Library OF CA -PARTMENT OF CIVIL Ubuuu>- . ENGINEERING U. otC. ASSOCIATION LIBRARY WORKS OF PROFESSOR MERRIMAN PUBLISHED BY JOHN WILEY & SONS TREATISE ON HYDRAULICS. 8vo, 593 pages, $5.00 MECHANICS OF MATERIALS. 8vo. 518 pages. $5.00 PRECISK SURVEYING AND GK<. DESY. 8vo, 261 pages, $2.50 METHOD OF LEAST SQUARFS 8vo, 238 pa.ces. $2.00 ELEMENTS OF SANITARY ENGINEEMNG. 8vo, 252 pages, net. '2.00 ELEMENTS OF MECHANICS, 12010, 172 pages, mt, $1.00 STRENGTH OF MATERIALS. 12010, 15' pages. ;,, 382 pas.es. $2.50 PART IV. HIGHER STRUCT uRt*. Svo, 385 paj.es, $2.50 By Professors MERRIMAN and BROOKS HANDBOOK FOR SuKVt-.YxKS. 12111 ., ^40 pages $2.00 Edited by Professors MERRIMAN and WOODWARD SERI s OF MATHEMATICAL MONOGRAPHS. Eleven Volumes, Svo. Each, $1.00 ELEMENTS ijjM - PF PRECISE SURVEYING AND GEODESY. BY MANSFIELD MERRIMAN, PROFESSOR OF CIVIL ENGINEERING IN LEHIGH UNI\ SECOND EDITION, REVISED. SECOND THOUSAND. NEW YORK; JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1907. v,i /f^o^ijA.! Engineering Library Copyright, 1899, BY MANSFIELD MERRIMAN. ROBERT DRUMMOND, PRINTER, NEW YORK. jl PREFACE. THIS volume is designed as an introduction to those precise methods of surveying which are coming more and more into use in America as land becomes more valuable. The theory at the foundation of these methods is that of the principle of least squares, which shows both how to obtain the most proba- ble value of a measured quantity and how to ascertain its degree of precision. Four chapters are devoted to this theory and to its application in common triangulations, base-line measurements, and leveling. Then follows the consideration of the earth as a sphere and as a spheroid, with methods for determining azimuth and for the execution of triangulations which 1 require the figure of the earth to be taken into account. The treatment has been made as concise and elementary as possible, the volume being designed as a text-book for engi- neering students and a manual for surveyors rather than as a treatise for geodeticians. In this edition all known errors have been corrected, several new problems have been inserted, and new matter on map projections and on American geodetic work has been added in Arts. 68 and 86. M. M. 793981 CONTENTS, CHAPTER I. THE METHOD OF LEAST SQUARES. ART. PAGE 1. ERRORS OF OBSERVATIONS 7 2. LAW OF PROBABILITY OF ERROR .* . ^ 3. THE PRINCIPLE OF LEAST SQUARES 12 4. WEIGHTED OBSERVATIONS 14 5. OBSERVATION EQUATIONS 16 6. INDIRECT OBSERVATIONS OF EQUAL WEIGHT 19 7. INDIRECT OBSERVATIONS OF UNEQUAL WEIGHT . . . . 22 8. SOLUTION OF NORMAL EQUATIONS 24 9. THE PROBABLE ERROR . 25 10. PROBABLE ERRORS FOR INDIRECT OBSERVATIONS .... 29 11. PROBABLE ERRORS OF COMPUTED VALUES 31 12. CRITICAL REMARKS .... .-' 33 CHAPTER II. PRECISE PLANE TRIANGULATIOM. 13. COORDINATES AND AZIMUTHS 36 14. MEASUREMENT OF ANGLES : . . -. ' 39 15. PROBABLE ERRORS AND WEIGHTS OF ANGLES 43 1 6. THE STATION ADJUSTMENT . .' . . . . . * ,- . 46 17. ERRORS IN A TRIANGLE . v . . . . .. . 49 18. ADJUSTMENT OF A TRIANGLE . . . . . . . . ; . 51 19. TRIANGLE COMPUTATIONS . t . v * 53 20. Two CONNECTED TRIANGLES 57 3 4 CONTENTS. PACK 21. DIRECT OBSERVATIONS WITH ONE CONDITION 59 22. INTERSECTIONS ON A SECONDARY STATION 61 23. THE THREE-POINT PROBLEM .64 24. GENERAL CONSIDERATIONS 67 CHAPTER III. BASE LINES. 25. PRINCIPLES AND METHODS 71 26. PROBABLE ERROR AND UNCERTAINTY . . . . . . 72 27. BASES AND ANGLES .74 28. STANDARD TAPES .<.'.'..'. . . 76 29. MEASUREMENT WITH A TAPE . . 79 30. BROKEN BASES . .'..'. . . 83 31. REDUCTION TO OCEAN LEVEL 84 CHAPTER IV. LEVELING. 32. SPIRIT LEVELING . . . ': . . . .. /.'. . . . . 87 33. DUPLICATE LINES . . . ... . . . . . . . 89 34. PROBABLE ERRORS AND WEIGHTS 91 35. ADJUSTMENT OF A LEVEL NET . 94 36. GEODETIC SPIRIT LEVELING 96 37. REFRACTION AND CURVATURE \. . . 99 38. VERTICAL ANGLES . . . ... . . . ... .101 39. LEVELING BY VERTICAL ANGLES 104 CHAPTER V. ASTRONOMICAL WORK. 40. FUNDAMENTAL NOTIONS ......;-.-. 107 41. AZIMUTH WITH THE SOLAR TRANSIT no 42. AZIMUTH BY AN ALTITUDE OF THE SUN 114 43. AZIMUTH BY POLARIS AT ELONGATION 117 44. AZIMUTH BY POLARIS AT ANY HOUR-ANGLE 121 45. LATITUDE BY THE SUN 124 CONTENTS. 5 PAGE 46. LATITUDE rv A STAR 127 47. TIME . . . ., . . | f ?, . ... . , % _ . . . 129 48. LONGITUDE . . . . .4, 132 49. PRECISE DETERMINATIONS ". '." .134 CHAPTER VI. SPHERICAL GEODESY. 50. EARLY HISTORY .... . ,, . ........ 137 51. HISTORY FROM 1300 TO 1750 . . 140 52. MEASUREMENT OF MERIDIAN ARCS 144 53. THE EARTH AS A SPHERE 147 54. LINES ON A SPHERE . . .' . . . 149 55. ANGLES AND TRIANGLES . . . . . * . . . . . 151 56. LATITUDES, LONGITUDES, AND AZIMUTHS 153 CHAPTER VII. SPHEROIDAL GEODESY." 57. PROPERTIES OF THE ELLIPSE . 156 58. DISCUSSION OF MERIDIAN ARCS .,-... ... . . . . 158 59. PLUMB-LINE DEFLECTIONS. , . ......... 164 60. DIMENSIONS OF THE SPHEROID 168 61. LENGTHS OF MERIDIAN AND PARALLEL ARCS . . . .170 62. NORMAL SECTIONS AND GEODESIC LINES 172 63. TRIANGLES AND AREAS. . . . . . . . ..- . . . 175 64. LATITUDES, LONGITUDES, AND AZIMUTHS . . . .*..-. 177 CHAPTER VIII. GEODETIC COORDINATES AND PROJECTIONS. 65. THE COORDINATE SYSTEM. . . ... . * . . . 181 66. LMZ COMPUTATIONS . . . . , . ^ . , . 183 67. THE INVERSE LMZ PROBLEM 186 68. MAP PROJECTIONS . . . . . . . 189 69. THE POLYCONIC PROJECTION * . 192 70. RECTANGULAR SPHERICAL COORDINATES 194 O CONTENTS. CHAPTER IX. GEODETIC TRIANGULATION. PAG3 71. RECONNAISSANCE. ". 198 72. STATIONS AND TOWERS 200 73. SIGNALS 204 74. HORIZONTAL ANGLES ...... 206 75. THE STATION ADJUSTMENT 210 76. TRIANGLE COMPUTATIONS 212 77. THE FIGURE ADJUSTMENT 216 78. CONDITIONED OBSERVATIONS 219 79. ADJUSTMENT OF A POLYGON 222 80. ADJUSTMENT OF A QUADRILATERAL . .225 81. FINAL CONSIDERATIONS 228 CHAPTER X. THE FIGURE OF THE EARTH. 82. THE EARTH AS A SPHEROID 232 83. THE EARTH AS AN ELLIPSOID 235 84. THE EARTH AS AN OVALOID 239 85. THE EARTH AS A GEOID 242 86. CONCLUSION 245 CHAPTER XI. TABLES. 87. EXPLANATION OF THE TABLES 249 I. MEAN CELESTIAL REFRACTION 251 II. LENGTHS OF ARCS OF THE MERIDIAN . . . . . . .252 III. LENGTHS OF ARCS OF PARALLELS 253 IV. LOGARITHMS FOR GEODETIC COMPUTATIONS 254 V. LOGARITHMS FOR THE LMZ PROBLEM . . . . .255 VI. CONSTANTS AND THEIR LOGARITHMS 256 INDEX 257 PRECISE SURVEYING AND GEODESY. CHAPTER I. THE METHOD OF LEAST SQUARES. 1. ERRORS OF OBSERVATIONS. The Method of Least Squares furnishes processes of com- putation by which the most probable values of quantities are found from the results of measurements. The simplest case is that of a quantity which is directly measured several times with equal precision ; here it is universally agreed that the arithmetic mean of the several values is the most probable value of the quantity. When a quantity is measured the result of the operation is a numerical value called an observation. If Z be the true value of a quantity and J/ t and J/ a be two observations upon it, then Z M l and Z M^ are the errors of those ob- servations. Constant or systematic errors are those which result from causes well understood and which can be computed or eliminated. As such may be mentioned: theoretical errors, like the effects of refraction upon a vertical angle, or the effects of temperature upon a steel tape, which can be com- puted when proper data are known and hence need not be classed as real errors; instrumental errors, like the effects of 7 A s:-?*-:*--*^ 8 THE METHOD OF LEAST SQUARES. I. an imperfect adjustment of an instrument, which can be removed by taking proper precautions in advance; and per- sonal errors which are due to the habits of the observer, who may, for example, always give the reading of a scale too great. All these causes are to be carefully investigated and the resultant errors removed from the final observations. Mistakes are errors due to such serious mental confusion that the observation cannot be regarded with any confidence, as for instance writing 53 when 35 is intended. Observa- tions affected with mistakes must be rejected, although when these are of small magnitude it is sometimes not easy to distinguish them from errors. Accidental errors are those that still remain after all con- stant errors and mistakes have been carefully investigated and eliminated. Such, for example, are the errors in leveling arising from sudden expansions of the bubble and standards, or from the effects of the wind, or from irregular refraction. They also arise from the imperfections of human touch and sight, which render it difficult to handle instruments delicately or to read verniers with perfect accuracy. These are the errors that exist in the final observations and whose discus- sion forms the subject of this chapter. However carefully the measurements be made, the final observations do not agree; all of these observations cannot be correct, since the quantity has only one value, and each of them can be regarded only as an approximation to the truth. The absolutely true value of the quantity in question cannot be ascertained, but instead of it one must be deter- mined, derived from the combination of the observations, which shall be the " most probable value," that is to say, the value which is probably nearest to the true value. The difference between the most probable value of a quantity and an observation is called the * residual error ' of that observation. Thus, if z be the most probable value of 2. LAW OF PROBABILITY OF ERROR. 9 a quantity derived from the observations M l and M and v l and v t be the residual errors, then v l = z - Mi, v, = z - M t . (i) When the measurements are numerous and precise the most probable value z does not greatly differ from the true value Z, and the residuals do not greatly differ from the true errors. Prob. i. Eight measurements of a line give the values 186.4, 186.3, 186.2, 186.3, l8 6-3, 186.2, 185.9 an( * 186.4 inches, and its most probable length is their arithmetic mean. Compute the eight residual errors ; find the sum of the positive residuals and the sum of the negative residuals. 2. LAW OF PROBABILITY OF ERROR. The probability of an error is the ratio of the number of errors of that magnitude to the total number of errors. If there be 100 observations of an angle which give 27 errors lying between i" and 2" the probability that an error lies between these limits is 0.27. Probabilities are thus measured by numbers lying between o (impossibility) and I (certainty). A marksman firing at a target with the intention of hitting the center may be compared to an observer, the position of a shot on the target to an observation, and its distance from the center to an error. If the marksman be skilled and all horizontal errors, like the effect of gravity, be eliminated in the sighting of the rifle, it is recognized that the deviations of the shots, or errors, are quite regular and symmetrical. First, it is noticed that small errors are more frequent than large ones; secondly, that errors on one side are about as frequent as on the other; and thirdly, that very large errors do not occur. Moreover, it is known that the greater the skill of the marksman the nearer are his shots to the center of the target. As an illustration a record of one thousand shots fired 10 THE METHOD OF LEAST SQUARES. I. from a battery gun at a target six hundred feet distant may be considered. The target was a rectangle fifty-two feet long by eleven feet high, and the point of aim was its central hori- zontal line. All the shots struck the target, and the record of the number in the eleven horizontal divisions, each one foot in width, is as follows: In top division In second division In third division In fourth division In fifth division In middle division In seventh division In eighth division In ninth division In tenth division In bottom division Total 1 shot 4 shots 10 shots 89 shots 190 shots 212 shots 204 shots 193 shots 79 shots 16 shots 2 shots i ooo shots The figure shows by means of ordinates the distribution of these shots; A being the top division, O the middle, and B the bottom division It will be observed that there is a slight preponderance of shots below the middle, and there is reason to believe that this is due to a constant error of gravitation not entirely elim- inated in the sighting of the gun. If this series of shots A Q ' ' ' ' were to be repeated again under exactly similar condi- tions, it might be fair to infer that 0.212 would be the prob- ability of a given shot striking the middle division, and that o.ooi would be the probability of striking the top division. Thus the probability of an error decreases with the magnitude of the error, 2. LAW OF PROBABILITY OF ERROR. II In treatises on the Method of Least Squares the theory of mathematical probability is f applied to the deduction of the relation between an error x, and its probability y. The equation deduced is y = -*-, (2) where c and h are constants that depend upon the precision of the measurements and e is the number 2.71828- . This equation expresses the law of probability of accidental errors of observations. It shows that y has its greatest value when x is zero, that y becomes very small when x is very large, and that the same value of y is given by equal positive and negative values of x. The figure shows the curve expressed by the equation, x and y being parallel to the axes OX and OY, and OM being any error whose probability y is given by the ordinate MN. This law is deduced under the supposition of a very large number of errors, and hence in a particular case close agree- ment is not to be expected. For any series of errors the values of c and //* can be computed and the theoretic number of errors can then be compared with those actually observed. For example, in the above case of the shots on the target, the value of c can be found to be 0.234 and that of h* to be 0.173, and the following comparison shows the agreement of practice with theory: For division No. i 2 3 45 6 7 8 91011 Actual shots i 4 10 89 190 212 2^4 193 79 16 2 Theoretic shots 3 15 50 118 197 234 197 118 50 15 3 The dotted curve on the graphic representation shows the theoretic distribution of the shots. In general it may be said 12 THE METHOD OF LEAST SQUARES. I. that the results -of observation are in good agreement with the theoretic law, and that this agreement is closer the greater the number of errors considered. Prob. 2. Given the equation y = o.234? + + pv H * = a minimum. (4) Hence a more general statement of the principle of least squajj is: In observations of unequal precision the most probable values of the observed quantities are those which render the sum of the weighted squares of the residual errors a minimum. Here it is seen that the term " weighted square of a re- sidual " means the product of the square of the residual by its weight. An application of this principle to the case of weighted observations on a single quantity will now be made. Let z be the most probable value of the quantity whose observed values are M 1 , J/, , . . . M n having the weights/, ,/,,.. ./. Then the residuals are z M l , z M 9 , . . . z M n , and from the general principle of least squares given by (3), p,(z - M^+pte - M$ + . . . +/(> - M H )* = a minimum. The first derivative of this, placed equal to zero, gives /,(* - M,) +ft,(g -M,)+ . . . + fm (s - M,) = o, from which the most probable value of z is j.A The value of z thus found is sometimes called the weighted arithmetic mean, and the method of computing it is fre- quently expressed by the rule: Multiply each observation by its weight and divide the sum of the products by the sum of the weights. Prob. 4. Prove the principle (4) directly from the law of prob- l6 THE METHOD OF LEAST SQUARES. I. ability of error given by (2), assuming that J? represents the weight of the observation whose error is x. 5. OBSERVATION EQUATIONS. When observations are taken of several related quantities, the measurements are usually made upon functions of those quantities. Thus the sum and differ- ence of two quantities might be ob- served instead of the quantities them- selves. Such measurements produce "indirect observations" which are generally represented by equations called "observation equations." To illustrate how they arise, let the following practical case be considered. Let O represent a bench-mark, and X, Y, Z, three points whose elevations above O are to be determined. Let five lines of levels be run, giving the results Observation i. X above O = 10.35 ^ eet ' Observation 2. Y above X = 7.25 feet. Observation 3. Y above O = 17.63 feet. Observation 4. Y above Z = 9.10 feet. Observation 5. Z below X = 1.94 feet. Here it will be at once perceived that the measurements are discordant; if observations I and 2 are taken as correct, the elevation of X is 10.35 feet, and that of Y is 17.60 feet; if 2 and 3 are correct, then X is 10.38 feet and Y is 17.63 feet; and it will be found impossible to deduce values that will exactly satisfy all the observations. Let the elevations of the points X, Y, and Z above O be denoted by ;r, y, and s, then the observations furnish the following equations: 1. *== 10.35, 2. y-x= 7.25, 3. y = 17.63, 4. y- z = 9.10, 5. x z = 1.94. 5. OBSERVATION EQUATIONS. 1 7 The number of these equations is five, the number of the unknown quantities is th,re~e, and hence an exact solution cannot be made. The best j;hat can be done is to find values for x, y, and z which are the most probable, and these will be found in the next Article by the help of the principle of least squares. Observations are called " direct " when made upon the 1 quantity whose value is sought, and " indirect " when made \ upon functions of the quantities whose values are required. J Thus in the above example the first and third observations are direct, and the others are indirect, being made upon differences of elevation instead of upon the elevations them- selves. Indirect observations are of frequent occurrence in the operations of precise surveying. Quantities are said to be " independent" when each can vary without affecting the value of the others; thus in the above example the elevation of any one station above the bench-mark O is entirely independent of the elevations of the others, or in other words there is no necessary relation between the values of x^ y, and z. Quantities are said to be " conditioned " when they are so 1 related that a change in one necessarily affects the values of \ the others; thus if the three angles of a plane triangle be called x, y, and z t it is necessary that x -\- y -f- s = 180 and the values to be found for the angles must satisfy this condi- tion. In stating observation equations it will often be found best to select the quantities to be determined in such a way that they shall be independent; thus if the three angles of a triangle are observed to be 62 20' 43", 36 14' 06", and 81 25' 08", let x and y denote the most probable values of the first and second angles, then the observation equations are x = 62 20' 43", y 36 14 06, 180 x y 81 25 08, 1 8 THE .^METHOD OF LEAST SQUARES. I. the last of which may be written 9834' 52", and here x and y are independent quantities. Thus by properly limiting the number of unknown quantities these can always be rendered independent of each other. As a second example of the statement of observation equations take the following values of the angles measured at North Base, Keweenaw Point, on the United States Lake Survey : CNM=. 55 57' 58".68, MNQ= 48 49 13 .64, CNQ = 104 47 12 .66, QNS= 54 38 15 .53, MNS = 103 27 28 .99. The object of these observations is to find the values of the four angles around the point N\ but if x^ y, z> and w repre- sent these angles, then x-\-y-\-z-\-w= 360 and the quantities are conditioned. To make the quantities inde- pendent only three unknowns should be taken; thus let CNM x, MNQ y, and QNS = 2, then the observation equations are *= 55 57' 58".68, y = 48 49 13 .64, = 104 47 12 .66, ~ 54 38 15 -53, = 103 27 28 .99, and in the next Article it will be shown how the most prob- able values of x, y, and z are to be found. Thus, in general, observations upon several quantities lead to observation equations whose number is greater than that of the unknown quantities, and no system of values can be found that will exactly satisfy the observation equations. They may, however, be approximately satisfied by many 6. INDIRECT OBSERVATIONS OF EQUAL WEIGHT. 19 systems of values; and the problem is to determine that system which is the most probable and hence the best. ProK 5. State observation equations for the above example, tak- ing SNQ = j, SNM-= t, Stfb = u. 6. INDIRECT OBSERVATIONS OF EQUAL WEIGHT. When observation equations have been written so that the unknown quantities have no necessary relation to each other, the case is called that of indirect independent observations. Let M l , My , . . . M n be n observations of equal weight made upon functions of the unknown quantities x, y, z, etc. Let the observations give the following observation equations: a n x -\- b n y + c n z + . . . = M n , in which a t , a^ , . . . a n , b l , b^ , . . . b n , etc., are known co- efficients of the unknown quantities. The most probable values of x, y, z^ etc., when found and inserted in the equations will not exactly satisfy them, but leave small resid- ual errors, v l , z/ a , . . . v n ; thus strictly a^x -f- b^y -\- c^z -\- . . . M^ = i\ , a n x + b n y + c n z + M n v n , and, by the principle of least squares given by (2) in Art. 2, the sum of the squares of these residuals must be a minimum in order to give the most probable values of x y y, and z. In order to find the condition for the most probable value of x let the terms independent of x in the equations be denoted by N^ , N t , . . . N n ; then they may be written a n x + N n = v 2O THE METHOD OF LEAST SQUARES. I. Squaring both terms of these equations, and adding, gives J + . . . + (a n x + and this is to be made a minimum to give the most probable value of x. Differentiating it with respect to x, and placing the first derivative equal to zero, there results ,(,* + N t ) + a,(a^ + /V,) + . . . + a,(a nX + TV.) = o, (6) and this is the condition for the most probable value of x. In like manner a similar condition may be stated for each of the other unknown quantities. The conditions thus stated are called " normal equations, " and their solution will furnish the most probable values of the required quantities. The following is hence the rule for the adjustment of observations of equal weight involving several independent quantities: For each of the unknown quantities form a normal equa- tion by multiplying each observation equation by the coefficient of that unknown quantity in that equation and adding the results. Then the solution of these normal equations will furnish the most probable values of the unknown quantities. In forming the normal equations it should be particularly noticed that the signs of coefficients are to be observed in performing the multiplications, and also that when the unknown quantity under consideration does not occur in an observation equation its coefficient is o. As an example the five observation equations at the begin- f the last Article will be taken. They may be written 1. x = 10.35, 2. X+y = 7.25, 3-- y = 17.63, 4. y z 9.10, 5. x * = 1.94. 6. INDIRECT OBSERVATIONS OF EQUAL WEIGHT. 21 Now to form the normal equation for x> the first equation is to be multiplied by I, the second by I, and the fifth by I ; and adding these, ^ ~3* - y - * = 5-04. In like manner to find the normal equation for 7, the second equation is multiplied by I, the third by i, and the fourth by i, whence - X + & - 2 = 33.98. Lastly, to find the normal equation for #, the fourth equation is multiplied by i and the fifth by I, and adding, ^ x y -\- 2z II .04. These three normal equations contain three unknown quan- tities, and their solution gives x = 10.372, y =r 17.61, z = 8.47 feet, which are the most probable values of the three elevations. As a second example the three observation equations near the middle of the last Article are x = 62 20' 43", y = 36 14 06, x + y = 9 8 34 52. Applying the rule, the two normal equations are 2x + y = 1 60 55 35, x + 2y = 134 48 58, and the solution of these gives x =. 62 20' 44", y = 36 14' 07", whence the third angle of the triangle is 180 degrees minus the sum of these, or 81 25' 09". By comparing these with the observed values it will be seen that each observation is cor- rected by the same amount; this is because the observations are of equal weight and each angle is similarly related to the other two. 22 THE METHOD OF LEAST SQUARES. I. Prob. 6. Form and solve the normal equations for the observa- tion equations of Prob. 5. 7. INDIRECT OBSERVATIONS OF UNEQUAL WEIGHT. The two preceding Articles give the method of adjusting indirect observations of equal weight upon several independ- ent quantities; now is to be investigated the case of indirect observations of different weights upon such quantities. Let /, , / a , . . . p n be the weights of the n observations M l , M , ... M n , so that the observation equations are a^x + b^y + c^z -f- . . . = M l , with weight/, , + b,y + cjs + . . . = M, , with weight / 3 , a n x -f- b n y -\- c n z -f- . . . = M n , with weight p n . Now if the first equation were written/, times, the second / a times, etc., all the equations would have the same weight and the rule of the last Article would apply. That is, if each of the above equations be multiplied by the coefficient of x in that equation, and also by its weight, the sum will be the condition for the most probable value of x\ and in like manner is found the condition for the most probable value of each of the other unknowns. These conditions are the normal equations. The following is hence the rule for the adjustment of observations of unequal weight upon several independent quantities: For each of the unknown quantities form a normal equa- tion by multiplying each observation equation by the coefficient of that unknown quantity in that equation, and also by its weight, and adding the results. The solution of these normal equations will furnish the most probable values of the unknown quantities. In applying this rule the same precautions are to be observed regarding signs of the coefficients as before stated. 7, INDIRECT OBSERVATIONS OF UNEQUAL WEIGHT. 23 An algebraic expression of the normal equations can be made by introducing the following abbreviations: [X] ==. A*," '-k, A".' + , + \_pab~] = //z A + A^A -[- . . . + [paM] = AM*i and then the normal equations can be written \_pcf\x + \_pab~\y-\- \_pac\s + etc. \_paM~\, * + etc. - , + ]Jbc\y + [X] + etc. - ' Here it will be seen that the coefficients of the unknown quantities in the first vertical column are the same as those in the first horizontal line, those in the second column the same as those in the second line, and so on. This is a char- acteristic of normal equations and serves as a check when they are deduced by direct application of the rule. If the observations are of equal weight, p is to be made unity throughout, and the method reduces to that of the last Article. As a numerical illustration let five observations produce the five observation equations \.-\-x = o, with weight 3, 2. ~\~ y > with weight 19, 3. -f- 2 = o, with weight 13, 4. + x + y + 0.34, with weight 17, 5. -f- y -f- z = o. 18, with weight 6. From these the normal equations, formed either by the rule or by help of the algorithm, are + i?y = + 5.78, + 427 + 6s = + 4.70, i. 08, 24 THE METHOD OF LEAST SQUARES. I. whose solution furnishes the results x = + 0.285, y = + 0.005, * = 0.059, which are the most probable values of the required quantities. Prob. 7. In a plane triangle six observations give A 42 17' 35", three observations give B = 56 40' 09", and two observations give C = 81 02' 10". Compute the adjusted values of the angles. 8. SOLUTION OF NORMAL EQUATIONS. The normal equations which arise in the adjustment of observations may be solved by any algebraic process. It is desirable, however, to use methods which will furnish the value of each unknown quantity independently of the others, as the liability to error is thus lessened. When there are but two normal equations let them be expressed in the form A,x + B,y = A , A^ + Bj = >., then the solution by any method gives the formulas AA - AA A A - A A ~ B,A, - B,AC ~ A & - A,B^ ' which can easily be kept in mind by noting the order of the letters and subscripts. It may be observed also that the two denominators are equal numerically but of contrary sign. For three normal equations let them be written in the form A,x + B,y + O = A , and the solution leads to the formulas 1 ~ - (A.C.-AtC^+^C.-A.C^+^Ct-A^)*. Q. THE PROBABLE ERROR. 25 in which the three denominators have the same value. After a little practice it will be easy to use these formulas with great rapidity in the solutiori of normal equations. When the number of normal equations is greater than three, general formulas for solution are too lengthy to be written, and the systematic method of substitution devised by Gauss is generally employed. This is explained and exemplified in text-books on the Method of Least Squares, but lack of space forbids its presentation here. Prob. 8. Solve the normal equations 3* y + 2z = 5, x + 4y + z= 6, 2* -f y + 53 = 3, and check the solution by showing that the values of the roots satisfy the equations. 9. THE PROBABLE ERROR. The Method of Least Squares comprises two tolerably dis- tinct divisions. The first is the adjustment of observations, or the determination of the most probable values of observed quantities. The second is the investigation of the precision of observations and of the adjusted results. The first is done by the application of the principle of least squares given in Art. 3 ; the second is done by the determination of the probable error, the rules for which will now be presented. The following may be stated as a definition of the term " probable error " : In any large series of errors the probable error is an error of such a value that the number of errors less than it is the same as the number greater than it. The probable error is hence an error which is as likely as not to be exceeded. In the figure if the ordinate MN be drawn so as to divide the area on each side of O Y into two equal parts, then OM is the probable error. Here the total area between the curve and the Jf-axis is unity (certainty), and the area MN YNM is 0.5 ; thus the probability that an THE METHOD OF LEAST SQUARES. I. error is greater than OM is 0.5, and that it is less than OM is also 0.5. To render more definite the conception of probable error let two sets of observations made upon the length of a line X MOM X be considered. The first set, made with a chain, gives 634.7 feet with a probable error of 0.3 feet. The second set, made with a tape, gives 634.64 with a probable error of 0.06 feet; thus, /, = 634.7 0.3 and / = 634.64 0.06; and it is an even chance that 634.7 is within 0.3 of the truth, and also an even chance that 634.64 is within 0.06 of the truth. The probable error thus gives an absolute idea of the accuracy of the results; it also serves as a means of compar- ing the precision of different observations, for in the above case the precision of the second result is to be taken as much cnrgflter than that of the first. It is a principle of the Method of Least Squares that weights of observations are reciprocally proportional to the squares of their probable errors. Thus, for the above numeri- cal example, ' I/25 ' Hence the second observation has a value about 25 times that of the first when it is to be used in combination with other measurements. Weights and probable errors are con- stantly used in the discussion of observations. Weights are usually determined from the number of measurements or from knowledge of the manner in which they are made, but prob- able errors are computed from the observations themselves. 9. THE PROBABLE ERROR. , 2/ For the case of direct observations on the same quantity, all being of equal precision^ the arithmetic mean is the most probable value (Art. 3). Subtracting each observation from the mean gives the residuals v l , v^ , . , . v n , and the sum of the squares of these is represented by JSV 2 . Then = 0.6745Y - (9) is the probable error for a single observation, and, since n is the weight of the arithmetic mean, is the probable error of the arithmetic mean. For example, let six observations of an angle be taken with equal care and let these be arranged as below in the column headed M. The sum of these values divided by 6 gives 48 06' 14" './ as M IS *U 4 8o6 / 12' '. 5 + 2".2 4.84 15 .0 o .3 0.09 r, == 2-58 20 3 -5 .6 31.36 08 9 + 5 .8 33.64 z = 48 06' '5 .1 o .4 0.16 16 4 i .7 2.89 r = i".os 14". 7 i z = 48 06' 14". 7 o".o 73.98 =J?f? the most probable value of the angle, the second column gives the residuals, and the third their squares. Then by the use of the formula the probable error of a single observa- tion is found to be 2". 58 and that of the arithmetic mean, to be I /; .O5. Thus if another observation were to be taken it is as likely as not that it will deviate 2^.58 from the truth. For the case of n observations of different weights on one quantity the weighted mean is the most probable value (Art. 9). Subtracting each observation from this gives the residuals, and the square of each of these is to be multiplied 28 THE METHOD OF LEAST SQUARES. 1. by its weight to give the sum of the weighted squares, which may be represented by 2pv*. Then r, = 0.6745^^ (9)" is the probable error of an observation of the weight unity. and if 2f represent the sum of the weights, ' - r= 7F " is the probable error of the weighted mean. As an example, let the observations in the first column of the following table be the results of the repetition of an angle at different times, i8".26 arising from five repetitions, i6".3O from four, and so on, the weights of the observations being taken the same as the number of repetitions. Then the general mean z has M p 32 of i8". 2 6 5 16 .30 . 4 21 .06 I '7 -95 4 16 .20 3 20 .85 4 V Z/* pv> o".io O.OIO 0.05 + i .86 3.460 13-84 2 .90 8.410 8.41 + .21 0.044 0.18 + I .96 3.842 "53 **. 2 .69 7.236 28.94 z 32 07' i8".i6 2i=2f the weight 21, the sum of the several weights or the number of single measures. Subtracting each M from z gives the residuals in the column v\ next from a table of squares the numbers in the column v* are found, and multiplying each of these by the corresponding weight gives the quantities pv* whose sum is 62.95. Then, since n is 6, the probable error of an observation whose weight is unity is found from the formula to be r l = 2". 39 and that of the weighted mean to be r = o".52. Hence the final value of the angle may be written z 32 of i8".l6 o".52, which indicates a high degree of precision. 10. PROBABLE ERRORS FOR INDIRECT OBSERVATIONS. 29 Prob. 9. Four measurements of a base line give the results 922.220 feet, 922.197 feet, 922.221 feet, and 922.217 feet. Compute the probable error of the most probable value. 10. PROBABLE ERRORS FOR INDIRECT OBSERVATIONS. x- It is sometimes required to find the probable errors of the observed quantities M l , M^ , . . . M n , and the probable errors of the quantities x, y, 2, etc., whose values have been obtained by the methods of Arts. 7 and 8. These may be found by first deducing the probable error of an observation of the weight unity and then dividing this by the weights A A pn and p x , p y , p z , etc. If n is the number of observations, q the number of unknown quantities, and "2pv* the sum of the weighted squares of the residuals, then, as shown in treatises on the Method of Least Squares, (10) is the formula for the probable error of an observation of the weight unity, and r _ 12- r - Jl I/A *# are the probable errors of M l and of x respectively. The weights /, , /,,.../ are known, but the weights p x , p y , etc., are to be derived by preserving the absolute terms of the normal equations in literal form during the solu- tion. Then the weight of any unknown quantity is the reciprocal of the coefficient of the absolute term which belongs to the normal equation for that unknown quantity. For instance, take the normal equations 1* - y - z = A , - *+ ar * = A, x y + 2z = A . 30 THE METHOD OF LEAST SQUARES. The solution of these by any method gives Hence the weight of x is |, that of y is J-, and that of z is I. If it be only desired to find the weight of x, the terms D^ and D^ need not be retained in the computation; if only to find the weight of z t the terms D^ and D^ can be omitted. As a numerical example the observation equations given at the beginning of Art. 5 may again be considered. These may be written, if x, y y and z denote the most probable elevations, x 10.35 = v l9 y x 7.25 = v t , y - 17.63 = v t , y z 9. 10 = v tt x z - 1.94 v t t in which v t , v t , etc., are the residual errors. Now in Art. 6 the most probable values were derived, x = 10.37, y = 17.61, and z = 8.47 feet, and substituting these, the residuals are found to be v t = +0.02, z/, = o.oi, v 3 = 0.02, v^= + o -4 v = 0.04. Now, as the weights are equal, ~2pv* becomes 2v*, and its value is 2v* = o.o 041. Then, since n is 5 and q is 3, /O.OO4I A / = o.< V 5-3 r, = 0.6745^7 _^ = 0.031 feet, which is the probable error of a single observation. y 1 (By the method above explained it will be found that the weight of x is 1.6, whence its probable error is 0.031 ~\ , = = = 0.024 feet, r 1. 6 II. PROBABLE ERRORS OF COMPUTED VALUES. 31 and in a similar manner the probable errors of y and z are 0024 feet and 0.031 feet. .The final adjusted values may then be written x = 10.37 - 02 > " y I'fkl 0.02, z = 8.47 0.03. Prob. 10. Four measurements give the observation equations -|- x = 12.27, w i tn weight 2, x + y = 1.04, with weight 2, y -|- z 3.30, with weight i, z 16.67, with weight i. Find the most probable values of #, y, and z, their weights and their probable errors. 11. PROBABLE ERRORS OF COMPUTED VALUES. The determination of the precision of quantities which are computed from observed quantities is now to be discussed. For instance, the area of a field is computed from its sides and angles; when the most probable values of these have been found by measurement, the most probable value of the area is computed by the rules of geometry, and the precision of that area will depend upon the precision of the observed quantities. Let 2^ and 2 t be two adjusted values whose probable errors are r l and r a ; it is required to find the probable error r of the sum 2 = 2^ -f- z y If z//, z;,", etc., be residual errors for z l and z> 3 ', v 9 ", etc., be residual errors for # , then the corre- sponding errors for z are v' = z>/ -f- z>/, v" = v" + v" t etc. Squaring each of these and adding the results gives and for a large number of errors 2vj> t is zero, since each product vjj^ is as likely to be positive as negative. Now .2V, 2v*, and 2v^ are proportional to r\ r*, and r, a as seen by (9), and accordingly 32 THE METHOD OF LEAST SQUARES. I. gives the probable error of the sum , -|~ a . In like manner it may be shown that the probable error of the difference z l #, is also give.n by Vr* -f- r a *. Further, if z = z t z % . . . s* , then r* = r l * + rS + . . . + rj (11) determines the probable error of . For example, if a base line be measured in three parts giving 250.33 0.05, 461.29 0.07, and 732.40 o. 10 feet, then r = 0.13 feet, and the total length may be written 1444.02 0.13 feet. If x be an observed quantity whose probable error is r, then the probable error of ax is ar. Thus, if the diameter of a circle be observed to be 42 feet 2 inches 0.5 inches, the circumference is 132.47 i 0.13 feet. If X be any function of x, then the error dx in x produces the error dX in X, and the error r in x produces the error 7 y r-j in X. For example, let x be the observed diameter of a circle and r its probable error; then X = \nx* is its area, and dX = ^nx-dx, whence the probable error of X is T'^TTX. Thus, if x is 42 feet 2 inches 0.5 inches, the area is 1396.46 2.76 square feet. Lastly, let X be any function of the independently observed quantities x, y, z, etc., and let it be required to find the probable error of X from the probable errors r l , r, , r , , etc., of the observed quantities. If the measurements are made with precision, so that the probable errors are small, it can be shown that determines the probable error of X. For example, let x and y be the sides of a rectangular field and X = xy its area. Then the probable error r l in x gives the probable error r^y in X, and the probable error r, in y gives the probably error 12. CRITICAL REMARKS. 3$ rjc in X, so that (r,^) 2 -f- (r^x)* is the square of the resulting probable error of X. Ttyus, if x 50.00 o.oi feet and y 200 O.O2 feet, the ara is 10000 2.24 square feet. Formula (ii) / will be frequently used in the following pages, it being a general rule that includes all cases. As another illustration let A and B be two points whose hori- zontal distance apart is /, and let 6 be the vertical angle of elevation of B above A ; let r l be the probable error of /, and r a the probable error of 0. The height of B above A is given by X = / tan B, and, by the application of the formula, regarding h as x and 6 as y, there results r 2 = (r, tan fff + (r t //cos' ff)\ If / = 1035.2 1.3 feet, and B = 3 10' 02', then r l = 1.3 feet, but, to make the computation, r 2 must be expressed in the same unit as cos 2 0, that is, in radians; since 3438 minutes make one radian, the numerical value of r a is 2/3438. Then are found r, tan 6 = 0.072 feet, (r a //cos" 6) = 0.604 f eet > whence r = 0.608 feet. The value of X being 1035.2 tan 3 10' = 57.27 feet, this may be written 57- 2 7 0.61 feet, and thus it is as likely as not that the error in the computed height is less than 0.61 feet. Here it is seen that the probable error in the small vertical angle produces the greater part of the probable error in the com- puted result. Prob. ii. In a plane triangle ABC let A = 90, C = 16 04' 45" 30", and a = 6256.8 0.7. Compute the length of the side c and its probable error. 12. CRITICAL REMARKS. The most important processes for the adjustment and comparison of observations have now been presented, but the brief space at command has forbidden extended theoretic discussions like those found in treatises on the Method of Least Squares. The student has been obliged to take for UNIVERSITY OF CALIFORNIA DEPARTMENT OF CIVIL ENGINEERS 34 THE METHOD OF LEAST SQUARES. I. granted the law of probability of error and the formulas for probable errors, but otherwise the subject has been developed in logical manner. Legendre, in announcing the principle of least squares in 1805, gave no proof of its correctness or validity; he notes, however, that this principle balances the errors, so that the effect of the extreme ones is neutralized. In mechanics the center of gravity is a point about which all the particles of the body balance; so the arithmetic mean gives a value about which all the errors balance, the sum of their residuals being zero. The moment of inertia of a body is a minimum for an axis passing through the center of gravity; so the sum of the squares of the residual errors is to be made a minimum in order to find the most probable values of an observed quantity. The radius of gyration with respect to an axis through the center of gravity bears also an analogy to the probable error. Thus the Method of Least Squares may be justified by the mechanical principles of equilibrium. Numerous applications of the adjustment of observations will be given in the following Chapters, and a simplification will be introduced whereby the formation of normal equations from observation equations may be rendered numerically easier. A treatment of conditioned observations by the use of " correlate equations" will also be presented, whereby the work of computation may often be materially shortened. As measurements become more and more precise the neces- sity for rational processes of adjustment and comparison becomes greater and greater. In physics, astronomy, geodesy, and wherever precise observations are taken, the Method of Least Squares is now universally used, and there is little doubt but that in future years all books on surveying will treat more or less of its principles and processes. A list of writings on errors of observations and on the Method of Least Squares from 1722 to 1876 will be found in Transactions of the Connecticut Academy, 1877, vol. IV, pp. 12. CRITICAL REMARKS. 35 151-222. Many of these, together with others from 1877 to 1888, are given in Gore's ,Bfbliography of Geodesy, published in Report of the U. S. Coajjt and Geodetic Survey for 1887, pp. 313-512. Prob. 12. A base line was measured in three parts, the values found for these being 126.74, 219.18, and 270.40 meters. The total length was then measured and found to be 616.39 meters. Find the adjusted length of the base, the weights of the four observations being 17, 9, 8, and 3. Prob. i2a. A plane triangle has the angle A measured ten times, v7 measured five times, and C measured once. The sum of the iliree mean values is found to differ d seconds from 180. How should this d be divided among the three angles ? Prob. \2b. Solve the following normal equations, and find the weights of the values of x and w\ 2x y z w = + 0.53, x + 4y w = 0.27, x + 32: + w = - 0.50, * y + * + 5^ ^ + 0 so that y = 48 49 13 .64 +7,, z = 54 38 15 .53 + * t , represent the most probable values of the quantities x, y, and 2. Then substituting these in the observation equations the latter become + x, = o".oo, + /i =0 .00, -}- z, = O .00, + ^ 4.^ = - o .18. Next, by the rule of Art. 6, the normal equations are r'. + S/', + *, = + 0.16, y, + 2a t = o.i 8, l6. THE STATION ADJUSTMENT. 47 the solution of which gives the corrections ;r, = + o". 15, 7/=+o.04, , = 0.11, and hence the most -probable' values of x, y, and z are * - 55 57' 58".8 3 - 67W, 7 = 48 49 13 .68 = MNQ, z = 54 38 15 .42 = g^VS, and from these by addition the most probable values of the other observed angles are x + / = 104 47 12 .51 = Wg, j -j- z = 103 27 29 .10 = MNS. The residuals for the five observation equations, found by substituting the most probable values, are + o". 15, + o".O4, 0.15, o.i I, -fo. 11, and the sum of their squares is 0.0708, which is smaller than can be obtained by any other values of x, y, and z. From (10) the probable error of each of the given observations may now be found to be o". 13. When the weights are unequal the method of Art. 7 is to be followed. As an example, let the following be three angles measured at the station O: MO A = 46 53' 29". 4 with weight 4, MOC = 135 27 II .1 with weight 9, AOC 88 33 41 .1 with weight 2. Now let x and z be the most probable values of any two angles, say of MO A and MOC. Then the observation equa- tions are x = 46 53' 29".4, weight 4, z =. 135 27 ii . i, weight 9, z x 88 33 41 .1, weight 2. Next let x l and z^ be the most probable corrections to the observed values of x and 2, so that *= 4 653 / 2 9 ".4 + *i, z = 135 27 ii .1 +* 48 PRECISE PLANE TRIANGULATION. II. are assumed probable values of x and z. Let these be sub- stituted in the observation equations, which thus reduce to x^ = o".oo, weight 4, z l = o .00, weight 9, x l z l = -f- 0.60, weight 2. From these the normal equations are formed ; they are 6x, 2z l = + 1.20, 2X l -\~ IlZ l = 1. 2O, from which the most probable corrections are Finally, the adjusted values of the three angles are x = 46 53' 2 9 ".6 = MOA, z = 135 27 ii .o = MOC, z x = 88 33 41 .4 = AOC. Here it is seen that the observation having the largest weight receives the least correction, which should of course be the case. It is well to note that the numerical part of the assumed probable values may be anything that is convenient; thus in the last example 46 53' oo".o + x l might be taken for x, and 135 27' oo".o + z v for z, then the values for x l and z l would be found to be + 29". 6 and + u".o. The object of introducing x l , y l , and z l is, however, to make the numbers in the right-hand members of the observation and normal equations as small as possible, and this is generally secured by taking the corrections as additions to observed values. After the adjustment is made the azimuths of all the lines radiating from the station are easily found by simple addition or subtraction, provided the azimuth of one line is known. Thus for the last example let the azimuth of OM be given as 279 04' i8".4, then the azimuth of OA is 325 57' 48".O, and the azimuth of OC is 54 31' 29".4. I/. ERRORS IN A TRIANGLE. 49 P"ob. 1 6. Angles measured at the station O between the stations Z>, K } M, and C gave the following results: DDK 66 33' 43"-7o, weight 2, KOM~= 66 14 22 .10, weight 2, KOC 108 02 2 9 .62, weight i, MOC = 4i 48 07 .02, weight 2, COD = 185 24 47 .65, weight 2. State the observation equations, form and solve the normal equa- tions, find the adjusted angles, and show that the adjusted value of COD is 185 24' 47". 41 with a probable error of o".2o. 17. ERRORS IN A TRIANGLE. The simplest triangulation is a single triangle in which one side and the three angles are measured in order to find the lengths of the other sides. The precision of the values found for these sides will depend upon the probable error of the base and the probable errors of the measured angles. The best triangle is one whose angles are each about 60 degrees, and a triangle having one angle less than 30 degrees is not a good one, as will now be shown. In a triangle whose sides are a, b, and c, let the angles A, B, and C and the side a be obtained by measurement. The sides b and c then are s'm sinC b = a . A , c = a -r. sin^ smA Now suppose each angle to have a probable error r\ then by the use of (n)' the probable errors in b and c are found to be r b = br i/cotM + cot 3 ^, r e = cr Vcot'A + cot'C. (17) If A, B, or C is a small angle its cotangent is large and accordingly r b and r c may be great. As far as b is concerned the smallest value of r b will obtain when A = B, and as far as c is concerned the smallest probable error results when A = C' t or the three angles should be equal and each be 60 50 PRECISE PLANE TRIANGULATION. II. degrees in order that the precision of b and c should be the same and each be as small as possible. As a numerical example let a = I ooo feet, A = 90, B = 10, C = 80, and let the probable error in each angle be i'. Here by computation b= 173.65 feet, c= 984.81 feet, and then r t = I73-65 X 5-6; X r t r c = 984.81 X 0.176 X r. The value of r to be used here is i' expressed in radians, or r = 7r/(i8o x 60) = 0.000291. Accordingly the probable error of b is 0.29 feet and that of c is 0.06 feet, so that the computed values of b and c have a large degree of uncertainty. It will be noticed that b, which is opposite the small angle B, is liable to a far greater error than is c. For a second example take the triangle in which a = i ooo feet, A = 60, B = 60, C ~ 60, and let the probable error in each angle be i'. Here b = i ooo feet, c i ooo feet, and r = 0.000291 ; then from (17) there is found r b = r c = 0.24 feet, so that the probable error of the computed side b is less than in the previous case. The uncertainty of a line is the ratio of its probable error A 1 ' A. 1 I T"*1 "-*_1/* * 1 1,1 \ to its length. J Thus in the first numerical example the uncertainty of the computed value of b is 0.29/173.65 = -g-J-g- nearly, and that of the computed value of c is 0.06/984.81 = rs --J lnr nearly. In the second example, however, the uncertainties of b and c are 0.24/1 ooo = f-faf nearly. An uncertainty of -^-j^ is greater than that of a rough linear measurement, and an uncertainty of y^-g^ is greater than should occur in the lengths of the lines computed in precise triangulations. In primary geodetic triangulation work the uncertainty of the computed sides of the triangles is usually about ^nj-VoT' ^us tne probable error in a line 30 ooo meters long would be o. I meters. From formula (17) it is seen that the uncertainties in the computed values of b and c are 18. THE TRIANGLE ADJUSTMENT. 5 1 + cot'^, u e = r Vcot'A + cot'C, ( i ;)' and hence these may be computed without knowing the lengths of the sides b and & If the probable errors of A, B, and C are different, let them be represented by r l , r^ , and r t ; then from (I i)', u b = Vr* co? A + r, 1 cot*j9, u e = Vr, 1 cotM + r* cot a <^, (i 7)" are the uncertainties in the computed lengths of b and c. If the base a has a probable error r a , this may also be taken into account by (n)', and it will be . found that the term (rjti) 2 must be added to the other terms under the radical signs in formula (17)". In laying out a triangulation it is not possible to locate the stations so that each angle may be approximately 60 degrees, but it should be kept in mind that this is the best possible ar- rangement and that it should be secured whenever feasible. Angles less than 30 degrees should not be used except in un- usual cases, or when the distances computed from them are not to be used for the computation of other distances. Prob. 17. In a triangle the adjusted values of the observed angles are 25 18' 07", 64 01' 26", and 90 40' 27", each having a probable error of i". The length of the side opposite the smallest angle is 3 499.39 feet, and its uncertainty is j-g-fanr. Find the uncer- tainties in the computed values of the other sides. 18. THE TRIANGLE ADJUSTMENT. When the three angles of a plane triangle have been meas- ured their sum should equal 180 degrees, but as this is rarely the case they are to be adjusted so as to fulfil this condition. This is readily done in any particular case by the methods of Arts. 6, 7, and 16, but more convenient rules for doing it will now be deduced. First, let the three observed values be of equal weight, and 52 PRECISE PLANE TRIANGULATION. II. let these be A, B y and C. Let x and y be the most probable values of A and B\ then the observation equations are x = A, y = B, 180 x -- y = C. Now let v^ and v^ be the most probable corrections to be applied to A and B in order to give the most probable values of x and jy, or x = A + v, , y = B + v t . Substituting these in the observation equations, the latter reduce to v t = o, v t = o, ^4-^=1 80 A B C. Letting d represent the small quantity 180 (A + B + C) the normal equations are found to be 27 'i + v* = d> v i + 2v i = d, whose solution gives v l = \d and v 9 = \d, which are the corrections to be applied to A and B. Then the correction to be applied to C is also \d* Hence the rule: Subtract the sum of the angles from 180 and apply one third of the dis- crepancy to each of the measured values. For instance, if the three measured angles are 64 12' 19" '.3, 80 o/ 47". o, and 35 39' 55". 8, their sum is 180 oo' 02".!, and the dis- crepancy d is 02".!. Then o"./ is to be subtracted from each angle, giving 64 12' i8".6, 80 07' 46". 3, and 35 39' 5 5". I as the most probable values. Secondly, let the three observed values be of unequal weight. Let these be A with weight /, , B with weight/,, and C with weight /,. The observation equations are the same as before, but are weighted, namely, Vi = O, with weight /, , ?; a = o, with weight/,, z/, -f- v^ = d, with weight / 3 . From these the normal equations are (P, + A)v> + A", = A", + (A -f AK - IQ. TRIANGLE COMPUTATIONS. 53 whose solution gives the corrections v l and v^ and then the correction v t is d z>, v^ " Accordingly the results are d * d d in which, for abbreviation, P represents 1 \- These A A A formulas show that the corrections are inversely as the weights, so that the angle having the smallest weight receives the largest correction. For example, let the weights of A, B, and C be 10, 5, and I ; then v l = ^d, v^ = T 2 g^, and v t = ^%d, so that the correction for C is ten times that for B and five times that for A. If only two angles of a triangle are measured there can be no adjustment made. If A and B are given by observations these are the most probable values of those angles, and the most probable value of C is 180 A B. In all precise primary work the third angle should be measured as a check, as also to show the precision of the observations, whenever it is practicable. Spires and other inaccessible points may, however, be used as stations in secondary triangulation. Prob. 18. The observed angles of a triangle are 74 19' i4"-3 with weight 3, 35 10' 42^.6 with weight 7, and 70 30' O9".4 with weight 9. Find the adjusted values of the angles. 19. TRIANGLE COMPUTATIONS. The computation of the sides of a triangle is a simple matter, one side having been measured as a base line or being known from preceding computations. The theorem used is that the sides are proportional to the sines of their opposite angles; thus in the triangle ABC let the side AB be known, then log CA = log AB log smC -(- log sinB, log CB = log AB log s'mC -f- log sinA. 54 PRECISE PLANE TRIANGULATION. II. In making these computations it is desirable that a uniform method should be followed, and the following form for arranging the numerical work is recommended, it being similar to that used by the U. S. Coast and Geodetic Survey. COMPUTATION OF A PLANE TRIANGLE. Lines and Stations. Distances and Angles. Logarithms. B \ AB 2753.53 3.4398898 C 49 04' 49". 28 0.1216914 \ ^^^C A 90 21 24 .66 1.9999916 \ >/ B 40 33 46 .06 I.8I3IOII \y/ CB 3643.95 3.5615728 A CA 2369.64 3.3746823 | Here the stations are arranged in the order of azimuth, and that is placed first which is opposite to the given side, the length of this and its logarithm being put on the top line. Opposite the second and third angles are written their logarithmic sines, and opposite the first angle the arithmetical complement of its logarithmic sine. Now, to find the log of CB the logarithm opposite B is to be covered with a lead- pencil and the other three logarithms added. So to find the log of CA the logarithm opposite A is to be covered and the other three logarithms added. Lastly, the distances corre- sponding to these logarithms are taken from the table. If the precision of angle work extends to seconds or tenths of seconds, as it does on primary triangulation, a seven-place table of logarithms will be needed. Six-place tables are rarely found conveniently arranged for rapid and accurate computation. For a large class of secondary work five-place tables are sufficiently precise. In taking a log sin from the tables the student should note that the characteristics 9. and 8. mean 1. and 2. and should write them in the latter manner in his computations. 19. TRIANGLE COMPUTATIONS. 55 When the above triangle ABC is connected with a coordi- nate system the azimuth of AB is known from previous computations. Then, fro&i this and the angles A and B, the azimuths of AB and BC are easily found. Let the azimuth of AB be 149 42' 55".68; then that of BA is 329 42' 5 5 ".68, and accordingly Azimuth AC = azimuth AB + angle A 240 04' 20". 34, Azimute BC = azimuth BA angle B = 289 09 09 .62. As a check on these azimuths it may be noted that the second minus the first should be equal to the angle C. The next computation is that of finding the coordinates of C from those of A and B. For the above triangle suppose that the coordinates of A have been assumed and that those of B have been computed from (13), the values being Station. A B Latitude 10000.00 12377.76 Longitude 8 ooo.oo 9 388.59 and let it be required to compute the latitude and longitude of C. These should be found in two ways by the formulas in (13), so as to check the correctness of the results, and the form below shows how the numerical work may be arranged in a systematic manner. In the first column / denotes the length of AC or BC, the logarithm of the former being put in the third column and that of the latter in the fifth column. Similarly Z denotes the azimuth of AC or BC whose values are given in the second and fourth columns; adjacent to these are written the values of log cosZ and log sinZ, 1. being written instead of the 9. in the tables. Then log / added to log cosZ gives log / cosZ, and log / added to log sinZ gives log / s'mZ. The values of / cosZ and / s'mZ are next taken from the logarithmic tables and placed in the second and fourth columns. Opposite L l and J/, are placed the lati- tudes and longitudes of A and B, and the values of / cosZ and / s'mZ are added to or subtracted from them as required PRECISE PLANE TRIANGULATION. II. by the signs of cosZ and sinZ. It will be better, however, for the student to determine whether these are to be added or subtracted by drawing figures at the top of the table. COMPUTATION OF COORDINATES. C i / c C \ COMPUTED \ s' COMPUTED FROM A\ FROM ^^\^^ Symbols. 4. \ B. ^\ 1 J L Distances and Azimuths. Logarithms. Distances and Azimuths. Logarithms. , 3.3746823 3-56I5728 log c< sZ = 1.6980292 log cosZ = I.5I59884 z 240 04' 20". 34 289 09' 09". 62 log sinZ = 1.9378462 log sinZ = 1.9752699 /cosZ I 182.26 3.0727H5 I 195-53 3.0775612 /sinZ 2053.66 3.3125285 3442.25 3.5368427 1 10000.00 12377.76 Mv 8 ooo.oo 9388.59 lat. of C II 182.26 II 182.23 long. off 5946.34 5946.24 If the computations be correctly made the two values of the latitude of C must exactly agree, as also the two values of the longitude of C. In this case there is a discrepancy of 0.03 in the latitudes and of o.io in the longitudes and hence the numerical work must be revised so as to detect and remove the errors of computation. Prob. 19. Revise all the computations in this Article and find the correct values of the coordinates of C. Also make the computations for the triangle DEF, in which F = 95 24' oi".o, E = 54 58' o8".6, & = 29 37' 5"-4> I>E=6 584.20 feet, lat. D = -f 15 328.75 feet, long. D = -f- 12 047.05 feet, azimuth DE = 216 17' 05". 6, and determine the coordinates of E\ finding lastly the coordinates of F in two ways. 2O. TWO CONNECTED TRIANGLES. 57 20. Two CONNECTED TRIANGLES. Let two triangles ABC and CD A have the side AC in common and let all the angles be measured, the observations being as follows and all of equal weight: ^,=45 19' 07", C.= 50 19' 37", A, = 48 07 15 , C, = 37 46 50 , = 96 06 15 54 19 y2 = 93 26 28 , B = 81 33 18 , Here it is seen that the sum of A l and A, Is 06" less than A, that the sum of C l and 7 2 is 12" greater than. C, that the sum of A l , B, and T a is 10" greater than 1 80, and that the sum of ^4 a , C 19 and Z> is 06" greater than 180. It is re- quired to find the most prob- able values of the angles which entirely remove these dis- crepancies. The number of observed angles is eight, but these are subject to the four conditions just mentioned, and accord- ingly there are really but 8 4 = 4 independent angles to be used in the computation. Take A lt A,, C i , and C 9 as these independent angles and let a t , tf, , c l , and c^ be the most probable corrections to be applied to the observed values. The observation equations then are a, = oo", c l = oo", #, oo , c t = oo , , i + a * = + 06 , Cl + c, = -- 12 , a, +<:,= 10 , a, + c, = - 06 . From these the normal equations are formed ; they are + 4- Ci + 3'. = 00 , 18 , 22-T CIVIL ENGINEERING U. of C. ASSttCUTION LIBRARY = 48 o; 15 .1, = 93 26 23 .6, = 81 33 13 -8,. C, = 50" 19' 31'VI, C, = 37 46 45 -5, C = 88 06 16 .6, ./? = 96 54 06 .0, 58 PRECISE PLANE TRIANGULATION. II. and the solution of these gives a, = + 00". i, ^ = + 01". 5, ^, = -04". 5, *,= os".9, as the most probable values of the corrections to the four angles. Then from the geometric conditions the corrections to the other angles are = *, + *,- 06" = - 4". 4, c = Cl + r, + 12" = + 1.6, b = a t c l 10" = 4.2, d = a l c^o6" = 3.0, and applying these to the observed values they become A B which are the most probable values of the angles and which at the same time satisfy the geometry of the figure. When the observations are of unequal weights these are to be used in forming the normal equations from the observation equations. If one or more angles are unmeasured these do not appear in the observation equations and their values are to be derived from the adjusted results. If the angles A and C are not measured, but all the others are, then the only adjustment required is that of each triangle by the method of Art. 1 8. If the length and azimuth of AB and the coordinates of A be given, the lengths and azimuths of the other lines of the figure, as also the coordinates of B, C, and D, may be com- puted by the methods of Art. 19. Thus a simple triangula- tion is established. When more than two triangles are con- nected the station adjustments are usually made first, and afterwards the triangle adjustments ; cases of this kind are discussed in Chapter IX. Prob. 20. In the above figure let the observed values be as given except that of D y which is not measured. Find the adjusted values of all the angles. 21. DIRECT OBSERVATIONS WITH ONE CONDITION. 59 21. DIRECT OBSERVATIONS WITH ONE CONDITION. In Art. 1 8 are given examples where direct observations on several quantities are connected by a single conditional equation, and as other cases are to be discussed in future Articles it will be well to derive a general method of procedure which will simplify the numerical work. Let x and y be two quantities whose values have been fo-und by observation, these having the weights/, and /,. Let these quantities be connected by the conditional equation in which q l and q^ are known coefficients, and D is a known quantity. Let z^ and v^ be the most probable corrections to the observed values so that the observation equations are v, = o, weight/,; ^ = o, weight/,, and the conditional equation reduces to W + 4W = d. Now let the value of one of these corrections be found from the last equation and be substituted in the observation equa- tions, and then let the normal equations be formed and solved, and finally let the other correction be found from the conditional equation. The results will be q,d q^d *-$* v * = f,r' in which, for abbreviation, the letter P represents the quantity + A ^ A The same process may be extended to any number of unknown quantities and similar formulas result. Thus if z>, = o, v a = o, . . . v n = o, with weights /,,/,, . . . / , and if the conditional equation is 60 PRECISE PLANE TRIANGULATION. II. then let P= y + q -j + . . . + ~- t and the most probable values are q^ d q^ d q n d which also exactly satisfy the conditional equation. Formula (21) hence gives a general solution of this important case. As a numerical example let there be measured at a station O the three angles AOB = 97 18' 20" with weight 5, BOC = 135 20' OS" with weight 3, COA =127 21' 29" with weight 6. Let x, y, and 2 be the most probable values, then must x -\- y -f- z = 360. Take v l , z' a , and v s as the correc- tions to the observations, and the conditional equation reduces to v^ + ^ + v* = + 06". Here q l = q^ = q^ = i and d = 6"; also / A = 5, / a 3, / 3 = 6, and hence P 0.7 and d/P =i + 8.57. Accordingly from (21) the values of the corrections are v l = -\- i" '.7, z> 4 = -f- 2^.9, z/, = -|- i /7 .4, so that the most probable values of the three angles which satisfy the conditional equation are 97 18' 21". 7, 135 20' 07". 9, and 127 21' 30". 4. The above is the simplest application of the method of correlates which is extensively used in the adjustment of geodetic triangulations; further examples of it will be given in Chapter IX. For the case of equal weights the/'s dis- appear from the above formulas and P becomes the sum of the squares of the ^'s. For instance, if the three observed angles of the last paragraph be of equal weight, then P = i -\- i -)- i = 3 and hence v l = v^ = v^ = \d, a result which agrees with the rule established in Art. 18 ; accordingly the adjusted values are found to be 19 18' 22", 135 20' 07", and 127 21' 3i v , the sum of which is 180. Prob. 21. The five interior angles of a pentagon, as found by measurement, are 80 19', 120 57', 107 04', 141 35', and 90 oo'. Compute the adjusted angles, taking the weight of the last value as three times that of each of the others. 22. INTERSECTIONS ON A SECONDARY STATION. 6l 22. INTERSECTIONS > ON A SECONDARY STATION. & After a triangulafion has been established any side may be used as a base from which to locate a secondary station by means of two measured angles. If, however, a third station is also used another computation may be made, and in general the results will not exactly agree with the first one owing to errors of observation. An adjustment is hence to be made in order to obtain the most probable position of the secondary station. Let ABC be a triangle whose angles are known, it being a part of an established triangulation. At the three corners let the angles A l , B iy and C l be measured m order to locate a secondary station 5. The lines determined by these angles do not in general meet at the same point, and hence the ob- servations are to be adjusted to secure this result. The con- dition that the three lines shall meet in S is established by equating the expressions for the length of one as found from another in two ways; thus let BS be found, first by the triangle ABS and secondly through the triangles ASC and BSC; the values are sine, and hence the conditional equation is sin^ t sin^ s'mC l = smA^ sin^ 3 sin(7 2 , which must be exactly satisfied by the most probable values to be found for A, , B, , and C t . This is called a side equa- tion because it expresses the necessary relation between the 62 PRECISE PLANE TRIANGULATION. IL three lines or sides which meet at 5. By taking the logarithm of each member it becomes log sin^4i + log sin2?i + log sinCi logsin^a log sin^ 2 logsinCa=o, which is the form for practical numerical work. As an example let the given angles of the triangle be A = 83 39' 01", B = 57 19' 42", and C= 39 01' if. Let the three angles, as measured to locate 5, be A l = 41 05' 10" with weight 2, B, 30 15' 12" with weight 3, and \ = 1 8 46' of with weight I ; it is required to adjust these so that the three lines may meet in 5 and so that the values found may be the most probable. Let a" , b" ', and c" be the corrections expressed in sec- onds to be added to the observed values A l , B l , and C,. Then 41 05' 10" + a" is to be substituted for A l in the above conditional equation and similarly for B^ and C } . Now log sin (A, +*") = log sin A, + 0"-diff. i"', where diff. i' is the tabular difference for one second corresponding to the value of A^\ thus log sin (41 05' 10" + a") is 1.81769 -\- o.2^a" ', where 0.23 is in units of the fifth decimal place of the logarithm. In this manner the following tabulation is made : Observed Angles. Log. Sines. A, = 41 05' 10" 1.81769 + 0.230" ^ = 30 15 12 1.70228 -f 0.35^" C l = 18 46 07 1.50751 -{- o.6i , B^ , and C 9 being those found by subtract- ing A lt BI, and (7, from A, B, and C\ and their corrections being the negatives of a", b" , and c" . Equating the two members of the conditional equation, it reduces to 0.450" + 0.75^" + \.\gc" = 17, 22. INTERSECTIONS ON A SECONDARY STATION. 63 while the observation equations are a" = o, b" = o, and c" = o, whose weights are 2, 3, and i, respectively. By the method of Art. 21 are now found ,'//, = o. 101, q?/p^ 0.187, ?,V/ 3 1.440, and P = 1.728, whence d/P _f_ 9 .8 3 . Then *" = 0.225 X 9.83 = + 2 // .2, b" = -f- 2". 4, " = + l i"-8, and finally the logarithmic corrections are o.2^a" = -f- i , 0.358" = + I, etc. Accordingly, the most probable values of the angles and of their logarithmic sines are found to be Adjusted Angles. Log. Sines. A l = 4i05' 12" 1.81770 -#1 3 I 5 J 4 1.70229 C } 18 46 19 1.50758 1.02757 ^, = 42 33 49 1.83021 B^ 27 04 28 1-65815 C, = 20 14 58 1-539 21 1.02757 and these satisfy the geometric conditions of the figure as closely as can be done by the use of five-place logarithms. From these angles and the ^iven lengths of AB, BC, and CA the distances AS, J3S, and CS may now be computed. The above method also applies when the point 5 is without the given triangle. Thus, if j> be situated as shown in the figure, the above notation can be used by making BAS = A t , CAS = A, , SCB = C, , and SCA C r If the three points A, B, and C fall in the same straight line, the method fails, as then the conditional side equation is satisfied identically; in this case the distances AB and BC are known and a differ- ent side equation aiises which involves these lengths. 6 4 PRECISE PLANE TRIANGULATION. II. If in the last figure there be given the distances AB and BC and the angle B, and if A l , B lt and C t be observed, the condition that the lines AS, BS, and CS shall meet in one point is AB . sinA, sm(^ + Q = BC . sinC, sin^, + B,\ which may be used in a manner similar to that of the above example. Thus let there be given AB = i 067.950 meters, BC= 883.839 meters, B = 135 50' 51". 6, and let there be observed ^, = 75 56' oo".5, , = 68 34' 15". 2, and C t = 81 06' 35". o, all of equal weight. Then by a similar process it will be found that the adjusted values of these angles are A, = 75 56' 08". 8, B, = 68 34' 08". 6, and C* = 81 06' 37 // .o, and that the two values of BC, computed from these, are equal. Prob. 22. Let FG and Zf be two parts of a straight line, each 800 feet long. At F, G, and H are measured the angles which lines from a station 6" make with the base, namely, SFG 40 i2 f ', 1?GS= 92 58', and GHS = 43 55'. Compute the length of GS in two ways, and, if they are not equal, find the most probable values of the angles which will effect an agreement. 23. THE THREE-POINT PROBLEM. In secondary triangulation the position of a station 5 is sometimes determined by measuring the angles S, and S 9 subtended at it by three stations A, B, and C whose positions are known. It is well to measure the three angles at 5 and then by the station adjustment find the most probable values 23. THE THREE-POINT PROBLEM. 65 of 5, and 5 a . The data of the three known points give the distances AB and BC which will be called a and b, and also the angle CBA which wilfrbe called B. The problem is to determine the distances SA, SB, and SC. These distances can be found as soon as the angles A and C are known. Since the sum of the interior angles of the quadrilateral is 360 degrees, A+C= 360 - B - S, - S, ; and since the side >S is common to two triangles, the expres- sions for its length when equated give s'mA b sin^S", sinC a sinS/ Thus two equations are established whose solution will give A and C. Let A -\- C = 2m and A C = 2. The value of m is known, namely, m = i8o-i(^ + 5 1 + 5' i ), (23) and that of is to be found. Let V be such an angle that a sinS 2 then since A = m -f- n and C = m n, the second equation becomes which is readily reduced to the form , tanw = tanw cot(F+ 45), (23) x/ from which n is computed. The solution is hence made by first finding m from (23), secondly finding V from (23)', thirdly finding n from (23)", and lastly the value of A is wi n and that of C is m . 66 PRECISE PLANE TRIANGULATION. II. As a numerical example let the following be the given data for three stations, as determined by triangulation: Line. Azimuth. Distance. Station. Latitude. Longitude. ID 327 06' 49" 9 on.o ft. 7 34 !04- 2 5 2 58i.5 DJ 74 56 58 5 794-5 D 26537.2 47 688.9 Jl 184 25 52 9 098.9 J 25 03 2 .5 53 284.5 At a station 5, within the triangle IDJ, there are measured the angles ISD = 127 47' 33", 7XS/ = 87 & 18", and /57 = 144 34' 09". It is required to compute the lengths and azimuths of 57, SD, 5/, and also the coordinates of 5. Let station / correspond to A and station D to C\ then drawing a figure and comparing it with that above, the data are 5, = 144 34' 09", 5, = 87 38' 18", B = 74 5$' 58" - 4 25' 52" = 70 31' 06", a = 9098.9 feet, = 5794-5 feet. Next A + C=S7 1 6' 27" = 2i, and 01 = 28 38' 14". From (23)' log tan Fis found, whence V 69 43' 13", and then from (23)" log tan n is found, whence n = 14 06' 42". Accordingly ^ = 14 31' 32" = 57/, and C" = 42 44' 56" = JDS. From the triangle ISJ are computed the dis- tances 5/=56oo.6 feet and / = 3936.6 feet; from the triangle JSD are found SJ = 3936.6 feet and SD = 4417.3 feet. The azimuth of SD is 74 56' 58" + 42 44' 55" + 180 297 41' 54", and that of 57 is 169 54' 20". Lastly, the lengths of 57 and SD are multiplied by the sines and cosines of their azimuths, giving the differences of latitude and longitude, which being added to or subtracted from the latitudes and longitudes of 7 and D furnish the coordinates of 5 in two ways. The latitude of 5 is found to be 28 590.4 feet and its longitude 51 600.0 feet. A theoretic ambiguity is found in the above solution, since Fand n may each have two different values corresponding to the values of tan Fand tan n. This may be removed by always taking Fas less than 90 and positive, and then tak- ing n as less than 90 but making it positive or negative according as tan n is positive or negative. 24. GENERAL CONSIDERATIONS. When the point 6" in the above figure falls upon the cir- cumference of a circle passing through A, B, and C, the solu- tion is indeterminate, as should be the case. When 5 lies very near this circumference the results of the computation will be uncertain. In such an event a fourth station should be used in the field work. When more than three stations are observed from 5 there arises the Appoint problem, in which three different locations for 5 can be computed by taking the stations three at a time. In this case a process of adjustment by the Method of Least Squares is to be followed so that the four lines may intersect in one point. This process will not be developd here, as it is of infrequent application and the numerical work is lengthy. Prob. 23. Make the computations for the triangle IDJiioio. the above data, letting station / correspond to A and station J to C. 24. GENERAL CONSIDERATIONS. A series of connected triangles with one or more measured bases may be called a triangle net. The purpose of the triangulation and the topography of the country will deter- mine the location of the stations and the size of the triangles. A chain net is one suitable for a river survey, a polygonal net where the triangles from one or more polygons is some- 68 PRECISE PLANE TRIANGULATION. II. times used for a city survey, and a net composed of quadri- laterals each formed by four overlapping triangles is often used in geodetic work. The three types are, however, frequently combined together, single triangles, polygons, or quadrilaterals being used in different parts of the same net. A chain net is the simplest in adjustment, since no side equation arises if there be but one base. In the other kinds there will be one side equation for each polygon and one for each quadrilateral, but increased labor in computation counts for little when precision is demanded. A quadrilateral is a_ figure securing high precision, and the polygon takes almost rank with it, since the side equation eliminates accidental errors that otherwise might be propagated along the net. In the preceding pages only an introduction to the methods of adjustment has been given. The subject, however, will be continued in Chapter IX, where cases involving more than one conditional equation will be discussed. It may have been noticed in the use of the side equation in the preceding Articles that the smallest angles receive the largest corrections if the weights of the observations are equal. It hence appears to be important in conducting the field work to measure angles less than 30 or greater than 150 with a higher degree of precision than those between 30 and 150. By so doing the weight of the smaller angle will overbalance the error due to the large tabular difference in its sine, and the corrections will be more uniformly dis- tributed among the measured values. Geodetic triangulation nets differ from plane ones only in the greater size of the triangles and in the fact that the sum of the angles of each triangle is greater than 180. All the preceding methods are hence directly applicable in geodetic work. When, however, a plane net is extended for some distance east or west of the meridian where the initial azimuth was determined, the computed azimuths become less or 24. GENERAL CONSIDERATIONS. greater than the true ones owing to the curvature of the earth. In geodetic work this discrepancy is removed by introducing a correction whteh renders the back azimuth of a line different from its front azimuth, each value being the angle which the line makes with a meridian drawn through the end considered. When the plane coordinates of two stations are known the length and azimuth of the line joining them is readily com- puted. Thus, let L l and Z, be the given latitudes, then the latitude difference L^ L l is known; also let M t and M^ be the given longitudes, then the longitude difference M^ M t is known. From (13) it is seen that the azimuth from the first point to the second is found by (24) and the distance / ma'y then be computed from j\/r n/T T _ T i'J-n ^^ LrJ- i J-^ ^^ J-^i 1 = s'mZ or / = cosZ As an example, let the latitudes of two stations F and G be given as 15420.72 feet and 18 115.13 feet, and their longi- tudes as 20 347.19 feet and 14 739.08 feet; here the latitude difference is -|- 2 694.41 feet and the longitude difference is AZIMUTH AND DISTANCE FOR FG. Symbols. Distances and Azimuth. Logarithms. M t - J/i -f 5608.11 3.7488165 z a -z a 2 694.41 3.4304637 z 244 20' 1 7". I 0.3183528 tans . 1.9549007 siru 1.6365480 cosz / 622I.8O 3-7939I58 3-7939I57 70 PRECISE PLANE TRIANGULATION. II. 5 608. II feet. The computation may be arranged in the form as shown. The second logarithm subtracted from the first gives log tanZ and then Z is taken from the table; as tanZ is positive 64 20' ij" .1 is the azimuth of GF and 244 20' if. I is the azimuth of PG. Then log sinZ and log cos^ are taken out, and the subtraction of these from the first and second logarithms gives two values of log / which must agree within one unit of the last decimal. Lastly / is taken from the table. Thus the distance and azimuth between two stations which are not connected by a side of one of the triangles may be quickly computed in a plane system of coordinates. Prob. 24. The latitudes of two stations M and TV are 12 900.21 and 9 883.85 feet, and their longitudes are 27 333.16 and 35 640.93 feet. Compute the distance and azimuth from M to N* Prob. 240. In Art. 22 it should be noted that the signs of the corrections to a log sin are to be taken as negative for an angle between 90 and 180. What is the correction to a log sin for an observed value A l = 123* 15' 30"? 25. PRINCIPLES AND METHODS. CHAPTER III. BASE LIN ES. 25. PRINCIPLES AND METHODS. The principle involved in the measurement of a base line is the same as that in common chaining, the unit of measure being applied successively from one end of the line to the other. It is very important that length of the measuring unit should be accurately known in terms of the standard linear foot or meter, for otherwise its absolute error may be multiplied so as to give an erroneous length for the base. As the measuring bars or tapes are of metal they expand or contract as the temperature rises or falls and hence the coefficient of expansion of the metal must be known in order to eliminate errors due to this source. Other systematic errors, like those due to pull and sag in a tape and those due to the inclination of the base to the horizontal, must also be eliminated by computation. Accidental errors due to indefi- nite causes still remain in each result and, in order that the final length may be largely free from these, the measurement must be repeated several times and their mean be taken. Metallic bars from 10 to 20 feet in length have been extensively used for base measurements. These are of two classes, end measures and line measures. With end measures the distance between the extremities of the ends is a unit, and measurement is made by contact, one bar being placed in position and another brought into line so that the ends of the two touch each other; these ends are usually rounded to a radius equal to the length of a bar. With line measures but 72 BASE LINES. III. one bar is required, the distance between two marks engraved upon its upper surface being a unit; a microscope being placed on a movable frame over one mark, the bar is moved forward until the other mark comes into the same position, and then the microscope is moved forward to the first mark. In each case the number of bar-lengths multiplied by the length of one gives the length of the base. End measures are more convenient than line measures, but are generally not as precise. In order to eliminate effects of temperature, compound bars composed of metals whose rates of expansion are different have been devised and used; in these one bar expands more than another, so that by the use of a compensating lever the distance between the marks or ends is supposed to remain invariable. Since 1885 the long steel tape has been extensively used in the measurement of base lines, and has been shown to give results of a high degree of precision. As such a tape can readily be bought and standardized, as its use involves little expert knowledge, and as a base can be measured with it very cheaply, a full explanation of the method of procedure will be given in later Articles. Prob. 25. Consult Report of U. S. Coast Survey for 1897, and de- scribe the duplex base apparatus, and ascertain the character of its work. 26. PROBABLE ERROR AND UNCERTAINTY. As a line is measured by the continued application of a unit of measure the probable error in a result found for its length should increase with that length. The law of this increase is found from formula (11); thus if r l be the prob- able error of the unit of measure and / be the length of the line, the probable error of / is r = r,Vl (25) that is, the probable error in a measurement of a line 26. PROBABLE ERROR AND UNCERTAINTY. 73 increases with the square root of its length. Thus if two lines are measured with e,qual care and the second is four times as long as the first, t&e probable error of the second measurement is twice that of the first one. Since weights are inversely as the squares of probable errors it follows that the weights of linear measurements made with equal care are inversely as the lengths of the lines. Thus, a measurement of I ooo feet must be twice repeated and the mean of the results be taken in order to be worth as much as a single measurement of 500 feet. In combining linear measures, therefore, the weights of observations should be taken as the reciprocals of the distances. The most convenient'way to find the value of r l is to make duplicate measures of lines of different lengths. Let the lengths of the lines be / t ,/,,.../, the differences of the duplicate measures be d v , d^ , . . . d n , and n be the number of lines. Then, as shown in treatises on the Method of Least Squares, the probable error of a linear unit is r, = o.476 9 V T*J-. (26) For example, in order to find the probable error of measure- ment with a steel tape four lines were measured as follows: 7=427.34 854.21 1281.71 i 708.40 feet 7=427.37 854.20 1281.74 i 708.33 feet d = 0.03 -f- o.oi . 0-03 +0.07 feet p = 0.00234 0.00117 0.00078 0.00059 Here the weights are taken as the reciprocals of the lengths, since the weight of a line one foot long is taken as unity. Then by the use of the formula the probable error of a measurement one foot long is found to be 0.00058 feet, and accordingly that of one 100 feet long would be 0.0058 feet. Of course a larger number of observations than four is required to deduce a reliable value of this probable error. The uncertainty in the length of a line is expressed by the 74 BASE LINES. III. ratio of its probable error to its length (Art. 26), and is hence given by rj Vl y where r, is the probable error of a line one unit in length. Accordingly, if a certain line has an uncer- tainty of T7nnnn> tne uncert ainty of a line four times as long and measured in the same manner is $-$%-$-$. It thus follows that greater errors in the computed sides of triangles might result from a long base than from a shorter one. Prob. 26. Let the probable error of measurement with a steel tape be 0,005 feet for 100 feet. A square city lot is laid out with this tape so as to contain 43 560 square feet. Compute the prob- able error of this area. 27. BASES AND ANGLES. The uncertainty in the length of a computed side of a triangle is caused by a combination of the errors in the base with those in the angles, and the influence of the angles is usually greater than that of the base. Let the base a in the triangle ABC be measured with a probable error r a , and let r be the probable error of the angle measurements expressed in radians. Then by (n), r b = b V(r a /af + r 2 cotM + r' co?B is the probable error in the computed value of b. Now in Art. 17 it was shown that the best-shaped triangle is an equilateral one, and for this case the formula gives u b = u r as the uncertainty in the computed value of b. Let the probable error of the angle observations be one second or 0.000004848 radians. Then, if the base were without error, the uncertainty in b would be ^-g-g-Vrnp but if the base have also an uncertainty of ^jVmr the uncertainty in b will be It is not easy to carry on a triangulation so that the mean probable error of the adjusted angles shall be less than one 27. BASES AND ANGLES. 75 second, but it is very easy to measure a base of moderate length so that its uncertainty shall be less than ^j^Voif ^ n geodetic work bases have been, measured with an uncertainty of less than y fruTTnnr*' ^ thus appears that even in the best- proportioned triangle the precision of the base measurement can be rendered greater than that of the angle work. The difficulty of finding good locations for bases and the expense of measuring them renders it customary, however, to use only one or two in a triangulation net of moderate extent. When the sides of the triangles are from one to ten miles in length a base line about a mile long may be used. Care must be taken that the triangles connecting it with the main net are well proportioned, no angle being less than 30 degrees. The topography of the country will determine the location of the stations to a great extent, but the figures show two methods of gradually increasing the lengths of the sides away from a measured base AB\ the second method is the better one. In geodetic work bases several miles in length have been used. For example a base of the U. S. Coast and Geodetic Survey in Massachusetts is nearly lof miles long, its meas- urement occupying three months in 1844. The final result, reduced to the ocean level, -was 17326.376 0.036 meters, giving an uncertainty of ^g-yVcro- About 295 miles north- easterly is the Epping base, and 230 miles southwesterly is the Fire Island base, which were also measured with similar precision. The length of the Massachusetts base as computed 76 BASE LINES. III. through the triangulation from the Epping base was found to be 17 326.528 meters, and its length computed from the Fire Island base was found to be 17 326.445 meters. The actual uncertainties between the measured and computed values are hence TTT Vorr and yy^Vinr respectively, the effect of the errors in the angles being four times that of the base errors in the first case. In general it is found that angle errors do not increase the uncertainties of computed lengths to the extent that might be inferred from the preceding discussion, and this is probably due in part to the fact that they are largely eliminated in the adjustment of the triangu- lation. Prob. 27. Four measures of a base line give the values 922,220, 922.197, 922.221, and 922.217 feet. Show that the uncertainty of the mean of these measures is about 28. STANDARD TAPES. A long steel tape is the most convenient apparatus for measuring the base line of a river or city survey, and it has also been used for geodetic bases with excellent results. It is necessary that it should be compared with a standard, and this can be done for a small fee by the Bureau of Weights and Measures at Washington. The certificate furnished will state the error of its length for a certain temperature and pull, or it will state that it is correct at a given temperature and pull. The coefficient of expansion, or the relative change in length for i Fahrenheit, should also be stated, in order that the effect of temperature may be eliminated. The coefficient of stretch, or the relative change in length for one pound of pull, must also be known. A tape thus standard- ized becomes itself a standard with which other tapes may be compared. To compare another tape with the standard tape the coefficient of expansion of the latter must be known. To 28. STANDARD TAPES. 77 determine this the tape is stretched out on the floor of a large room whose temperature* can be varied. With a spring balance at each end it is Bulled to a certain tension, the thermometer noted and a certain length marked on two tin plates temporarily fastened on the floor. The temperature is then raised or lowered and the operation again repeated under the same pull. The change of length as marked on the tin plates is accurately measured, and this is divided by the total length and by the number of degrees to give the coefficient of expansion. The work should then be repeated several times using different lengths in each case, and the mean of the results be taken for the final coefficient. If a tape is to be used under different tensions its coeffi- cient of stretch should also be determined. The operation of doing this is similar to that above described, except that the temperature should be kept constant and the pull be varied. The change of length divided by the difference of the pulls and by the total length is the coefficient of stretch. Sometimes a tape is stretched over two supports A and B, and thus owing to the sag the measured distance is too long. Let /be the distance read on the tape under a pull P, let d be deflection or sag at the middle, and w the weight of the tape per linear unit. The curve of the tape is closely that of a parabola, and if L be the n T horizontal distance, L = / j very nearly. Also taking moments about the middle of the span, Pd = \wl-\l nearly. Eliminating d from these two equations there results from which the true distance L can be computed from the observed distance /. If the distance AB be subdivided into 78 BASE LINES. III. n equal parts by stakes whose tops are on the same level as those at A and B, then gives the horizontal distance between A and B. It thus appears that any observation of a distance read on a steel tape may contain three systematic errors due to tem- perature, pull, and sag. Let / be the temperature and / the pull at which the tape is a standard, let T be the temperature and P the pull at which a measurement / is taken, let e be the coefficient of expansion, and s the coefficient of stretch, let w be the weight of the tape per linear unit, and if sag exists let n be the number of equal spaces in the distance /. Then the reading / is to be corrected by applying the follow* ing quantities: Correction for temperature = -\- e(T i)l, Correction for pull = -{- s(P /)/, Correction for sags = As an illustration, let t = 56 degrees, / = 16 pounds, ^ = 0.00000703, s = 0.00001782, w = 0.0066 pounds per linear foot. Let a horizontal distance 309.845 feet be read at a temperature of 49^ degrees under a pull of 20 pounds, there being 7 subdivisions in that distance. Then the cor- rection for temperature is 0.0142 feet, that for pull is -f- 0.0221, and that for sag is 0.0028 feet. The corrected measured distance is then 309.850 feet. Lastly, if the measurement is made on a slope it must be reduced to the horizontal. For this purpose the difference of elevation of the two ends is found by leveling. Let // be this distance and L the length on the slope, then the hori- zontal distance is L\ / I -^. For instance if the length 29. MEASUREMENT WITH A TAPE. 79 309.850 feet has 2.813 feet as the difference of level of the ends, then the horizontal distance is 309.838 feet. Steel tapes used in base-ljjie work usually vary in length from 300 to 500 feet. They have division marks at every 50 feet, but near the ends the marks are one foot apart, and a finely graduated rule is used for reading decimal parts of a foot. Prob. 28. A tape is a standard at 41 F. when under 16 pounds pull and no sag, its coefficient of expansion being 0.0000069 an ^ its coefficient of stretch 0.00000195. Find the pull P so that no cor- rections will be necessary when measurements are made at a tem- perature of 38 degrees and with no sags. 29. MEASUREMENT WITH A TAPE. When a base is to be measured with precision it should be laid out into divisions, each shorter than the length of the tape, and stout posts be set at its ends and at the points of division. In these posts are placed metallic plugs, each having drawn upon it a fine line at right angles to the direc- tion of the base. The elevations of these plugs should be carefully determined by leveling. Each division is then subdivided into several equal parts by light stakes set in line and on grade, the distance between the stakes being fifty feet or less. The tops of these stakes should be smooth and rounded so that friction may not prevent the transmission of a uniform tension throughout the tape; on the top of each stake two small nails may be driven to keep the tape in position. Instead of stakes special iron pins are sometimes used each having a hook to hold the tape. The measurement should be done on a cloudy day with little wind in order to avoid errors due to change in tempera- ture. The tape is suspended over two plugs and upon the intermediate stakes and pulled at both ends by spring balances to the desired tension. At one plug a graduation 80 BASE LINES. Ill mark of the tape is made to coincide with the fine line on the plug, and at the other end the distance between the fine line and the nearest graduation mark is read by a closely grad- uated rule. Several measures of each division should be made at different times and with different pulls and the tem- perature be noted at each reading. FIELD NOTES. BASE LINE EG. OCT. 3, 1888, P.M. Divisions. No. of Sub- divisions. Diff.in Eleva- tion of Ends. Temperature. Pull. Measured Distance. Remarks. feet o Ibs. feet Ill 7 2.813 51 16 309.865 50.5 18 309.857 50.5 20 309.842 5 16 309 . 8/0 5 18 309.857 Cloudy. 49-5 20 309.845 II 7 5.6l8 48 16 332.736 47-5 18 332.727 No Wind. 47-5 20 332.712 47 16 332.740 47 18 332.726 47 20 332.715 I 6 7.9 2 4 47 16 279.850 47 18 279.843 47 20 279.832 48 16 279.848 48.5 18 279.840 48 20 279.837 The field notes of one measurement of a short base line EG, about 922 feet long, will illustrate the method of opera- tion. There were three divisions, designated as I, II, and III, the first having six and the others seven subdivisions. 2 9 . MEASUREMENT WITH A TAPE. 81 The steel tape used was about 400 feet long, and stated by its makers to be a standard at 56 Fahrenheit when under a pull of 1 6 pounds ^and having no sag. Its coefficient of expansion had been determined to be 0.00000703, its coeffi- cient of stretch 0.00001782, and its weight per linear foot 0.0066 pounds. In order to correct the field results the expressions deduced in the last Article become Correction for temperature = 0.00000703(56 Correction for pull = -j- o. 00001782^ i6)/; I 3 Correction for sag 0.00000181 5-^; from which the corrections are computed. For example, in division III, where n = 7, the mean of the observed distances is 309.856 feet and this is taken as the value of / in all the corrections. These being computed the corrected inclined distances are found and their mean gives 309.851 feet as the inclined length. Lastly, this is reduced to the horizontal, and 309.838 feet is the final length of division III. COMPUTATIONS, DIVISION III, BASE EG. 1 Corrections. Temp. Pull p Measured jDistance. Corrected Distance. Notes. Temp. Sag. Pull. Ibs. feet. feet. feet. feet. feet. 5i. 16 309.865 .0109 - .0043 O 309 8498 = 7 50.5 18 309.857 .OI2O -0034 -f- .0110 309.8526 50 5 20 309.842 .0120 .0028 + .0221 309-8493 h 2.813 ft. 50. 16 309.870 .0131 - .0043 O 309.8526 50. 18 309.857 .0131 .0034 -j- .0110 309.8515 C f = 0.0128 ft. 49-5 20 309.845 .0142 .0028 -f .0221 309.8501 Mean inclined distance = 309 851 ft. Mean horizontal distance = 309 838 ft. Proceeding in the same manner the corrections were found for Divisions I and II, and the sum of the three mean hori- 82 BASE LINES. III. zontal distances is 922.223 feet, which is the most probable length of the base line EG as determined from the observa- tions of one day. Four other measurements of this base, made on four different days, gave the results 922.220, 922.221, 922.226, and 922.217 feet. The mean of the five is 922.221 feet, whose probable error is o.ooi feet nearly, and accordingly the uncertainty of this final mean is about STroViFir- I fc i s thus seen that work of a high degree of pre- cision can be done with a long steel tape whose constants are known. The greatest errors in tape-line measurements are those due to errors in comparison with the standard and those due to the fact that ^he temperature of the metal is not the same as that of the air. The latter error may be removed by making some measurements when the temperature is rising and others when it is falling, and methods have also been devised of finding the exact temperature of the tape by means of an electric current passing through it; the former error cannot be removed except by the use of different tapes which have been independently compared with the official standard. An account of the measurement of a geodetic base of 3780 meters, or about 2.3 miles, by steel tapes is given by Woodward in Transactions of American Society of Civil Engineers for October, 1893. It is concluded that the prob- able uncertainty in the final result, arising from all sources except that of error in the tape, cannot exceed ^--frorinnr- This precision was secured by four days' work with twelve men, most of the measurements being made at night. In general it seems to be an established conclusion that precision in base measurements may be secured more cheaply by the use of tapes than by any other method. Prob. 29. Correct the measurements on Division I of the above base line JEG, and compute the most probable value of its final length and its probable error. 30. BROKEN BASES. 83 30. BROKEN BASES. ' -j A base line should be perfectly straight and its ends be intervisible, but cases sometimes arise where obstructions, like a river or swampy land, render direct measurement im- practicable. In geodetic work such a location should not be selected for a base line, but in secondary plane triangulation it may be used if expense is thereby avoided. The first case is where the base AB is computed from two distances a and b, measured along the lines BC and AC. The three angles of the triangle are also measured and adjusted. The length of the base is then computed from = b cosA -(- a cos^, or from = (a + )' - 4ab sin*$C. It might at first be thought that A ' the small angles would introduce a high uncertainty in the computed length, but on reflection it is seen that this is not the case because two sides of the triangle are given, and accordingly the uncertainty due to the angles decreases with their sines. For instance, if a = b and if C = 170, it will be found that a probable error of one minute in C produces an uncertainty of only T ou\w m t ^ le computed length of the base. A second case is where a stream crosses the base line between B and C. Here four points are selected on the line, two on each bank, and at these the angles are read which the base makes with lines drawn to an auxiliary station 5. From these angles and the measured distances AB and CD the distance BC is computed in two ways, namely, - A 5 ' mA sin ^' + ^ sinP sinQg. + == sinC, sm(A 4- 84 BASE LINES. III. and the angles should be measured with such precision that these agree in the last decimal used in the numerical work. Another method of procedure in the last case is to measure only the angles at the station 5. Let these be called 5, , 5, , and 5, , as shown in the figure, and let AB and CD be called a and b. Then the distance BC may be computed from pr - ~ Jj Cx , 2 COSJ 2 where a b is to be taken as always positive and where y is an angle whose value is found from tan> _ _^ sin(5 1 + 5.) sin(5 a + 5.) ~ (a - b)" sinS, sinS, This is the method recommended by the U. S. Coast and Geodetic Survey. The demonstration of these formulas may be easily made by applying the second equation of Art. 23 to the three points ABC and then to BCD, and equating the two expressions each of which contains the unknown distance BC. In order to verify the result another station S' may be selected, the angles be measured there, and another compu- tation be made. |l |; ) ' Prob. 30. For the last case let'jkhere be given a = 90.0242 meters, b = 120.0316 meters, S 1 = 19 41^4". 6, S^ = 39 20' 45". 2, and S 1 ,, = 26 19' 32". 8. Using seven-mace logarithms show that the length of C\$ 107.8408 meters. 31. REDUCTION TO OCEAN LEVEL. Geodetic base lines must be reduced to mean ocean level in order that perfect agreement may obtain in the sides of triangles computed from different bases. Let AB be the base whose measured length is / and whose mean elevation above mean ocean level is h. Let ab represent this ocean level whose radius of curvature Ca or Cb is R. Then, SI- REDUCTION TO OCEAN LEVEL. from the two similar sectors, the value of ab i R' (30 and therefore the correction to, be subtracted from the adjusted measured length is Ih/R. For a long base this correction will be appreciable even when the base is but a few feet above the mean ocean level. In Chapter VII it is shown how the radii of curvature have been found for different lati- tudes; it is there seen that for common cases the logarithm of R may be found by taking the mean of the logarithms of R^ and R^ given in Table IV at the end of this volume. When the azimuth of the base is given and great accuracy is required R should be computed from R in which Z is the azimuth of the base line and R l and R t are taken from Table IV. For example, let the adjusted measured length of the base be 1 8 207.3267 meters, its mean height above ocean level 523.2 meters, and its mean latitude 40 36' '. From Table IV the logarithm of R is 6.8044705 and the correction Ih/R is found to be 1.4943 meters, so that the length on the ocean level is 18 205.8324 meters. If the azimuth of the base be 75 40', the more accurate formula gives the logarithm of R as 6.8052175, from which the correction Ih/R is 1.4917 meters, so that the final length on ocean level is 18 205.8350 meters. As the lengths of the base lines have been reduced to ocean level it follows that all distances computed in a geodetic tri- angulation are really the projections of the actual distances on 86 BASE LINES. III. the surface of a spheroid coinciding with ocean level. Thus, if /! represent the computed geodetic distance between two stations whose elevation above ocean level is //, it is plain from (31) that the true distance between those stations is For example, let ! 1 = 100 miles and h = 2\ miles ; then, using for R the mean value 3959 miles, the true distance / is 100.06315 miles. Here it is seen that the difference / / t is 1/1583.6 of the distance l lt so that the error in considering a computed geodetic distance as the actual distance may often be a large one. Prob. 31. A base line measurement, made 374 feet above ocean level, gives 1725.065 feet. What is the length of the base when reduced to ocean level? 32. SPIRIT LEVELING. 8/ CHAPTER IV. LEVELING. 32. SPIRIT LEVELING. The method of determining differences of elevation by an engineer's level and rod is called spirit leveling to distinguish it from the method in which vertical angles are used. In common work the telescope is made level by bringing the bubble into the middle of the attached scale. In geodetic work a sensitive bubble is used and readings of its ends taken on the scale, corrections to the rod readings being applied according to the distance of the rod from the instrument. A level surface is one parallel to that of a fluid at rest, and the process of leveling consists in rinding the elevations of points above the mean surface of the ocean. The line of collimation of the telescope of a properly adjusted and leveled \A' / IB' /C A -x C instrument, when revolved around the vertical axis, gen- erates a plane which is tangent to a level surface. The line of sight, however, is depressed below that plane owing to refraction, and it lies between the tangent plane and the level surface, but nearer to the former. Thus if / be the telescope of the instrument, the straight line A'B' represents the tangent plane, and the curved line ab the level surface, while the actual line of sight is a'b' y the points a f and b r , in conse- 88 LEVELING. IV. quence of refraction, appearing to be in the tangent plane at A 1 and ff. The rule that front and back sights should be of equal length in order to secure precision is one that is well known, and the figure shows the reasons for it. Let rods be set at A, B, and C in order to find the heights of B and C above A ; then the observer will set the targets at a ', b r , and c', and the readings of the rods will be Aa', Bb' ', and Cc' . The height of B above A will be given by Bb' Aa! , and that of C above A will be given by Cc' Aa' . Now, owing to the combined effect of curvature of the level surface and of refraction of the air, the errors aa' , bb 1 ', and cc' have been made in the rod readings, but the difference Bb' Aa' is the same as Bb Aa if the horizontal distances from B and A to the instrument are equal, while the difference Cc' Aa r is not the same as Cc Aa if the rod C is further from the instrument than the rod A. It will be shown in Art. 37 that the deviation of a tangent plane from a level surface is about two-thirds of a foot at a distance of one mile and -72* feet at a distance of n miles, also that the deviation of the tangent plane from the refraction surface is one-seventh of that of the level surface. The com- bined effect of curvature and refraction is hence to cause an elevation of the line of sight above the level surface amount- ing to about 0.57 feet in one mile or 0.57^' feet in ;/ miles; a more exact rule is 0.000206 feet in 100 feet and 0.000206^* feet in loon feet. Thus, in the above figure, if the rods A and B be at 500 feet from the instrument, aa' and bb' are each 0.0051 feet, but the difference of level between A and B is free from error. If the rod at C be I ooo feet from the instrument cc' is 0.0206 feet, and hence the difference Cc' Aa' is 0.0155 feet in error, for cc* is 0.0155 feet greater than aa' . Another class of errors that is largely removed by taking backsights and foresights of equal length are those due to- 33- DUPLICATE LINES. 89 lack of perfect adjustment of the instrument. Thus if the line of collimation be not, exactly parallel to the level bubble the reading on the rod at An may be too great, but when the sight is made on B the reading there is also too great, and hence these equal errors disappear in taking the difference of the rod readings. It is not desirable to try to do precise work with an instrument that is not in good adjustment, but it is essential to note that precise work cannot be done with back and front sights of unequal length, unless the lengths of these be measured and a correction be applied for the com- bined effect of curvature and refraction. In common work pacing, or even estimation, may be sufficient to prevent the introduction of these errors, but in precise work the distances should always be measured to the nearest foot. Prob. 32. Let the rod readings at A, B, and C be 1.073, 3- I 37> and 9.271 feet, the distances from the instrument being 200, 250, and 400 feet. Find the elevations of B and C above A. 33. DUPLICATE LINES. In common work with an engineer's level the precision of the elevations of the bench marks may be increased by running a second line between them and then taking the mean of the differences of level. This precaution can never be neglected in good work, for one measurement affords no data for estimating the precision of the results, It is better to run the two lines in opposite directions rather than in the same direction. Semi-duplicate lines are those run in the same direction, having the same bench marks and heights of instruments but different turning points. Two sets of notes are kept which are not compared until a check is made on a bench mark. Thus in the figure let M and N be two bench marks and /, , 7 a , /, , and 7 4 the points where the level is set up, while AU A t , and A 3 are the turning points on line A, and B l , 90 LEVELING. IV. BV and 7?, are the turning points on line B. The instrument being set at I I a backsight is taken on M and recorded in the ^ __A__ M ^ :== ir :: ^ /4 notes for line A ; then another backsight is taken on M and recorded in the notes for line B. The two turning points A l and B l having been selected, foresights are taken upon them in succession and the readings recorded in the notes for lines A and B respectively. Then the instrument is moved to 7 3 and backsights taken on A, and B l which are recorded in the separate notes for A and B. On arriving at 7 4 backsights are taken upon A 3 and B t and two foresights upon N. Thus two lines MA^A^A^N and MB,B^B t N have been run between the bench .marks M and N; if the elevation of TV is to be deter- mined from the given elevation of M, two sets of observa- tions are at hand from whose comparison and combination it can be obtained with a higher degree of precision than by a single line. Another method of running semi-duplicate lines is to have the same turning points but different heights of instruments. Thus, in the above figure, if 7, be a bench mark the level is set at A l , a backsight taken upon I I and a frontsight upon 7,; then the instrument is set at B^ , a backsight taken upon 7, and a frontsight upon 7, . This method is not as conven- ient or expeditious as that above described, since it involves two rodmen, and it would be better to run two independent duplicate lines in opposite directions between the bench marks. By taking proper precautions to preserve equality in the lengths of back and front sights, shading the instrument from the rays of the sun, and keeping the rod truly vertical, semi- duplicate lines may be run with an engineer's level so that the probable error of differences in elevation shall be less than 0.005 feet for bench marks one mile apart. In precise level- 34- PROBABLE ERRORS AND WEIGHTS. Ql ing where readings are taken to ten-thousandths of a foot, the probable error may be, made much smaller. The adjusted elevations of the benches arfi of course the mean of the values found by the two lines. It is sometimes observed that the elevations found by one line tend to be greater than those found by the other. For example, a line of semi-duplicate levels run from Bethlehem to Allentown, Pa., by students of Lehigh University in 1894 may be briefly noted. The total distance was 32 750 feet, this being divided into 27 sections with 28 bench marks. Computing the 27 differences of level for lines A and B it was found that nine were the same for both, that line A had nine greater and also nine less than line B\ computing the eleva- tions of the 27 benches from that of the Bethlehem bench it was found that 25 of these were greater on line B than on line A. The discrepancy between the two lines reached a maximum of 0.009 ^ eet at 18000 feet from the Bethlehem bench, then decreased to o.ooi feet, and afterwards increased until it became 0.005 ^ eet at ^ e Allentown bench. The probable error of the difference of level between the end benches, computed by the method of the next Article, was found to be 0.004 f eet - This is perhaps a little smaller than would be found by independent duplicate lines run in opposite directions. Prob. 33. The difference of level of two points P l and /*, was found, by setting the level half-way between them, to be 6.438 feet. A second observation gave 6.436 feet, and a third one gave 6.437 feet. Show that the probable error of a single observation was 0.0007 feet. 34. PROBABLE ERRORS AND WEIGHTS. The probable error of the difference in elevation of two bench marks increases with the number of times the instru- ment is set up between them, and will hence be greater in a hilly region than in a prairie country. It will also depend 92 LEVELING. IV. upon the precision of the instrument and upon the skill of the leveler and rodman, so that different classes of work will have different probable errors. Assuming that the instrument is set up about the same number of times in a distance of one mile or one kilometer, it will be clear that the probable error in leveling is governed by the same law as that for linear measurements, namely that it increases as the square root of the distance. Thus if r l is the probable error in leveling a distance of unity, say one mile or one kilometer, then the probable error in leveling the distance / is r = r l V~l. Thus if the probable error for a line one mile long is 0.006 feet the probable error for a line four miles long is 0.012 feet. By means of duplicate lines of levels the probable error r, may be obtained by the application of formula (26), the weights being taken as the reciprocals of the lengths of the lines. Semi-duplicate lines, like those described in the last Article, may be used for the same purpose, but probably the value of r, found from them is somewhat smaller than from two lines run in opposite directions. As an example of the method, let D a and D b be the differences of elevation be- tween two bench marks as determined by the two lines, d the differences, or discrepancies, between these, / the dis- tance between the benches, and p the weight of d in terms of the weight of the unit of distance. * Taking the following five measurements, and regarding 1000 feet as the unit of distance, the sum of the five values of pd* is 0.0000404 D a 3.801 13.429 0.363 + 5.528 + 9-657 feet D b - 3-803 13.426 0.365 + 5.532 -f 9.653 feet d -f O.OO2 0.003 + 0.002 O.OO4 + 0.004 feet I 0.400 0.840 I.5OO 1.800 2.000 feet/ 1 ooo p 2.50 1.19 0.67 0.56 0.50 pd* 0.0000 1 00 0.0000107 0.0000027 0.00000 90 0.000000080 and then from formula (26) the value of r l is found to be 0.0014 f eet - Thus, for this class of work, the probable error 34- PROBABLE ERRORS AND WEIGHTS. 93 in leveling a distance of I ooo feet is 0.0014 ^ eet an d hence the probable error in leveling' any distance is 6.0014 1/7, where /is the distance in thousand of feet. To find the probable error for one mile /Is to be taken as 5.28, and thus 0.0032 Vn expresses the probable error of a line of levels n miles in length. As weights are inversely proportional to the squares of probable errors it follows that the weights of differences of elevation are inversely proportional to the distances over which the leveling is extended. For example, let there be run three routes from P to Q giving the results Route. Miles. P above Q. 1 5 37-407 feet 2 6 37-392 feet 3 10 37.4H feet If the precision of the work per mile is the same, the value of r l being the same for the three lines, then the weights of the three results are to be taken as ^, ^-, and -^. The adjusted elevation of P above Q is then found by the rule of Art. 4 to be 37.403 feet. The probable error r l may be also computed from lines run between two benches by different routes, as in the last example. The method to be followed is that of formula (9)" '. Thus, taking the weight of a line one mile long as unity, the residuals v are found and the sum 2pv* is V V 0.004 0.000016 -f-O.OII O.OOOI2I O.OI I O.OOOI 2 I 0.20 0.10 0.00000320 0.00002057 O.OOOOI2IO z = 37.403 0.000258 0.00003587 = formed. Then, from the formula, r l = 0.0029 ^ eet which is the probable error of a difference of level found from a line CIVIL ENGINEERING 94 LEVELING IV. one mile long. Finally, the probable errors of the three observed differences of level are found from the square-root rule to be 0.0065, 0.0070, and 0.0092 feet, while the probable error of the adjusted elevation is 0.0039 feet, so that 37.403 0.004 feet may be written as the final result. Prob. 34. If the probable error in leveling one mile is 0.003 feet, what is the probable error in a line one kilometer long, and also in a line 100 kilometers long? 35. ADJUSTMENT OF A LEVEL NET. When a closed circuit is made by running from A around to A, leaving the benches^, C, and D, the adjusted eleva- tions of these are to be made by distributing the error of closure in direct proportion to the distances ) between the benches. For example, starting from A with the correct elevation of 420.317 feet above mean ocean level, the following elevations of other benches are found, and on returning to A its elevation is 420.467 feet, showing a discrepancy of o. 150 feet. The distances between the benches being 6, 3, 4, and 2 miles, T 6 T of the discrepancy is to be subtracted from the elevation of B, T 9 y from that of C, and so on. This method of adjustment is one that would Correction. o.ooo 0.060 0.090 o. 130 o. 150 be naturally used by every one, and it will be seen that it agrees with the results obtained by the application of the rule in Art. 21 to the determination of the most probable differ- ences of the elevations between the benches. Bench. Miles from A. Observed Elevation. Adjusted Elevation. A 420.317 420.317 B 6 532.918 532.858 C 9 607.200 607 . i 10 D '3 5!-3T5 510. 185 A 15 420.467 420.317 35. ADJUSTMENT OF A LEVEL NET. 95 A net of levels consists of several lines connecting benches in such a manner that the, elevation of one can be deduced from another by several different routes. An example of the method of adjustment is given in Arts. 5 and 6, where, how- ever, the weights of the different results are taken as equal. By introducing the weights according to the method of Art. 7, taking them as inversely proportional to the lengths of the lines, the same process may be applied to any given case. For example, take the case shown in the figure where eight differences of elevation between six benches are ob- served in a net consisting of three closed figures. These three figures give three geomet- ric conditions and accordingly there can be but five independ- ent quantities in the observation equations. This is perhaps seen more clearly by noting that, if the elevation of one bench be given, the elevations of the five others are to be obtained. In general the number of independent quantities in any net of level lines is one less than the number of benches. For example, let the eight observed differences of elevation be as given below, their weights being taken as the reciprocals of the distances between the benches. Let v. , v~ , v. , v* , No. Benches. Observed h. Feet. Distance. Miles. Weight. Adjusted h. Feet. I B above A 12 .02 4.0 0.25 12.039 2 C above B 23.06 7.2 o. 14 23.012 3 D above C 14.30 5.0 0.20 14.340 4 D above F 29.44 6-3 o. 16 29.389 5 C above F 15 .02 2.0 0.50 15.049 6 F above E 9-34 4.8 O.2I 9-372 7 B above E '45 3-5 0.29 1 .410 8 E above A 10 67 8-3 0.12 10.630 96 LEVELING. IV. and v, be corrections to be applied to the observed values ^i > h, 1 ^ 3 > h b > and // 7 in order to give the most probable values. Then the eight observations are expressed in terms of these five quantities, the condition //, /z 4 -f- h b = o giving the fourth observation in terms of z>, and t/ 4 ; thus, 1. + v, = o, 2. + 7.', = O, 3. + ^ 8 = o, 4- + V t + V. = + 0.12, 5. + v. o, 6. + v, - v b - v, = - 0.15, 7. + v, = o, 8. + ^i ^T = + o. 10. From these the normal equations are formed by the rule of Art. 7, using the given weights, and their solution furnishes the values v[ = +0.019, v t = 0.048, etc., from which the above adjusted most probable values of the five quantities are found. Then the values of 7z 4 , h t , and h 6 immediately result. The probable error for a line of levels one mile long can now be computed by formula (10). Each of the corrections being squared and multiplied by its weight, 2pv* is found to be 0.00246; then as n = 8 and g = $, there results r l = 0.019 f eet as tne probable error in leveling one mile, and accordingly 0.019 ^ ls tne probable error in leveling / miles. The degree of precision of the levels in this net is hence quite low compared with that required for city work. Another method of stating observation equations in the above case is to take the elevations of five benches as the quantities to be found. Thus, if the elevation of A be given, approximate values of the elevations of the others are readily found, and the corrections to be applied to these may be called v bt v e1 etc. Then each observation is expressed in terms of these corrections, and their most probable values are found by the solution of the resulting normal equations. 36. GEODETIC SPIRIT LEVELING. 9? The adjusted elevations will be the same as those derived from the adjusted differences that are given above. Thus if the elevation of A fee 312.724 feet, that of B will be 324.763 feet, that of E will be 323.353 feet, and so on. Prob. 35. The elevation of a bench P is 725.038 feet. Level lines run between it and the benches Q, -tf, and S, give the following observations: No. Benches. Difference in Elevation. Distance. Feet. Miles. 1 B above A 35.080 3 2 C below A 8.698 6 3 D below A 1 9-9S 4 4 C above D 11.212 3 5 C below B 43-7 8 3 6 B above D 54 "995 6 State the observation equations, form and solve the normal equa- tions, and find the adjusted elevations of the benches. 36. GEODETIC SPIRIT LEVELING. Engineers' levels are of two types, the Y level and the dumpy level, the former being easier of adjustment while the latter is more precise. A dumpy level with two vertical and three horizontal wires in the diaphragm of its telescope, and having also a sensitive bubble, may be called a geodetic instru- ment. The rod is to be brought into the field between the two vertical wires, and readings taken upon it by each of the horizontal wires or by the help of a micrometer screw. The limits of the ends of the bubble are read upon the attached scale. The rod is provided with attached levels for securing verticality and it is set on a foot-plate planted in the ground. The distances from the instrument to the backsight and front- sight positions of the rod are measured. Each instrument must be tested at intervals in order to determine the angular distance between the wires and the angular value of one division of the bubble scale. The usual 98 LEVELING. IV. adjustments for the level bubble and collimation axis are to be made, as also a series of measurements for determining the small errors still remaining in them. With these data tables can be made out for reducing the readings of each wire to the middle wire, for eliminating the error of inclination as determined by the readings of the ends of the bubble, and for eliminating the error of collimation. As precise leveling for geodetic surveys is generally done in the metric system, the constants of Art. 32 are not directly applicable for the elimination of errors caused by unequal lengths of back and fore sights. If these distances be in meters and /, and /, their values, the former being the greater, then, for usual atmospheric conditions, d = 0.0000675 (A' - '.') (36) is the correction in millimeters to be subtracted from the difference in elevation h l h^. For instance, if /, = 200 meters and / = 170 meters, then the difference h l // as found from the rod readings is 0.75 millimeters, or 0.00075 meters, too large. If the distances be in feet, then d = 0.000206 (/,' /,') (36)' is the correction in thousandths of a foot to be subtracted from the difference h v //,; thus if /, = 656 feet and / = 558 feet, d is 2.5 thousandths of a foot or 0.0025 f ee *. These formulas are demonstrated in Art. 37. The running of a line of geodetic levels is necessarily slow work, for daily tests of the instrument must be made and corrections applied to every rod reading in order to remove the errors above mentioned. The line is divided into sections from five to ten miles in length, each of which is leveled in opposite directions. The probable error of the elevations of the bench marks found by combining the two sets of observa- tions has been made less than two millimeters for a distance of one kilometer, which is equivalent to about 0.008 feet for a distance of one mile. 37. REFRACTION AND CURVATURE. 99 Notwithstanding the apparent accuracy of leveling by one of the instruments above described, the item of cost is so high that it cannot be usecf except on government work. It may be remarked, further, that the probable errors deduced from the discussion of such level lines are but little, if any, less than those that can be obtained by good work under the common method. By rerunning the sections several times by the semi-duplicate plan of Art. 34, using a good engineer's dumpy level, and eliminating the systematic errors by making equal the lengths of backsights and foresights, it is not diffi- cult to secure results whose probable errors shall be as low or lower than those of the so-called geodetic method, while the cost of the work per mile will be less than half as great. Prob. 36. Consult Wilson's paper on " Spirit Leveling " in Trans- actions of American Society of Civil Engineers, 1898, Vol. xxxix, and Molitor's paper in Vol. XLV, 1901. Compare methods used by the different U. S. Government Surveys, and collect facts regarding the cost of running long lines of levels. 37. REFRACTION AND CURVATURE. When light travels through air of varying density, its path is a curved line. If the surface of the earth were a plane a ray of light moving horizontally would suffer no refraction since the air would be of uniform density at all points in its path. Owing to the curvature of the earth a ray of light passing from c' to 7, in the figure of Art. 32, travels through air of increasing- density because c is further than / from the level surface; similarly light passing from c to /tends to do so in a straight line, but encountering denser air its path becomes a curve which lies between the chord cl and the arc cl. Hence refraction is a consequence of curvature. To develop formulas for the effect of curvature and refrac- tion, it is necessary to take for granted that the earth is 9 globe whose mean radius R is about 3 959 miles or 6371 kilo' 100 LEVELING. IV. meters. Let AO and BO be this radius in the exaggera- ted figure, AC a short distance / which is sensibly equal to the tangent AB y and bA the path in which light travels from b to A, thus making b appear at B to an observer at A. The de- viation due to curvature in the dis- tance / is hence represented by BC and that of refraction by Bb, their difference bC being the combined de- viation ; let these be called c, kc, and d respectively, k being an abstract number less than unity whose value will be shown later to be about ^. Thus d is expressed by (i k)c. The value of c is readily found, from the right-angled tri- angle ABO, to be given by 2Rc + S = /", or since c is very small its square may be neglected, and thus r C =TR is the deviation of the tangent plane from the level surface. The combined deviation due to curvature and refraction, of the distance bC, is then *= (l -*>5- (37) From this, using for k the mean value ^, there is found d= 0.00000006757', (d and / in meters), ers), ) . ., , ) ' d = 0.00000002 o6/ a , (d and / in feet), from which the formulas in Art. 38 directly result. If the elevation of the eye of an observer above the ocean is known the distance to the sea horizon may be deduced from (38). Thus, for different systems of measures, /(in kilometers) = 3.85 Vd(\n meters), /(in statute miles) =1.32 Vd (in feet), / (in nautical miles) =1.13 \' d (in feet). 38. VERTICAL ANGLES. IOI These results, like all in .this Article, are mean ones, as curvature varies in different latitudes, and refraction varies under different atmospheric^'conditions. Prob. 37. If the elevation d above the sea horizon is given in meters, what is the formula for / in nautical miles ? 38. VERTICAL ANGLES. The effect of refraction on any vertical angle is to render the measured value too large or too small according as it is an angle of elevation or angle of depression, while curvature produces the opposite effect. In the figure let A and B be two stations whose horizontal distance apart is /, the station B being higher than A. In order to find the difference in elevation the vertical angle of elevation BAG, or the vertical angle of depression ABD, is needed. Let an instrument be set at A and its horizontal limb be made tangent to the level surface ARC in the direc- tion Ae\ in consequence of refraction the station B appears to be in the direction Af, and fAe is the measured angle of elevation. The measured value is thus too large by the refraction angle fAB and too small by the curvature angle eAC\ the true required angle BAC is hence/>^ fAB -f- eAC. In the same manner, when the instrument is set at B y the measured angle of depression isf'Be', which is too small by the refraction angle ABf and too large by the curvature angle DBe' . These effects of refraction and curvature are small, and sensibly the same at A and B under similar atmos- pheric conditions. Thus the combined effect of refraction and curvature renders the measured angle at A too small and that at B too large by the same number of seconds. Let a be the angle of elevation at A and ft the angle of 102 LEVELING. IV. depression at B, and let d' r be the correction in seconds, so that BAG ex + d" and ADD = ft d" are the true required angles. In the last Article the linear correction d normal to /was found. The corresponding angle in radians is d/l and the corresponding value in seconds is 206 265^/7, and accordingly is the error in latitude show that dZ = tanz 44. AZIMUTH BY POLARIS AT ANY HOUR-ANGLE. Polaris or any other circumpolar star may be used at any position for the determination of azimuth, if the observer's watch indicates correct time, either local or standard, and if the latitude and longitude of the place are known. From the time of observation and the data given in the nautical almanac the hour-angle of the star is to be found and the solution of a spherical triangle then gives the azimuth of the star. Let Z be the zenith, P the pole, 5 the place of the star, and N the north point of the horizon. In the spherical triangle PZS the angle Z is the azimuth of the star east of the meridian, and the angle at P is its hour-angle minus 180; the side ZP is the co-latitude 90 0, the side PS is the co- declination 90 6. Let / denote the hour-angle of the star, that is the obtuse spherical angle NPS\ then the solution for the angle Z gives sin/ cos/ - cos0 taneT (44) 122 ASTRONOMICAL WORK. V. from which Z is to be computed after the hour-angle t has been determined. The field operations may be conducted exactly like those explained in the last Article. Another method of observation preferred by many observers is illustrated in the notes be- low. Pointing is first made on the mark at B and the horizontal circle is read; then four pointings and readings are made on the star, two with the telescope in the direct position and two with it in the reverse position ; finally a point- ing and reading on the mark is taken again; each reading is of course the mean of the two verniers. The time as indicated by the watch must be noted for each pointing on the star, and the mean of these times is that to be used to find the hour-angle /. The process of finding t by the help of the nautical almanac is shown in the lower part of the table. Then from the formula tanZ is found to be negative and hence the star was west of the meridian; accordingly Z is o 59' 06". 3, and finally the geodetic azimuth of the line AB is 175 57' 47". 6, the probable error of which may be estimated at 10" or 15". Making a number of observations on different parts of the circle and taking their mean, a fair determination of azimuth may be obtained by one night's work'. To secure the elimination of the instrumental errors of the transit more completely, one half of the pointings on the star may be made by looking at its reflection in a dish of mercury placed near the objective end of the telescope. When a geodetic theodolite is used corrections for the error of level in the telescope standards are to be applied, unless this be eliminated by taking half the observations in a mercury hori- 44- AZIMUTH BY POLARIS AT ANY HOUR-ANGLE. 123 zon. If a sidereal chronometer is at hand the time should be noted by it, as thus the reduction of local mean solar time to sidereal time is avoided.^ AZIMUTH BY POLARIS, APRIL 29, 1897. Watch Time P.M. Tel. Readings on Mark. Readings on Star. Data. D 25 II' 20" -0 = 40 36' 24" R 25 II 20 A. = 75 22' 50" 8 h i6 m 30 s R 30 14' 30" S =-. 88 45' 35"-6 8 18 30 D 30 13 05 a = i h 20 m 19". 2 8 20 30 D 30 II 50 8 22 30 R R 25 ii oo 30 10 40 Watch 93 s . 3 faster than Eastern standard time. D 25 ii 10 J. Doe, observer. 8 h I9 m 30" 25 ii' I2 /X 5. 30 12' 3i".2 _ oi 33.3 = Watch error. 30 12' 31". 2 8 h I7 m s6 8 .7 = Eastern standard time. oi 3^ .3 = Longitude correction. 25 ii 12 .5 BAH 1 = 5 oi 18 .7 Z= o 59 06 .3 8 16 25 .4 + 01 21 .7 = Loc al mean solar time. luction to sidereal nterval. = Rec i BAN 4 02 12 .4 2 30 48 .5 + oo 49 .4 = Sidereal time Greenwich mean noon. = Longitude correction. = Sidereal time. 1 80 Azimuth AB = 175 57' 47". 6 R. Roe, computer. io h 49 m 25' .0 I 20 19 .2 = Right Ascension of Polaris. = Hour-angle in time. 9 U 29 m 05". 8 142 16' 27". 3 =/ Any circumpolar star may be used by this method, but preference is generally given to Polaris as it is of second magnitude and easily identified. Other stars sometimes used are d Ursae Minoris and 51 Cephei, which are of fifth magni- tude and hence not so easily located as Polaris. 124 ASTRONOMICAL WORK. V. Although theoretically the observation may be taken at any time, yet a discussion of equation (44) will show that the conditions most favorable to precision occur when t is either about 90 or 270, that is when the star is near elongation. When the star is at either the upper or lower culmination small errors in and / may produce large errors in Z, and hence a star should not be observed when near its meridian passage. Errors due to either or 6 may be eliminated by observing the star at symmetrical positions east and west of the meridian, and taking the mean of the two computed results. Prob. 44. If an error of 15 seconds had been made in the mean time of the watch readings in the above example, what error would have been produced in the resulting azimuth of the line AJ 45. LATITUDE BY THE SUN. When the sun is on the meridian it is at its maximum altitude very nearly, and if this be measured with a sextant the latitude of the place becomes known. Thus in the figure let the circle be a meridian section of the celestial sphere, P the pole, Z the zenith, Q the equator, 5 the sun, and //'//'the horizon. The arc SH' is the altitude of the sun, SQ is its declina- tion, and ZQ is the latitude of the place of observation A. Let // be the meridian altitude, corrected for refraction and parallax, 6 the declination of the sun, and the latitude of the place. Then, from the figure, = 90 " h + <$, (45) in which <$" is positive when the sun is north of the equator and negative when it is south of the equator. On the ocean the altitude is taken by bringing down the Q 45 LATITUDE BY THE SUN. 12$ image of the sun until its lower limb touches the sea horizon. On land the image is brought down until the lower limb touches its reflection as seen in a dish of mercury, and thus the double altitude is read. The operations are begun several minutes before apparent noon and a number of measurements made which give altitudes gradually increasing to a maximum and then decreasing. The proper corrections are then to be applied to the maximum altitude, and finally the above formula gives the latitude, d being taken from the nautical almanac. For example on October i, 1897, the maximum double altitude of the sun's lower limb, observed with a sextant by a student at Lehigh University and corrected for index error and eccentricity of the instrument, was 91 51' if .2, one-half of which, or 45 55' 38". 6, is the apparent altitude of the lower limb. To this is to be added 16' oi".6 for the sun's semi-diameter, giving 46 n' 40". 2 apparent altitude of the sun's center. The refraction correction to be subtracted is o' 53 // . I and the parallax correction to be added is 06". 2, and thus the true altitude h was 45 $4' 51". 6. The declination of the sun being S. 3 28' 44". 3, as found from the nautical almanac, taking account of the difference in longitude, the latitude of the place by (45) is 40 36' 24", a result whose probable error is 5" or- more, since but a single reading was taken on the sextant. A more precise determination can be made by taking about six altitudes at intervals of one minute, half being before and half after the time of maximum altitude. It is well also that three of them should be taken by bringing the sun's lower limb into coincidence with its image in the mercury, and three by bringing the upper limb to coincide with its image j the mean of the six then gives the double apparent altitude of the sun's center. This altitude, after correction for the errors of the instrument and for parallax and refraction, may be safely used to give a latitude determination with a prob- UNIVERSITY OF CALIFORNIA NT OF CIVIL ENGINEERS ECKKEi-EY. CALIFOi-tMIA T26 ASTRONOMICAL WORK. V. observations upon it with the sextant or with the engineers' transit. If the altitude k l be observed at upper and h^ at lower culmination, then the mean of these, each being corrected for refraction, gives the lati- tude, or = (/*, +./*,). Here h l may be itself the mean of several altitudes taken at equal intervals before and after the upper culmination and h^ may be the mean of several similarly taken before and after the lower culmination. With a good sextant the latitude may be found by a few series of observa- tions with a probable error of one or two seconds of angle, and with a transit to a less degree of precision. Formula (45)' of the last Article may be used to find the latitude from an observed altitude of any star in any position, if its hour-angle t is known. When a sidereal chronometer is 128 ASTROKOiMICAL WORK. V. at hand the sidereal time of taking the altitude, diminished by the right ascension of the star, gives the hour-angle in sidereal time, and fifteen times this is the value of t in angular measure. For example on December 8, 1897, ten altitudes of Polaris were taken with a sextant in about thirteen minutes at Lehigh University, the time of each being noted on a sidereal chronometer. The mean apparent altitude, after correction for index error and eccentricity, was 41 47' 46". 3, and applying the refraction correction i' 06". 3, the mean true altitude is // 41 46' 40". o. The mean sidereal time of the times of the ten measurements was o h io m 54 s . 9, from which is subtracted the right ascension of Polaris or i h 50'" 33 s . 6 to give its hour-angle 22 h 50 33 S .6, whose equivalent in degree measure is / = 342 38' 24''. o. The declination of Polaris being # = 88 46' n".g t the auxiliary V is found to be 88 49' 33". 5, and then from (45)' there results for the latitude the value = 40 $6' 17".$. The probable error of a single determination of latitude made in this manner is much less than that of one found from observations on the sun, say about i" or 2" . When a common watch is used its error and rate should be known so that the time correspond- ing to the mean altitude may be converted into local mean solar time and then, with the help of the nautical almanac, into sidereal time, from which the hour-angle t is found as before. It is preferable that the student should use a com- mon watch rather than a sidereal chronometer, since the former is more generally at hand in actual work. The best time for making this observation is when the star is near culmination, since then an error in h produces the smallest error in 0. In the above example the star was about i h io m from the lower culmination and hence in a favorable position. Prob. 46. Deduce sin h = sin sin d + cos cos 6 cos /, and show that an error dh gives the least error d(t> when / = o or when /= 1 80. 47- TIME. 129 * 47. TIME. A watch may he set to"focal apparent solar time by noting the instant when the sun attains its maximum altitude, and then, applying the equation of time, local mean solar time is approximately known. At any telegraph station in the United States a watch may be closely set to standard time, Eastern standard time being the mean solar time of the 75th meridian and hence five hours slower than Greenwich mean solar time, while Central, Mountain, and Pacific standard times are the mean solar times of the QOth, iO5th, and i2Oth meridians respectively. The mean solar time at any other meridian is found from standard time by applying to standard time the correction for difference of longitude, 15 degrees corresponding to one hour of time. When greater precision is required an altitude of the sun or of a star is to be taken and from this the error of the watch can be computed, if the latitude of the place is known. In the figure of Art. 42 let 5 be the place of the sun or star; the arc SZ is the co-altitude or zenith distance , the arc SP is the co-declination 90 - d, the arc ZP is the co-latitude 90 0, and the angle SPZ is the hour-angle of the star, which is designated by /. The solution of the spherical triangle gives i cos-sr = sin0 sintf -f- cos0 costf cos/, from which / can be computed; it is, however, customary to reduce this equation to the form tan*/ = A / sin * + - V sin *<* ~ *+~^ (47) V cosi(* + + d) cosi(* - 6) From this / is found in degrees, minutes, and seconds, and this value is then changed into time by dividing it by 15. When the sun is observed this result is apparent solar time; when a star is observed its sidereal time interval is to be reduced to mean solar time. 130 ASTRONOMICAL WORK. V. For example, take the data of Art. 42 where observations on the sun gave the mean corrected co-altitude z = 46 02' 32" at 8 h 43 ra oo s A.M. by the watch, the sun's declination being d 19 54' 05" and the latitude of the place = 40 36' 27". Inserting these values in the above formula there is found t = 48 32' 50", which is the hour-angle between the sun and the meridian; this reduced to time gives 3 h 14 n s .3 as the interval between the time of the mean altitude and that of apparent solar noon. Hence 8 h 45 m 58 8 .7 was the local apparent solar time, and subtracting the equation of time for the given day, there results 8 h 42 I5 8 .5 as the local mean solar time which corresponded to 8 h 43 oo 8 of the watch. Hence the deviation of the watch from local mean solar time was o m 44 s . 5 fast. Further as the place of observation was o 22' 35" west of the 75th meridian, and as this corresponds to o h oi m 3O 8 .3, it follows that the deviation of the watch from Eastern standard time was 45 s . 8 slow. The probable error of this determination may be several seconds. Far better work can be done by observing a star, and a good sextant is always to be preferred to an engineer's transit for taking the altitudes, the image being brought down to coincide with its reflection in a dish of mercury. The follow- ing is a record of an observation made at Lehigh University on May 9, 1899, by this method. The watch was supposed to carry Eastern standard time and it was required to deter- mine its error. The sextant was set successively at even 10 seconds of arc and the watch time of each noted; thus the observed mean double altitude 90 05' oo" occurred at 8 h 02 m 36 8 .64 by the watch. This is corrected for index error and eccentricity, and the apparent double altitude found, to which a refraction correction, computed from a formula that takes barometer and thermometer into account, is applied. Thus the true zenith distance z is found, and from this and the given latitude of the place and declination of the star the hour-angle t is computed from the above 47- TIME. TIME BY a GEMINORUM (CASTOR), MAY 9, 1899. Obs. No. Double Altitude. Watch Time P.M. **' Data and Remarks. I 90 50' 8 1 ' oo m 36*. 5 J. H. O., observer. C. L. T., recorder. 2 90 40 , oi 03 .5 Pistor and Martin's Prismatic Sextant 3 90 30 oi 30 .5 with mercury horizon and glass cover. 4 90 20 oi 56 .5 Watch carrying approximate Eastern standard time. 5 6* 90 10 90 oo 02 23 .5 O2 5O .2 = 4o' 3 6'2 3 ". 2 Index Error, from Arcturus. A = 75 22' 57".3 , 7 89 50 03 16 .6 10 45 a = 7 h 28 ii. 30 , 8 89 40 03 42 .5 d - 32 06' 36". 7 9 89 30 04 ii .4 I0 55 Barometer, 29' .550 10 89 20 04 35 -8 io 55 Aftir'Tirrl tTi C-T m f\f\ Q r\ P Detached therm., 64.7 F. Means 90 05' oo" 8 h 02 m 36 s . 64 * Horizon cover reversed. Observed zh = 90 05' oo" True sidereal time /-(- a= ii h i2 m io 8 .63 Index error = 16 51.25 Sidereal time mean noon = 3 09 09 .27 Eccentricity - - 48.85 Sidereal interva i after 2k = 89 47 25.90 mean n n = 8 03 oi .36 Apparent ^=44 53 42. 9 5 Correction to mean solar ^ _ ^ ^ ^ J^gfj""* **+ * '"** * f ** 33-t Local mean solar time = 8 oi 42 .07 True/&=44 52 47-49 Reduction to 75 th meridian = + oi 31.82 Hour-angle/- 55 59' 50" Eastern standard tims = 803 13.89 Hour/= 3 h 43 M 59'.33 Watch time = 802 36.64 Watch error (slow) = 37*-25 C. L. T., computer. formula, and converted into time measure. The true sidereal time then results by adding the right ascension of the star, and this is converted into local mean solar time and then into Eastern standard time, from which finally the watch error is found to have been 37.25 seconds slow. The probable error of this determination is less than one-quarter of a second. The most favorable position of a star for this work is when 132 ASTRONOMICAL WORK. V. it is on the prime vertical. For, if dz be an error in 2, the corresponding error dt in /, obtained by differentiating the first equation of this Article, is sin^-^/cos^ costf sin/, and since sin^/costf sin/ = sinzT, this gives dt = dz/cvsty s'mZ. Accordingly the azimuth Z should be 90 or 270 in order that dt may have its smallest value. In the same manner it is shown that errors in the assumed latitude produce the least effect when the star is on the prime vertical. Prob. 47. Using the data of the above observation on Castor, find the error in the computed Eastern standard time which would be caused by an error of 01" in the altitude h ; also the error which would be caused by an error of 01" in the latitude 0. 48. LONGITUDE. When accurate standard time is at hand the comparison of It with the local mean time gives the longitude. Thus, if the local mean solar time of a place has been found by a star observation to be I4 m o8 s .4 faster than Central standard time, the place is 3 22' 04" east of the 90 meridian and hence its longitude is 86 27' $6" west of Greenwich. This method is used at sea, where daily observations for local mean solar time are made on the sun or stars when the weather permits, this local time being compared with a chronometer which indicates either Greenwich mean solar time or that of a port whose longitude is known. As one second of time is equiva- lent to fifteen seconds of angle, it is seen that this method is not very precise, particularly when it is considered that the best watches are liable to vary one or more seconds per day. The method of lunar distances is extensively used at sea for finding the Greenwich time. In the nautical almanac will be found the true angular distances between the moon's center and several stars and planets for every day in the year and for three-hour intervals, these distances being stated as they would appear from the center of the earth. If one of these apparent distances be measured at any place, as also 48. LONGITUDE. 133 the apparent altitudes of the star and moon, the data are at hand for computing the true distance as seen from the center of the earth at the same instant, and thus from the almanac Greenwich mean time is known. Then, the difference between local and Greenwich time gives the longitude of the place. This method involves laborious computations unless special tables are at hand. Anothec method is that of lunar culminations which re- quires that azimuth and time should have been determined. The instant of the passage of the moon's bright limb across the meridian is observed, and a correction applied to find the local mean time of passage of the moon's center. This local mean time, converted into sidereal time, furnishes the right ascension of the moon, while the Greenwich mean time corresponding to the same right ascension can be found from the almanac. Lastly the difference between local and Green- wich mean time gives the longitude of the place. As an example of the method of lunar culminations the following rough observation with an engineer's transit, made at Lehigh University on May 23, 1899, may be of interest. On that day the moon crossed the meridian at about io h 55 P.M., and it was accordingly arranged to determine azimuth by pointing on Polaris a few minutes previous. By a simple computation it was determined that the azimuth of Polaris at io h 45 m local mean time was 180 36' 23". 4 and at that instant the cross-hair of the transit telescope was set on the star. Then the angle o 36' 20" was turned off toward the west and the telescope reversed, thus pointing southward in the plane of the meridian. When the moon's west limb touched the vertical cross-hair the time was noted as io h 53 m 34 s . i. Reducing this to sidereal time with the assumed longitude 5 h , and adding a correction for the time required for the semi-diameter of the moon to pass the meridian, the right ascension of the moon's center when crossing the meridian of the place is found to be I5 h oo m 53 8 .44, and the 134 ASTRONOMICAL WORK. V. corresponding mean local time i~o h 54' 55 8 .59. From the nautical almanac the Greenwich mean time at which the moon's center had this right ascension is found to be I5 h 54 27 s . 84, and consequently the longiti^de of the place of obser- vation is 4 h 59 m 42 8 .25 in time or 74 55' 34" in arc, a result which is in error by nearly half a degree, the true value being 75 22' 23". It thus appears that no close determination of the longitude of a place can be made by the method of moon culminations with an engineer's transit. Nevertheless in an unexplored region the method is of value in making an approximate determination to be used in time observations and in taking quantities from the nautical almanac. Prob. 48. At a certain place on December 5, 1900, the moon's right ascension was observed as 4 h 32 O5 s .3i at 8 P.M. local mean time. From the nautical almanac it is found that the right ascen- sions 4 h 31 03 9 .69 and 4 h 33 37 S .82 occurred at 3 A.M. and 4 A.M., Greenwich mean time, on December 6, 1900. Find the longitude of the place of observation. 49. PRECISE DETERMINATIONS. The methods set forth in the preceding pages give results whose precision is far lower than that needed for the astro- nomical work of a geodetic survey. When it is required to determine azimuth, latitude, and longitude at one of the stations of a geodetic triangulation such methods are generally used to furnish preliminary approximate values, for it has been seen that each of these elements depends upon the others, and hence rough methods must precede precise ones. These preliminary values may be supposed to give the lati- tude within one or two seconds, the longitude within ten or twenty seconds, and the azimuth within six or eight seconds. To make a precise determination of azimuth a direction theodolite, having a circle divided to 5 minutes and reading by microscopes to seconds or less, is used. The observations 49- PRECISE DETERMINATIONS. 135 are made on close circumpolar stars by the method of Art. 35, great pains being taken to eliminate the error of level in the horizontal axis of the telescope. By making a sufficient number of measurements the azimuth of a line running from the station to a signal may be found with a probable error of i" or less, and by measuring the angle between this line and one of the sides of the triangulation the azimuth of the latter is known with almost equal precision. To make a precise determination of latitude a zenith tele- scope is to be set up in the plane of the meridian and the difference of the meridian zenith distances of two stars that cross the meridian near the zenith, but on opposite sides of it, is observed. Let #, and multiplyed by 360 sithes; and than thei ben 31 500 Myles, every of 8 Furlonges, aftre Myles of oure Contree. So moche hathe the Earthe in roundnesse, and of heighte enviroun, aftre myn opynyoun and myn undir- stondynge." These views of Mandeville appear to have produced but little influence, for it was not until the fifteenth century, when the first gleams of light broke in upon the darkness of the middle ages, that men began to think again about the shape and size of the earth. Navigators began to doubt that its surface was a level plane, and here and there one, like Columbus, asserted it to be globular. In the sixteenth cen- tury, the doctrine of the spherical form of the earth was again generally accepted, and one of the ships of Magellan, after a three years' voyage, accomplished its circumnavigation. With the acceptance of this idea arose also the question as to the size of the globe, and Fernel, in 1525, made a meas- urement of an arc of a meridian by rolling a wheel from Paris to Amiens to find the distance and by observing the latitudes with large wooden triangles, from which he deduced about 57050 toises for the length of one degree. At this time methods of precision in surveying were entirely unknown. In 1617 Snellius conceived the idea of triangulating from a known base line, and thus, near Leyden, he measured a meridian arc which gives 55020 toises for the length of a degree. Norwood, in 1633, chained the distance from London to York, and deduced 57424 toises for a degree. Picard, who was the first to use spider lines in a telescope, remeasured, in 1669, the arc from Paris to Amiens, using a base line and triangulation, and found one degree to be 142 SPHERICAL GEODESY. VI. 57 060 toises. This was the result that Newton used when making his famous calculation which proved that the moon gravitated toward the earth. The toise, it should here be noted, was an old French measure, approximately equal to 6.3946 English feet or 1.949 meters. It is of classic interest on account of its use in all the early meridian arcs and in the surveys for deciding upon the length of the meter. From 1690 to 1718 Cassini carried on surveys in France, more precise probably than any preceding ones, and in 1720 'he published the following results regarding three meridian arcs: Arc. Mean Latitude. Toises in One Degree. 1 49 56' 56 970 2 49 22 57 060 3 47 55 57098 From these it appeared that the length of a degree of latitude increased toward the equator, or that the earth was flatter at the equator than at the poles. In other words he claimed that the earth was not spherical but spheroidal, and that the spheroid was a prolate one. From the time men had ceased to believe in the flatness of the earth, and had begun to regard it as a sphere, their investigations had been directed toward its size alone; now, however, the inquiry assumed a new phase, and its shape came up again for discussion. A prolate spheroid is generated by an ellipse revolving about its major axis, and an oblate spheroid by an ellipse revolving about its minor axis. The first diagram of the figure represents a meridian section of the earth regarded as a prolate spheroid, and the second shows the section of an oblate spheroid. In each diagram PP is the axis, Q.Q the equator, and A a place of observation whose horizon is AH> zenith Z, latitude ABQ, and radius of curvature AR. Now if the earth be regarded as a sphere and its radius be found from a meridian arc near A, the value AR will result. In HISTORY FROM 1300 TO 1751, H3 the prolate spheroid the radius of curvature is least at the poles and greatest at the equator, and the reverse in the oblate. Hence if the lengths of the degrees of latitude decrease from the equator to the poles, it shows that the earth is prolate; but if they increase from the equator toward the poles, it is a proof that it is oblate in shape. It is now necessary to go back to the year 1687, the date of the publication of the first edition of Newton's Principia. In Book III of that great work are discussed the observations of Richer, who, having been sent to Cayenne, in equatorial South America, on an astronomical expedition, noted that his clock, which kept accurate time in Paris, there continually lost two seconds daily, and could only be corrected by shortening the pendulum. Now, the time of oscillation o a pendulum of constant length depends upon the intensity of the force of gravity, and Newton showed, after making due allowance for the effect of centrifugal force, that the force of gravity at Cayenne, compared with that at Paris, was too small for the hypothesis of a spherical globe; in short, that Cayenne was further from the center of the globe than Paris, or that the earth was an oblate spheroid flattened at the poles. He computed, too, that the amount of this flattening at both poles was between T J T and -gfa of the whole diameter. Now it will be remembered that Newton's philosophy did 144 SPHERICAL GEODESY. VI. not gain ready acceptance in France; this investigation, in particular, called forth much argument, and when Cassini's surveys were completed, indicating a prolate spheroid, the discussion became a controversy. Then the French Academy resolved to send expeditions to measure two meridian arcs that would definitely settle the matter, one near the equator and another as far north as possible. Accordingly two parties set out in 1735, one for Lapland, the other for Peru. The Lapland expedition measured its base upon the frozen surface of a river, executed its triangu- lation and latitude observations, and returned in two years with the results / = 92 778 toises, B = I . 6221. The Peru- vian expedition measured two bases, executed its triangula- tion and latitude work, and returned in seven years with the results 7=176875 toises, 0=3. 1176. From these the values of the length of one degree were found, and then the following results could be written: Arc. Mean Latitude. Toises in One Degree. Lapland N. 66 20' 57438 France N. 49 22 57 060 Peru S. i 34 56 728 These figures decided the question. Since that time every one has granted that the earth is an oblate spheroid rather than a sphere or an prolate spheroid. Prob. 51. From the above data compute the radius of curvature for the Lapland arc and for the Peruvian arc. 52. MEASUREMENT OF MERIDIAN ARCS, The general principles regarding the measurement of a meridian arc have been given in Art. 50, but it is now to be noted that the successful execution of the work demands accurate instruments, good observers, and long-continued labor. The latitude observations are now made by the zenith telescope method of Art. 49, the bases, angles, and azimuths MEASUREMENT OF MERIDIAN ARCS. 145 are measured with corresponding precision, while the adjust- ment by the Method of JLeast Squares reduces the residual errors to a minimum. In. the last century these precise methods were unknown, yet the results deduced gave valu- able information and progress was constantly made in methods of observation and computation. It will be of historic interest, perhaps, to give a brief account of the firs^ meridian arc measured in the United States. In 1763 the proprietors of Pennsylvania and Maryland employed two astronomers named Mason and Dixon to locate the boundary lines between their respective possessions. This occupied several years, and while engaged upon it, Mason and Dixon noted that several of the lines, particularly the one between Maryland and Delaware, were well adapted to the determination of the length of a degree, being on low and level land, and deviating but little from the meridian. Representing this to the Royal Society of London, of which they were members, they received tools and money to carry on the work. The measured lines are shown in the annexed sketch. AB is the boundary between Delaware and Maryland, about 82 miles long and making an angle of about four degrees with the meridian; BD is a short line running nearly east and west; CD andVW are meridians about five and fifteen miles in length respec- tively ; CP is an arc of the parallel, the same in fact as that of the southern boundary of Pennsylvania. In 1766 Mason and Dixon set up a portable astronomical instrument at A, the southwest corner of the present State of Delaware, and by observing equal altitudes of certain stars, determined the local time and the meridian, after which the azimuth of the line AB was measured, and the latitude of A found by observing the zenith distances of several stars as they crossed 146 SPHERICAL GEODESY. VI. the meridian. At ^V, a point in the forks of the river Brandywine, the zenith distances of the same stars were also measured, from which it was easy to find the latitude of N t and the difference of latitude between A and N. In 1768 they made the linear measurements by means of wooden rectan- gular frames 20 feet in length. All the lines had in previous years been run in the operation for establishing the boun- daries, and along each of them " a vista" cut, which " was about eight or nine yards wide, and, in general, seen about two miles, beautifully terminating to the eye in a point." Toward this point they sighted the wood frames, made them truly level and noted the thermometer in order to correct for the effect of temperature. Through the swamps they waded with the wooden frames, but across the rivers they found the distance by a measured base and triangle. The results of this field work, as sent to England in 1768, were as follows: latitude of A = 38 27' 34", latitude of N= 39 56' 19", azimuth of AB at A = 176 16' 30", angle BDCgf 2f 30", ^ = 434011.6 feet, BD = 1489.9 feet, DC = 26608.0 feet, PN = 78 290.7 feet, DC and PN being true meridians while CP was an arc of the parallel. From these results the difference of latitude between A and N is = I.479I7. To find the linear distance /, an approximate value of the radius of the earth was assumed and each of the measured lines projected upon the meridian AN' by arcs of parallels NN', PP' , etc. Thus were found ^#=433078.8 feet, '>' = Sg.S feet, D' P' = 26 608.0 feet, and P'N' = 78 290.7 feet, whose sum is / = 538 067.3 feet. The length of one degree of the meridian now is 1/6 = 363 764 feet = 68.894 miles, from which the radius of curvature is R t = 57.2958/70 = 3947.4 miles. These are the final results of the measurement of the 53. THE EARTH AS A SPHERE, 147 meridian arc made by Mason and Dixon; they are now k.iown to be too, small, v the present accepted values for the mean latitude of the arc^being 68.984 miles and 3 952.4 miles, but in view of the primitive methods employed it is .surprising that the agreement is so close. During the fifty years following 1750 a number of meridian arcs were measured, one in South Africa, one in Italy, one in Hungary, one in Lapland, while in France and England geodetic surveys furnished the data for computing other arcs. Most important of all was the triangulation executed in France and Spain about 1800 for determining the length of the meter, which embraced an arc of ten degrees in length. All these arcs confirmed the conclusion that the earth is not a sphere, but an oblate spheroid flattened at the poles. Prob. 52. Compute the length of a quadrant of the meridian in meters, using the results of Mason and Dixon and supposing the earth to be a sphere. 53. THE EARTH AS A SPHERE. Although the earth is not a sphere it is sufficient in many investigations to regard it as such, since the amount of flattening at the poles is not large. In fact, if the earth is represented by a globe sixteen inches in equatorial diameter the polar diameter would be 15.945 inches, so that the differ- ence between the two diameters would not be perceptible to the eye. The question now arises as to what value shall be taken for the radius of the earth and what is the mean length of a degree of latitude on its surface. This question cannot be answered without anticipating to a certain extent some of the conclusions of the next chapter. The mean length of a degree of latitude is the mean of the lengths of all the degrees from the equator to the poles, or one-ninetieth of the elliptical quadrant. The value adopted for the quadrant in this book is q 10 ooi 997 meters = 32 814886 feet, 148 SPHERICAL GEODESY. VI. and from this is deduced the following useful table of mean lengths of arcs on the earth's meridian: One degree = in 133 meters = 364610 feet, One minute = I 852.2 meters = 6076.8 feet, One second = 30.87 meters = 101.28 feet. The mean length of one degree may also be stated in round numbers, easy to remember, as 69 statute miles or 60 nautical miles, one nautical mile thus being one minute of latitude. The mean radius of the earth, considered as a sphere, must be the arithmetical mean of all the radii of the spheroid. This is evidently the same as the radius of a sphere having a volume equal to the volume of the spheroid. Let a be the equatorial and b the polar radius of the oblate spheroid, whose accepted values are 6378278 and 6356654 meters respectively; its volume is ^ncfb. Let R be the radius of the sphere whose volume is ^nR*. Equating these values, there is found R = 6371 062 meters = 20 902 416 feet, or, in round numbers, R = 6371 kilometers 3 959 statute miles, for the mean radius of the earth. This mean value of the radius is, however, incongruous with the above mean length of a degree of latitude, for the quadrant of a circle corresponding to a radius of 6 371 kilo- meters is nearly six kilometers greater than the true elliptical quadrant. In certain cases it might be more logical to use the radius of a circle whose quadrant is equal to the true quadrant; this requires the equation \nR = 10001997 meters, from which R 6 369 kilometers = 3 957 statute miles, and this is less by two miles than the mean radius of* the- sphere. This discrepancy is unavoidable, since the proper- 54- LINES ON A SPHERE. 149 ties of a sphere and a spheroid are not the same. Thus it is impossible, when precisian is demanded, to regard the earth as a sphere. Prob. 53. Taking the area of the earth's spheroidal surface as 196 940 400 square miles, find the radius of a sphere having the same area. 54. LINES ON A SPHERE. The intersection of a plane and a sphere is always a circle. When the plane passes through the center of the sphere the circle is called a great circle, its radius P being R and its circumference 2nR. When the plane does not pass through the B > center the radius of the circle is less than R, say r, and its circumference is 2nr. All great circles cut out by planes passing through the axis of the earth are called meridians and these, of course, converge and meet at the poles. All small circles cut out by planes perpendicular to the axis are called parallels. Latitude is measured north and south on the meridians from the equator toward the poles, while longitude is measured east and west on the parallels from the meridian of Greenwich. Using the mean figures of the last Article, one minute of latitude corresponds to 1852 meters or 6077 feet. One minute of longitude on the equator has the same value, but one minute of longitude on any parallel circle is smaller the nearer the circle is to the pole. Thus if A be a point on a parallel whose radius AC is r, and whose latitude AOQ is 0, and if R be the radius of the sphere, then r = R cos0, and accordingly 2nr = 27r^-cos0, that is, the length of the parallel circle is equal to the length of a great circle multiplied 150 SPHERICAL GEODESY. VL by the cosine of the latitude. Hence the length of one degree or one minute of longitude at any latitude is found by multiplying the values of the last Article by the cosine of the latitude. Thus, using I 852 meters or 6 077 feet for the length of one minute at the equator, the length of one minute of longitude at latitude 40 is I 419 meters or 4655 feet, while at latitude 80 it is 322 meters or I 055 feet. The above figure shows two orthographic projections of the meridians and parallels of a sphere, the first being a projec- tion on a plane through the axis, and the other a projection on the plane of the equator. The parallels appear as straight lines in the first diagram and as circles in the second. The shortest distance on the surface of a sphere between any two points is along an arc of a great circle joining them. This can be rigidly demonstrated by establishing a general expression for the length of a line on the spherical surface and making it a minimum, but it will be just as well for the student to satisfy himself of the truth of the proposition by actually drawing and measuring lines on such a surface. As an illustration, the distance from A to B in the above figure may be computed by the route ACB along the parallel and by the route APB on the great circle. The length of the first is TtR cos0 and that of the second is nR 2cj)R. Thus, if be 45 degrees or \it radians, the first route has the length 2.22R while the second has the length 1-57^. In like manner the distance from A to C is i.nR along the parallel, but I.O$R along a great circle passing through A and C. The azimuth of a line on a sphere is estimated, as in a plane, from the south around through the west; thus the northward azimuth of all meridians is 180 degrees. As all meridians converge at the poles the back azimuth of an oblique line is not equal to its front azimuth plus 1 80. A great circle passing through C with an east and west direction at that point cuts the neighboring meridians at different angles and finally crosses the equator and attains the same 55. ANGLES, TRIANGLES, AND AREAS. 151 southern latitude as C on the opposite side of the sphere. All the meridians cut the equator at right angles, but they cut other parallels at smaller Angles. An oblique line crossing all meridians at the same angle is of a spiral nature and is called a loxidrome. Prob. 54. What part of the surface of a sphere is north of north latitude 60 degrees ? 55. ANGLES, TRIANGLES, AND AREAS. A spherical angle is the plane angle between the tangents to the arcs of the great circles at their point of intersection; thus the spherical angle BAG is the same as the plane angle bAc. When a horizontal angle is measured at a station A on the sur- face of the earth, the limb of the instrument is made level or tangent to the spherical surface, and hence when pointing is made upon B and C the plane angle bAc is the result of the work. If the triangle be of sufficient size it will be found that the sum of the three measured angles is greater than 180 degrees. A spherical triangle is one included by three arcs of great circles. It is a well-known geometrical theorem that the sum of the angles of a spherical triangle is greater than two right angles, and that the excess above two right angles bears the same ratio to a right angle as the area of the triangle bears to the area of the tri-rectangular triangle. The tri- rectangular triangle, shown by PQO in the figure of the last Article, is one-eighth of the surface of the sphere or %7rR*. Thus from the theorem the spherical excess is given by Excess in right angles = area of triangle/7r^ J , or, since there are 90 X 60 X 60 seconds in a right angle, Excess in seconds 648000 area/7rtf a . (55) I$2 SPHERICAL GEODESY. VL Taking for R the mean value of the radius of the earth con- sidered as a sphere (Art. 53), this becomes Excess in seconds = area in square kilometers/197 * ,, = area in square miles/76, which are convenient approximate rules for practical use. Thus a triangle has one second of spherical excess for each 197 square kilometers or 76 square miles of area. The same rule applies to quadrilaterals or polygons on the earth's surface bounded by great circles, the word excess meaning the excess of the sum of the interior spherical angles over the theoretic sum for a plane figure. Thus a polygon or triangle of the size of the State of Connecticut has a spherical excess of about 64 seconds; this amount is rarely exceeded in the triangles of geodetic triangulations and is usually much smaller. A geodetic triangle is necessarily small since the stations must be intervisible, and hence its curved surface does not sensibly differ in area from that of the plane triangle formed by lines equal in length to the spherical arcs. These are the distances computed from the triangulation work, and the corresponding plane angles are found by subtracting one-third of the spherical excess from each spherical angle. For instance, let two sides of a triangle be a = 36 440 meters, b =^ 23 700 meters, and their included angle (7 = 49 05'; then the area is ^ab s'mC = 326.3 square kilometers, and by (55)' the spherical excess is oi".66. It will be seen later that the above equation (55) is directly applicable to triangles on a spheroid by taking for R the radius of the sphere osculatory to the spheroid at the center of gravity of the triangle. In many common cases, however, the rough rules of (55)' will give the spherical excess correctly to hundredths of a second. The area of a zone of the sphere bounded by the parallel circles whose latitudes are Z, and Z a is easily derived. The 5 6. LATITUDES, LONGITUDES, AND AZIMUTHS. 153 differential expression is 2nrRdL, where r is the radius of the parallel and R that of thevsphere. But r = R cosZ, and hence A = 27tR*~l cosLcfL = 2 7i R* (sin Z, sinZj) is the area of the zone between the upper latitude Z,, and the lower latitude L^ i Thus to find the area between latitude 30 and the equator, Z a = 30 and L l = o, whence A = nR* or one-fourth of the surface of the sphere. The area of a trapezoidal degree, that is, of a surface bounded by two parallels one degree apart and by two meridians one degree apart, may be readily deduced from the last equation and will be found o. 000304167 ^ 2 cosZ, in which L is the middle latitude of the trapezoid. Thus, taking R = 3 959 miles and L = 45, the area of the trapezoidal degree is 3 376 square miles. Prob. 55. In a spherical triangle two angles are observed to be 79 03' 4i".93 and 59 35' 44".38, and the included side is 23 700 meters. Compute the spherical excess and find the other spherical angle. 56. LATITUDES, LONGITUDES, AND AZIMUTHS. Let A and B be two points on the surface of the sphere, L and M being the latitude and longitude of A, and L' and M' those of B. The latitudes are estimated northward from the equator and the longitudes westward from the meridian of Greenwich, both in degrees, min- utes, and seconds of arc. Let a great circle connect the points A and B, and let its angular length be 5. Let the meridians through A and B be produced to meet at the north pole P and to cross the equator at Q and Q.' 154 SPHERICAL GEODESY. VI. The azimuth of AB is the angle QAB, which is called Z, and the azimuth of BA is the obtuse angle Q ' BA, which is called Z' . Let the latitude and longitude of A be given, together with the length and azimuth of AB. It is re- quired to find the latitude and longitude of B and the azi- muth of BA. In the spherical triangle ABP the side PA is 90 Z, the side PB is 90 L ', and the side AB is S; the angle A is 180 Z, the angle B is Z' 180, and the angle P is M' M. Writing the formula of spherical trigonometry for the cosine of PB in the notation here used, it becomes cos (90-Z')=cos (9o-Z) cosS+sin (90-Z) sin S cos (i8o-Z), which reduces immediately to sinZ' = sinL cosS cosL s'mS cosZ, (56) and the latitude L' is hence expressed in terms of known quantities. In the same triangle, using the theorem that the sines of the sides PB and AB are proportional to the sines of the opposite angles, sin (90 - L')/s'mS = sin (i 80 - Z)/sin (M f - M), or sin (M f M) = sinS sinZ/cosZ', (56)' from which the longitude M f can be computed after the lati- tude L' has been found. To deduce Z' one of the formulas known as Napier's analogies will be most convenient in numerical work, namely, tan \(A + )/cotK = cos|(/>- PA)/cos%(PB + PA), and, reducing this to the notation in hand, it becomes cot (Z'- Z) = tan \(M' - M} sin \(L -f Z')/cos \(L' - L), (56)" from which the back azimuth Z f can be computed. These formulas apply to a spherical arc of any size on any sphere. For example, let L = 40 45^ and M = 73 58', these being values for New York City, and let it be required 56. LATITUDES, LONGITUDES, AND AZIMUTHS. 155 to find L', M f , and Z' , for a point whose angular distance is 5=35 and whose azimuth at New York is Z = 90. From formulas (56) and (6)' log s'mL' = 1.72812, L f = 32 19', log sin (M'-M)= 1.83172, M' = u6'43 / , which indicates that the point is located in the vicinity of San Diego, California. To find Z' the formula (56)" gives log coti(Z - Z') ^ 1.36852, Z = 243 42', which shows that if a great circle be drawn between the two places the direction of this is due west at New York, but at San Diego its direction is N. 64 E. As the earth is not a sphere, these results may be a degree or more in error. Then a line AB runs due north its azimuth Z is 180 ; th :i ( .-6) reduces to Z/ = Z, + S, and (56)' gives M f = M, vh. ( ')" shows thit Z' = Z + 180 = o. If it runs due L. irjj r :., c ; then (56) gives L' = L - S and (56)' gives M' - M, > !i "c (56)" shows that Z' = Z + 180 = 180. If the two ;, L.tJ arc on the same parallel of latitude, then L' = L and S h th.ir angular distance on a great circle. For most geodetic triangles the lengths of the sides are so small compared with the radius of curvature R that it is suffi- cient to take cosS = I and sinS = l/R, where / is the length of the arc or chord joining A and B. Then the above formulas may be directly applied to such triangles, and in Art. 64 it will be shown how they are further simplified. Prob. 56. Given L = 40 36' 22^.452, M = 75 22' 5i".i5o, Z= 193 56' 28". i, and I 1726.60 meters. Taking ^ = 6371 kilometers, compute latitude L ', longitude J/', and azimuth Z' '. 156 SPHEROIDAL GEODESY. VII, CHAPTER VII. SPHEROIDAL GEODESY. 57. PROPERTIES OF THE ELLIPSE. Since an oblate spheroid is generated by the revolution of an ellipse about its minor axis, the equator and all the sec- tions of the spheroid parallel to the equator are circles, and all sections made by planes passing through the axis of revolution are equal ellipses. Let a and b represent the lengths of the semi-major and semi-minor axes of this meridian ellipse, which are the same as the semi-equatorial and semi-polar diameters of the spheroid; when the values of a and b have been found all the other dimensions of the ellipse and the spheroid become known. It is necessary first to deduce several equations expressing the properties of the ellipse, and then by discussing them in connection with the results of measurements of meridian arcs the form and size of the spheroid is to be found. The eccentricity of an ellipse is the ratio of the distance between focus and center to the semi-major axis, and the ellipticity is the ratio of the flattening of one pole to the semi-major axis. Let e be the eccentricity and f be the ellipticity, then - b* a- b The relation between these two fractions is 57- PROPERTIES OF THE ELLIPSE. '57 and b may be expressed in terms of e and/ in two ways, b = e = *(!-/). Thus two quantities determine an ellipse; those generally used are a and ^, and When these have been found b and f are known. The equation of the ellipse referred to the rectangular axes QQ and PP is ay + V = # 2 2 , in which j/ and x are the ordinate and abscissa of any point A. Let L be the latitude of A, that is, the angle ABQ, and let it be required to find the relation between x and L. At A draw the tangent AT, and, since the angle A TB is 90 L, tan( 9 o whence tfx y = ^ tanZ. Inserting this in the rectangular equation of the ellipse, and replacing & by its value a\i e 9 ), there is found, after reduction, x a cosL/(i e* sin'Z)*, (57) which is an equation of the ellipse in terms of the variables x and L. If L = o then x = a, and if L 90 then x = o. To find the radius of curvature of the ellipse at A the rect- angular equation of the ellipse may be written in the form y = *-(*> - *)'. 158 SPHEROIDAL GEODESY. VII. and the first and second derivatives of y with respect to x are dy _ bx d*y ab d* ~~ ' a(a - *')*' ~dx* = " (a* - x*f and then, replacing tf by a\\ e*) and x by its value from (57), which is the required radius of curvature. If L = o then RI = &/a\ if L = 90 then ^ = c?/b. To find the length of an arc of the ellipse the differential element R^dL is to be integrated between the limits o and L. This furnishes an elliptic integral which cannot be evaluated except by a series; thus / _ a (i - /) C L (i + $S sin'Z + Y^ 4 sin4/ - + -X I/O the integration of which gives / = a(i - e*)l(i + t* a + ll^ 4 + ...)/(f' a + IS-'* + -) sin* + (^V 4 + ...) sin 4 /.-...] ( 57 )" for the length of a meridian arc from o to L. If L = 90 = %TT, then /, = i(i -i^-- T V'-rf^'-- ). which is the length of a quadrant of the ellipse. Prob. 57. Given e = 0.082271, and a = 6 378 206 meters, to com- pute the length of the quadrant .to the nearest meter. 58. DISCUSSION OF MERIDIAN ARCS. Since a spheroid is determined by the two elements a and e of the generating ellipse two equations are required to find their values. These may be established by the discussion of two meridian arcs in different latitudes. Let /, and / be their lengths, l and # a their amplitudes or the number of degrees of latitude between their northern and southern ends, Z,, and L t the latitudes of their middle points, an 1 R l and ^? t 5 8. DISCUSSION OF MERIDIAN ARCS. 159 the radii of curvature at these points. Regarding these arcs as arcs of circles, their radii of curvature are 1804 but considering the middle points as lying upon the circum- ference of an ellipse their radii, as given by (57)', are R - -fCiJiLfV R - a(l ~ ^ 1 ~ ( , J 2 T \' 2 / 9 1 T \$* Equating the values of R^ and also the values of R^ there are found two equations whose solution gives I u i8o/,(i - S sin'Z,)' i8o/,(i - / sin'Z,)* " (59) by which the eccentricity of the ellipse and its major axis may be computed from the data of two measured meridian arcs. It is plain that these elements wiii be most accurately determined when one arc is as near the pole as possible while the other is at the equator. These conditions exist in the Lapland and Peruvian arcs (Art. 51), the results of which became known about 1745. The data for these arcs are as follows: Peruvian Arc. /, = 176 875.5 toises Lapland Arc. A 9 2 777-9 8 toises 0, = i 37' V-57 0* = 3 7' 3"-46 Z, = + 66 20' io".o5, Z, = i 31' oo".34 Reducing the amplitudes to degrees, and substituting in (59), there results * = 0.00643506, whence 0.080219, CIYIL ENGINEERING U. ofC. ASSOCIATION LIBRARY l6o SPHEROIDAL GEODESY. VII. and then substituting the value of e* in either of the values of a there is found a 3 271 652 toises. These two values completely determine the ellipse and the oblate spheroid generated by it. Then, from the expressions of the last Article, /= 0.003223, and the length of the quadrant is 5 130817 toises, or 10000 150 meters. It is often customary to state the values of e and f as vulgar fractions, since thus a clearer idea of the oblateness of the spheroid is presented. For this case the rough values are 1 f l ~ 12.$' ~ 310.3' or the distance of the focus of the ellipse from the center is T ^.-g-th and the flattening at one of the poles is .^.^th of the equatorial radius. About the year 1745 the results of the surveys instituted by the French Academy became known ; these have been given in Art. 51 in toises. The length of one degree of lati- tude is //#, if be in degrees, and thus these data give every- thing necessary for computing u, e, and a from the above formulas. From these three arcs three computations were made by the above method, and these gave results about as folljws for the ellipticity of the spheroid: From Lapland and French arcs, f From Lapland and Peruvian arcs, f = From French and Peruvian arcs, f = ^ T . Now, if the earth be a spheroid of revolution, and if the measurements be precise, these values of the ellipticity should be the same. Since, however, they disagree the conclusion was easy to make either that the assumption of the spheroid was incorrect or that the surveys were lacking in precision. After the year 1750, when the results of the Lapland and Peruvian arcs had become known, great interest was mani- fested in securing additional data by the measurement of 5 8. DISCUSSION OF MERIDIAN ARCS. 161 other meridian arcs in order to determine whether or not the earth was a true ellipsoid of revolution. The following table MERIDIAN ARCS. No. Locality of Arc. Middle Latitude. Length of One Decree. I Lapland, + 66 20' Toises. 57405 2 Holland, + 5 2 4 57H5 3 France, + 49 23 57 O74i 4 Austria, + 48 43 57086 5 France, + 45 43 57034 6 Italy, + 43 OI 56979 7 Pennsylvania, + 39 12 56888 8 Peru, - i 34 56753 9 Cape of Good Hope, - 33 18 57037 gives the results of nine arcs which were measured during the eighteenth century and discussed by Laplace in 1799. For this purpose he took the expression for the radius of curva- ture given in (57)', developed it by the binomial formula, and divided it by iSo/TT, thus obtaining d = - I oO sin'Z as an expression for the length of one degree of latitude. It thus appeared that the length of a degree could be expressed by d = M + N sin'Z + P sin'Z + . . ., in which M = na(i - *')/i8o, N = %e'M, P = - 1 / e 4 M, and Laplace in discussing the above data concluded that it was unnecessary to retain the term containing P since its value is small. Accordingly he wrote d = M + N sin'Z, and then proceeded to find probable values of M and N from 162 SPHEROIDAL GEODESY. VII. the nine observations of the above table, and from these to deduce the values of a and e. At that time the Method of Least Squares was unknown, but Laplace wrote the nine observation equations, and then used the two conditions that the algebraic sum of the errors should be zero and that the sum of the same errors all taken positively should be a minimum. He thus obtained two resulting equations from which he found M = 56 753 toises, N = 613.1 toises, and accordingly d = 5 6 753 + 613.1 sin'Z is an empirical formula for the length of one degree. From these values of J/and -/V he found e* = 2N/$M = 0.007202, and then f = ^ g . The last step in Laplace's investigation is the comparison of the observed values of the lengths of the degrees with those computed from his empirical formula. For the Lapland arc, for instance, observation gives d = 57 405 toises, while the formula gives d = 57 267 toises, the difference, or residual error, being 138 toises, a distance equal to nearly 900 feet, or to nearly 9 seconds of latitude. These errors, says Laplace, are so great that they cannot result from the in- accuracies of surveys, and hence it must be concluded that the earth deviates materially from the elliptical figure. At the beginning of the nineteenth century it was the pre- vailing opinion among scientists, founded on investigations similar to that of Laplace, that the contradictions in the data derived from meridian arcs, when combined on the hypothesis of an oblate spheroidal surface, could not be attributed to the inaccuracies of surveys, but must be due in part, at last, to deviations of the earth's figure from the assumed form. This conclusion, although founded on data furnished by sur- veys that would nowadays be considered rude, has been confirmed by all later investigations, so that it can be laid down as a demonstrated fact that this earth is not an oblate 58. DISCUSSION OF MERIDIAN ARCS. 163 spheroid. Yet it must never be forgotten that the actual deviations from that form are very small when compared with the great size of the glob* itself. In some of the practical problems into which the shape of the earth enters it is suffi- cient to regard it as a sphere, in many others a spheroid must be used, while in only a few cases is it required to regard the deviation from the spheroidal form. Now it was agreed by all in the early part of the nineteenth century, that for the practical purposes of mathematical geography and geodesy it was highly desirable to determine the elements of an ellipse agreeing as closely as possible with the actual meridian section of the earth, or, in other words, that the most probable spheroid should be deduced from the data of observation. This search after the most probable ellipse resulted in the discovery by Legendre, in 1805, of the Method of Least Squares, and the first problem to which this method was applied was a discussion of the elements of the ellipse result- ing from five portions of the French meridian arc. Important geodetic work was carried on in France and Spain by Delambre and Mechain for determining the length of the meter, which, with the accompanying office work, lasted from 1792 to 1807. The meridian arc embraced an amplitude of nearly ten degrees, and the methods for the measurement of bases and angles were greatly improved, in fact approaching for the first time to modern precision. The results were combined with those of the Peruvian arc to find the eccentricity, and this gave for the ellipticity -g^ and for the quadrant 5 130740 toises. This was equivalent to looooooo meters, since by the French law the meter had been defined to be one ten-millionth part of the quadrant. It is now known that this length of the quadrant is too small by nearly 2 ooo meters (Art. 60). Prob. 58. Explain how the method of least squares is to be applied to the deduction of an empirical formula for the length of one degree from the data in the above table. 164 SPHEROIDAL GEODESY. VII. 59. PLUMB-LINE DEFLECTIONS. During the nineteenth century many investigations of the size and shape of the elliptical meridian of the oblate spheroid were made. The most important of these gave the results for the ellipticity and for the length of the quadrant which are stated in the following table: DIMENSIONS OF SPHEROIDS. Year. By whom. Ellipticity. Quadrant in Meters. 1810 Delambre 1/334 10 000 000 1819 Walbeck 1/302.8 10 000 268 1830 Schmidt 1/297.5 10 ooo 075 1830 Airy 1/299.3 10 ooo 976 1841 Bessel 1/299.2 10 ooo 856 1856 Clarke 1/298.1 10 ooi 515 I86 3 Pratt 1/295.3 10 ooi 924 1866 Clarke 1/295.0 10 ooi 887 1868 Fischer 1/288.5 10 ooi 714 1878 Jordan 1/286.5 10 ooo 681 1880 Clarke 1/293.5 10 ooi 869 Mcst of these results were determined by a discussion of the da* a of several meridian arcs by the Method of Least Squares, ar//l a brief explanation is now to be given as to how such ^inputations are made. The principle of the Method of Least Squares (Art. 3) equires that the sum of the squares of the errors of observa- tion shall be rendered a minimum in order to give the most probable values of the observed quantities. The first inquiry then is as to where the errors of observation in a meridian arc lie; are they in the linear distance / or in the angular amplitude 6 ? The error in a linear distance that is computed from a good triangulation is known to be very small, say PLUMB-LINE DEFLECTIONS. 165 less than -^^nrV^-g-th part of its length (Art. 26). The error in an observed latitude found by the zenith-telescope method cannot exceed half a second (Art. 49). Neither of these errors can account for even a small part of the discrepancy that is found between the observed and computed length of a degree of latitude (Art. 59). Early in the nineteenth century it was suspected that the cause of these discrepancies was due to deflections of the plumb lines from the normal to the spheroid. To illus- trate let the sketch represent a very small part of a meridian section of the earth. O is the ocean, M a mountain, and A a latitude station between them ; ece is a part of the meridian ellipse coinciding with the ocean surface; Ac repre- sents the normal to the ellipse, and Ah, perpendicular to Ac, the true level for the station A. Now owing to the attraction of the mountain M, the plumb line is drawn southward from the normal to the position AC, and the apparent level is depressed to AH. If AP be parallel to the earth's axis, and hence pointing toward the pole, the angle PAh is the latitude of A for the spheroid eee; but as the instrument at A can only be set for the level AH, the observed latitude is PAH, which is greater than the former by the angle hAH. These differences or errors are usually not large, rarely exceeding ten seconds, yet since a single second of latitude corresponds to about 31 meters or 101 feet, it is evident that the error due to these plumb-line deflections may be very great com- pared with any accidental error in the measured length of the meridian arc. It is not necessary, of course, that there should be any plumb bob on an instrument for determining latitude, but whatever affects a plumb line affects the level bubble by 166 SPHEROIDAL GEODESY. VI I. which the graduated limb is made horizontal. Even when a sextant is used the effect of gravity upon the mercury in the artificial horizon may make its surface deviate from parallelism to the tangent plane of the spheroid. Thus, the term plumb-line deflection means really the elevation or depres- sion of the astronomical horizon with reference to the plane tangent to the spheroid. Astronomical latitude is determined with reference to a vertical line at the place of observation (Art. 40), but geodetic latitude is with reference to a normal to a spheroid at that point, and the difference of these is called the plumb-line deflection. A plumb-line direction and an astronomical latitude are real things, but a normal to a spheroid and the corresponding geodetic latitude are artificial things, and hence a plumb-line deviation depends upon the particular spheroid to which it is referred. In deducing the elements of an ellipse from the data of meridian arcs, the lengths are hence to be taken as without error, and the sum of the squares of the errors in the latitudes is to be made a minimum. For this purpose let / be the length of an arc and 6 its amplitude in seconds; the radius of curvature, regarding it as an arc of a circle, is 206 26$l/8, and equating this to the expression for the radius of curvature given by ($?)' there is found = 2062657(1 e* sin'Z) f /rf(i - e*), in which L is the middle latitude. Now if L' and L" be the latitudes of the north and south ends of the arc, this expres- sion becomes, after developing the parenthesis and neglecting powers of e higher than the second, + 206 265/(l _ s sin . z This equation contains the two elements a and e 1 whose values are to be found, while the other quantities are the data of the meridian arc. Now let v^ and v t be corrections to be applied to the observed latitudes L 1 and L" ', these being the plumb- 59. PLUMB-LINE DEFLECTIONS. l6/ line deviations from the normals to the spheroid ; also let x be a correction to be appHed to an assumed value of a y and y a correction to be applied teo an assumed value of e 1 . Then this equation may be put into the form v l :>, = mx -\-ny-\-p, where m, n, and / are known functions of the observed quantities. Now if there be three meridian arcs, each having two latitude stations, there will be six plumb-line deviations; thus for the first arc the two corrections may be written v t = v z -f- mx -f- ny -\- p, v* = v t , and similarly for the other arcs. If the left members be made zero these are the six observation equations which contain the five unknown quantities z/ a , z/ 4 , v t , x, and y. The normal equations are now formed, and their solution gives the most probable values of x and j, from which those of a and e* are found, and also the most probable values of the plumb-line deviations. Such is a brief outline of the process of determining the size of the earth from several measured meridian arcs. In practice the numerical work is abbreviated by using the method of correlates (Art. 78), but even then is very lengthy, several weeks being required to form and solve the normal equations when many arcs are used. Each arc, moreover, generally has several latitude stations, so that the number of observation equations is more than twice as many as there are meridian arcs. The spheroid thu^ deduced is the most probable one that can be derived from the given data, for the sum of the squares of the errors in the latitudes has been made a minimum. Prob. 59. Consult Clarke's investigation of 1866 in Comparison of Standards, published by the British Ordnance Survey, and ascertain the number of meridian arcs used, the number of normal equations, and the greatest values of the plumb-line deflections. 168 SPHEROIDAL GEODESY. VIL 60. DIMENSIONS OF THE SPHEROID. The most important investigations for the determination of the size of the spheroid are those made by Bessel in 1841 and by Clarke in 1866. The data employed by Bessel included ten meridian arcs, namely, one in each of the countries Lapland, Denmark, England, France, and Peru, two in Germany, and two in India. The sum of the ampli- tudes of these arcs is about 50.5 degrees, and they include 38 latitude stations. In the manner briefly described above, there were written 38 observation equations, from which 12 normal equations containing 12 unknown quantities were deduced. The solution of these gave the elements of the meridian ellipse, and also the residual errors in the latitudes due to the deflections of the plumb lines. The greatest of these errors was 6". 45, and the mean value 2 ".64. The spheroid resulting from this investigation is often called the Bessel spheroid, and the elements of the generating ellipse, Bessel's elements; the values of these will be given below. In 1866 Clarke, of the British Ordnance Survey, published a valuable discussion, which included a minute comparison of all the standards of measure that had been used in the various countries. The data were derived from six arcs, situated in Russia, Great Britain, France, India, Peru, and South Africa, including 40 latitude stations, and in total embracing an amplitude of over 76. The mean value of the plumb-line deflections, or latitude errors, was found to be I ".42. This investigation is generally regarded as the most important one of the last quarter of a century, and the values derived by it as more precise than those of Bessel. The Clarke spheroid, as it is generally called, has been used in most of the geodetic work done in America since 1880, as it is found to represent the earth's true figure in this continent somewhat better than the spheroid of Bessel. All the results and computations in the following pages of o. DIMENSIONS OF THE SPHEROID. 169 this volume will be based on the Clarke spheroid of 1866, but it is well for the studer\t "to be acquainted with the Bessel spheroid also, since it is gxtensively used in Europe. The following table gives the complete elements of the two spheroids, and it will be noted that the spheroid of Bessel is smaller than that of Clarke and also less elliptical or oblate. In order to form an idea of the precision of these results it may be noted that the probable error of Bessel's quadrant is 498 meters and that of Clarke's quadrant slightly less. ELEMENTS OF THE SPHEROID. Bessel, 1841. Clarke, 1866. Semi-major axis a j meters \ feet 6 377 397 20923597 6 378 278 20926062 Semi-minor axis b j meters 1 feet 6356079 20853654 6356654 20855 121 Meridian quadrant in meters 10000856 10001 997 Eccentricity D ' which, for numerical work, is better written cos'Z 174 SPHEROIDAL GEODESY. VIII For example, let it be required to find the radius of curvatuie of a normal section at a point A in latitude 39, its azimuth being 45. Taking the logarithms of R, and R^ from Table IV and performing the operations there is found logTc = 6.8051230, whence ^ = 6384443 meters. This formula is useful in reducing base measurements to ocean level (Art. 31). When an instrument is leveled at a station A and pointed to a second station B y points set out in the line of sight fall in the curve AaB which is cut from the spheroid by the normal section at A. When the instrument is leveled at B and pointed at A, the curve BbA will result, this being cut from the spheroid by the normal section at B. These two curves differ very slightly in azimuth; for a line 100 miles long the difference cannot exceed o". I, so that it is of slight importance in common geodetic triangulation. These normal sections are plane curves. The alignment curve between two stations A and B is a curve traced by starting at A, setting out a point in the direction of B, then moving the instrument to that point, backsighting on A, setting a second point in the direction of B, and so on. The broken line AcB in the figure represents this curve, which is one of double curvature; at any point c the vertical tangent plane to the curve passes through both A and B. The alignment curve is, of course, a shorter path between A and B than that on either of the normal sections. The shortest line between two stations on the spheroid is called a geodesic line, or simply a geodesic. It is not shown in the above figure, but may be closely represented by a curve of less curvature than AcB and crossing it near c\ it is, like the alignment curve, a line of double curvature. The geodesic has the property that the plane containing any element of the curve is normal to the spheroid at that 63. TRIANGLES AND AREAS. 175 element. The differential equation of the geodesic can be deduced and its properties be studied, but this is not expedient or necessary in^n elementary book of this kind, particularly as the" line is of no importance in the practical operations of geodesy. On a sphere the two normal sections and the alignment and geodesic curves between A and B coincide in an arc of a great circle. On a spheroid they also coincide when the two stations are on the same meridian; but in other cases they are separate and distinct. For any two intervisible points on the earth's surface, however, they do not appreciably differ in length, and it is only in the case of the longest lines that a difference in their azimuths can be detected. Prob. 62. A base line 8046.74 meters long has an azimuth of about 60 and its elevation above ocean level is 1609.35 meters. What is the length of the base reduced to ocean level ? 63. TRIANGLES AND AREAS. A triangle on the surface of a spheroid has the sum of its three angles greater than two right angles. An exact expres- sion for this spheroidal excess might be es f ablished, but, since only triangle sides between intervisible points can be used in geodesy, it is always sufficiently accurate to regard these points as lying on the surface of a sphere osculatory to the spheroid at the middle point of the triangle. The radius of this osculatory sphere is i/T^A:, , where R^ and R^ are the radii of curvature of the two principal normal sections through the point. Accordingly the formula (55) becomes Excess in seconds = 206 265 area/^,^ 2 , (63) in which logT?, and log^ 3 may be computed from (62) and (62)' or be taken directly from Table IV, while the logarithm of 206 265 is found in Table VI. As this computation is one that is frequently required the CIVIL ENGINEERING U.oiC ttSMUTMN UPiSI 176 SPHEROIDAL GEODESY. VII. quantity 206 26$/R l R. t may be called 2m and the logarithms of values of m for different latitudes be tabulated as is done in the last column of Table IV. Thus the practical formula for the computation of the excess in seconds is Excess = m . 2 Area = m . ab sin 7, (63)' in which ab sin7 is double the area of the triangle, a and b being two sides and C their included angle; here the area must be in square meters, or a and b must be in meters. For instance, let it be required to compute the spherical excess for a triangle whose area is 197 square kilometers, the latitude of its middle point being 37-J- . From Table IV the logarithm of m for the given latitude is taken and this added to the logarithm of double the area in square meters gives 0.00026 as the logarithm of the excess in seconds, whence excess = i".ooi, which differs by only o".ooi from the value given by the rough rule of Art. 55. Among the many interesting questions relating to the spheroid is that of the areas of zones and the areas of trape- zoids bounded by meridians and parallels. The differential expression for the area of a zone is 2nrR l dL, where r is the radius of the parallel and R l the radius of curvature of the meridian at the latitude L. The values of r and R l are given by (57) and (57)', and thus is found A = 27ta*(i V) / (i / sin 2 Z)~ 2 cosZ dL t which, when integrated between the limits L l and Z a , gives the area of the zone between those latitudes. The integral contains a hyperbolic or logarithmic function and hence is rather tedious in computation, but tables have been made giving its values. Among the best of these is Woodward's Geographical Tables, published by the Smithsonian Institu- tion, where the areas of trapezoids bounded by meridians and parallels are given in square miles. Prob. 63. Prove that the entire surface of the spheroid is expressed 64., LATITUDES, LONGITUDES, AND AZIMUTHS. 177 by 27ta i ( i H '- log e j, and show that this reduces to 471-0* for the sphere whose radius is*tf. 64. LATITUDES, LONGITUDES, AND AZIMUTHS. The formulas established in Art. 56 for the spherical triangle may be adapted to any practical case arising on the spheroid with any required degree of precision. The problem to be solved is as follows: given the lati- tude and longitude of a point A, the azimuth of AB and its length, to find the latitude and longitude of B and the azimuth of BA. The notation will be the same as that in Art. 56, the given latitude and longitude being designated by L and M, the given distance and azimuth by / and Z, while the required quantities are L', M', and Z '. Let 6L be the difference in latitude L' Z, and dM be the difference in longitude M' - M\ also let dZ be the angle by which the meridian at B deviates from parallelism to that at A, so that dZ =. Z' - Z 180. Then when dL, dM, and 6Z are known, the required quantities will be given by }-dZ. (64) The problem now is to find formulas for computing dL, dM, and dZ. The solution here given will be sufficient to furnish the results correctly to thousandths of seconds for all cases when the length / does not exceed about 20 kilometers or 12 miles. Resuming formula (56), and writing L -f- dL in place of L, it becomes, after developing the first member and dividing by cosZ,, i;8 SPHEROIDAL GEODESY. VII. sm6L = sinS cosZ-f- (cosdL cosS) tanL. Now as both 6L and 5 are small arcs their sines may be taken as equal to the arcs themselves; and also cosS I S 2 and similarly for cosSL. Accordingly the equation reduces to - dL = ScosZ-f S a tanZ, ()' tan. Here the first term of the second member is an approximate value of 6L, the second term being small since it contains S\ Accordingly, dL in the third term may be replaced by S cosZ. Further, since 6L and 5 are in radians the value of 5 is l/R, if R be the radius of curvature at the locality. Accordingly the equation becomes / cosZ r sin'Z tanZ - 6L = - ^ . The next question to be considered is regarding the value of R. With regard to the first term, which is the controlling one, it plainly should be the radius of curvature of the meridian passing through the middle of the arc, but as the latitude of that point is not known it is to be taken as that of the meridian at A. With regard to the second term it is not important, since its value is small, what radius should be taken, and it is customary to take that of the osculatory sphere at A. Now let 7^ and R^ be the radii of curvature of the principal normal sections at A y the values of these being as given in Art. 62 ; then the equation becomes / cosZ r sin'Z tanZ RT ~^R, ' This value is a close approximation, but it can be rendered clpser by adding a term to reduce it to the radius of curva- ture of the meridian at the middle point of the line /. This r> r> term will be 6L ! p -, where R m denotes that radius. MB Introducing the general values of the radius from (57)' for 64. LATITUDES, LONGITUDES, AND AZIMUTHS. 179 the latitudes L and L m , replacing L m by L %<5L, developing, and neglecting terms containing powers of e higher than the square, the additional term is found to be V i - e * sin 1 /:) 1 and accordingly the final formula for the difference of latitude is - 6L = / cosZ. B + /' sin'Z . C + tf . D, (64)' in which h denotes the value of SL as found from the first and second terms, and in which the letters B, C, and D are factors depending only on the dimensions of the spheroid' and on the given latitude L. In order that dL may be found in seconds the above expressions for the constants are to be multiplied by the number of seconds in a radian, and thus 206 265 206265 tan }^ a sinZ, rosZ, B = TS > C = _ , . . - , D = 206265(1 -e's'm'L)*' are the final factors which can be computed and tabulated for different values of the latitude L. In order to find the difference of longitude between A and /?, formula (56)' may be used, M f M being replaced by <$M. Since this is small the sine may be replaced by the arc, giving 6M = Ss Here, as before, 5 may be replaced by l/R and the value of the radius should be that of the prime-vertical normal section through B. Introducing this, and reducing from radians to seconds, it becomes /sinZ. A' 206 A = -- in which A' is to be used for the latitude L '. To find the difference in azimuth dZ, formula (56)" is 180 SPHEROIDAL GEODESY. VII. resumed, and replacing Z' Z by 180 -j- $2, and M' M by <5M, it becomes tan^tfZ = ta.n%6M sin(Z, -\- L')/c.os%(L f L). Also, since the differences of azimuth and longitude are very small, their tangents are proportional to the number of seconds in their arcs, and is final formula for computing the difference in azimuth. The above formulas were derived by Hilgard in 1846, and together with values of the logarithms of the factors A f , B, C y and D, will be found in Appendix No. 7 of the Coast and Geodetic Survey Report for 1884; an abridgment of those tablfes is given in Table V at the end of this book, the proper change being made for the fact that the ratio of meter of that Appendix to the meter of this book is i.ooooii. By the help of these the computations may be expeditiously made, as will be illustrated in Art. 66. When using the formulas in connection with these tables it should be remembered that the distance / must be taken in meters. As a simple example of one application of the formula for dL let it be required to find the number of seconds in a meridian arc whose length is 1 1 076.4 meters and whose southern end has the latitude 26. Here Z = 180, cosZ = I, whence dL l-B + (IB)*D, and by the use of Table V there is found + dL = 360". 953 o".OQ2, so that the latitude of the north end of the arc is 26 10' oo".95 I. Prob. 64. Given L = 42, M = 80, / = 1000 meters, and Z = 90 for the point^. Compute L', M ', and Z' for the point B. 65. THE COORDINATE SYSTEM. l8l I 4 CHAPTER VIII. v GEODETIC COORDINATES AND PROJECTIONS. 65. THE COORDINATE SYSTEM. The system of coordinates used in geodesy is generally the angular one employed in geography, latitudes being estimated north and south from the equator and longitudes east and west from the meridian of Greenwich. In North America both latitudes and longitudes are taken as positive and the signs of the coordinates of a point are hence the same as in the linear system of Art. I. Thus, if a point is determined to have the latitude 40 19' 04" '.237 and the longitude 85 07' 3 5 ".026, it can be at once located roughly on a small- scale map or be precisely located on a large-scale map upon which the meridians and parallels are accurately drawn in a certain system of map projection. It is well to keep in mind the approximate rules of Art. 53 regarding the lengths of one degree, one minute, and one second of latitude. One second of latitude being about 31 meters or 101 feet, one-tenth of a second is about 3 meters or 10 feet, one-hundredth of a second is 0,3 meters or I foot, and one-thousandth of a second is 0.03 meters or o. I feet. Precise geodetic work should hence carry the latitudes to thousandths of a second of angle in order to secure a precision comparable with that of precise plane triangulation. A second of longitude is nearly the same as that of latitude on the equator, but at any other place it is smaller, a rough rule being that it is equal to a second of latitude multiplied 182 GEODETIC COORDINATES AND PROJECTIONS. VIII. by the cosine of the latitude. Thus at latitude 40^, since cos4c is 0.774, the length of one second on the parallel is about 78 feet. The formulas of Art. 61 furnish expressions by which the lengths of one degree, one minute, and one second of both latitude and longitude can be computed for any given lati- tude L, and values of 'some of these will be found in Tables II and III, These are sometimes of service in changing angular differences of latitude and longitude between the stations of a secondary triangulation into linear differences, but a more extended table is necessary in order to make such computations with rapidity. The sketch below gives a representation of the coordinate system of geodesy, the meridians and parallels being roughly drawn on the polyconic projec- -4050 tion method which is explained in Art. 69. Station Pis located at latitude 40 45' and at longi- tude 86 43', while station P' is located at latitude 40 36' and longitude 86 58'. The straight 87lo 87oo' 86so' 86V 86 b 3o' line, or geodesic, joining the points is very slightly curved in the projection. The azimuth of PP' is about 53 43', this being measured from the south around toward the west. The azimuth of P'Pis about 233 33', this being also measured from the south around through the west and north. Owing to the convergence of the meridians that pass through P and ' P', the back azimuth of P'P differs by 10' from the azimuth of PP' plus 180 degrees. This coordinate system is not a convenient one for the use of local surveyors, but in an area of considerable extent it is a necessary one for the location of points in their relative positions on the spheroid. For an area of moderate size it may be modified in many ways, one of these being the well- known system of the public land surveys of the United 66. LMZ COMPUTATIONS. 183 States, while another andjnore satisfactory system is that of linear rectangular spherical coordinates which is described in Art. 70. It is well to note again that the latitudes and longitudes used in geodetic work do not generally agree with the lati- tudes and longitudes obtained by astronomical observations. Thus, if these coordinates be found astronomically for an initial station P together with the azimuth of PP', and if the distance PP' be directly measured or be found by computa- tion from a measured base, then the latitude and longitude of P' and the azimuth of P'P can be computed by the methods of the last Article, and these computed values are called geodetic ones. If further astronomical observations be made at P' , the results will generally differ from the geodetic ones owing to the plumb-line deflection at P' '. That is to say, the Clarke spheroid passed through P and oriented by the astronomical work done there, has a tangent plane at P' which is not parallel to the astronomical horizon at that point (Art. 59). The differences of latitude and longitude as found by geodetic triangulation are, in fact, always far more precise than those derived from astronomical observations, and it is only by the field operations of geodesy that coordinates of stations can be found so as to form a reliable basis for accu- rate surveys. Prob. 65. From the above data for stations P and P' determine roughly, with the help of Tables II and III, the length of the line PP' in miles. 66. LMZ COMPUTATIONS. When the latitude L and longitude M of a station P are given, together with the distance / and azimuth Z from it to a second station P', the latitude L' and longitude M' 9 together with the back azimuth Z' , can be computed. The formulas of Art. 64 will determine the latitudes and longi- tudes correctly to thousandths of a second when the length 184 GEODETIC COORDINATES AND PROJECTIONS. VIII. of the line does not exceed about 20 kilometers or 12 miles, and correctly to hundredths of a second for much longer dis- tances. These formulas will now be exemplified. Let the given station be one called Bake Oven, whose known latitude is 40 44' 54". 109 and whose known longitude is 75 44' 02". 222. Let the distance and azimuth to a second station called Packer Spire be 33 932.55 meters and 297 36' 49". 42. In the form below these data are seen in italic type, together with the logarithm of / in two places; then log s'mZ and log cosZ are found and put in their places, while the logarithms of / a and s'm'Z are found by doubling those of / and sinZ, and the logarithms of the factors B, C, and D are' taken from Table V. By addition the logarithm of IcosZ-B, or//, is found, and its double is the logarithm of tf '. Then the logarithms of the second and third terms are found, and thus the final value of dL is 5 1 i".8, from which the latitude of Packer Spire at once results, as also the mean latitude %(L -f- L'). The longitude computation is now made as indicated by the formula, the factor^' being taken for the latitude L '. Lastly, 6Z is computed, and the back azimuth from Packer Spire to Bake Oven is determined by the rule Z' = Z + 180 + dZ. A second or check computation should always be made whenever there is another station that furnishes sufficient data. For this case the latitude of a station called Smith's Gap is 40 49' 2 1 ".787 and its longitude 75 25' 21 ".906, while the distance and azimuth from it to Packer Spire are 24 332.28 meters and 351 il' 08". 84. Inserting these data in another form, and carrying out the computations in the same manner, the value of L' will be found to agree within o".oo6, or 0.6 feet, with that of the first computation, while the value of M f will be found to agree within o".oo2. These discrepancies are due to the fact that both lines exceed 20 kilometers in length. The back azimuth from Packer Spire to Smith's Gap is found to be 171 12' 52". 29, and the 66. LMZ COMPUTATIONS. FORM FOR LMZ COMPUTATION. i8 S PACKER SPIRE computed from BAKE OVEN. ** ABAKE OVEN - 8L - tcosZ. B'+l 1 sin'Z. C+ tf.D ^""^X^ -j- SAf = tsinZ.A'/cosL' ^^^^ dZ = d M sm\(L -$- L') PACKER SPIRED Z Bake Oven SZ 180 to Packer Spire 2 97 36' 49" .42 + 13 53 -59 180 Z' Packer Spire to Bake Oven 117 50' 43". 01 L 40* 44' 34". 1 09 Bake Oven M 7 5 44' 02" .222 SL 08 31 .859 / = 3} 932.55 meters dM 21 18 .904 L' 40 36' 22". 250 Packer Spire M' 75 22' 43".3i8 / 4,3306166 cosZ 1.6660576 B 2.5107900 /* 9.06123 sin'Z 1.89496 C 9.33974 h* 5.4149 D 8.3882 h 2.7074642 ist term + 509". 8755 2d and 3d terms i .9831 0.29593 2d term i".9767 8.8031 3d term o".oo64 / 4.3306166 sinZ 1.9474790 A' 2.5090982 cosL'(a.f) 0.1196432 dM 3.1068370 s\n\(L+L') 1.8141128 - SL + 5ii .859 \$L _ 04 ' I5 ". 93 l(L + L f ) 40 40' 38". 18 3.1068370 + dM - I278".904 2.9209498 -dZ - 8 33 ".59 difference between this and the back azimuth to Bake Oven is 53 22 7 09". 28, which furnishes a final check on the work, as this is the value of the spherical angle at Packer Spire. The best way to carry on these two computations is to enter the data in both, find log sinZ and log cosZ for both tt the same time, take out the factors B, C, and D for both, 186 GEODETIC COORDINATES AND PROJECTIONS. VIII. and thus finish the computation of 6L in both at the same time. If these values agree, as they should unless the lines are too long, the two computations for SM may be made, and lastly the two for dZ. The signs of 6L and 6M can in all cases be found from the signs of cos^ and sinZ, but it will be just as well for the student to determine them from the figure that should always be drawn at the top of each com- putation sheet. The above formulas may be used in finding coordinates to tenth of seconds or to single seconds for primary lines of almost any length, but when these are required to thousandths of seconds additional terms are needed. These terms and the form for computation will not be presented in this ele-x mentary book, but they may be found in the paper of the Coast and Geodetic Survey cited in Art. 64. Prob. 66. Using the above latitudes and longitudes of the stations Bake Oven and Smith Gap, and also the data that the distance and azimuth from Bake Oven to Topton are 30433.63 meters and 351 48' 49". n, and from Smith Gap to Topton are 44 239.59 meters and 29 54' i7".84, make the two LMZ computations for Topton, and check the back azimuths by comparison with the spherical angle at Topton, whose value is 37 53' i9".oi. 67. THE INVERSE LMZ PROBLEM. When the latitudes and longitudes of two stations are given, it is possible, if they are not too far apart, to compute the length of line joining them and the front and back azimuths of that line. This is readily done in a plane system of coordinates, as illustrated in Art. 24, but in a geodetic system it is more difficult. This is called the inverse LMZ problem, and it will now be shown how the formulas of Art, 64 are applied to its solution. Since the latitudes L and L' are given, as also the longi- tudes M and M', the values of 6L and dM are known. Then formulas (64)' and (64)" may be written in the form 7^ ^ A *-/' 67. THE INVERSE ZJ/Z PROBLEM. 187 - tfZ = (/ cosZ)^ + (I sinZyC + (6Df . D, (67) + dM = (I s&Z)A y /cosL', (67)' in which /and Z are two Unknown quantities to be found. From (67)' the value of (/sinZ) at once results and this inserted in (67) gives the value of (/cosZ); then, by dividing the former by the 'latter tanZ is found and hence Z. Also dividing (/sinZ) by sinZ the value of / results. Lastly 6Z is computed by (64)'" and Z f by (64). The form used in Art. 66 may be advantageously employed in making the computations, as will now be exemplified. Let the latitudes and longitudes of the stations Smith Gap and Bake Oven be given as stated in the last Article, and let it be required to compute /, Z, and Z'. These, with the resulting values of SL and SM, are first inserted in the form as seen in italic type. From Table V the factors A', B, C, and D are taken. The logarithm of dM is found, then those of cosZ' and A', and accordingly the logarithm of (/ sinZ) results. From dL the logarithm of (dL)* is found, and that of (/sinZ) 2 being also known, the second and third terms of the value of dL are determined, and finally the first term whose logarithm is then known and from which the logarithm of (/ cosZ) results. Then tanZ is obtained as expjained above whence Z is found; then the logarithm of/ and its value in meters is determined. Lastly, the computation of dZ is made and the back azimuth Z' is obtained. A check computation for this case should also be made by changing the order of the stations; thus the values of L and M may be taken for the station Bake Oven and L' and M' for the station Smith's Gap. If the lengths of the lines do not exceed 20 kilometers the values of the lengths and azimuths should exactly agree with those of the first com- putation. This inverse solution is often advantageous in field work in determining the directions between stations which are not-connected by a triangle side. 188 GEODETIC COORDINATES AND PROJECTIONS. VIII. INVERSE LMZ COMPUTATION. A A SMITHS GAP/A - dL = (/cosZ)^ + (/sinZ)C + (<5Z) 2 Z> .^^\ + 3M=(isinZ)A'/cosL' ''^^^ Z \ CTBAKE OVEN - 3Z = dM sin(Z + Z') Z Smith's Gap to Bake Oven SZ 180 Z' Bake Oven to Smith's Gap 72 39' 07".24 12 II .82 1 80 252 26' 55".42 L $L L' 40 49' 2i".787 Smith's Gap M 75 23' 21". god 04 27 .678 /= 27535.63 meters dM 18 40 .j/6 40 44' 54 '.109 Bake Oven M' 75 44' 02". 222 2d (cosz) 3.9I436I9 B 2.5107842 (cok) 8 ' 8 3935 C .34087 (Z) 48552 D 8.3884 h 24251461 ist term 266". 162 and 3d terms + I .516 - SL 267". 678 logdL 2.42762 $(+') 40 47' 07". 9 0.18022 2d term -f- 1.5143 3 2436 3d term -f o".ooi8 (sinz) 4.4196765 A' .5090946 cosZ'(a . f) 0.1205694 dM 3.049340 s\nl(L+L') 1.815066 3.0493405 + dM 1 120". 316 2.864406 -SZ - 73i.82 (/sinZ) 4.4196765 (/sinZ) 4.4196765 (/cosZ) 3.9143619 sinZ 1.9797815 tanZ 0.5053146 / 4.4398950 Prob. 67. Make the inverse LMZ computation for the above data, taking L and M for the station Bake Oven and L' and M' for the station Smith's Gap. 68. MAP PROJECTIONS. I8 9 68. MAp PROJECTIONS. p As a surface of double curvature cannot be developed on a plane it is impossible to devise any method of representing a large area on a map without some distor- tion. The method of orthographic pro- jection is perfectly satisfactory for a small area, but when applied to the whole earth, or even to a large county, the features near the edges of the map are crowded together so as to appear un- natural. For instance, in the lower dia- gram of the figure, which shows an ortho- graphic projection of the northern hemisphere on the plane of the equator, it is seen that the distance between parallels of latitude near the outer parts of the map is much less than near the central part; in the upper diagram, which is an orthographic projection on the plane of one of the meridians, a similar distortion is also observed. A projection devised by Flamsted to avoid this distortion consists in dividing the central meridian NS into parts pro- portional to the distance be- tween the parallels, and through these points drawing straight lines to represent those paral- lels. Each parallel is then divided into the same number of equal parts, and the merid- ians are drawn through these rm \\ \ \ \ \ \ \ Q points of division. All the trapezoids between two given parallels thus have the same area, and the distortion is less than in the orthographic projection. GEODETIC COORDINATES AND PROJECTIONS. VIII. A projection devised by Bonne is constructed in a similar manner to that of Flamsted, except that the parallels are concentric circles. The center of these circles is in the middle meridian and at a distance of a co(L from the middle parallel whose latitude is L\ thus in representing half of the hemisphere the radius of the middle parallel is equal to the y equatorial radius a, and the radius of any other parallel is a d, where d is its distance from the middle parallel. This method gives trapezoids of equal area and an orthographic projection along the middle meridian and parallel, but the shape becomes rather awkward where a large area is repre- sented. The cylindrical map projection or development is made by projecting the parallels upon a circumscribing cylinder by lines drawn from the center of the sphere, and then develop- ing the cylindrical surface. If QPQ be a meridian section of the spheroid and d any point upon it, then d is projected at D on the cylinder, and thus the parallel through d is projected upon the cylinder in a circle whose diameter is D' D. The cylinder being developed on a plane tangent to the cylinder, the circle D' D rolls out into the straight line D,D^ while the equator rolls out in its true length on the line Q^. If L be the latitude of any parallel and R the radius of the sphere, the distance of the parallel from the equator on the development is R tanZ. Thus the distances between parallels increases toward the poles and the poles themselves cannot be shown on this projection. The equator being divided into equal parts, representing degrees of longitude, 68. MAP PROJECTIONS. the meridians are drawn parallel to each other, and accord- ingly one degree of longitude has the same length on all parts of the map. Along the eqtfator the projection is orthographic, Di D 2 19 50 35" 25 50" 75 Qi Q 2 !o 12 61 1 ( 6 K 11 and a distance may be laid off in its true length. At any latitude L a distance / has the length / secZ when laid off along a parallel and the length / sec 2 Z, when laid off along a meridian. In the polar regions the distortion is so great that the projection is unsatisfactory. Mercator's map projection is a favorite one among naviga- tors, since the course of a ship may be plotted upon it in a straight line as long as the ship sails on the same true bearing. It resembles the cylindrical projection in having parallel straight lines for the meridians, while the circles of latitude are also straight lines normal to the meridians, but the dis- tance on the projection of any circle of latitude from the equa- tor is derived from the condition that the spiral line, or loxo- drome, on which the ship sails, shall be a straight line on the projection. Let the left-hand diagram in the following figure represent a part of the sphere, QC and Q 1 C 1 being meridians and CC l a circle of latitude, while CE 1 E 2 is the loxodrome which makes the same angle with all meridians. Let the right-hand diagram represent the Mercator projection, where GEODETIC COORDINATES AND PROJECTIONS. VIII. the corresponding meridians are qc and q^c^ while cc l is the circle of latitude and cee^ the straight line representing the loxodrome. The! condition of equal azimuths for the loxo- drome in the two figures is that the ratio c l ejcc l shall equal the ratio C^E /CC l when the two meridians are very near together. Then."^,^, may be called dy where y is the distance qc which is to be determined. Now cc l is equal to qq l or to QQ lt and, if R be the radius of the sphere and L the latitude of C, the value of C l E l is R.dL 9 while that of CC, is QQ^cosL. Q 2 Accordingly dy = RdL/cosL is the condition that the azi- muths of CEfii and ce t e^ shall be equal and that ce t e % shall be a straight line. The integration of this gives or = 2.3025851 R log tan(45 where the logarithmic tangent is in the common system. Values of y for different" values of L may be computed from this formula, and the Mercator projection of the surface of the sphere is then easily constructed. For the Mercator projection of the surface of a spheroid a formula for the distance qc or y may be obtained in a similar manner from the condition c l e l /cc l = C^EjCC^. Here c l e l is dy and cc l is a.d\, where a is the semi-major axis of the spheroid and d\ is the difference in longitude between the two meridians; also C l E l is R r dL, where R^ is the radius of curva- ture of the meridian at the latitude L, and CC t is r.d\ where 68. MAP PROJECTIONS. \g\b r is the radius of the circle of latitude. The values of r and R^ are given in (57) and (57)', and the condition reduces to - y ~ cosL(i e 2 sin 2 Z) * This expression may be integrated, and, neglecting terms con- taining powers of e higher than the fourth, the integral is y 2.3025851 a log tan (45 -|- \L] ae 2 sinZ, \ae* sm 3 L, from which values of y are readily computed. Mercator's projection is mostly used on the ocean, where the nautical mile is the unit of measure. Taking a and e for the Bessel spheroid, and using 1855.11 meters as the length of a nautical mile, the value of y in nautical miles is y 791 5.705 log tan (45 + JZ) 22.945 sinZ 0.05 I sin 3 Z. For example, the computed values of y at intervals of 5 from latitude 30 to latitude 50 are as follows : 30 35 40 45 50 7=1876.9 2231.1 2607.9 30*3.7 3456.9 4y= 354-2 376.8 405.8 443.2 and by using the differences in the last line a Mercator pro- jection is readily constructed, the distance between two merid- ians 5 degrees apart being exactly 300 nautical miles. Books on navigation give values of y for every degree and minute of latitude, as they are needed both for the construction of charts and in nautical computations. Prob. 68. For a cylindrical projection show that the distance of any parallel of latitude from the equator is given by y = a tanZ a? sinZ(i ) = 14.6130, log (206 26$/R*) = 9.7055, log (206 26$/2R*) = 9.4045- Now take /= 27516.0 metres, M, = +42585.934 meters, L, = + 51 449.866 meters, and Z, = 16 47' 06". 38, while the length of the line Pf* is / = 27516.0 meters. Then m = + 26 343.669 meters, n = + 7 946.133 meters, and J/ a = 42 585.9344- 7 946.i33+o- 2 33-o.o23 =+5o 532.277 meters, L t = 51 449.866 26 343.669 o.i 23 -{-0.007 = + 25 106.081 meters, Z a = 16 47' o 9 ". 3 8 + 180 - o 3 ".66-f oo // . 53 = 196 47' o6".2 5 , and thus the point P^ is completely determined. One of the great advantages of this system is that the difference of the front and back direction-angles of a line differs but little from 180, and either may be used by a local ?O. LINEAR SPHERICAL COORDINATES. 197 surveyor to check his topographic work. With geodetic azimuths, on the other , hand, the orientation of such local work may be more accurately made at the starting station, but when checking on a second station the large difference in direction is liable to lead to confusion. Undoubtedly the system of linear spherical coordinates must in time come into use in America, and by it or some similar method the results of the geodetic triangulations can be made more generally available for use in precise detailed surveys of large areas. For a fuller account of the system, as also for the method of rinding the distance and direction-angle between two stations whose latitudes and longitudes are given, reference is made to Vol. II of Jordan's Handbuch der Vermessungskunde. Prob. 70. Given L v = -j- 50 ooo, J/, 10 ooc, Z, = + 60 ooo, and J/,, = 20 ooo meters, to find the distance from P l to P % and the direction-angles Z l and Z a . 198 GEODETIC TRIANGULATION. IX. CHAPTER IX. GEODETIC TRIANGULATION. 71. RECONNAISSANCE. The first thing to be done in a reconnaissance for selecting the stations of a geodetic triangulation is to make a careful study of all existing maps. Small-scale sketch maps should be prepared, showing the principal watersheds and mountain ranges as far as they are known, and these are to be taken into the field by the reconnaissance party. Such a party consists of two or three men and it is equipped with aneroid barometers, prismatic pocket compasses, binocular field glasses, and photographic cameras, together with apparatus for climbing trees. Ascending to one of the highest elevations in the region a series of sketches showing the visible horizon and intermediate hill ranges is made. On this are marked the magnetic bear- ings and the estimated distances to all prominent peaks and gaps. Photographs of the portions of the horizon where it seems probable that stations may be located should also be taken, and the names of all mountain ranges and peaks be ascertained. Then, ascending to another elevation several miles away, a similar series of sketches is made, and after several of these observations the party obtains a fair idea of the topography of the country. The heights of all the posi- tions occupied in this work are to be obtained as closely as can be done with the aneroid. The results of these operations are to be plotted from day 71. RECONNAISSANCE. 199 to day on the sketch maps, the visible horizon as seen at each station being roughly drawn, and the intersection of these horizon lines, together with the observed heights, will give information regarding the approximate positions to be selected for the primary stations. In many cases there are one or two peaks so prominent that no doubt exists as to their avail- ability for stations, while regarding others much additional field work must be done before a final decision can be made. The intervisibility of adjacent stations must of course be insured, and in a prairie country where high towers are to be erected this requires the application of the rules of Art. 37 regarding curvature of the earth and refraction. The primary stations are to be so selected as to secure the best triangle, polygon, or quadrilateral nets to cover the given area under the prescribed conditions of precision and cost, care being taken to avoid angles less than 30 degrees, except in quadri- laterals (Art. 17). As a general rule for primary triangula- tion the longest possible lines are to be obtained which are consistent with the formation of well-proportioned triangles. As an example of- one of the field computations, suppose that two stations 28 miles apart are 65 and 105 feet, respec- tively, above the ocean level, and that the highest point between them is on a ridge 10 miles from the first station and 20 feet above ocean level. It is required to find the height of towers at the two stations so that the line of sight may pass 10 feet above the top of the ridge. Assuming that the line of sight is parallel to a tangent to the level surface at the ridge, the combined effect of curvature and refraction is 0.57 X io 2 = 57 feet for the first station and 0.57 X i8 2 = 185 feet for the second station. Hence the height of the tower at the first station should be 57 + 30 65 = 22 feet, and at the second station 185 + 30 105 = no feet. Reference may be made to the Reports of the Coast and Geodetic Survey for 1882 and 1885 for a full account of the rules of reconnaissance, and to Final Results of the Triangu- 200 GEODETIC TKIANGULATION. IX. lation of the New York State Survey (Albany, 1887) for an interesting description of the detailed field work. After the reconnaissance party has established a few stations a triangulation party may start at work in the measurement of angles. It is the duty of this party to mark the stations, erect the towers and signals, and make the observations of the horizontal and vertical angles. Sometimes the reconnais- sance and triangulation work are done by the same party, this method usually saving expense. Base-line measurement and astronomical work are, however, usually done by specially trained parties. Prob. 71. A station B is 325 feet above A, but between them, at a distance of 15 miles from A and 25 miles from B, is a ridge which is 10 feet above A. If no tower is built at B and one 50 feet high at A y how much above the ridge does the line of sight pass ? 72. STATIONS AND TOWERS. The marking of a station in a permanent manner is usually done by the first triangulation party which takes the field, the reconnaissance party merely selecting and describing the approximate location. It is believed, however, that if the responsibility of marking the station were assigned to the reconnaissance party, a better location would often be made. The name of the station is usually assigned by the reconnais- sance party, and this should be the same as the local name of the peak or ridge on which it is situated. The stations are marked by bolts set into the rock, or by stone monuments set in the ground. In the latter case it is customary to bury beneath the monument a bottle or crock whose center marks the center of the station. When this is done the knowledge of the bottle or crock should be con- cealed from the people of the neighborhood, and it should be covered with a large flat stone having a hole drilled in its upper surface. The bottom of this flat stone should be about /2. STATIONS AND TOWERS. 2OI six inches above the crock, its top about three feet below the surface of the ground, arid upon it the foot of the monument may be set. The, centersr'of the underground mark, of the hole in the flat stone, and of the top of the monument should be in the same vertical. Near the top of the monument " U. S.," or other appropriate letters should be cut. Detailed instructions regarding the methods of marking stations may be found in the Reports above cited. Reference points should be located on surrounding rocks, or by auxiliary monuments, from which bearings and distances are to be measured to the station. The geodetic surveyor should always make his description of the station clear and full, so that it may be found after the lapse of many years. For this purpose it is well to run a traverse line to the nearest public road, if there is one within a reasonable distance, and erect there a monument which may serve as a starting point for future parties. A tower is a structure erected over a station for the support of the theodolite and observer. It consists of two parts, an interior tripod to carry the instrument, and an exterior scaffold entirely surrounding the tripod but unconnected with it. The interior tripod is usually made of three posts braced together, while the outside scaffold is a structure like a braced pier having four posts. Rough towers made of timber cut on the spot can be built for about $1.00 per vertical foot up to heights of 30 feet, exclusive of the cost of the timber. Towers higher than 50 feet are usually built of sawn timber bolted together, and one 150 feet high makes a heavy item in the expense of triangulation. At some stations no towers are required as the instrument may be directly upon the ground. Even in such cases a low tower ten feet in height is to be recommended, as it adds much to the comfort of the observer in warm weather, and has the advantage of elevating the line of sight above the surrounding earth. / The four posts of the exterior scaffold should be extended 202 GEODETIC TRIANGULATION. IX. about eight feet above the platform so as to allow canvas to be spread to protect the instrument from the sun and wind. The effect of the sun on the interior tripod is in high towers often very marked, the top moving in a lateral direction so as to describe an ellipse. To lessen this effect, and also for protection against wind, it is often screened by a canvas covering placed around the upper part of the scaffold. The views of two triangulation stations of the U. S. Coast and Geodetic Survey here given may be of interest to students. The first shows a tower 130 feet in height erected by Mosman at Tate, Ohio, and the second the method used by the author at Port Clinton, Pa., where no tower was recuired. In the first view the theodolite is not visible, but in the second it ij seen mounted upon the tripod. In both views heliotropes for flashing to the adjacent stations may be distinguished. Detailed information regarding the erection of towers will be found in the Report of U. S. Coast and Geodetic Survey for 1882, pages 199-208. Sometimes a church spire, or other inaccessible point, is used as a station a|nd angles are measured at other stations by sighting upon it. This is of frequent occurrence in second- 72. STATIONS AND TOWERS. 203 ary triangulation, but should be avoided in primary work. Sometimes in primary work- an eccentric station is occupied near the true one, the angles observed there, and~their values then reduced to the true sta- tion. Let A be the true sta- tion and a the eccentric one, and let it be required to find the true angle MAN from the observed angle MaN. To do this the distance Aa must be carefully measured and also the angle AaM, and the distances AM and AN must be found from the triangulation. Let Aa = d, AaM = #, AM = m, AN = n, MaN a, and MAN = A. Then, as the opposite angles made by the crossing lines are equal, A -f- M equals a -\- N, and accordingly the required angle is A = a - M + N, (72) in which M and N are to be computed from s'mM = sin#, sinN = -sin(0 + a), (72)' or, for most primary work, since m and n are large, M 206 265 sin0, N - 206 265 - sin(# + a\ (72)" wi fi where M and N will be found directly in seconds. For example, let ^2.2145 meters, logm 3 90891, log;* = 3.95713, = 28 07', a = 64 18' 20". 1 3. Then M = 26" . 550 and N 50". 372, whence A 64 18' 43". 95. The angle 6 is here measured from the fixed line aA around to the left-hand station and its value may range from o to 360 ; hence the signs of M and N will depend upon the signs of sin# and sin(# + a). Thus if M and N were located at the left of a in the figure, 6 would be over 180 ind sin# would be negative. With regard to the use of (72)' and (72)" it 204 GEODETIC TRIANGULATION. IX. may be said that the former need not be employed unless M and TV are greater than 15 minutes. Prob. 72. Draw the figure for the case where d '= 2.2145 meters, log/?/ = 2.90891, logtf = 2.95713, 6 = 208 07', a = 96 07' c>3".72, and compute the true angle A. 73. SIGNALS. A signal is a pole, target, or other object erected at a station upon which the observer at another station points in measuring the angles. The simplest signal is a pole, but its use involves a liability to error 1.1 sighting upon the illumi- nated side, and hence for the most accurate work plane targets are preferred. These are made of a wooden frame- work covered with either black or white muslin. For a dis- tance of fifteen or twenty miles good dimensions for a target are 2 feet in width and 12 feet in height. The target has the disadvantage of requiring to be set anew whenever the observer changes his station, but it has the advantage of being more easily seen than a pole. The old practice of putting a tin cone on a pole and of sighting on the illuminated side cannot be recommended, except for reconnaissance work. For long lines neither pole nor target can be recognized, and the heliotrope must be used. This instrument consists essentially of a mirror which reflects the sunlight to the observer's station. The usual size of the mirror is about two inches in diam- eter, and it should be mounted so that it has a motion about a vertical and a horizontal axis. The mirror may be placed at one end of a board about three feet long upon which are two sights in the same line with the center of the mirror. The sights being pointed at the distant station, the mirror is constantly turned by an attendant, called a 73- SIGNALS. 205 heliotropec, so that the shadow of the rear sight falls upon the front one, and the* sunlight then is reflected to the observer, who sees jt as a 9*ar twinkling in the horizon. As the apparent diameter of the sun is about half a degree, the reflected rays form a cone having the same angle, so that it is only necessary to point the heliotrope within a quarter of a degree of an object in order that the light may reach it. The light of a heliotrope may be seen through haze of moderate intensity if the observer knows where to point his telescope in order to find it. Lines from ten to fifteen miles in length are usually observed with pole or target signals. For lines from fifteen to forty miles a combination of target and heliotrope is advantageous, the former being used on cloudy days and the latter in sunshine; in this case the heliotroper erects the target over the station and places his instrument in line in front of it. For lines exceeding fifty miles in length the heliotrope is the only feasible signal unless the atmosphere be unusually clear. Probably the longest side yet observed is one of 192 miles in California, where the heliotrope had a mirror of 77 square inches. Night signals have been successfully used. These are generally large kerosene lamps with reflectors, which are placed in position and lighted by the heliotropers on leaving their stations in the evening. A magnesium tape whose burning is regulated by clockwork has been also employed. Night work should be usually combined with day work, the observer being on duty from noon to midnight. The best time for measuring horizontal angles is from six until nine o'clock in the morning, and from three in the afternoon until after sunset, as then the air is the clearest and the lateral refraction disturbances are the smallest. For vertical angles, on the other hand, the best time is during the two hours preceding and following noon, the vertical refractiqn^ then the least variable. 206 GEODETIC TRIANGULATION. IX In measuring horizontal angles it is sometimes necessary that a signal should be set at a short distance to one side of the center of the station. This is called the case of an eccentric signal, and a correc- tion is to be applied to the observed angle to reduce it to the true angle. For instance, in 1878 an observer at the station O measured the angles COa and aOB, where the heliotrope had been set at a instead of at the true station A. The distance Aa was reported as 16 feet 2 inches, and the angle OaA as 129 35'. Later, in 1883, the work had pro- gressed so that OA was found to be 29 556 meters. The value of the small angle AOa in seconds is computed from 206265^0 sinOaA/AO and will be found to be 26" '.63, and this is the correction to be added to aOB and to be sub- tracted from COa. An eccentric signal should be avoided. Indeed it is best that heliotropers should not know that it can be used, other- wise they will be often tempted to set their heliotropes eccentrically from considerations of personal comfort and may neglect to take the measurements that are necessary for cor- recting the angles. Prob. 73. Compute the correction AOa when the side OA is not very large compared with Aa, say when OA = 295.56 meters. 74. HORIZONTAL ANGLES. Two classes of triangulation are always recognized in geodetic work; the primary series, which connects directly with the bases and has the longest possible lines, and the secondary series, which locates stations within the primary triangles. To these are ultimately added a tertiary series for establishing stations at closer intervals for the special use of plane-table and stadia parties. It is generally required that 74- HORIZONTAL ANGLES. 207 the probable error of an observed value of a horizontal angle shall not exceed o".3Q on primary work and o /r .8o on secondary work. On primary work repeating and direction theodolites are used, on secondary work repeating theodo- lites, while for the tertiary work the engineers' transit gives all the precision desirable. In fact a good engineer's transit will give as precise results as those required for secondary triangulation, provided the length of the lines.be such that the signals can be clearly seen with its telescope. A repeating theodolite does not differ in principle from an engineers' transit. The telescope, however, is so long that it cannot be turned over on its axis, -but must be lifted out of the standards in order to be reversed in position. The graduated limb is usually from 8 to 12 inches in diameter, is divided into ten-minute divisions, and reads by three verniers to 3" or 5". Circles 16 and 20 inches in diameter were formerly used, but it is now known that the precision of these is little if any superior to those of 8 and 10 inches. The method of observation, in order to eliminate systematic and accidental errors, is in all respects the same as that described in Art. 14. Owing to the atmospheric disturbances on long lines of sight it is important that the work on each angle should be distributed over several days, and this is easy to arrange, since the rarity of good weather usually requires a party to remain two or three weeks at a sta- tion when several lines concentrate there. The following is a record of the work done with a repeating theodolite at Bear's Head station in Pennsylvania from July 20 to July 30, 1885. During these eleven days there were only eight when the weather permitted observations, and on five of BakeOven Port Clinton 208 gEODETIC TRIANGULATION. ANGLES AT BEAR'S HEAD. IX. Name of Angle. No. of Reps. Observed Value. Adjusted. Penobscot Knob 48 51" 19' 5 9 ".7i 60''. 15 Penobscot Bake Oven 40 85 52 32 .39 33 -22 Penobscot Port Clinton 48 139 24 06 .97 06 .91 Penobscot White Horse 40 1 80 39 48 .43 47 -24 Knob Bake Oven 48 34 32 33 -16 33 -07 Knob Port Clinton 4 8 88 04 05 .75 c6 .76 Knob White Horse 48 129 19 47 .57 47 -09 Bake Oven Port Clinton 4 8 53 3i 34 -27 33 -69 Bake Oven White Horse 40 94 47 12 .71 14 .02 Port Clinton White Horse 4 8 41 15 39 -97 40 -33 these no measurements could be made until about three o'clock in the afternoon. The total number of measures is seen to be 456, .or an average of 1 14 for each independent angle. The station adjustment being made by the method of Aft. 16, the average probable error of a single observed value is found to be o" .fi and that of an adjusted value about o".6o. It is thus seen that the adjustment has greatly increased the precision. A direction theodolite has no verniers, but is read by three or more micrometer microscopes placed around the limb. The circle in the figure represents the field of view of one of the microscopes in Which three divisions of the graduated limb are seen. By turning the micrometer screw the cross-hair is moved to a or b, thus reading the distance ac or be in seconds. When pointing on the first station the cross- hair may be set at a graduation mark, and when pointing at the second the reading is taken as just described. Such theodolites have large circles so that the limb may be divided to 5 minutes while the micrometers will read to seconds, and by taking the mean of all the micrometer readings a close 74- HORIZONTAL ANGLES. 2OQ determination of the angle can be made. No repetitions are possible by this method* but different series of readings are taken, on different parts f the limb in order to eliminate errors of graduation, measures are made both from left to right and from right to left in order to eliminate errors due to clamping and twist, and the work is distributed over differ- ent days to eliminate atmospheric influences. There are two methods of measuring the angles at a station with a direction theodolite. The first, called the method of single angles, is to determine each angle independently by the process above described ; thus in the case of four lines meeting at O each angle is measured by reading first on the left-hand line and second on the right-hand line; thus the value found for BOC or BOD is independent of any reading made on AOB. In this method all the results are to be treated and adjusted exactly as if they had been made by a repeating theodolite. In the second method of observation, called the method of directions, a line OA is taken as a reference line and pointing and reading taken on it; then the limb is turned and readings taken on B, C, and D in succession. Another line OB is then taken as a reference line, and readings taken on C, D, and A in succession. Here it is seen that the values found are not independent, as the initial reading enters into all the results of each series; consequently the adjustment is more complicated than that of the other method. 210 GEODETIC TRIANGULATION. IX. Prob. 74. Regarding the above observations at Bear's Head sta- tion as of equal weight, compute the probable error of a single observed value. 75. THE STATION ADJUSTMENT. The station adjustment for all cases except the method of directions is made by the method of Art. 16, which need not be further explained here. When the weights are so nearly equal as those of the case given in the last Article, it is an unwarrantable refinement to take them into account. With regard to the probable errors it is to be noted that those of the adjusted values need rarely be computed by the method of Art. 10 except in special scientific investigations. It is well, however, to find the probable error of a single observa- tion by formula (10), and this ought to have a reasonable agreement with the average probable error of the observed values as computed from (9)' '. Some observers prefer to measure the n angles included between the n lines meeting at a station instead of combining the lines to make \n(n i) angles as in the example of the last Article. This case is called "closing the horizon/' and thus the conditional equation is introduced that the sum of the n single angles shall be 360 degrees. The adjustment may be made by the method of Art. 16, employing only n I independent quantities, but the numerical work will usually be shorter by the method of Art. 21. The method of directions requires a slightly different process of station adjustment. To explain it take the case where the three lines OA, OB, \j and OC meet at the station O, / /$ and let x and y be the most prob- able values of the angles A OB and AOC. Suppose that OM denotes the direction of the tele- scope when the mean reading of the three microscope microm- 75. THE STATION ADJUSTMENT. 211 eters is o oo' oo".oo, and let m denote the most probable value of MO A. Then let the three readings on A, B, and C give the three observation Equations m = 60 18' 20". 5, m + x = 85 04 13 .0, / + j = 119 50 14 .2. Next let the circle be turned so that ON gives the zero direc tion and let n be the angle NO A. Then three readings being taken on A t B, and C again, there are three additional observation equations n = 120 if 05". o, n + x - 145 02 53 .o, n + y = 179 48 59 5. Again if the circle be turned about 60 degrees further and three readings be taken upon A, B, and C there will be three more observation equations, while a fourth, fifth, and sixth set will each give three others. Thus for six positions of the circle there will be 18 observation equations involving 8 unknown quantities; from these the normal equations are formed by the rule of Art. 6, or, if they are of unequal weight, by the rule of Art. 7, and their solution will furnish the most probable values of x and y. The quantities m, n, . . . may be eliminated from the normal equations, before solving for x and y, as their numerical values are not required. The numerical work may be abbreviated by introducing corrections to assumed values of the quantities. Thus, for the above case, let m l and n l be corrections to the observed values of m and n\ also let ^ = 24 45' 52". 5 -[-*", and y = 59 31' 5 3". 7 +.1V Then the six observation equations reduce to m l o, *, + *, = o, m l + y, o, a, = o, n v + x, = - 04". 5, n l +y 1 = + o".8, and from these the four normal equations are 212 GEODETIC TRIANGULATION. IX. 3*. + *i + ^i = - 3 -7, ;;/, + ;/, + 2*, = 04 .5, ^ + t + 27, = + oo .8. Taking the values of ;, and ;, from the first and second equations and substituting them in the others, these become 4*i 2r, = 09". 8, 2*1 + 4/i = + 06". I, from which ..#,.= 02" '.2$ and j^ = + 00^.40 are the most probable 'corrections, whence x = 24 45' 50". 2 5 and / = 59 31' 54". 10 are the adjusted values of the angles AOB and AOC, and accordingly the most probable value of BOC is 24 36'03".85. The angles found by the station adjustment are spherical angles, because the graduated circle is made level, that is parallel to a tangent plane to the spheroid at the station. Strictly speaking the level position of the graduated limb is an astronomical and not a geodetic one (Art. 59), but this slight discrepancy of a few seconds can produce no measur- able effect on the observed angles. It should be borne in mind that it is of great importance to avoid inaccuracy of level when measuring angles, since this renders their values too large, and there is no method of eliminating its influence. Prob. 75. Given the observed angles AOB = 86 07' 17" with weight 6, BOC 89 10' 35" with weight 4, and COA 184 41' 55" with weight' i. Compute the most probable values of the angles. 76. TRIANGLE COMPUTATIONS. After the angles have been measured at a number of stations and the length. of one side has been obtained, either by connecting with an adjacent triangulation or by measuring it as a base, computations of the lengths of the triangle sides are to be made. The three angles of a triangle do not add up to 1 80 degrees and hence the results obtained for the sides 76. TRIANGLE COMPUTATIONS. . 213 are only approximate, but they are more than sufficiently accurate to compute the spherical excess of the triangle. These computations are tile same in every respect as those explained in Art. 19, except that five-place logarithms should be used, the logarithmic sines taken to the nearest 10" of angle, and the lertgths determined only to the nearest 10 meters./ The formula for spherical excess established in (6$)' of Art. 63 may now be used and the excess be found for each tri- angle. In order to take the factor m from Table IV the mean latitude must be known roughly. In the first instance this may be estimated, but in later work it will be found from the results of the LMZ computations. Then, Spherical excess m . ab sindT, in which a and b are any two sides of the triangle and C is the angle included between them. As a numerical example of the computation of spherical excess the following data of a triangle will be used : Stations. Angles adjusted at Stations. Approximate Distances. Approximate Latitudes. Pimple Hill 49 04' 50' '.13 27 540 meters 4i 02' Smith's Gap 9 21 2 5 53 36 440 meters 40 49 Bake Oven 40 33 46 .91 23 700 meters 40 45 Sum = 180 po' 02''. 57 Mean L. = 40 52' Now C can be taken as any one of these angles and a and b as the two adjacent sides. It is advisable to make two check computations for the excess, thus: Numbers. Logarithms. Numbers. Logarithms. factor m 9.40441 factor m 9.40441 a = 36 440 4-56158 * B= 56, B,= 30, *.= 27, C=40, >i9> C t = 2i, 5, = 106, ,= 135, S.= i20, where 5, is the angle subtended by AB, and 5, and S t those subtended by BC arid CA. Then, the station adjustments give the four conditions A=A,+A, , B=B l +B* , C=C,+C, , S, +,+, = 360. The figure adjustment, supposing the triangles to be plane ones, requires the three angle conditions ^,+^+5,= 1 80, ^ + C,+S,= 180, ^+^ 1 +5,= i8o, and also, as shown in Art. 22, the side condition sin//! sin^, sin7, = sinA^ sin.Z? 2 sin(7 2 . The problem now is to determine the most probable values of the twelve observed angles which at the same time satisfy the eight conditional equations. This problem is capable of rigorous solution, but when a figure contains many triangles it leads to very laborious com- putations. The custom has hence arisen of dividing the work into two parts; first, the station adjustments are made, each independently of the others, and secondly the values found by these station adjustments are then corrected so as to satisfy all the conditions of the figure adjustment. The station adjustments are generally made in the field, but the figure adjustment, which is far more lengthy, is reserved for the office, and is made by the method of correlates that is explained in the next Article. In applying these principles to a triangle net consisting of a chain of simple triangles, having one side AB measured as a base and B all angles observed, it is seen that the figure adjustment has as many angle equations as there are triangles, and no side equations. 2l8 GEODETIC TRIANGULATION. IX. The figure adjustment is hence very simple, each triangle being treated in succession by the method of Art. 76, and the spherical angles and plane angles thus found are the final adjusted values. If, however, another side HK be also measured as a base, then a conditional side equation is to be introduced to express the requirement that the length HK as computed from AB shall be the same as the measured length; an illustration of this case for two triangles is given in the last paragraph of Art. 22. In primary geodetic triangulation all stations are occupied and all lines sighted over in both directions. In secondary work a few of the stations may not be occupied, these being church spires or other inaccessible points. Thus in the last figure if one of the stations between B and H be not occupied the number of angle equations will be diminished by three, because there will be three triangles, in each of which one angle has not been observed and hence its value is to be found from those of the observed angles. In stating the conditional equations that enter into a figure adjustment care should be taken to introduce no unnecessary ones, and the following rules will be useful for that purpose; these rules suppose only one base to have been measured. Let n be the total number of lines and n f the number of lines sighted over in both directions, let s be the total number of stations and s' the number of stations occupied for angle measurements. Then, in the figure adjustment, Number of angle equations = ' / -j- i, . , Number of side equations = n 2s + 3. For instance, in the figure ABCS, at the beginning of this Article, n' = n = 6, s' = s = 4, and hence there are three angle equations and one side equation; if, however, the station S had not been occupied, then n' = 3, n = 6, A' = 3, s = 4, and accordingly there would be one angle equation and one side equation. 78. CONDITIONED OBSERVATIONS. 2 19 P/ob. 77. How many angle and side equations are there in the figure adjustment of each of, the triangle nets shown in Art. 24, one base and the angles being measured 7 78. CONDITIONED OBSERVATIONS. By the proper selection of the unknown quantities it is generally possible to state observation equations so that these quantities will be independent (Art. 25), but a shorter method of adjustment, known as the method of correlates, may be established. In this method each observed quantity is repre- sented by a letter and all the conditional equations are written, as in the illustration of the last Article. Let x, y, z, etc., represent the quantities whose values are to be found, and let the conditional equations be *.* + &*y + - - = #> bs + b.y + . . . = b, c& + c* y + - = c* in which the coefficients and constant terms are theoretic numbers. Now let M } , M^ , M t , ... be the values found by the observations for x> y, z y . . . ; if these values be in- serted in the conditional equations they will not reduce to zero, owing to the errors of the measurements. Hence, let v\ , ^ a , ^ 3 , . . . be small corrections which when applied to M l , J/ a , M 9 , . . . will render them the most probable values. Then if x, y, . . . be replaced by M l -\-v lt M t + v t , . . . the conditional equations reduce to in which d l , d t , d t , . . . are small quantities called discrepan- cies. The problem now is to find values of v t , z/, , v t , . . . which exactly satisfy these equations and which at the same time are the most probable values. 220 GEODETIC TRIANGULATION. IX. The following is the solution of this problem which is deduced in treatises on the Method of Least Squares. Let Pi A A De tne weights of the observations M { , J/ s , J/, , . . . and let , , a , k t , . . . be quantities which are determined by the solution of the normal equations rtf 3 ~] I ab~^ rac ~\ l-jl*. + Lj]*. + l-j]*. + '-' = 4. LAP + ' ' ' = d " (78)' P**IA i - i A ' i A -r * -I- 1 r J*. H- 1 T I* / These equations are the same in number as the number of conditional equations, , being known as the correlate of the first equation, a of the second, and so on. The brackets indicate summation in accordance with the same notation as that employed in Art. 7, namely, ^i ^L__j-f^ \ ab ~\ - a ^ i *A i /J-A" h A " L/J SS A " A + and the coefficients have similar properties to those in the normal equations for independent observations. By the solution of these normal equations the values of k l , k^ , , , . . . are found; then the corrections are and these added to M l , J/ a , . . . give the most probable values of AT, jj/, ... which exactly satisfy the theoretic con- ditions. When there is but one conditional equation there is but one normal equation and one correlate, k^ whose value is d.j I , and thus the values of z/, , z/ 3 , . . . agree with 78. CONDITIONED OBSERVATIONS. 221 those deduced in Art. 21, where d l is called d, and q is used instead of a. f * As an illustration of the nðod, let there be five measure- ments on five quantities, giving the observation equations, 1. x = 47.26, with weight 3, 2. y 39-O4, with weight 19, 3. z = 6.35, with weight 13, 4. w= 86.64, with weight 17, 5. u = 35.21, with weight 6, which are subject to the two theoretical conditions, x + y w = o, y -\- z u = 10. Let v l , v^ , z/ s , v^ , and z/ be the most probable corrections - to the observed values, so that the observation equations become i ' Jr\ ~ J , 2. z> a = O, p^ = 19, 3- v, = o, /, = 13, 4- ^ = o, / 4 = 17, 5- ^ = o, /. = 6, and the conditional equations reduce to Vl + Vt - v,' = + 0.34, v* + v s - v, = - 0.18. Now, by comparison with the notation in (78), and thus the normal equations of (78)' become 0.445^ + 0.053^ = + 0.34, -53^i + 0.296^ = o. 18, whose solution gives k, = -(-0.855 and k^ 0.759. Then by (78)" the values of the corrections, or residual errors, are and hence the adjusted values of the observations are * = 47-545, 7 = 39-045, 3 = 6.291, w = 86.590, u = 35.336, 222 GEODETIC TRTANGULATION. IX. which are the most probable results that exactly satisfy the two conditional equations. The probable error of an observation of the weight unity may be computed by the formula r l = 0.6745*7 ^ ,. Y n - q + n > in which n is the number of observation equations, q the number of unknown quantities, and n' the number of condi- tional equations. For the above example the residuals are already found; squaring them, multiplying each by its weight, and adding, gives ^pv 1 = 0.428, whence r^ = 0.309. Accordingly the probable error of the first observation is 0.309/4/3 = 0.18, and the weight of the adjusted value of that observation must be somewhat smaller than o. 18. Prob. 78. Four lines OA, OB y OC, and OD meet at a station O, and the following angles are observed, all of equal weight : AOB = 19 47' 13", BOC = 40 38' 04", COD = 6 5 i2'io", DO A = 54 22' 2 10 be corrections in seconds to be added to the observed values of the angles. Let d l be the difference in seconds between the theoretic sum and the sum of the ten observed values; then ^ + *', + ^ 3 + "4 + V, + V< + V, + V % + V, + V 19 = d, is the conditional angle equation. To state the second con- dition let expressions for the side SD, as computed from SA by two routes, be written; if these be equated there results sin^, sin s sinC 5 sinZ>, sin^ = sin^ 2 sin# 4 sinC 6 sinZ> 8 sin 10 as the conditional side equation. This is to be expressed in terms of the corrections in a similar manner to that used in Art. 22, log (A, + v,) being written as log A, + v t diff. i" ', where diff. \" is the tabular logarithmic difference of the logarithmic sine corresponding to the angle A lt Let the observed values be those written below, all being of equal weight. As the sum of these is 540 oo' 10" the discrepancy d l is 10", the angle equation is known, while Observed Angles. Log. Sines. A l = 25 47' 23" 1 6385588 + 43.52;, 3 5 6 3 1 22 1.9212208 -f I4.ov 3 C*. = 85 28 57 1.9986487+ i. 7v A = 8 3 12 39 1.9969439+ 2 .5z/ 7 -^9 = 4i l6 15 1.8192933 + 23.9^ 1.3746655 A t = 50 12' 54" 1.8856162 + 17.52;, BI 48 52 12 1.8769214 + 18.4^ C % = 61 58 02 1.9458027 -f- n.2v t >* = 3 8 2 5 7 1-7933543 + 26.52;. ^io=48 15 19 1.8728079 -f i8.8z; 10 l-3745 2 5 the side equation is found by equating the two sums of the logarithmic sines. Thus 224 GEODETIC TRIANGULATION. IX. 1 4- *B 4" *l 4- 7'4 + Vf, 4- 7' 8 + Z/ 7 + 8 4- 7/j 4 /, = lo", 43-5 t 'i- I 7'5 z '94-i4-*'s 18.4^4+1 .jr,, 11.2^,4-2.5^ 26.57/84-23.9^9 i8.8z/ IO =- 1630, where the second member of the last equation is in units of the seventh decimal place of logarithms. By the method of the last Article the solution is now readily made, placing #, = -f- I, # a = -(- i, . . . d^ = 10 and b, = + 43.5, ^ = 17.5, . . . ^ a = + 1630. The two correlative normal equations are found to be -|- io/, 6.8/ a =: 10, 6.8/, -j- 4494.5^ = -- 1630, from which , = 1.248 and a = 0.364. Then by (78)" ^=-17".!, *. = -6". 3 , =-i". 9 , *,=-*"*, *.=- 9 ".9, , = + 5"-i, * 4 =+5"-5, ^ = + 2".8, ^. =+8".4, ^ 10 - + 5" 6, are the most probable corrections to the observed values, and applying them the sum of the adjusted values will be found to be exactly 540, and then the angles at 5 may be obtained. Also, multiplying each v by its tabular difference, the correc- tions to the logarithms may be found, and the sums of the two sets should then be exactly equal. For a large polygon where the spherical excess of the tri- angles can be detected the method of adjustment is the same, the two conditional equations being slightly modified. First, the theoretic sum of the ten angles exceeds 540 de- grees by two-thirds of the spherical excess of the entire poly- gon; thus if this excess be i8".o the discrepancy d l will be i8".o io".o =+ 8".o. Secondly, each observed angle is to be diminished by one-third of the spherical excess of its triangle before placing it in the side equation ; for instance, if the spherical excess of the triangle ABS is 03". o, then the value of A l to be used in the side equation is 25 47' 22" . The solution is now made as before and the corrections v l , v^ , . . . z> 10 found; these, added to the angles used in the side equations, give the adjusted plane angles, or, added to the observed values, they give the adjusted spherical angles. 80. ADJUSTMENT OF A QUADRILATERAL. 225 When the station 5 is occupied and all the angles there are observed there will be .five angle equations and one side equation in the figure adjustment. The side equation is the same as before and the five angle equations may be taken as those expressing the conditions that the sum of the angles in each triangle shall equal its theoretic value. Thus, for the triangle ABS, if the sum of the observed angles be 180 oo' 05" and the spherical excess 'be 03". o, the angle equation is v * + ^s + v n = +02". These six equations lead to six correlative normal equations, by whose solution the six cor- relatives are found, and then the fifteen corrections are obtained. Lastly, the adjusted spherical angles result by adding these corrections to the observed values, and the adjusted plane angles are found by subtracting from the spherical angles the proper amount for spherical excess. It may be remarked, however, that this solution can be abbre- viated by an artifice similar to that used in the next Article. Prob. 79. In the first diagram of Art. 77 let there be given A l = 4 oi, A t = 43 i, B, = 2 9 f, B % = 2 6|, C, = 19, C, = 21. Adjust these observations so that the results shall satisfy all the con- ditions of the figure adjustment. 80. ADJUSTMENT OF A QUADRILATERAL. In the quadrilateral ABCD let the two single angles at each corner be equally well measured. The rule of Art. 77 shows that the figure adjustment c requires three angle equations and one side equation. The three angle equations may be written by taking any three of the triangles and im- posing the conditions that in each the sum of the adjusted values shall equal the theoretic sum; the three triangles that have the point B in common will be chosen for this purpose. Let d l , d^ , d^ be the discrepancies for these triangles, d^ being that for the triangle whose large 226 GEODETIC TRIANGULATION. IX. angle is A, while d, and d^ are those for the triangles whose large angles are B and C\ also let d^ be the discrepancy for the fourth triangle CD A. Let v l , v, , . . . v % be the correc- tions in seconds to be applied to the observed values. Then the three conditional angle equations are *> + *>. + % + >. = ^3 , and the conditional side equation is sin^4 t sin^ B sin(7 7 sinZ\ = sin^4, sin^ 4 sin" 6 sinZ> 8 . For given numerical values of the eight angles the adjustment may now be made by the method of Art. 78, there being four correlatives and four normal equations. It is, however, frequently required to make an approximate adjustment, whereby the three angle equations will be satis- fied, the side equation not being used. Taking, then, three correlatives, the normal equations (78)' are, since all weights are unity, 4k, + 2k, = d, , 2k, + 4^ + 2k, = d, , 2k, + 4k, = d, . Solving tliese and substituting the values in (78)", the cor- rections are found, and, remembering that d, -\- -d< = */, + d t , these may be written *>> = v, - K + M - 4), which are very easy in numerical application. For instance, let the three triangles have the spherical excesses .$, = O 7/ .48, s t = i".05, s s i x/ .4i, and s t = 0^.84, and let the observed values of the eight angles be arranged in four sets, one for each triangle. The sum of the observed angles for the first set subtracted from the theoretic sum gives the discrepancy 80. ADJUSTMENT OF A QUADRILATERAL. 22? d l = -J-O2". 73, and similarly for the other sets. Then by (80) the corrections are ;-{- o".62, -f-"'74 -j- o".67, and -|- o".56, whence result the ^adjusted values of the spherical angles. The sum of these is 360 oo' oi".88, which is a check on the work, since the spherical excess of the figure is jj -|- j a = j, -|- J 4 = I'/.SQ, the error of one unit in the second decimal being due to the lost digits in the third decimal. Lastly, the plane angles are found by subtracting the proper amounts from the spherical angles, these amounts being com- puted from (80) by using the given excesses instead of the discrepancies. Observed Di = 58 ' 44' At = 25 18 A a = 58 54 B* = 37 02 Bs = 27 38 a = 56 24 a = 33 53 Angles. A 38".98 16 .80 57 .54 04 -43 B 57"-54 04 .43 46 .48 09 .77 C 9 .77 35 -14 D 16 .80 35 -14 Adjusted Angles. Spherical. Plane. 39". 60 39.51 17 .42 17.33 58 .28 58.14 05 .17 O5.O2 47 .15 46.77 10 .44 10.06 35 -70 35.38 Dt, = 62 03 27 56 27 56 28 .12 27.80 179 59 57 -75 58 .22 58 95 58 .48 01 .88 00.01 180 00 oo .48 or 05 01 .41 oo .84 01 .89 oo.oo -{- O2 .73 -f- 02 .83 -\- O2 .46 -j- O2 .36 OO .OO OO.OO The adjusted values thus found will not, in general, satisfy the side equation, but by the following process a second series of corrections may be obtained that will insure this result. Let z/j , z/ a , . . . z/ be the additional corrections to be applied to the above values of the plane angles. Then the angle equations are Vi + V i + Vi + V* = > v, + v, + v, + v. = o, ^i + ^. + ^ + ^ 8 = o, and the side equation takes the form aj,\ + aj>^ + a,v, + a t v, a,v, a.v t a,v, a t v e = 2 5 6 TABLES. TABLE VI. CONSTANTS AND THEIR LOGARITHMS. Name. (Radius of circle or sphere = i.) Symbol. Number. Logarithm. Area of circle it 3 141 592654 0.497 149873 Circumference of circle 2ir 6.283 185 307 0.798 179868 Surface of sphere 471 12.566 370614 1.099 209864 I* 0.523 59 s 776 1.7l8 998 622 Quadrant of circle 0.785 398 163 1.895089881 Area of semicircle \" 1.570796327 o. 196 119 877 Volume of sphere 4.187 790205 0.622088 609 7t* 9.869 604401 0.994299745 n* 1.772453851 0.248 574936 Degrees in a radian I80/7T 57-295 779513 1.758 122632 Minutes in a radian IO8OO/7T 3437.746771 3.536273883 Seconds in a radian 648ooo/7T 2O6 264.806 5.314425 133 I/^ 0.318 309 886 1.502 850 127 0.564 189 584 1.751 425064 i/* 2 o.ioi 321 184 1.005 700255 Circumference/36o arc 0.017453293 2.241 877 368 sin 0.017452406 2.241855318 Circumference/2 1600 arc o.ooo 290 888 4.463 726 117 sin o.ooo 290 888 4.463 726 III Circumference/I296ooo arc ' 0.000004848 6.685 574867 sin o . ooo 004 848 6.685 574867 Ba.se Naperian system of logs e 2.718 281 828 0.434 294 482 Modulus common system of logs M 0.434294482 1.637 784311 Naperian log of 10 i/M 2.302 585 093 0.362 215 689 hr 0.4769363 1.6784604 Probable error constant hr yT 0.6744897 1.8289754 Feet in one meter m/ft. 3.2808333 0.5159841 Miles in one kilometer km/mi. 0.621 369 9 1-7933502 INDEX. 257 INDEX. Accidental errors, 8, 9 Adjustment of angles, 18, 43, 209, 211 observations, 25, 219 triangles, 17, 51, 213 Adjustments, complex, 229 figure, 216 levels, 94 polygon, 222 quadrilateral, 225 station, 18, 51, 209 Alignment curve, 174 Almanac, nautical, 109 Anaximander, 138 Angles, 18, 28, 39, 206 and bases, 74 7 at stations, 18, 51, 60, 209 horizontal, 39, 206 geodetic, 215 on sphere, 151 vertical, 33, 101, 104, 105 Archimedes, 138 Arcs on meridian, 160, 170, 252 parallel, 171, 253 Areas, error in, 32 spherical, 151 spheroidal, 175 Aristotle, 138 Arithmetic mean, 7, 13, 27, 34 Astronomical latitude, 108, 166, 244 notions, 108 work, 107-135 Atmosphere, 98, 249 Axis of earth, 136, 169, 247 Azimuth, astronomical, 107, 109, 134 by altitude of sun, 114 Azimuth, by Polaris, 117, 121 by solar transit, no geodetic, 36, 177 on sphere, 150, 153 on spheroid, 177, 215 plane, 36, 55, 69 Base lines, 71-86 Bars, measuring, 71 Bessel's spheroid, 168 Bonne's projection, 190 Boundary lines, 230 British Ordinance Survey, 168, 237 Broken bases. 83 Bruns, 247 Cassini, 142 Celestial refraction, 251 Centrifugal force, 233 Central standard time, 129 Circles, 148, 256 Circuit of levels, 94 Chain triangle nets, 67 Clairaut's theorem, 232 Clarke's ellipsoids, 238 spheroid, 168, 237 Coefficient of expansion, 76 refraction, 102 stretch, 76 Columbus, 141 Conditional equations, 59, 218 Conditioned observations, 17, 59,219 Constant errors, 7 Constants, table of, 256 Coordinates, 36, 55, 69, 181, 194 2 5 8 INDEX. Coordinate systems, 36, 182, 194 Correlates, method of, 60, 220 Cost of towers, 201 Culmination of Polaris, 127 moon, 133 Curvature of earth, 88, 99 radii of, 157, 173, 254 Curve of probability of error, n, 26 Declination of Polaris, 118 sun, in Deflection of plumb-line, 164, 243 Degree of meridian, 141, 147,181,252 parallel, 149, 171, 253 Degree, trapezoidal, 153, 176 Delambre, 163 Direct observations, 17, 59 Direction-angle, 195 Direction theodolite, 39, 207 Directions, method of, 209, 210 Dumpy level, 97 Duplicate level lines, 89 Earth, figure of, 137-149, 232-248 orbit of, 240 radius o.f, 148 radii of curvature, 173, 254 Eastern standard time, 129 Eccentric signals, 205 stations, 202 Elements of ellipsoid, 238 spheroid, 159, i66 t 168 Ellipse, equation of, 157 properties of, 156 radii of curvature, 158 Ellipsoid, earth as, 235, 238 Ellipticity, 156, 168, 232 Elongation of Polaris, 118 End measures, 71 Engineers' level, 97 transit, 114 Equations, normal, 20, 22 observation, 16, 19, 22 side, 61, 218 Eratosthenes, 138 Error, law of probability, 9 probable, 26 Errors, accidental, 8 constant, 7 residual, 8 Errors in angles, 40 base lines, 78, 82 triangles, 49 Eccentricity of ellipse, 156, Excess, spherical, 151, 156, 176, 213 Expansion, coefficient of, 76, 78 Fernel, 141 Field notes, 41, 44, 80 Figure adjustment, 60, 216 Figure of earth, 137, 140, 147, 232-248 Flamsted's projection, 189 Functions of observations, 32 Gauss, 25 Geodesy, 137-256 Geodesic lines, 174 Geodetic bases, 75 coordinates, 181 latitude, 183, 245 leveling, 87, 97 projections, 181, 189 tables, 194, 249 triangulation, 50,68,85,105, 198-231 Geography, 36, 108, 181, 240 Geographical tables, 176, 194 Geoid, 242, 243, 245, 247 Gore, 35 Greeks, speculations of, 138 Heights, determination of, 87, 103, 105 Heliotrope, 202, 204 Hilgard, 180 History of geodesy, 137-144, 160, 232, 242 Horizon, 108 Horizontal angles, 39, 44, 109, 206 Hour angle, 109 INDEX. 259 Independent angles, 42, 210 observations, 17 Indirect observations, 17, 19, 22] 29 Inverse LMZ problem, 186 Jordan, 197 Latitude, 36, 108, 165 astronomical, 107 by a star, 127 by the sun, 126 ellipsoidal, 236 geodetic, 157, 183, 245 spherical, 153 spheroidal, 177 variation of, 136 Lamp signals, 205 Lapland arc, 144, 160 Laplace, 161, 233 Law of error, 9, u Least squares, 7-35 history of, 163 literature of, 34 principle of, 12, 15 theory of, 33 Legendre, 34, 163 Lengths of arcs of meridian, 170, 252 parallel, 171, 253 quadrant, 164,169 Level surface, 87, 100 Leveling, 16, 87-106 Linear coordinates, 37, 194 measurements, 26, 71 Literature of geodesy, 35 LMZ computations, 183, 186, 255 constants, 180, 255 Logarithms for geodetic computa- tions, 254, 255 Logarithms of constants, 254-256 Longitude, 36, 132, 135, 153, 177 Loxodrome, 191 Lunar measurements, 132, 133 Magellan, 141 Mandeville, 140 Map projections, 189 Mason and Dixon, 145 Measurement of angles, 39, 44 lines, 79 meridian arcs, 139, 144 with tape, 79 Mercator's projection, 191 Meridian arcs, 139, 144, 147, 158, 160. 167, 168, 170, 252 Metallic bars, 71 tapes, 76 Meter, 163, 169, 180 Method of least squares, 7-35, l6s Mistakes, 8 Monuments, 200 Mosman, 202 Most probable value, 8, 13, 15, 33^ N- point problem, 67 Nautical almanac, 109 Navigation, 126, 191 Net of level lines, 95 triangles, 67, 109, 217, 229 Newton, 142, 143, 232 New York state survey, 200 Normal equations, 20, 22, 24 sections, 172 Norwood, 141 Oblate spheroid, 143 Observation equations, 16, 19 Observations, conditional, 17 direct, 17 independent, 17 indirect, 17-22 weighted, 14 Ocean level, reduction to, 84 surface, 138, 241, 242 Orbit of earth, 240 Orthographic projections, 150, 189 Osculatory sphere, 175 Ovaloid, 239 Parallax of sun, 116 260 INDEX. Parallel of latitude, 149 lengths of, 171, 234, 253 Pendulum observations, 232, 241 Pentagon, 60 Peruvian arc, 144 Photographs, 198 Picard, 141 Plane triangles, 54 triangulation, 36-70 Plato, 138 Plumb-line deflections, 164 Polaris, azimuth by, 117. 121 declination of, 118 latitude by, 127 magnitude of, 123 Posidonias, 140 Polyconic projection, 192 Precise astronomical work, 134 leveling, 97 triangulation, 36 Precision of measurements, 12, 25 Primary triangulation, 206, 218 Prime vertical, 132, 173 Probable error, 25 of angles, 43, 206 azimuth, 113, 116 bases, 75 conditioned observations, 222 computed values, 31 level lines, 91, 93, 96 linear measures, 72 indirect observations, 29 Probability curve, n, 13 Projections, 181, 189 Prolate spheroid, 143 Quadrant of ellipsoid, 238 spheroid, 163, 164 Quadrilateral, 67, 152, 225 Radian, 33, 256 Radii of curvature, 85, 173, 175, 254 Radius of earth, 148 Reconnaissance, 198 Rectangular coordinates, 37, 183, 193 Refraction, celestial, 112, 115, 199, 249, 251 terrestrial, 8, 88, 99, IOI Rejection of observations, 44 Repeating theodolite, 206 Repetition of angles, 41 Residual errors, 8, 19, 30 Sag of tape, 77 Schott, C. A., 248, 249^ Secondary stations, 61, 64, 202 triangulation, 53, 64, 202 Semi-duplicate level lines, 89 Sextant, 125, 128, 130 Shooting at target, 10 Side equations, 61, 68 Signals, 203 Slope, correction for, 78 Smithsonian tables, 176, 249 Snellius, 141 Solar transit, no Soldner, 196 Solution of equations, 24 Sphere, lines on, 149 areas on, 151 Spherical angles, 151, 212 coordinates, 154, 194 excess, 152, 213 geodesy, 137-155 Spheroid, dimensions of, 168, 232,246 oblate, 143 prolate, 143 Spheroidal geodesy, 156-180 Spirit leveling, 87, 94 Standard time, 129 Star, azimuth by, 117 latitude by, 116 Station, adjustment at, 18, 42, 46, 209, 2ii Stations, 200 Sun, azimuth by, no, 114 declination of, ill latitude by, 124 Systematic errors, 7, 40 Tables, 249-256 INDEX. 26l Tapes, standard, 71, 76 Target shooting, 9 Temperature, 7, 71, 76, 78 . s Time, determination of, 169 standard, 161 Theodolites, 206, 207 Three-point problem, 64 Todhunter, 234 Toise, 142 Topography, 198 Towers, 200, 202 Transit, engineers', 39 Trapezoidal degree, 176 Triangle adjustment, 17, 21, 49, 51 computation, 53,57,212,215 net, 67 on sphere, 151 spheroid, 175 Triangulation, geodetic, 198-231 plane, 3^-70, 107 Uncertainty of a base, 72, 74 line, 50 U. S. Bureau of Equipment, 109 Weights and Meas- ures, 76 U. S. Coa-st and Geodetic Survey, 75, 84, 102, 180, 194, 199, 202, 231, 234, 249 Lake Survey, 18, 245 Northern Boundary Survey, 231 Variations in geoid, 247 latitude, 136 of ocean surface. 241 Verniers, 40 Vertical angles, 33, 101, 230 Weighted mean, 28 observations, 14 Weights, 19, 22, 25 of angles, ,43, 45. 214 levels, 91 lines, 73 Wilson, 99 Woodward, 82, 176, 194 Yard, 169, 249 Zenith telescope, 135 Zone, spherical, 152 spheroidal, 176 CIVIL ra.\EE8LNG (I of t SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILtEY & SONS, NEW YORK, LONDON: CHAPMAN & HALL, LIMITED. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE. Armsby's Manual of Cattle-feeding i2mo. Si 75 Principles of Animal Nutrition 8vo, 4 oo Budd and Hansen's American Horticultural Manual: Part I. 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(Mercur.) 8vo, half morocco, 7 50 Manual for Courts-martial. i6mo, morocco, i 50 * Mercur's Attack of Fortified Places i2mo, 2 oo * Elements of the Art pf War 8vo, 4 oo Metcalf's Cost of Manufactures And the Administration of Workshops. .8vo, 5 oo * Ordnance and Gunnery. 2 vols I2mo, 5 oo Murray's Infantry Drill Regulations i8mo, paper, 10 Nixon's Adjutants' Manual 241110, i oo Peabody's Naval Architecture 8vo, 7 50 * Phelps's Practical Marine Surveying 8vo, 2 50 Powell's Army Officer's Examiner i2mo, 4 oo ;Sharpe's Art of Subsisting Armies in War i8mo, morocco, i 50 * Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing. 24mo, leather, 50 Weaver's Military Explosives 8vo, 3 oo "Wheeler's Siege Operations and Military Mining 8vo, 2 oo Winthrop's Abridgment of Military Law 121110, 2 50 Woodhull's Notes on Military Hygiene i6mo, i 50 Young's Simple Elements of Navigation i6mo, morocco, 2 oo ASSAYING. 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Baker's Engineers' Surveying Instruments i2mo, 3 oo Bixby's Graphical Computing Table Paper 19^X24! inches. 25 Breed and Hosmer's Principles and Practice of Surveying ,8vo, 3 oo * Burr's Ancient and Modern Engineering and the Isthmian Canal 8vo, 3 50 Comstock's Field Astronomy for Engineers 8vo, 2 50 * CortheU's Allowable Pressures on Deep Foundations I2mo, i 25 CrandalTs Text-book on Geodesy and Least Scjuares 8vo, 3 oo Davis's Elevation and Stadia Tables 8vo, i oo Elliott's Engineering for Land Drainage i2mo, i 50 Practical Farm Drainage i2mo, i oo *Fiebeger's Treatise on Civil Engineering 8vo, 5 oo Flemer's Phototopographic Methods and Instruments 8vo, 5 oo Folwell's Sewerage. 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(Wood.) 8vo, 5 oo * Descriptive Geometry 8vo i 50 Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 Merriman and Brooks's Handbook for Surveyors i6mo, morocco, 2 oo Nugent's Plane Surveying. '. 8vo, 3 50 Ogden's Sewer Design 12 mo, 2 oo Parsons's Disposal of Municipal Refuse 8vo, 2 oo Patton's Treatise on Civil Engineering 8vo half leather, 7 50 Reed's Topographical Drawing and Sketching 4to, 5 oo Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo Riemer's Shaft-sinking under Difficult Conditions. (Corning and Peele.) . .8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. 8vo, 2 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Tracy's Plane Surveying I6mo, morocco, 3 oo * Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 oo Venable's Garbage Crematories in America 8vo, 2 oo Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo Sheep, 5 50 Law of Contracts 8vo, 3 oo Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50 Webb's Problems in the Use and Adjustment of Engineering Instruments. i6mo, morocco, i 25 Wilson's Topographic Surveying 8vo, 3 50 BRIDGES AND ROOFS. Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo Burr and Falk's Influence Lines for Bridge and Roof Computations 8vo, 3 oo Design and Construction of Metallic Bridges 8vo, 5 oo Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 oo Foster's Treatise on Wooden Trestle Bridges 4to, 5 oo Fowler's Ordinary Foundations 8vo, 3 50 Greene's Roof Trusses 8vo, i 25 Bridge Trusses 8vo, 2 50 Arches in Wood, Iron, and Stone 8vo, 2 50 Grimm's Secondary Stresses in Bridge Trusses. (Tn Press.) Howe's Treatise on Arches 8vo, 4 oo Design of Simple Roof -trusses in Wood and Steel 8vo, 2 oo Symmetrical Masonry Arches 8vo, 2 50 Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of Modern Framed Structures Small 4to, 10 oo Merriman and Jacoby's Text-book on Roofs and Bridges: Part I. Stresses in Simple Trusses 8vo, 2 50 Part II. Graphic Statics 8vo, 2 50 Part III. Bridge Design 8vo, 2 50 Part IV. Higher Structures 8vo, 2 50 7 Morison's Memphis Bridge 4to, 10 oo Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . i6mo, morocco, 2 oo Specifications for Steel Bridges i2mo. 50 Wright's Designing of Draw-spans. Two parts in one volume 8vo, 3 50 HYDRAULICS. Barnes's Ice Formation 8vo, 3 oo Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an Orifice. (Trautwine.) 8vo, 2 oo Bovey's Treatise on Hydraulics 8vo, 5 oo Church's Mechanics of Engineering 8vo, 6 oo Diagrams of Mean Velocity of Water in Open Channels paper, i 50 Hydraulic Motors 8vo, 2 oo Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 Flather's Dynamometers, and the Measurement of Power 12 mo, 3 oo Folwell's Water-supply Engineering 8vo, 4 oo FrizelFs Water-power 8vo, 5 oo Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works I2mo, 2 50 Ganguillet and Kutter's General Formula for the Uniform Flow of Water in Rivers and Other Channels. 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(Considered Principally from a Sanitary Standpoint.) 8vo. 4 oo Merriman's Treatise on Hydraulics 8vo, 5 oo * Michie's Elements of Analytical Mechanics 8vo, 4 oo Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- supply Large 8vo, 5 oo * Thomas and Watt's Improvement of Rivers 4to, 6 oo Turneaure and Russell's Public Water-supplies 8vo, 5 oo Wegmann's Design and Construction of Dams. 5th Edition, enlarged. . 4to, 6 oo Water-supply of the City of New York from 1658 to 1895 410, 10 oo Whipple's Value of Pure Water Large i2mo, i oo Williams and Hazen's Hydraulic Tables 8vo, i 50 Wilson's Irrigation Engineering Small 8vo, 4 oo Wolff's Windmill as a Prime Mover 8vo, 3 oo Wood's Turbines 8vo, 2 50 Elements of Analytical Mechanics 8vo, 3 oo MATERIALS OF ENGINEERING. Baker's Treatise on Masonry Construction 8vo, 5 oo Roads and Pavements . 8vo, 5 oo Black's United States Public Works Oblong 410, 5 oo * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 5 Byrne's Highway Construction 8vo, 5 oo Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 oo Church's Mechanics of Engineering 8vo, 6 oo Du Bois's Mechanics of Engineering. Vol. I Small 4to, 7 50 *Eckel's Cements, Limes, and Plas ers 8vo, 6 oo 8 Johnson's Materials of Construction Large 8vo, 6 oo Fowler's Ordinary Foundations 8vo, 3 50 Graves's Forest Mensuration 8vo, 4 oo * Greene's Structural Mechanics 8vo, 2 50 Keep's Cast Iron 8vo, 2 50 Lanza's Applied Mechanics v .1 8vo, 7 50 Martens's Handbook on Testing Materials. 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Non-metallic Materials of Engineering and Metallurgy 8vo, 2 oo ' Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Tillson's Street Pavements and Paving Materials 8vo, 4 oo Turneaure and Maurer's Principles of Reinforced Concrete Construction.. .8vo, 3 oo Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor., 2 oo * Specifications for Steel Bridges i2mo, 50 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation of Timber 8vo, 2 oo Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and SteeL 8vo, 4 oo RAILWAY ENGINEERING. Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, i 25 Berg's Buildings and Structures of American Railroads 4to, 5 oo Brook's Handbook of Street Railroad Location i6mo, morocco, i 50 Butt's Civil Engineer's Field-book i6mo, morocco, 2 50 Crandall's Transition Curve i6mo, morocco, I 50 Railway and Other Earthwork Tables* 8vo, i 50 Crockett's Methods for Earthwork Computations. (In Press) Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo. morocco, 5 oo Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo Fisher's Table of Cubic Yards Cardboard, 25 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . . i6mo, mor., 2 50 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, i oo Molitor and Beard's Manual for Resident Engineers i6mo, i oo Nagle's Field Manual for Railroad Engineers i6mo, morocco, 3 oo Philbrick's Field Manual for Engineers i6mo, morocco, 3 oo .Raymond's Elements of Railroad Engineering. (In Press.) 9 Searles's Field Engineering i6mo morocco, 3 oo Railroad Spiral. i6mo, morocco, i 50 Taylor's Prismoidal Formulae and Earthwork. ... .... 8vo, i 50- * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo> 2 oa The Field Practice of Laying Out Circular Curves for Railroads. i2mo, morocco, 2 50- Cross-section Sheet Paper, 25 Webb's Railroad Construction i6mo, morocco, 5 oo> Economics of Railroad Construction . , Large i2mo, 2 50 Wellington's Economic Theory of the Location of Railways Small 8vo, 5 oo DRAWING. Barr's Kinematics of Machinery 8vo, 2 50 * Bartlett's Mechanical Drawing 8vo, 3 oo * " ' Abridged Ed 8vo. i 50 Coolidge's Manual of Drawing 8vo, paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical Engi neers Oblong 4to, 2 50 Durley's Kinematics of Machines 8vo, 4 oa Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50- Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 oo Jamison's Elements of Mechanical Drawing 8vo, 2 50 Advanced Mechanical Drawing 8vo, 2 oa Jones's Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oa MacCord's Elements of Descriptive Geometry 8vo, 3 oo Kinematics; or, Practical Mechanism Svo, 5 oo Mechanical Drawing 4to, 4 oa Velocity Diagrams Svo, i 50 MacLeod's Descriptive Geometry Small Svo, i 50 * Mahan's Descriptive Geometry and Stone-cutting Svo, i 50 Industrial Drawing. (Thompson.) Svo, 3 50 Moyer's Descriptive Geometry Svo, 2 oa Reed's Topographical Drawing and Sketching 4to, 5 oa Reid's Course in Mechanical Drawing Svo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 oo Robinson's Principles of Mechanism Svo, 3 oo Schwamb and Merrill's Elements of Mechanism Svo, 3 oo Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo, 2 50 Smith (A. W.) and Marx's Machine Design Svo, 3 oo * Titsworth's Elements of Mechanical Drawing Oblong Svo, 25 Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo, oo Drafting Instruments and Operations i2mo, Manual of Elementary Projection Drawing i2mo. Manual of Elementary Problems in the Linear Perspective of Form and oo Shadow ............................................... i2mo, Plane Problems in Elementary Geometry ...................... i2mo, Elements of Descriptive Geometry, Shadows, and Perspective ....... 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Design, Construction, and Operation 8vo, 6 oo Handbook of Engine and Boiler Trials, and the Use of the Indicator and the Prony Brake 8vo, 5 oo Stationary Steam-engines 8vo, 2 50 Steam-boiler Explosions in Theory and in Practice i2mo, i 50 Manual of Stpam-boilers, their Designs, Construction, and Operation. 8vo, 5 oo Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) 8vo, 4 oo Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 oo Whitham's Steam-engine Design 8vo, 5 oo Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 oo MECHANICS AND MACHINERY. 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(Pope, Haven, and Dean.). 8vo, 4 oo MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo Velocity Diagrams 8vo, i 50 * Martin's Text Book on Mechanics, Vol. I, Statics i2mo, i 25 * Vol. 2, Kinematics and Kinetics . .I2mo, l 50 Maurer's Technical Mechanics 8vo, 4 oo Merriman's Mechanics of Materials 8vo, 5 oo * Elements of Mechanics i2mo, i oo * Michie's Elements of Analytical Mechanics. . . 8vo, 4 oo * Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 50 Reagan's Locomotives : Simple, Compound, and Electric. New Edition. Large 12 mo, 3 5o Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism 8vo, 3 oo Ryan, Norris, and Hoxie's Electrical Machinery. 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