first on every other set-up. In the precise 
leveling of the U. S. Coast and Geodetic Survey a correction for 
excess of sights is applied for curvature and refraction and also 
for bubble error, together with corrections for absolute length 
of rod and average temperature of rod. An example illustrating 
the keeping of the notes is given on the next page. 



GEODETIC LEVELING 



157 



o 



o 



fi^ 



in 



■3-i 
















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Oi O OS 


CO CO to 


to CO (M 


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o o o 


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158 GEODETIC SURVEYING 

91. Rods and Turning Points. Various types of rods and 
turning points have been used in precise level work, with details 
changing from time to time. The notes here given are intended 
to briefly cover the points of interest to engineers. 

Rods. Precise leveling rods are now generally made of wood, 
sometimes soaked in melted paraffin to eliminate changes of length 
by absorption of atmospheric moisture, cross or T-shaped in 
section, about 3.5 meters in length, graduated metrically, pro- 
vided with a plumb line or level, and designed to be used with- 
out targets. The Coast Survey rod is cross-shaped in section, 
of pine wood which has absorbed about 20 per cent of its original 
weight of paraffin, graduated to centimeters and read by estima- 
tion to millimeters, and provided with a circular level for making 
it vertical. Target rods were abandoned by the Coast Survey 
in 1899. For a description of Coast Survey rods see Appendix 
No. 8, Report for 1895, and Appendix No. 8, Report for 1900. The 
precise rods used by the Corps of Engineers, U. S. A., are similar 
to the above, but T-shaped in cross-section. The Molitor 
rod (designed by Mr. David S. Molitor, and described in Trans. 
Am. Soc. C.E., Vol. XLV, page 12) is illustrated in Fig. 45, and 
is a precise rod of the highest class. The smallest divisions are 
two millimeters wide, and the reading is taken to millimeters or 
closer by estimation. 

Rod constant and adjustment. The precise leveling rod has one 
constant, and one adjustment. The rod constant is its absolute 
length between extreme divisions, which may differ slightly 
from its designated length, and which should be examined at 
least once a year. If the rod is long or short a self-evident 
correction is required, which only affects the final difference 
of elevation between two points. The rod adjustment is the 
adjustment of its level, which should be examined daily by 
making the rod vertical with a plumb line, and corrected if 
necessary. 

Turning points. Both foot-plates and foot-pins have been 
used for turning points. Cast iron foot-plates about six inches 
in diameter have been used extensively by the Coast Survey, 
but were practically abandoned in 1903 as inferior to pins. Fig. 45 
shows a style of foot-pin first used by Prof. J. B. Johnson in 1881, 
and meeting every requirement of a good pin. It is driven nearly 
flush with the ground with a wooden mallet. Such a pin is 



GEODETIC LEVELING 



159 



- 


t 

1 


TI 


« 


1 




bl 






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3 












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t. 






0} 






p 






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h 


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o 












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o 






iH 








1 

i 




i_J 






o 



Fig. 45. — ^Molitor's Precise-level Rod and Johnson's Foot-pin. 



IGO GEODETIC SUEVETING- 

best made of steel. The little groove in the head is to prevent 
dust or sand from settling on the bearing point. 

92. Adjustment of Level Work. In running level lines of 
any importance the work is always arranged so as to furnish 
a check on itself, or to connect with other systems, and a cor- 
responding adjustment is required to eliminate the discrepancies 
which appear. The problem may always be solved by the method 
of least squares when definite weights have been assigned to the 
various lines. When the work is all of the same grade the lines 
are weighted inversely as their length. This rule requires an 
error to be distributed uniformly along any given line to adjust 
the intermediate points. A common rule for intermediate points 
on a line or circuit is to distribute the error as the square root 
of the various lengths; but as this rule is inconsistent with itself 
it is not recommended. The following rules for the adjustment 
of level work will usually be found sufficient and satisfactory. 

Duplicate lines. A duplicate line is understood to mean a 
line run over the same route, but in the opposite direction and 
with different turning points. This is the best way of checking 
a single line of levels. The discrepancy which usually appears 
is divided equally between the two lines. 

Simultaneous lines. These are lines run over the same route 
in the same direction, but with different turning points. In 
this case the final elevation is taken as the mean of the elevations 
given by the different lines. 

Multiple lines. This is understood to mean two or more 
lines run between two points by different routes. In this case 
the difference of elevation as given by each line is weighted inversely 
as the length of that line, and the weighted arithmetic mean 
is taken as the most probable difference of elevation. Thus if 
the difference of elevation between A and B is 9.811 by a 6-mile 
line, 9.802 by an 8-mile line, and 9.840 by a 12-mile line, we have 

Mean difference of elevation 

_ (9811 X i) + (9.802 X i) + (9.840 X jS) 



+ J- 



= 9.S14. 

T2 



Intermediate points. These may occur on a line whose ends 
have been satisfactorily adjusted or on a closed circuit. In 
either case the required adjustment is distributed uniformly 
throughout the line, making the correction between any two 



GEODETIC LEVELING 



161 



points directly proportional to the length between those two 
points. 

Level nets. Any combination of level lines forming a series 
of closed circuits is called a polygonal system or level net. Fig. 46 
represents such a system. If the true difference of elevation 
were known from point to point, then the algebraic sum of the 
differences in any closed circuit would always equal zero, the 
rise and fall balancing. In practical work the various circuits 
seldom add up to zero, and an adjustment has to be made to 
eliminate the discrepancies. A rigor- 
ous adjustment requires the use of 
the method of least squares, but the 
approximate adjustment here described 
will generally give very nearly the same 
results. Pick out the circuit which 
shows the largest discrepancy, and 
distribute the error among the differ- 
ent lines in direct proportion to their 
length. Take the circuit showing the 
next largest discrepancy, and distribute 
its error imiformly among any of its 
lines not previously adjusted in some 
other circuit, continuing in this way 
until all the circuits have been ad- 
justed. The circuits here intended are 

the single closed figures, as BEFC, and not such a circuit as 
ABEFCA; and no attention is to be paid to the direction or 
combination in which the lines may have been run. 

93. Accuracy of Precise Spirit Leveling. The accuracy 
attainable in precise spirit leveling may be judged by noting the 
discrepancies between duplicate lines (Art. 92). On the U. S. 
Coast and Geodetic Survey the limit of discrepancy allowed 
between duplicate lines is 4mm. vX, meaning 4 millimeters 
multiplied by the square root of the distance in kilometers between 
the ends of the lines; if this limit is exceeded the line must be 
rerun both ways until two results are obtained which fall within 
the specified limits. In various important surveys the allowable 
limit has ranged from 5mm. v'i? to 10mm. Vif, or 0.021ft. ^/M 
to 0.042ft. Vikf where M is the distance in miles. The probable 
error of the mean result of a pair of duplicate lines is practically 




162 GEODETIC SUEVEYING 

one-third of the discrepancy, and in actual work of the highest 
grade falls below Imm.Vif. The adjusted value of the eleva- 
tion above mean sea level of Coast Survey bench mark K in 
St. Louis has a probable error of only 32 millimeters or about 
li inches, and it is almost certain that no amount of leveling 
will ever change the adopted elevation as much as 6 inches. 

A much more severe test of the accuracy of leveling is obtained 
from the closures of large circuits running up sometimes to 1000 
or more miles in circumference. The greatest error indicated 
by the circuit closures in any line in about 20,000 miles of precise 
spirit leveling executed by the U. S. Coast and Geodetic Survey 
and other organizations, is about one-tenth of an inch per mile. 
With the Coast Survey level of Art. 90 very much closer results 
have been reached. 



CHAPTER VII 

ASTRONOMICAL DETERMINATIONS 

94. General Considerations. The astronomical determina- 
tions required in practical geodesy are Time, Latitude, Longitude 
and Azimuth. The precise determination of these quantities 
requires special instruments as well as special knowledge and skill, 
and falls within the province of the astronomer or professional 
geodesist rather than that of the civil engineer. A fair deter- 
mination, however, of one or more of these quantities is not 
infrequently required of the engineer, so that a partial knowledge 
of the subject is necessary. A complete discussion of the sub- 
jects of this chapter may be found in Doolittle's Practical Astron- 
omy, or in Appendix No. 7, Report for 1897-98, U. S. Coast and 
Geodetic Survey. As the work of the fixed observatory is out- 
side the sphere of the engineer, the following articles are intended 
to cover field methods only. 

The instruments used by the engineer will generally be limited 
to the sextant, the engineer's transit, one of the higher grades 
of transits, or the altazimuth instruments of Chapter III. All 
of these instruments are suitable for either day or night observa- 
tions, except that the ordinary engineer's transit is not usually 
fiurnished with means for illuminating the cross-hairs at night. 
This difficulty may be overcome by substituting in place of the 
sunshade a similar shade of thin white paper, a flat piece of bright 
tin bent over in front of the object glass at an angle of about 45° 
and containing an oblong hole having a slightly less area than 
that of the lens, or a special reflecting shade which may be bought 
from the maker of the instrument. The light of a bull's-eye 
lantern thrown on any of these devices will render the cross-hairs 
visible. 

In astronomical work the observer is assumed to be at the 
center of the earth, this point being taken as the center of a great 

163 



164 GEODETIC SUEVEYING 

celestial sphere on which all the heavenly bodies are regarded as 
being projected. Any appreciable errors arising from the assumption 
that the earth is stationary or that the observer is at its center, 
are duly corrected. All vertical and horizontal planes and the 
planes of the earth's equator and meridians are imagined extended 
to an intersection with the celestial sphere, and are correspond- 
ingly named. Fig. 47, page 166, is a diagram of the celestial 
sphere, and the accompanying text contains the definitions and 
notation used in the discussions. A thorough study and compre- 
hension of the figure and text are absolutely essential for an 
understanding of what follows. The necessary values of the 
right ascensions, declinations, etc., required in the formulas, are 
obtained from the American Ephemeris, commonly called the 
Nautical Almanac, which is issued yearly (three years in advance) 
by the Government. 



Time 

95. General Principles. Time is measured by the rotation 
of the earth on its axis, which may be considered perfectly uniform 
for the closest work. The rotation is marked by the observer's 
meridian sweeping around the heavens. The intersection of 
this meridian with the celestial equator furnishes a point whose 
uniform movement aroimd the equator marks off time in angular 
value. The angle thus measured at any moment between the 
observer's meridian and the meridian of any given point (which 
may itself be moving) is the hour angle of that point at that 
moment. These angles are, of course, identical with the cor- 
responding spherical angles at the pole. When 360° of the equa- 
tor have passed by the meridian of a reference point (whether 
moving or not) the elapsed time is called twenty-four hours, so 
that any kind of time is changed from angular value to the hoiu- 
system by dividing by 15, and vice versa. There are two kinds 
of time in common use, mean solar time and sidereal time, based 
on the character of the reference point. Mean solar time is the 
ordinary time of civil life, and sidereal time is the time chiefly used 
in astronomical work. 

96. Mean Solar Time. The fundamental idea of solar time is to 
use as the measure of time the apparent daily motion of the sun 



ASTRONOMICAL DETEEMINATIONS 166 

around the earth; this is called apparent solar time, the upper transit 
of the sun at the observer's meridian being called apparent noon. 
Apparent solar time, however, is not uniform, on account of a 
lack of uniformity in the apparent annual motion of the sun 
around the earth. This is due to the fact that the apparent 
annual motion is in the ecliptic, the plane of which makes an angle 
with the plane of the equator, and the further fact that even in 
the ecliptic the apparent motion is not uniform. To overcome 
this difficulty, a fictitious sun, called the mean sun, is assumed to 
move annually around the equator at a perfectly uniform rate, 
and to make the circuit of the equator in the same total time that 
the true sun apparently makes the circuit of the ecliptic. Mean 
solar time is time as indicated by the apparent daily motion of 
the mean sun and is perfectly uniform. The difference between 
apparent solar time and mean solar time is called the equation 
of time, varies both ways from zero to about seventeen minutes, 
and is given in the Nautical Almanac for each day of the year. 
Local mean time for any meridian is the hour angle of the mean 
sun measured westward from that meridian, local mean noon 
being the time of the upper transit of the mean sun for that 
meridian. 

96a. Standard Time. This time, as now used in the United 
States, is mean solar time for certain specified meridians, each 
district using the time of one of these standard meridians instead 
of its own local time. The meridians used are the 75th, 90th, 
105th and 120th west of Greenwich, furnishing respectively 
Eastern, Central, Mountain and Pacific standard time. Standard 
time for all points in the United States differs only by even hours, 
with very large belts having exactly the same time, the variation 
from local mean time seldom exceeding a half hour. In the lat- 
itude of New York local mean time varies about four seconds 
for every mile east or west. Standard time may be obtained at 
any telegraph station with a probable error of less than a second. 
In all astronomical work standard time must be changed to local 
mean time. 

96b. To Change Standard Time to Local Mean Time and vice 
versa. The difference between standard time and local mean 
time at any point equals the difference of longitude (expresed 
in time units. Art. 113) between the given point and the standard 
time meridian used. For points east of the standard time 



166 



GEODETIC SURVEYING 



North 




Fig. 47.— The Celestial Sphere. 

EXPLANATION 

^2^^ = meridian of observer; 
Z, IF, Af =points on prime vertical; 
M, TO = projection of azimuth marks on celestial sphere; 
Z = observer's zenith; 
A'' = observer's nadir; 
Angles at Z, and corresponding horizontal angles at 0, are azimuth angles; 
Angels at P, and corresponding equatorial angles at 0, are hour angles. 

Conversion op Arc and Time 



Arc. Time. 

1° = 4 minutes 
1' =4 seconds 
1" = tV second 



Time. Arc. 

1 hour = 15° 

1 minute = 15' 

1 second = 15" 



ASTRONOMICAL DETERMINATIONS 167 



DEFINITIONS 

The zenith (at a given station) is the intersection of a vertical line with 
the upper portion of the celestial sphere. 

The nadir is the intersection of a vertical line with the lower portion 
of the celestial sphere. 

The meridian plane is the vertical plane through the zenith and the celes- 
tial poles, the meridian being the intersection of this plane with the celestial 
sphere. 

The prime vertical is the vertical plane (at the point of observation) at 
right angles with the meridian plane. 

The latitude of a station is the angular distance of the zenith from the 
equator, and has the same value as the altitude of the elevated pole. Lati- 
tude may also be defined as the declination of the zenith. North latitude 
is positive and south latitude negative. 

Co-ZaiiSude = 90°— latitude. 

Right ascension is the equatorial angular distance of a heavenly body 
measured eastward from the vernal equinox. 

Declination is the angular distance of a heavenly body from the equator. 
North declination is positive and south dechnation negative. 

Co-declination or polar distance = 90°— declination. 

The hour angle of a heavenly body is its equatorial angular distance 
from the meridian. Hour angles measured towards the west are positive, 
and vice versa. 

The azimuth of a heavenly body (or other point) is its horizontal angular 
distance from the south point of the meridian (unless specified as from the 
north point). Azimuth is positive when measured clockwise, and vice 
versa. 

The altitude of a heavenly body is its angular distance above the horizon. 

Co-altitude or zenith distoKce = 90°— altitude. 

Refraction is the angular increase in the apparent elevation of a heavenly 
body due to the refraction of light, and is always a negative correction. 

Parallax (in altitude) is the angular decrease in the apparent elevation 
of a heavenly body due to the observation being taken at the surface instead 
of at the center of the earth, and is always a positive correction. 

NOTATION 

^= latitude (-|- when north, — when south); 
Of = right ascension; 

S= declination (-|- when north, — when south); 
i = hour angle (-|- to west, — to east); 
A = azimuth from north point (-|- when measured clockwise); 
Z = azimuth from south point (-|- when measm-ed clockwise); 
ft = altitude; 
z=zenith distance; 
r= refraction; 
p= parallax. 



168 GEODETIC SURVEYING 

meridian local mean time is later than standard time, and vice 
versa. 

Example 1. New York, N.Y., uses 75th-meridian standard time. Given 
the longitude of Columbia College as 73° 58' 24". 6 west of Greenwch, what 
is the local mean time at 10'' 14™ 17^.2 p.m. standard time? 

75° 00' 00".0 loll 14™ 173.2 p.m. 

73 58 24 .6 4 06 .4 



1 5) 1° 01' 35".4 Ans. =10'> 18™ 23^.6 p.m. 
•i™ 06^.4 



Example 2. Philadelphia, Pa., uses 75th-meridian standard time. Given 
the longitude of Flower Observatory as 5^ 01™ 06^.6 west of Greenwich, what 
is the standard time at 9'' 06™ 18^.1 a.m. local mean time. 

15 )75° 00' 00".0 9'' 06™ 18M a.m. 

Sh 00™ 00^0 1 06 .6 

5 01 06 .6 

Ans. =9'' 07™ 24''.7 A.M. 



1™ 06^.6 

97. Sidereal Time, In this kind of time a sidereal day of 
twenty-four hours corresponds exactly to one revolution of the 
earth on its axis, as marked by two successive upper transits 
of any star over the same meridian. The sidereal day for any 
meridian commences when that meridian crosses the vernal 
equinox, and runs from zero to twenty-four hours. The sidereal 
time at any moment is the hour angle of the vernal equinox at 
that moment, counting westward from the meridian. As the 
right ascensions of stars and meridians are counted eastward 
from the vernal equinox, it, follows that the sidereal time 
for any observer is the same as the right ascension of his 
meridian at that moment. Hence when a star of known 
right ascension crosses the meridian the sidereal time 
becomes known at that moment. The right ascension 
of the mean sun at Greenwich mean noon (called sidereal 
time of Greenwich mean noon) is given in the Nautical 
Almanac for every day of the year, and is readily found 
for local mean noon at any pther meridian by adding the 
product of 9.8565 seconds by the given longitude west of Green- 
v,dch expressed in hours. 



ASTRONOMICAL DETERMINATIONS 169 

98. To Change a Sidereal to a Mean Time Interval, and vice 
versa. Owing to the relative directions in which the earth rotates 
on its axis and revolves around the sun the number of sidereal 
days in a tropical year (one complete revolution of the earth 
around the sun) is exactly one more than the number of solar 
days. According to Bessel the tropical year contains 365.24222 
mean solar days, hence 365.24222 mean solar days = 366.24222 
sidereal days, and therefore 

1 mean solar day= 1.0027379 sidereal days; 
1 sidereal day = 0.9972696 mean solar days; 

whence if Is is any sidereal interval of time and Im the mean solar 
interval of equal value, we have 

/<, = /m + 0.0027379 /„, (log 0.0027379 = 7.4374176 - 10) 
Im=Is - 0.0027304 I, (log 0.0027304 = 7.4362263 - 10) 

Where there is much of this work to be done the labor of computa- 
tion is lessened by usin^ the tables found in the Nautical Almanac 
and books of logarithms. 

99. To Change Local Mean Time or Standard Time to Sidereal. 
For local mean time this is done by converting the mean time 
interval between the given time and noon into the equivalent 
sidereal interval (Art. 98), and combining the result with the 
sidereal time of mean noon for the given place and date. Since 
the right ascension of the mean sun increases 360° or twenty- 
four hours in one year, the increase per day will be 3™ 56^.555, 
or 9^.8565 per hour. The sidereal time of mean noon for the 
given place is therefore found by taking the sidereal time of Green- 
wich mean noon from the Nautical Almanac and adding thereto 
the product of 9^8565 by the longitude in hours of the given 
meridian, counted westward from the meridian of Greenwich. 
If standard time is used it must first be changed to local mean 
time (Art. 966) before applying the above rule. 

Example. To find the sidereal time at Syracuse, N. Y., longitude 
76° 08' 20" .40 west of Greenwich, when the standard (75th meridian) time 
is 10" 42"! 00' A.M., January 17th, 1911. 



170 GEODETIC SURVEYING 



76° 08' 20". 40 

75 


IQH 42™ 00 '.00 standard time 
- 4 33 .36 


15) 1° 08' 20". 40 
4m 33=. 36 


10 37 26 . 64 local mean time 

12 


log 4953.36 =3.6948999 
log . 0027379 = 7 . 4374176 


111 22" 338. 36 = 49533.36 
+ 13 .56 


log (13^56) =1.1323175 


1 22 46 . 92 sidereal interva 


15)76° 08' 20". 40 
5h 04^338.36 


log 9.8565 = 0.9937227 
log 5.0759=0.7055131 


= 5.0759 hrs. 


log (503.03) = 1.6992358 



Sidereal time of Greenwich mean noon 19'' 43™ 093.48 
Reduction to Syracuse meridian + 50 .03 



Sidereal time of Syracuse mean noon 19 43 59 . 51 
Sid. int. from Syracuse mean noon — 1 22 46 .92 



Sidereal time at given instant IS^ 21™ 12^. 59 

100. To Change Sidereal to Local Mean Time or Standard 
Time. This is the reverse of the process in Art. 99, and consists 
in finding the difference between the given time and the sidereal 
time of mean noon for the given place and date, changing this 
interval to the corresponding mean time interval (Art. 98), and 
combining the result with twelve o'clock (mean noon) by addi- 
tion or subtraction as the case requires. The result is local mean 
time, and if standard time is wanted it is then obtained as 
explained in Art. 966. 

Example. To find the local mean time and standard (75th meridian) 
time at Syracuse, N. Y., longitude 76° 08' 20".40 west of Greenwich, when 
the sidereal time it 181 21™ 12s.59, January 17, 1911. 

76° 08' 20".40-75° = l° 08' 20". 40 = 4™ 333.36 
log 9.8565 =0.9937227 
15)76° 08' 20".40 log 5.0759 =0.7055131 



5t 04m 33s. 36 

= 5 . 0759 hrs. log (50^ . 03) = 1 . 6992358 



ASTEONOMICAL DETERMINATIONS 171 

Sidereal time of Greenwich mean noon 19'' 43™ 09^.48 

Reduction to Syracuse meridian + 50 .03 



Sidereal time of Syracuse mean noon 19 43 59 . 51 

Sidereal time at given instant 18 21 12 . 59 



Sidereal interval before Syracuse mean noon P 22™ 46^.92 



Ih 22™ 46^.92 = 4966^92 

log 4966. 92 =3.6960872 
log . 0027304 = 7 . 4362263 

log (133.56) =1.1323135 



Reduction to 1 1" 22™ 46^92 



mean time 
interval 



- 13 .56 

1 22 33 .36 
12 



Local mean time at given instant (mormng) 10'' 37™ 26^.64 
Reduction to standard time +4 33 . 36 

Standard time at given instant (morning) 10'' 42™ 00^.00 



101. Time by Single Altitudes. The altitude of any heavenly- 
body as seen by an observer at a given point is constantly chang- 
ing, each different altitude corresponding to a particular instant 
of time which can be computed if the latitude and longitude are 
approximately known. In finding local mean time or sidereal 
time it is sufficient to know "the latitude to the nearest minute 
and the longitude within a few degrees. In changing from local 
to standard time, however, an error of V will be caused by each 
15" error of longitude. If the latitude is not known it may 
generally be scaled sufficiently close from a good map, or it may 
be determined as explained in Arts. 107 or 108. By comparing 
the observed time for a certain measured altitude of sun or star 
with the corresponding computed time the error of the observer's 
timepiece is at once determined. The observation may be 
made with a transit (or altazimuth instrument), or with a sextant 
(and artificial horizon), the latter being the most accurate. In 
either case several observations ought to be taken in imme- 
diate succession, as described below, and the average time and 
average altitude used in the reductions. The probable error of 
the result may be several seconds with a transit, and a second 
or two with the sextant. The actual error is apt to be larger on 
account of the uncertainties of refraction. The observation is 
commonly made with the sextant and on the sun. 



172 GEODETIC SURVEYING 

101a. Making the Observation. The best time for making 
an observation on the sun is between 8 and 9 o'clock in the 
morning and between 3 and 4 o'clock in the afternoon, in order 
to secure a rapidly changing altitude and at the same time avoid 
irregular refraction as far as possible. The altitude of the 
center of the sim is never directly measured, but the observations 
are taken on either the upper or lower limb, or preferably an equal 
number of times on each limb. Star observations may be made 
at any hour of the night, selecting stars which are about three 
hours from the meridian and near the prime vertical, and hence 
changing rapidly in altitude at the time and place of observation. 
If two stars are observed at about the same time having about 
the same declination and about the same altitude, but lying on 
opposite sides of the meridian, the mean of the two results (de- 
terminations of the clock error) will be largely free from the errors 
due to the imcertainties of refraction. 

In taking the observation an attendant notes the watch 
time to the nearest second at the exact moment the pointing 
is made. // the transit is used, an equal number of readings 
should be taken with the telescope direct and reversed, the plate 
bubble parallel to the telescope being brought exactly central 
for each individual pointing in order to eliminate the instrumental 
errors of adjustment. If a star or one limb of the sun is observed 
there should be not less than 3 direct and 3 reversed readings. 
If both limbs of the sun are observed there should be not less 
than 2 direct and 2 reversed readings on each limb, or 3 direct 
on one limb and 3 reversed on the other limb. If the sextant and 
artificial horizon are used, and the pointings are made on a star 
or on one limb of the sim, not less than 5 readings of the double 
altitude should be taken; if both limbs of the sun are observed, 
not less than 3 readings should be taken for each limb. These 
double altitudes are always corrected for index error and some- 
times for eccentricity. It is considered better not to use the 
cover on the artificial horizon, but if it has to be done it should 
be reversed on half of the readings. If as much tin foil is added 
to commercial mercury as it will unite with, an amalgam is formed 
whose surface is not readily disturbed by the wind, thus rendering 
the cover unnecessary. When the mercury is poured in its 
dish it must be skimmed with a card to clean its reflecting surface. 
In all of the above methods of observing, the work is supposed 



ASTRONOMICAL DETERMINATIONS 173 

to be C9,rried on with reasonable regularity and expedition when 
once started. With any method it is desirable to take at least 
two sets of readings and compute them independently as a check, 
the extent of the disagreement showing the quality of the work 
that has been done, while the mean value is probably nearer the 
truth than the result of any single set. 

101b. The Computation. The first step in the computation 
of any set of observations is to find the average value of the meas- 
ured altitudes and the average value of the recorded times, these 
average values constituting the observed altitude and time for 
that set. This observed altitude is then reduced to the true 
altitude for the center of the object observed. The reductions 
which may be required are for refraction, parallax, and semi- 
diameter. The apparent altitude of all heavenly bodies is too 
large on account of the refraction of Ught; Table VIII gives the 
average angular value of refraction, which is a negative correc- 
tion for all measured altitudes. Parallax is an apparent dis- 
placement of a heavenly body due to the fact that the observer 
is not at the center of the earth; star observations require no 
correction for parallax; all solar observations require a positive 
correction for parallax, the amount being equal to 8". 9 multiplied 
by the cosine of the observed altitude. The correction for 
semi-diameter is only required in solar work, and not even then 
for the average of an equal number of observations on both limbs; 
when the average altitude refers to only one limb a self-evident 
positive or negative correction is required for semi-diameter, 
the value of which is given in the Nautical Almanac for the me- 
ridian of Greenwich for every day of the year, and can readily 
be interpolated for the given longitude. Letting h equal true 
altitude for center, h' equal measured altitude, r equal refrac- 
tion, p equal parallax, and s equal semi-diameter, we have 

h (for a star) = h' — r; 

h (sun, both limbs) = h' — r + p; 

h (sun, one limb) = h' — r + p ± s. 

In the polar triangle ZPS, Fig. 47, page 166, the three sides are 
known. ZP, the co-latitude, is found by subtracting the observer's 
latitude from 90°. PS, the polar distance or co-declination, is 



174 GEODETIC SUEVEYING 

found by subtracting the declination of the observed body from 90°. 
In the case of the sun the declination is constantly changing and 
must be taken for the given date and hour (the time being always 
approximately known). The sun's declination for Greenwich 
mean noon is given in the Nautical Almanac for every day in the 
year, and can be interpolated for the Greenwich time of the observa- 
tion; the Greenwich time of the observation differs from the 
observer's time by the difference in longitude in hours, remember- 
ing that for points west of Greenwich the clock time is earlier, and 
vice versa. ZS, the co-altitude, is found by subtracting the 
reduced altitude h from 90°. Using the notation of Fig. 47, 
we have from spherical trigonometry 

cos 3 = sin ^ sin I? + cos ^ cos § cos t, 

whence 

cos z — sin <i sin d 

cos t = -7 — ^—s , 

cos <p cos 

which for logarithmic computation is reduced to the form 



/ sin i[z + (^ - g)] sin i[z - (^ - g)] 
^"""^ '' ^'cos i[z + (96 + 8)] cos i[z -i^ + d)]- 

The value of t thus found is the hour angle of the observed body, 
or angular distance from the observer's meridian. Dividing t 
by 15 changes the angular value to the corresponding time interval. 

For a solar observation the time interval is subtracted from or 
added to 12 o'clock according as the sun is east or west of the 
meridian, giving the apparent solar time of the observation. 
This apparent time must be reduced to mean time by applying 
the equation of time for the given date and hour, taken from the 
Nautical Almanac in the manner above described for finding the 
declination. The local mean time of the observation as thus 
found may be changed to standard time (Art. 96&), or sidereal 
time (Art. 99), if so desired. 

For a star observation the time interval is subtracted from or 
added to the star's right ascension according as the star is east 
or west of the meridian, giving the sidereal time of the observation. 
This may be changed to local mean time (Art. 100), and thence 
to standard time (Art. 966), if so desired. 



ASTEONOMICAL DETEEMINATIONS 175 

EXAMPLE.— TIME BY SINGLE ALTITUDES OF THE SUN 



Chicago, III., June 1, 1911. 
Latitude =41° 50' 01". N. 

Longitude = 87° 36' 42".0 = 5h 50™ 26^.8 = 5.84 hrs. W. of Greenwich. 
Uses 90th meridian (Central Standard) time = 6.00 hrs. W. of Greenwich. 
Localtime— Standard time = 6" 00™ 00^0 -5i 50™ 26^.8 = 9™ 33^.2. 

h and e. 
Par. = 8". 9 X cos 48° 20' = 6". 7 

App. altitude = 48° 20' 00". 
Refraction = — 51 .3 
Parallax = + 6 .7 

;i=48° 19' 15". 4 



Sun. 
Lower limb • 


h' 

■ 47° 00' 

47 20 

L47 40 


Watch, A.M. 
gh 46™ 11^ 
8 48 06 
8 50 03 


Upper limb 


■49 00 

49 20 

.49 40 


8 54 45 
8 56 41 
8 58 38 


6)290° 00' 


6)53" 14™ 24^ 


Average 


48° 20' 


St 52™ 24^.0 



z = 41° 40' 44". 6 



Approximate interval after Greenwich mean noon 

= 8''52™24s.0+6hrs-12hrs = 2i 52™ 24^.0 = 2.87 hours. 

Time. d dd Eq. of Time. 

At Greenwich mean noon +21° 56' 33".6 +21".23 2™ 31^.87 

Reduction for 2.87 hrs. + 1 00 .8 - .11 - 1 .04 

At time of observation +21° 67' 34".4 +21".12 2^303.83 

Equation of time subiractive from apparent time (on given date). 

0.96X2.j_7^^„^^ 2_1,23 + 2^12^^^„ ^^ 

0.363X2. 87 = 1=. 04 21.18X2.87 =60". 8 

^-5 =19°52' 26".6 (j> + 3 = 63° 47' 35" .4 

z+(^-5) = 61 33 11 .2 z+{'p + 5)= 105 28 20 .0 

z-{(j>-d) = 21 48 18 .0 z-(^ + d) =~22 06 50 .8 



tan i.= v ^ f °° f' y '•;; -'- , ^!f '': ;r-;! -21° 58' 37".7 

^ V cos (52 44 10 .0) cos ( — 11 03 25 .4) 
« = 43° 57' 15".4 = 2i» 55™ 49^.0. 

Local apparent noon 12i> 00™ 00^0 

Hour angle of sun — 2 55 49 .0 

Apparent solar time 9^ 04™ 11^.0 

Equation of time — 2 30 . 8 

Local mean time of observation Q^ 01™ 40^.2 

Watch time of observation 8 52 24 .0 

Watch slow by mean time 9"^ 16^ . 2 

Reduction to standard time — 9 33 .2 

Watch fast by standard time 0™ 17= . 



176 GEODETIC SURVEYING 

In either case the error of the observer's timepiece (as deter- 
mined by any given set of observations) is obtained by comparing 
the observer's average time for the given set with the computed 
true time for the same set. 

102. Time by Equal Altitudes. In this method the clock 
time is noted at which the sun (or a star) has the same altitude 
on each side of the meridian, from which the clock time of meridian 
passage (upper or lower transit or culmination) is readily obtained. 
By comparing the clock time with the true time of meridian 
passage the error of the observer's clock is at once made known. 
The advantages of this method over the method of single altitudes 
are as follows: the results are in general more reliable; the com- 
putation is simpler, as it does not involve the solution of a spherical 
triangle; no correction is required for refraction, parallax, semi- 
diameter, nor instrumental errors; the latitude need not be known 
at all for star observations, and only very approximately for 
solar work. The observations may be made with a transit or a 
sextant (with artificial horizon), the latter being the most accurate. 
In either case several observations ought to be taken in immediate 
succession, as described below, and the average time used in the 
reductions. The probable error of the result should not exceed 
about two seconds with the transit nor about one second with 
the sextant. The actual error may be greater on account of the 
uncertainties of refraction. The method evidently assumes that 
the refraction will be the same for each of the equal altitudes, 
but on account of the lapse of time between the observations 
this is not necessarily true. The observation is commonly made 
with the sextant and on the sun. 

102a. Making the Observation. As with the previous 
method, the best time for making an observation on the sun is 
between 8 and 9 o'clock in the morning and between 3 and 4 
o'clock in the afternoon. The observations may be taken entirely 
on one limb of the sun or an equal number of times on each limb. 
The equal altitudes may be taken on the morning and afternoon 
of the same day, or on the afternoon of one day and the morning 
of the next day. For star observations a star should be selected 
which will be about three hours from the meridian and near the 
prime vertical at the times of observation. Since the equal 
altitudes observed must be within the hours of darkness, a star 
is required whose meridian passage occurs within about three 



ASTRONOMICAL DETERMINATIONS 177 

hours after dark and three hours before daylight. The sidereal 
time of meridian passage is always known, since it is the same 
as the star's right ascension, and the corresponding values of 
mean time and standard time are readily found by Arts. 100 and 
966. The equal altitudes may be taken during the same night, or 
on the morning and evening of the same day. 

In taking the observation the attendant notes the watch 
time to the nearest second at the exact moment the pointing is 
made. If the transit is used the telescope is not reversed, but the 
plate bubble parallel to the telescope is brought exactly central 
for each individual pointing; no corrections are made to the result- 
ing reading for any instrumental errors. If the sextant and 
artificial horizon are used no corrections are applied to the result- 
ing double altitude as measured. There is no great objection 
to using the cover of the artificial horizon in this method, and 
when used it is not reversed (as in Art. 101a); it is necessary, 
however, to use it in the same position at both periods of equal 
altitudes. 

If a star or one limb of the sun is observed there should be 
not less than 5 readings taken at each period of equal altitudes. 
If both limbs of the sun are observed there should be not less than 
3 readings (at each period) for each limb. The angular readings 
in this method are always equally spaced, the instrument being 
set in turn for each equal change of altitude and the time noted 
when the event occurs. In commencing operations the observer 
measures the approximate altitude, sets his vernier to the next 
convenient even reading, and watches for that altitude to be 
reached; the next setting is then made and that altitude waited 
for, and so on. At the second period the same settings must be 
used, but in reverse order. The size of the angular interval 
will depend on the abihty of the observer to make each setting 
in time to catch the given occurrence, and can best be found by 
trial; under average conditions a good observer would not find 
it difficult to use 10' settings on the transit and 20' on the sextant. 
It is desirable to take at least two independent sets of observa- 
tions, and compute them separately as a check and as an indica- 
tion of the reliability of the results; the adopted value would 
then be taken as the mean of the several determinations. 

102b. The Computation. In this method there is no object 
in finding the average of the observed altitudes, the method 



178 GEODETIC SURVEYING 

being based on the equality of the corresponding altitudes in- 
stead of their value. For each set of observations, however, 
it is necessary to find the average of the time readings for 
each of the two periods of equal altitudes. From these values 
the middle time (half-way point between the two average time 
readings) is found for star observations, and the middle time and 
elapsed time (interval between average time readings) for solar 
observations. For star observations the middle time is the 
observer's time of meridian passage. For solar observations 
a correction must be applied to the middle time to obtain the 
observer's time of meridian passage, on account of the changing 
declination of the sun. 

For solar observations on the same day, expressed in mean time 
units, we have from astronomy 



TT — M dd ■ t fisin. <ji tan ^\ 
15 \sin t tan tj' 



in which 



TJ = observer's time at apparent noon (upper transit of sun) ; 
M = middle time of the observations; 
t = one-half elapsed time, in hours to three places outside 

of parentheses and angular value inside of parentheses; 
(p = observer's latitude (approximate), + for north and — 

for south latitude; 
8 = sun's declination at mean noon for given date and longi- 
tude, + for north and — for south declination; 
d8 = hourly change of declination at mean noon for given 
date and longitude, + when north declination is increasing 
or south declination decreasing, and — when north declina- 
tion is decreasing or south declination increasing. 

The values for 8 and dd for the given date are found in the 
Nautical Almanac for Greenwich mean noon and interpolated for 
the given meridian. If a sidereal chronometer is used it is neces- 
sary to convert t into a mean time interval before inserting in the 
corrective term in the above formula, and the value of this term 
must then be reduced to a sidereal interval before subtracting 
from M. 

The true mean time of apparent noon is found by applying 



ASTRONOMICAL DETERMINATIONS 179 

to 12 o'clock (the apparent time) the equation of time for the given 
date and longitude. The equation of time (with directions for 
applying) is found in the Nautical Almanac for Greenwich 
apparent noon of the given date, and interpolated for the given 
meridian. The true mean time of apparent noon is then reduced 
to standard time (Art. 966), or sidereal time (Art. 99), if so desired. 

By comparing the observer's time, U, with the corresponding 
true time of apparent noon, the error of the observer's timepiece 
at apparent noon is made known. 

For solar observations on an afternoon and following morning, 
expressed in mean time units, we have from astronomy 

T — M 4- ^^"^A^^ ^ I tan 8 
15 \ sin t tan t 

in which L is the observer's time at apparent midnight (lower 
transit of sun), d and dd the declination and hourly change for 
mean midnight of initial date, and the other quantities remain 
as before. This problem is worked out as in the preceding case 
except that §, dd, and the equation of time must be interpolated 
for twelve hours more than the given longitude, and the clock 
error is determined for apparent midnight of the initial date. 

For star observations during the same night, taken on the same 
star, the middle time represents the observer's time for the star's 
upper transit. The true sidereal time of this transit equals the 
star's right ascension, as given in the Nautical Almanac, and this 
is changed to local mean time (Art. 100), and thence to standard 
time (Art 966), if so desired. 

By comparing the observer's middle time with the true time 
of upper transit, the error of the observer's timepiece is deter- 
mined for the moment at which it indicated the middle time. 

For star observations on morning and evening of same day, taken 
on the same star, the middle time represents the observer's time 
for the star's lower transit. The true sidereal time of this transit 
equals the star's right ascension plus twelve hours, and this is 
changed to local mean time (Art. 100), and thence to standard 
time (Art. 966), if so desired. 

By comparing the observer's middle time with the true time 
of lower transit, the error of the observer's timepiece is determined 
for the moment at which it indicated the middle time. 



180 



GEODETIC SURVEYING 



EXAMPLE.— TIME BY EQUAL ALTITUDES OF THE SUN 



Albany, N. Y., May 10, 1911. 

Latitude =42° 39' 12". 7 N. 

Longitude = 73° 46' 42". = 4" 55"068. 8 = 4. 92hrs.W. of Greenwich. 
Uses 75th meridian (Eastern Standard)time = 5.00 hrs. W. of Greenwich. 
Local time -Standard time = 5" 00" 008.0-4" 55" 068.8 = 4" 538.2 



Time. 
At Greenwich mean noon 
At Greenwich app. noon 

Reduction for 4 . 92 hrs. 
At Albany mean noon 
At Albany app. noon 



+ 17° 23' 56". 7 



dS • Eq. of Time. 



+ 
+ 17' 



3 

27' 



15 .8 
12". 5 



+ 39". 87 

- .15 

+ 39". 72 



3" 41=. 2 
+ .6 

3" 418.8 



Equation of time suhtractive from apparent time{on given date). 





0,73X4 
24 


^=0". 


39.87 + 39.72 
15 = 39". 80 


0.126X4 


92 = 08.62 39.80X4.92 =195". 8 


Sun. 


App. alt. 


Watch, A.M. Watch, p.m. 


Upper limb 


r 45° 00' 
] 45 20 
[45 40 


8" 58" 228 211 54m 133 
9 00 18 2 52 18 
9 02 12 2 50 24 


Lower limb 


(-45 40 

46 00 

.46 20 


9 05 14 2 47 20 
9 07 10 2 45 25 
9 09 05 2 43 29 


t 


f = 11" 56" 178 
= 2" 52" 348 
= 2.88hrs. 
= 43° 08' 30" 


6)54" 22" 21= 6)16" 53" 098 

5 9"03"43s.5 ( + 12)2" 48" 518.5 

( + 12) 2 48 51 .5 9 03 43 .5 

2)23l» 52" 358.0 2)5'' 45^088.0 

° lit 56" 178.5 2^52" 348.0 


tan 
sin 


4, (42° 39' 12" 
t (43 08 30 


log. log. 
. 7) = 9 . 9643882 tan 5 (17° 27' 12" . 5) = 9 . 4974948 
.0) =9.8349320 tan t (43 08 30 .0) = 9.9718084 



(1.3473) 



=0.1294562 



(0.3355) 



= 9 . 5256864 



1.3473-0.3355 = 1.0118 

M = llh 56" 178.5 
- 7 .7 


d^ (39.72) 

i(2.88) 

15 (a.c.) 

1.0118 


= 1 . 5990092 
= 0.4593925 
= 8.8239087 
= 0.0050947 


(7 = 11" 56" 098. 8 

Local apparent noon 

Equation of time 
Local mean time at apparent noon 

Watch time at apparent noon 
Watch slow by mean time 

Reduction to standard time 
Watch fast by standard time 


(78.7) 

12" 00" 008 
- 3 41 


= 0.8874051 

.0 

.8 


lit 56m 188 
11 56 09 
0"08s 
- 4 53 

4m 443 


.2 

.8 
.4 
2 
8 



ASTRONOMICAL DETERMINATIONS 181 

103. Time by Sun and Star Transits. The true time at which 
any heavenly body crosses the meridian is always known; in the 
case of the sun the upper transit is apparent noon, the mean 
time of which is determined by the equation of time (Art 96); 
in the case of a star the sidereal time at upper transit is the 
same as the star's right ascension; and (by Arts. 966, 99, and 100) 
sidereal time, mean time and standard time are mutually 
convertible. If the observer notes his own clock time when any 
heavenly body crosses the meridian, the error of his timepiece 
is made apparent by comparison with the corresponding known 
true time. In order that the observation may be made it is 
necessary to loiow the location of the true meridian from a pre- 
vious azimuth determination. (Astronomers have other ways 
of obtaining the meridian.) With the telescope in the plane of 
the true meridian, and set at a suitable vertical angle, it is only 
necessary to note the time when the given transit occurs. 

103a. Sun Transits with Engineering Instruments. The instru- 
ments used for determining time by transits of the sun may be 
the ordinary engineer's transit or the altazimuth instruments of 
Chapter III. A prismatic eyepiece will be required if the 
meridian altitude exceeds about 60°. The instrument (and 
striding level, if there be one) should be in good adjustment. 
The instant at which the advancing edge of the sun reaches the 
meridian is noted with the telescope direct, and the instant at 
which the following edge reaches the meridian is noted with the 
telescope reversed, the mean of the two time readings being the 
observer's time of meridian passage. When the telescope is 
reversed it will be necessary to revolve the instrument on its 
vertical axis, and the telescope must' be again brought into the 
plane of the meridian by sighting at the meridian mark as before. 
If the instrument has no striding level the plate bubble parallel 
to the horizontal axis of the telescope is to be kept exactly central 
while each observation is being made. If the instrument has a 
striding level it must not be reversed when the telescope is 
reversed, but the bubble must be kept central, as before, for 
each observation. If the instrument has three leveling screws 
it should be set with two screws parallel to the meridian and 
the bubble kept central with the remaining screw; if there are 
four leveling screws, place one pair in the meridian and hold the 
bubble central with the other pair. Time determined in the 



182 GEODETIC SURVEYING 

above maimer should not be in error by more than a second with 
an altazimuth instrument, nor by more than a couple of seconds 
with an engineer's transit. 

The above method is not adapted to precise time determina- 
tions, so that when the larger astronomical instruments are avail- 
able the observations are usually made on the stars. 

103b. Star Transits with Engineering Instruments. The 
instruments used by the engineer for determining time by star 
transits may be the ordinary transit or the altazimuth instru- 
ments of Chapter III. The instrument (and striding level, if 
there be one) should be in good adjustment. The stars have no 
appreciable diameter, so that only one observation is obtained 
for each star. Since the true time of each star transit will be 
needed in the reductions it is desirable to tabulate these values 
beforehand, in order to be ready to watch for each transit near 
the proper time, as a star occupies only about a minute or two 
in crossing the field of view. As previously explained (Art. 97) 
the sidereal time of transit for each star is the same as its right 
ascension; if the observer's timepiece records mean or standard 
time it will be necessary to reduce the sidereal time of transit 
accordingly, as explained in Art. 100. In order to eliminate instru- 
mental errors the stars are observed in pairs, the two stars of 
each pair having about the same declination; the second star 
of each pair is then observed with the telescope reversed. The 
instructions in the preceding article concerning the reversing and 
releveling of the instrument must be strictly adhered to. Only 
one result is obtained from each pair of stars, the average true 
time of transit for each pair being compared with the middle 
observed time for that pair to obtain the clock error for that 
instant of time. Not less than three pairs of stars should be 
observed and the results averaged If the clock rate is not known 
the middle times for the several pairs should not differ greatly, 
the average of the error determinations being considered as the 
true value at the average of the middle times. If the clock rate 
is known the several error determinations are first reduced to the 
same instant of time before averaging. 

Selection of stars. If several pairs of stars are observed it 
makes no difference in what order the stars come to the meridian 
so long as they are properly paired in the reductions. If the 
stars are so selected that all the first stars of the several pairs 



ASTRONOMICAL DETEEMINATIONS 183 

will cross the meridian before any of the second stars, but one 
reversal of the instrument will be required; this will be the case 
if all the first stars have less right ascensions than any of the 
second stars. In order to have ample time between observations 
for releveling, etc., stars should not be selected having right 
ascensions differing by about less than five minutes. Having 
decided on the period of the night during which it is desired to 
make the observations, the mean time for the beginning and end 
of this period must be converted into approximate sidereal time, 
and stars must be selected whose right ascensions lie within these 
limits. The approximate sidereal time for any mean time instant 
is found by adding the mean time interval from the preceding 
noon to the sidereal time of Greenwich mean noon for the same 
dat^. Stars near either pole are not suitable for time stars on 
account of their apparent slow movement across the meridian; 
it is not desirable to use stars whose declination is more than 60° 
either way from the equator. On accoimt of the uncertain state 
of the atmosphere at low altitudes stars should not be selected 
which will cross the meridian less than 30° above the horizon. Thus 
in 40° north latitude (see Fig. 47, page 166), the horizon will lie 50° 
south of the equator, and hence stars should not be taken lying 
over 20° south of the equator, so that for this latitude the stars 
selected should lie between 60° north declination and 20° south 
declination. The altitude of any star while crossing the meridian 
is readily obtained when it is remembered that the meridian 
altitude of the equator equals the observer's co-latitude, and that 
the star's distance from the equator is given by its declination. 
A prismatic eyepiece will be required for meridian altitudes 
exceeding about 60°. 

It is best to use the brightest stars available for the given 
time and place, as it is not easy to identify or observe the fainter 
stars; satisfactory results may be obtained with stars ranging 
from the first (brightest) magnitude to about the fifth magnitude, 
depending on the size of the instrument. A large list of stars 
from which to choose, with all necessary data, will be found 
in the Nautical Almanac. 

103c. Star Transits with Astronomical Instruments. The 
most accurate determinations of time are made by observing 
star transits with large portable astronomical transits or the 
still larger fixed observatory transits, in conjunction with an 



184 GEODETIC SUEVEYING 

astronomical clock beating seconds or a sidereal chronometer 
beating half seconds. A portable transit is illustrated in Fig. 48. 
A chronograph is generally used in the observatory, and 
sometimes in the field, for recording the observations. A chrono- 
graph is a clock-like device for moving a sheet of paper uniformly 
under a pen which automatically registers each second as indicated 
by the clock or chronometer; by breaking an electric circuit 
the observer causes the pen to record the star transits on the same 
sheet of paper; the time of transit is then obtained very accurately 
by scaling the distance from the nearest recorded second. When 
the chronograph is not used the observer listens to the chronom- 
eter beats and estimates the time of each transit to the 
nearest tenth of a second. The details of the instruments 
used, and the refinements in the methods of observation 
and computation, are beyond the scope of this treatise, but 
the principles involved are the same as those already given. 
The accuracy attainable is to about the nearest one-hundredth 
part of a second. 

104. Choice of Methods. Though other methods have been 
devised for determining time, those above given are the ones in 
most general use. The engineer may use any of the methods from 
Art. 101 to Art. 103&, inclusive. Engineers generally prefer to 
work in the daytime, taking their observations on the sun. The 
transit may be used, but the sextant is to be preferred. If the 
transit is used the method based on the meridian passage of the 
sun (Art. 103a) is likely to be the most satisfactory, while if the 
sextant is used the method of equal altitudes (Arts. 102, 102a, 
1026) will generally give the best results. Any of the methods 
will determine the true time as closely as the engineer will need 
it in any of his operations. 

105. Time Determinations at Sea. There are several methods 
of finding local time at sea, the method by single altitudes (Art. 
101) being most commonly used. The object observed may be 
the sun or one of the brighter stars. The observations are made 
with the sextant, the altitudes being measured from the sea 
horizon. This horizon is not the true horizon on account of the 
height of the observer above the surface of the sea and the effects 
of refraction. The result of this condition is to make all measured 
altitudes too large by an angle depending on the height of the 
observer and known as the dip of the horizon. The correction 



ASTRONOMICAL DETERMINATIONS 



185 




Fig. 48.— Portable Transit. 
From a photograph loaned by the U. S, C. and G. S. 



186 GEODETIC SUEVEYING 

for dip is always subtractive, and is in addition to the corrections 
required by Art. 101. Its value is given by the formula 

log D = 1.7712700 + i log h, 

in which D is the dip in seconds of arc and h is the observer's 
height in feet above the sea. The latitude required in the for- 
mula of Art. 101 is obtained sufficiently close by dead reckoning 
from the nearest observed latitude. Time at sea may be deter- 
mined in this manner with a probable error running upwards 
from a few seconds, depending on the circumstances surrounding 
the observations. 

Latitude 

106. General Principles. The latitude of a point on the 
surface of the earth is its angular distance from the equator in a 
meridional plane. In Fig. 49 the ellipse WNES represents a 
meridian section of the earth (Arts. 65, 66, 67), in which NS is 
the polar axis, or minor axis of the ellipse; WE, the equatorial 
diameter, or major axis of the ellipse; n, the position of the 
observer; nt the tangent at n; nl, the normal at n, it being noted 
that the normal at any point n does not pass through the center 
c (except when n is at the poles or on the equator) ; Zn, the direc- 
tion of the plumb line at n, frequently deviating a few seconds 
(Art. 75) from the direction of the normal nl; Z, the zenith, 
or intersection of the direction of the plimab line with the celestial 
sphere (Art. 94). 

Astronomical latitude is the angular distance of the zenith 
from the equator, or the angle between the plmnb line and the 
equatorial plane. In Fig. 49 the astronomical latitude of the 
point n would be shown by prolonging the line Zn to an intersec- 
tion with the line WE, the intersection commonly falling slightly 
to one side of the point I and making the angle a few seconds greater 
or less than the angle ^. The latitude as determined by observa- 
tion is always the astronomical latitude. Latitudes obtained at 
sea are of this kind. 

Geodetic latitude is the angle between the normal and the 
equator; in Fig. 49 the geodetic latitude of the point n is the 
angle <f>. The geodetic latitude can never be directly observed, 
nor can the deviation of the plumb line be found by direct meas- 



ASTEONOMICAL DETEEMINATIONS 



187 



urement. If, however, the latitude of the point n be found 
by computation (Chapter V) from the astronomical latitudes 
measured at various other triangulation stations, and these 
values be averaged in with its own astronomical latitude, the 
result may be assumed to be free from the effects of plumb line 




deviation and to represent the true geodetic latitude. In geodetic 
work geodetic latitude is always understood unless otherwise 
specified. 

Geocentric latitude is the angle between the equator and the 
radius vector from the center of the earth; in Fig. 49 the geo- 
centric latitude of the point n is the angle /?. The geocentric 



188 GEODETIC SURVEYING 

latitude can never be directly observed. It is computed from the 
geodetic latitude by the formula 



in which (Art. 69) 



62 

tan /? = "2 tan <{>, 



log ^ = 9.9970504 - 10. 



At the equator the geodetic and geocentric latitudes are each 
equal to zero. At the poles they are each equal to 90°. At any 
other point the geocentric latitude is less than the geodetic 
latitude. By the calculus we have, 

tan 96 (for ^ - /? = max.) = ^, or (j> = 45° 05' 50".21; 

tan /? (for 4> - ^ = max.) = -, or /? = 44 54 09 .79; 

or a maximum difference of 11' 40".42. The popular conception 
of latitude is geocentric latitude, but published latitudes are 
usually astronomical latitudes or geodetic latitudes. 

107. Latitude from Observations on the Sun at Apparent 
Noon. Latitude sufficiently close for many purposes may be 
obtained by measuring the altitude of the sun at apparent noon, 
or the moment when it crosses the meridian. The local mean 
time of apparent noon is found by applying to 12 o'clock (the 
apparent time) the equation of time as taken from the Nautical 
Almanac for the given date, interpolating for the given meridian; 
the corresponding standard time may then be found by Art. 96a. 
If the correct time is not known the altitude is measured 
when it attains its greatest value, which soon becomes evident 
to the observer who is following it up. A good observer can obtain 
an observation on each limb of the sun before there is any appre- 
ciable change of altitude, the mean of the readings being the 
observed altitude for the center; if only one limb is observed 
the reading must be reduced to the center by applying a correc- 
tion for semi-diameter as found in the Nautical Almanac for the 
given date, the result being the observed altitude. In either 
case '(the observed altitude is too large on account of refraction, 
and must be corrected by an amount which may be taken from 



ASTRONOMICAL DETERMINATIONS 189 

Table VIII for the given observed altitude. Theoretically all 
solar altitudes are measured too small on account of parallax 
(due to the observer not being at the center of the earth), the 
necessary correction being equal to 8".9 multiplied by the cosine 
of the observed altitude. The correction for parallax is a useless 
refinement with the engineer's transit, but may be applied, if 
desired, when a sextant or altazimuth instrument is used. 

The observation. Single altitudes of the sun may be measured 
with a transit or with an altazimuth instrument, but a pris- 
matic eyepiece will be required if the altitude exceeds about 60°. 
The instrument must be very carefully leveled at the moment of 
taking the observation, and if two readings can be secured the 
second reading should be taken on the other limb of the sun with 
the telescope reversed and the instrument carefully releveled, 
so as to eUminate the instrumental errors. If only one reading 
is seciu'ed it should be corrected for index error if one exists. If 
the altitude is not greater than about 60° an artificial horizon 
may be used and the double altitude measured with either of the 
above instruments or a sextant. If a transit or altazimuth 
instrument is used it is not reversed on any of the observa- 
tions, and it must not be releveled between the pointing to 
the sun and the pointing to its reflected image. If a sextant is 
used the correction for index error must be applied. 

The computation. Having applied the appropriate correc- 
tions to the measured altitude, as described above, the true 
altitude of the sun is obtained within the capacity of the instru- 
ment used. This value being subtracted from 90° gives the zenith 
distance of the sun. The declination of the swa. is taken from the 
Nautical Almanac for the given date and meridian, and this 
value is the distance of the sun from the equator. Knowing thus 
the distance from the equator to the sun, and from the sun to 
the zenith, an addition or subtraction (as the case requires) 
gives the zenith distance of the equator, and this value (Art. 106) 
is the observer's latitude. If an ordinary transit is used the 
latitude thus obtained should be correct to the nearest minute. 
If a sextant or an altazimuth instrument is used the result is 
generally much closer to the truth. Theoretically the result 
should be as accurate as the instrument will read, but there is 
always a doubt as to the precise value of the refraction, and the 
latitude obtained is subject to the same uncertainty. 



190 GEODETIC SURVEYING 

108. Latitude by Culmination of Circumpolar Stars. Stars 
having a polar distance (90° —declination) less than the observer's 
latitude never set, but appear to revolve continuously around 
the pole, and are hence called circumpolar stars. Such stars 
cross the observer's meridian twice every day, once above the 
pole (upper culmination) and once below the pole (lower culmina- 
tion). By referring to Fig. 47, page 166, it will be seen that the 
latitude of any place is always the same as the altitude of the 
elevated pole. By observing the altitude of a close circumpolar 
star at either upper or lower culmination, and combining the 
result (minus correction for refraction. Table VIII) with the star's 
polar distance (added for lower culmination, subtracted for 
upper culmination), the altitude of the elevated pole is obtained, 
and hence the observer's latitude. The polar distance must be 
based on the declination for the given date as found in the 
Nautical Almanac. The latitude as thus determined is much 
more reliable than that obtained by solar observations. 

In the northern hemisphere the best star to observe is Polaris 
(a Ursse Minoris), on account of its brightness (2nd magnitude) 
and its small polar distance (about 1° 10' in 1911). About the 
middle of the year both culminations of Polaris occur during 
daylight hours, rendering it unsuitable for observation. The next 
best star to observe is 51 Cephei, which also has a small polar dis- 
tance (about 2° 48' in 1911), but whose brightness (5th magnitude) 
is not equal to that of Polaris. As these two stars differ about 
five and one-half hours in right ascension, at least one of them 
must culminate during the hours of darliness. The sidereal time 
of upper culmination for either star is the same as its right ascen- 
sion (the exact value for the given date being taken from the 
Nautical Almanac), and this is converted into mean time by 
Art. 100. By a study of Fig. 50, which shows the arrangement of 
a number of stars in the vicinity of the north pole of the heavens, 
it will not be difficult to identify Polaris and 51 Cephei. The 
polar distances of these stars are so small that but little change 
of altitude occurs when they are near the meridian, so that several 
observations may be obtained and averaged. If the observations 
are taken within five minutes each side of the meridian the error 
in assuming the altitudes unchanging will not exceed 1" with 
Polaris and 2".5 with 51 Cephei, and may be ignored when observ- 
ing with engineering instruments. Within fifteen minutes either 



ASTRONOMICAL DETERMINATIONS 191 





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192 GEODETIC SURVEYING 

way from meridian passage the change in altitude (within 1" 
error) may be fomid, if desired, by multiplying the square of the 
time (in minutes) from culmination by 0".044 for Polaris and 
0".104 for 51 Cephei. If this correction is applied it is to be added 
to observations near upper culmination and subtracted from 
observations near lower culmination, to obtain the corresponding 
culminating altitude. 

In making the observation the altitude maybe directly measured 
with a transit or an altazimuth instrument. In order to eliminate 
instrumental errors at least two readings should be averaged 
together, one taken with telescope direct and one with telescope 
reversed. The instrument must be releveled after reversing, 
as it is necessary to have the bubbles exactly central at the moment 
each reading is taken. If by any accident only one reading is 
secured it must be corrected for index error, if one exists. The 
two readings should be obtained as near together and as near 
culmination as the skill of the observer will permit; two readings 
not over three minutes each way from the meridian are easily 
obtained. A better result will be obtained if four readings are 
averaged together, taking one direct reading, then two reversed 
readings, and then one direct reading, both bubbles being kept 
exactly central while taking each reading; this program is 
easily accomplished within five minutes each side of the meridian. 
If an artificial horizon is available it is better to measure the double 
altitude between the star and its image in the mercury, using 
either of the above instruments or a sextant. Angles measured 
with a sextant are always correicted for index error and sometimes 
for eccentricity. If a transit or altazimuth instrument is used the 
double altitude is obtained by reading on the star and then on 
its image, without reversing or releveling between the pointings. 
Two such double altitudes are easily obtained within three minutes 
each way from the meridian, using either of these instruments 
or a sextant. Latitudes obtained by the methods of this article 
should theoretically be correct within the reading capacity of 
the instrmnent, but may be further in error on account of the 
uncertainties of refraction. 

109. Latitude by Prime Vertical Transits. Stars whose 
declination is less than the observer's latitude apparently cross 
the pHme vertical (true east and west vertical plane) twice dur- 
ing each revolution of the earth on its axis. If the time elapsing 



ASTRONOMICAL DETERMINATIONS 



193 



between the east and west transit of any star is noted the observ- 
er's latitude may be found by com- 
putation. Referring to Fig. 51, F is 
the elevated pole of the celestial 
sphere; PZS', the observer's meridian; 
Z, the observer's zenith; SZS", the 
prime vertical; SS'S", the star's ap- 
parent path; PS, the star's polar 
distance; and PZ, the observer's co- 
latitude. In the spherical triangle 
PZS, right-angled at Z, the side PS 
and the angle SPZ are known; the 
side PS being the star's polar distance, 
and the angle SPZ equal to half the 

elapsed time changed to angular units by multiplying by 15. 
Hence, solving for the latitude ^, we have 




tan <p = 



tan 8 
cos t ' 



In this method the uncertainties of refraction are largely elim- 
inated because the times of transit are observed instead of the 
altitudes. The success of the method depends on the precision 
with which the meridian is determined and the prime vertical 
located therefrom, and the accuracy with which the telescope 

is made to describe a vertical plane. 
The method, though not much used 
in the United States, is one of the 
best, and with suitable instruments 
and refinements will determine lati- 
tude within a fraction of a second. If 
a close determination of latitude has 
to be made with an altazimuth instru- 
ment without a micrometer eyepiece, 
but which is furnished with a good 
stridinglevel,thismethod will probably 
give better results than any other. 

110. Latitude with the Zenith 
Telescope. This method (otherwise 
known as the Harrebow-Talcott 
method) is the one which the U. S. Coast and Geodetic Survey 




194 GEODETIC SURVEYING 

always uses for the precise determination of latitude, the probable 
error of the results being readily kept below a tenth of a second. 
Referring to Fig. 52, page 193, PEP'E' is a meridian section of the 
celestial sphere; PP', the polar axis; EE', the equator; C, the 
observer; Z, the zenith; jS and *S', two stars with nearly equal 
(within about 15') but opposite meridian zenith distances, and 
with a sufficient difference of right ascension to enable each one 
to be observed in turn as it crosses the meridian. 

Let <!> = EZ = observer's latitude; 

8 = ES = declination of S (from Nautical Almanac) ; 
d' = ES' = declination of S' (from Nautical Almanac) ; 
z = apparent zenith distance of S; 
z'= apparent zenith distance of S'; 
r = refraction correction for z (from Table VIII) ; 
r' = refraction correction for z' (from Table VIII) ; 
then 

z +r = ZS = true zenith distance of S; 
z' + r' = ZS' = true zenith distance of iS'; 
whence 

(f) = § + z + r 

= d'- {z'+r') 

2?^ = (5 + d'} + {z-z') + {r-r'y 
and we have for the latitude 

^ = hlid + d') +{z- z') + (r - r')\. 

In this equation the quantities {d + 8') and (r — r') are known, 
so that it is only necessary to obtaiu {z— z') by observation to 
determine the latitude. The quantity {z — z') is the difference 
between the zenith distances of the two stars S and S', and if 
this quantity is not over about 15' it can be measured with great 
accuracy by means of the zenith telescope (see Fig. 53). The 
instrument illustrated has an aperture of about three inches, 
a focal length of nearly four feet, and a magnifying power of 100. 
The telescope being set at a proper vertical angle for a given pair 
of stars is not changed thereafter, but each star is brought into 
the field of view by revolving the instrument on its vertical 
axis, and the difference of zenith distance is measured entirely 




Fig. 53. — Zenith Telescope. 

From a photograph loaned by the XJ. S. C. and G. S, 



196 GEODETIC SURVEYING 

with the micrometer eyepiece. Many pairs of stars are observed, 
and many refinements in observation and computation are required 
in the highest grade of work. For a complete discussion of the 
method the reader is referred to Appendix No. 7, Report for 
1897-98, U. S. Coast and Geodetic Survey. An altazimuth 
instrument with a micrometer eyepiece will give very good 
results by the above method, if used with proper precautions. 

111. Latitude Determinations at Sea. Many methods have 
been devised for determining latitude at sea. Greenwich time 
may or may not be required, according to the method used, 
but is generally available from the ship's chronometers. In any 
case the observation consists in measuring with the sextant the 
altitude of one or more of the heavenly bodies above the sea 
horizon. All such altitudes are reduced to the true horizon by 
applying a correction for dip, as explained in Art. 105, this cor- 
rection being in addition to any others which the observation 
requires to determine the true altitude. The most common 
observation for latitude is for the altitude of the sun at apparent 
noon, as explained in Art. 107. The meridian altitude of the pole 
star or other bright star is also often observed, the result in either 
case being worked out as explained for circumpolar stars in 
Art. 108. The error of a latitude determination at sea may range 
upwards from a fraction of a mile, depending on the circumstances 
surrounding the observation. 

112. Periodic Changes in Latitude. It is now known that the 
earth has a slight wabbling motion with respect to the axis about 
which it rotates. In consequence of this motion the north and 
south poles do not occupy a fixed position on the surface of the 
earth, but each one apparently revolves about a fixed mean 
point in a period of about 425 days. The distance between 
the actual pole and the mean point is not constant, but varies 
(during a series of revolutions) between about 0".16 (16.3 ft.), 
and about 0".36 (36.6 ft.). As the equator necessarily shifts 
its position ia accordance with the movement of the poles, it 
follows that the latitude at every point on the smf ace of the earth 
is subject to a continual oscillation about its mean value, the 
successive oscillations being of different extent and ranging from 
0".16 to 0".32 each way from the middle. In precise latitude 
work, therefore, the date of the determination is an essential 
part of the record. 



ASTRONOMICAL DETERMINATIONS 197 

Longitude 

113. General Principles. The longitude of any point on the 
surface of the earth is the angular distance of the meridian of that 
point from a given reference meridian, being positive when reclconed 
westward and negative when reckoned eastward. The meridian 
of Greenwich has been universally adopted (since 1884) as the 
standard reference meridian of the world, but other meridians 
(Washington, Paris, etc.) are often used for special work. Since 
time is measured by the uniform angular movement of the earth 
on its axis (west to east), it follows that longitude may be 
expressed equally well in either angular units or time units. As 
360° of arc correspond to twenty-four hours of time (mean or 
sidereal. Art. 95), the change from the angular to the time system 
is evidently made by dividing by 15, and vice versa; thus the 
longitude of Washington west from Greenwich may be written 
as 77° 03' 56".7, or 5^ 08"" 15'.78, as preferred. At the same 
absolute instant of time the true local time of any station differs 
from the true local time of any other station by the angular 
divergence (expressed in time units) of the meridians of these 
two stations; the difference of longitude of any two stations, 
therefore, is identical with the difference of local time. At the 
same instant of time, the difference between the local mean time 
and the sidereal time at any station is the same for all points in 
the world, so that the difference of local time between any two 
given stations is always numerically the same whether the com- 
parison is based on local mean time or sidereal time. From the 
nature of the case, it is evident that standard time (Art. 96a) 
bears no relation to the longitude of a station. 

Longitude as described above is geodetic longitude. Longitude 
obtained from observations on heavenly bodies, or astronomical 
longitude, is identical with geodetic longitude except where local 
deviation of the plumb line (Art. 75) exists. The geodetic long- 
itude of a point can never be directly observed, nor can the devia- 
tion of the plumb line be found by direct measurement. If, 
however, the longitude of any point be found by computation 
(Chapter V) from the astronomical longitudes measured at 
various other triangulation stations, and these values be averaged 
in with its own astronomical longitude, the result may be assumed 
to be free from the effects of plumb line deviation and to represent 



198 GEODETIC SUEVEYING 

the true geodetic longitude. In geodetic work geodetic longitude 
is always understood unless otherwise specified. 

The longitude of any given point is ordinarily obtained by 
finding how much it differs from that of some other point whose 
longitude has already been well determined. The finding of 
this difference of longitude is essentially the finding of the dif- 
ference of local time between the two points, the westerly 
point having the earliest time, and vice versa. The local time 
is found by the methods heretofore given, and the comparison 
is made as about to be explained. 

114. Difference of Longitude by Special Methods. These 
methods are rarely used any more, but are of considerable scientific 
interest, and hence are here briefly mentioned. 

By special phenomena. Certain astronomical phenomena, 
such as the eclipses of Jupiter's satellites, occur at the same instant 
of time as seen at any point on the earth from which they may 
be visible. These eclipses usually occur several times in the course 
of a month, the Washington mean time of the event being given 
in the Nautical Almanac. The observer notes the true local time 
at which the eclipse occurs, the error and rate of his timepiece 
having been previously determined. The difference between 
the Washington mean time and the local mean time of the eclipse 
is the observer's longitude from Washington. Eclipses of the 
moon may also be used in the same manner. Longitude obtained 
by these methods is apt to be several seconds of time in error, 
or a minute or more in arc. 

By flash signals. Two observers, having obtained their own 
local time by proper observations, may each note the reading of 
their own clock at the same instant of time, this instant being 
determined by an agreed signal visible to both. Such a signal 
may be the flash of a heliotrope by day, or any suitable fight 
signal by night. The difference of local time is then the difference 
of longitude. The error by this method may be kept below a 
second of time by averaging the results of a number of signals. 
This method usually requires one or more intermediate stations 
to be established to overcome the lack of intervisibilityj and is 
generally an expensive one. 

115. Longitude by Lunar Observations. If an observer notes 
his true local time (expressed as mean time) for any particular 
position of the moon, and obtains from the Nautical Almanac 



ASTEONOMIOAL DETERMINATIONS 199 

the Greenwich mean time when the moon occupied suCh a posi- 
tion, the longitude from Greenwich is given by 'the corresponding 
difference of time. Many methods have been devised on this 
basis, requiring laborious computations in their application, and 
in many of the methods not leading to very accurate results. 
Lunar methods are therefore not generally used except on long 
sea voyages or long exploration trips. A few of the methods are 
given below, but only in the roughest outline. 

By lunar distances. The angle between a star, the center 
of a planet, or the near edge of the sun, and the illuminated edge 
of the moon may be measured by a sextant, and reduced to 
what it would have been if it had been observed at the center of 
the earth and measured to the center of the moon. The Green- 
wich time of this position can be determined from the Nautical 
Almanac and compared with the local time at which the observa- 
tion was made. The accuracy attainable is about five seconds 
of^time. 

By lunar culminations. The local time of meridian passage 
of the moon's illuminated limb may be noted, expressed as sidereal 
time and corrected for semi-diameter, giving the moon's right 
ascension at the given instant, and Greenwich mean time for 
this value of the right ascension be compared with the observed 
local time. The accuracy attainable is about five seconds of time. 

By lunar occuUations. The occultation (covering) of a star 
by the moon may be observed, noting the local time of immersion 
(disappearance), or emersion (reappearance), or both, in which 
case the apparent right ascension of the corresponding edge of 
the moon at the given instant is the same as the right ascension 
of the given star. When proper correction has been made for 
refraction, parallax, semi-diameter, etc., the true right ascension 
becomes known for the given instant, and the corresponding 
Greenwich time is compared as before with the local observed time. 
This method, with the exception of telegraphic methods, is one of 
the best that is known for longitude work. When a number 
of such determinations are averaged together, an accuracy approx- 
imating a tenth of a second of time is attainable. 

116. Difference of Longitude by the Transportation of Chro- 
nometers. When this method is used a number of chronometers 
(from 5 to 50) are carried back and forth (from about 5 round trips 
upwards) between the two points whose difference of longitude 



200 GEODETIC SURVEYING 

is desired. On reaching each station the traveling chronometers 
are compared with the local chronometers. The errors of the 
local chronometers are determined astronomically at or near 
the time of comparison. The various values thus obtained for the 
difference of time between the two stations are averaged together 
and the result taken as the difference of longitude. Owing to 
the fact that each round trip furnishes two determinations that 
are oppositely affected by similar errors, and also to the refinements 
of method and reduction that are used in practice, the errors 
due to chronometer rates and irregularities are largely eliminated 
from the average result. The accuracy attainable (in time imits) 
may range between a few tenths of a second and less than a single 
tenth of a second, depending on the distance between stations, 
the number of trips made, and the number of chronometers 
transported. Longitude determinations by this method are now 
rarely made, except where telegraphic connection is not available. 
In order to make an accurate comparison of two mean time 
chronometers each one is independently compared with the same 
sidereal chronometer, and two sidereal chronometers are sim- 
ilarly compared by mutual reference to a mean time chronometer. 
Sidereal chronometers continually gain on mean time chronom- 
eters, the beats or ticks (half seconds) gradually receding from and 
approaching a coincidence that occurs about every three minutes. 
When the beats exactly coincide the chronometers differ precisely 
by the value in half seconds indicated by the subtraction of their 
face readings. As the ear can be trained to detect a lack of coin- 
cidence as small as the one-hundredth part of a second, a com- 
parison can be made with this degree of precision. 

117. Difference of Longitude by Telegraph. Where tele- 
graphic connection can be established between two stations it 
furnishes the best means of exchanging time signals, both on 
account of the great accuracy attainable and the comparative 
inexpensiveness. Difference of longitude obtained in this manner 
can be made more accurate than is possible by any other known 
method. The lines of the telegraph companies ramify in all 
directions, and the temporary use of a suitable wire can usually 
be obtained at reasonable cost, so that it is only necessary to 
erect short connecting lines between the observing stations and the 
telegraph stations. The most important applications of the 
method are as outlined below. 



ASTRONOMICAL DETERMINATIONS / 201 

By standard time signals. This method furnishes a quick 
means for an approximate longitude determination. Standard 
time can be obtained at any telegraph station with a probable 
error of less than a second. The observer's true local mean time 
is obtained by any of the simpler methods of observation. The 
difference of these times is the difference of longitude between 
the given standard time meridian and the meridian of the ob- 
server's station. 

By star signals. The difference of longitude of any two 
stations is identical with the sidereal time which elapses between 
the transit of any given star over the meridian of the easterly 
station, and the transit of the same star over the meridian of the 
westerly station; so that it is only necessary to observe how long 
it takes for any star to pass between the meridians of two stations 
to know their difference of longitude. In making use of this 
principle a chronograph (Art. 103c) is placed at each station, 
and these chronographs are connected by a telegraph line. A 
break-circuit chronometer, which may be placed anywhere in 
this line, records its beats on both chronographs. As the selected 
star crosses the meridian of the easterly observer he records this 
instant of time on both chronographs by tapping his break- 
circuit signal key. When the same star crosses the meridian of the 
westerly observer he likewise records this new instant of time 
on both chronographs. Each chronograph, therefore, contains 
a record of the time between transits, but the records are not 
identical, as it takes time for the signals to pass between the 
stations; in other words, each signal is recorded a little later on 
the distant chronograph than it is on the home chronograph. 
The record of the easterly chronograph thus becomes too great, 
and the record of the westerly one correspondingly too small; 
but the mean of the two records eliminates this error and gives 
(when corrected for chronometer rate) the true difference of 
longitude between the stations. In actual work the transits of 
many stars are observed at each station, so as to obtain an average 
value for the difference of longitude. The accuracy attainable 
is about 0.01 of a second of time. This method is one of the 
best, and was formerly largely used by the Coast Survey. The 
objection to the method is the difficulty of securing the monopoly 
of the telegraph line during the long period while the observa- 
tions are in progress, so that it is no longer much in use. 



202 GEODETIC SUEVEYING 

By arbitrary signals. This is the standard method of the 
Coast Survey at the present time, and requires the use of the 
telegraph line for only a few minutes during an arbitrary period 
(previously agreed upon) on each night that observations are in 
progress. In this method a chronometer and chronograph are 
installed at each station, and each chronometer records its beats 
on the home chronograph only. Each observer makes his own 
time observations, which are likewise recorded on his own chrono- 
graph alone. Observations at each station are taken both before 
and after the exchange of signals in order to determine the cor- 
responding chronometer's rate as well as its error. As far as 
possible the same stars are observed at each station, in order 
to avoid introducing errors of right ascension. In the most 
precise work the observers exchange places on different nights, 
in order to eliminate the effects of personal equation, and numerous 
other refinements are introduced. The chronograph sheet at 
each station enables the true time at that station to be computed 
for any instant within the range of the record, and the difference 
of these true times at any one instant of time is the difference of 
longitude between the stations. The whole object of the exchange 
of signals, therefore, is to identify the same instant of time on 
both chronograph sheets. At the agreed time for the exchange 
of signals the two stations are thrown into circuit with the main 
telegraph line, with connections so arranged that signals (momen- 
tary breaking of circuit) sent by either station are recorded on 
both chronographs. No signal, however, is recorded at exactly 
the same instant at both stations, on account of the time required 
for its passage between them. The difference of longitude as 
based on the signals from the western station is hence too large, 
and that based on the eastern station's signals correspondingly 
too small. The mean of the two values is taken as the true 
difference of longitude, while the difference of the two values 
represents double the time of signal transmission. In the Coast 
Survey program two independent sets of ten pairs of stars 
each are observed on five successive nights, the observers then 
exchanging places and continuing the observations in the same 
manner for five more nights. Signals are exchanged once each 
night at about the middle time for the work of both stations, 
the western station sending thirty signals at intervals of about 
two seconds, followed by thirty similar signals from the eastern 



ASTRONOMICAL DETERMINATIONS 203 

station. These signals were formerly sent by the chronometers, 
but are now sent by tapping a break-circuit signal key. The 
accuracy attainable, as in the case of star signals, is about 0.01 
of a second of time. 

118. Longitude Determinations at Sea. Every sea-going vessel 
carries one or more chronometers, the error and rate of each being 
determined before leaving port, so that the Greenwich time of 
any instant is always very closely known. The local time for 
the ship's position having been determined for any instant 
(Art. 105), and the corresponding Greenwich time being obtained 
from the chronometers, it is only necessary to take the difference 
of these times to have the ship's longitude from Greenwich. 
The result thus obtained is expressed in time units, but is readily 
converted into angular units by multiplying by 15 (Art. 113). 
In case of failure of the chronometers, longitude at sea can still 
be determined in a number of ways not requiring a previous 
knowledge of Greenwich time, such as the method of lunar dis- 
tances (Art. 115). Discussions and explanations of these methods 
can be found in all works on Navigation and Nautical Astronomy. 
A longitude determination at sea may be in error from a fraction 
of a mile to a number of miles, depending on the surrounding 
circumstances. 

119. Periodic Changes in Longitude. As explained in Art. 112, 
the poles of the earth are not fixed in position, but each one 
apparently revolves about a mean point in a period of about 425 
days, the radius-vector varying (during a series of revolutions) 
between about 0".16 and 0".36. The result of this shifting of 
the poles is to cause the longitude of any point to oscillate about 
a mean value, the amplitude of the oscillation depending on the 
location of the point. In precise longitude work, therefore, the 
date of the determination is an essential part of the record. 

Azimuth 

120. General Principles. By the azimuth of a hne (or a 
direction) from a given point is meant its angular divergence from 
the meridian at that point, counting clockwise from the south 
continuously up to 360°. From any intermediate point on a 
straight line the azimuths towards the two ends always differ by 
exactly 180°, so that in any case it is only necessary to determine 



204 GEODETIC SURVEYING 

the azimuth in one direction. In passing along a straight line 
the azimuth varies continuously from point to point, unless the 
line be the equator or a meridian. The cause of this change and 
the methods for computing it are explained in detail in Arts. 68 
to 73, inclusive. The following articles are concerned solely 
with the determination of azimuth (and hence of the meridian) 
at any one given point. 

Geodetic azimuth is that in which the angular divergence from 
the meridian is measured in a plane which is tangent to the 
spheroid at the given point. Azimuth obtained from observations 
on heavenly bodies, or astronomical az muth, is identical with 
geodetic azimuth except where local deviation of the pliunb line 
(Art. 75) exists. The geodetic azimuth of a line from a given 
point can never be directly observed, nor can the deviation of 
the plumb line be found by direct measurement. If, however, 
the azimuth of a line from a given point be fovmd by computa- 
tion (Chapter V) from the azimuth determinations made at 
various other triangulation stations, and these values be averaged 
in with the observed value, the result may be assumed to be free 
from the effects of plumb line deviation and to represent the true 
geodetic azimuth. In geodetic work geodetic azimuth is always 
understood unless otherwise specified. 

121. The Azimuth Mark. This is the signal which gives the 
direction of the line whose azimuth is being determined. An 
azimuth mark should not be placed less than about a mile from 
the observer, otherwise a change of focus will be required between 
the heavenly body and the mark. Experience has shown that 
refocussing during an observation is very undesirable. When 
azimuth is obtained by solar observations any of the usual day- 
time signals (Art. 19) may be used, being located at a special 
azimuth point or a regular triangulation station as circumstances 
may require. When azimuth is obtained by stellar observations 
a special azimuth point is generally located one or more miles 
from the instrument. The azimuth mark should be moimted 
on a post or otherwise raised about five feet above the ground, 
and generally consists of a bull's-eye lantern enclosed in a box 
or placed behind a screen, a small circular hole being provided 
for the light to pass through on its way to the observer. If the 
diameter of the hole does not subtend over a second of arc (0.3 
of an inch per mile) at the eye of the observer, the light will 



ASTRONOMICAL DETERMINATIONS 205 

closely resemble a star in both apparent size and brilliancy, which 
is the object sought. The face of the box or screen is often painted 
with stripes or other design so that it may also be observed in the 
daytime. 

122. Azimuth by Stm or Star Altitudes. The altitude of any 
heavenly body as seen by an observer at a given point is con- 
stantly changing, each different altitude corresponding to a par- 
ticular azimuth which can be computed if the latitude and longi- 
tude are approximately known. For the degree of accuracy 
sought by this method it is sufficient to know the latitude to the 
nearest minute and the longitude within a few degrees. The 
difference in azimuth of any two lines from the same point is 
always exactly the same as their angular divergence. If, therefore, 
the horizontal angle between the azimuth mark and the given 
heavenly body is measured at the same moment that the altitude 
is taken, the azimuth of the line to the azimuth mark is obtained 
by simply combining the computed azimuth of the heavenly body 
with this measured horizontal angle. The observation may be 
made with a transit or an altazimuth instrument. The probable 
error of a single determination should not exceed a minute of 
arc with the ordinary engineer's transit, nor a half minute with the 
larger instruments. The actual error may be larger than the 
probable error on account of the uncertainties of refraction. 

122a. Making the Observation. The best time for making 
an observation on the sun is between about 8 and 10 o'clock in 
the morning and 2 and 4 o'clock in the afternoon. The sun should 
not be observed within less than two hours of the meridian 
because its change in azimuth is then so much more rapid than 
its change in altitude; nor when it is much more than four hours 
from the meridian on account of the uncertain refraction at low 
altitudes. In the latitude of New York it is not desirable to 
observe the sun for azimuth in the winter time because its dis- 
tance from the prime vertical during suitable hours results in such 
a rapid movement in azimuth as compared with its movement 
in altitude. Star observations may be made at any hour of the 
night, selecting stars which are about three hours from the meridian 
and near the prime vertical, and hence changing but slowly in 
azimuth as compared with the change in altitude. The observa- 
tions are made in sets of two, taking one reading with the tele- 
scope direct and the other with the telescope reversed, the mean 



206 GEODETIC SURVEYING 

horizontal and the mean vertical angle constituting the observed 
values for that set. Several independent sets should be taken and 
separately reduced, the mean of the resulting azimuths being the 
most probable value. The instrument should be in perfect 
adjustment and be leveled up with the long bubble or the striding 
level, and should not be releveled except at the beginning of each 
set. The center of the sun is not directly observed, but the read- 
ing is taken with the image of the sun tangent to the horizontal 
and vertical hairs. A complete set is made up as follows: Sight 
on the mark and read the horizontal circle; unclamp the upper 
motion and bring the sun's image tangent to the horizontal and 
vertical hairs in that quadrant where it appears by its own motion 
to approach both hairs; note the time to the nearest minute and 
read both circles; unclamp the upper motion, invert the telescope, 
and bring the sun's image tangent in that quadrant where it appears 
to recede from both hairs; note the time and read both circles; 
unclamp the upper motion, sight on the mark and read the hori- 
zontal circle. A star set is taken in the same manner except that 
in each pointing the image of the star is bisected by both hairs. If 
the instrument does not have a full vertical circle the telescope 
is not inverted between the observations, but an index correction 
must be applied to the observed altitudes. The values used in the 
computations of the next article are those which correspond to 
the center of the observed object. If for any reason only one 
observation is secured on the sun, thus leaving the set incomplete, 
the observed altitude is reduced to the center by applying a 
correction for semi-diameter, and the observed horizontal angle 
is reduced to the center by applying a correction found by divid- 
ing the semi-diameter by the cosine of the altitude. The semi- 
diameter is taken from the Nautical Almanac for the given 
time and date, and the correction is added or subtracted 
in accordance with the particular limb of the sun which was 
observed. 

122b. The Computation. It is best to reduce each set inde- 
pendently and average the final results. The observed altitude 
must first be reduced to the true altitude. The apparent altitude 
of all heavenly bodies is too large on account of refraction, the 
required correction being found in Table VIII. The apparent 
altitude of the sun is also too small on account of parallax, the 
amount being equal to 8". 9 multiplied by the cosine of the 



ASTRONOMICAL DETERMINATIONS 207 

observed altitude, but this correction is so small it would seldom 
be applied in this method. 

In the polar triangle ZPS, Fig. 47, page 166, the three sides are 
known. ZP, the co-latitude, is found by subtracting the observer's 
latitude from 90°. PS, the polar distance or co-declination, is 
found by subtracting the dechnation of the observed body from 
90°. In the case of the sun the declination is constantly changing 
and must be taken for the given date and hour (the time being 
always approximately known). The sun's declination for Green- 
wich mean noon is given in the Nautical Almanac for every day 
in the year, and can be interpolated for the Greenwich time of 
the observation; the Greenwich time of the observation differs 
from the observer's time by the difference in longitude in hours, 
remembering that for points west of Greenwich the clock time 
is earlier and vice versa. ZS, the co-altitude, is found by sub- 
tracting the true (reduced) altitude of the observed body from 90°. 
Using the notation of Fig. 47, we have from spherical trigonometry, 

sin d = cos z sin (f> + sin z cos (J) cos A, 

whence 

, sin d — cos z sin d) 

cos A = ; J , 

sin z cos <p 

which for logarithmic computation is reduced to the form 

J tos i[z + {,p + d)] sin i[z + {i> -d)] 
tan iJL M^^^ ^^^ -{4, + d)\ sin ^[z -{<}>- d)]' 

The value of A thus found is the azimuth angle (from north 
branch of meridian) of the given heavenly body at the moment 
of observation. If the observed body was east of the meridian 
its azimuth (from the south point) equals 180° + A ; if west of 
the meridian, 180° — A. The azimuth of the azimuth mark is 
then found by combining the azimuth of the observed body with 
the corresponding angle between the azimuth mark and the 
observed body, the combination being made by addition or 
subtraction as the case requires. 

123. Azimuth from Observations on Circumpolar Stars. The 
simplest and most accurate method of determining azimuth is 
by suitable observations on close circumpolar stars, furnishing 
any desired degree of precision up to the highest attainable. In 



208 GEODETIC SUEVEYING 

northern latitudes the best available stars are a Ursse Minoris 
(2nd magnitude), d Ursee Minoris (4th magnitude), 51 Cephei 
(5th magnitude), and A Ursae Minoris (6th magnitude). Of these 
four a UrsiE Minoris, commonly known as Po aris, is usually 
chosen by engineers on account of its brightness, the other three 
being barely visible to the naked eye. The four stars named may 
be identified by reference to Fig. 50, page 191. 

Owing to the rotation of the earth on its axis the azimuth 
of any star, as seen from a given point, is constantly changing, 
but the value of the azimuth may be computed for any given 
instant of time when the position of the observer is known. The 
most favorable time for the observation of a close circumpolar 
star is at or near elongation (greatest apparent distance east or 
west of the meridian), as its motion in azimuth is then reduced 
to a minimum; but entirely satisfactory results may be obtained 
from observations taken at any time within about two hours 
either way from elongation; the only point involved is that time 
must be known with increasing accuracy the greater the interval 
from elongation, in order to secure the same degree of precision 
in the azimuth determination. In any case the actual observation 
consists in measuring the horizontal angle between an azimuth 
mark and the given star, and noting the time at which the star 
pointing is made. The azimuth of the mark is then obtained 
by combining the measured angle (by addition or subtraction 
as the case requires) with the computed azimuth of the star. 
The details of the observation will depend on the instrument 
available and the degree of precision desired, in the result. The 
instruments used may be the ordinary engineer's transit, the 
larger transits equipped with striding levels, the repeating instru- 
ment, or the direction instrument. Close instrumental adjust- 
ments are necessary for good work. The methods ordinarily 
used are the direction method, the repeating method, and the 
micrometric method. Certain formulas enter more or less into 
all the methods. 

123a. Fundamental Formulas. The following symbols are 
involved in the formulas as here given: 

A = azimuth of star (at any time) from north point, 

+ when east, — when west ; 
Ae = azimuth of star at elongation; 



ASTEONOMICAL DETERMINATIONS 209 

Ao = azimuth of star at mean hour angle of n pointings; 
n = number of pointings to star; 

t = hour angle of star (at any time), + when star is 
west, — when east, or may be counted westward up 
to 24 hours or 360°; 
te = hour angle of star at elongation; 
M = interval of any one hour angle from the mean of 

n given hour angles; 
C = curvature correction in seconds of arc; 
D = correction for diurnal aberration La seconds of arc; 
De = ditto for a close circumpolar star at elongation; 
4> = latitude, + when north, — when south; 
S = declination of star, + when north, — when south; 
Am = azimuth of mark from north point, + to east, 
— to west; 
Z = azimuth of mark from south point; 
h = mean altitude of star; 

d = value of one division of bubble tube in seconds; 
w, w', etc. = readings of west end of bubble tube when sighting 
on star; 
W = mean value of w, w' , etc. ; 
e, e', etc. = readings of east end of bubble tube when sighting on 
star; 
E = mean value of e, e', etc. ; 
■ 6 = mean inclination of telescope axis in seconds when 

sighting on star; 
X = angle correction in seconds due to inclination of 

telescope axis; 
oi = star's right ascension; 
8 = sidereal time at any instant; 
Se = sidereal time of star's elongation. 

a. Hour angle at any instant. The hour angle of a star 
(in time units) at any instant of sidereal time is given by the 
formula 

t=S -a. 

The corresponding value of t in angular units is obtained 
(Art. 95) by multiplying by 15. The particular unit in which t is 
to be expressed is always apparent from the formula in which it 
occurs. If local mean time or standard time is used it must be 



210 GEODETIC SURVEYING 

reduced to sidereal time (Art. 99) before being used in the formula 
for t. . 

b. Hour angle at elongation. In the polar triangle ZPp, 
Fig. 47, page 166, p may be taken to represent Polaris or any 
other star at elongation, or greatest apparent distance from the 
meridian for the observer whose zenith is at Z. In this triangle 
the side PZ is the observer's co-latitude, the side Pp is the star's 
co-declination, and the angle ZpP equals 90° on account of the 
tangency at the point p. Solving for the angle ZPp, or the star's 
hour angle at elongation, we have 

tan d) 

cos te = 7 f-. 

tan 

c. Time of elongation. Having found te from the formula 
in (&), the sidereal time of elongation is given by the formulas 

Se = a + te (western elongation), 
Se = ct — te (eastern elongation). 

The sidereal time thus obtained is changed to local mean time or 
standard time by Art. 100 when so desired. 

d. Azimuth at elongation. If the above triangle (6) be solved 
for the angle PZp, or the star's azimuth at elongation, we have 

. sin polar distance cos 8 

sm Ae = 1 ,., J = X- 

cos latitude cos p 

e. Reduction to elongation. If the angle between the azimuth 
mark and a close circumpolar star is measured within about 
thirty minutes either way from elongation, 'the measured angle 
may be reduced very nearly to what it would have been if measured 
at elongation by applying the following correction: 

, 2 sin2 i{te - t) 
A. - A = ta,ii Ae ^ ' 



sin 1" 

The quantity (ie ~ is equivalent to the sidereal time interval 
from elongation, and may be substituted directly without com- 
puting the hour angle represented by t. If the mean or standard 



ASTEONOMICAL DETEEMINATIONS 211 

time interval is thus used the value which the formula gives for 
(Ae— A) must be increased by t-Iit P^'^ of itself. 

/. -Azimuth at any hour angle. If the star is observed at any- 
other hour angle than that which corresponds to elongation, a polar 
triangle will be formed similar to ZPp, Fig. 47, page 166, but 
with all the angles oblique. In this case the azimuth A at the 
given hour angle t is given by the formula, 

. sin t 

tan A = 



sin ^ cos t — cos ^ tan d 

_ cot d sec (j) sin t 

1 — cot d tan ^cos t 

= — cot d sec ^ sin tlr. 1, 

in which 

a = cot d tan ^ cos t. 

g. The curvature correction. If a series of observations are 
taken on a star the hour angle and corresponding azimuth must 
necessarily be different for each pointing. The mean value of 
such azimuths is frequently desired, and may of course be found 
by computing each azimuth separately and averaging the results. 
The same value, however, may be obtained much more simply 
by computing the azimuth corresponding to the mean of the 
several hour angles, and then applying the so-called curvature 
correction to reduce this result to the mean azimuth desired. 
The reason that such a correction is required is because the motion 
of a star in azimuth is not uniform, but varies from zero at elonga- 
tion to a maximum a1 culmination. In the case of a close circum- 
polar star, and a series of observations not extending over about 
a half hour, the curvature correction is given by the formula 

„ ^ . 1 „ 2 sin2 iJt 
C = tan 4o-S . J, , 
n sm 1 

in which Jt is expressed in angular value, or 

C = tan 4o^^ sin l"-5:(ii)2 
2 n ' 



212 GEODETIC SURVEYING 

in which Jt is expressed in sidereal seconds of time. If Jt is 
expressed in mean-time seconds the value of C thus obtained 
must be increased by rhr part of itself. 



log— ^ sm 1" 



= 6.7367275 - 10. 



The sign of the curvature correction C is known from the fact that 
the true mean azimuth always lies nearer the meridian than the 
azimuth that corresponds to the mean hour angle. From the 
nature of the case it is evident that the several values of M in 
time units may be obtained directly from the observed times 
(without changing them to hour angles) by taking the differences 
between each observed time and the mean of all the observed times. 
h. Correction for inclination of telescope axis. If the axis 
of the telescope is not horizontal the line of sight will not describe 
a vertical plane when the telescope is revolved on this axis, and 
hence the measured angle between the star and the mark will be 
in error a corresponding amount. The inclination of the axis 
is found from the readings of the striding level. If the level is 
reversed but once the usual formula is 

b =^[{w+w') -(e + e')]; 

but if the level is reversed more than once it is more convenient 
to write 

b=^iW-E). 

So far as the present purpose is concerned these formulas are 
equally applicable whether the level is actually reversed on the 
pivots, or reversed in direction because the instrument is turned 
through 180°. In one case the value obtained is the actual 
average inclination of the axis, while in the other case it is the 
net inclination. By the east or west end of the bubble tube is 
meant literally the end which happens to be east or west when the 
reading is taken. The correction required on account of the 
inclination b, due to the altitude of the star, is 

X = b tan h. 



ASTRONOMICAL DETERMINATIONS 213 

The value of x thus obtained is to be subtracted algebraically 
from the computed azimuth of the mark. Ordinarily a similar 
correction for inclination due to altitude of mark is not required, 
as the mark is generally nearly in the horizon of the instrument. 
If, however, the angular elevation (+ altitude) or depression 
( — altitude) of the mark is reasonably large, the striding level 
should be read when pointing to the mark and a similar correction 
computed. In this case the correction is to be added algebraically 
to the computed azimuth of the mark. 

i. Correction for diurnal aberration. Owing to the rotation 
of the earth on its axis and the aberration of light thereby caused, 
the apparent position of any star is always more or less east of 
its true position, the amount of the displacement depending on 
the position of the observer and the position of the star. A 
corresponding correction is required for all azimuths based on 
the measurement of a horizontal angle between a mark and a 
star, and is given by the formula 

cos h 

which for a close circumpolar star at elongation reduces to 

Z)e=0".32 cos A. 

In obtaining azimuth from a north circumpolar star it is evident 
that the azimuth of the mark (counting clockwise from either 
the north or south point) must be increased by the amount of 
the above correction. 

j. Reduction of azimuth to south point. In making azimuth 
determinations by observations on north circumpolar stars it is 
customary to refer all results to the north point until the azimuth 
of the mark is thus expressed. The azimuth of the mark from the 
south point is then given by the formula 

Z = 180° + Am, 

in which proper regard must be had to the negative sign of A „ if 
it is taken counter-clockwise. 

123b. Approximate Determinations. It is frequently desirable 
to make approximate determinations of azimuth, either because 
the work in hand does not call for any greater accuracy, or as a 
preliminary to the more accurate location of the meridian. Such 



214 GEODETIC SURVEYING 

determinations may be made by measuring sun or star altitudes, 
as explained in Art. 122, but observations on Polaris (or other 
circumpolar stars) give more reliable results without any increase 
in either field or office labor. The ordinary engineer's transit 
may be used for such work, and with proper care will give correct 
results within the smallest reading of the instrument. Since 
the observation is best made at or near elongation the time of 
elongation (c, Art. 123a) is computed beforehand, so that proper 
preparation may be made. Assuming the instrument to be in 
good adjustment and carefully leveled, the observation consists 
in reading on the mark with telescope direct, reading on the star 
with telescope direct, reading on the star with telescope reversed, 
and ending with a reading on the mark with telescope reversed. 
The lower motion must be left clamped and all pointings made with 
the upper motion alone. The instrument must not be releveled 
during the set. Both plate verniers should be read at each pointing. 
The four pointings should be made in close succession, but with- 
out undue haste or lack of care. If the observation is being made 
at elongation the first pointing to the mark is made a few minutes 
before the computed time of elongation, and the two star point- 
ings as near as may be to the time of elongation. If time is not 
accurately known the star is followed with the telescope until 
elongation is evidently reached, when the necessary observations 
are quickly taken. For five minutes each side of elongation the 
motion of the star in azimuth is scarcely perceptible in an engineer's 
transit. If the observations are not taken at elongation time must 
be accurately known and read to the nearest second at each star 
pointing. The observations having been completed the mean 
angle , between the mark and the star is obtained from the four 
readings taken, and it only remains to compute the mean azimuth 
of the star to know the azimuth of the mark. If the star point- 
ings were made within about ten minutes either way from elonga- 
tion the azimuth of the star may be taken as equal to its azimuth 
at elongation (rf. Art. 123a). If the star pointings were made 
within about a half hour either way from elongation the angle 
between the mark and the star may be reduced to what it would 
have been at elongation by use of the formula for reduction to 
elongation (e. Art. 123a), the quantity (te—t) being taken as 
the angular value of the time interval between the time of elonga- 
tion and the average time of the star pointings. If the observa- 



ASTEONOMICAL DETERMINATIONS 215 

tions are taken over about a half hour from elongation it is better to 
compute the true aziinuth of the star for the average time of the 
star pointings (/, Art. 123a). 

123c. The Direction Method. In this method the angle 
between the mark and the star is measured with a direction 
instrument (Arts. 42-47), the process being substantially the same 
as there described for measuring angles between triangulation 
stations. Owing to the fact that the star is in motion during the 
observations, however, the angle being measured is constantly 
changing, and the reductions must be correspondingly modified. 
Owing to the altitude of the star serious errors are introduced 
by any lack of horizontality in the telescope axis, and a cor- 
responding correction muSt be made in accordance with the read- 
ings of the striding level. If the mark is more than a few degrees 
out of the horizon a similar correction will be required for the same 
reason. The observations may be made at any hour angle, good 
work requiring time to be known to the nearest second. A good 
program for one set is to read twice on the mark with telescope 
direct; then read twice on the star with telescope direct, noting 
the exact time of each pointing and the reading of each end of 
the striding level at each pointing; then read twice on the star 
with telescope reversed, noting time and bubble readings as 
before; then read twice on the mark with telescope reversed. 
The striding level is left with the same ends on the same 
pivots throughout the observations. Thb mean azimuth of the 
star for the four pointings is then found by computing the 
azimuth corresponding to the average time of these pointings 
(/, Art. 123a), and then applying the curvature correction 
(g, Art. 123a). The apparent azimuth of the mark is then found by 
combining the computed star azimuth with the mean measured 
angle. The true azimuth of the mark (as given by this set) is 
then found by applying to the apparent azimuth the level cor- 
rection and the aberration correction {h and i, Art. 123a), and 
reducing the result to the south point (j, Art. 123a). By taking 
a nimiber of sets each night for several nights, and averaging 
the different results, a very close determination of azimuth 
may be secured. With skilled observers the probable error of a 
single set should not exceed about a half a second of arc, and this 
may be reduced to a tenth of a second by averaging about twenty- 
five sets. 



216 



GEODETIC SUEVEYING 



EXAMPLE.— AZIMUTH BY DIRECTION METHOD *— RECORD 



Station: Mount Nebo, Utah. 
Instrument: 20-m. Theodolite No. 5. 
Star: Polaris, near lower culmination. 



Date: July 21, 1887. 

Observer: W. E. 

Position X. 



Object. 



Chron. 
Time. 



Pos. 

of 

Tel. 



Mic. 



Circle Heading. 



Forw. 
d. 



Back, 
d. 



Mean 
d. 



Corr. 
for 
Run. 



Cor'd 
Mean 



Levels and 
Remarks. 



k, m. 



Az. mark 



Az. mark 



Star 



Star 



Star 



Star 



Mean of 
4 times 



Az. mark 



Az. mark 



15 06 47.0 



IS 10 23.3 



IS IS 57.8 



15 19 41.8 



15 13 12 4 



140 



136 



53 



53 



09 



11 



15 



14.8 
14.6 
32.3 



25.6 



14.2 
13.4 
29.7 



20.6 


19.1 


14.7 
14.4 
32.1 


14.2 
13.5 
30.0 


20.4 


19.2 


45.3 
44.3 
60.7 


43.0 
43.8 
59.2 


50.1 


48.7 


07.0 
07.2 
22.6 


06.5 
06.3 
21.0 


12.3 


11.3 


41.3 
32.0 
44.0 


40.5 
30.3 
43.7 


39.1 


38.2 


09.5 
57.5 
10.5 


08.5 
57.3 
10.0 


05.8 


05.3 


27.0 
17.8 
29.0 


26.0 
16.5 
27.5 


24.6 


23,3 


28.3 
18.7 
29.7 


26.5 
16.7 
28.7 



24.0 



19.8 



19.8 



49.4 



11.8 



38.6 



24.0 



24.8 



-0.2 



-0.5 



19.6 



19.6 



-0.2 



-1-0.5 



-0.2 



-0.2 



12.1 



38.4 



06.1 



23.8 



"\V. B. 
43.5 27.0 



53.7 17.5 



97.2 44.5 
4-52.7 



39.5 32.3 



27.4 44.6 



66.9 76.9 
-10.0 



Mean circle 
reading: 

On star: 
136''12'26".3! 

On mark; 
140°63'21".90 



* Abridged from example ia Appendix No. 7, Report for 1897-98, U. S. Coast and 
Geodetic Survey. 



ASTRONOMICAL DETEEMINATIONS 



217 



AZIMUTH BY DIRECTION METHOD— COMPUTATION 



Mount Nebo, Utah, July, 1887. 


= 39° 48' 33". 44 


* Explanation. 






Date and position 


July 21, X 


July 21, XI 


Mean chronometer time 


IShlSm 12^.44 


0'»55m 10=. 06 


Chronometer correction 


-35 .40 


-34 .62 


Sidereal time 


15 12 37 .04 


54 35 .44 


a of polaris 


1 17 58 .16 


1 17 58 .48 


t of polaris (time) 


13 54 38 .88 ' 


-0 23 23 .04 


t of polaris (arc) 


208 °39' 43". 20 


-5-50' 45". 60 


d of polaris 


88 42 06 . 13 


88 42 06 .20 


log cot S 


8.35532 


8.35532 


log tan <l> 


9.92087 


9.92087 


log cos t 

logo 


9.94323 n 


9.99773 


8.21942 n 


8.27392 


log cot d 


8.355325 


8.355319 


log sec <l> 


0.114537 


0.114537 


log sin t 


9.680917 n 


9.007983 n 


log 1/1 -a 
log (—tan A) 


9.992861 


0.008237 


8.143640 n 


7.486076 n 


A 


+0° 47' 51". 02 


+0° 10' 31". 68 




6"25s,4 81". 


7'°08^8 100". 3 


, 2 sin^ iJt 


2 49 .2 15 .6 


3 23 .1 22 .5 


J<and ^^^„ 


2 45 .3 14 .9 


3 19 .4 21 .7 




6 29 .3 82 .6 


7 12 .4 102 .0 


194". 1 


246". 5 




48 .5 


61 .6 


, 1^2sin=ii« 
log • — ^ ~ — : — TT, — 


1,68574 


1.78958 


^ n sm 1" 






log (curvature correction) 


9.82938 


9.27566 


Curvature correction 


+0".68 


+0".19 


Mean azimuth of star 


+0° 47' 50". 34 


+0° 10' 31". 49 


Circle reads on star 


136 12 26 .38 


151 14 14 .30 


Circle reads on north 


135 24 36 .04 


151 03 42 .81 


Circle reads on mark 


140 53 21 .90 


156 32 25 .95 


Approx. azimuth of mark 


+ 5 28 45 .86 


43 .14 


Level correction 


-3 .94 


-0 .73 


Azimuth of mark 


5 28 41 .92 


42 .41 



218 GEODETIC SURVEYING 

123d. The Repeating Method. In this method the angle 
between the mark and the star may be measured with any of the 
usual engineering transits or with the regular geodetic repeating 
instrvmient (Arts. 38^1), the process being substantially the same 
as there described for measuring angles between triangulation 
stations. The observations and reductions are best made as 
described in Arts. 40, 40a, and 40&, ignoring for the time being 
the fact that the angle which is being repeated is constantly 
changing in value on account of the apparent motion of the star. 
Time must be correctly known and noted to the nearest second 
for each star pointing, but only the total angle readings are taken, 
as with terrestrial angles. The striding level (if the instrument 
has one) may be kept with the same ends on the same pivots 
throughout the observations, and both ends should be read imme- 
diately after the 1st, 3d, 4th and 6th star pointings in each series 
of six pointings. If the mark is more than a few degrees out of 
the horizon similar readings of the striding level are also required 
for its pointings. The observations may be made at any hour 
angle, but it is preferable to work within a couple of hours of 
elongation. 

In making the reductions the azimuth of the mark from the 
north point is deduced separately from each series of six pointings, 
applying the level correction Qi, Art. 123a) in each case, but 
omitting the aberration correction. The two results obtained from 
the two series of 6 D. and R. pointings are averaged together to 
obtain the value of the determination as given by that set. Two 
or more complete sets may be taken and averaged together as 
desired. The true azimuth of the mark (as given by these sets) 
is then found by applying the aberration correction (i. Art. 123a) 
to this final mean, and reducing this result to the south point 
(j. Art. 123a). In reducing each series of six pointings the accum- 
ulated angle is divided by six exactly as if the star had remained 
entirely stationary. The mean angle thus obtained is the same 
as it would have been if the star had remained all the time at the 
mean point of its six separate positions. The corresponding 
azimuth of this mean point is found by computing the azimuth 
for the mean of the six times at which the star pointings were 
made (/, Art. 123a) and applying the curvature correction 
{g, Art. 123a). 

The accuracy attainable by this method depends on the char- 



ASTRONOMICAL DETEEMINATIONS 



219 






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r 


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(N 












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a 






















§ 






















CQ 


^ 


^ 


lO 


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o 






03 


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(N 


o 












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h 






















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r-i 
















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to^ 


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o "^ ic o ^ t> 


coco 


CO coco CO 




l^rji COOT 




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COCO 


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r-l .-1 




ta ■ 












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ril 115 


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220 



GEODETIC SURVEYING 



acter of the instrument with which the work is done. The probable 
error of the average value obtained from a complete double set 
of twenty-four pointings should not exceed about five seconds 
with a good engineer's transit, nor a single second with a 
12-inch repeater; and these probable errors may be much further 
reduced by averaging many determinations together. 



AZIMUTH BY REPETITIONS— COMPUTATION 
Kahatchee, Ala. 



Explanation. 



Date 

Chronometer time 

Chronometer correction . 
Sidereal time 



Hour-angle (i) 

t in arc 

log sin (^ 

log cos t 

log sin <j> cos t 

sin <j> cos t 

cos ^ tan d 

cos ^ tan 3 — sin <(> cos i . . . . 

log sin i 

log (cos (fi tan 5 — sin ^ cos t) 
log ( — tan A) 

A 



Jt and 



2 sin' yt 
sinl" ■ 



June 6 
14h54m 17s. 7 

-31 .1 

14 53 46 .6 
1 21 20 .3 

13 32 26 .3 

203° 06' 34". 5 
9.73876 
9.96367 n 
9.70243 n 

- 0.5040 

+ 38.7399 

+ 39.2439 
9.593830 n 
1 . 593772 
8.000058 n 

+0° 34' 22". 7 



7m47s,7 
5 09 .7 
1 26 .7 
1 52 .3 
4 54 .3 
7 37 .3 



119". 3 

52 .3 

4 .1 

6 .9 

47 .2 



114 .0 



1 2 sin2 i^t 
n sm 1" 

log (curvature correction) 

Curvature correction 

Mean azimuth of star. . . . 

Angle star-mark 

Level correction 

Corrected angle 

Azimuth of mark E. of N 



343 .S 

57 .3 

1.7582 

9.7583 
+ 0.6 
+ 0°34' 22". 1 

72 57 50 .2 

- 1 .6 
48 .6 

73 32 10 .7 



June 6 
15hll" 48^2 
-31 .1 
15 11 17 .1 
1 21 20 .3 
13 49 56 .8 
207° 29' 12". 
9.73876 
9.94798 n 
9.68674 n 
- 0.4861 
+ 38.7399 
+ 39.2260 
9.664211 n 
1 . 593574 
8.070637 n 
0°40' 26". 9 



7m04s,2 
4 30 2 

1 54 .2 

2 26 .8 
4 25 .8 
6 35 .8 



98". 1 
39 .8 
7 .1 
11 .8 
38 .5 
85 .4 



280 .7 

46 S 

1.6702 

9.7408 

+ 0.6 

+ 0° 40' 26". 3 

72 51 46 .7 

- 1 .8 
44 .9 

73 32 11 .2 



ASTRONOMICAL DETERMINATIONS 221 

123e. The Micrometric Method. In this method the angle 
between the mark and the star is measured with an eyepiece 
micrometer, no use whatever being made of the horizontal-limb 
graduations. Any form of transit or theodohte rnay be used 
that contains an eyepiece micrometer arranged to measure 
angles in the plane defined by the optical axis and the horizontal 
axis of the telescope. An eyepiece micrometer is essentially 
the same as the micrometer found on the microscopes of direc- 
tion instruments and described in Art. 45. When the observing 
telescope is fitted with an eyepiece micrometer the moving hairs 
lie in the focal plane of the objective and pass across the images 
of the objects viewed. When the angle between two' objects is 
small (about two minutes or less) it may be assumed with great 
exactness to be proportional to the distance between the corre- 
sponding images in the telescope, and this distance is measured 
by the micrometer screw with great precision. In applying this 
method to the determination of azimuth the mark is placed nearly 
in the vertical plane through the star, and the small horizontal 
angle between the mark and the star is determined from measure- 
ments made entirely with the micrometer, leaving all the hori- 
zontal motions of the instrument clamped in a fixed position. 
The azimuth of the mark is then obtained by combining this 
angle with the computed azimuth of the star. 

In the eyepiece micrometer the value of the angle measured 
is not given directly by the readings taken, as these indicate 
only the number of revolutions made by the screw. The reading 
is commonly taken to the nearest thousandth of a revolution, 
the whole number of revolutions being read from the comb scale, 
the t(3nths and hundredths from the graduations on the head, 
and the thousandths by estimation. In order to convert the read- 
ing into angular value it is necessary to know the angular value 
of one turn of the micrometer screw. The value of one turn of 
the screw is foimd by measuring therewith an angle whose value 
is already known. The value of such an angle may be found by 
measuring it directly with the horizontal circle, or by computing 
it from linear measurements. The value of one turn of the screw 
piay also be obtained by observations on a close circumpolar star 
near culmination, since the angle between any two positions of 
the star is readily computed from the times of observation, and the 
necessary reductions are then easily made. 



222 GEODETIC SURVEYING 

As already stated, the eyepiece micrometer measures angles 
in the_ plane defined by the optical axis and the horizontal axis 
of the telescope, and the corresponding horizontal angle must 
hence be obtained by a suitable reduction for the given altitude. 
To measure the horizontal angle between two objects at different 
elevations, therefore, it is necessary to find the micrometer value 
for the distance of each, object from the line of coUimation, reduce 
each value to the horizontal for the corresponding altitude, and 
combine the results for the complete horizontal angle. The reduc- 
tion in each case is effected by multiplying the micrometer value 
by the secant of the altitude. In the case of azimuth determina- 
tions the reduction must necessarily be made for the star, but 
need not be made for the mark unless it is several degrees out of 
the horizon. 

The micrometric method may be used at any hour angle, 
but unless the star is near elongation it will pass out of the safe 
range of the micrometer after but two or three sets of observa- 
tions have been secured. If the mark is placed about one or 
two minutes nearer the meridian than the star at elongation, 
the observations may be carried on within an hour or more each 
way from elongation, and a small error in time will have little 
or no effect on the result. In Coast Survey Appendix No. 7, 
Report for 1897-98, the following procedure is recommended: 
" The micrometer line is placed nearly in the line of coUimation of 
the telescope, a pointing made upon the mark by turning the 
horizontal circle, and the instrument is then clamped in azimuth. 
The program is then to take five pointings upon the mark; 
direct the telescope to the star; place the striding level in posi- 
tion; take three pointings upon the star with chronometer times; 
read and reverse the striding level; take two more pointings upon 
the star, noting the times; read the striding level. This com- 
pletes a half-set. The horizontal axis of the telescope is then 
reversed in the wyes; the telescope pointed approximately to 
the star; the striding level placed in position; three pointings 
taken upon the star with observed chronometer times; the strid- 
ing level is read and reversed; two more pointings are taken 
upon the star, with observed times; the striding level is read; and 
finally five pointings upon the mark are taken." In reducing 
such a set of observations the micrometer reading for the line of 
coUimation is taken as the mean of all the readings on the mark, 



ASTRONOMICAL DETERMINATIONS 



223 



05 



o 



O 



O 

O 
O 

S 

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P3 
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ft 


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rt 


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224 GEODETIC SURVEYING 



AZIMUTH BY MICROMETRIC METHOD— COMPUTATION 



CoUimatlon reads J(18 . 3^34 + 18 . 2808) = 18' . 2971 

Markeastofcollimation, 18.3134-18.2971 =0.0163= 02". 02 
Circle E., star E. of collimation 

(18. 4042-18. 2971)(1. 1690)= .1252 
Circle W., star E. of collimation 

(18. 2971-18. 0912)(1. 1695)= .2408 
Mean, star E. of collimation = 0.1835= 22 .70 



Mark west of star = 20 . 68 

Level correction (1 . 55) (0 . 92) (0 . 606) = - . 86 



Mark west of star, corrected = 19 . 82 



Mean chronometer time of observation = 211" 10™ 36^ .6 
Chronometer correction =—2 11 28.2 

Sidereal time = 18 59 08 . 4 

a = 1 20 07 .4 



Hour-angle, t, m time 17^' 39™ 01= .0 

Hour-angle, i, in arc 264° 45' 15". 



log cot d 
log tan 96 
log cos t 


= 


8.34362 
9.78436 
8.96108 n 




log a 

log cot d 
log sec ^ 
log sin t 
log 1/1 -a 


= 


7.08906 n 

8.343618 
0.068431 
9.998177 n 
9.999467 




log ( — tan A) 

A 

log 12.67 


= 


8.409693 n 
+ 1° 28' 16" 

1 . 10278 


.91 


log curvature correction 
Curvature correction 
Diur. aber. corr. 


= 


9.51247 
-0 
+0 


.33 

32 


Mean azimuth of star 
Mark west of star 


= 


+ 1° 28' 16" 
19 


.90 

.82 



Azimuth of mark, E. of N. = + 1° 27' 57" .08 



ASTEONOMICAj. •!•- , . ' n N vTIONS 225 

and all micrometer readings are rett ^ue. Since the 

star is changing rapidly in altitude th. ter readings 

are reduced to the horizontal for the meu .' ai f-ach half- 

set, the altitude of the star being occasioii r. '.:■ inter- 
polated for any desired time. The mean a^- i ' ' r tar 
for each set is found by computing the azimuth < K^,- 
to the average time of the pointings (/, Art. 123a), a. 
the curvature correction (gr. Art. 123a). The apparent 
of the mark is then found by combining the computed star aZi. ii 
with the measured angle (reduced to the horizontal). Theti < 
azimuth of the mark (as given by this set) is finally found by apply- 
ing to the apparent azimuth the level correction and the aberra- 
tion correction (/i and i, Art. 123a), and reducing the result to the 
south point (j, Art. 123a). 

The time occupied in taking a set of observations in the man- 
ner above specified should not average over fifteen minutes, 
so that a number of sets may be taken in a single night. By 
averaging the results of a number of nights' work a very close 
determination of azimuth may be secured. The method is more 
accurate than the direction method or the repeating method. 
With skilled observers the probable error of the mean of 25 or 30 
sets should be less than a tenth of a second. 

124. Azimuth Determinations at Sea. It is sometimes neces- 
sary to make an azimuth determination at sea in order to test 
the correctness of the ship's compasses. The method commonly 
employed is to measure the altitude of the sun or one of the brighter 
stars, and at the same instant take its bearing as shown by the 
compass to be tested. The azimuth of the given heavenly body 
is then computed from its observed altitude and the result reduced 
to a bearing. The difference between the observed bearing and 
the computed bearing is the error of the compass. The method 
and reductions for the azimuth observation are the same as 
explained in detail in Arts. 122, 122a, and 1226, except that the 
observation consists in measuring the altitude above the sea 
horizon by means of a sextant, and that a correction for dip 
(Art. 105) must be made. The latitude and longitude of the ship's 
position are always sufficiently well known for use in the reduc- 
tions. The computed bearing should not be in error over a few 
minutes, which is very much closer than it is possible to take the 
compass bearing. 



226 ■ IBk KEYING 



SBflb^ 



125. 


Pe- 


c CU 


the pole! 


^ 


k; ear^n » 


entlv 




:A ak/','' 



.tzimuth. As explained in Art. 112, 
■ ,<; fixed in position, but each one appar- 
xiean point in a period of about 425 days, 
ib s-v-ecior ying (during a series of revolutions) between 

/uui 0".'* '**' j"-36. The result of this shifting of the poles 
iB to a»'i# - azimuth of a line from a given point to oscillate 
aboiJ*" p -in value, the amplitude of the oscillation depending 
OB i' ycation of the point. In precise azimuth work, therefore, 
■:t, ate of the determination is an essential part of the record. 



CHAPTER VIII 
GEODETIC MAP DRAWING 

126. General Considerations. The object of a geodetic u. fs or 
chart is to represent on a flat surface, with as much accuracy ■ f 
position as possible, the natural and the artificial features of a given 
portion of the earth's surface. It is presumed that the engineer 
is familiar with the lettering of maps and the usual methods of 
representing the natural or topographical features, and such mat- 
ters are not here considered. The artificial features of a map 
are the meridians and parallels, the triangulation system or other 
plotted lines of location, and any lines which may be drawn to 
determine latitude, longitude, azimuth, angles, distances, or areas. 

In an absolute y perfect map the meridians and other straight 
lines (in the surveying sense), would appear as straight lines; the 
meridians would show a proper convergence in passing towards 
the poles; the parallels of latitude would be parallel to each other 
and properly spaced, and would cross all meridians at right angles; 
all points would be properly plotted in latitude and longitude; and 
azimuths, angles, distances and areas would everywhere scale 
correctly. On account of the spheroidal shape of the earth, it 
is evident that such a map is an impossibility, except for very 
limited areas. Some form of distortion must necessarily exist 
in any representation of a double curved surface on a flat sheet. 
By selecting a type of projection depending on the use to be made 
of the map, however, the distortion may be minimized in those 
features where accuracy is most desired, and entirely satisfactory 
maps produced. The principal types of map projection, as 
explained in the following articles, are the cylindrical, the trape- 
zoidal, and the conical, these terms referring to the considerations 
governing the plotting of the meridians and parallels. 

In the work of plane surveying the areas involved are usually 
of such small extent that no appreciable error is introduced in 
plotting by plane angles and straight line distances, drawing all 

227 



/ 

/ 

228 GEODE'^i' Si i -NG 

/ 
meridians or other nr «tsul ■ ^^ lines perfectly straight and 
parallel, and all pp- " oil. ^/6ast and west lines also straight 

and parallel anr' ,^s with the meridians. On account 

of the larp- y/ed in geodetic work it is generally- 

necessary ; ;,. , it VcA /fidians and parallels first (in accordance 
with f '-f'' >i--' ji4 of projection and the scale of the map), 
and !. , >, 1 1 .i-yfiindamental point of the survey by means of 
i-' . !.' ^ yix)ngitude without regard to angles or distances. 

■■ihf. > "la ^cfetails may then be plotted as in plane surveying. 
li% .<ietic map thus plotted the unavoidable distortion is 

j'and distributed as much as possible, 
ilie true lengths of 1° of latitude and longitude at the latitude 
/axQ given by the formulas 

1° of latitude ) na{l - e^) 



atthelat. ^ j 180(1 - e2 sin^ ^)!i' 
1° of longitude ] _ na cos 4> 



atthelat. ^ j 180(1 - e^ sin^ 96)*' 

in which formulas the letters have the significaiice and values of 
Arts. 67 and 69. The values of one degree of latitude and longitude 
are given for a number of latitudes in Table IX, and may be 
interpolated for intermediate latitudes. 

Since the radius of curvature of the meridian section increases 
from the equator to the poles it follows that the above formula 
for the length of a degree of latitude can only be correct in the 
immediate vicinity of the given latitude. The true length L 
of a meridian arc extending from the equator to any latitude 4> 
is given by the formula 

L = a(l - e)2(Af^ ~ N sm2(j) + P sin A.4, - etc.), 

in which 

ill = 1 + |e2 + l^e* +. . ., 
N =fe2 + Me* +. . ., 

p = a^e^ + . . ■ • 

For the length I of a meridian arc from the latitude <p to the lati- 
tude (j)', therefore, we have practically 

Z = a(l - e)2[M(9^' - 96) - iV(sin 2<j}' - sin 20) 

+ P(sin A,j>' ~ sin 40)]. 



GEODETIC MAP DRAWING 



229 



Substituting the values of a and e from Art. 67, and reducing 
the formula to its simplest form, we have 

I = A{(j)' - <j>) - B sin (96' - <j)) cos {4>' + 0) 

+ C sin 2 (^' - (j)) cos 2(^' + 96), 

in which <j) and <f>' in the first term of the second member are to 
be expressed in degrees and decimals, and in which 



A = 



B = 



metric, 111133.30 
feet, 364609.84 

metric, 32434.25 
feet, 106411.37 



„ _ ( metric, 
\ feet. 



34.41 
112.89 



log A = 
log 5 = 
logC = 



metric, 5.0458443 
feet, 5.5618285 



4.5110039 
5.0269881 



metric 
feet, 

metric, 1.5366847 
feet, 2.0526689 



127. Cylindrical Projections. The distinguishing feature of 
all cylindrical projections consists in the projection of the given 
area on the surface of a right cylinder (of special radius) whose 
axis is the same as the polar axis of the earth. The flat map 
desired is then produced by the development of this cylinder. 
In all forms of this projection the meridians are projected by the 
meridional planes into the corresponding right line elements of 
the cylinder, so that after development the meridians appear as 
equidistant parallel straight lines. The parallels of latitude 
are projected into the circular elements of the cylinder in a nim:iber 
of different ways, but in any case, after development, appear as 
parallel straight lines crossing the meridians everywhere at right 
angles. The three most common types of this projection are 
explained in the following articles. 

127a. Simple Cylindrical Projection. In this type of pro- 
jection, as illustrated in Fig. 54, page 230, the cylinder is so taken 
as to intersect the spheroid at the middle latitude of the area to be 
mapped, the parallels of latitude being projected into the cylinder 
by lines taken normal to the surface of the spheroid. It is evident 
from the figure that the parallels will not be represented by equi- 
distant lines, but will separate more and more in advancing towards 
the poles. This distortion in latitude is offset to a certain extent 
by a similar error in longitude, caused by the lack of convergence 
in the plotted meridians, so that the various topographical features 
remain approximately true to shape. On account of the varying 



230 



GEODETIC SURVEYING 



distortion in both latitude and longitude no single scale can be 
correctly applied to all parts of such a map. For the true lengths 
of one degree of latitude or longitude see Table IX or Art. 126. 
The projected distance x between the meridians, per degree of 
longitude, due to the middle latitude </>', is given by the formula 



_ 7ra r cos <j>' 1 

~ 180 [ (1 - e2 sin2 <6')* J ' 



and the projected distance y, from the equator to any parallel 
4>, by the formula 



y=atan^[ ^^_J^.^;^,^, ] - 



ae^ sin <f) 



(1 - e^&ui?4>)^' 




X 


X 


X 


X 


1 


X 















I 
I 



Fig. 54. — Simple Cylindrical Projection. 



in which formulas the letters have the significance and values 
of Arts. 67 and 69. 

When the cylinder is taken tangent to the equator (making 
<p' = 0), the factor in the brackets reduces to unity, and we have 



z = 



Tza 
l80 



and 



y = a tan <f) 



ae^ sin 4> 
(1 — e^ sin^ ^)i' 



In making a map by this method the meridians and parallels 
are spaced in accordance with the above formulas, and the fimda- 
mental points of the survey are then plotted by latitudes and 
longitudes. For small areas (10 square miles) within about 45° 
of the equator there is not much distortion in such a map. The 
amount of the distortion in any case is readily obtained by com- 



GEODETIC MAP DRAWING 



231 



paring the results given by the true formulas and the formulas 
used for the projection. 

127b. Rectangular Cylindrical Projection. In this type of 
projection, as illustrated in Fig. 55, the cylinder is so taken as to 
intersect the spheroid at the middle latitude of the area to be 
mapped, and the meridians are correctly developed on the ele- 
ments of the cylinder, so that in the finished map the parallels 
are spaced true to scale. The error due to the lack of convergence 
of the meridians still remains, so that the same scale can not be 
applied to all parts of the map. The distortion in longitude is more 
apparent than in the preceding projection, because no distor- 
tion exists in latitude. As in the previous case the meridians 
are spaced true to scale along the central parallel. 



•^daieLat.0' 



Equator 



X 


X 


X 


X 


X 


X 















Fig. 55. — Rectangular Cylindrical Projection. 



In makiug a map by this method the central meridian and 
parallel are first drawn and graduated to scale, using Table IX 
or the formulas of Art. 126. The remaining parallels and meridians 
are then drawn, and the survey plotted by latitudes and long- 
itudes. For small areas (10 square miles) within about 45° of 
the equator there is not much distortion in such a map, straight 
lines on the ground being straight on the map, and angles and 
distances scaling correctly. The plotting for such an area may 
therefore be done by latitudes and longitudes, or by angles and 
distances, as in plane surveying. 

127c. Mercator's Cylindrical Projection. This type of pro- 
jection, which is largely used for nautical maps, is illustrated 
in Fig. 56, page 232. As in the simple cylindrical projection, 
the space between the parallels constantly increases in advancing 
from the equator towards the poles, but the spacing is governed 
by an entirely different law. In Mercator's cylindrical projec- 
tion the cylinder is taken as tangent at the equator, so that the 



232 



GEODETIC SUEVEYINa 



spacing of the meridians along the equator is true to scale in the 
finished map. As the plotted meridians fail to converge, the 
distance between them is too great at all other points, the extent 
of the distortion becoming more and more pronounced as the 
latitude increases. To offset this condition the distance between 
the parallels is also distorted more and more as the latitude 
increases, making the law of distortion exactly the same in both 
cases. In that part of the map where the distance between the 
meridians scales twice its true value, for instance, the distance 
between the parallels should also scale twice its true value. 
Since this distortion factor changes with the slightest change of 





^ 


/ 
/ 
/ 
/ 
/ 
— ~~^ / '' 


- 








































k 




/ 


Any Lat,0\^'' 








\J 


1 














V 




/ 


\ 


1 
1 




' 


Eguator 


X 


X 


X 


X 


X 


1 
1 

\ 



Fig. 56. — ^Mercator's Cylindrical Projection. 

latitude, however, it is evident that a satisfactory map will require 
the meridian to be built up of a great many very small pieces, 
each multiplied in length by its own appropriate factor. A per- 
fect map on this basis requires an infinitesimal subdivision of the 
meridian, and a summation of these elements by the methods of 
the integral calculus. Using the notation and the formulsis of 
Arts. 67 and 69, and remembering that the distortion of any 
parallel is inversely proportional to its radius, we have for the 
distortion factor s at any latitude 0, 



a 
r 



(1 



N cos i 



sina^)^ 



cos ^ 



Multiplying the meridian element, Rd4>, by the distortion factor 
s, we have for dy, the projected meridian element. 



dy = s{Rd<f>) 



o(l - e2)# 
cos ^(1 — e^ sin2(/))4' 



GEODETIC MAP DRAWING 233 

whence, by integration, 

.-I.I5..8.5a[l„.(i±|||)-e.og([±||^)], 

in which y is the projected distance from the equator to any 
parallel of latitude 4^, and in which the formula is adapted to the 
use of common logarithms. The value of x per degree of longitude, 
for the spacing of the meridians, is given by the formula 

_ 7ca 

In making a map by this method the meridians and parallels 
are spaced in accordance with the above formulas, and the fun- 
damental points of the map are then plotted by latitudes and 
longitudes. It is evident that such a map will be true to scale 
only in the vicinity of the equator, and that different scales must 
be used for every part of the map. If it is desired, however, to 
have the map true to any given scale along the central parallel ^', 
it is only necessary to divide the above values of x and y by the 
distortion factor s' corresponding to the latitude ^'. 

A rhumb line or loxodrome between any two points on a spheroid 
is a spiral line which crosses all the intermediate meridians at the 
same angle. Except for points very far apart such a line is not 
very much longer than the corresponding great circle distance. 
Great circle sailing is sometimes practised by navigators, but 
ordinarily vessels follow a rhumb line, keeping the same course 
for considerable distances. A rhumb line of any length or angle 
will always appear in Mercator's projection as an absolutely 
straight line, crossing the plotted meridians at exactly the same 
angle as that at which the rhumb line crosses the real meridians. 
When a ship sails from a known point in a given direction, there- 
fore, its path is plotted on a Mercator chart by simply drawing a 
straight line through the given point and in the given direction. 
The distance traveled by the ship is plotted in accordance with the 
scale suitable to the given part of the map. Similarly the proper 
course to sail between any two points can be scaled directly from 
the map with a protractor. It is for these reasons that this type 
of projection is so valuable for nautical purposes. 



234 



GEODETIC SURVEYING 



128. Trapezoidal Projection. In this type of projection, 
as illustrated in Fig. 57, the meridians and parallels form a series 
of trapezoids. All the meridians and parallels are drawn as 
straight lines. The central meridian is first drawn and properly- 
graduated in degrees or minutes. The parallels of latitude are 
then drawn through these points of division as parallel lines at 
right angles to this meridian. Two parallels, at about one-fourth 
and three-fourths the height of the map, are then properly gradu- 
ated, and the corresponding points of division coimected by a series 
of converging straight lines to represent the meridians. For 
the correct distances required in making the graduations see 



Graduated 




Correctly 




Graduated 



Correctly 



Graduated 
Correctly 



Fig. 57. — Trapezoidal Projection. 

Table IX or Art. 126. From the nature of the construction it is 
plain that the central meridian is the only one which the parallels 
cross at right angles. The fundamental points of such a map 
are plotted by latitudes and longitudes. For small areas 
(25 square miles) the distortion in distance is very slight in 
this type of map. 

129. Conical Projections. The distinguishing feature of the 
conical projections consists in the projection of the given area 
on the surface of one or more right cones (of special dimensions) 
whose axes are the same as the polar axis of the earth. The 
flat map desired is then pr duced by the development of the 
cone or c nes thus used. In some forms of this projection the 
meridians are projected into the right line elements of the cones, 
while in other forms a different plan is adopted ; so that in some ' 
forms the meridians become straight lines after development, 



GEODETIC MAP DEAWING 



235 



while in other forms they appear as curved lines. The parallels 
of latitude are always projected into the circular elements of the 
cone or cones, and after development always appear as circular 
arcs. The four most common types of this projection are explained 
in the following articles. 

129a. Simple Conic Projection. In this type of projection, 
as illustrated in Fig. 58, the projection is made on a single cone 
taken tangent to the spheroid at the middle latitude of the area 
to be mapped. The meridians are projected into the right line 
elements of the cone by the meridional planes, and appear as 
straight lines after development. The meridians are correctly 
developed on the elements of the cone, so that the parallels are 
all spaced true to scale on the finished map, The parallels are 

A 





A 




^^\b 


/Middle Lat.(>' 


\C 




\d 


I Equator 


^\ 




Fig. 58. — Simple Conic Projection. 

drawn as concentric circles from the center A, the distance AC 
being the tangent distance for the middle latitude. The central 
parallel is graduated true to scale, and the meridians are drawn as 
straight lines from the center A through the points of division. 
For the tangent distance AC we have, from Art. 69, 

a cot <f> 



AC = T = N cot (f> 



(1 - e2 sin2 ^)i- 



The correct values for graduating the meridian and central 
parallel may be taken from Table IX or computed by the formulas 
of Art. 126. 

When it is impracticable to draw the arc EH from the center 
A it may be located by rectangular coordinates from the point 
C, as indicated by the dotted lines. To find the coordinates of 



236 GEODETIC SUKVEYINa 

any point H (see Fig. 59) let d equal the angular difference of 
longitude subtended by the arc CH (radius = r), and 8' equal the 
developed angle subtended by the same arc CH (radius =iV cot. <j>). 
Then, since equal lengths of arc in different circles subtend angles 
inversely as the radii, we have 

d' _ r _ N cos (}) _ . , 

I ~ N cot<p ~ N coi<i> ~ ^^'^ ^' 
giving 

8'= d sm(f>; 
whence 

X = AH sin S'= N cot <p sin (d sin (/>), 
and 

y =^AH vers d'= 2N cot ^ sw?(d^^^\. 

These values of x and y are readily computed by means of the 
data given in Table IX. In this projection the coordinates of 
the different arcs vary directly as their radii, 
so that the coordinates of the remaining parallels 
may be found by a simple proportion. As a 
check on the work the meridians should be 
straight and uniformly spaced. 

In making a map by this method the merid- 
ians and parallels are spaced in accordance with 
the above rules, and the fundamental points of 
the survey are then plotted by latitudes and 
longitudes. In this projection the meridians and 
Fig. 59. parallels intersect at the proper angle of 90°, and 

the parallels are properly spaced; but the spacing 
of the meridians is exaggerated everywhere except along the 
central parallel, and all areas are oo large. Such a map is satis- 
factory up to areas measuring several hundred miles each way. 

129b. Mercator's Conic Projection. In this type of pro- 
jection, as illustrated in Fig. 60, the projection is made on a single 
cone, taken so as to intersect the spheroid midway between the 
middle parallel and the extreme parallels of the area to be mapped. 
The remaining parallels may be considered as projected into the 
cone so that the spacing along the line BF is exactly proportional 
to the true spacing along the meridian GHK; or mathematically 

BC _CD_ ^ _ chord CE 

GC ~ CH ~ arc CE ' 




GEODETIC MAP DRAWING 



237 



After development the entire figure is then proportionately- 
enlarged until the spacing of the parallels is again true to scale; 
following which the developed angle and its subdivisions are 
correspondingly reduced in size, in order to make the projected 
parallels C'C" and E'E" true to the same scale. The distances 
B'C = arc GC, CD' = arc CH, etc., are found from Art. 126 or 
Table IX. The radius A'C is then computed from the formula 

A'C _ cos <j>il -e^ sin2 ^")i 

A'C + arc CE cos (j)" (1 - e^ sin2 ^)i" 

The remaining radii are found from A'C by a proper combina- 
tion of the known distances along the line A'F'. The parallel 

A A' 




Fig. 60. — ^Mercator's Conic Projection. 

E'E" is then graduated both ways from the central meridian by 
means of the values found from Art. 126 or Table IX, and the 
mer dians are drawn as straight lines from the point A'. 

The parallels may be plotted by rectangular coordinates 
when it is impracticable to use the center A', but the values given 
in Table IX are not correct for this type of projection. The 
individual angles at the apex A' are readily obtained from the 
radius A'E' and the subdivisions along the arc E'E", and the 
coordinates are then found for this arc and proportioned for the 
other arcs directly as their radii. 

In making a map by this method the meridians and parallels 
are drawn in accordance with the above rules, and the fundamental 



238 



GEODETIC SURVEYING 



points of the survey are then plotted by latitudes and longitudes. 
In this projection the meridians are straight lines, the meridians 
and parallels cross at the proper angle of 90°, and the parallels 
of latitude are properly spaced. The meridians are properly 
spaced on the parallels C'C" and E'E", but are a little too widely 
spaced outside of these parallels, and a Uttle too closely spaced 
within these parallels. Areas outside of these same parallels are 
too large, while areas within them are too small; but the total 
area is nearly correct. Mercator's conic projection is suitable 
for very large areas, having been used for whole continents. It 
has also been largely used for the maps in atlases and geographies. 
129c. Bonne's Conic Projection. In this type of projection, 
as illustrated in Fig. 61, the projection is made on a single cone 





A 


/^^ 


■ \\n 


/Middle Lat.?i' 


\n 




\e 




\\f 


1 Equator 


1 ' 



/ 

/ 
/ 
/ 

/ 
/ 
/ 


qA-V 




E ^ 




F _- 







Fig. 61- — Boime's Conic Projection. 



taken tangent to the spheroid at the middle latitude of the area 
to be mapped. The central meridian is projected into the straight 
line AF, with the parallels spaced true to scale and drawn as 
concentric circles, in accordance with the rules and formulas for 
simple conic projection (Art. 129a). Each parallel is then gradu- 
ated true to scale (see Art. 126 or Table IX), and the meridians 
are drawn as curved lines through corresponding divisions of the 
parallels. 

In making a map by this method the fundamental points of 
the survey must be plotted by latitudes and longitudes. In this 
projection the meridians and parallels fail to cross at right angles 



GEODETIC MAP DRAWING 



239 



but the same scale holds good for all the meridians and all the 
parallels. Bonne's conic projection is suitable for very large 
areas, having been used for whole continents. It has also been 
largely used for the maps in atlases and geographies. 

129d. Polyconic Projection. In this type of projection, as 
illustrated in Fig. 62, a separate tangent cone is taken for each 
parallel of latitude, and made tangent to the spheroid at that 
parallel. Each parallel on the map results from the development 
of its own special cone, appearing as the arc of a circle, with a 
radius equal to the corresponding tangent distance. The parallel 




Fig. 62. — Polyconic Projection. 



through the point G, for instance, is drawn as a circular arc with 
a radius equal to the tangent distance BG, and so on. The 
central meridian is drawn as a straight line, on which all the 
parallels are spaced true to scale, so that the division EF equals 
the arc EF, the division FG equals the arc FG, and so on. The arcs 
representing the various parallels are then drawn through these 
division points with the appropriate radii, and with the centers 
located on the central meridian. Each parallel as thus represented 
is then graduated true to scale, and the meridians are drawn as 
curved lines connecting the corresponding divisions. 

In making a map by this method the meridians and parallels 
are plotted in accordance with the data given in Table IX, or 



240 GEODETIC SURVEYING 

from corresponding values computed by the rules and formulas 
of Arts. 126 and 129a, remembering that each parallel is here 
equivalent to the central parallel of the simple conic projection. 
The plotting is customarily done by rectangular coordinates, 
the meridians and parallels being taken so close together that the 
intersection points may be connected by straight lines. The 
fundamental points of the survey are then plotted by latitudes 
and longitudes. 

This type of projection is suitable for very large areas. The 
meridians are spaced true to scale throughout the map and cross 
the parallels nearly at right angles. The parallels are spaced 
true to scale only along the central meridian, and diverge more 
and more from each other as the distance from the central merid- 
ian increases. The whole of North America, however, may be 
represented without material distortion. The U. S. Coast and 
Geodetic Survey and the U. S. Geological Survey have adopted 
the polyconic system of projection to the exclusion of all others. 
For further information on this subject see " Tables for the 
Projection of Maps, Based upon the Polyconic Projection of 
Clarke's Spheroid of 1866, and computed from the Equator to 
the Poles; Special Publication No. 5, U. S. Coast and Geodetic 
Survey, U. S. Government Printing Office, 1900." 

The above type of polyconic projection is sometimes called 
the simple polyconic, to distinguish it from the rectangula/ poly- 
conic, in which the scales along the parallels are so taken as to 
make all the meridians and parallels cross at right angles. When 
not -otherwise specified the simple polyconic is in general under- 
stood to be the one intended. 



PART II 

ADJUSTMENT OF OBSERVATIONS BY THE 
METHOD OF LEAST SQUARES 



CHAPTER IX 

DEFINITIONS AND PRINCIPLES 

130. General Considerations. In various departments of 
science, such as Astronomy, Geodesy, Chemistry, Physics, etc., 
numerous values have to be determined either directly or indirectly 
by some process of measurement. When any fixed magnitude, 
however, is measured a number of times imder the same apparent 
conditions, and with equal care, the results are always foimd to 
disagree raore or less amongst themselves. With skillful observers, 
and refined methods and instruments, the absolute values of 
the discrepancies are decreased, but the relative disagreement 
often becomes more pronoxmced. The conclusion is obviously 
reached that all measurements are affected by certain small and 
unknown errors that can neither be foreseen nor avoided. The 
object of the method of Least Squares is to find the most probable 

'values of imknown quantities from the results of observation, 
and to gage the precision of the observed and reduced values. 

131. Classification of Quantities. The quantities observed 
are either independent or conditioned. 

An independent quantity is one whose value is independent of 
the values of any of the associated quantities, or which may be 
so considered during a particular discussion. Thus in the case 
of level work the elevation of any individual bench mark is an 
independent quantity, since it bears no necessary relation to the 
elevation of any other bench mark. While in the case of a triangle 

241 



242 GEODETIC SUEVEYING 

we may consider any two of the angles as independent quantities 
in any discussion in' which the remaining angle is made to depend 
on these two. 

A conditioned quantity (or dependent quantity) is one whose 
value bears some necessary relation to one or more associated 
quantities. In any' case of conditioned quantities we may regard 
these quantities as being mutually dependent on each other, or 
any number of them as being dependent on the remaining ones. 
Thus if the angles of a triangle are denoted by x, y, and z, we 
may write the conditional equation 

x + y + z = 180°, 

and regard each angle as a conditioned quantity; or we may write, 
for instance, 

z = 180° -X- y, 

and regard z as conditioned and x and y as independent. 

132. Classification of Values. In considering the value of 
any quantity it is necessary to distinguish between the true value, 
the observed value, and the most probable value. 

The tru£ value of a quantity is, as its name implies, that value 
which is absolutely free of all error. Since (Art. 130) all measure- 
ments are subject to certain unknown errors, it follows that the 
true value of a quantity may never be known with absolute pre- 
cision. In any case such a value would seldom be any exact 
number of units, but could only be expressed as an unending 
decimal. 

The observed value of a quantity is technically understood to 
mean the value which results from an observation when correc- 
tions have been applied for all known errors. Thus in measuring 
a horizontal angle with a sextant the vernier reading must be 
correcited for the index error to obtain the observed value of the 
angle; in measuring a base line with a steel tape the corrections 
for horizontal and vertical alignment, pull, sag, temperature, and 
absolute length, are understood to have been applied; and so on. 

The most probable value of a quantity is that value which is 
most likely to be the true value in view of all the measurements 
on which it is based. The most probable value in any case is 
not supposed to be the same as the true value, but only that value 
which. is more likely to be the true value than any other single 
value that might be proposed. 



DEFINITIONS AND PRINCIPLES 243 

133. Observed Values and Weights. The observations which 
are made on unknown quantities may be direct or indirect, and 
in either case of equal or of unequal weight. 

A direct observation is one that is made directly on the quan- 
tity whose value is desired. Thus a single measurement of an 
angle is a direct observation. 

An indirect observation is one that is made on some function 
of one or more unknown quantities. Thus the measurement 
of an angle by repetition represents an indirect observation, 
since some multiple of the angle is measured instead of the single 
value. So also in ordinary leveling the observations are indirect, 
since they represent the difference of elevation from point to 
point instead of the elevations of the different points. 

By the weight of an observation is meant its relative worth. 
When observations are made on any magnitude with all the con- 
ditions remaining the same, so that all the results obtained may 
be regarded as equally reliable, the observations are said to be of 
equal weight or precision, or of unit weight. When the condi- 
tions vary, so that the results obtained are not regarded as equally 
reliable, the observations are said to be of unequal weight or pre- 
cision. It has been agreed by mathematicians that the most 
probable value of a quantity that can be deduced from two obser- 
vations of unit weight shall be assigned a weight of two, from three 
such observations a weight of three, and so on. Hence when an 
observation is made under such favorable circmnstances that the 
result obtained is thought to be as reliable as the most probable 
value due to two observations which would be considered of unit 
weight, we may arbitrarily assign a weight of two to such an 
observation; and so on. As the weights apphed in any set of 
observations are purely relative, their meaning will not be changed 
by multiplying or dividing them all by the same munber. The 
elementary conception of weight is therefore extended to include 
decimals and fractions as well as integers, since any set of wei2,hts 
could be reduced to integers by the use of a suitable factor. 

134. Most Probable Values and Weights. In any set of 
observations the most probable value of the unknown quantity 
will evidently be some intermediate or mean value. There are 
many types of mean value, but manifestly they are all subject 
to the fundamental condition that in the case of equal values the 
mean value must be that common value. Three of the common 



244 GEODETIC SURVEYING 

typms of mean value are the arithmetic mean, the geometric mean, 
and the quadratic mean. If there are n quantities whose respective 
values are Mi, M2, etc., we have, 

= the arithmetic mean; 

n 



'^MiM2 . . . Mn = the geometric mean; 
= the quadratic mean; 



S 



(1) 



all of which satisfy the fundamental condition of a mean value. 
In the case of direct observations of equal weight it has been 
universally agreed that the arithmetic mean is the most probable 
value. In accordance with this principle, and the definition of 
weight as given in Art. 133, it is evident that the weight of the 
arithmetic mean is equal to the number of observations. Sim- 
ilarly, an observation to which a weight of two has been assigned 
may be regarded as the arithmetic mean of two component obser- 
vations of unit weight, and so on, provided no special assimiption 
is made regarding the relative values of these components. 
For direct observations of unequal weight, therefore. 

Let z = the most probable value of a given magnitude; 

Ml, M2, etc. = the values of the several measurements; 

Pi, P2, etc. = the respective weights of these measurements; 
api, ap2, etc. = the corresponding integral weights due to the 
use of the factor a; 
f^i', mi", etc. = the api imit weight components of Mi when con- 
sidered as an arithmetic mean 
m2', m2", etc. = similarly for M2, and so on; 

then we may write as equivalent expressions 

,, mi' + mi" . . . Smi 

Ml = = 

api api 



whence 



^T wi2' -I- m2" . . . 2m2 , 

jy-12 = = , etc; 

ap2 ap2 

Smi = apiMi, 
Sm2 = ap2M2, etc.; 



DEFINITIONS AND PRINCIPLES 245 

and, since the various values of m are of unit weight, 

_ Smi + Sm2 . . . 

api + ap2 . . . ' 
or 

^ Sap Sap ~ Sp ' • ■ • ■ ^' 

from which we have the general principle : 

In the case of direct observations of unequal weight the most 
probable value is found by multiplying each observation by its weight 
and dividing the sum of these products by the sum of the weights. 
The result thus obtained is called the weighted arithmetic mean. 
In the above discussion the value of z is found by taking the 
arithmetic mean of Sap quantities whose sum is Sm, so that the 
integral weight of z is Sap. Dividing by a in order to express this 
result in accordance with the original scale of weights, we have 

Weight of 3 = Sp; ....... (3). 

or, expressed in words, the weight of the weighted arithmetic 
mean is equal to the sum of the individual weights. 

135. True and Residual Errors. It is necessary to distin- 
guish between true errors and residual errors. 

A true error, as its name implies, is the amount by which any 
proposed value of a quantity differs from its true value. True 
errors are generally considered as positive when the proposed 
value is in excess and vice versa. Since (Art. 132) the true value 
of a quantity can never be known, it follows that the true error 
is likewise beyond determination. 

A residual error is the difference between any observed value 
of a quantity and its most probable value, in the same set of 
observations. The subtraction is taken algebraically in which- 
ever way is most convenient in the given discussion. In the case 
of indirect observations the most probable value of the observed 
quantity is found by substituting the most probable values of the 
individual unknowns in the given observation equation (Art. 158). 
Residual errors are frequently called simply residuals. 

In the case of the arithmetic mean the sum of the residual errors 
is zero. This is proved as follows: 



246 



GEODETIC SURVEYING 



Let 
Ml, M^, 



Vl, V2, 



then 



n = the number of observations; 
Mn = the observed values; 
2 = the arithmetic mean; 
. . Vn= the residual errors; 

vi = z — Ml 
V2 = z — M2 



but 



or 



from which 
whence 



Vn = Z - Mn 

Hv = m — STIf ; 

SM 

z = , 

n 

nz = SAf, 

nz - SM = 0; 

2i) = 0, ... 



(4) 

which was to be proved. 

In the case of the weighted arithmetic mean the sum of the weighted 
residuals equals zero. This is proved as follows: 

Let n = the number of observations; 

Ml, M2, ,. . Mn = the observed values; 

Pi, P2, ■ ■ ■ Pn = the corresponding weights; 

z = the weighted arithmetic mean; 
1^1, V2, . . . Vn = the residual errors; 



then 



but 



or 



pivi = pi(z - Ml) 
P2V2 = P2(z - M2) 

PnVn = Pn{Z — Mn) 
Upv = Sp-2 - I,pM; 



Z = 



Sp-z = I,pM, 



DEFINITIONS AND PEINCIPLES 247 

from which 

2p-z - 2pM =0; 
whence 

^pv =0, (5) 

which was to be proved. 

136. Sources of Error. The errors existing in observed values 
may be due to mistakes, systematic errors, accidental errors, or 
the least count of the instrument. 

A mistake is, as its name implies, an error in reading or record- 
ing a result, and is not supposed to have escaped detection and 
correction. 

A systematic error is one that follows some definite law, and is 
hence free from any element of chance. Errors of this kind may 
be classed as atmospheric errors, such as the effect of refraction 
on a vertical angle, or the effect of temperature on a steel tape; 
instrumental errors, such as those 'due to index errors or imperfect 
adjustments; and personal errors, such as individual peculiarities 
in always reading a scale a little too small, or in recording a star 
transit a little too late. Systematic errors usually affect all the 
observations in the same manner, and thus tend to escape detec- 
tion by failing to appear as discrepancies. Such errors, however, 
are in general well understood, and are supposed to be eliminated 
by the method of observing or by subsequent reduction. 

An accidental error is one that happens purely as a matter of 
chance, and not in obedience to any fixed law. Thus, for instance, 
in bisecting a target an observer will sometimes err a little to 
the right, and sometimes a little to the left, without any assignable 
cause; a steel tape will be slightly lengthened or shortened by a 
momentary change of temperature due to a passing current of 
air, and so on. 

An error due to the least count of the instrument is one that is 
caused by a measurement that is not capable of exact expression 
in terms of the least count. Thus an angle may be read to the 
nearest second by an instrument which has a least count of this 
value, but the true value of the angle may differ from this reading 
by some fraction of a second which can not be read. 

137. Nature of Accidental Errors. Errors of this kind are 
due to the limitations of the instruments used; the estimations 
required in making bisections, scale readings, etc., and the con- 



248 GEODETIC SURVEYING 

stantly changing conditions during the progress of an observa- 
tion. Each individual error is usually very minute, but the 
possible number of such errors that may occur in any one measure- 
ment is almost without limit. In general it may be said that any 
single observation is affected by a very large number of such errors, 
the total accidental error being due to the algebraic sum of these 
small individual errors. Thus in measuring a horizontal angle 
with a transit the instrument is seldom in a perfectly stable posi- 
tion; the leveling is not perfect; the lines and levels of the instru- 
ment are affected by the wind and varying temperatures; the 
graduations are not perfect; the reading is affected by the judg- 
ment of the observer; the target is bisected only by estimation; 
the line of sight is subject to irregular sidewise refraction due to 
changing air currents; and so on. As long as the component 
errors are all accidental, however, the total error may be regarded 
as a single accidental error. 

133. The Laws of Chance. The errors remaining in ol)served 
values after all possible corrections have been made are presumed 
to be accidental errors, and must therefore be assumed to have 
occurred in accordance with the laws of chance. By the laws of 
chance are meant those laws which determine the probability of 
occurrence of events which happen by chance. 

By the ■probability of an event is meant the relative frequency 
of its occurrence. It is not only a reasonable assumption but also 
a matter of common experience, that in the long run the relative 
frequency with which a proposed event occurs will closely approach 
the relative possibilities of the case. Thus in tossing a coin 
heads may come up as one possibility out of the two possibilities 
of heads or tails, so that the probability of a head coming up is 
one-half; and in a very large number of trials the occurrence of 
heads will closely approximate one-half the total nvunber of trials. 
Probabilities are therefore represented by fractions ranging in 
value from zero to unity, in which zero represents impossibility of 
occurrence, while unity represents certainty of occurrence. 

The three fundamental laws of chance are those relating to 
simple events, compound events, and concurrent events. 

139. A Simple Event is one involving a single condition which 
must be satisfied. The probability of a simple event is equal to 
the relative possibility of its occurrence. Thus the probability of 
drawing an ace from a pack of cards is iV, since there are four 



DEFINITIONS AND PRINCIPLES 249 

such possibilities out of 52, and -f^ = xV; but the probability 
of drawing an ace of clubs, for instance, is only jV, since there 
is only one such possibility out of 52. 

140. A Compound Event is one involving two or more con- 
ditions of which only one is required to be satisfied. The proba- 
bility of a compound event is equal to the sum of the probabilities of 
the component simple events. This law is evidently true, since the 
number of favorable possibilities for the compound event equals 
the sum of the corresponding simple possibilities, and the total 
number of possibilities remains unchanged. Thus the probability 
of getting either a club or a spade in a single draw from a pack 
of cards is one-half, because the probability of getting a club is one- 
quarter, and the probability of getting a spade is one-quarter, and 
i -|- i = i; or in other words the 13 chances for getting a club 
are added to the 13 chances. for getting a spade, making 26 favor- 
able possibilities out of a total of 52. The probability of draw- 
ing either a club, spade, heart, or diamond, equals i -|- i -|- i -|- i, 
which equals unity, since the proposed event is a certainty. 

141. A Concurrent Event is one involving two or more con- 
ditions, all of which are required to be satisfied together. The 
probability of a concurrent event is equal to the product of the prob- 
abilities of the component simple events. This law is evidently 
true, since the number of favorable possibilities for the concurrent 
event is equal to the product of the corresponding simple pos- 
sibilities; while the total niraiber of possibilities is equal to the 
product of the corresponding totals for the component simple 
events. Thus the probability of cutting an ace in a pack of cards 
is ^V, so, that the probability of getting two aces by cutting two 
packs of cards is -^ X -gV = *^ ^ /a ~ tV X yV = ttt- It is evi- 
dent that the required condition will be satisfied if any one of the 
four aces in one pack is matched with any one of the four aces in the 
other pack, so that there are 4X4 favorable possibilities. Also 
the cutting may result in getting any one of 52 cards in one pack 
against any one of 52 cards in the other pack, so that there are 
52X52 total possibilities. Multipljang the two probabihties, 
therefore, gives the relative possibility and therefore the required 
probability for the given concurrent event. Similarly the propo- 
sition may be proved for a concurrent event involving any 
number of simple events. Thus in throwing three dice the 
probability of getting 3 fours, for instance, will be |XiXi=2Tir; 



250 GEODETIC SUEVEYING 

the probability of drawing a deuce from a pack of cards at 
the same time that an ace is thrown with a die, will be 
iVXt = tV; and so on. 

In figuring the probability of a concurrent event it is neces- 
sary to guard against two possible sources of error. In the 
first place the probabilities of the simple events involved in a 
concurrent event may be changed by the concurrent condition. 
Thus the probability of drawing a red card from a pack is |f , 
but the probability of drawing two red cards in succession from 
a pack is not MXfl, but ff X|t, since the drawing of the first 
card changes the conditions under which the second card is drawn. 
In the second place, the probability of a concurrent event may 
be modified by the sense in which the order of simple events 
may be involved. Thus in cutting two packs of cards the prob- 
ability that the first pack will cut an ace and the second a king 
is tVXi\ = tb'5; but the probability that the first pack will cut 
a king and the second an ace is also rTXTV = Ti¥; so that the 
probability of cutting an ace and a king without regard to 
specific packs becomes y^ti and not xb^t, as might be inferred. 

142. Misapplication of the Laws of Chance. The probability 
of a given event is the relative frequency of its occurrence in 
the long run, and not in a limited number of cases. It is not 
to be expected that every two tosses of a coin will result in one 
head and one tail, since other arrangements are possible, and 
the laws of chance are founded on the idea that every possible 
event will occur its proportionate number of times. Thus in 
the case of a coin we have for all possible events in two tosses. 

Probability of 2 heads = J 

" 1 head and 1 tail = 4 

" 1 tail and 1 head = | 

2 tails = i 

Some one of these events must happen, so that the total prob- 
ability is i+i+4.+ i, which equals unity, as it should in a 
case of certainty. The probability of two tosses including a 
head and a tail (which may occur in two ways) is i+i=|, so 
that the proposed event is not one that occurs at every trial, 
as is often inferred. 

An event whose probability is extremely high will not neces- 
sarily happen on a given occasion, and this failure to happen 



DEFINITIONS AND PEINOIPLES 251 

does not imply an error in the theory of probabilities. The 
very fact that the given probability is not quite unity indicates 
the chance of occasional failures. Similarly an event with a 
very small probability will sometimes happen, otherwise its 
probability should be precisely and not approximately zero. 

The probability of a future event is not affected by the 
result of events which have already taken place. Thus if a tossed 
coin has resulted in heads ten times in succession it is natural 
to look on a new toss as much more likely to result in tails than 
in heads; but mature thought will show that the probabilities are 
still one-half and one-half for any new toss that may be made. The 
confusion in, such a case comes from regarding the ten successive 
heads as an abnormal occurrence, whereas, being one of the 
possible occurrences, it should happen in due course along with 
all other possible events. If tails were more likely to come up 
than heads in any particular toss, it would imply some difference 
of conditions instead of any overlapping influence. If the toss 
of a coin is ever regarded as a matter of chance, it must always 
be so regarded. 



CHAPTER X 

THE THEORY OF ERRORS 

143. The Laws of Accidental Error. The mathematical 
theory of errors relates entirely to those errors which are purely 
accidental, and which therefore follow the laws of probability. 
Mistakes or blunders, which follow no law, and systematic 
errors, which follow special laws for each individual case, can 
not be included in. such a discussion. If a sufficient number of 
observations are taken it is found by experience that the accidental 
errors which occur in the results are governed by the four fol- 
lowing laws : 

1. Plus and minus errors of the same magnitude occur with 
equal frequency. 

This law is a necessary consequence of the accidental char- 
acter of the errors. An excess of plus or minus errors would 
indicate some cause favoring that condition, whereas only acci- 
dental errors are under consideration. 

2. Errors of increasing magnitude occur loith decreasing frequency. 
This law is the result of experience, but for mathematical 

purposes it is replaced by the equivalent statement that errors 
of increasing magnitude occur with decreasing facility. For 
reasons yet to appear (Art. 146) the facility of an error is rated 
in units that make it proportional to the relative frequency with 
which that error occurs instead of equal thereto. 

3. Very large errors do not occur at all. 

This law is also the result of experience, but it is not in 
suitable form for mathematical expression. It is satisfactorily 
replaced by the assumption that very large errors occur with 
great infrequency. 

4. Accidental errors are systematically modified by the cir- 
cumstances of observation. 

This law is a necessary consequence of the first three laws, 
and emphasizes the fact that these three laws always hold good 

252 



THE THEORY OF ERROES 



253 



however much the absolute values of the errors may be modified 
by favorable or unfavorable conditions. The chief circumstances 
affecting a set of observations are the atmospheric conditions, 
the skill of the observer, and the precision of the instruments. 

144. Graphical Representation of the Laws of Error. The 
four laws of error are graphically represented in Fig. 63, in which 
the solid curve corresponds to a series of observations taken 
under a certain set of conditions, and the dotted curve to a 
senes of observations taken under more favorable conditions. 
For reasons which will appear in due course any such curve is 
called a probability curve. The line XX, or axis of x, is taken 
as the axis of errors, and the line AY, or axis of y, as the axis 
of facility, the point A being taken as the origin of coordinates. 
Thus in the case of the solid curve, if the line Aa represents any 





,^'' 


'^•^ 








-<\. 


















/ 




I 


\. 




^^•^y 








^ 


'^V 


















^ — "^^r^^ 












"e 


^ 


















:::=^^^^^^^'' 

















A a d 

Pig. 63. — Probability Curves. 



proposed error, then the ordinate db represents the facility with 
which that error occurs in the case assimied. The first law is 
illustrated by making the curves symmetrical with reference to 
the axis of y, so that the ordinates are equal for corresponding 
plus and minus values of x. The second law is illustrated by the 
decreasing ordinates as the plus and minus abscissas are increased 
in length. The third law does not admit of exact representation, 
since a mathematical curve can not have all its ordinates equal 
to zero after passing a certain point; a satisfactory result is 
reached, however, by making all ordinates after a certain point 
extremely small, with the axis of x as an asymptote to the curves. 
The fourth law is illustrated by means of the solid curve and the 
dotted curve, both of which are consistent with the first three 
laws, but which have different ordinates for the same proposed 
error. Thus small errors, such as Aa, occur with greater frequency 
(or greater facility) in the case of the dotted curve than in the 



254 GEODETIC SURVEYING 

case of the solid curve, as shown by the ordinate ac being longer 
than the ordinate ab\ while large errors, such as Ad, occur with 
less frequency (or less facility) in the case of the dotted curve 
than in the case of the solid curve, as shown by the ordinate 
de being shorter than the ordinate d/. 

145. The Two Tjrpes of Error. The recorded readings in 
any series of observations are subject to two distinct types of 
error. The first type of error includes all those errors involved 
in the making of the measurement, such as those due to imper- 
fect instrumental adjustments, unfavorable atmospheric conditions, 
imperfect bisection of targets, imperfect estimation of scale 
readings, etc. The second type of error is that involved in 
the reading or recording of the result, which must be done in 
terms of some definite least count which excludes all inter- 
mediate values. 

A given reading, therefore, does not indicate that precisely that 
value has been reached in the process of measurement, but only 
such a value as must be represented by that reading; so that 
a given reading may be due to any one of an infinite number 
of possible values lying within the limits of the least count. 
Similarly, the error in the recorded reading does not indicate 
that precisely that error has been made in the process of measure- 
ment, but only such an error as must be represented by that 
value; so that the error of the recorded reading may in fact 
be due to any one of an infinite number of possible errors Ijang 
within the limits of the least count. The first type of error 
is the true type or that which corresponds to the accidental 
conditions under which a series of observations are made, while 
the second type is a false type or definite condition or limitation 
under which- the work must be done. Thus in sighting at a target 
a nimiber of times the angular errors of bisection may vary 
among themselves by amounts which can only be expressed in 
indefinitely small decimals of a second. If the least count 
recognized in recording the scale readings is one second, however, 
the recorded readings and the corresponding errors will vary among 
themselves by amounts which differ by even seconds. The 
probability curve of the preceding article is based on the first 
type of error only, and is therefore a mathematically con- 
tinuous curve, since all values of the error are possible with this 
type. In speaking of the errors of observations, however, the 



THE THEORY OF ERRORS 265 

errors of the recorded values are in general understood, and these 
must necessarily differ among themselves by exactly the value 
of the least count. 

146. The Facility of Error. If an instrument is correctly 
read to any given least count, no reading can be in error by more 
than plus or minus a half of this least count; or, in other words, 
each reading is the central value of an infinite number of 
possible values lying within the limits of the least count. If 
a great many observations are taken on a given magnitude, each 
particular reading will be found to repeat itself with more or 
less frequency, since all values lying within a half of the least 
count of that particular reading must be recorded with the 
value of that reading. If the same instrument, however, carried 
finer graduations, with the least count half the previous value, 
each reading would represent only those values within half 
the previous limits. There would then be twice as many repre- 
sentative readings, with each one standing for half as many 
actual values as with the coarser graduations. It is thus seen 
that the relative frequency with which a given reading (and 
the corresponding error) occurs, is directly proportional to the 
least count of the instrmnent, or least count used in recording 
the readings. Just as each reading is taken to represent an 
infinite number of possible values within the limits of the least 
count, so that reading must correspond to an infinite number of 
possible errors within the same limits, each possible error having 
a different facility of occurrence. Since in the long run, however, 
each reading will be practically the average of all the values 
that it represents, so the fa-cility of the error due to that reading 
may be taken practically as the average facility of all the corre- 
sponding errors. By definition (Art. 143) the facility of a given 
accidental error is proportional to the frequency of its occurrence. 
It is thus seen that the relative frequency with which a given 
error (representing all possible errors due to a given reading) 
occurs, is proportional to the facility of that error. Since the 
relative frequency with which a given error occurs is proportional 
to both its facility and the least count, it is proportional to 
t^heir product, and is always made equal to this product by using 
a suitable scale of facility. The facility of a given error is hence 
equal to the relative frequency of occurrence of that error divided 
by the least count. 



256 



GEODETIC SURVEYING 



147. The Probability of Error. By the prohdbility of an 
error is meant the relative frequency of its occurrence. Thus 
in the measurement of an angle, if a given error occurred (on 
the average) 27 times in 1000 observations, then the probability 
that an additional measurement would be in error by that same 
amount would be xHir- The probability of a given error being 
identical with its relative frequency of occurrence is hence (Art. 
146) equal to the product of the facility of that error by the least 
count. The probabihty of error for a certain set of conditions 
is illustrated in Fig. 64. In this figure the spaces da, ae, eb, and 
bf are each equal to one-half of the least count. The probability 
that an error Aa will occur is hence, in accordance with the 
above principles, equal to the product of am (the facility) by 





> 


f 


s 














\ 




» 

7 






l\ 






__^^^^^ 






1 


1 


^v 


"r--__ 



A d a e h f a 

Fig. 64. — The Trobability of Error. 



de (the least count). As the least count is always very small, 
we may write without appreciable error, 

Probability of error Aa = amXde = area dste. 

But (Art. 145) the error .4 a in the recorded reading includes all 
the possible errors lying between Ad and Ae, that is, within 
half the least count each way from Aa. The area dste therefore 
represents the probability that the actual error committed lies 
between the values Ad and Ae. Similarly the area etuf represents 
the probability of an actual error between the values Ae and 
Af. The probability that an actual error shall lie either between 
Ad and .4e or between Ae and Af (compound event. Art. 140), 
or in other words between Ad and Af, is equal to the sum of the 
two separate probabilities, that is, to the combined area dsuf. 
Or, in general, the probability that an error shall fall between 
any two values Ac and Ag, is represented by the area included 
between the corresponding ordinates cr and gv. On account 



THE THEORY OF ERRORS 257 

of this characteristic property the curve of facilities is commonly 
called the probability curve. Strictly speaking the ordinates 
limiting the area can only occur at certain equally spaced intervals 
depending on the least count, but no material error is ever intro- 
duced by drawing them at any points whatever. 

148. The Law of the Facility of Error is that law which con- 
nects all the possible errors in any set of observations with their 
corresponding facilities, and is expressed analytically by the 
equation of the probability curve. The law which governs the 
occurrence of errors in any particular set of observations is 
necessarily unknown and beyond determination, being the com- 
bined result of an uncertain number of variable and unknown 
causes. Fortunately, however, it is found by experience that 
there is one particular form of law which (with proper constants) 
very closely represents the facility of error in all classes of obser- 
vations. This form of law is that which is in accordance with 
the assiunption that the arithmetic mean of the observed values 
is the most probable value when the same magnitude has been 
observed a large number of times under the same conditions. 
The same form of law being accepted as satisfactory in all cases, 
therefore, the law for any particular case is determined by the 
substitution of the proper constants. 

149. Form of the Probability Equation. If x represents any 
possible error and y the facility of its occurrence, we may write 

y = 4>ix), (6) 

which is read y equals a function of x. When the form of this 
fimction has been determined the expression will be the general 
equation of the probability curve. Since the probability that 
the error x (of a recorded reading) will occur is equal (Art. 147) 
to its facility multiplied by the least count, we have 

P = yJx = ^{x)Jx, (7) 

in which P is the probability of the occurrence of the error x, 
and dx is the least count. If xi, X2, . . . Xn are the true errors in 
the observed values of any magnitude Z, and Pi, P2, . . . Pn 
are the corresponding probabilities of occurrence, we thus have 

Pi = (j}(xi)Jx, P2 = (I>{x2)dx, etc. 



258 GEODETIC SURVEYING 

The probability P of the occurrence of this particular series 
of errors, xi, X2, etc., in a set of observations of equal weight, 
being a concurrent event (Art. 141), is equal to the product 
of the individual probabilities, giving 

P =4>{x{)-<f>{x2)...4>{xn)-{^xy; .... (8) 
whence 

log P = log 4>{xi) + log 4>^X2). . . + log <j){xn) + n log Ax. (9) 

The true value of the unknown quantity Z, and the errors 
Xi, X2, etc., can never be known. Any assumed value of Z will 
result in a particular series of values Vi, V2, etc., for the errors 
of the several observations. That value of Z will be the most 
probable which produces the series of errors which has the 
highest probability of occurrence. Replacing the true errors 
Xi, Xn, etc., in Eq. (9) by the variable errors V\, V2, etc., and 
making the first differential coefficient equal to zero to obtain 
a maximiun value of P, we have 

d\og(i>{vi) ^ dl0g4>{V2) _^ d log 4>{Vn) ^ Q ^Q, 

dvi dv2 ' ' ' dv„ ' 

which may be written 

/ d log ^(.0 \ ^ ^ / d log j^fe) \_^/ d log ^(.„) \ 

\ VidVi / V V2dV2 J \ VndVn ) 

But it has already been decided (Art. 134) that the arithmetic 
mean of such a series of observed values is the most probable 
value of the quantity observed. The adoption of the arith- 
metic mean as the most probable value, however, requires 
the algebraic sum of the residuals (Art. 135) to reduce to zero; 
whence 

t)i + ■U2 . . . + v« = (12) 

Since v\, V2, etc., are the result of chance, and hence independent 
of each other, it follows from Eq. (12) that the coefficients of 
Vi, etc., in Eq. (11) must all have the same value. Representing 
this unknown value for any particular set of observations by the 



THE THEOEY OF EEEOES 259 

constant k, we have as the general condition which makes the 
arithmetic mean the most probable value, 

vdv ' 

whence by transposition 

d log (f> (v) = kvdv. 
Integrating this equation 

log <p(v) = ikiP + log c, 

in which log c represents the unknown constant of integration. 
Passing to numbers, we have 

<f>iv) = ce**"', (13) 

in which e equals the base of the Naperian system of logarithms. 
It is necessary at this point to remember that the probability 
of the occurrence of a given error does not involve the question 
as to whether we are right or wrong in assuming that an error of 
that value has occurred in a particular observation. Thus in 
the preceding discussion the probabilities assigned to the assumed 
values of vi, V2, etc., are the probabilities for true errors of these 
values, regardless of whether such errors have or have not occurred 
in the given case. It is of the utmost importance, therefore, to 
realize that Eq. (13) is not based on the assmnption that the 
error v has occurred, but is ^ general statement of fact concern- 
ing any true error whose magnitude is v. Replacing v in Eq. (13) 
by X, the adopted symbol for true errors, we have 



but from equation (6) 
whence 



^{x) = ce*'^^'; 



y =^(x); 



y = ce^kx\ 



260 GEODETIC SURVEYING 

Since the facility y decreases as the numerical value of x increases, 
it follows that \k is, essentially negative, and it is therefore 
commonly replaced by — hj^. Making this substitution, we have 

V = ce-^"'\ (14) 

in which y equals the facility with which any error x occurs, 
c and h are unknown constants depending on the circumstances 
of observation, and e is the base of the Naperian system of log- 
arithms. Though correct in apparent form, Eq. (14) must not 
yet be regarded as the general equation of the probability curve, 
since the quantities c and h appear as arbitrary constants, 
whereas t wi 1 be shown in the next article that these values are 
dependent on each other. 

150. General Equation of the Probability Curve. The proba- 
bility that an error shall fall between any two given values 
(Art. 147) is equal to the area between the corresponding ordi- 
nates of the probability curve. The probability that an error shall 
fall between — oo and -|- oo is therefore equal to the entire area 
of the curve. But it is absolutely certain that any error which 
may occur will fall between these extreme limits, and the proba- 
bility of a certain event (Art. 138) is equal to unity. The entire 
area of any curve represented by Eq. (14) must therefore be equal 
to unity. Since all probability curves have the same total area, 
it follows that any change in h will require a compensating change 
in c; or, in other words,''c must be a function of h. The general 
expression for the area of any plane curve is 



=jydx 



Substituting the value of y from Eq. (14) 

The probability P that an error x will fall between the limits a 
and h, is therefore 



P 



= f ce-f^'^'dx, (15) 



THE THEORY OF ERRORS 261 

and between the limits — oo and + oo , is 

P =f ce-f^'^'dx =cf"^ e-f^'^'dx. 
But this probability, being a certainty, equals unity; whence 

«-' — 00 

or 



-/: 



The second member of this equation is a definite integral whose 
evaluation by the methods of the calculus (for which such works 
should be seen) gives 



£ 



hence 

c ~ h ' 
and 

h 

which substituted in Eq. (15) gives for the probabiUty P that an 
error x will fall between any limits a and b, 

P =^(V^''^'dx (16) 

Also substituting the above value of c in Eq. (14) we have for the 
general equation of the probability curve 

y=^e-^^^\ (17) 

in which y is the facility with, which any error x occurs, e 
( = 2.7182818) is the base of the Naperian system of logarithms, 
and h (called the precision factor) is a constant depending on the 
circumstances of observation. The constant h is the only element 



262 GEODETIC SUEVBYING 

in Eq. (17) which can vary with the precision of the work, and 
therefore of necessity becomes the measure of that precision. 
151. The Value of the Precision Factor. The general equa- 
tion of the probability curve is given by Eq. (17), but the definite 
equation for any particular set of observations is not known 
until the corresponding value of h has been determined. The 
probability that an error x will occur (Art. 149) is 

P = yJx = ^(x)Jx. 

Substituting the value of y from Eq. (17), 

P =-^e-^''''Jx =^{x)Jx (18) 

With an infinite number of observations any residual v^ would be 
infinitely close to the corresponding true error xi, and the relative 
frequency with which vi occurred would not differ appreciably 
from Pi. The value of h for any particular case could thus be 
found from Eq. (18) by substituting these values for P and x. 
As the number of observations is always limited, however, the 
best that can be done is to find the most probable value of h 
for the given case. The probability that a given set of errors 
has occurred is, by Eq. (8), 

P = <j){Xi) ■4>{X2). . . 4>{Xn) • (ix)". 

But from Eqs. (6) and (17) 



so that 



and 



<i>{x{) = — = e-'''-^>', etc.; 



P = (-^ye-'''^^'(ia;)«, 



7t 

log P -^ n log h — h^Hx^ + n log Ax — ^-Tog ;:; 



whence by making the first derivative with respect to h equal to 
zero 

^ -2llx^-h =0. 
n 



THE THEOEY OF EERORS 263 

Solving for h we have 

...... (19) 



- P^ 

"^223 



2Sa;2 ' • 

in which n is the number of observations taken, and Sa^ is the 
sum of the squares of the true errors which have occurred. The 
true errors, however, can never be known, and formula (19) must 
therefore be modified so as to give the most probable value of h 
that can be determined from the residual errors. A discussion 
of this condition is beyond the scope of this book, but for observa- 
tions of equal (or unit) weight results in the formula 






^=yl2^' (20) 

in which n as before is the number of observations that have been 
taken, and HiP is the sum of the squares of the residual errors. 
For observations of unequal weight (Art. 133) formula (19) 
becomes 



'-4 



2^pv^'' 



(21) 



in which Spy^ is the sum of the weighted squares of the residuals, 
and h as before is the precision factor for observations of unit 
weight. 

For the general case of indirect observations (Art. 168) on inde- 
pendent quantities, that is, with no conditional equations (Art. 131), 
formula (19) becomes 



^=yli 



2Spi;2 



(22) 



in which n is the number of observation equations, q is the number 
of imknown quantities, ^piP is the sum of the weighted squares 
of the residuals, and h is the precision factor for observations 
of unit weight. 

For the general case of indirect observations involving con- 
ditional equations, formula (19) becomes 



h = ^^ ovLI " ' (23) 



jn — q + c 
2J:piP 



264 



GEODETIC SURVEYING 



in which c is the number of conditional equations, n is the number 
of observation equations, q is the number of unknown quantities, 
Spy2 is the sum of the weighted squares of the residuals, and h is 
the precision factor for observations of unit weight. As will be 
understood later (Art. 166), the number of independent unkno-mis 
is always reduced by an amount which equals the number of 
conditional equations, so that q in Eq. (22) is simply replaced by 
(q - c) in Eq. (23). 

152. Comparison of Theory and Experience. In the Funda- 
menta Astronomice Bessel gives the following comparison of theory 
and experience. In a series of 470 observations by Bradley on 
the right ascensions of Sirius and Altair the value of h was found 
to be 1.80865, giving rise to the following table: 







Probability of 
Errors. 


Number of Errors 




By Theory. 


By Experience. 


0.0 


0.1 


0.2018 


94.8 


94 


0.1 


0.2 


0.1889 


88.8 


88 


0.2 


0.3 


0.1666 


78.3 


78 


0.3 


0.4 


0.1364 


64.1 


58 


0.4 


0.5 


0.1053 


49.5 


51 


0.5 


0.6 


0.0761 


35.8 


36 


0.6 


0.7 


0.0514 


24.2 


26 


0.7 


0.8 


0.0328 


15.4 


14 


0.8 


0.9 


0.0194 


9.1 


10 


0.9 


1.0 


0.0107 


5.0 


7 


1.0 


00 


0.0106 


5.0 


8 


Totals 1.0000 


470.0 


470 



The last column in this table tacitly assumes that the true errors 
do not differ materially from the residual errors, the true errors 
being of course unknown. The agreement of theory and expe- 
rience is very satisfactory. 

There are two important points to be observed in applying 
the theory of errors to the results obtained in practical work. 



THE THEOEY OE EERORS 265 

In the first place, the theory of errors presupposes that a very large 
number of observations have been made. It is customary, how- 
ever, to apply the theory to any number of observations, however 
limited. It is evident in such cases that reasonable judgment 
must be used in interpreting the results obtained by the applica- 
tion of the theory. In the second place, the theory of errors is 
the theory of accidental errors. It is in general impossible to 
entirely prevent systematic errors in a process of observation; 
and such errors can not be discovered or eliminated by any num- 
ber of observations, however great, if the circumstances of observa- 
tion remain unchanged. The theory of errors, therefore, makes 
no pretense of discovering the truth in any case, but only to 
determine the best conclusions that can be drawn from the observa- 
tions that have been made. 



CHAPTER XI 

MOST PROBABLE VALUES OE INDEPENDENT QUANTITIES 

153. General Considerations. In accordance with the dis- 
cussions of the previous chapter it is evident that the true value 
of an observed quantity can never be found. Adopting any 
particular value for the observed quantity is equivalent to assum- 
ing that a certain series of errors has occurred in the observed 
values. Manifestly the most probable value of the observed 
quantity is that which corresponds to the most probable series 
of errors; or, in other words, that series of errors which has the 
highest probability of occurrence. It is therefore by means of 
the theory of errors (Chapter X) that rules are established for 
determining the most probable values of observed quantities. 

154. Fundamental Principle of Least Squares. For the general 
equation of the probability curve, Eq. (17), Art. 150, we have 

Vtt 

in which y is the facility of occurrence of any error x under the 
conditions represented by the precision factor h. The probability 
that any error x will occur (Art. 147) is equal to its facility multi- 
plied by the least count, or 

P = yAx. 

Hence if x^, X2, . ■ ■ Xn are the errors in the observed values 
of any magnitude Z, and Pi, P2, . . . P„ are the corre- 
sponding probabilities of occurrence, we have 

h h 

2/1 =-^:^e-'''^'^ 2/2 = ^= e~'''^^', etc., 
Vtt Vtt 

and 

Pi = yiJx, P2 = 2/2^^, etc. 

266 



PROBABLE VALUES OF INDEPENDENT QUANTITIES 267 

The probability P of the occurrence of this particular series of 
errors Xi, x^, etc., in the given set of observations, being a con- 
current event (Art. 141), is equal to the product of the individual 
probabilities, giving 

-P = (2/12/2 . . . VnWxY = i^j^^ e-^'^^\AxY. 

This equation is true for any proposed series of errors, and 
hence for that series of residual errors t'j, vi, . . . Vn, which 
results from assigning the most probable value to the observed 
quantity. In this case Sx^ becomes Sw^, and we have 

P = (-^''e-'''^^'{Ax)« (24) 

But (Art. 153) the most probable value of the observed quantity 
corresponds to that series of errors which has the highest prob- 
ability of occurrence. The most probable value z of any observed 
quantity Z, therefore, requires P in Eq. (24) to be a maximum, 
and this in turn requires l!,v^ to be a minimum. We thus have the 
following 

Pkinciple: In observations of equal precision the most probable 
values of the observed quantities are those that render the sum of the 
squares of the residual errors a minimum. 

It is on account of this principle that the Method of Least 
Squares has been so named. 

155. Direct Observations of Equal Weight. A direct observa- 
tion (Art. 133) is one that is made directly on the quantity whose 
value is to be determined. When the given magnitude is measured 
a number of times under the same conditions (as represented 
by the same precision factor h in the probability curve), the results 
obtained are said to be of equal weight or precision. In such a case 
the most probable value of the quantity sought mu^t accord with 
the principle of the previous article, that is, the sum of the squares 
of the residual errors must be a minimum. 

Let z = the most probable value of a given magnitude; 
n = the number of measurements taken; 
Ml, M% . . . Mn = the several measured values; 

then (Art. 154) 

(Ml - z)^ + (Ma - s)2 . . . + (M„ - 2)2 = a minimum. 



268 GEODETIC SURVEYING 

Placing the first derivative equal to zero, 

2(M, -z) +2{M2-z)... + 2{M„ - z) =0; 

whence 

(Ml + Mg . . . + M„) -nz = 0, 
and 

Ml + Ms . . . + M„ I:M 



(25) 



or, expressed in words, in the case of direct observations of equal 
weight the most probable value of the unknown quantity is equal 
to the arithmetic mean of the observed values. The above 
discussion, however, must not be regarded as a proof of this 
principle of the arithmetic mean, since (Art. 149) this very prin- 
ciple was one of the conditions under which the equation of 
the probability curve was deduced. Eq. (25) therefore simply 
shows that the equation of the probability curve is correct in form 
and consistent with this principle. 

Example. The observed values (of equal weight) of an angle A are 
29° 21' 59".l, 29° 22' 06".4, and 29° 21' 58".l. What is the most probable 
value? 

29° 21' 59".l 
29 22 06 .4 
29 21 58 .1 
3 )88 06 03 .6 
29 22 01 .2 

The most probable value is therefore 29° 22' 01".2. 

156. General Principle of Least Squares. When' a given 
magnitude is measured a number of times under different con- 
ditions (so that the precision factor corresponding to some of the 
observations is not the same for all of them) , the results obtained 
are said to be of unequal weight or precision. In accordance with 
the sense in which weights are understood (Art. 133), an observa- 
tion assigned a weight of two means it is considered as good a 
determination as the arithmetic mean of two observations of 
unit weight, and so on. It is immaterial whether any one of the 
observed values is considered of unit weight, as this is merely a 
basis of comparison. 



PROBABLE VALUES OF INDEPENDENT QUANTITIES 269 

Let 2 = the most probable value of a given magni- 

tude; 
Ml, M2, etc. = the values of the several measurements; 
Pi, P2, etc. = the respective weights of these measure- 
ments; 
api, ap2, etc. = the corresponding integral weights due to 

the use of the factor a; 
mi', mi", etc. = the api unit weight components of Mi 
when considered as an arithmetical 
mean; 
W2', m2", etc. = similarly for M2, and so on; 

vi, V2, etc. = the residuals due to Mi, M2, etc.; 

then, as in Art. 134, we have 

^ ^ mi' + mi" . . . __ Umi 
^ api ~ api ' 

, , mi' + mi" . . . S?re2 

M2 = ■ = , 

ap2 api 

Sm _ ^ap.M _ S£M 

Sap ~ Sap ~ Sp ^^^' 

The value of z thus obtained is evidently independent of any 
particular set of values that may be assigned to the components 
mi', mi", etc., the components m2', m2", etc., and so on. Since 
these various components are all of equal weight we must have 
in accordance with Art. 154, 

S(2; — mi)2 + S(z — m2)2 . . . + S(2— m„)2 = a minimum, (27) 

as a criterion that must be satisfied when z is the most probable 
value of the quantity Z. But, in accordance with Eq. (26), 
this criterion must determine the same value of z no matter what 
particular sets of values may be substituted for the components 
mi', mi", etc., mi, m-i' , etc., and so on. Adopting, therefore, 
the particular sets of values 



mi = mi" = . 


, . . = Ml, 


m2' = mi" = . 


.. = Mi, 


etc. 


etc., 



270 GEODETIC SURVEYING 

whence 

^(z — mi)2 = api (z — MiY = api-vi^, 

2(z — m2)^ = ap2 (z — M^'^ = ap2-V2, 
etc. etc., 

and substituting in Eq. (27), we have 

api-v-^ + ap2V^. . . + apn'Vn = a minimum; 

or, dividing out the common factor a, 

PiVi^ + P2V2^- . . + PnV„^ = Si minimum. . . (28) 

We thus have the following 

General Pbinciple: In observations of unequal precision the 
most probhble values of the observed quantities are those that render 
the sum of the weighted squares of the residual errors a minimum. 

157. Direct Observations of Unequal Weight. When a given 
magnitude is directly measured a. number of times it may be 
necessary to assign different weights to the results obtained, on 
account of some change in the conditions governing the measure- 
ments. In such a case the most probable value of the quantity 
sought must accord with the principle of the previous article, 
that is, the sum of the weighted squares of the residual errors 
must be a minimum. 

Let z = the most probable value of a given magnitude; 
Ml, M2, ■ . Mn = the several measured values; 
Pi, p2, ■ ■ ■ Pn = the corresponding weights; 

then (Art. 156) 

Pi(M 1 - zy + P2{M2 - zy . . .+ PniMn — z)^ = 3, minimum. 

Placing the first derivative equal to zero, 

2pi(Mi - z) + 2p2'M2 - 2) ... + 2p„(M, - 2) = 0; 



PROBABLE VALUES OF INDEPENDENT QUANTITIES 271 

whence 

(pi Ml + P2M2 ...+ VnMn) - (Pi + P2 • ■ ■ + P«) 3 = 0, 
and 

Z _ Pl^l +P1M2. . . +VnMn ^ 2pM. 

Vl + V2 . . . + Vn Hip ' ' ' ' ^ ' 

or, expressed in words, in the case of direct observations of unequal 
weight the most probable value of the unknown quantity is equal 
to the weighted arithmetic mean of the observed values. The 
above discussion, however, must not be regarded as a proof of 
this principle of the weighted arithmetic mean, since Eq. (29) 
is deduced from a principle based in part on the truth of Eq. (26), 
which is identical with Eq. (29). As the truth of Eq. (26) is 
established in Art. 156, however, Eq. (29) shows that the general 
principle of least squares leads to a correct result in a case where 
the answer is already known. 

Example. The observed values for the length of a certain base line are 
4863.241 ft. (weight 2), and 4863.182 ft. (weight 1). What is the most 
probable value? 

4863.241 X 2 = 9726.482 
4863.182 X 1 = 4863.182 



3)14589.664 



4863.221 
The most probable value is therefore 4863.221 ft. 

158. Indirect Observations. An indirect ohservation is one 
that is made on some function of one or more quantities, instead 
of being made directly on the quantities themselves. Thus in 
measuring an angle by repetition the observation is indirect, as 
the angle actually read is not the angle sought, but some multiple 
thereof. Similarly when angles are measured in combination 
the observations are indirect, since the values of the individual 
angles must be deduced from the results obtained by some pro- 
cess of computation. 

An observation equation is an equation expressing the function 
observed and the value obtained. Thus if x, y, etc., represent 
the unknown quantities whose values are to be deduced from the 



272 GEODETIC SURVEYING 

observation, we may have as observation equations such expres- 
sions as 

6a; = 185° 19' 40", 
or 

7x + 102/ - 3z = 65.73, 

according to the function observed. 

In general the observation equations which occur in geodetic 
work may be written in the following form : 



aix + hiy + CiZ . . . = Mi (weight pi) 
a^x + 622/ + C23 . . . = M2 (weight P2) 

a„x + hnV + c„3 . . . = Mn (weight p„) 



(30) 



in which ai, 02, &i, ^2 etc., are known coefficients; x, y, etc., are 
the unknown quantities; Mi, M2, etc., are the observed values; 
and pi, p2, etc., are the respective weights of these values. If the 
number of observation equations is less than the number of unknown 
quantities, the values of x, y, z, etc., can not be found, nor even 
their most probable values. If the number of observation equa- 
tions equals the nvunber of unknown quantities, the equations 
may be solved as simultaneous equations, and each equation will 
be exactly satisfied by the values obtained for x, y, z, etc., even 
though these values are not the true values sought. If the num- 
ber of observation equations exceeds the number of unknown 
quantities there will in general be no values of x, y, z, etc., which 
will exactly satisfy all the equations, on account of the unavoidable 
errors of observation. Hence if the most probable values of the 
imknown quantities be substituted the equations will not be 
exactly satisfied, but will reduce to small residuals vi, m, vz, etc. 
If, therefore, x, y, z, etc., be understood to mean the most probable 
values of these quantities, we will have 



a-iX + biy + CiZ . . . — ilf 1 = Vi (weight p{) 
a^x + 622/ + C2Z . . . — M2 = «'2 (weight p^) 

a„x + 6„2/ + c„z . . . - M„ = «;„ (weight p„) 



(31) 



PEOBAELE VALUES OF INDEPENDENT QUANTITIES 273 

By a consideration of these equations, together with any special 
conditions which must be satisfied, rules may be established for 
finding the most probable values of the unknown quantities in 
all cases of indirect observations. 

159. Indirect Observations of Equal Weight on Independent 
Quantities. An independent quantity is one whose value is 
independent of the value of any other quantity under considera- 
tion. Thus in a line of levels the elevation of any particular 
bench mark bears no necessary relation to the elevation of any 
other bench mark; whereas in a triangle the three angles are not 
independent of each other, as their sum must necessarily equal 
180°. 

In the case of indirect observations of equal weight on inde- 
pendent quantities, the most probable values of the tmknown 
quantities are found by a direct application of the method of 
normal equations. A normal equation is an equation of condi- 
tion which determines the most probable value of any one unknown 
quantity corresponding to any particular set of values assigned to 
the remaining unknowns. A normal equation must therefore 
be specifically a normal equation in x, or in y, etc. By forming 
a normal equation for each of the unknowns there will be as many 
equations as unknown quantities. The solution of these equa- 
tions as simultaneous will give a set of values for the unknowns 
in which each value is the most probable that is consistent with 
the remaining values, which can only be the case when all the 
values are simultaneously the most probable values of the unknown 
quantities. 

To establish a rule for forming the normal equations in the 
case of equal weights let us re-write Eqs. (31), omitting the 
weights, thus: 



aix + biy + ciz ... — Mi = Vi 

0,2^ + b2y + C2Z . . . ~ M2 - V2 

o»a; + b„y + CnZ . . ■ — Mr, = K 



(32) 



In accordance with Art. 154 the most probable values of the 
unknown quantities are those which give 

vi^ + V2^ ■ ■ ■ + Vr? = & minimum. 



274 GEODETIC SURVEYING 

Since (in lormmg the normal equations) the most probable value 
of X is desired for any assumed set of values for the remaining 
unknowns, we place the first derivative with respect to x equal 
to zero ; whence, omitting the common factor 2, we have 

But from Eqs. (32), under the given assimiption of fixed values 
for all quantities excepting x, we obtain 

dvi dv2 , 

-dx = """ 'dx = "'' "*°- 

whence by substitution, 

aivi + 02^2 . ■ . + a-nVn = = normal equation in x. 
In a similar manner we have 

bivi + b2V2 . . ■ + b„Vn = = normal equation in y; 

ciwi + 02^2 • • • + CnV„ = = normal equation in z; 
etc., etc.; 

and hence for forming the several normal equations in the case 
of indirect observations of equal weight on independent quan- 
tities, we have the following 

Rule : To form the normal equation for each one of the unknown 
quantities, multiply each observation equation by the algebraic 
coefficient of that unknown quantity in that equation, and add the 
results. 

Having formed the several normal equations, their solution 
as simultaneous equations gives the most probable values of the 
unknown quantities. 

Examph 1. Given the observation equation 

6x = 90° 15' 30"- 

In applying the above rule to this case we would have to multiply the whole 
equation by 6, and then divide by 36 to obtain the most probable value 
of X. It is evident that we would obtain the same value of x by dividing 
the original equation by 6, so that in the case of a single equation with a 
single unknown quantity the most probable value of that quantity is obtained 
by simply solving the equation. 



PROBABLE VALUES OF INDEPENDENT QUANTITIES 275 

Example 2. Given the observation equations 

2x = 124.72, 

X = 62.31, 

7x = 439.00. 

Multiplying the first equation by 2, the second by 1, and the third by 7, 
we have 

4x = 249.44; 

X = 62.31; 

49x = 3073.00; 

whence by addition we obtain the normal equation 

54x= 3384.76, 
the solution of which gives 

X = 62.68, 

which is hence the most probable value that can be obtained from the given 
set of observations. The student is cautioned against adding up the obser- 
vation equations and solving for x, as this plan does not give the most 
probable value in such cases. 

Example 3. Given the observation equations 

2x+ y = 31.65, 
X - 3i/ = 5.03, 
X - y = 11.26. 

Following the rule for normal equations, we have 

ix + 2y = 63.30 
X - 3y = 5.03 
X - y = 11.26 
&x — 2y = 79.59 = normal equation in x; 
and 

2x + y = 31.65 

- 3x + 92/ = - 15.09 

- X + y = - 11-26 

- 2x -\- lly = 5.30 = normal equation in y. 

It is absolutely essential in forming the normal equations to multiply by 
the algebraic coefficients as illustrated above, and not simply by the numerical 
value of the coefficient. Bringing the normal equations together, we have 

Ox - 2y = 79.59, 
- 2x + 112/ = 5.30. 

Attention is called to the fact that the coefficients in the first row and first 
column are identical in sign, value, and order, and that the same is true of 
the second row and second column. The same law would hold good if there 
were a third row and a third column, and so on (Art. 162); and this is a 
check that must never be neglected. Solving the two normal equations as 
simultaneous equations, we have 

X = 14.29 and y = 3.08, 
and these are hence their most probable values. 



276 GEODETIC SUEVEYING 

160. Indirect Observations of Unequal Weight on Independent 
Quantities. In the case of indirect observations of unequal 
weight on independent quantities, the most probable values of 
the unknown quantities are found by the solution of one or more 
normal equations which involve the different weights in their 
formation. 

To establish a rule for forming the normal equations in the 
case of unequal weights let us re-write Eqs. (31), thus: 



aix + biy + ciz . . . — Mi = vi (weight pi) 
a2X + 62?/ + C2Z . . . — M2 = V2 (weight ^2) 

anX + bny + CnZ ■ . . - Mn = Vn (weight Pn) 



(33) 



In accordance with Art. 156 the most probable values of the 
imknown quantities are those which give 

Piwi^ + P2«'2^ . . . + PnVr? = a minimum. 

Since (in forming the normal equations, Art. 159) the most 
probable value of x is desired for any assumed set of values for 
the remaining imknowns, we place the first derivative with 
respect to x equal to zero; whence, omitting the common 
factor 2, we have 

But from Eqs. (33), under the given assumption of fixed values 
for all quantities excepting x, we obtain 

dvi dv2 

whence by substitution, 

(aipi)vi + {a2P2)V2 . . . + {a„Pn)Vn = = normal equation in x. 

In a similar manner we have 

(bipijvi + (62P2)w2 . . . + {bnPn)Vn = = uormal equation in y; 

(cipi)wi + {c2P2)V2 . . . + {cnPn)Vn = = normal equation in z; 

etc., etc.; 



PEOBABLE VALUES OF INDEPENDENT QUANTITIES 277 

and hence for forming the several normal equations in the case 
of indirect observations of unequal weight on independent quan- 
tities, we have the following 

Rule : To form the normal equation for each one of the unknown 
quantities, multiply each observation equation by the product of the 
weight of that observation and the algebraic coefficient of that unknown 
quantity in that equation, and add the results. 

Having formed the several normal equations, their solution 
as simultaneous equations gives the most probable values of the 
unknown quantities. 

Exam-pk 1 . Given the observation equations 

3x = 15° 30' 34" .6 (weight 2), 
5x = 25 50 55 .0 (weight 3). 

Multiplying the first equation by 6 ( = 3 X'2), and the second equation by 
15 ( = 5 X 3), we have 

18x = 93°03'27".6; 
75x = 387 43 45 .0; 

whence by addition we obtain the normal equation 

93x = 480° 47' 12".6, 

the solution of which gives 

x = 5° 10' ll".l, 

which is hence the most probable value that can be obtained from the given 
set of observations. 

Example 2. Given the observation equations 

x+ y = 10.90 (weight 3), 

2x — y = 1.61 (weight 1), 

X + 32/ = 24.49 (weight 2). 

Following the rule for normal equations, we have 

3x + 32/ = 32.70 
4x - 22/ = 3.22 
2x + 62/ = 48.98 

9x + 72/ = 84.90 = normal equation in x; 
and 

3x + 32/ = 32.70 
-2x + y =- 1.61 
6x + I82/ = 146.94 
7x + 222/ = 178.03 = normal equation in y. 

Solving these two normal equations as simultaneous, we have 

X = 4.172, and y = 6.765, 

and these are hence their most probable values. 



278 GEODETIC SURVEYING 

161. Reduction of Weighted Observations to Equivalent 
Observations of Unit Weight. To establish a rule for this pur- 
pose let us re-write Eqs. (30) , thus : 

aix + biy + ciz . . . = Mi (weight pi), 

a2X + hiV + C2Z . . . = M2 (weight P2) , 

a„x + bny + CnZ . . . = Mr, (weight p„). 

Let C be such a factor as will change the first of these equations 
to an equivalent equation of unit weight, so that we may write 

Caix + Cbiy + Cciz . . . = CMi (weight 1), 
a2X + 622/ + C2Z ■ ■ ■ = M2 (weight P2) , 

o-nX + bnV + c„z . . . = M„ (weight p„) ; 

in which the most probable values of x, y, z, etc., are to remain the 
same as in the original equations; or, in other words, the two 
sets of equations are to lead to the same normal equations. In 
accordance with the rule of Art. 160, we have from the first set 
of equations 



Normal 

equation 

in x 



(piai^x+piaibiy+piaiciz . . . =piOiMi) 

+ (p2a2^X+p2a2b2y + P2a2C2Z . . . =^202^2) 

-|- ( etc., etc ) 

and from the second set of equations 



(34) 



Normal 

equation 

in X 



(C^aiH+C^aibiy+C^aiciz . . . =C^aiMi) 
+ ip2a2^x+p2a2b2y+p2a2C2Z . . . =p2a2M2) 
+ ( etc., etc ) 



(35) 



Comparing Eq. (34) with Eq. (35), term by term, we find they are 
in all respects identical provided we write 

whence 

C = V^ (36) 



PEOBABLE VALUES OF INDEPENDENT QUANTITIES 279 

From the symmetry of the equations involved it is evident that 
the same conclusion would result from a comparison of the nor- 
mal equations in y, z, etc. Hence it is seen that an observation 
equation of any given weight may be reduced to an equivalent 
equation of unit weight by multiplying the given equation by the 
square root of the given weight. Evidently the converse of this 
proposition is also true, so that an equation of unit weight can be 
raised to an equivalent equation of any given weight by dividing 
the given equation by the square root of the given weight. The 
general laws of weights, as given in Art. 53, are readily derived 
by an application of these two principles. The new equations 
formed in the manner described, and taken in conjunction with 
the new weights, may be used in any computations in place of the 
original equations, whenever so desired. 

Example 1. Given the observation equation 

3a; = 8.66 (weight 4). 

What is the equivalent observation equation of unit weight? 
Since the square root of 4 is 2, we have 

&x = 17.32 (weight 1) 
as the equivalent equation. 

Example 2. Given the observation equation 

Zx + Qy = 11.04 (weight 1). 

What is the equivalent observation equation of the weight 9? 
Since the square root of 9 is 3, we have 

s + 2)/ = 3.68 (weight 9) 

as the equivalent equation. 

Example 3. Given the observation equation 

X + y — 2z = a (weight 3). 

What is the equivalent observation equation of the weight 7? 
Multiplying by ^3 and dividing by V?, we have 

\/f a; + Vf V - 2\/f z = Vf a (weight 7) 

as the equivalent equation. 



280 GEODETIC SURVEYING 

162. Law of the Coefficients in Normal Equations. In accord- 
ance with Art. 158, we may write in general for any set of 
observations 

aix + hiy +ciz . . . = Mi (weight pi) , 
a2X + bzy + c^z . . . = M2 (weight P2) , 



arfl + hny + CnZ. . . = Mn (weight p„). 

Forming the normal equation in x in accordance with the rule of 
Art. 160, the multiplying factors are piai, ^2^2, etc., giving 

piaiH + piaibiy + piaiaz . . . = piaiMi 

P2a2^X + P20.2h2y + P2a2C2Z . . . = ^202^2 



S(pa^)x+2i(pa6)2/ + S(pac)3 . . . =S(paM')= normal equation in x. 

Similarly, for the normal equation in y, the multiplying factors 
are pi6i, P2&2, etc., giving 

T,{pah)x + 'S^{pb^)y + 'L{phc)z . . .=Il(p&M) = normal equation in j/. 

Similarly, for the normal equation in z, the multiplying factors 
are pici, P2C2, etc., giving 

I!(pac)a;+S(p&c)?/ + 2(pc2)z . . .=S(pcikf) = normal equation in z ; 

and so on for any additional unknown quantities. Collecting 
the several normal equations together, we have 

S(pa2)a; + ^{pah)y + ll{pac)z . . . = HipaM); 
^{pab)x + S(p&2)2/ + '^{phc)z . . . = 2(p6ilf); 
S(pac)a; + ll{phc)y + 2(pc2)z . . . = S(pcM); 
etc., etc. 

An examination of these equations shows that the coefficients in 
the first row and in the first column are identical in sign, value, 
and order. The same proposition is true of the second row and 
second column, the third row and third column, and so on. This 
is hence the general law of the coefficients in any set of normal 
equations, and furnishes a check on the work that should never 
be neglected. 



PEOBABLE VALUES OF INDEPENDENT QUANTITIES 281 

Example. Let the following observation equations be given: 

2x - z = 8.71 (weight 2), 

x-2y + Zz = 2.16 (weight 1), 

2/ - 2z = 1.07 (weight 2), 

X -Zy =- 1.93 (weight 1). 

I 

[The corresponding normal equations are 

IQx - 

— 5a; - 

— X - 

from which we have 



lOx — by — z = 38.93 = normal equation in x; 

— 5x + 15y — lOz = — 7.97 = normal equation in y; 

— X - lOy + 19z = - 15.22 = normal equation in z; 



„ ^ . , . / Fu:st row are + 10, — 5,-1. 

Coemcients m i ,,. , , , r. ^ 

I First column are +10,-5,-1. 

„„.... / Second row are — 5, + 15, — 10. 

(Joefncients m i c j i c ic ^n 

L becond column are — 5, + 15, — 10. 

ri a- ■ i ■ / Third row are - 1, - 10, + 19. 

Coefficients m [ ^^^^ ^^j^^^ ^^^ - 1, - loi + 19. 

163. Reduced Observation Equations. Such observation equa- 
tions as are likely to occur in geodetic work may be written imder 
the general form 

ax + by + CZ + etc. = M (37) 

Substituting 

X = Xi -\- Vi 



y = yi + V2 

Z = Zl + V3 



(38) 



in which xi, yi, 21, etc., are any assumed constants,, and vi, V2, V3, 
etc., ate new imknowns, the equation takes the reduced form 

avi + bv2 +CV3 + etc. = M — {axi + byi + czi + etc.). (39) 

In this new equation it will be noticed that the first member is 
identical in form with the first member of the original equation, 
the only change being the substitution of the new variables for 
the old ones; and that the second member is what the original 
equation reduces to when the assumed constants are substituted 
for the corresponding variables. The reduced observation 
Eq. (39) may therefore be written out at once from the observa- 



282 GEODETIC SUEVEYING 

tion Eq. (37), without going through the direct substitution of 
Eqs. (38). Particular attention is called to the second member 
of Eq. (39), in which it is seen that the result due in any case to 
the use of the assumed values of x, y, etc., must always be sub- 
tracted from the corresponding measured value, and not vice 
versa, as any error in sign will render the whole computation 
worthless. It is also to be noted that the original weights apply 
also to the reduced observation equations, since these are simply 
different expressions for the original equations. 

In view of the meaning of the terms in Eqs. (38) it is evident 
that the most probable value of x is that which is due to the most 
probable value of vi, and correspondingly with all the other 
unknowns. We may, therefore, in any case, reduce all the 
original observation equations to the form of Eq. (39), determine 
from these reduced equations the most probable values of vi, V2, 
etc., and then by means of Eqs. (38) determine the most probable 
values of x, y, z, etc. The object of this method of computation 
is to save labor by keeping all the work in small numbers. This 
result is accomplished by assigning to xi, yi, etc., values which 
are known to be approximately equal to x, y, etc., as this will 
evidently reduce the second term of equations like Eq. (39) to 
values approximating zero. Approximate values' of the unknowns 
are always obtainable from an inspection of the observation, 
equations, or by obvious combinations thereof. 

Example 1. Given the following observation equations: 

X = 178.651, 

y = 204.196, 

X + 2/ = 382.860, 

2x + y = 561.522; 

to find the most probable values of the unknowns by the method of reduced 
observation equations. 

Assuming for the most probable values 

x = 178.651 + vi, 
y = 204.196 + v^, 

we have by substitution in the observation equations, or directly in accord- 
ance with Eq. (39), 

«i = 0.000; 

D2 = 0.000; 

vi + Vi = 0.012; 

2wi +V2 = 0.024. 



PROBABLE VALUES OF INDEPENDENT QUANTITIES 283 

Forming the normal equations from these reduced observation equations, 
we have 

6di + Swa = 0.060; 

3!)i + 3v2 = 0.036; 
whose solution gives 

vi = 0.008 and V2 = 0.004; 

whence for the most probable values of x and y we have 

X = 178.651 + 0.008 = 178.659; 
y = 204.196 + 0.004 = 204.200. 

These results are identical with what would have been obtained if any other 
values had been used for xi and yi, or if the normal equations had been 
formed directly from the original observation equations. 

Example 2. Given the following observation equations: 

21 + 2/ = 116° 38' 19".7 (weight 2), 

a; + 2/ = 73 17 22 .1 (weight 1), 

X - y = 13 24 28 .3 (weight 3), 

x + 2y = 103 13 47 .7 (weight 1); 

to find the most probable values of the unknowns by the method of reduced 
observation equations. 

It is readily seen that the first two of these equations are exactly satisfied 
if we write 

X = 43° 20' 57".6; 

2/ = 29 56 24 .5. 

Adopting these as the approximate values we have for the most probable 
values 

a; = 43° 20' 67".6 + Vi; 

y = 29 56 24 .5 + v^; 

whence by substitution in the observation equations, or directly in accord- 
ance with Eq. (39), we have 

2i;i + f 2 = 0".0 (weight 2); 
vi + V2 = .0 (weight 1); 
vi — «)2 = — 4 .8 (weight 3) ; 
vi +2v2 = 1 .1 (weight 1). 

Forming the normal equations from these reduced observation equations, 
we have 

132)1 + 4j)2 = - 13".3; 
4t;i + 10t>2 = 16 .6; 
whose solution gives 

vi 1".75 and «2 = + 2".36; 

whence for the most probable values of x and y we have 

X = (43° 20' 57".6) - 1".75 = 43° 20' 55".86; 
- y = (29 66 24 .5) + 2 .36 = 29 66 26 .86. 

As in the previous example these results are identical with what would have 
been obtained if any other values had been used for Xi and yi, or if the normal 
equations had been formed directly from the original observation equations. 



CHAPTER XII 

MOST PROBABLE VALUES OF CONDITIONED AND COMPUTED 
QUANTITIES 

164. Conditional Equations. The methods heretofore given 
determine the most probable values in all cases where the quanti- 
ties observed are independent of each other. In many cases, how- 
ever, certain rigorous conditions must also be satisfied, so that any 
change in one quantity demands an equivalent change in one 
or more other quantities. Thus in a triangle the three angles 
can not have independent values, but only such values as will add 
up to exactly 180°. When quantities are thus dependent on each 
other they are called conditioned quantities. By an equation of 
condition or a conditional equation is meant an equation which 
expresses a relation that must exist among dependent quantities. 
Thus if X, y, and z denote the three angles of a triangle we have 
the corresponding conditional equation 

x + y + z = 180°. 

In such a case the most probable values of x, y, and z are not 
those values which may be individually the most probable, but 
those values which belong to the most probable set of values that 
will satisfy the given conditional equation. In accordance with 
the principles heretofore established that set of values is the most 
probable which leads to a minimum value for the sum of the 
weighted squares of the resulting residuals in the observation 
equations. 

In the problems which occur in geodetic work the conditional 
equations may in general be expressed in the form 



aix -1-022/ ■ ■ ■ + aj = E^ 
bix + b2y . . . + bj = E^ 

mix + m2y . . . + mj, = E„ 



(40) 



284 



PROBABLE VALUES OF CONDITIONED QUANTITIES 285 

in which x, y, t, etc., are the most probable values of the unknown 
quantities, and u is the number of such quantities. It is evident 
that the number of independent conditional equations must be 
less than the number of unknown quantities. For if these equa- 
tions are equal in number with the unknown quantities their 
solution as simultaneous equations will determine absolute values 
for the unknowns, so that such quantities can not be the subject 
of measurement. While if the number of these equations exceeds 
the number of unknowns, such equations can not all be inde- 
pendent without some of them being inconsistent. On the other 
hand the total mmiber of equations (sum of the observation and 
the independent conditional equations) must exceed the number 
of unknown quantities. For if the total number of equations is 
equal to the number of unknown quantities, their solution as 
simultaneous equations will furnish a set of values which will 
exactly satisfy all the equations, without involving any question 
of what values may be the most probable. While if the total 
number of equations is less than the number of unknown 
quantities the problem becomes indeterminate. 

There are in general two methods of finding the most probable 
values of the unknown quantities in cases involving conditioned 
quantities. In the first method the conditional equations are 
avoided (or eliminated) by impressing their significance on the - 
observation equations, which reduces the problem to the cases 
previously given. In the second method the observation equa- 
tions are eliminated by impressing their significance on the con- 
ditional equations, when the solution may be effected by the 
method of correlatives (Art. 167). The first method is the most 
direct in elementary problems, but the second method greatly 
reduces the work ,of computation in the case of complicated 
problems. 

165. Avoidance of Conditional Equations. In a large num- 
ber of problems it is possible to avoid the use of conditional 
equations by the manner in which the observation equations are 
expressed. The conditions which have to be satisfied in any 
given case are never alone sufficient to determine the values of 
any of the unknown quantities, as otherwise these quantities 
would not be the subject of observation. It is only after definite 
values have been assigned to some of the unknown quantities 
that the conditional equations limit the values of the remaining 



286 GEODETIC SURVEYING 

ones. In any problem, therefore, a certain number of values 
may be regarded as independent of the conditional equations, 
whence the remaining values become dependent on the independent 
ones. Thus in a triangle any two of the angles may be regarded 
as independent, whence the remaining one becomes dependent 
on these two, since the total sum must be 180°. In any elementary 
problem it is generally self evident as to how many quantities 
must be regarded as independent, and which ones may be so taken. 
In such cases the conditional equations may be avoided by 
writing out all of the observation equations in terms of the 
independent quantities. The most probable values of these 
quantities may then be found by the regular rules for independent 
quantities, whence the most probable values of the remaining 
quantities are determined by the surrounding conditions that 
must be satisfied. 

Example 1. Referring to Fig. 65, the following angular measurements 
have been made; 

X = 28° 11' 52".2; 
y = 30 42 22 .7; 
z = 58 64 17 .6. 

What are the most probable values of these angles? 
It is evident from the figure that these angles are sub- 
ject to the condition 

X + y = z. 

If, however, we write the observation equations in the 
form 

X = 28° 11' 52".2; 

y = 30 42 22 .7; 

x+y = 58 54 17 .6; 

the conditional equation is avoided, since x and y are manifestly inde- 
pendent angles. The second set of observation equations must lead to 
exactly the same figures for the most probable values of x and y (and hence 
for z) as the first set, since it is only another way of stating exactly the 
same thing. Since x and y are independent angles we may write for the 
most probable values 

X = 28° 11' 52".2 + vi; 

y = 30 42. 22 .7 + V2; 

whence the reduced observation equations are 

vi = 0".0; 

!)2 = .0; 

vi + vt = 2 .7. 




Fig. 65. 



PROBABLE VALUES OF CONDITIONED QUANTITIES 287 
The corresponding normal equations are 



2wi + V2 

Vi + 2t)2 



2".7; 
2 .7; 



whose solution gives 

vi= + 0".9 and 1^2=+ 0".9. 

The most probable values of the given angles are therefore 

X = 28° 11' 53".l; 
2/ = 30 42 23 .6; 
z = 58 54 16 .7. 

Example S. Referring to Fig. 66, the following angular measurements 
have been made: 

X = 80° 46' 37".6 (weight 2); 
y = 135 08 14 .9 (weight 1); 
z = 144 06 10 .8 (weight 3). 

What are the most probable values of these angles? 
It is evident from the figure that these angles are 
subject to the condition 

x + y + z = 360°. 

Any two angles at a point, such as x and y, may 
be regarded as independent, so that the conditional 
equation is avoided by writing all the observation 
equations in terms of these two quantities. Thus we 
write: 

X = 80° 45' 37".6 (weight 2); 

y = 135 08 14 .9 (weight 1); 

360° - (a; + 2/) = 144 06 10 .8 (weight 3) ; 




whence by substituting 



we have 



X = 80° 45' 37".6 + vi, 
y = 135 08 14 .9 + V2, 



Vi = 0".0 (weight 2); 

!)2 = .0 (weight 1); 

t/i + f 2 = — 3 .3 (weight 3) ; 

from which the normal equations are 

5«i + Zv2= - 9".9; 

3!)i -\-ivi 9 .9; 

whose solution gives 

vi= - 0".9 and v^ = - 1".8. 
The most probable values of the given angles are therefore 

X = 80° 45' 36".7; 
y = 135 08 13 .1; 
2 = 144 06 10 .2. 



288 GEODETIC SUEVEYING 

166. Elimination of Conditional Equations. If the con- 
ditional equations can not be directly avoided, as in the 
preceding article, the same result may be indirectly accomplished 
by algebraic elimination, as about to be explained. The number 
of unknown quantities (Art. 164) necessarily exceeds the number' 
of independent conditional equations. The number of dependent 
unknowns, however, can not exceed the number of independent 
conditional equations, since any values whatever may be assigned 
to the remaining unknowns and still leave the equations capable 
of solution. Thus if there are five unknowns and three independent 
conditional equations, any values may be assigned to any two of 
the unknowns, leaving three equations with three imknowns and 
hence capable of solution. The unknowns selected as arbitrary 
values thus become independent quantities on which all the others 
must depend, and the nmnber of unknowns which may be thus 
selected as independent quantities is evidently equal to the 
difference between the total number of unknowns and the number 
of independent conditional equations. If the most probable 
values are assigned to the independent quantities, the most 
probable values of the dependent quantities then become known 
by substituting the values of the independent quantities in the 
dependent equations. The general plan of procedure is as 
follows: 

1. Determine the nmnber of independent unknowns by sub- 
tracting the number of conditional equations from the number 
of unknown quantities. 

2. Select this number of unknowns as independent quantities. 

3. Transpose the conditional equations so that the dependent 
quantities are all on the left-hand side and the independent quan- 
tities on the right-hand side. 

4. Solve the conditional equations for the dependent unknowns, 
which will thus express each of these dependent unknowns in 
terms of the independent unknowns. 

5. Substitute these values of the dependent unknowns in the 
observation equations, which will then contain nothing but 
independent imknowns. 

6. Find the most probable values of the independent unknowns 
from these modified observation equations by the regular rules 
for independent quantities. 

7. Substitute these values of the independent unknowns in 



PEOBABLE VALUES OF CONDITIONED QUANTITIES 289 

the expressions for the dependent unknowns, alid thus determine 
the most probable values of the remaining quantities. 

Example. Given the following data, to find the most probable values 
of X, y, and «: 



[ X = 17.82 (weight 2); 

• y = 15.11 (weight 4); 

z = 29.16 (weight 3). 



Observation equations ] y = 15.11 (weight 4); 



Conditional equations {Hfl ^ ^Z'lS. 

Th,e solution is as follows: 

Number of observation equations = 3. 
Number of conditional equations = 2. 
Number of independent quantities = 1. 

Let X be the independent quantity, and y and z the dependent quantities. 
Transpose the conditional equations so as to leave only the dependent 
quantities on the left hand side, thus: 

5y = 112.00 - 2x; 
y - z = 39.00 - 3a;. 

Solve for the dependent quantities, giving the dependent equations 

y = 22.40 - 0.4x; 
z = - 16.60 + 2.6a;. 

Substitute in the observation equations, giving 

X = 17.82 (weight 2); 

22.40 - 0.4x = 15.11 (weight 4); 

- 16.60 + 2.6x = 29.16 (weight 3); 

whence 

X = 17.82 (weight 2); 
0.4x = 7.29 (weight 4); 
2.6x = 45.76 (weights); 

in which x is an independent unknown. Forming the normal equation 
by multiplying the above equations respectively by 2, 1.6, and 7.8, we have 

2.00X = 35.640, 

0.64x = 11.664, 

20.28X = 356.928 



22.92X = 404.232; 
X = 17.637; 



which, substituted in the first dependent equation, gives, 
y = 22.40 - 0.4(17.637) = 15.345, 



290 GEODETIC SURVEYING 

and substituted in the second dependent equation, gives 

z = - 16.60 + 2.6(17.637) = 29.255; 

so that for the most probable values of the unknown quantities, we have 

X = 17.637; 
y = 15.345; 
z = 29.255. 

As a check on the work of computation, we may substitute these values in 
the conditional equations, giving 

2x + 5y = 35.274 + 76.725 = 111.999; 

3x+ y - z = 52.911 + 15.345 - 29.255 = 39.001; 

from which it is seen that each equation checks with the corresponding 
conditional equation within 0.001, which is an entirely satisfactory check. 
The essential feature of the above method is the eUmination of the con- 
ditional equations. In Art. 167 the same problem is worked out by eUm- 
inating the observation equations. The results obtained are of course 
identical. 

167. Method of Correlatives. The general method of correla- 
tives is beyond the scope of the present volume. The case here 
given is the only one that is likely to be of service to the civil 
engineer. In this case the observations are made directly on 
each unknown quantity, and the number of observation equations 
equals the number of unknown quantities. Let u be the number 
of unknown quantities, for which the observation equations may 
be written 

X = Ml (weight pi); 

y = M 2 (weight p^) ," 

t = M„ (weight p J ; 
and for which (Art. 164) the conditional equations may be written 

aix -\-a2y...-\-aJ, = E^ 
bix + b2y . . . + bj = Eb 



mix + may . . . + mj, = E„ 



(41) 



If, as heretofore, x, y, t, etc., be understood to represent the most 
probable values of the unknown quantities, and vi, V2, Vu, etc., 



PEOBABLE VALUES OF CONDITIONED QUANTITIES 291 



represent the corresponding residuals in the given equations, we 
may write 

X = Ml + vi (weight pi) 



y = M2 + V2 (weight P2) 



(42) 



t = M^ + Vu (weight pj . 
which, substituted in Eq. (41), give the conditional equations 
aivi + a2V2 • ■ . + a^Vu = Ea — ^aM 
bivi + b2V2 . . . + M« = Eb - 26M 



miv\ + in2V2 . ■ . + myVy, = E^ — ^mM 



(43) 



As explained in Art. 164, these conditional equations must be 
less in number than the number of unknown quantities. The 
values of vi, V2, etc., thus become indeterminate, and an infinite 
number of sets of values will satisfy the equations. The values 
in any one set (called simultaneous values) are not independent, 
however, as they must be such as will satisfy the above equations. 
If 2^1, V2, etc., in Eqs. (43) are assumed to vary through all 
possible simultaneous values due to any set of values dvi, dv2, 
etc., and all possible sets of values dwi, dv2, etc., are taken in turn, 
the most probable set of values vi, V2, etc., for the given set of 
observations will eventually be reached. The values dvi, dv2, 
etc., in any one set, however, can not be independent, as it is 
evident that dependent quantities can not be varied indepen- 
dently. Differentiating Eqs. (43), we have 



aidvi + a2dv2 • . . + ciudvu 







bidvi + b2dv2 



+ b^dvy, = 



m\dvi + m2dv2 . . • + mudVy, = 



(44) 



and these new equations of condition show the relations that must 
exist among the quantities dvi, dv2, etc. Since the number of 
equations is less than the number of quantities dvi, dv2, etc., it 
follows that an infinite number of sets of simultaneous values of 
dvi, dv2, etc., is possible. In order to involve Eqs. (44) simul- 



292 



GEODETIC SURVEYING 



taneously in an algebraic discussion it is necessary to replace 
them by a single equivalent equation, meaning an equation so 
formed that the only values which will satisfy it are those which 
will individually satisfy the original equations which it replaces. 
This is done by writing 



ki(aidvi + a2dv2 . . . + audvy) 
+ k2{hidvi + h2dv2 . ■ - + 6„dt;„) 

+ k^imidvi + m2dv2 . ■ ■ + m^dvu) 



^ = 0; 



(45) 



in which ki, k2, etc., are independent constants which may have 
any possible values assigned to them at pleasure. Since Eq. (45) 
must by agreement remain true for all possible sets of values 
ki, k2, etc., its component members must individually remain 
equal to zero. But these component members are identical with 
the first members of the original conditional equations, so that 
no set of values dvi, dv2, etc., can satisfy Eq. (45) unless it can 
also satisfy each of Eqs. (44). The values in any such set are 
called simultaneous values. 

In order to determine the most probable values of vi, V2, etc., 
we must have (Art. 156) 

pivi^ + P2V2^ . . . + Pu^u^ = a minimum. 

In accordance with the principles of the calculus for the case of 
dependent quantities the first derivative of this expression must 
equal zero for every possible set of values dvi, dv2, etc. Hence, 
by differentiating, and omitting the factor 2, we have 



pividvi + p2V2dv2 . . . + PuVudvu = 0, 



(46) 



in which dvi, dv2, etc., must be simultaneous values. Since these 
values are also simultaneous in Eq. (45), we may combine this 
equation with Eq. (46) and write 



PlVidVi + p2V2dV2 . 



+ PuVudVu 

ki{aidvi + a2dv2 
+ k2(bidvi + b2dv2 



. + aJLvu) 
. + hydvy) 



+ k^imidvi + m2dv2 . . . + m^dv^) 



PROBABLE VALUES OF CONDITIONED QUANTITIES 293 
whence, by rearranging the terms, we have 



[piVi — (AiiOi + k2bi 
+ [p2i>2 — {kia2 + ^262 



+ kmmi)]dvi 

+ kmm2)]dV2 



■ '+ [puVu — {kiUu + A;2&« . . . + kmmu)]dvu 



= 0. 



(47) 



Since ki, ^2, etc., are independent and arbitrary constants, it is 
evident that this equation can not be true unless its component 
members are each equal to zerb, so that 



[piVi — (fciai + ^261 . . 
etc., 

. from which we have 

piVi = kitti + fe&i 
P2V2 = kia2 + k2b2 

PuK = hau + k2hu 



+ k^mi)]dvi = 0; 
etc.; 



. + fc„m2 



+ kj^in^^ . 



(48) 



as the general equations of condition for the most probable 
values of v\, V2, etc. 

It is evident that Eqs. (48) can not be solved for vi, V2, etc., 
until definite values have been assigned to ki, fe, etc. In the 
general discussion of the problem the values of ki, /c2, etc., have 
been entirely arbitrary, since the numerical requirements of 
Eqs. (43) vanished in the differentiation. In any particular case, 
however, the m conditional Eqs. (43) must be numerically satisfied 
in order to satisfy the rigid geometrical conditions of the case, 
while the u conditional Eqs. (48) must be satisfied in order to have 
the most probable values for vi, V2, etc. There are thus m + u 
simultaneous equations to be satisfied. But there are also m -\- u 
unknown quantities, since the m unknown quatities ki, k2, etc., 
corresponding to the m conditional Eqs. (43), have been added 
to the u unknown quantities vi, V2, etc. In any particular case, 
therefore, there is but one set of values for the m unknown quan- 
tities ki, k2, etc., and the u unknown quantitites vi, V2, etc., that 
will satisfy the m + u equations consisting of Eqs. (43) and (48). 
The auxiliary quantities ki, k2, etc., are called the correlatives 



294 



GEODETIC SURVEYING 



(or correlates) of the corresponding conditional Eqs. (43), and the 
quantities vi, V2, etc., are the most probable values of the residual 
errors in the observation equations. Substituting in Eqs. (43) 
the values of vi, V2, etc., due to Eqs. (48), we have 



kill— + feli — 
P V 



,ani 



+ km^—- = E, - T.aM 
V 



JfciS- + k2^- ... + k,Jl— =Et - llbM 



in which 



fciS 


am 
~P 


+ k2^ 


bm 

y ■ 


.+fc,„S 


m2 
P 


= 


E^ 






p 


Pi 


P2 




+ 


a^ 

Pu' 






s^ = 


aibi 


, a2&2 




+ 


ttubu 






p 


Pi 


P2 ■ 






Pu 






etc 


) 






etc. 



HmM 



(49) 



Attention is called to the fact that the law of the coefficients 
in Eqs. (49) is the same (Art. 162) as the law of the coefficients 
for normal equations, and this is a check that should never be 
neglected. It is evident that fci, ^2, etc., can be found by solving 
the simultaneous Eqs. (49). Then, from Eqs. (48), we have 



n 



V2 



ki"^ + k2^ 

Pi Pi 

kl h k2 — 

P2 P2 



Vu = kl— + ^2-" 



Pu 



Pu 



and from Eqs. (42), 



+ k, 



mi 



P2 






Pu J 



X = Ml + vi 
y = M2 + V2 

t = M^ + Vu 



(50) 



(51) 



PEOBABLE VALUES OF CONDITIONED QUANTITIES 295 



in which x, y, t, etc., are the most probable values of the quantities 
whose observed values were Mi, M2, M^, etc. 

Example. Given the following data, to find the most probable values of 
X, y, and Z''. 

fx = 17.82 (weight 2); 

Observation equations ] y = 15.11 (weight 4); 

[z =29.16 (weight 3). 

„ ,.,. , ,. (2x + 5y =112.00; 

Conditional equations 1 3^ _j_ y_^ ^ Z^m. 

In this case we have 



Ea 

l^aM = 


112.00 
111.19 






Ei 


= 39.00 
SbM = 39.41 


Ea - SaM = 


0.81 


E,- 


S6M = - 0.41 


Ml = 17.82 
Ml = 15.11 
Ml = 29.16 


oi = 2 
Oa = 5 
as =0 




61 = 

62 = 
6s = 


3 pi =2 

1 P2 = 4 
- 1 Pa = 3 


y"^ 33 

"p -T 


Pi 


= 1 




61 ^3 
Pi 2 


p 4 


£2 

P2 


_ 5 
4 




62 1 

P2 4 


^6^ _ 61 
p 12 


as 

P3 


= 




6s _ _1 
Ps 3 


33, , 17, 

4 4 

17, , 61, 


0.81 
-0.41 




giving 


ffci = +0.2454. 
[fcz = -0.2859. 


We thus have 













Vi = 0.2454 X 1 - 0.2859 X I = - 0.183; 
V2 = 0.2454 X f - 0.2859 Xi = + 0.235; 
t)3 = 0.2454 X + 0.2859 X i = + 0.t)95; 

whence, for the most probable values of x, y, and z, we have 

a; = Ml + wi = 17.82 - 0.183 = 17.637; 
y = M2+V2 = 15.11 + 0.235 = 15.345; 
s = Ms + fs = 29.16 + 0.095 = 29.255. 

As a check on the work of computation we may substitute these values in 
the conditional equations, giving 

2x+5y = 35.274 + 76.725 = 111.999; 

Zx+ 2/ - « = 52.911 + 15.345 - 29.255 = 39.001; 

from which it is seen that each equation checks with the corresponding con- 
ditional equation within 0.001, which is an entirely satisfactory check. The 



296 GEODETIC SURVEYING 

essential feature of the above method is the elimination of the observation 
equations. In Art. 166 the same problem is worked out by eliminating the 
conditional equations. The results obtained are of course identical. 

168. Most Probable Values of Computed Quantities. By a com- 
puted quantity is meant a value derived from one or more observed 
quantities by means of some geometric or analytic relation. 
The most probable values of computed quantities are found from 
the most probable values of the observed quantities by employ- 
ing the same rules that are used with mathematically exact quan- 
tities. Thus the most probable value of the area of a rectangle 
is that which is given by the product of the most probable values 
of its base and altitude; the most probable value of the circum- 
ference of a circle is equal to t: times the most probable value of 
its diameter; and so on. 



CHAPTER XIII 

PROBABLE ERRORS OF OBSERVED AND COMPUTED QUANTITIES 

A. Op Observed Quantities 

169. General Considerations. The most probable value of 
a quantity does not in itself convey any idea of the precision of 
the determination, nor of the favorable or unfavorable circum- 
stances surrounding the individual measurements. Any single 
measurement tends to lie closer to the truth the finer the instru- 
ment and the method used, the greater the skill of the observer, 
the better the atmospheric conditions, etc. The accidental errors 
of observation tend to be more thoroughly eliminated from the 
average value of a series of measurements the greater the number 
of measurements which are averaged together. Some criterion 
or standard of judgment is therefore necessary as a gage of pre- 
cision. Since the probability curve for any particular case shows 
the facility of error in that case, and thus represents all the sur- 
rounding circumstances under which the given observations 
were taken, it is evident that some suitable function of the proba- 
bility curve must be adopted as an indication of the precision 
of the results obtained. The function which is commonly adopted 
as the gage of precision is called the probable error. 

170. Fundamental Meaning of the Probable Error. By the 
probable error of a quantity is meant an error of such a magnitude 
that errors of either greater or lesser numerical value are equally 
likely to occur under the same circumstances of observation. 
Or, in other words, in any extended series of observations the 
probability is that the number of errors numerically greater than 
the probable error will equal the number of errors numerically 
less than the probable error. The probable error of a single 
observation thus becomes the critical value that the numerical 
error of any single observation is equally likely to exceed or fall 
short of. Similarly the probable error of the arithmetic mean 

297 



298 GEODETIC SURVEYING 

becomes the critical value that the numerical error of any iden- 
tically obtained arithmetic mean is equally likely to exceed or 
fall short of. Thus if the probable error of any angular measure- 
ment is said to be five seconds, the meaning is that the probability 
of the error lying between the limits of minus five seconds and plus 
five seconds equals the probability of its lying outside of these 
limits. The probable error is always written after a measured 
quantity with the plus and minus sign. Thus if an angular 
measurement is written 

72° 10' 15".8 ± 1".3, 

it indicates that 1".3 is the probable error of the given determina- 
tion. The probable error of a quantity can not be a positive 
quantity only, or a negative quantity only, but always requires 
both signs. It is important to note that the probable error is an 
altogether different thing from the most probable error. Since 
errors of decreasing magnitude occur with increasing frequency, 
the most probable error in any case is always zero. 

171. Graphical Representation of the Probable Error. The 
probability that an error will fall between any two given limits 
(Art. 147) is equal to the area included between the corresponding 
ordinates of the probability curve. The probability that an error 
will fall outside of any two given limits must hence be equal to 
the sum of the areas outside of these limits. If these two proba- 
bihties are equal, therefore, each such probability must be 
represented by one-half of the total area. The probable error 
thus becomes that error (plus and minus) whose two ordinates 
include one-half the area of the probability curve. Referring 
to Fig. 67, the solid ciu-ve corresponds to a series of observations 
taken under a certain set of conditions, and the dotted curve 
to a series of observations taken under more favorable conditions. 
The ordinates yi, yi, correspond to the probable error n of an 
observation of unit weight taken under the conditions pro- 
ducing the solid probability curve, and include between them- 
selves one-half of the area of that curve. The ordinates y', y', 
correspond to the probable error r' of an observation of unit 
weight taken under the conditions producing the dotted proba- 
bility curve, and include between themselves one-half of the 
area of the dotted curve. The area for any probability curve 
(Art. 150) being always equal to unity, it follows that yi, j/i. 



PROBABLE ERROES OF OBSERVED QUANTITIES 299 



and y', y', include equal areas. Hence as the center ordinate at 
A grows higher and higher with increasing accuracy of observation, 
so also must the ordinates yi, yi, draw closer together. It is 
thus seen that the probable error n grows smaller and smaller 
as the accuracy of the work increases, and therefore furnishes a 
satisfactory gage of precision. 

Y 




Fig. 67.— The Probable Error. 






172. General Value of the Probable Error. The area of any 
probability curve (Art. 150) equals unity. The area between 
any probable error ordinates yi, yi (Art. 171), is equal to half 
the area of the corresponding probability curve. But the area 
between the ordinates yi, yi (Art. 147), is equal to the probability 
that an error will fall between the values x = — ri and x = + ri. 
Hence from Eq. (16) we have 



1 ^ f '•■ 

2 VnXrl 



e-^'^'dx. 



(52) 



Since (Art. 150) the precision of any set of observations depends 
entirely on the value of h, it follows that the probable error ri 
must be some fimction of h. The last member of Eq. (52) is not 
directly integrable, so that the numerical relation of the quan- 
tities h and n can only be found by an indirect method of suc- 
cessive approximation which is beyond the scope of this volume. 
As the result of such a discussion we have, 



ri = 



0.4769363 



h 



(53) 



It is thus seen that for different grades of work the probable error 
n varies inversely as the precision factor h. 



300 GEODETIC, SURVEYING 

By more or less similar processes of reasoning it is also estab- 
lished that the probable error of any quantity or observation 
varies inversely as the square root of its weight. Thus if ri is 
the probable error of an observation of unit weight, then for the 
probable error rp of any value with the weight p, we have 

r-p = -> (54) 

173. Direct Observations of Equal Weight. From Eq. (20) 
we have 



^ ~\ 25:*>2 • 



Substituting this value of h in Eq. (53) and reducing, we have 
n = 0.6745 ^-^^zTi' ^^'"^ 

in which r\ is the probable error of a single observation in the 
case of direct observations of equal weight on a single unknown 
quantity, and n is the number of observations. 

Since in this case (Art. 134) the weight of the arithmetic 
mean is equal to the number of observations, we have (Art. 172), 



Ta -77= =0.6745. , '^■,. , . . . (56) 
Vn y^nin — 1) 

in which r^ is the probable error of the arithmetic mean in the 
case of direct observations of equal weight on a single unknown 
quantity, and n is the number of observations. 

Example. Direct observations on an angle A : 

Observed values v 

29° 21' 59".l - 2".l 

29 22 06 .4 +5 .2 

29 21 58 .1 - 3 .1 
3 )88 06 03 .6 
z = 29 22 01 .2 

The probable error of a single observation is therefore 

n = 0.6745"^-^ = 0.6745 V^= ± 3".06; 





»2 




4.41 




27.04 




9.61 


Sw^ 


= 41.06 


n 


= 3 



PROBABLE EREOES OF OBSERVED QUANTITIES 301 

and of the arithmetic mean, 

n 3.06 



whence we have 



= ± 1".76; 



Most probable value of A = 29° 22' 01".2 ± 1".76. 

174. Direct Observations of Unequal Weight. From Eq. (2l) 
we have 

Substituting this value of h in Eq. (53) and reducing, we have 






n = 0.6745 J— ^, (57) 



in which ri is the probable error of an observation of unit weight 
in the case of direct observations of unequal weight on a single 
unknown quantity, and n is the number of observations. The 
value of n thus becomes purely a standard of reference, and it is 
entirely immaterial whether or not any one of the observations 
has been assigned a unit weight. Having found the value of 
ri we have, from Eq. (54), 

_ '"1 

in which rp is the probable error of any observation whose weight 
is p. 

Since in the case of weighted observations (Art. 134) the weight 
of t^e weighted arithmetic mean is equal to the sum of the indi- 
vidual weights, we have (Art. 172), 



^^=vk^'-''^'4^^^)' ■ ■ ■ ^''^ 



in which rpa is the probable error of the weighted arithmetic mean 
in the case of direct observations of unequal weight on a single 
imknown quantity. 



302 GEODETIC SURVEYING 

Example. Direct base-line measurements of miequal weight: 

Observed values p pM v v^ pifl 

4863.241ft. 2 9726.482 0.020 0.000400 0.000800 
4863.182 ft. 1 4863.182 - 0.039 0.001521 0.001521 
Sp = 3 )14589.664 Spu' = 0.002321 

2=4863.221 ft. n = 2. 

The probable error pf an observation of miit weight is therefore 



0.6745 J^^= 0.6745x1 "-""^'^^^ ±0.032 ft.; 
■ 'n — 1 ' ^ 1 



of an observation of the weight 2, 



r, 0.032 
rj = -4= = — = = ± 0.023 ft.; 
Vp V2 

and of the weighted arithmetic mean, 

r, 0.032 „„ , 
rpa = ^=i= y=^ ± 0.019 ft.; 

whence we have 

Most probable value = 4861.221 ± 0.019 ft. 

175. Indirect Observations on Independent Quantities. From 
Eq. (22) we have 



»=j; 



- Q 



Substituting this value of h in Eq. (53) and reducing, we have 



n =0.6745^-^, (59) 

in which ri is the probable error of an observation of unit weight 
in the case of indirect observations on independent quantities 
(that is with no conditional equations), n is the number of observa- 
tion equations, and q is the number of unknown quantities. 
Having found the value of n, we have, from Art. 172, 

n n n . 

rv = —i=, r^ = —.= , Ty = —=, etc., 
vp Vp^ Vpy 

in which rp is the probable error of a;ny observation whose weight 
is p, and r^ is the probable error of any unknown, x, in terms of 
its weight p^, and so on. 



PEOBABLE EERORS OF OBSERVED QUANTITIES 303 

The weights px, Pv, etc., of the unknown quantities are found 
from the normal equations by means of the following 

Rule: In solving the normal equations preserve the absolute 
terms in literal form; then the weight of any unknown quantity is 
contained in the expression for that quantity, and is the reciprocal 
of the coefficient of the absolute term which belonged to the normal 
equation for that unknown quanti y. 

In applying the above rule no change whatever is to be made 
in the original form of any normal equation until the absolute 
term has been replaced by a literal term. If the normal equations 
are correctly solved the coefficients in the literal expressions for 
the unknown quantities will follow the same law (Art. 162) as 
the coefficients of normal equations, and this check must never 
be neglected. 

Example. Given the foUowing observation equations to determine the 
most probable values and the probable errors of the unknown quantities: . 

z + y = 10.90 (weights); 

2x - y = 1.61 (weight 1); 

X + 3y -=- 24.49 (weight 2). 

Forming the normal equations, we have 

9a; + 7y = 84.90 = N^ = normal equation in x; 
7x + 22y = 178.03 = JVj, = normal equation in y; 
whence 

X = tANx - TTsNy = 4.172, nearly; 

y ThNx + iTwNy = 6.765, nearly; 

and, by the rule, ' 

Weight of a; = W = 6.773, nearly = p^; 
" 2/ = i|^ = 16.556 " =py. 

Substituting in the original equations the values obtained for x and y, there 
results 

x+ y = 10.937; 
2x - y = 1.579; 

x + 3y = 24.467; 

whence, for the residuals, we have, 

Vi = 10.90 - 10.937 = - 0.037 (weight 3); 
V2 = 1.61 - 1.579 = + 0.031 (weight 1); 
V, = 24.49 - 24.467 = + 0.023 (weight 2). 

We therefore have for the probable error of aa observation of unit weight. 



0.6745 J-^^^ = 0.6745a/^555^ = ± 0.053; 
'71 — a '3—2 



304 GEODETIC SURVEYING 

for the probable error of x, 

■Ti 0-053 

r, = — F= = , = ± 0.020; 

Vp^ V6773 

and for the probable error of y, 

tx 0.053 „„,„ 

rj, = — ^ = , = ± 0.013; 

Vpy V16.556 

whence we write 

X = 4.172 ± 0.020 and y = 6.765 ± 0.013. 

176. Indirect Observations Involving Conditional Equations. 

From Eq. (23) we have 






g + c 



Substituting this value of In. in Eq. (53) and reducing, we have 



'^/^ 



r-i = 0.6745^/^^^, .... (60) 



in which r\ is the probable error of an observation of unit weight 
in the case of indirect observations involving conditional equa- 
tions, n is the number of observation equations, q is the niunber 
of unknown quantities, and c is the number of conditional equa- 
tions. Having found the value of r\, we have, from Art. 172, 

_ ri _ ri _ ri 

Tf — ,_, r^ — , — , Ty -=, etc., 

in which, as in the previous article, r^ is the probable error of any 
observation whose weight is p, and r^ is the probable error of any 
unknown, x, in terms of its weight 'p^, and so on. 

In order to find the value of the weights p^,, p^, etc., the con- 
ditional equations are first eliminated (Art. 166), and the normal 
equations due to the resulting observation equations are then 
treated by the rule of the preceding article. By repeating the 
process with different sets of unknowns eliminated, the weight 
of each unknown will eventually be determined. 

177. Other Measures of Precision. The measures of precision 
thus far introduced are the precision factor h,, and the probable 
error r. Two other measures of precision are sometimes used, 



PROBABLE EEEOES OF OBSERVED QUANTITIES 305 

and are of great theoretic value. These are known as the mean 
error, &nd.\hQ mean absolute error. _ - -^ 

By the'mean error is meant the square root of the sum of the^ 
squares^ of the true errors. -- — 

By the mean absolute error (often called the vwan of the errors) 
is meant the arithmetic mean of the absolute values (numerical 
values) of the true errors. _ 

Referring to Fig.' 68, the precision factor h is equal to Vtt 
times the central ordinate AY. Considering either half of the 



Poiat of Inflection 



Point of Inflection 




Fig. 68. — Measures of Precision. 



curve alone, the ordinate for the probable error r bisects the 
included area, the ordinate for the mean absolute error i) passes 
through the center of gravity, and the ordinate for the mean 
error e passes through the center of gyration about the axis A Y. 
The ordinate for e also passes through the point of inflection 
of the curve. 

The measure of precision most commonly used in practice is 
the probable error r, but as the different measures bear fixed 
relations to each other a knowledge of any one of them determines 
the value of all the others, as shown in the following summary: 



Precision factor = h. 
Probable error = r = 



0.4769363 



Mean absolute error = rj 



Mean error 



hVn 
1 



= 1.1829 r. 
= 1.4826 r. 



306 GEODETIC SURVEYING 



B. Of Computed Quantities 

178. Typical Cases. When the probable error is known 
for each of the quantities from which a computed quantity is 
derived, the probable error of the computed quantity may also 
be determined. Any problem which may arise will come under 
one or more of the five following cases : 

1. The computed quantity is the sum or difference of an observed 
quantity and a constant. 

2. The computed quantity is obtained from an observed quantity 
by the use of a constant factor. 

3. The computed quantity is any function of a single observed 
quantity. 

4. The computed quantity is the algebraic sum of two or more 
independently observed quantities. 

5. The computed quantity is any function of two or more inde- 
pendently observed quantities. 

The fifth case is general, and embraces all the other cases. 
The first four cases, however, are of such frequent occurrence that 
special rules are developed for them . Any combination of the rules 
is therefore admissible that does not violate their fundamental 
conditions, since the first four rules are only special cases of the 
fifth rule. 

179. The Computed Quantity is the Sum or Difference of an 
Observed Quantity and a Constant. 

Let u and r-„ = the computed quantity and its probable error; 
X and r^ = the observed quantity and its probable error; 
a = a constant; 



then 
and 



u — ± X ± a; 



ru = r^ (61) 



It is evidently immaterial whether x is directly observed or 
is the result of computation on one or more observed quantities. 
The only essential condition is satisfied if r^ is the probable error 
of X. If a; is a computed quantity the probable error r^ may be 
derived by any one of the present rules. 



PROBABLE ERRORS OF COMPUTED QUANTITIES 307 

Example. Referring to Fig. 69, the most probable value of the angle x is 

X = 30° 45' 17".22 ± 1".63. 

What is the most probable value of its supplement y, and the probable error 
of this value 7 

From the conditions of the problem we 
have 

y = 180° - x; 

whence 




?•„ = r^ = ± 1".63, F:g. 69. 

and 

y = 149° 14' 42".78 ± 1".63. 

180- The Computed Quantity is Obtained from an Observed 
Quantity by the Use of a Constant Factor. 

Let u and r„ = the computed quantity and its probable error; 
X and Tx = the observed quantity and its probable error; 
a = a constant; 



then 
and 



u = ax 



Tu = ar^ (62) 



Evidently, as in the previous case, x may be any function of one 
or more observed quantities, provided that r^is its correct probable 
error. The rule of this article is only true when the constant a 
represents a strictly mathematical relation, such as the relation 
between the diameter and the circumference of a circle. Staking 
out 100 feet by marking off successively this number of single 
feet is not such a case, as the total space staked out is not neces- 
sarily exactly 100 times any one of the single spaces as actually 
marked off. In all probability some of the feet will be too long 
and others will be too short, so that (owing to this compensating 
effect) the total error will be very much less than 100 times any 
single error, and the probable error must be found by Art. 182. 
In the case of the circle, however, the circumference is of neces- 
sity exactly equal in every case to n times the diameter. 



308 GEODETIC SURVEYING 

Example. The radius of a circle, as measured, equals 271.16 ± 0.04 ft. 
What is the most probable value of the circumference, and the probable 
error of this value? 

Circumference = 271.16 X 2x = 1703.75 ft.; 
r^= r^X 2x = ± 0.04 X 2x = ± 0.25 ft.; 

whence we write 

Circumference = 1703.75 ± 0.25 ft. 

181. The Computed Quantity is any Function of a Single 
Observed Quantity. 

Let u and r-„ =the computed quantity and its probable error; 
X and r^ = the observed quantity and its probable error; 
then 

u = (j>(x); 
and 

'-='^t (^^) 

Evidently, as in the two previous cases, x may be any function 
of one or more observed quantities, provided that r^ is its correct 
probable error. 

Example. The radius 5 of a circle equals 42.27 ± 0.02 ft. What is 
the most probable value and the probable error of the area? 

M = «e' = (42.27)^ X X = 5613.26; 

du = 2xxdx, -— = 2xa;, 

dx 

■ru = 'rx~ = rx{2%x) = ± 0.02 X 2x X 42.27 = ± 5.31; 
dx 

whence we write 

Area = 5613.26 ± 5.31 sq.ft. 

182. The Computed Quantity is the Algebraic Sum of Two or 
More Independently Observed Quantities. 

Let u and r„ = the computed quantity and its probable 
error; 
X, y, etc. = the independently observed quantities; 
Tx, Ty, etc. = the probable errors of x, y, etc. ; J 

then 

w = ± a; ± 2/ ± etc.; 
and 

/•u-VrJ+V + etcT ...... (64) 



PROBABLE EEEOES OF COMPUTED QUANTITIES 309 

The observed quantities x, y, z, etc., may each be a different 
function of one or more observed quantities, but the absolute 
independence of x, y, z, etc., must be maintained. In other 
words, X must be independent of any observed quantity involved 
in y, z, etc.; y independent of any observed quantity involved 
in X, z, etc. ; and so on. Thus, for instance, we can not regard 
2x as equal to a; + a;, and substitute in the above formula, since 
X and X in the quantity 2x are not independent quantities. 
Attention is also called to the fact that the signs under the 
radical are always positive, whether the computed quantity is 
the result of addition or subtraction or both combined. 

Example 1. Referring to Fig. 70, given 

X = 70° 13' 27".60 ± 2".16; 
2/ = 40 67 19 .32 ± 1 .07; 

to find the most probable value and the probable error of z. 
In this case 

2 = x + y = 111° 10' 46".92; 



whence we write 



r„ = V (2.16)^ +(1.07)' = ± 2".41; 



2 = 111° 10' 46".92 ± 2".41. 



^ 

<^1^ V|^ 



Fig. 70. 



Fig. 71. 



Example 2. Referring to Fig. 71, given 

X = 70° 13' 27".60 ± 2".16; 
3/ = 40 57 19 .32 db 1 .07; 

to find the most probable value and the probable error of z. 
In this case 

2 = a; - 2/ = 29° 16' 08".28; 

r„ = V'(2.16)' + (1.07)' = ± 2".41; 

whence we write 

z = 29° 16' 08".28 ± 2".41. 



310 GEODETIC SURVEYING 

183. The Computed Quantity is any Function of Two or More 
Independently Observed Quantities. 

Let u and r„ = the computed quantity and its probable error; 
X, y, etc. = the independently observed quantities; 
r^, Ty, etc. = the probable errors of x, y, etc.; 
then 

u = ^{x, y, etc.); 
and 

^" = V(^^^)'+('"«S)' + ^*^- • • • ^^^^ 

All the remarks under the previous case apply with equal force 
to the present case. 

Example 1. The measured values for the two sides of a rectangle are 

X = 55.28 ± 0.03 ft. 
y = 85.72 ± 0.05 ft. 

What is the most probable value of the area and its probable error? 

u = xy = 55.28 X 85.72 = 4738.60; 

du _ du _ 

dx ' dy ' 

Tu = ^{rxyY + (ryxY 



= V(0.03 X 85.72)^ + (0.05 X 55.28)^ = ± 3.78; 

whence we write 

Area = 4738.60 ± 3.78 sq.ft. m 

Example 2. Referring to the right-angled 
triangle in Fig. 72, given 

x = 38.17 ± 0.06 ft.; Fig. 72. 

y = 19.16 ± 0.04 ft.; 

to find the most probable value of the hypothenuse u and its probable error. 




M = Va;^ + 2/2 = V(38.17)2 + (19.16)' = 42.71; 
du _ X du _ y 



dx Va:' + 2/2 'dy V x^ + y^' 



{rxxy + (r„yy 
x' + y' 



r = K j'^. y + ( 'yy V = /' 

/ (38.17 X 0.05)2 + (19.16 X 0.04)' ^ 
V (38.17)2 + (19.16)2 ±0.05; 



PROBABLE ERRORS QF COMPUTED QUANTITIES 311 

whence we write 

Hypothenuse = 42.71 ± 0.05 ft. 

Example 3. Referring to Pig. 73, in which the horizontal distance x 
and the vertical angle <j) have been measured, 
given 

X = 489.11 ±0.32 ft.; 

<(> = 12° 17' ± 1'; Fig. 73. 

to find the most probable value of the elevation u and its probable error 

w = a; tan (ji = 106.49; 

du X 




du 
dx 



tan 0, 



d<p cos''^ ' 
■„= Ltan^)^ + /^ ^^ 

\ \C0S I 



It is necessary at this point to remember that expressing an angle in degrees, 
minutes, and seconds, is only a trigonometrical convenience, and that the 
true measure of an angle is the ratio of the subtending arc to its radius. 
An arc expressed in minutes must therefore be compared with a radian ex- 
pressed in minutes (that is, an arc whose length equals that of the describing 
radius) in order to complete its angular meaning. 

1 radian = 3438', nearly. r,^ = 



3438 



r„ = J(o.; 



whence we write 



32 tan ,!.)^ + (-^ X ,, 
\3438 cos'^ 



u = 106.49 ± 0.16 ft. 



11\2 



\ = ± 0.16; 



CHAPTER XIV 

APPLICATION TO ANGULAR MEASUREMENTS 

184. General Considerations. In the adjustment of angular 
measurements three classes of problems may arise, known as 
single angle adjustment, station adjustment, and figure adjust- 
ment. 

By single angle adjustment is meant the determination of the 
most probable value of an angle which can be obtained from the 
measurements made directly upon it. 

By station adjustment is meant the determination of the most 
probable values of two or more angles at a single station, in order 
to meet the condition of being geometrically consistent. 

By figure adjustment is meant the determination of the most 
probable values of the angles involved in any geometric figure, 
in order to meet the condition of being geometrically consistent. 

In trigonometric work of any importance each individual 
angle is always measured a large number of times, and the most 
probable value due to these results is considered as its measured 
value. The station adjustment or figure adjustment is then 
made in accordance with the conditions of the given case. 

Single Angle Adjustment 

185. The Case of Equal Weights. In this case (Art. 155) 
the most probable value is the arithmetic mean of the individual 
measurements. 

Example. Three equally reliable measurements of the angle x give 
29° 21' 59".l, 29° 22' 06" A, 29° 21' 58". 1. What is its most probable 
value? 

29° 21' 59".l 
29 22 06 .4 
29 21 58 .1 
3 )88 06 03 .6 
29° 22' 01".2 

The most probable value is therefore 29° 22' 01".2. 

312 



APPLICATION TO ANGULAE MEASUREMENTS 



313 



186. The Case of Unequal Weights. In this case (Art. 157) 
the most probable value is the weighted arithmetic mean of the 
individual measurements. 

Exampk. Three measurements of an angle x give 38° 15' 17".2 (weight 1), 
38° 15' 15".5 (weight 3), and 38° 15' 18".0 (weight 2). What is its most 
probable value? 



38° 15' 


17".2 X 1 = 38° 


15' 


17".2 


38 15 


15 .5X3 = 114 


45 


46 .5 


38 15 


18 .0 X 2 = 76 


30 


36 .0 




6)229_ 


31 


39 .7 



38° 15' 16".6 
The most probable value is therefore 38° 15' 16".6. 

Station Adjustment 

187. General Considerations. All cases of station adjust- 
ment necessarily imply one or more conditional equations. In 
the determination of the most probable values of the several 
angles these equations may be avoided (Art. 165), eliminated 
(Art. 166), or involved in the computa- 
tion (Art. 167), as found most convenient. 
The angles at a station are in general ' 
measured under similar conditions, so 
that in making the adjustment it is 
customary to give to each angle a weight 
equal to the number of observations (or 
the sum of the weights in the case of 
weighted observations) on which it de- 
pends. Angles are seldom measured a 
sufficient number of times to make it 
justifiable to weight them inversely as 

the squares of their probable errors, as would be required by 
the last paragraph of Art. 172. The following cases of station 
adjustment show the general principles involved: 

188. Closing the Horizon with Angles of Equal Weight. 
Referring to Fig. 74, 

Let X, y, z, . . . w = the angles measured; 
a,b,c,...m = their measured values; 
n = the number of angles measured; 
d=(a + b-{-c...-\-in)— 360° = the discrepancy to 
be adjusted; 




Fig. 74. 



314 GEODETIC SUEVEYING 

then the observation equations are 

X = a; 
y = b; 
z = c: 



w = m; 

and the conditional equation is 

x + y + z . . . +w = 360°. 

It is evident from the figure, however, that this conditional 
equation may be avoided (Art. 165) by regarding all the angles 
except w, for instance, as independent, and involving the required 
condition by expressing this angle in terms of the others. The 
observation equations thus become 

X = a; 
y = b; 
z = c; 

360° - (x+ 2/ + z . . .) = m. 

Passing to the reduced observation equations (Art. 163) by sub- 
stituting for the most probable values of the unknown quantities, 



we have 



X 


= 


a + 


«'i; 


y 


= 


b + 


V2; 


z 




c + 
etc. 


vs; 




Vl 


= 0; 






V2 


= 0; 






V3 


= 0; 





vi+V2 + va . . . = 360° - (a + b + c . . . + m) = - d; 

giving the normal equations 

2wi + V2 + V3 + Vi + vs . . . = — d; 

i)i + 2w2 + Vs + Vi + Vs . . . = — d; 

Vl + V2 + 2v3 + Vi + Vs . . . = - d; 

etc., etc. 



APPLICATION TO ANGULAR MEASUREMENTS 315 
Subtracting the second equation from the first, we have 

Z)l — 2)2 = 0, or Vi = V2. 

Subtracting the third equation from the second, we have 
!;2 — fa = 0, or V2 = vs. 



Or, in general, 








Ul = ^2 = fa = «'4 = fs 


= etc 


But 








fl + f2 + t)3 . . . = 


-d; 


whence 








Ui = f 2 = Vs = etc. = 


d 

n 



. . . (66) 

Eq. (66) shows that when angles of equal weight are arranged 
around a point so as to close the horizon, the most probable value 
for each angle is found by a uniform dis- 
tribution of the discrepancy. 

Example. Referring to Fig. 75, the following 
observations are to be adjusted: 



X = 45° 
y = 151 
z = 162 


20' 
52 
46 


19' 

48 
58 


'.3 
.6 

.4 


(weight 1); 
(weight 1); 
(weight 1). 


360° 
360 


00' 
00 


06' 
00 


'.3 
.0 






d= + 06".3 Fig. 75. 

In accordance with the above principle each angle must be reduced by 2".l, 
giving for the most probable values 

X = 45° 20' 17".2; 
y = 151 52 46 .5; 
z = 162 46 56 .3. 

189. Closing the Horizon with Angles of Unequal Weight. 

Referring to Fig. 74, page 313, 

Let x,y,z,...w = the angles measured; 
a,b,c,...m = their measured values; 
pi, P2, Vz, ■ ■ ■ pn = their respective weights; 

n = the number of angles measured; 
d ^ (a. + 6 + c . . . + m) - 360° = the discrepancy to 
be adjusted; 



316 GEODETIC SURVEYING 

then the observation equations are 

X = a (weight pi) ; 
y = h (weight P2) ; 
z = c (weight ps) ; 

w = m (weight pn) ; 

and the conditional equation is 

x-{-y + z . . . + 10 = 360°. 

It is evident that this conditional equation may be avoided, as in 
Art. 188, by writing the observation equations in the form 

a; = a (weight pi); 
y = h (weight P2) ; 
z = c (weight ps) ; 

360° -{x-\- y + z. . .) =m (weight p„). 

Passing to the reduced observation equations, as before, by 
substituting 

X = a + vi; 

y = b + V2; 
etc.; 

we have 

wi = (weight pi) ; 
^2 = (weight P2) ; 
t)3 = (weight Ps) ; 

wi + W2 + f 3 • ■ • = — d (weight p„) ; 
giving the normal equations 

Pl«l + Pn(Vi + V2 + V3 . . .) = - Pnd; 
P2^2 + Pn(Vl + V2 + V3 . . .) = - Pnd; 
PZVS + Pn{V] + V2 + V3 . . .) = - Pnd. 

etc., etc. 



APPLICATION TO ANGULAR MEASUREMENTS 317 

Subtracting the second equation from the first, the third equation 
from the second, and so on, we have 

Pivi — P2V2 = 0, or pivi = P2V2; 

P2V2 — P3V3 = 0, or P2V2 = psvs; 

etc., ete.; 

or, in general, 

PiVi = P2V2 = P3V3 = PiVi = etc. . . . (67) 

Eq.(67) shows that when angles of unequal weight are arranged 
around a point so as to close the horizon, the most probable value 
for each angle is found by distributing the discrepancy inversely 
as the corresponding weights. 

Example. Referring to Fig. 75, page 315, the following observations are 
to be adjusted: 

x= 50° 49' 27".6 (weight 2); 
y = 149 22 22 .8 (weight 1); 
z = 159 48 05 .9 (weight 3). 

359° 59' 56".3 
360 00 00 .0 



d = - 03".7 



In accordance with the above principle this discrepancy is to be distributed 
as 

111. 
2 ■ 1 • 3' 

which, cleared of fractions, equals 

3:6:2. 

The three corrections are thus 

2.7 Xtt = 1".01, 3.7 X A = 2".02, and 3.7 X it = 0".67. 
The most probable values are therefore 

X = 50° 49' 28".61; 
y = 149 22 24 .82; 
z = 159 48 06 .57. 

190. Simple Summation Adjustments. Referring to Fig. 76, 
page 318, let x, y, z, etc., represent a series of angles at the point C, 



318 



GEODETIC SURVEYING 



and let w represent the corresponding summation angle. Then 
we must have geometrically, 

w = x-i-y + z + etc. 

But the measured values of these angles will seldom satisfy this 
conditional equation, and an adjustment becomes necessary to 
remove the discrepancy. In making the 
adjustment it is evidently immaterial 
whether we regard w or w' as the angle 
actually measured, since these values 
are mutually convertible and only differ- 
ent expressions for the same fundamental 
idea. The adjustment may therefore be 
made in any case by subtracting the 
measured value of w from 360° to obtain 
the apparent value of w', and then 
applying the rule of Arts. 188 or 189, 
as may be necessary. Since the correc- 
tion to w' will have the same sign as all the remaining corrections, 
it is evident that the correction to w must have the opposite 
sign. We are thus led to the following conclusions : 

In the case of equal weights the most probable values of the 
measured angles are obtained by an equal numerical distribu- 
tion of the discrepancy, with opposite signs for the summation- 
angle correction and all the remaining corrections. 

In the case of unequal weights the most probable values of the 
measured angles are obtained by a numerical distribution of the 
discrepancy inversely proportional to the several weights, with 
opposite signs for the summation-angle correction and all the 
remaining corrections. 




Example 


1. Referring to 


Fi^ 


;. 77, the foUo 


adjusted : 
















X 


= 39° 


12' 


32" 


.6 


(weight 1); 




y 


= 44 


47 


59 


.3 


(weight 1); 




X + y 


= 84 


00 


35 


.8 


(weight 1). 






39° 


12' 


32" 


■-6 








44 


47 


59 


.3 








84 


00 


31 


.9 








84 


00 


35 


.8 








3 


)03 


.9 
.3 






1 






APPLICATION TO ANGULAR MEASUREMENTS 319 

In accordance with the above principles the most probable corrections to 
the measured angles are 

+ 1".3, + 1".3, - 1".3; 

giving as the most probable values, 

X = 39° 12' 33".9; 

2/ = 44 48 00 .6; 

a; + 2/ = 84 00 34 .5. 

> Example 2. Referring to Fig. 77, the following observations are to be 
adjusted ; 

X = 40° 16' 23".7 (weight 2); 

2/ = 46 36 48 .5 (weights); 

a; + 2/ = 86 53 08 .0 (weight 4). 

40° 16' 23".7 
46 36 48 .5 
86 53 12 .2 
86 53 08 .0 
d = 04 .2 

In accordance with the above principles this discrepancy is to be distributed 
numerically as 

1 . i . 1. 

2 '3 "4' 

which, cleared of fractions, equals 

6:4:3; 
giving as the most probable corrections . 

- 4.2 X fV = - 1".94 

- 4.2 X T^ = - 1".29 
+ 4.2 X A = + 0".97 

and therefore as the most probable values 

X = 40° 16' 21".76; 

2/ = 46 36 47 21; 

a; + 2/ = 86 53 08 .97. 

191. The General Case. The cases given in Arts. 188, 189, and 
190, are the only ones in which it is desirable to establish special 
rules. Any case of station adjustment may be solved by writ- 
ing out the observation and conditional equations and then apply- 
ing the principles developed in Chapters XI and XII. 



320 



GEODETIC SURVEYING 



Example 1. Referring to Fig. 78, find the most probable values of the 
angles x, y, and z, from the foUowing observations : 



s + 2/ = 53 
X + y + z =86 



X = 25° 17' 10".2 (weight 1) 

2/ = 28 22 16 .4 (weight 2) 

z = 32 40 28 .5 (weight 2) 

39 23 .1 (weight 2) 

19 57 .8 (weight 1). 



Letting v,, vi, v,, be the most probable corrections for x, y, and z, we may 
write (Art. 163) the reduced observation equations 

Vi = 0".0 (weight 1) 

V2 = .0 (weight 2) 

v, = .0 (weight 2) 

Vi + Vi = — 3 .5 (weight 2) 

V, + vi + V, = + 2 .7 (weight 1) 





Fig. 78. 



Fig. 79. 



from which we have the normal equations 

» vi + 3w2 + 1)3 = - 4.3 

3wi + 51)2 + V3 = — 4.3 

vi+ V2+ Svs = 2.7 
whose solution gives 

Vi = - 1".04, V2= - 0".52, Ds 



+ 1".42. 



The most probable values of the given angles are therefore 

X = 25° 17' 09".16; 
y = 28 22 15 .88; 
« = 32 40 29 .92. 

Example 2. Referring to Fig. 79, find the most probable values of the 
angles x, y, and z, from the following observations: 

X = 14° 11' 17".l (weight 1); 

y = 19 07 21 .3 (weight 2); 
X + y = 33 18 43 .4 (weight 1); 

z = 326 41 18 .2 (weight 2); 
y + z = 345 48 39 .2 (weight 3). 



APPLICATION TO ANGULAR MEASUREMENTS 321 

As the angles x, y, and z close the horizon they must satisfy the conditional 
equation 

x + y + z = 360°. 

Avoiding this conditional equation by subtracting all angles containing 
z from 360°, we have 

X = 14° 11' 17".l (weight 1); 

2/ = 19 07 21 .3 (weight 2); 

X + 2/ = 33 18 43 .4 (weight 1); 

X + 2/ = 33 18 41 .8 (weight 2); 

I = 14 11 20 .8 (weight 3); 

in which x and y may be regarded as independent quantities. 

Letting v^ and Vi be the most probable corrections for x and y, and 
writing the reduced observation equations in accordance with Art. 163, 
we have 

wi = 0".0 (weight 1); 

tij = .0 (weight 2); 
vi + Vi = 5 .0 (weight 1); 
vi + vi = Z A (weight 2); 
vi = 3 .7 (weights); 

from which we have the normal equations 

7j)i + 3f2 = 22.9; 
3»i + 5v2 = 11.8; 
whose solution gives 

vi= -\- 3".04, wj = + 0".53. 

The most probable values of x and y are therefore 

X = 14° 11' 20".14; 
y = 19 07 21 .83; 

and hence the most probable value for z must be 

z = 326° 41' 18".03, 

in order to make the sum total of 360° 

Figure Adjustment 

192. General Considerations. All cases of figure adjust- 
ment necessarily imply one or more conditional equations. In 
the determination of the most probable values of the several 
angles these equations may be avoided (Art. 165), eliminated 
(Art. 166), or involved in the computation (Art. 167), as found 
most convenient. The angles in a triangulation system are in 
general measured under similar conditions, so that in making the 
adjustment it is customary to give to each angle a weight equal to 
the number of observations (or the sum of the weights in the case 
of weighted observations) on which it depends. Angles are sel- 
dom measured a sufficient number of times to make it justifiable 
to weight them inversely as the squares of their probable errors, 



322 GEODETIC SUEVEYING 

as would be required by the last paragraph of Art. 172. In work of 
moderate extent any required station adjustment may be made 
prior to the figure adjustment, but in very important work it may 
be desirable to make both adjustments in one operation. Except 
in very important work, the triangles, quadrilaterals, or other 
figures in a system may be adjusted independently. In work of 
the highest importance the whole system would be adjusted in 
one operation. The following cases of figure adjustment show 
the general principles involved, assimiing that the reduction for 
spherical excess (Arts. 66, 57, 58) has already been made. 

193. Triangle Adjustment with Angles of Equal "Weight. 
Referring to Fig. 80, 




Fig. 80. 

Let X, y, z = the unknown angles; 
a,b,c = the measured values; 

d = (a + b -{- c) — 180° = the discrepancy to be 
adjusted. 

Avoiding the conditional equation (Art. 163) for the sum of the 
three angles by writing the observation equations in terms of 
X and y as independent quantities, we have 

X = a; 
y = b; 
x + y = 180° - c. 

Substituting for the most probable values 

X = a -\- vi; 
y = b + V2; 
we have 

vi =0; 

V2 = 0; 
i^i + «'2 = 180° - (a + b + c) = - d; 



APPLICATION TO ANGULAE MEASUEEMENTS 323 

giving the normal equations, 

2^1 + V2 = — d; 
vi + 2^2 = — d; 
whence by subtraction, 

v\ — V2 = 0, or vi = Vi- 
la, a similar manner it may be shown that vi or v^, is equal to 
vz, or in general, 

Vi = V2 = V3. 

Bat evidently, 

vi -{■ V2 -[- Vz = — d; 
whence, 

i;i = ■y2 = ■y.s = - (68) 

Equation (68) shows that when the measured angles of a tri- 
angle are considered of equal weight, the most probable values of 
these angles are found by adjusting each angle equally for one-third 
of the discrepancy. 

Example. The measured values (of equal weight) for the three angles 
of a triangle are 92° 33' 15".4, 48° 11' 29".6, and 39° 15' 12".3. What are 
the most probable values? 

Measured Values Most Probable Values 

92° 33' 15".4 92° 33' 16".3 

48 11 29 .6 48 11 30 .5 

39 15 12 .3 39 15 13 .2 

179° 59' 67".3 180° 00' 00" .0 

180 00 00 .0 
3) - 02".7 
- 0".9 ' 

194. Triangle Adjustment with Angles of Unequal Weight. 

Referring to Fig. 80, 

Let X, y, z = the unknown angles; 
a,b,c== the measured values; 
Pi) P2, Pz — the respective weights; 

d = (a + 6 -f- c) — 180° = the discrepancy to be 
adjusted. 
Avoiding the conditional equation as before by making x and y 
the independent quantities, we have 

a; = a (weight pi); 

y = b (weight ^2); 

X + y = 180° — c (weight pz). 



324 



GEODETIC SUEVEYING 



Substituting, as before, 






we have 



= (weight pi); 

j;2 = (weight p2); 

VI+V2 = 180° -{a + b + c) = -d (weight ps); 

giving the normal equations 

Pivi + psivi + V2) = — Pad; 
P2V2 + psivi + V2) = — p-id; 

whence, by subtraction, 

piVi — P2V2 = 0, or piVi = P2V2- 

In a similar manner it may be shown that pivx or P2V2 is equal to 
P3V3. Hence, in any case, 

vi + V2 + 1)3= — d] 

PlVi = P2V2= P3V3. 



(69) 



Eqs. (69) show that when the measured angles of a triangle are 
considered of unequal weight, the most probable values of these 
angles are found by distributing the discrepancy inversely as the 
corresponding weights. 

Example. The measured values for the three angles of a triangle axe 
97° 49' 56" .8 (weight 2), 38° 06' 05".0 (weight 1), and 44° 04' 01".l (w-eight 3). 
What are the most probable values? 

97° 49' 56".8 

38 06 05 .0 

44 04 01 .1 

180° 00' 02".9 

180 00 00 .0 



i:| = S,e.; 



d = + 02".9 

3+6 + 2 = 11; 



+ 02.9 X A = + 00".79, + 02.9 X 1^ = + 01".58, 

+ 02.9 X A = + 00".53. 

The most probable values are therefore 

97° 49' 56".01 
38 06 03 .42 
44 04 00 .57 



180° 00' 00".00 



APPLICATION TO ANGULAB MEASUEEMENTS 325 



195. Two Connected Triangles. A simple case of figure 
adjustment is illustrated in Fig. 81. Two triangles are here 
connected by the common side AB, and the eight indicated 
angles are measured. It is evident from the figure that four 
independent conditional equations must be satisfied by the 
adjusted values of the angles, for the summation angles at A and B 
must agree with their component angles, and the angles in each 
of the two triangles must add up to 180°. The problem may be 
worked out by the methods of Arts. 165, 166, or 167. The fol- 

A 




Fig. 81. — ^Two Connected Triangles. 

lowing example is worked out by the algebraic elimination of the 
conditional equations (Art. 166) in order to illustrate this method. 

fee. 

Exampk. Referring to Fig. 81, given the following observed values of 
equal weight, to find the most probable values of the measured angles: 

observed Values of Angles 



Ai = 65° 25' 18".l; A = 141° 
A2 = 75 43 45 .1; B = 100 
Bs- = 47 26 11 .9; C = 67 
Bi = 53 19 51 .8; D = 50 


09' 
46 
08 
56 


02' 
06 

28 
25 


•2; 

.6; 

•4; 
.2. 


t the four conditional equations, we have 








A = Ai+ Ai\ 
B = B,+ Bi-, 
C+Ai+B, = 180°; 
D +A2 + Bi = 180°. 









In accordance with Art. 166, any four of the imknowns which may be 
considered as independent may be found from these equations in terms 
of the remaining unknowns. It is evident from an inspection of either the 
figure or the conditional equations that A, B, C, and D may be thus con- 
sidered as independent. These four are selected in preference to any other 



326 



GEODETIC SURVEYING 



four because they are so easily found from the given conditional equations. 
Solving for these quantities, we have 

A = Ai+ Ai, 

B = Bs -\- Bt; 

C = 180° - (A1 + B3); 

D = 180° - (A2 + B4). 

Substituting in the observation equations and reducing, we have 



Ai = 65° 


25' 


18".l; 


Ai + ^2 = 141° 


09' 


02".2; 


Ai = 75 


43 


45 .1; 


B3 + B, = 100 


46 


06 .6; 


B3 =47 


26 


11 .9; 


Ai + B3 = 112 


51 


31 .6; 


Bi = 53 


19 


51 .8; 


A2 + B4 = 129 


03 


34 .8. 



Letting Vi, vi, Vs, Vi, be the most probable corrections for Ai, A2, B3, B4, 
respectively, we may write the reduced observation equations (Art. 163) 
as follows: 



Vl = 0".0 

!)2 = .0 

Us = .0 

U4 = .0 



fl + W2 = - 1".0 

fa + ^4 = + 2 .9 
vi + va = +\ .6 
Vi -\- Vl = — 2 .1. 



In a simple case like this the reduced observation equations would usually 
be written directly from the figure instead of going through the above alge- 
braic work. Having decided on the proper independent quantities, these 
equations are simply written so as to represent the apparent discrepancy 
in each observation, always subtracting the independent quantities from 
the values they are compared with. Forming the normal equations, we have 



whose solution gives 



Using these corrections to find Ai, A2, B,, and B4, and then the conditional 
equations to find A, B, C, and D, we have for the most probable values 



Svi + Da + W3 
Vl + 3Vi + 1)4 = 

Vl + 3W3 + 2)4 = 
2)2 + 1)3 + 32)4 = 


= + 0".6; 
= -3 .1; 

= +4 .5; 
- + .8; 


Vl = + 0".10, 2)3 = 
!)2 = — 1 .13, Vl = 


-- + 1".41, 
= + .17. 



Ai = 65° 25' 18".20 

A2 = 75. 43 43 .97 

B3 = 47 26 13 .31 

Bi = 53 19 51 .97 



A =141° 09' 02".17; 

B = 100 46 05 .28; 

C = 67 08 28 .49; 

D = 50 56 24 .06. 



196. Quadrilateral Adjustment. The best method to use in 
adjusting a geodetic quadrilateral, Fig. 82, is the method of 
correlatives, Art. 167. In accordance with Art. 58 the adjusted 
angles must satisfy the following three angle equations: 

a + b + c + d+e+r+g + h=360° ] 

a+b=e+f ' .... (70) 

c+d=g+h ^ J 



APPLICATION TO ANGULAR MEASUREMENTS 327 



and also the following side equation : 

sin a sin c sin e sin 5f_ 
sin 6 sin d sin /sin A ' 

which naay be written in the logarithmic form 

S log sin(a, c, e,g)— S log sin(6, d,j, h) 



(71) 




(72) 



Fig. 82. — ^The Geodetic Quadrilateral. 

Letting Ma, Mj,, etc., represent the measured values of the 
angles a, b, etc., and h, h, h, h, represent the discrepancies in 
these equations due to the errors in the measured angles, we have 

S(M„toMA)-360° = Zi 

S log sin (ikf„, M„ Me, Mg) - 2 log sin {M^, Ma, Mf, M^) = h - 

The corrections Va, v^, etc., to be added algebraically to the 
measured values Ma, Mj,, etc., must reduce these equations to 
zero in order that the conditional equations (70) and (71) may be 
satisfied. Therefore we must have 

Va-\- Vi+ Vo + Vd+ Ve+ V/+ Vg+ Vh= -li ' 
Va+ Vb - Ve- Vf = -h 

Vc + Vd — T^g— ^h= -h 

daVa—diVb + dcVc—ddVd + deVe-dfV/+dgVg-dhVh= —h ■ 

in which Va, Vf,, etc., are to be expressed in seconds, and in which 
da, db, etc., are the tabular differences for one second for the 



(73 



328 



GEODETIC SUEVEYING 



log sin Ma, log sin Mi,, etc. If any angle is greater than 90° it 
is evident that the corresponding tabular difference must be 
considered negative, since the sine will then decrease as the angle 
increases in value. The conditional Eqs. (73) being in the form 
of Eqs. (43), the most probable values of Va, Vt, etc., may now 
be found by the method of correlatives (Art. 167), by means of 
Eqs. (49) and (50). Re-writing these equations with the symbols 
used in the present article, and remembering that there are four 
conditional equations and hence four correlatives required, we 
have in the general case, from Eqs. (49) and (73), 



, ^a^ J y^ab „ac „ad 

P P P P 



■h 



k^^"b b^ be bd^_ 

P P P P 

kii:-+k2 2- + ksJl- + ki^- = -h 

p p P p 



kX- + k2i:^-^ + ksii'-^ + k,i:'^=^ 



p p 

and from Eqs. (50) and (73), 

Va 



Vb 



h 



= «! h k2— 

Pa Pa 


Pa 


= ki \- A2 — 

Pb Pb 


-kj^ 

Pb 


Pc 


+ Ala- + ki^ 

Pc Pc 


Pd 


+ k,^ - k,^ 

Pd yd 


7 ^ 7 1 

= ki k2— 

Pe Pe 


+%* 


7 1 7 1 


-'4 

Pf 


Pa 


- kS + kiii 

Po Pg 



Vh = ki~- 
Ph 



k„- 



Ph J 



(74) 



(75) 



APPLICATION TO ANGULAR MEASUREMENTS 329 



(76) 



in which p„ represents the weight of lf„, p^ the weight of M^, and 
so on. 

In the case of equal weights we have, from Eqs. (73) and (74), 
8h + [ida + d,+de + dg) - {di, + dd + df+dh)]k4: = -h 

4k2+ {da -db -de + d/)ki = — h 

4A;3 + {dc-dd-dg+dh)ki = -h 

[(da + dc+de + dn) - {di,+dd + df+dk)]h 

+ {da-db-d^ + df)k2+{dc-da-dg+dh)k3+'2d^k4= -Z4 . 
and from Eqs. (75), 

Va = kl + k'j + daki 

1-% = ki + k2 — diki 
Vc = ki + ks + dcki 
'Vd = ki + ks — ddki 

Ve = ki — /C2 + deki 

Vf = ki — k^ — dfki 
Vo = ki — ks + dgki 
Vh = ki - ks - dhki 

Having found the values of Va, Vi, etc., we have in any case for 
the most probable values of the angles a, b, etc., 

a = Ma + Va-, e = Me + v^; 

b=Mb + vi,; f=M,+Vr; 

c = Mc + Vc-, g = Mg + Vg; 

d= Md + Va; h= Mu + Vh. 



(77) 



\ 



(78) 



197. Other Cases of Figure Adjustment. There is evidently 
no limit to the number of cases of figure adjustment that may be 
made the subject of consideration, but few of them are likely to 
be of interest to the civil engineer. Any case that may arise may 
be adjusted by the method of correlatives (Art. 167), similarly to 
the quadrilateral adjustment (Art. 196), provided the observa- 
tion equations and conditional equations are properly expressed. 
In any case the conditional equations must cover all the geo- 
metrical conditions which must be satisfied, and at the same time 
must be absolutely independent of each other. The number of 



330 



GEODETIC SURVEYING 






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APPLICATION TO ANGULAR MEASUREMENTS 331 

independent conditional equations can always be ascertained by 
subtracting the number of independent quantities from the number 
of observed quantities. The number of independent quantities 
is in general easily determined by an inspection of the given figure 
being that number of independent values which fixes a singl, 
location for each angular point. A study of the following exam 
pies will illustrate the principles involved. 



Example 1. Referring to Fig. 83, the base A B and the indicated angles 
have been measured; determine the number and nature of the independent 
conditional equations. 

It is evident from the figure that it will take two angles from the fixed 
points A and B to locate either C or D, and that these four angles are 
independent. We may therefore select Ai, Ai, B^, Bi, as independent 
angles, and as this will fix the points C and D it wiU also fix the values of 
the angles Ci and Cj, so that we can not have more than four independent 
angles. In this particular case any four of the angles can be taken as the 
independent ones, but this freedom of choice is not a general rule. As there 
are six observations of which only four are independent, it follows (Art. 166) 
that two independent conditional equations must be involved. Starting 
from any known side, A B, we may in general compute any other hne of a 
system through two different sets of triangles, and the requirement that these 
two results shall be identical will always lead to a corresponding side equa- 
tion. In the present case, therefore, the two conditional equations must 
consist of one angle equation and one side equation. The angle equation is 
evidently, 

Ai + ^2 + Bi + Bj + Ci + C2 = 180°. 

Taking C Z) as a convenient line from which to determine the side equation, 
and equating its values as computed through the triangles A B D and A C D, 
and through the triangless A B D and BCD, the side equation is easily 
found to be, 

sin A, sin Bi sin Ci = sin Ai sin Bi sin C2. 





Fig. 83. 



Fig. 84. 



Example B. Referring to Fig. 84, the base A B and the indicated angles 
have been measured; determine the number and nature of the independent 
conditional equations. 

In this case, as in the previous one, four independent angles will fix the 
whole figure, so that the fact that nine angles have been measured demands 
the existence of five independent conditional equations, as nine minus four 



332 



GEODETIC SURVEYING 



equals five. In regarding any four of the angles as independent, it is evident 
that no three of them must lie in any one triangle, as this would at once 
destroy the independence of these three angles by setting a condition on 
their sum. Since, as explained in Example 1, there must be one side equa- 
tion, on account of the one known Une A B, it follows that the present case 
must involve four independent angle equations to make up the total of 
five independent conditional equations required. An examination of the 
figure, however, furnishes five angle equations, as follows; 

Ai+Ci-\-Di = 180° 

Aj + Bi + Ds = 180° 

B2 + C, + D, = 180° 

Ai + ^2 + Bi + B2 + Ci + C2 = 180° 

Di-\-Di+D3 = 360° 

As there can be but four independent angle equations, it follows that 
any one of these five must be dependent on the other four. An examination 
of the equations will show at once that any one of them may be derived 
from the remaining four. We may therefore choose any four of these five 
equations for our four angle equations. Since the figure is identical with 
the one in Example 1, our side equation as before will be, 

sin Ai sin Bi sin Ci = sin Ai sin Bj sin C2. 

Example 3. Referring to Fig. 85, the base A B and the indicated angles 
have been measured, the interior station being a random point not purposely 

faUing on any diagonal of the figure; 
determine the number and nature of the 
independent conditional equations. 

In this case the angles in any five of 
the six triangles will fix the whole figure; 
and since there can be but two indepen- 
dent angles in each of the five triangles 
so selected, it foUows that we must have 
ten independent angles. As there are 
eighteen measured angles and ten inde- 
pendent angles, we must have eight inde- 
pendent conditional equations. As before 
there must be one side equation, leaving 
seven angle equations required. Eight such 
equations may be formed, to meet the con- 
ditions that six triangles must each contain 
180°, that the corner angles of the hexagon must add up to 720°, and that 
the central angles must add up to 360°. Any seven of these eight angle 
equations may be taken as the independent ones, when the requirement of 
the other one will also be satisfied. For the side equation we may com- 
pute any side, such as E D, by going around the figure in both directions 
from AB, from which it will appear, as in the previous examples, that 
the product of the sines of one set of alternate corner angles must equal 
the product of the sines of the other set of alternate corner angles. 




FiQ. 85. 



CHAPTER XV 

APPLICATION TO BASE-LINE WORK 

198. Unweighted Measurements. If a base line is measured 
from end to end a number of times in the same manner, and 
under such conditions that the different determinations of its 
length may be regarded as of equal weight, then (Art. 155) the 
arithmetic mean of the several results is the most probable value 
of its length. The probable error of a single measurement 
(Art. 173) is given by the formula 

n = 0.6745^/-^^ , (79) 

^n — 1 

and the probable error of the arithmetic mean (Art. 173) of n 
measurements by the formula 



^. = -^ = 0-6745^ ^,"^,, (SO) 



Vn ' \«(w - 1)' 



Example. Direct base-line measurements of equal weight: 
Observed Values v v^ 

6717.601ft. -0.025 0.000625 

6717 . 632 ft. + . 006 . 000036 

6717 . 645 ft. + . 019 . 000361 



3)20152 . 878 ft. Sw^ = . 001022 



z = 6717.626 ft. n = S 

± 0.0152 ft. 



n = 0.6745a/- 



001022 



r^ = 2:2152 ^ ^ ^^^^ ^^ 
V3 
Most probable value = 6717.626 ± 0.0088 ft. 

199. Weighted Measurements. >If a base line is measured 
from end to end a number of times in the same manner, but under 

333 



334 



GEODETIC SURVEYING 



such conditions that the different determinations of its length 
must be regarded as of unequal weight, then (Art. 157) the weighted 
arithmetic mean of the several results is the most probable value 
of its length. The probable error of a single measurement of 
unit weight (Art. 174) is given by the formula 



r-i = 0.6745 






(81) 



the probable error of any measurement of the weight p (Art. 174) 
by the formula 



rp=—^ = 0.6745 
Vp 



4. 



lipv^- 
p{n — 1) 



(82) 



and the probable error of the weighted arithmetic mean (Art. 174) 
by the formula 



rj,„ = -^ = 0.6745 



4 



Tipv^ 



Sp. {n — ly 



(83) 



Example. Direct base-Hne measurements of unequal weight: 



Observed Values 
7829.614 ft. 
7829.657 ft. 
7829.668 ft. 
7829.628 ft. 



pU 

7829.614 
15659.314 

7829.668 
23488,884 



- 0.026 
+ 0.017 
+ 0.028 
-0.012 



0.000676 
0.000289 
0.000784 
0.000144 



0.000676 
0.000578 
0.000784 
0.000432 



2p 



7 )54807.480 
^z= 7829.640 ft. 

n = 0.6745, 
0.0194 



Upv^ = 0.002470 



n = 4 



^/' 



002470 



± 0.0194 ft. 



n = 



rpa = 



V2 
0.0194 



± 0.0137 ft. 



^ = ± 0.0112 ft. 



V3 
0.0194 

VY 



± 0.0073 ft. 



Most probable value = 7829.640 ± 0.0073 ft. 

200. Duplicate Lines. In work of ordinary importance or 
moderate extent it is sufficient to measure a base line twice and 
average the results for the adopted length. When the same line 



APPLICATION TO BASE-LINE WORK 335 

is measured twice with equal care it is called a duplicate line. The 
rules of Art. 198 necessarily include duplicate lines, but this 
case is of such frequent occurrence that special rules are found 
convenient for the probable errors. Letting d represent the dis- 
crepancy between the two measurements, and remembering that 
the arithmetic mean is the most probable value, we have 

d , d 

''i = + 2 ^^^ ''s = - 2 • 

Substituting these values in Eq. (79) and replacing n with Vi for 
the case of duplicate lines, we have for the probable error of a 
single measurement of the length I, 

n = 0.4769Vd2 = ± 0.4769d. . . . (84) 

Substituting the same values in Eq. (80), we have for the probable 
error of the arithmetic mean, 

r„ = ± 0.3348 d; (85) 

Ta (approximately) = ± ^d (86) 



whence 



Example. Measurement of a duplicate base line: 

Observed Values 
4998.693 ft. 0.4769 X 0.034 = 0.0162. 

4998.659 ft. 0.3348 X 0.034 = 0.0114. 

d = 0.034 ft. 

r, = ± 0.0162 ft. ra= ± 0.0114 ft. 

Most probable value = 4998.676 ± 0.0114 ft. 

201. Sectional Lines. A base line may be divided up into 
two or more sections, and each section measured a number of 
times as a separate line. Each section, on account of its several 
measurements, will thus have a most probable length and a prob- 
able error independent of any other section of the line. If 
lijh, ■ • • ^K)be the most probable lengths of the several sections, 
then (Art. 168) the most probable length L for the whole hne, is 

L = h + h ... + ln = ^l (87) 

And if ri, /•2, . . . »•„, be the probable errors of the several values 
li, h, etc., then (Art. 182) the probable error tl for the whole 
line, is 

r-i = Vri2 + r-gZ . , . + r„2 = VSr^. . . . (88) 



336 GEODETIC SURVEYING 

Example. Sectional base-line measurement. Given 

k = 3816.172 ± 0.022 ft. 
k = 4122.804 ± 0.019 ft. 
h = 3641.763 ± 0.017 ft. 

L = 3816.172 + 4122.804 + 3641.763 = 11580.739 ft. 



r^ = \/(0.022)' + (0.019)^ + (0.017)^ = ± 0.034 ft. 
Most probable value L = 11580.739 ± 0.034 ft. 

202. General Law of the Probable Errors. In measuring a 
base line bar by bar or tape-length by tape-length, the case is 
essentially one of sectional measurement (Art. 201), in which 
each section is measured a single time, and in which each full 
section is of the same measured bar- or tape-length. If the con- 
ditions remain unchanged throughout the measurement, therefore, 
the probable error will be the same for each full section. As 
explained in Art. 180, however, this is not a case of computed 
values depending on a constant factor, so that the probable error 
of the whole line will not follow the law of that article. 
Let L = the total length for a line of full sections; 
tl = theprobableerror of this line; 
t = the length of the measuring instrument; 
Tt = the probable error for each length measured; 
n = the number of lengths measured; 



then (Art. 201) 






r^ = \/Sr2 = Vnr?. 


But evidently 






L 

" = T = 


whence 






- = Vr^^^^^ 



(89) 



Eq. (89) is derived on the assumption that only full bar- or tape- 
lengths are used. The fractional lengths that occur at the ends 
of a base (or elsewhere) form such a small proportion of the total 
length, however, that no appreciable error can arise by assuming 
Eq. (89) as generally true. A consideration of the various 
methods and instruments used in measuring base lines also shows 



APPLICATION TO BASE-LINE WORK 337 

that there is nothing in any case which can materially modify 
the truth of this equation. We may therefore write as a 

General Law : Under the same conditions of measurement the 
probable error of a base line varies directly as the square root of its 
length. 

From the manner in which this law has been derived it is 
evident that it is theoretically true whether the length assigned 
to;a base line is the result of a single measurement, or. the average 
of a number of measurements, so long as the lines being compared 
have all been measured in the same way. In cases where the 
given lines have been measured more than once, so that each 
line has its own direct probable error, we can not expect an exact 
agreement with the law. But this /-elation of the probable 
errors is more likely than any other that can be assigned, and 
hence shows the relative accuracy that may be reasonably expected 
in lines of different length. The chief point of interest in the law 
lies in the fact that the error in a base line is not likely to increase 
any faster than the square root of its length, so that the probable 
error where a line is made four times as long should not be more 
than doubled, and so on. 

Example. A base line measured imder certain conditions has the value 
7716.982 ± 0.028 ft. What is the theoretical probable error of a base line 
15693.284 ft. long, measured under the same conditions? 



0.028 J 



16693.284 ^^„_„3gg_ 



7716.982 
Theoretical probable error of new Une = ± 0.0399 ft. 

203. The Law of Relative Weight. In accordance with 
the law of the previous article, we may write for the probable 
error of a base line of any length 

rL = mVL, (90) 

in which m is a coefficient depending on the conditions of measure- 
ment. Also in accordance with the law of Art. 172, we may write 

1 

rz, = s — ^, 
Vp 

in which p is the weight assigned to the line and s is a coefficient 
depending on the unit of weight and the conditions of measure- 



338 GEODETIC SURVEYING 

ment. Since the xinit of weight is entirely arbitrary we may assign 
that value to p which will make s equal m, and write 

rL==m^ (91) 

Combining Eqs. (90) and (91), we have 

m\/L = m—i=L; 
Vp 



from which 



p = ^; (92) 



whence we have the 

General Law: Under the same conditions of measurement the 
weight of a base line varies inversely as its length. 

From the manner in which this law has been derived it is 
evident that it is theoretically true whether the length assigned 
to a base line is the result of a single measurement, or the average 
of a nvmiber of measurements, provided the lines compared have 
all been measured in the same way. 

If two or more base lines are measured under different con- 
ditions, they may be first weighted so as to offset this circum- 
stance, and then weighted inversely as their lengths. The 
relative weight of each line will then be the product of the weights 
applied to it. 

204. Probable Error of a Line of Unit Length. The probable 
error of an angular measurement conveys an absolute idea of its 
precision without regard to the size of the angle. The probable 
error of a base line, however, conveys no idea of the precision 
of the work imless accompanied by the length of the line. It is 
therefore convenient to reduce the probable error of a base line 
to its corresponding value for a similar line of unit length. A 
unit of comparison is thus established for different grades or 
pieces of work which is independent of the length of the bases. 
Such a unit has no actual existence, but is purely a mathematical 
basis of comparison. 

From Eq. (89) we have 

rL = -^VL. 

Vt 



APPLICATION TO BASE-LINE WOEK 339 

Hence, when L equals 1, we have for Tq, the probable error of a 
unit length of line, 

_ ''« 



whence in general 



tl = roVZ, (93) 



in which all the values refer to single measurements. From this 
equation we see that the probable error of any base line is equal 
to the square root of its length multiplied by the probable error 
of a imit length of such a line. If r^ is well determined for 
given instruments, conditions, and methods, Eq. (93) informs us 
in advance what is a suitable probable error for a single measure- 
ment, and hence (Art. 198) for the average of any number of 
measurements of a line of the given length L. The base-line 
party therefore knows whether its work is up to standard, or 
whether additional measurements are required. 

205. Determination of the Numerical Value of the Probable 
Error of a Line of Unit Length, From Eq. (93) we have, 



r-jr = Tq-ZL; 



whence 



vi <"'" 



So that in any case where the length of a line and the correspond- 
ing probable error are known, the formula determines a value 
for ro. In order for the value of ro to be reliable it must be based 
on many such determinations, but the expense prohibits many 
measurements of a long base line. As the law is known, however, 
which connects the values of the probable error for all lengths 
of line, it is just as satisfactory to determine ro from much shorter 
lines, which may be quickly and cheaply measured many times. 
The usual plan is to measure a series of duplicate lines, so that the 
probable error for a single measurement is known in each case 
from the discrepancy in each pair of lines. Since all results are 
reduced to the same unit length it is immaterial whether the 
different duplicate lines are, of equal length or not. 



340 GEODETIC SUEVEYING 

In accordance with Eq. (84) we have, for any single measure- 
ment of the duplicate line I, 

n = 0.4769\/d2; 
whence, in accordance with Eq. (94), 

but, in accordance with Eq. (92), we have for any length of line I 

1 ■ 

whence 

Vq = QA769Vpd^, (95) 

when determined from a single duplicate line. If a number of 
duplicate hues are measured we will have a corresponding number 
of values {ro)i, (''0)2, etc., based on the discrepancies di, d2, etc., 
of the several duplicate lines. It might as first be supposed that 
the average value of these determinations of ro would best repre- 
sent the result of all the measurements. What is really wanted, 
however, is that value of rg which gives equal recognition to the 
conditions which caused its different values. A just recognition 
of each value of ro, therefore, will require us to consider equal 
sections of any line as having been measured respectively under 
those conditions that produced the several values of Tq. The 
probable error for the whole line is then found from the probable 
errors of the different sections, and this result reduced to the 
probable error of a unit length. 

Let n = the number of values (ro):, (ro)2, etc.; 
L = the length of any given line; 

whence the required equal sections will be 

a = (^) = etc. ^^, 
\n/i \nj 2 n 

and, in accordance with Eq. (93), 

r ^^^ = (r-o) lyg, r ^^ = {r,) 2 ^^, etc. ; 



APPLICATION TO BASE-LINE WORK 
whence, in accordance with Eq. (88), 

and, in accordance with Eq. (94), 

""o- V~rr' 

but, in accordance with Eq. (95), 

(ro)i = 0.4769V pd?, (ro)2 = 0.4769 V^"?, etc.; 

Sro2 = (0.4769)2Spd2; 



341 



(96) 



so that 
whence 



r-Q = 0.4769 






(97) 



when determined from a number of duplicate lines. In using 
formulas (95) and (97) it is to be remembered that d is the dis- 
crepancy in any duplicate line, p is the weight (reciprocal of the 
length) of that line, n is the number of duplicate lines, and ro 
is the probable error of a single measurement of a line of unit 
length. 

Example. Determination and application of the probable error of a base 
line of unit length : 



Duplicate Lines 
512.017 ft. 
512.011 " 




d 
0.006 


d2 
0.000036 


p 


0.0000000703 


619.184 ft. 
619.176 " 




0.008 


0.000064 


eio 


0.0000001034 


750.962 ft. 
750.971 " 




0.009 


0.000081 


Th 


0.0000001079 


619.180 ft. 
619.184 " 




0.004 


0.000016 


619 


0.0000000258 


750.960 ft. 
750.972 " 




0.012 


0.000144 


1 
761 


0.0000001917 


from which we have 












S 
whence 


0.4 


= 0.0000004991 < 


md n 
= ± O.OC 


= 5; 


To = 


^cn../ 0.0000004991 
769'V 


)0151 ft., 



342 GEODETIC SURVEYING 

which is therefore the probable error for a single measurement of one foot 
made under the given conditions. For a single measurement of a base 
line of any length L, therefore, made under these same conditions, the 
probable error would be, in accordance with Eq. (93), 

Tl = 7-0 Vl= ± 0.000151 Vl ft. 

Thus if L is 10,000 feet, we would have 



ri. = ± 0.000151 X VlOOOO = ± 0.0151 ft. 

And if such a line were measured four times we should have, theoretically, for 
the probable error of the average length, 

ra= ± 0.0151 -r- vT = ± 0.0076 ft. 

It thus becomes known in advance what probable error is to be expected 
under the given conditions. 

206. The Uncertainty of ^ Base Line. By the uncertainty 
of a base line is meant the value obtained by dividing its probable 
error by its length. In accordance with Art. 202, the probable 
error of a base line varies as the square root of its length, so that 
the probable error increases much more slowly than the length 
of the line. On account of the greater opportunity for the 
compensation of errors, therefore, long lines are relatively more 
accurate than short lines. While the unit probable error r^ 
very satisfactorily indicates the grade of accuracy, whether a 
line be long or short, it does not furnish any idea of the degree of 
accuracy with which the length of a given line is known. The 
uncertainty of a base line, however, shows at once the precision 
attained in its measurement. If ri be the probable error of a 
single measurement of a base line whose length is I, then for the 
uncertainty C/j of a single measurement, we have 

and for the uncertainty U^ of the arithmetic mean of n measure- 
ments, 

f/„ = r? = _!i_. 

But, in accordance with Eq. (93), 

n = roVT; 



APPLICATION TO BASE-LINE WORK 343 

whence 



and 



so that we may write, 



and 



_r-o\/r_ rp ' 
' I VJ' 



IV n Vrii 



^^=l = Vf' (^^) 



C/„ = ^=-^ (99) 

' Vnl 

Example 1. Three measurements of a base line under the same con- 
ditions give z = 6716.626 ± 0.0088 ft. and n = ± 0.0152 ft. What is the 
uncertainty of a single measurement and also of the arithmetic mean? 

rr Ti 0.0152 1 



Ua 



I 6717.626 441949' 

ra ^ 0.0088 ^ 1 
I 6717.626 763366' 



Example 2. A base line of 10,000 ft. length is to be measured four times 
under conditions which make the probable error of a unit length of line 
equal ± 0.000316 ft. What should be the uncertainty of each measurement 
and of the average of the four measurements? 

„ To 0.000316 1 

fi = 



Ua = 



VT VlOOOO 316456' 
To 0.000316 1 



vW V 40000 632912 



CHAPTER XVI 
APPLICATION TO LEVEL WORK 

207. Unweighted Meastirements. If the difference of ele- 
vation of two stations is msiasured a number of times in the same 
manner, over the same length of line, and under such conditions 
that the different determinations may be regarded as of equal 
weight, then (Art. 155) the arithmetic mean of the several results 
is the most probable value of this difference of elevation. The 
probable error of a single measurement (Art. 173) is given by the 
formula 



r-i = 0.6745 J;^73^, (100) 

and the probable error of the arithmetic mean (Art. 173) of n 
measurements by the formula 



■a = ~ = 0.674:5 J .^"^ ,, . . . . (101) 



i(n - 1)' ■ ■ 
Example. Difference of elevation by direct observations of equal weight 



Observed Values 


V 


1|2 


11.501ft. 


+ 0.009 


0.000081 


11.509 ft. 


+ 0.017 


0.000289 


11.480 ft. 


- 0.012 


0.000144 


11.478 ft. 


-0.014 


0.000196 


4)45.968 ft. 




Sd^ = 0.000710 


= 11.492 ft. 




n = 4 



n = 0.6745 J' 



'0-0007J0^^„^„^^^^^_ 



., = 0^4 ^±0.0052 ft. 

\/4 

Most probable value = 11.492 ± 0.0052 ft. 

344 



APPLICATION TO LEVEL WOEK 345 

208. Weighted Measurements. If the difference of eleva- 
tion of two stations is measured a number of times in the same 
manner, and over the same length of line, but under such condi- 
tions that the different determinations must be regarded as of 
unequal weight, then (Art. 157) the weighted arithmetic mean of 
the several results is the most probable value of this difference of 
elevation. The probable error of a single measurement of unit 
weight (Art. 174) is given by the formula 






n = 0.6745^^^, (102) 

the probable error of any measurement of the weight p (Art. 174) 
by the formula 



r, = -^ = 0.6745 /_^F!!^, .... ao3) 
vp \ pin — 1) 

and the probable error of the weighted arithmetic mean (Art. 174) 
by the formula 



;^ = 0.6745 J^.^^. . . . (104) 



VSp ■ \ ^Pin - 1) 



Example. Difference of elevation by direct observations of unequal 
weight: 

Observed Values p pM v v^ pv^ 

17.643 ft. 1 17.643 -0.028 0.000784 0.000784 

17.647 ft. 1 17.647 -0.024 0.000576 0.000576 

17 679 ft. 2 35.358 +0.008 0.000064 0.000128 

17.683 ft. 3 53.049 +0.012 0.00 0144 0.000432 

Sp = 7 ) 123.697. tpv^ = 0.001920 

z = 17.671 n = 4 



^4- 



n = 0.6745. /^^^5^5?2. = ± 0.0171 ft. 



r, = ''-:^=±Qm2Ut. 



n = 



^•na 



V2 
0.0171 

0.0171 



= ± 0.0099 ft. 



= ± 0.0064 ft. 



Most probable value = 17.671 ± 0.0064 ft. 



346 GEODETIC SUEVEYING 

209. Duplicate Lines. In precise level work a duplicate line 
of levels is understood to mean a line which is run twice over the 
same route with equal care, but in opposite directions. The 
object of running in opposite directions is to eliminate from the 
mean result those systematic errors which are liable to occur in 
leveling, due to a risiag or settling of the instrument or tiu-ning 
points during the progress of the work. As explained in Art. 88 
the details of the work are so arranged that these errors tend to 
neutralize each other to a large extent as the work progresses, so 
that no material error is committed by assuming that the results 
obtained are affected only by accidental errors. The most prob- 
able value for the difference of elevation of any two stations, 
based on a duplicate line, is equal to the average of the two results 
furnished by such a line. Letting d represent the discrepancy 
between the result obtained from the forward line and that 
obtained from the reverse line, we thus have 

d , d 

vi = +-2 and V2 = - ^. 

Substituting these values in Eq. (100) and replacing n with r, 
for the case of duplicate lines, we have for the probable error 
of a single determination (forward or reverse) by a line of the 
length I, 

ri = 0.4769\/d2 = 0A769d (105) 

Substituting the same values in Eq. (101), we have for the 
probable error of the arithmetic mean of the results obtained by 
the forward and reverse lines, 



whence 



r„ = 0.3348d; (106) 

Ta (approximately) = id (107) 



Example. Duplicate liiie of levels: 

Observed Values 
29.648 ft. 0.4769 X 0.028 = 0.0134. 

29.676 ft. 0.3348 X 0.028 = 0.0094. 

d = 0.028 ft. 

r; = ± 0.0134 ft. Ta = ± 0.0094 ft. 

Most probable value = 29.662 ± 0.0094 ft. 



APPLICATION TO LEVEL WORK 347 

210. Sectional Lines. Every line of levels which includes 
one or more intermediate bench marks may be regarded as made 
up of a series of sections connecting these bench marks. In 
general the work will be done by the method of duplicate leveling 
(Art. 209), so that a value for the difference of elevation of any two 
successive bench marks (limiting a section) will be obtained from 
the forward line, and another value from the reverse line. From 
these two values (Art. 209) we will have a most probable value 
and a probable error for any given section, which will be independ- 
ent of all other sections. In whatever manner the leveling may 
be done, however, the subsequent treatment of the results will be 
the same, provided the determinations for each section are kept 
independent. If ei, 62, . . . en, be the most probable values for 
the difference of elevation between the successive bench marks, 
then (Art. 168) the most probable difference of elevation E 
between the terminal bench marks, is 

^ = ei + 62 . . . +en = 2e. . . . (108) 

And if ri, r2, . . . Vn, be the probable errors of the several values 
ei, 62, etc., then (Art. 182) the probable error r^j for the total dif- 
ference of elevation E, is 



rE 



Vri2 + r-a^ . . . + r-„2 = vTr^. . . . (109) 



Example. Level work on sectional lines. Given 

ei = 9.116 ± 0.008 ft. 

62 = 31.659 ± 0.031 ft. 

63 = 22.427 ± 0.018 ft. 

E = 9.116 + 31.659 + 22.427 = 63.202 ft. 

r^ = V(0.008)2 + (0.031)'! + (o.018y = ± 0.037 ft. 

Most probable value E = 63.202 ± 0.037 ft. 

211. General Law of the Probable Errors. In measuring 
the difference of elevation between any two bench marks by pass- 
ing (in the usual way) through a series of turning points, the case 
is essentially one of sectional measurement (Art. 210), in which the 
difference of elevation for each section is measured a single time, 
and in which under similar conditions the average distance 
between turning points may be assumed to be the same for any 
length of line. Running a line of levels is thus entirely analogous 



348 GEODETIC SURVEYING 

to measuring a base line, and hence the same laws must hold good. 
In accordance with Art. 202, and without further demonstration, 
we may therefore write as a 

General Law: Under the same conditions of measurement 
the probable error of a line of levels varies as the square root of its 
length. 

From the considerations on which this law is based it is evident 
that it is theoretically true whether the difference of elevation 
assigned to the terminals of a line is the result of a single measure- 
ment, a number of measurements, or a duplicate measurement, so 
long as the lines being compared are all identical in these details. 

Example. A line of levels 10 miles long has a probable error of ± 0.156 ft. 
What is the theoretical value of the probable error for a Mne 60 miles long, 
run under the same conditions? 

0.156 V|5 = 0.156 V6"= ± 0.382 ft. 

Theoretical probable error of new line = ± 0.382 ft. 

212. The Law of Relative Weight. As explained in the 
previous article, the laws derived for base-line work are equally 
applicable to level work. In accordance with Art. 203, and with- 
out further demonstration, we may therefore write as a 

General Law : Under the same conditions of measurement the 
weight of the result due to any line of levels varies inversely as the 
length of the line. 

From the considerations on which this law is based it is evident 
that it is theoretically true whether the difference of elevation 
assigned to the terminals of the line is the result of a single meas- 
urement, a number of measurements, or a duplicate measurement, 
so long as the lines being compared are all identical in these 
details. 

If two or more level lines are run under different conditions, 
they may be first weighted so as to offset this circumstance, and 
then weighted inversely as their lengths. The relative weight of 
each line will then be the product of the weights applied to it. 

213. Probable Error of a Line of Unit Length. The probable 
error corresponding to a given line of levels conveys no idea of the 
precision of the work unless accompanied by the length of the line. 
It is therefore convenient to reduce the probable error of a line of 
levels to its corresponding value for a similar line of unit length. 



APPLICATION TO LEVEL WORK 349 

A unit of comparison is thus established for different grades or 
pieces of work which is independent of the length of the lines. 
Such a unit has no actual existence, but is purely a mathematical 
basis of comparison. 

As explained in Art. 211, the laws derived for base-line work 
are equally applicable to level work. In accordance with Art. 204, 
and without further demonstration, we may therefore write 

r^ = ?-oVL, (110) 

in which tl is the probable error for a given line of levels of the 
length L, Vq is the probable error for a unit length of such a line, 
and in which all the values refer to single measurements. This 
equation indicates that the probable error of any given line of 
levels is equal to the square root of its length multiplied by the 
probable error for a unit length of such a line. If Tq is well deter- 
mined for given instruments, conditions, and methods, Eq. (110) 
informs us in advance what is a suitable probable error for a 
single line of levels, and hence (Art. 207) for the average result 
obtained by re-running such a line any number of times. In 
accordance with this article the probable error in the mean result 
of a duplicate line is equal to the second member of Eq. (110) 
divided by V2. In any case, therefore, the level party knows 
whether its work is up to standard, or whether additional measure- 
ments are required. 

214. Determination of the Numerical Value of the Probable 
Error of a Line of Unit Length. As explained in Art. 211, the 
laws and rules for base-line work are equally applicable to level 
work. The method of Art. 205 is consequently adapted to the 
present case by running ofte or more duplicate level lines of 
moderate length, and noting the length of line (one way) and the 
discrepancy for each duplicate line. In accordance with Eq.(97), 
and without further demonstration, we may therefore write 



0.4769 



\IS., (Ill) 

\ n 



in which Tq is the probable error in running a single line of levels 
of unit length, d is the discrepancy in any duplicate line, p is 
the weight (reciprocal of the one way length) of that line, and n 
is the number of duplicate lines. 



350 



GEODETIC SURVEYING 



Example. Determination and application of the probable error of a 
level line of unit length: 



Difference of Elevation d 


d' 


I 


p 


pd2 


16.298 ft. 
16.314" 


0.016 


0.000256 


810 


^ 


0.0000003160 


16.308 ft. 
16.296" 


0.012 


0.000144 


810 


8^0 


0.0000001778 


18.540 ft. 
18.549" 


0.009 


0.000081 


560 


1 

560 


0.0000001446 


18.552 ft 
18.542" 


0.010 


0.000100 


560 


beo 


0.0000001786 


21.663 ft. 
. 21.648" 


0.015 


0.000225 


782 


ih 


0.0000003085 


21.661ft. 
21 649 " 


0.012 


0.000144 


782 


^ 


0.0000001841 


21.664 ft. 
21.650" 


0.014 


0.000196 


782 


Th 


0.0000002506 



from which we have 



whence 



Xpd" = 0.0000015602 and n = 7; 



r„ = 0.4769 



V°= 



1.0000015602 



= ± 0.000225 ft., 



which is therefore the probable error in running a single line of levels for 

a distance of one foot under the given conditions. For a single line of levels 

of any length L, run under the same conditions, the probable error would 
be, in accordance with Eq. (110), 

TL =roVL = ± 0.000225^1" ft. 

Thus if L is 10,000 feet, we would have 

Tl = ± 0.000225-^/10000 = ± 0.0225 ft. 

And if such a line of levels were run four successive times we should have, 
theoretically, for the probable error of the average difference of elevation, 

r-a = ± 0.0225 -^ Vi" = ± 0.0113 ft. 

It thus becomes known in advance what probable error is to be expected 
under the given conditions. 

215. Mtiltiple Lines. By a multiple line of levels is meant a 
set of two or more lines connecting the same two bench marks 
by routes of different length. In order to find the most probable 
value for the difference of elevation between the terminals of a 
multiple line, it is neces.sary (Art. 212) to weight each constituent 
line inversely at its length. If the character of the work requires 
any of the lines to be also weighted for other causes, then the 



APPLICATION TO LEVEL WOEK 351 

final weight of such hne must be taken as the product of its indi- 
vidual weights. Having weighted the several lines as thus explained 
the case becomes identical with any case of weighted measure- 
ments (Art. 208), and hence the probable error of a single measure- 
ment of unit weight is given by the formula 



'•i 



= '-''^'yl^'' ^112) 



the probable error of any of the lines of the weight p by the 
formula 

and the probable error of the weighted arithmetic mean by the 
formula 



^^ = 0.6745 Jy^^y . 

V:^p MZpin - 1) 



V = -^1= = 0.6745 V ^^,: _,, . . . (114) 

._ 5 Miles 

.2i^ Miles. 





~ 3% MUes - 
Fig. 86. 

Example. Three lines of levels, as shown in Fig. 86, give the following 
results : 

A to B, 5 mile line, + 95.659 ft. 
A to B, 2i mile Hne, + 95.814 ft. 
A to B, 3i mile Hne, + 95.867 ft. 

The elevation of A is 416.723 feet. What is the most probable value for 
the elevation of B, and the probable error of this result? 



M 
95.659 
95.814 
95.867 


p pM 

0.2 19.1318 
0.4 38.3256 
0.3 28.7601 


V 

- 0.138 
+ 0.017 
+ 0.070 


»2 

0.019044 
0.000289 
0.004900 


pu2 

0.0038088 
0.0001156 
0.0014700 


Sp 


= 0.9J86.2175 
95.797 

rpa = 0.6745-1 




n = 
± 0.0369 ft. 


: 0.0053944 
3 




/0.OO53944 _ 





416.723 + 95.797 = 512.520 ft. 
Most probable value for elevation of B = 512.520 ± 0.0369 ft. 



352 



GEODETIC SURVEYING 



216. Level Nets. When three or more bench marks are 
interconnected by level lines so as to form a combination of 
closed rings, the resulting figure is called a level net. Fig. 87 
represents such a level net, involving nine bench marks. The 
elevation of any bench mark is necessarily independent of any 
other bench mark, but the differences between the elevations of 
adjacent bench marks are not independent quantities, since in 
any closed circuit their algebraic sum must equal zero. In the 
given figure there are evidently fifteen observation equations, 
namely, the observed difference of elevation between A and B, 
B and C, etc. But there are also seven closed rings, ABCD, ADA, 

etc., forming seven independent condi- 
tional equations. Fifteen minus seven 
leaves eight, so that (Art. 166) there 
can be but eight independent quanti- 
ties involved in the fifteen observation 
equations. The number of indepen- 
dent quantities must evidently be one 
less than the number of bench marks, 
since one of these must be assumed as 
known or fixed, and nine minus one 
gives eight as before. It sometimes 
happens that more than one line con- 
nects the same two points, as between 
A and D in the fi ure; but this fact 
makes no difference in the method of 
computation. Sometimes a point B 
occurs on a line without being coimected with any other point. 
Such a point has no influence on the adjustments of any other 
point, and may be included or omitted, as preferred, in making 
such other adjustments. If omitted ^in adjusting the other 
points its own most probable value can be found afterwards 
by Art. 217. 

There are two general methods of making the computations 
for the adjustments of a level net, each of which may be modified 
in a number of ways. In the first method the most probable 
values are found for the several differences of elevation between 
the bench marks, the most probable values for the elevations of 
the different bench marks being then found hy combining these 
differences. In the second method the computations are arranged so 




APPLICATION TO LEVEL WOEK 



353 



as to lead directly to the most probable values for the elevations 
of the bench marks. In any case each of the connecting lines 
must be properly weighted. If the lines are all run singly they 
are weighted inversely as their lengths unless some special con- 
dition requires some of these weights to be modified. If all the 
lines are duplicate lines, the average difference of elevation in 
each case may be treated as if due to a single line, and weighted 
inversely as its length. If special conditions exist the weights 
must be made to correspond. The manner 
in which each method is worked out is 
illustrated by the following example. 

Example. Referring to the level net indicated 
in Fig, 88, the field notes show the following 
results: 

AtoB = + 11.841 ft. 



= + 



Bto C 
C toD 

Dto E = - 

EtoA 

B to E -= - 

C toE = + 



5.496 ft. 
8.207 ft. 
6.720 ft. 
8.515 ft. 
3.218 ft. 
2.619 ft. 




The figures on the diagram are the lengths in miles 
of the various lines. The arrow-heads show the 
direction in which each hne was run. The eleva- 
tion of the point A is 610.693 ft. What are the 
most probable values for the elevations of the re- 
maining stations? 

First method. As there are but four unknown 
bench marks (5, C, D, E), there can be but four in- 
dependent unknowns in the observation equations. 

As the lines AB, BC, CD, DE, may evidently be selected as the independent 
unknowns, we may write for the most probable values of the corresponding 
differences of elevation 

AtoB=+ 11.841 + vi; 

B toC = - 5.496 4- W, 

C toD= + 8.207 + V3; 

DtoE = - 5.720 + U4. 

The conditional equations involved in the several closed circuits may then 
be avoided (Art. 165) by writing all the observation equations in terms of 
these quantities. Writing the reduced observation equations (Art. 163) 
directly from the figure, we have, by comparison with the observed values, 

(A to B) vi = 0.000 (weight 0.4) 

{B to C) Vi = 0.000 (weight 0.3) 

(C to D) Vi = 0.000 (weight 0.4) 

(D to E) Vi = 0.000 (weight 0.3) 

\e to A) -V - vi - V3 - Vi = + 0.317 (weight 0.2) 

(B to E) V2 + V3 + Vi= - 0.209 (weight 0.5) 

(C to E) V3+Vi = + 0.132 (weight 0.5) 



354 GEODETIC SURVEYING 

As an illustration of how these equations are formed let us consider the 
observed line CE. 

Most probable value, C to D = + 8.207 + Vt. 

Most probable value, Dto E = - 5.720 + Vt. 

Hence, by addition, 

Most probable value, C to ^ = + 2.487 + % + Vi. 
Observed value, C to E = + 2.619. 

Hence this observation equation requires 

Vi+Vi = + 0.132. 

No values of Vi, %, Vs, vt, can meet the requirements of all the observation 
equations, and hence to find the most probable values of Vi, v^, Va, vt, we 
form the normal equations in the usual way, giving, 

0.6wi + 0.2!;2 + 0.2W3 + 0.2^4 = - 0.0634 

0.2iii + l.Owj + 0.7w3 + 0.7u4 = - 0.1679 

0.2i;i + 0.7v2 + 1.6ws + 1.2w4 = - 0.1019 

0.2di + 0.7w2 + 1.2 vs+ 1.5K4 = - 0.1019 

whose solution gives 

vi = - 0.0556 ft.; «3 = + 0.0092 ft.; 

Vi= - 0.1718 ft.; vi = + 0.0123 ft.; 

whence, for the most probable values, we have 

AtoB = + 11.7854 ft. 

B to C = - 5.6678 " A = 610.693 ft. 

C to D = + 8.2162 " B = 622.478 " 

DtoE=- 5.7077" 0=616.811" 

EtoA = - 8.6261 " D = 625.027 " 

BtoE=- 3.1593" .E = 619.319" 

CtoE = + 2.5085" 

Second method. In this method we first find approximate values for the 
unknown elevations by combining the observed values in any convenient 
way, thus: 

A = 610.693 C = 617.038 (approx.) 

+ 11.841 + 8.207 



B = 622.534 (approx.) D = 625.245 (approx.) 

- 5.496 - 5.720 



C = 617.038 (approx.) E = 619.525 (approx;) 

and then write, for the most probable values, 

A = 610.693; 
B = 622.534 + Vi; 
C = 617.038 + t)2; 
D = 625.245 + us; 
E = 619.525 + Vi. 



APPLICATION TO LEVEL WOEK 



355 



Substituting these values in the observation equations, we have 
Ato B = + 11.841 +vi = + 11.841: 



BtoC = - 


5.496 


-«! + %= - 


5.496 


C to D = + 


8.207 


- f 2 + Va = + 


8.207 


DtoE = - 


6.720 


— Va + Vi = — 


5.720 


E toA = - 


8.832 


-Vi = - 


8.515 


BtoE = - 


3.009 


-Vi+Vi= - 


3.218 


CtoE = + 


2.487 


- % + Wd = + 


2.619. 



Eeducing and weighting inversely as the distances, we have 

vi = 0.000 (weight 0.4) 

-vi + Vi = 0.000 (weight 0.3) 

-vi + vi = 0.000 (weight 0.4) 

-V3 + Vi= 0.000 (weight 0.3) 

- v,= + 0.317 (weight 0.2) 

-Vi +Vi = - 0.209 (weight 0.5) 

-V2 +Vi = + 0.312 (weight 0.5) 

Forming the normal equations, we have 

1.2j)i - O.Sfz - 0.5w4 = + 0.1045 

- 0.3i;i + 1.2i;2 - OAvz - 0.5t)4 = - 0.0660 

- 0Av2 + 0.7v3 - 0.3w4 = 0.0000 

- 0.5vi - 0.5z)2 - 0.3% + 1.5w4 = - 0.1019 

whose solution gives 

1-1 = - 0.0566 ft.; V3 = - 0.2182 ft.; 

V2= - 0.2274 " Vi= - 0.2069 " 

whence, for the most probable values, we have (as before) 

A = 610.693 ft. 
B = 622.478" 
C = 616.811" 
D = 625.027" 
E = 619.319" 

217. Intermediate Points. By an inter- 
mediate point is meant one lying only on 
a single line of levels, and hence having 
n9 influence on the general adjustment. 
Thus in Fig. 89 the bench marks A and B 
are adjusted as a part of the complete level 
net ABCDEFG. The point I is an inter- 
mediate point, having no influence on the 
general adjustment, but simply lying be- 
tween the djusted bench marks A and B. 
In adjusting level net it s not necessary 
to separate the intermediate points from the others, as the 
results will come out the same whether any or all of the inter- 
mediate points are omitted or included. The work of compu- 




FiG. 89. 



356 GEODETIC SURVEYING 

tation may be reduced, however, where there are many inter- 
mediate points, by adjusting the main system first and the inter- 
mediate points afterwards. Referring to Fig. 89, page 355, 

Let I be an intermediate point lying between the adjusted 
bench marks A and B; 
a = the distance A to 7; 
b = the distance I to B; 

d = the discrepancy between the line AB as run and the 
difference between the adjusted values of A and B 
(+ if the line as run makes B too high) ; 
e = observed change in elevation from A to I; 
e' = observed change in elevation from I to B; 
then 

A+e + e' = B + d, 



or 
and 



e' = B - A - e + d; 

I (observed) = A + e (weight b) ; 

I (observed) =B — e' = A+e — d (weight a); 



or, taking the weighted arithmetic mean, 

bA + be + aA + ae — ad 



b + a 



I (most probable) 



As / represents any intermediate point, and a the corresponding 
distance from the commencement A of the given 
line, it follows from this equation that the most 
probable values for any intermediate points are 
U Miles, arrived at by adjusting for the discrepancy d in 
direct proportion to the distances from the initial 
point A. This law may be otherwise expressed 
by saying that the discrepancy is to be distributed 
uniformly along the line on the basis of dis- 
^^™«^:, tance. 

Example. In the line of levels indicated in Fig. 90 the 

field notes show the following changes in elevation: 
V 2 Miles. 

AtoB = + 2.626 ft. 
^A BtoC = - 3.483" 

Fig. 90. C to D = +6.915" 



APPLICATION TO LEVEL WOEK 



357 



The adjusted elevations at A and D are 

A = 28.655 ft. 
D = 34.317" 

What are the most probable elevations of the intermediate points B and C? 
28.655 
+ 2.626 

Discrepancy = + 0.396 ft. Total distance = 9 miles. 

31.281 0.396 X f = 0.088 ft. 0.396 X | = 0.220 ft. 

- 3.483 



27.798 
+ 6.915 

34.713 
34.317 



ition 


Apparent Elevation 


Correction 


Adjusted Elevation 


A 


28.655 


0.000 


28.655 ft. 


B 


31.281 


- 0.088 


31.193" 


C 


27.798 


- 0.220 


27.578" 


D 


34.713 


- 0.396 


34.317" 



+ 0.396 

218. Closed Circuits. By a closed circuit in level work is 
meant a line of levels which returns to the initial point, or, in 
other words, forms a single closed ring. The shape of such a circuit 
is entirely immaterial, whether approxi- 
mately circular, narrow and elongated, 
or irregular in any degree. A level net 
is in general a combination of closed 
circuits, but these circuits can not be 
adjusted separately, as they are not 
independent. So also if any part of 
the ring is leveled over more than once 
it becomes essentially a level net, and 
must be adjusted accordingly. If, how- 
ever, the circuit is independent of all 
other work, and has been run around but once under uniform 
conditions, it may be adjusted by a simpler process. Referring 
to Fig. 91, 

Let A, B, C, D, E be the bench marks on an independent 
closed circuit; 
A = the initial bench mark; 

a = distance A-B-C to any point C; 

b = distance C-D-E-A back to A ; 

d =1= discrepancy on arriving at A ( + if too high) ; 




Fig. 91. 



then 



e = observed change in elevation from A to C; 
e' = observed change in elevation from C to A; 

A + e + e' = A + d, 



358 



GEODETIC SURVEYING 



or 



and 



e'= — e + d; 



C (observed) = A + e (weight b) ; 

C (observed) = A —e' = A + e — d (weight a); 

or, taking the weighted arithmetic mean, 

bA + be + aA + ae — ad 



C (most probable) 



= {A + e)- 



b + a 
a 



a + b 



(116) 



As C represents any point in the circuit, and a the corresponding 
distance from the initial point A, it follows from this equation 
that the most probable values for the elevations of any points 
B, C, D, E, etc., are arrived at by adjusting the observed eleva- 
tions for the discrepancy d directly as the respective distances 
from the initial point. This law may be otherwise expressed by 
saying that the discrepancy is to be distributed uniformly around 
the circuit on the basis of distance. 

Example. In the closed line of levels indicated in Fig. 91, page 357, the 
field notes show the following changes in elevation: 



AtoB = - 2.176 ft., 
BtoC=+ 6.481 ft., 

C to D ^ 1.712 ft., 

DtoE = - 4.820 ft., 
EtoA = + 2.017 ft.. 

Given the elevation of A as 47.913 feet, what are the adjusted elevations 
around the line? 
47.913 
- 2.176 



distance = 3 miles, 
distance = 1 mile, 
distance = 2 miles, 
distance = 2 miles, 
distance = 3 miles. 



45.737 
+ 6.481 

52.218 

- 1.712 

50.506 

- 4.820 

45.686 
+ 2.017 

47.703 
47.913 



Discrepancy = — 0.210 ft. Total distance = 11 miles. 
0.210 X A = 0.057 ft 0.210 X t\= 0.105 ft. 

0.210 X A = 0.076 ft. 0.210X t\= 0.153 ft. 



Station Apparent Elevation Correction Adjusted Elevation 



A 


47.913 


0.000 


47.913 ft. 


B 


45.737 


+ 0.057 


45.794" 


C 


52.218 


+ 0.076 


52.294 " 


D 


60.506 


+ 0.105 


50.611 " 


E 


45.686 


+ 0.153 


45.839" 



0.210 



APPLICATION TO LEVEL WOEK 



359 



219. Branch Lines, Circuits, and Nets. Any level line, circuit, 
or net that is independent of another 
system except for one common point, 
is called a branch system. Thus in 
Fig. 92 the dotted lines represent the 
original system, ABCD a branch line, 
HKLMN a branch circuit, and PRSTV 
a branch net. In adjusting the main 
system the results will be the same 
whether any or all of the branch sys- 
tems are included or omitted. If 
there is much branch work, however, 
the labor of computation may be re- 
duced by adjusting the main system 
first and the branch systems after- 
wards. When the main system is 
adjusted the elevations ofA,H, P, etc., 

become fixed quantities which must not be disturbed in adjusting 
the branch systems. 




FiQ. 92. 



TABLES 



TABLES 



TABLE I.— CURVATURE AND REFRACTION (IN ELEVATION)* 





Difference in Feet for 




Difference in Feet for 


Dis- 
tance, 








Dis- 
tance. 




















Milea. 


Curvature. 


Refraction. 


Curvature 

and 
Refraction. 


Miles. 


Curvature. 


RSraction. 


Curvature 

and 
Refraction. 


1 


0.7 


0.1 


0.6 


34 


771.3 


108.0 


663.3 


2 


2.7 


0.4 


2.3 


35 


817.4 


114.4 


703.0 


3 


6.0 


0.8 


5.2 


36 


864.8 


121.1 


743.7 


4 


10.7 


1.5 


9.2 


37 


913.5 


127.9 


785.6 


5 


16.7 


2.3 


14.4 


38 


963.5 


134.9 


828.6 


6 


24.0 


3.4 


20.6 


39 


1014.9 


142.1 


872.8 


7 


32.7 


4.6 


28.1 


40 


1067.6 


149.5 


918.1 


8 


42.7 


6.0 


36.7 


41 


1121.7 


157.0 


964.7 


9 


54.0 


7.6 


46.4 


42 


1177.0 


164.8 


1012.2 


10 


66.7 


9.3 


57.4 


43 


1233.7 


172.7 


1061.0 


11 


80.7 


11.3 


69.4 


44 


1291.8 


180.8 


1111.0 


12 


96.1 


13.4 


82.7 


45 


1351.2 


189.2 


1162.0 


13 


112.8 


15.8 


97.0 


46 


1411.9 


197.7 


1214.2 


14 


130.8 


18.3 


112.5 


47 


1474.0 


206.3 


1267.7 


15 


150.1 


21.0 


129.1 


48 


1537.3 


215.2 


1322.1 


16 


170.8 


23.9 


146.9 


49 


1602.0 


224.3 


1377.7 


17 


192.8 


27.0 


165.8 


50 


1668.1 


233.6 


1434.6 


18 


216.2 


30.3 


185.9 


51 


1735.5 


243.0 


1492.5 


19 


240.9 


33.7 


207,2 


52 


1804.2 


252.6 


1551.6 


20 


266.9 


37.4 


229.5 


53 


1874.3 


262.4 


1611.9 


21 


294.3 


41.2 


253.1 


54 


1945.7 


272.4 


1673.3 


22 


322.9 


45.2 


277.7 


55 


2018.4 


282.6 


1735.8 


23 


353.0 


49.4 


303.6 


56 


2092.5 


292.9 


1799.6 


24 


384.3 


53.8 


330.5 


67 


2167.9 


303.6 


1864.4 


25 


417.0 


58.4 


358.6 


58 


2244.6 


314.2 


1930.4 


26 


451.1 


63.1 


388.0 


59 


2322.7 


325.2 


1997.6 


27 


486.4 


68.1 


418.3 


60 


2402.1 


336.3 


2065.8 


28 


523.1 


73.2 


449.9 


61 


2482.8 


347.6 


2135.2 


29 


561.2 


78.6 


482.6 


62 


2564.9 


359.1 


2205.8 


30 


600.5 


84.1 


516.4 


63 


2648.3 


370.8 


2277.5 


31 


641.2 


89.8 


551.4 


64 


2733.0 


382.6 


2350.4 


32 


683.3 


95.7 


587.6 


65 


2819.1 


394.7 


2424.4 


33 


726.6 


101.7 


624.9 


66 


2906.5 


406.9 


2499.6 



* From Appendix No. 9, Report for 1882, United States Coast and Geodetic Survey. 

363 



364 



GEODETIC SURVEYING 



TABLE II.— LOGARITHMS OF THE PUISSANT FACTORS* 
(In U. S. Legal Meters) 



Lat. 


A 


B 


C 


D 


E 


F 


o 


-10 


- 10 


- 10 


- 10 


— 20 


— 20 


20 


8.5095499 


8.512155s 


0.96732 


2 . 1996 


5.7574 


7.772 


21 


8 • 5095330 


8.5121049 


0.99036 


2.2170 


5. 77" 


7.787 


22 


8.5095155 


8.5120524 


I. 01252 


2 . 2333 


5-7851 


7.800 


23 


8 . 5094973 


8.5119979 


1.03389 


2.2485 


5.7997 


7.812 


24 


8.5094786 


8.5119416 


1.05455 


2.2627 


5. 8146 


7.823 


25 


8 . 5094592 


8. 51 18834 


1.07456 


2.2759 


5.8300 


7-832 


26 


8.5094392 


8,5118236 


1.09399 


2.2882 


5 . 8458 


7.841. 


27 


8 . 5094187 


8.5117620 


1.11289 


2.2997 


5 . 8620 


7.849) 


28 


8.5093977 


8.5116989 


1.13131 


2.3104 


5.8785 


7.855 


29 


8.5093761 


8.5116342 


1-14931 


2.3203 


5.8955 


7.861 


30 


8.5093,541 


8,5115682 


1.16691 


2.3294 


5.9127 


7.866 


31 


8.5093316 


8,5115007 


1.18415 


2.3379 


5.9304 


7.870 


32 


8 . 5093087 


8.5114321 


I. 20107 


2.3456 


5 9484 


7.873 


33 


8 . 5092854 


8.5113622 


1.21771 


2.3527 


5.9667 


7.87s 


34 


8.5092618 


8.5112912 


I . 23408 


2.3592 


5 9853 


7.877 


35 


8.5092378 


8.5112192 


1.25023 


2.3651 


6.0043 


7.877 


36 


8.5092135 


8.5111463 


I. 26616 


2.3704 


6.0237 


7.877 


37 


8.5091889 


8. 51 10725 


I .28192 


2,3750 


6.0433 


7.876 


38 


8.5091640 


8.5109980 


1,29752 


2.3792 


6.0633 


7.874 


39 


8.5091390 


8.5109228 


I. 31298 


2.3827 


6.0836 


7.872 


40 


8. 5091 137 


8.5108470 


1.32832 


2,3857 


6.1043 


7.869 


41 


8 . 5090883 


8.5107708 


I. 34357 


2.3882 


6.1253 


7.864 


42 


8.5090628 


8.5106942 


1.35874 


2.3901 


6.1467 


7.860 


43 


8.5090372 


8.5106173 


1,37385 


2,3914 


6.1684 


7.854 


44 


8. 50901 I 5 


8 . 5105402 


1.38893 


2.3923 


6.1905 


7.848 


4S 


8.5089857 


8.5104630 


1,40399 


2.3926 


6.2130 


7.840 


46 


8.5089600 


8.5103858 


I. 41905 


2,3924 


6.2359 


7.832 


47 


8 . SO89343 


8.5103087 


I. 43413 


2.3917 


6.2592 


7.824 


48 


8 . 5089086 


8.5102317 


1.44925 


2.3904 


6 . 2830 


7.814 


49 


8.5088831 


8.5101551 


1.46442 


2.3886 


6.3071 


7.804 


SO 


8.5088576 


8.5100788 


1.47967 


2.3862 


6.3318 


7.792 


51 


8.5088324 


8.5100029 


I. 49501 


2.3833 


6.3569 


7.780 


52 


8.5088073 


8 . 5099276 


I. 51047 


2.3799 


6.3826 


7.767 


53 


8.5087824 


8.5098530 


1.52607 


2.3759 


6 . 4088 


7. 753 


54 


8.5087577 


8.5097791 


I. 54182 


2.3713 


6.4355 


7.738 


55 


8.5087334 


8 .5097060 


1.55776 


2.3661 


6 . 4629 


7.723 


56 


8.5087093 


8.5096338 


1.57390 


2 . 3603 


6.4909 


7.706 


57 


8.5086856 


8 . 5095626 


1.59027 


2.3539 


6.5196 


7.688 


58 


8.5086622 


8 . 5094925 


I. 6069 I 


2.3469 


6.5490 


7.669 


59 


8 . S086393 


8.5094236 


1.62383 


2.3392 


6.5792 


7.649 


60 


8.5086167 


8.5093560 


I. 64108 


2.3309 


6.6102 


7.627 


61 


8 . 5085946 


8.5092897 


1.65868 


2.3218 


6.6422 


7.605 


62 


8.5085730 


8 . 5092248 


1.67667 


2.3120 


6.6750 


7.581 


63 


8.5085519 


8.5091614 


1.69509 


2.3014 


6.7089 


7.556 


64 


8.5085313 


8.5090996 


I. 71399 


2.2901 


6.7440 


7.529 


65 


8.5085112 


8.5090395 


1 ■ 73342 


2.2778 


6.7802 


7.501 


66 


8.5084917 


8. 508981 I 


I ■ 75343 


2.2647 


6.8177 


7.471 


67 


8.5084729 


8 . 5089245 


1.77409 


2.2506 


6.8567 


7.440 


68 


8.5084546 


8.5088698 


I . 79546 


2.2354 


6.8972 


7,406 


69 


8 . 5084370 


8.5088170 


I. 81762 


2.2192 


6,9395 


7.371 



* Based on tables in App. No. 9, Report for 1894, U. S. Coast and Geodetic Survey. 



TABLES 



365 



TABLE II.-LOGARITHMS OF THE PUISSANT FACTORS— 

(Continued) 

Log G=log diff. for (log 4^)— log diff. for (log s) 



log s 


log difference. 


log JX 


logs 


log difference. 


log JX 


3.876 


0.0000001 


2-. 385 


4.922 


0.0000124 


3.431 


4.026 


002 


2.535 


4.932 


130 


3.441 


4.114 


003 


2.623 


4.941 


136 


3.450 


4.177 


004 


2.686 


4.950 


142 


3.459 


4.225 


005 


2.734 


4.959 


147 


3.468 


4.265 


006 


2.774 


4.968 


153 


3.477 


4.298 


007 


2.807 


4.976 


160 


3.485 


4.327 


008 


2.836 


4.985 


166 


3.494- 


4.353 


009 


2.862 


4.993 


172 


3.502 


4.376 


010 


2.885 


5.002 


179 


3.511 


4.396 


Oil 


2.905 


5.010 


186 


3.519 


4.415 


012 


2.924 


5.017 


192 


3.526 


4.433 


013 


2.942 


5.025 


199 


3.534 


4.449 


014 


2.958 


5.033 


206 


3.542 


4.464 


015 


2.973 


5.040 


213 


3.549 


4.478 


016 


2.987 


5.047 


221 


3.556 


4.491 


017 


3.000 


5.054 


228 


3.563 


4.503 


' 018 


3.012 


5.062 


236 


3.571 


4.526 


020 


3.035 


5.068 


243 


3.577 ' 


4.548 


023 


3.057 


5.075 


251 


3.584 


4.570 


025 


3.079 


5.082 


259 


3.591 


4.591 


027 


3.100 


5.088 


267 


3.597 


4.612 


030 


3.121 


5.095 


275 


3.601 


4.631 


033 


3.140 


5.102 


284 


3.611 


4.649 


036 


3.158 


5.108 


292 


3.617 


4.667 


039 


3.176 


5.114 


300 


3 . 623 


4.684 


042 


3.193 


5.120 


309 


3.629 


4.701 


045 


3.210 


5.126 


318 


3.635 


4.716 


048 


3.225 


5.132 


327 


3.641 


4.732 


052 


3.241 


5.138 


336 


3.647 


4.746 


056 


3.255 


5.144 


345 


3.653 


4.761 


059 


3.270 


5.150 


354 


3.659 


4.774 


063 


3.283 


5.156 


364 


3.665 


4.788 


067 


3.297 


5.161 


373 


3.670 


4.801 


071 


3.310 


5.167 


383 


3.676 


4.813 


075 


3.322 


5.172 


392 


3.681 


4.825 


080 


3.334 


5.178 


402 


3.687 


4.834 


084 


3.343 


5.183 


412 


3.692 


4.849 


089 


3.358 


5.188 


422 


3.697 


4.860 


094 


3.369 


5.193 


433 


3.702 


4.871 


098 


3.380 


5.199 


443 


3.708 


4.882 


103 


3.391 


5.204 


453 


3.713 


4.892 


108 


3.401 


5.209 


464 


3.718 


4.903 


114 


3.412 


5.214 


474 


3.723 


4.913 


119 


3.422 


5.219 


486 


3.728 



Note. — The logarithms in the above table require s to be expressed in meters and JX in 
seconds of arc. If s is expressed in feet its logarithm must be reduced by 0.516 before using 
in this table. 



366 



GEODETIC SURVEYING 



TABLE III.— BAROMETRIC ELEVATIONS' 

30 



Containing H = 62737 log 



B 



B. 



Inches. 

11.0 

11.1 

.11.2 
11.3 
11.4 
11.5 
11.6 
11.7 
11.8 
11.9 
12.0 
12.1 
12.2 
12.3 
12.4 
12.5 
12.6 
12.7 
12.8 
12.9 
13.0 
13.1 
13.2 
13.3 
13.4 
13.5 
13.6 
13.7 
13.8 
13.9 
14.0 



H. 



Feet. 

27,336 
27,090 
26,846 
26,604 
26,364 
26,126 
25,890 
25,656 
25,424 
25,194 
24,966 
24,740 
24,516 
24,294 
24,073 
23,854 
23,637 
23,421 
23,207 
22,995 
22,785 
22,576 
22,368 
22,162 
21,958 
21,757 
21,557 
21,358 
21,160 
20,962 
20,765 



Dif. for 
.01. 



Feet. 

-24.6 
24.4 
24.2 
24.0 
23.8 
23.6 
23.4 
23.2 
23.0 
22.8 
22.6 
22.4 
22.2 
22.1 
21.9 
21.7 
21.6 
21.4 
21.2 
21.0 
20.9 
20.8 
20.6 
20.4 
20.1 
20.0 
19.9 
19.8 
19.8 

-19.7 



Inches. 

14.0 
14.1 
14.2 
14.3 
14.4 
14.5 
14.6 
14.7 
14.8 
14.9 
15.0 
15.1 
15.2 
15.3 
15.4 
15.5 
15.6 
15.7 
15.8 
15.9 
16.0 
16.1 
16.2 
16.3 
16.4 
16.5 
16.6 
16.7 
16.8 
16.9 
17.0 



H. 



Feet. 

20,765 

20,570 

20,377 

20,186 

19,997 

19,809 

19,623 

19,437 

19,252 

19,068 

18,886 

18,705 

18,525 

18,346 

18,168 

17,992 

17,817 

17,643 

17,470 

17,298 

17,127 

16,958 

16,789 

16,621 

16,454 

16,288 

16,124 

15,961 

15,798 

15,636 

15,476 



Dif. for 
.01. 



Feet. 

-19.5 
19.3 
19.1 
18.9 
18.8 
18.6 
18.6 
18.5 
18.4 
18.2 
18.1 
18.0 
17.9 
17.8 
17.6 
17.5 
17.4 
17.3 
17.2 
17.1 
16.9 
16.9 
16.8 
16.7 
16.6 
16.4 
16.3 
16.3 
16.2 

-16.0 



Inches. 

17.0 
17.1 
17.2 
17.3 
17.4 
17.5 
17.6 



H. 



17.7 


17.8 


17.9 


18.0 


18.1 


18.2 


18.3 


18.4 


18.5 


18.6 


18.7 


18.8 


18.9 


19.0 


19.1 


19.2 


19.3 


19.4 


19.5 


19.6 


19.7 


19.8 


19.9 


20.0 



Feet. 
15,476 
15,316 
15,157 
14,999 
14,842 
14,686 
14,531 
14,377 
14,223 
14,070 
13,918 
13,767 
13,617 
13,468 
13,319 
13,172 
13,025 
12,879 
12,733 
12,689 
12,445 
12,302 
12,160 
12,018 
11,877 
11,737 
11,598 
11,459 
11,321 
11,184 
11,047 



Dif. for 
.01. 



Feet. 

-16.0 
15.9 
15.8 
15.7 
15.6 
15.5 
15.4 
15.4 
15.3 
15.2 
15.1 
15.0 
14.9 
14.9 
14.7 
14.7 
14.6 
14.6 
14.4 
14.4 
14.3 
14.2 
14.2 
14.1 
14.0 
13.9 
13.9 
13.8 
13.7 
-13.7 



* From Appendix No. 10, Report for 1881, United States Coast and Geodetic Survey. 



TABLES 



367 



TABLE III.— BAROMETRIC ELEVATIONS— (Con^mweo) 

30 



Containing ff = 62737 log 



B 



B. 


H. 


Dif. for 
.01. 


B. 


H. 


Dif. for 

.01. 


B. 


¥■ 


Dif. for 
.01. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet. 


20.0 


11,047 


-13.6 


23.0 


7,239 


-11.8 


26.0 


3,899 


-10.5 


20.1 


10,911 


13.5 
13.4 


23.1 


7,121 


11.7 


26.1 


3,794 


10.4 


20.2 


10,776 


23.2 


7,004 


11.7 


26.2 


3,690 


10.4 


20.3 


10,642 


13.4 


23.3 


6,887 


11.7 


26.3 


3,586 


10.3 


20 4 


10,508 


13.3 


23.4 


6,770 


11.6 


26.4 


3,483 


10.3 


20.5 


10,375 


23.5 


6,654 




26.5 


3,380 








13.3 






11.6 






10.3 


20.6 


10,242 


13.2 


23.6 


6,538 


11.5 


26.6 


3,277 


10.2 


20.7 


10,110 


13.1 


23.7 


6,423 


11.5 


26.7 


3,175 


10.2 


20.8 


9,979 


13.1 


23.8 


6,308 


11.4 


26.8 


3,073 


10.1 


20.9 


9,848 


13.0 


23.9 


6,194 


11.4 


26.9 


2,072 


10.1 


21.0 


9,718 


12.9 


24.0 


6,080 


11.3 


27.0 


2,871 


10.1 


21.1 


9,589 


12.9 


24.1 


5,967 


11.3 


27.1 


2,770 


10.0 


21.2 


9,460 


12.8 


24.2 


5,854 


11.3 


27.2 


2,670 


10.0 


21.3 


9,332 


12.8 


24.3 


5,741 


11.2 


27.3 


2,570 


10.0 


21.4 


9,204 


12.7 


24.4 


5,629 


11.1 


27.4 


2,470 


9.9 


21.5 


9,077 


12.6 


24.5 


5,518 


11.1 


27.5 


2,371 


9.9 


2-1.6 


8,951 


12,6 


24.6 


5,407 


11.1 


27.6 


2,272 


9.9 


21.7 


8,825 


12.5 


24.7 


5,296 


11.0 


27.7 


2,173 


9.8 


21.8 


8,700 


12.5 


24.8 


5,186 


10.9 


27.8 


2,075 


9.8 


21.9 


8,575 


12.4 


24.9 


5,077 


10.9 


27.9 


1,977 


9.7 


22.0 


8,451 


12.4 


25.0 


4,968 


10.9 


28.0 


1,880 


9.7 


22.1 


8,327 


12.3 


25.1 


4,859 


10.8 


28.1 


1,783 


9.7 


22.2 


8,204 


12.2 


25.2 


4,751 


10.8 


28.2 


1,686 


9.7 


22.3 


8,082 


12.2 


25.3 


4,643 


10.8 


28.3 


1,589 


9.6 


22.4 


7,960 


12.2 


25.4 


4,535 


10.7 


28.4 


1,493 


9.6 


22.5 


7,838 


12.1 


25.5 


4,428 


10.7 


28.5 


1,397 


9.5 


22.6 


7,717 


12.0 


25.6 


4,321 


10.6 


28.6 


1,302 


9.5 


22.7 


7,597 


12.0 


25.7 


4,215 


10.6 


28.7 


1,207 


9.5 


22.8 


7,477 


11.9 


25.8 


4,109 


10.5 


28.8 


1,112 


9.4 


22.9 


7,358 


-11.9 


25.9 


4,004 


-10.5 


28.9 


1,018 


-9.4 


23.0 


7,239 




26.0 


3,899 




29.0 


924 





368 



GEODETIC 8UEVEYING 



TABLE III.— BAROMETRIC ELEVATIONS— Conimued 



30 
Containing H = 62737 log — . 



B. 


H. 


Dif. for 
.01. 


B. 


H. 


Dif. for 
.01. 


B. 


H. 


Dif. for 
.01, 


Inches. 


Feet. 


Feet. 


Inches. 


Feet. 


Feet 


Inches. 


Feet. 


Feet. 


29.0 
29.1 


924 
830 


-9.4 

9.4 

9.3 
9.3 
9.2 
9.2 
-9.2 


29.7 
29.8 


274 

182 


-9.2 

9.1 
9.1 
9.1 

9.0 

9.0 

-9.0 


30.4 
30.5 


-361 
451 


-9.0 
8.9 
8.9 

8.8 

8.8 

-8.8 


29.2 
29.3 


736 
643 


29.9 
30.0 


91 
00 


30.6 
30.7 


540 
629 


29.4 


550 


30.1 


- 91 


30.8 


717 


29.5 
29.6 
29.7 


458 
366 
274 


30.2 
30.3 
30.4 


181 

271 

-361 


30.9 
31.0 


805 
-893 



TABLE IV.— CORRECTION COEFFICIENTS TO BAROMETRIC 
ELEVATIONS FOR TEMPERATURE (FAHRENHEIT) AND 
HUMIDITY * 



t+v 


C 


t+t' 


C 


t+l' 


C 


0° 


-0.1025 


60° 


-0.0380 


120° 


+0.0262 


5 


-0.0970 


65 


-0.0326 


125 


+0.0315 


10 


-0.0915 


70 


-0.0273 


130 


+0.0368 


15 


-0.0860 


75 


-0.0220 


135 


+0.0420 


20 


-0.0806 


80 


-0.0166 


140 


+0.0472 


25 


-0.0752 


85 


-0.0112 


145 


+0.0524 


30 


-0.0698 


90 


-0.0058 


150 


+0.0575 


35 


-0.0645 


95 


-0.0004 


155 


+0.0626 


40 


-0.0592 


100 


+0.0049 


160 


+0.0677 


45 


-0.0539 


105 


+0.0102 


165 


+0.0728 


50 


-0.0486 


110 


+0.0156 


170 


+0.0779 


55 


-0.0433 


115 


+0.0209 


175 


+0.0829 


60 


-0.0380 


120 


+0.0262 


180 


+0.0879 



* Based on Tables I and IV, Appendix No. 10, Report for 1881, United States Coast 
and Geodetic Survey. 



TABLES 



369 



TABLE v.— LOGARITHMS OF RADIUS OF CURVATURE 
(In U. S. Legal Meters) 









Latitude. 






Azimuth. 












24° 


26° 


28° 


30° 


32° 


0° 


180° 


Meridian 


6.802484 


6.802602 


6 802726 


6.802857 


6.802993 


5 


175 


185° 


355° 


2503 


2620 


2744 


2874 


3009 


10 


170 


190 


350 


2558 


2674 


2796 


2924 


3057 


15 


165 


195 


345 


2649 


2761 


2880 


3005 


3135 


20 


160 


200 


340 


2771 


2880 


2995 


3116 


3241 


30 


150 


210 


330 


3098 


3197 


3301 


3410 


3523 


40 


140 


220 


320 


3501 


3585 


3676 


3771 


3869 


50 


130 


230 


310 


6.803928 


6.803999 


6.804075 


6.804155 


6.804238 


60 


120 


240 


300 


4330 


4389 


4451 


4517 


4585 


70 


110 


250 


290 


4658 


4707 


4758 


4812 


4868 


75 


105 


255 


285 


4781 


4827 


4874 


4923 


4974 


80 


100 


260 


280 


4872 


4914 


4958 


5004 


5052 


85 


95 


265 


275 


4928 


4968 


5011 


5054 


5101 


90 


Prime Vert. 


270 


4947 


4986 


5028 


5071 


5117 




34° 


36° 


38° 


40° 


42° 


0° 


180° 


Meridian 


6.803134 


6.803279 


6.803427 


6.803578 


6.803731 


5 


175 


185° 


355° 


3150 


3294 


3441 


3591 


3744 


10 


170 


190 


350 


3195 


3337 


3483 


3631 


3780 


15 


165 


195 


345 


3270 


3409 


3551 


3695 


3840 


20 


160 


200 


340 


3371 


3505 


3642 


3781 


3922 


30 


150 


210 


330 


3641 


3762 


3885 


4011 


4138 


40 


140 


220 


320 


3972 


4077 


4184 


4294 


4405 


50 


130 


230 


310 


6.804324 


6.804412 


6.804503 


6.804595 


6.804688 


60 


120 


240 


300 


4655 


4728 


4802 


4878 


4954 


70 


110 


250 


290 


4926 


4985 


5046 


5109 


5171 


75 


105 


255 


285 


5027 


5081 


5138 


5195 


5253 


80 


100 


260 


280 


5102 


5153 


5206 


5259 


5313 


85 


95 


265 


275 


5148 


5197 


5247 


5299 


5350 


90 


Frime Vert, 


270 


5164 


5212 


5261 


5312 


5363 




44° 


46° 


48° 


50° 


52° 


0° 


180° 


Meridian 


6.803885 


6.804040 


6:804194 


6.804347 


6.804498 


5 


175 


185° 


355° 


3897 


4050 


4204 


4356 


4506 


10 


170 


190 


350 


3931 


4082 


4233 


4383 


4531 


15 


165 


195 


345 


3987 


4135 


4282 


4428 


4573 


20 


160 


200 


340 


4064 


4206 


4348 


4489 


4629 


30 


150 


210 


330 


4267 


4396 


4524 


4652 


4778 


40 


140 


220 


320 


4516 


4628 


4740 


4851 


4960 


50 


130 


230 


310 


6.804782 


6.804876 


6.804970 


6.805063 


6,805155 


60 


120 


240 


300 


5030 


6109 


5186 


5262 


5338 


70 


110 


250 


290 


5234 


5298 


5362 


5425 


5487 


75 


105 


255 


285 


5312 


5369 


5428 


5486 


■ 5543 


80 


100 


260 


280 


5368 


5422 


5477 


5531 


6584 


85 


95 


265 


275 


5402 


5455 


5507 


5559 


5610 


90 


Prime Vert. 


270 


5414 


5465 


55-17 


5568 


5618 



370 



GEODETIC SUEVEYING 



TABLE VI.— LOGARITHMS OP RADIUS OF CURVATURE 

(In feet) 



Azimuth. 


Latitude. 


28° 


30° 


32° 


34° 


36° 


0° 


180° 


Meridian 


7.318711 


7.318841 


7.318978 


7.319118 


7.319263 


5 


175 


185° 


355° 


8728 


8858 


8993 


9134 


9278 


10 


170 


190 


350 


8780 


8908 


9041 


9179 


9321 


IS 


165 


195 


345 


8864 


8989 


9119 


9254 


9393 


20 


160 


200 


340 


8979 


9100 


9225 


9355 


9489 


30 


150 


210 


330 


9285 


9394 


9507 


9625 


9746 


40 


140 


220 


320 


9660 


9755 


9853 


9956 


320061 


50 


130 


230 


310 


7.320059 


7.320139 


7.320222 


7.320308 


7.320396 


60 


120 


240 


300 


0435 


0501 


0569 


0639 


0712 


70 


110 


250 


290 


0742 


0796 


0852 


0910 


0969 


75 


105 


255 


285 


0858 


0907 


0958 


1011 


1065 


80 


100 


260 


280 


0942 


0988 


1036 


1086 


1137 


85 


95 


265 


275 


0995 


1038 


1085 


1132 


1181 


90 


Prime Vert. 


270 


1012 


1055 


1101 


1148 


1196 




38° 


40° 


42° 


44° 


46° 


0° 


180° 


Meridian 


7.319412 


7.319562 


7.319715 


7.319869 


7.320024 


5 


175 


185° 


355° 


9425 


9575 


9728 


9881 


0034 


10 


170 


190 


350 


9467 


9615 


9764 


9915 


0066 


15 


165 


195 


345 


9535 


9679 


9824 


9971 


0119 


20 


160 


200 


340 


9626 


9765 


9906 


320048 


0190 


30 


150 


210 


330 


9869 


9995 


320122 


0251 


0380 


40 


140 


220 


320 


320168 


320278 


0389 


0500 


0612 


50 


130 


230 


310 


7.320487 


7.320579 


7.320672 


7.320766 


7.320860 


60 


120 


240 


300 


0786 


0862 


0938 


1014 


1093 


70 


110 


250 


290 


1030 


1093 


1155 


1218 


1282 


75 


105 


255 


285 


1122 


1179 


1237 


1296 


1353 


80 


100 


260 


280 


1190 


1243 


1297 


1352 


1406 


85 


95 


265 


275 


1231 


1283 


1334 


1386 


1439 


90 


Prime Vert. 


270 


1246 


1296 


1347 


1398 


1449 



TABLE VII.— CORRECTIONS FOR CURVATURE AND REFRACTION 
IN PRECISE SPIRIT LEVELING 





Correction 




Correction 




Correction 


Distance. 


to Rod 


Distance. 


to Rod 


Distance. 


to Rod 




Reading. 




Reading, 




Reading. 


Meters. 


mm. 


Meters. 


mm. 


Meters. 


mm. 


Oto 27 


0.0 


100 


-0.68 


200 


-2.73 


28 to 47 


-0.1 


no 


-0.83 


210 


-3.01 


48 to 60 


-0.2 


120 


-0.98 


220 


-3.31 


61 to 72 


-0.3 


130 


-1.15 


230 


-3.61 


73 to 81 


-0.4 


140 


-1.34 


240 


-3.94 


82 to 90 


-0.5 


150 


-1.54 


250 


-4.27 


91 to 98 


-0.6 


160 


-1.75 


260 


-4.62 


99 to 105 


-0.7 


170 


-1.97 


270 


-4.98 


106 to 112 


-0.8 


180 


-2.21 


280 


-6.36 


113 tolls 


-0.9 


190 


-2,47 


290 


-5.75 - 



TABLES 
TABLE VIIL— MEAN ANGULAR REFRACTION 



371 



Apparent 
Altitude. 


Refraction. 


Apparent 
Altitude. 


Refraction. 


Apparent 
Altitude, 


Kef r action. 


Apparent 

Zenith 
Distance. 


o / 


/ 


ft 


o 


/ // 


o 


/ 


It 


o 


00 


34 


54.1 


10 


5 16.2 


50 





48,4 


40 


10 


32 


49.2 


11 


4 48.6 


51 





46.7 


39 


20 


30 


52.3 


12 


4 25.0 


52 





45.1 


38 


30 


29 


03.5 


13 


4 04.9 


53 





43.5 


37 


40 


27 


22.7 


14 


3 47.4 


54 





41.9 


36 


50 


25 


49.8 




















16 


3 32.1 


55 





40.4 


35 


1 00 


24 


24.6 


16 


3 18.6 


56 





38.9 


34 


10 


23 


06.7 


17 


3 06.6 


57 


. 


37.5 


33 


20 


21 


55.6 


18 


2 65.8 


58 





36.1 


32 


30 


20 


50.9 


19 


2 46.1 


59 





34.7 


31 


40 


19 


51.9 














60 


18 


58.0 


20 


2 37.3 


60 





33.3 


30 








21 


2 29.3 


61 





32.0 


29 


2 00 


18 


08.6 


22 


2 21.9 


62 





30.7 


28 


10 


17 


23.0 


23 


2 16.2 


63 





29.4 


27 


20 


16 


40.7 


24 


2 08.9 


64 





28.2 


26 


30 


16 


00.9 














40 


15 


23.4 


26 


2 03.2 


65 





26.9 


26 


50 


14 


47.8 


26 


1 67.8 


66 





25.7 


24 








27 


1 52.8 


67 





24.5 


23 


3 00 


14 


.14.6 


28 


1 48.2 


68 





23.3 


22 


10 


13 


43.7 


29 


1 43.8 


69 





22.2 


21 


20 


13 


15.0 














30 


12 


48.3 


30 


1 39.7 


70 





21.0 


20 


40 


12 


23.7 


31 


1 35.8 


71 





19.9 


19 


50 


12 


00.7 


32 


1 32,1 


72 





18.8 


18 








33 


1 28.7 


73 





17.7 


17 


4 00 


11 


38.9 


34 


1 26.4 


74 





16.6 


16 


10 


11 


18.3 














20 


10 


58.6 


35 


1 22.3 


75 





15.5 


16 


30 


10 


39.6 


36 


1 19,3 


76 





14.5 


14 


40 


10 


21.2 


37 


1 16.5 


77 





13.4 


13 


50 


10 


03.3 


38 


1 13,8 


78 





12,3 


12 








39 


1 11,2 


79 





11,2 


11 


5 00 


9 


46.5 














30 


9 


01.9 


40 


1 08.7 


80 





10.2 


10 








41 


1 06,3 


81 





09.1 


9 


6 00 


8 


23,3 


42 


1 04.0 


82 





08.1 


8 


30 


7 


49.5 


43 


1 01.8 


83 





07.1 


7 








44 


59.7 


84 





06.1 


6 


7 00 


7 


19.7 














30 


6 


63.3 


45 


67.7 


85 





05.1 


5 








46 


55.7 


86 





04,1 


4 


8 00 


6 


29.6 


47 


53,8 


87 





03,0 


3 


30 


6 


08.4 


48 


51,9 


88 





02.0 


2 








49 


50.2 


89 





01.0 


1 


9 00 


6 


49.3 














30 


6 


32.0 


60 


48,4 


90 





00,0 






372 



GEODETIC SUEVEYING 



TABLE IX.— ELEMENTS OF MAP PROJECTIONS 



Lat. 


Logarith 


ns (U. S. Legal Meters). 


1° in Meters. 


Logarithm 
(1-e'sinZ^). 

(-10) 


B 


N 


r 


Latitude. 

(<^-30' to 

.#. + 30') 


Longitude. 
(On Par. of 
Latitude.) 


20° 
22 
24 
26 

28 


6.8022696 
3727 
4835 
6015 
7262 


6.8048752 
9096 
9465 
9859 

6.8050274 


6.7778610 
.7720755 
.7656767 
.7586461 
.7509623 


110700 
726 
754 
785 
816 


104650 
103265 
101755 
100121 
98365 


9.9996560 
5873 
5134 
4347 
3516 


30 
32 
34 
36 

38 


6.8028569 

9930 

6.8031339 

2788 
4271 


6.8050710 
1164 
1633 
2116 
2611 


6.7426016 
.7335369 
.7237375 
.7131692 
. 7017932 


110850 
884 
920 
957 
995 


96489 
94496 
92388 
90167 
87836 


9.9992645 
.1738 

0798 
9.9989832 

8843 


40 
42 
44 
46 

48 


6.8035781 
7309 
8849 

6.8040393 
1934 


6.8053114 
3623 
4136 
4651 
5165 


6.6895654 
.6764358 
.6623477 
.6472364 
.6310274 


111034 
073 
112 
152 
191 


85397 
82854 
80209 
77466 
74629 


9.9987837 
6818 
5792 
4762 
3735 


50 
52 
54 
56 

58 


6.8043463 
4975 
6460 
7913 
9326 


6.8055675 
6178 
6674 
7158 
7629 


6.6136350 
.5949598 
.5748861 
. 5532775 
. 5299726 


111231 
269 
307 
345 
381 


71699 
68681 
65579 
62396 
59136 


9.9982715 
1708 
0717 

9.9979749 
8807 


60 


6.8050691 


6.8058084 


6 . 5047784 


111416 


55803 


9.9977897 



Lat. 


Element 
of 


Coordinates of Developed Arcs. 1 


^ 


V 


Cone. 


for 1° of Long. 


for 71° of Long. 


for X? of Long. 


for 71°. 


Miles. 


Miles. 


Meters. 


Value for (1°) X 


Miles. 


Meters. 


(1°) X 


20° 


10893 


65.03 


104649 


n -003(0.19771°) 


0.1941 


312.3 


7l2 


22 


9814 


64.17 


103264 


n.cos (0.216«°) 


. 2098 


337.6 


71! 


24 


8907 


63.23 


101754 


n-C03 (0.235n°) 


0.2244 


361.2 


71' 


26 


8131 


62.21 


100120 


m-cos (0,253n°) 


0,2380 


383.0 


71' 


28 


7459 


61.12 


98364 


n-cos(0.271«°) 


0.2504 


403.0 


712 


30 


6870 


59.95 


96488 


n -008(0.28871°) 


0.2616 


421,0 


712 


32 


6349 


58.72 


94495 


n-oos (0.30571°) 


0.2715 


437,0 


71! 


34 


5882 


57.41 


92386 


71 -cos (0.322n°) 


0.2801 


450.8 


71! 


36 


5461 


56.03 


90165 


71 -cos (0.33971°) 


0.2874 


462,5 


712 


38 


5079 


54.58 


87834 


71 -cos (0.35571°) 


0.2932 


471,9 


712 


40 


4730 


53.06 


85395 


7i-cos(0.371?i°) 


0.2976 


479.0 


71' 


42 


4408 


51.48 


82852 


71 -cos (0.38671°) 


0.3006 


483.8 


7l2 


44 


4111 


49.84 


80207 


7i-oos(0.400n°) 


0.3021 


486.2 


712 


46 


3834 


48.13 


77464 


71 -cos (0.41471°) 


. 3022 


486.3 


712 


48 


3575 


46.37 


74627 


71 -cos (0.42871°) 


0.3007 


484.0 


712 


50 


3332 


44.55 


71697 


71 -cos (0. 44171°) 


. 2978 


479.3 


7l2 


52 


3103 


42.68 


68679 


n-cos (0. 45471°) 


0.2935 


472.3 


712 


54 


2886 


40.75 


65577 


71 - cos (0. 46671°) 


0.2877 


463.0 


712 


56 


2679 


38,77 


62394 


71 -cos (0.47871°) 


0.2805 


451.4 


712 


58 


2483 


36.74 


59134 


71. cos (0.48971°) 


0.2719 


437.6 


712 


60 


2294 


34.67 


55801 


71 -cos (0.49971°) 


0.2620 


421.7 


712 



TABLES 



373 



TABLE X.— CONSTANTS AND THEIR LOGARITHMS 



General Constants. 


Number. 


Logarithm. 


7r 


3.141592654 

0.318309886 

9.869604401 

0.101321184 

1.772453851 

0.564189584 

57.29577951 
3437,746771 
206264.8062 

0.017453293 
0.017452406 
0.000290888 
0.000290888 
0.000004848 
0.000004848 

2.718281828 
0.434294482 
0.434294482 
2.302585093 

3,2808693,. 
3.280833333 

0.621369949 
1609.347219 

0.4769363.. 
0.6744897.. 


0.4971498727 

9,5028501273 

0.9942997454 

9.0057002546 

0.2485749363 

9.7514250637 

1.7581226324 
3.5362738828 
5.3144251332 

8,2418773676 
8.2418553284 
6.4637261172 
6.4637261109 
4.6855748668 
4.6855748668 

0.4342944819 
9.6377843113 
9.6377843113 
0.3622156887 

0,5159889297 
0.5159841687 

9.7933502462 
3.2066497538 

9.6784604... 
9.8289754... 


-10 
-10 
-10 

-10 
-10 
-10 
-10 
-10 
-10 

-10 
-10 

-10 

-10 
-10 


1 


7^ 
TT^ 


1 


k2 

Vtt 


1 


sfk 

Degrees in a radian 


Minutes in a radian 


Seconds in a radian 


Arcl" 


Sin 1° 




Sin 1' 


Arc 1" 






Modulus of common logarithms (M) 


Natural log x -r-common log x 




1 U. S. legal meter = 3.2808333 + ft 

1 kilometer = five-eighths mile, nearly . . . 








Geodetic Constants. 
(Clarke's 1866 Spheroid.) 


Logarithms. 


U. S. Legal Meters. 


Feet. 1 


Semi-major axis = o . . . 


6.8047033 
6.8032285 
9.9985252 
6.8039665 

7.5302093 
8.9152513 
7.8305026 
9.9970504 
6.8017537 
6.8061781 


-10 
-10 
-10 
-10 
-10 


7.3206875 
7.3192127 
9.9985252 
7.3199507 

7.5302093 
8.9152513 
7.8305026 
9.9970504 
7.3177379 
7.3221623 


-10 
-10 
-10 
-10 
-10 


Semi-minor axis — 6~o ^\ — e- 


Ratio of axes-293.98-r 294,98 


Mean radius 


Ellipticity = = e 

/o2 -62 




L^=l_e2 


a2 

^-!=„(l_e,) 


o2 a 


6 Vi - e2 







BIBLIOGRAPHY 



REFERENCES ON GEODETIC SURVEYING 

Adjustment of Observations, Wright and Hayford. D. Van Nostrand & 

Co., New York, 1904. 
Elements of Geodesy, Gore. John Wiley & Sons, New York, 1893. Gillespie's 

Higher Surveying, Staley. D. Appleton & Co., New York, 1897. 
Johnson's Theory and Practice of Surveying, Smith. John Wiley & Sons, 

New York, 1910. 
Manual of Spherical and Practical Astronomy, Chauvenet. J. B. Lippin- 

cott & Co., Philadelphia, 1885. 
Practica' Astronomy as Apphed to Geodesy and Navigation, DooUttle. 

John Wiley & Sons, New York, 1893. 
Precise Surveying and Geodesy, Merriman. John Wiley & Sons, New 

York, 1899. 
Principles and Practice of Surveying, Breed and Hosmer. John Wiley & 

Sons, New York, 1906. 
Text Book of Field Astronomy for Engineers, Comstock. John Wiley & 

Sons, New York, 1902. 
Text Book of Geodetic Astronomy, Hayford. John Wiley & Sons, New York, 

1898. 
Text Book on Geodesy and Least Squares, Crandall. John Wiley & Sons, 

New York, 1907. 
Geodesic Night Signals, Appendix No. 8, Report for 1880, U. S. Coast and 

Geodetic Survey. 
Field Work of the Triangulation, Appendix No. 9, Report for 1882, U. S. 

Coast and Geodetic Survey. 
Observing Tripods and Scaffolds, Appendix No. 10, Report for 1882, U. S. 

Coast and Geodetic Survey. 
Geodetic Reconnaissance, Appendix No. 10, Report for 1885, U. S. Coast 

and Geodetic Survey. 
Relation of the Yard to the Meter, Appendix No. 16, Report for 1890, U. S. 

Coast and Geodetic Survey. 
Fundamental Standards of Length and Mass, Appendix No. 6, Report for 

1893, U. S. Coast and Geodetic Survey. 
Perfected Form of Base Apparatus, Appendix No. 17, Report for 1880, U. S. 

Coast and Geodetic Survey. 

374 



BIBLIOGEAPHY 375 

Description of a Compensating Base Apparatus, Appendix No. 7, Report 

for 1882, U. S. Coast and Geodetic Survey. 
The Eimbeck Duplex Base-bar, Appendix No. 11, Report for 1897, U. S. 

Coast and Geodetic Survey. 
Measurement of Base Lines (Jaderin Method) with Steel Tapes and with 

Steel and Brass Wires, Appendix No. 5, Report for 1893, U. S. Coast 

and Geodetic Survey. 
Measurement of Base Lines with Steel and Invar Tapes, Appendix No. 4, 

Report for 1907, U. S. Coast and Geodetic Survey. 
Run of the Micrometer, Appendix No. 8, Report for 1884, U. S. Coast and 

Geodetic Survey. 
Synthetic Adjustment of Triangulation Systems, Appendix No. 12, Report 

for 1892, U. S. Coast and Geodetic Siirvey. 
Formulas and Tables for the Computation of Geodetic Positions, Appendix 

No. 9, Report for 1894, and Appendix No. 4, Report for 1901, U. S. 

Coast and Geodetic Survey. 
Barometric Hypsometry, Appendix No. 10, Report for 1881, U. S. Coast 

and Geodetic Survey. 
Transcontinental Line of Levehng in the United States, Appendix No. 11, 

Report for 1882, IT. S. Coast and Geodetic Survey. 
SeH-registering Tide Gauges, Appendix No. 7, Report for 1897, U. S. Coast 

and Geodetic Survey. 
Precise Leveling in the United States, Appendix No. 8, Report for 1899, and 

Appendix No. 3, Report for 1903, U. S. Coast and Geodetic Survey.- 
Variations in Latitude, Appendix No. 13, Report for 1891, Appendix No. 1, 

Report for 1892, Appendix No. 2, Report for 1892, and Appendix No. 

11, Report for 1893, U. S. Coast and Geodetic Survey. 
Tables of Azimuth and Apparent Altitude of Polaris, Appendix No. 10, 

Report for 1895, U. S. Coast and Geodetic Survey. 
Determination of Time, Latitude, Longitude, and Azimuth, Appendix No. 7, 

Report for 1898, U. S. Coast and Geodetic Survey. 
A Treatise on Projections, U. S. Coast and Geodetic Survey, 1882. 
Tables for the Polyconic Projection of Maps (Clarke's 1866 Spheroid), 

Appendix No. 6, Report for 1884, U. S. Coast and Geodetic Survey. 
Geographical Tables and Formulas, U. S. Geological Survey, 1908. 
Bibliography of Geodesy (Gore), Appendix No. 8, Report for 1902, U. S. 

Coast and Geodetic Survey. 

REFERENCES ON METHOD OF LEAST SQUARES. 

Manual of Spherical and Practical Astronomy, Chauvenet. J. B. Lippincott 

Co., PhOadelphia, 1885. 
Approximate Determination of Probable Error, Appendix No. 13, Report 

for 1890, U. S. Coast and Geodetic Survey. 
Theory of Errors and Method of Least Squares, Johnson. John Wiley & 

Sons, New York, 1893. 
Practical Astronomy as Applied to Geodesy and Navigation, Doolittle. 

John Wiley & Sons, New York, 1893. 



376 BIBLIOGEAPHY 

Precise Surveying and Geodesy, Merriman. John Wiley & Sons, New 

York, 1899. 
Adjustment of Observations, Wright and Hayford. D. Van Nostrand & 

Co., New York, 1904.' 
Text Book on Geodesy and Least Squares, Crandall. John Wiley & Sons, 

New York, 1907. 



INDEX 



PAGE 

Aberration of light (diurnal) 213 

Absolute length, correction for 36 

Absolute locations 4 

Accidental errors: 

laws of . 252 

nature of 247 

theory of 252-265 

Accuracy attainable in 

angle measurements 78 

barometric leveling 129 

base-line measurement 45 

closing triangles 102 

precise spirit leveling 161 

trigonometric leveling 139 

Adjustment of 

angle measiu-ements 81, 100, 312—332 

base-line measurements. 333 

level work 160, 344-359 

observations 3, 241-359 

quadrilaterals 90-100, 327 

triangles 89, 322—326 

Adjustments of 

Coast Survey precise level 155-156 

direction instrument 65 

European type of precise level 146-152 

repeating instrument 59 

Alignment corrections: 

• horizontal 40 

vertical '^2 

Alignment curve 64 

Altazimuth iastnmient 48, 51 

Altitude 167 

American Ephemeris 164 

Aneroid barometer 126, 127 

377 



378 INDEX 

FAOB 

Angles: 

accuracy of measurements 78 

adjustment of 81, 100, 312-332 

eccentric 75 

exterior and interior , 53 

instruments for measuring 47, 52, 60 

measurement of 47-80 

Apparent time 165 

Arcs, elliptic 108 

Arithmetic mean 244 

Associations, geodetic 1 

Astronomical determinations 163-226 

See also Azimuth, Latitude, Longitude, and Time. 

Azimuth 4, 109, 167, 203 

astronomical 204 

geodetic 204 

lines, planes and sections Ill 

marlfs 204 

periodic changes in 226 

Azimuthal angles 109, 117 

Azimuth determinations 203-226 

approximate 214 

at sea 225 

by meridian altitudes of sun or stars 205 

by observations on circumpolar stars 207-225 

direction method 215 

fimdamental formulas 208 

micrometric method 221 

repeating method 218 

Back azimuth 109, 113, 122 

errors 122 

Barometers, aneroid and mercurial 126, 127 

Barometric leveling 125, 126-130 

Base-bars 24-29 

compensating 26 

Eimbeck duplex 26 

general features of 25 

standardizing 33 

thermometric 26 

tripods for 27 

Base-Une measurements 24-46, 333-343 

accuracy of 45 

adjustment of 333 

check bases 5 

corrections required 24, 35-44 

duplicate hnes 334 



INDEX 379 

PAGB 

Base-line measurements — {continued) 

gaps, computing length of 44 

general law of probable error 336 

law of relative weight 337 

probable error of . . . , 65 

probable error of lines of unit length . ". 338, 339 

sectional lines 335 

standardizing bars and tapes 33 

with base-bars 24r-29 

with steel and brass wires 32 

with steel and invar tapes 30-32 

uncertainty of 46, 342 

Bessel's solution of geodetic problem 118 

Bessel's spheroid 106 

Bibhography 374 

Board signals 20 

Bonne's map projection 238 

Celestial sphere 166 

Chance, laws of 248 

Changes, periodic: 

in azimuth 226 

in latitude 1.96 

in longitude 203 

Check bases 5 

Chronograph .' 184 

Circumpolar stars 190, 207 

Clarke's spheroid 106 

Clarke's solution of geodetic problem: 

direct 116 

inverse 118 

Closed level circuits 160, 357 

Closing the horizon 53, 313, 315 

Coast and Geodetic Survey, United States 1 

papers of 1 

precise level 153 

Coefficient of refraction 138 

Co-functions: 

altitude 167 

declination , 167 

latitude 167 

Comparator 34 

Compensating base-bars 26 

Computation of geodetic positions 103-124 

Bessel's solution 118 

Clarke's solution 116 

Helmert's solution 118 



380 INDEX 

PAOB 

Computations of geodetic positions — {continued) 

inverse problem • 118 

Puissant's solution 113 

Computed quantities: 

most probable values of 296 

probable errors of ' 306-311 

Conditional equations 284 

Conditioned quantities: 

definition of 242, 284 

most probable values of 284-295 

probable errors of 304 

Convergence of meridians 88, 111 

Corrections in base-line work 24, 35-44 

Correlative equations 290 

Cross-section of tapes 38 

Culmination, meaning of 190 

Curvature and refraction (in elevation) 12 

Declination 167 

Degree, length of: 

meridian 228 

parallel of latitude 228 

Dependent equations 284 

Dependent quantities: 

definition of 242, 284 

most probable values of 284-295 

probable errors of 304 

Deviation of plumb line 124 

Dip of horizon 184 

Direction instrument 47, 50, 60 

adjustments of 71 

Direct observations 243 

Distances, polar and zenith 167 

Diiu-nal aberration 213 

Duplex base-bars 26 

DupUcate base lines 334 

Duplicate level lines 160, 346 

Earth, figure of: 

general figure 104 

practical figure 106 

precise figure 105 

Eccentric signals 20, 78 

stations : 75 

Eimbeck duplex base-bar 26 

Elevation of stations 62 

Ellipsoid, definition of 105 



INDEX 381 

FAOE 

EUiptio arcs 108 

Elongation, definition of 208 

Ephemeris, American ■ 164 

Equation of time 165 

Equations : 

conditional 284 

correlative 290 

dependent 284 

normal 273, 275 

observation 271 

probability > 257 

reduced observation 281 

Errors: 

classification of 245, 247 

facility of 255 

in precise leveling 143 

laws of 252 

probability of 256 

theory of : 252-265 

types of 254 

European precise level 141, 145 

adjustments of 146-152 

Exterior angles 53 

Figure adjustment 81, 87, 100, 312, 321-332 

Figure of earth: 

analytical considerations 110 

constants of 106 

general figure 104 

geometrical considerations 106 

practical figure 106 

precise figure 105 

Filar micrometer: 

description of 66 

reading the micrometer , 67 

run of the micrometer 68 

Flattening of the Earth's poles 104, 105 

Foot pins and plates 158 

Gaps in base lines 44 

Geodesic liae 109 

Geodesy: 

definition of 1 

history of 1 

scope of 2 

Geodetic associations 1 

Geodetic leveling 125-162 



382 INDEX 

PAOE 

Geodetic map drawing 227-240 

Geodetic positions, computation of 103-124 

Geodetic quadrilateral 7, 90, 327 

Geodetic surveyiug 1-240 

Geodetic work in the United States , 1 

Geoid, definition of 106 

Geometric mean 244 

Harrebow-Talcott latitude method 193 

Heat radiation 47 

Height of stations 17 

Heliotropes 21 

Helmert's solution of geodetic problem 118 

History of plane and geodetic surveyiag 1 

Horizontal alignment 40 

Hour angle .• 164, 167 

Independent quantities: 

definition of 241 

most probable values of 266-283 

probable errors of 300-304 

Indirect observations 243 

Instruments, geodetic; see Angles, Astronomical determinations. Base- 
line measurements and geodetic leveUng. 

Interior angles 53 

Intermediate points in leveling 160, 217 

International Geodetic Association 1 

Intervisibility of stations 11, 14 

Invar tapes 32 

Inverse geodetic problem 118 

Jaderin base-Une methods: 

with tapes 31 

with wires 32 

Latitude 109, 167, 186 

astronomical 186 

geocentric 187 

geodetic 186 

locating a parallel of 120 

periodic changes in 196 

Latitude determinations 188-196 

at sea 196 

by circumpolar culminations 190 

by Harrebow-Talcott method 193 

by meridian altitudes of sun 188 

by prime- vertical transits 192 

by zenith telescope 193 



INDEX 383 

PAGE 

Law of 

coefficients in correlative equations 294 

coefficients in normal equations 280 

facility of error 257 

Laws of 

chance 248-250 

errors 252 

weights 82 

Least squares, method of 241-359 

Lengths of bars and tapes 24, 33 

Leveling: 

barometric 125, 126-130 

geodetic 125-162 

precise spirit 125, 139-162 

trigonometric 125, 130-139 

Level work: 

adjustments 160, 344-359 

branch hnes, circuits and nets 359 

closed circuits 160, 357 

duphcate hnes 160, 346 

general law of probable error 347 

intermediate points : 160, 355 

law of relative weight 348 

level nets 161, 352 

multiple hnes 160, 350 

probable error of lines of unit length 348, 349 

sectional hnes 347 

simultaneous lines 160 

Light, diurnal aberration of 213 

L. M. Z. problem 103 

Locating a parallel of latitude 120 

Locations, absolute and relative 4 

Longitude 109, 197 

astronomical 197 

geodetic 197 

periodic changes in *. 203 

Longitude determinations 197-203 

at sea 203 

by lunar observations 198 

lunar culminations 199 

lunar distances 199 

lunar occultations 199 

by special methods 198 

flash signals 198 

special phenomena 198 

by telegraph 200 

arbitrary signals 202 



384 INDEX 

PAGE 

by telegraph — (continued) 

standard time signals 201 

star signals 201 

by transportation of chronometers 199 

Loxodrome 233 

Map projections 227-240 

conical 234 

Bonne's projection 238 

Mercator's conic '. 236 

simple conic 235 

cylindrical ' .'229 

Mercator's cylindrical 231 

rectangular cylindrical 231 

simple cyUndrioal i 229 

polyconic 240 

rectangiilar polyconic 241 

simple polyconic 240 

trapezoidal 234 

Mean absolute error .....' 305 

Mean error 305 

Mean of errors 305 

Mean radius of the earth 44 

Mean sea level 43, 125 

Mean solar time 165 

Measures of precision 262, 304 

Mercator's projections: 

conic 236 

cylindrical 231 

Mercurial barometer 126, 127 

Meridian 167 

lengths 228 

line, plane, and section 167 

Meridians, convergence of 88, 111 

Method of least squares 241-359 

Micrometer: 

filar 66 

microscope 65 

reading of 67 

run of 68 

Mistakes 247 



INDEX 385 

FAQB 

Most probable values of — {continued) 

independent quantities 266-283 

observed quantities 242, 266, 295 

Multiple level lines 169, 360 

Nadir 167 

Nautical Almanac 164 

Night signals 23 

Normal ■ 110 

Normal equations 273, 276 

law of coefficients 280 

Normal tension 40 

Observation equations: 

definition of 271 

reduced 281 

reduction to unit weight 278 

Observations: 

adjustment of 3, 241-359 

classification of 243 

Observed quantities: 

most probable values of 266-295 

probable errors of 297-305 

Observed values, definition of 242 

OvalcSd, definition of 105 

Papers of U. S. Coast and Geodetic Survey 1 

Parallax (in altitude) 167, 171 

Parallel of latitude, location of 120 

Parallels, length of one degree 228 

Phase 20 

Phaseless targets 20 

Plane surveying, history of 1 

Plumb-line deviation 124 

Polar distance 167 

Pole signals 20 

Precise spirit leveling 125, 139-162 

accuracy attainable 161 

adjustment of results ; .' 160, 344-359 

n^Oof ail^TQTr Tll-OnJoO loTI-ol 1 d.9 IRS 



386 INDEX 

PAGE 

Precise spirit leveling — {continued) 

instruments used 139, 145, 153 

methods 143, 145 

rods and turning points 158 

sources of error 143 

Primary triangles and systems 9 

Prime vertical 110, 167 

Prime-vertical transits 192 

Probability: 

equation of 257, 260 

laws of chance 248 

Probable error: 

general value of 29*9 

meaning of 297 

Probable errors of 

angle measurements 79 

base-Une measurements 46 

computed quantities 306-311 

conditioned quantities 304 

dependent quantities 304 

independent quantities 300-304 

observed quantities 297-305 

Projection of maps 227-240 

See Map projections for list of types. 

Puissant' s solution of geodetic problem: 

direct 113 

inverse 118 

PuU, with tapes and wires 24, 30, 38 

Quadratic mean 244 

Quadrilateral, geodetic 7, 90, 327 

algebraic adjustment of 90-102 

approximate 92 

definition of rigorous 96 

least square adjustment of 327 

Quantities: 

classification of 241 

most probable values of 266-296 

computed quantities 296 

observed quantities 266-296 

probable errors of 297-311 

computed quantities 306-311 

observed quantities 297-305 

Radiation, heat 47 

Reading micrometers 67 

Reconnoissance 10 



INDEX 387 

PAGE 

Reduced observation equations 281 

Reduction to center 75 

Reduction to mean sea level 43 

Refraction: 

angular 167 

coefficient of ■ 138 

in elevation 12 

Relative locations 4 

Repeating instruments 47, 49, 62 

adjustments of 59 

Residual errors 245 

Residuals 245 

Rhumb line 233 

Right ascension 167 

Run of micrometer 68 

Sag 24, 30, 39 

Secondary triangles and systems 9 

Sectional lines: 

base hues 335 

level lines 347 

Sidereal time 165, 168 

Signals at stations 18 

board 20 

eccentric 20, 78 

hehotrope 21 

night 23 

phaseless 20 

pole 20 

Simultaneous level hnes 160 

Single angle adjustment 312 

Solar time 165 

Spherical excess 88, 89, 90 

Spheroid: 

Bessel's , 106 

Clarke's 106 

definition of 105 

Spirit leveling, see Precise spirit leveling. 

Standardizing bars and tapes 33 

Standard time 165 

Station adjustment 81, 84, 312, 313-319 

Stations: 

elevation of 14, 17 

height ,of 17 

intervisibility of 11, 14 

marks 17 

selection of 10 



388 INDEX 

PAGE 

Stations — {continued) 

signals and targets 18 

towers 17, 18 

triangulation 5 

Steel tapes 24, 30, 32 

corrections required in tape measurements 24, 33-39 

standardizing 33 

Steel and brass wires 32 

Systematic errors 247 

Tables 361-373 

Tangents 110, 120 

Targets 18 

Telegraphic determination of longitude 200 

Telescope, zenith 193 

Temperature corrections in base-line work 24, 31, 36 

Tension, tapes and wires 40 

Tertiary triangles and systems 9 

TheodoHte 48 

Theory of errors ' 252-265 

comparison of theory and experience 264 

Theory of weights 81, 243 

Thermometric base-bars 26 

Tide gauges: 

automatic 125 

staff 126 

Time 164 

conversion of 165, 169, 170 

general principles 164 

varieties of 165 

Time determinations 164-186 

at sea 184 

by equal altitudes of sun V . 176 

by siagle altitudes of sun 171 

by sun and star transits 181 

choice of methods 184 

Towers, station and signal 17, 18, 47 

Transit, astronomical 183, 185 

Triangles: 

accuracy in closing 102 

adjustment of 89, 322-326 

classification of 9 

computation of 102 

Triangulation: 

adjustments and computations 81-102, 312-332 

general scheme 4 

principles of , 4-23 



INDEX 389 

FAOB 

Triangulation — (continued) 

stations 5, 10 

systems 5-9 

Trigonometrical leveling 125, 130-139 

accuracy attainable 139 

observations at one station 133 

reciprocal observations 136 

sear-horizon method 131 

Tripods for 

angle-measuring instruments 18 

base-bars 27 

leveling instruments 143 

True errors 245 

True values 242 

Turning points 158 

Uncertainty of base-line measurements 46, 342 

United States Coast and Geodetic Survey 1 

papers of 1 

precise level 153 

Values, classification of 242, 244 

Variations, periodic: 

in azimuth 226 

in latitude 196 

in longitude 203 

Vertical alignment 42 

Weight: 

laws of 82 

theory of 81, 243 

Wires, steel and brass 32 

Zenith 167 

Zenith distance 167 

Zenith telescope 193